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Existence of a positive solution to a system of discrete fractional boundary value problems
Applied Mathematics and Computation 217 (2011) 4740–4753
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Existence of a positive solution to a system of discrete fractional boundary value problems Christopher S. Goodrich Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA
a r t i c l e
i n f o
a b s t r a c t
Keywords: Discrete fractional calculus Eigenvalues Positive solution Coupled system of boundary value problems Cone
We analyze a system of discrete fractional difference equations subject to nonlocal boundary conditions. We consider the system of equations given by Dmi yi ðtÞ ¼ ki ai ðtþ mi 1Þfi ðy1 ðt þ m1 1Þ; y2 ðt þ m2 1ÞÞ, for t 2 ½0; bN0 , subject to yi(mi 2) = wi(yi) and yi(mi + b) = /i(yi), for i = 1, 2, where wi ; /i : Rbþ3 ! R are given functionals. We also assume that mi 2 (1, 2], for each i. Although we assume that both ai and fi(y1, y2) are nonnegative for each i, we do not necessarily presume that each wi(yi) and /i(yi) is nonnegative for each i and each yi P 0. This generalizes some recent results both on discrete fractional boundary value problems and on discrete integer-order boundary value problems, and our techniques provide new results in each case. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction In this paper we are concerned with a pair of discrete fractional boundary value problems of the form
Dm1 y1 ðtÞ ¼ k1 a1 ðt þ m1 1Þf1 ðy1 ðt þ m1 1Þ; y2 ðt þ m2 1ÞÞ;
ð1:1Þ
Dm2 y2 ðtÞ ¼ k2 a2 ðt þ m2 1Þf2 ðy1 ðt þ m1 1Þ; y2 ðt þ m2 1ÞÞ; for t 2 ½0; bN0 , subject to the boundary conditions
y1 ðm1 2Þ ¼ w1 ðy1 Þ;
y2 ðm2 2Þ ¼ w2 ðy2 Þ;
y1 ðm1 þ bÞ ¼ /1 ðy1 Þ;
y2 ðm2 þ bÞ ¼ /2 ðy2 Þ;
ð1:2Þ ð1:3Þ bþ3
where ki > 0; ai : R ! ½0; þ1Þ; mi 2 ð1; 2 for each 1 6 i 6 2, and for each i we have that wi ; /i : R ! R are given functionals. We shall also assume that fi: [0, +1) [0, +1) ? [0, +1) and is continuous for each admissible i. We point out that we do not necessarily assume that either the wi’s or the /i’s are nonnegative for all y P 0 – i.e., for all nonnegative functions y(t). In spite of this, we shall still show that problem (1.1)–(1.3) can admit a positive solution for certain values of the ki’s (i.e., the eigenvalues) and some standard assumptions on the nonlinearities fi, for 1 6 i 6 2. We shall use some ideas introduced recently by Infante and Webb [1] to accomplish this. In addition to this, we shall show that by weakening the assumptions made on the functionals but at the same time imposing a nonnegativity condition on them, it remains possible to deduce the existence of at least one positive solution to (1.1)–(1.3). Thus, in the sequel we shall give two relatively distinct sets of conditions under which problem (1.1)–(1.3) will admit at least one positive solution. It should be noted, as will be described in the sequel, that our theorems provide completely new results and techniques even in the case when (1.1) is of integer-order – that is, m1 = m2 = 2. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.11.029
C.S. Goodrich / Applied Mathematics and Computation 217 (2011) 4740–4753
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To place problem (1.1)–(1.3) in an appropriate context, we first point out that a simpler, integer-order version of (1.1)– (1.3) was considered recently by Henderson, Ntouyas, and Purnaras [2]. In particular, they considered the problem
8 2 > < D uðn 1Þ þ kaðnÞf ðuðnÞ; v ðnÞÞ ¼ 0 D2 v ðn 1Þ þ lbðnÞgðuðnÞ; v ðnÞÞ ¼ 0 > :
ð1:4Þ
subject to the multipoint-type boundary conditions
uð0Þ ¼ buðgÞ;
uðNÞ ¼ auðgÞ;
v ð0Þ ¼ bv ðgÞ; v ðNÞ ¼ av ðgÞ;
ð1:5Þ
where g 2 {1, . . . , N 1}, n 2 {1, 2, . . . , N 1}, N P 4, and a, b, k, l > 0. They then deduced, under various assumptions on f, g, k, l, a and b, the existence of at least one positive solution to (1.4) and (1.5). Clearly, if we put m1 = m2 = 2, and appropriately choose wi(yi) and /i(yi), 1 6 i 6 2, then (1.1)–(1.3) reduces to problem (1.4) and (1.5). So, in particular, our results in this paper will provide a generalization of the results provided in [2]. Moreover, it should be noted that a similar problem in the continuous fractional calculus was recently considered by Su [3]. We next point out that discrete multipoint (or nonlocal) boundary value problems (BVPs) have been extensively studied in recent years. This is also true in the more general case of an arbitrary time scale T. As for but a few examples, we note that Anderson [4] considered the problem uDr(t) + f(t, u(t)) = 0, u(0) = 0, a u(g) = u(T), on a time scale T. Kaufmann and Raffoul [5] considered a closely related problem. Similarly, Cheung, et al. [6] considered a delta-nabla difference equation of the form rDu(k) + f(k, u(k)) = 0 together with a couple of a different specific nonlocal conditions. For some other papers on discrete nonlocal BVPs, we refer the reader to [7–13] and the references therein. On the other hand, the discrete fractional calculus is a relatively new and emerging area of mathematics. While the continuous fractional calculus has been studied for some time and put to good use in modeling and applied mathematics (see, for example, [14–22] and the references therein), the discrete fractional calculus has only recently begun to be developed seriously. In a series of very useful and interesting papers, namely [23–26], Atici and Eloe developed some of the basic theory of both discrete fractional IVPs and BVPs with delta derivative on the time scale Z. Moreover, a recent paper by Atici and Sßengül [27] has provided some analysis of discrete fractional variational problems; their paper also provided some initial attempts at using the discrete fractional calculus to model biological processes. Similarly, the present author has established in [28–34] some results on both discrete fractional IVPs and BVPs. Finally, recent papers by Bastos, et al. [35] and Anastassiou [36] have provided extensions of the discrete fractional calculus with delta derivative to more general time scales. Thus, the discrete fractional calculus has recently attracted increasing attention from a growing number of researchers. It is also worth noting that some progress made toward developing the discrete fractional calculus with the nabla derivative on the time scale Z. Three works come to mind here. Firstly, we are aware of a work by Atici and Eloe [37], in which certain of the basic theory of fractional nabla derivatives on the time scale Z was developed. Secondly, a recent paper by Anastassiou [38] has established some results regarding the fractional calculus on time scales with nabla derivative. Thirdly, in a recent work by the present author [34], a discrete fractional delta-nabla BVP of the form Dlrmy(t) = f(t + l + m 1, y(t + l + m 1)) subject to the boundary conditions y(l + m 2) = w(y), y(l + m + b) = 0 was considered for 0 < l 6 1, 0 < m 6 1, and l + m 2 (1, 2]. Since delta-nabla BVPs have proved to be interesting to study in the integer-order case, we feel that their study in the fractional case will continue to be worthwhile. Underlying many of these papers, at least the ones that consider FBVPs with delta derivative, is a consideration of the general discrete fractional boundary value problem (FBVP)
Dm yðtÞ þ f ðt þ m 1; yðt þ m 1ÞÞ ¼ 0; subject to the boundary conditions
yðm 2Þ ¼ wðyÞ;
yðb þ mÞ ¼ /ðyÞ;
for m 2 (1, 2], where w and / are given functionals. For example, in [26] the authors studied this problem in case w / 0 – that is, the so-called conjugate FBVP. Similarly, in [29], the author studied this problem in case w(y) 0 and /(y) = Dy(m + b) – that is, the so-called right-focal FBVP. With this context in mind, it is clear that problem (1.1)–(1.3), in addition to a generalization of [2], is a generalization of certain of the FBVPs considered in [26,29,30]. Indeed, our results in this work will generalize some of the previous results on discrete FBVPs as well as discrete integer-order BVPs. Specifically, we will analyze (1.1)–(1.3) by developing a representation for solutions of (1.1)–(1.3) in terms of the fixed point of an operator. Having fixed our framework, we shall then use some ideas presented by Infante and Webb [1] to deduce the existence of a positive solution (1.1)–(1.3) even if none of the wi’s and /i’s is nonnegative for all y P 0. In addition, we shall then present in Section 4 a different existence result that does not use the cone introduced by Infante and Webb but rather uses a more traditional cone together with some different assumptions about the each of the wi’s and /i’s. Finally, in Section 5 we shall give a specific numerical example of our result, something lacking in [2]. We conclude the paper in Section 6 with some thoughts for future research in the directions outlined in this paper.
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2. Preliminaries We first wish to collect some basic lemmas that will be important to us in the sequel. These and other related results and their proofs can be found, for example, in [23–26]. Definition 2.1. We define
Cðt þ 1Þ ; Cðt þ 1 mÞ for any t and m for which the right-hand side is defined. We also appeal to the convention that if t + 1 m is a pole of the tm :¼
Gamma function and t + 1 is not a pole, then tm = 0. Definition 2.2. The mth fractional sum of a function f defined on the set Na :¼ fa; a þ 1; . . .g, for m > 0, is defined to be
Dm f ðtÞ ¼ Dm f ðt; aÞ :¼
tm 1 X ðt s 1Þm1 f ðsÞ; CðmÞ s¼a
where t 2 fa þ m; a þ m þ 1; . . .g ¼: Naþm . We also define the mth fractional difference, where m > 0 and 0 6 N 1 < m 6 N with N 2 N, to be
Dm f ðtÞ :¼ DN DðNmÞ f ðtÞ; where t 2 NaþNm . Lemma 2.3. Let t and m be any numbers for which tm and tm1 are defined. Then
Dtm ¼ mt m1 : Lemma 2.4. Let 0 6 N 1 < m 6 N, where N 2 N and N 1P0. Then
Dm Dm yðtÞ ¼ yðtÞ þ C 1 tm1 þ C 2 t m2 þ þ C N t mN ; for some C i 2 R, with 1 6 i 6 N. We now wish to fix our framework for the study of problem (1.1)–(1.3). First of all, we let Bi represent the Banach space of all maps from ½mi 2; . . . ; mi þ bNm 2 into R when equipped with the usual maximum norm, k k. We shall then put i
X :¼ B1 B2 : By equipping X with the norm
kðy1 ; y2 Þk :¼ ky1 k þ ky2 k; it follows that ðX ; k kÞ is a Banach space, too – see, for example, [2,39]. Next we wish to develop a representation for a solution of (1.1)–(1.3) as the fixed point of an appropriate operator on X . To accomplish this, however, we present now some straightforward adaptations of results from [26] that will be of use here. Because the proofs of these adaptions are evident, we do not include them. Lemma 2.5 [26]. Let 1 < m 6 2 and h : ½m 1; m þ b 1Nm1 ! R be given. The unique solution of the FBVP Dmy(t) = h(t + m 1), P y(m 2) = 0 = y(m + b) is given by yðtÞ ¼ bs¼0 Gðt; sÞhðs þ m 1Þ, where G : ½m 2; m þ bNm2 ½0; bN0 ! R is defined by
8 m1 > t ðmþbs1Þm1 ðt s 1Þm1 ; 0 6 s < t m þ 1 6 b; 1 < ðmþbÞm1 Gðt; sÞ :¼ m1 ðmþbs1Þm1 t CðmÞ > : ; 0 6 t m þ 1 6 s 6 b: ðmþbÞm1
Lemma 2.6 [26]. The Green’s function G(t, s) given in Lemma 2.5 satisfies: 1. G(t, s) P 0 for each ðt; sÞ 2 ½m 2; m þ bNm2 ½0; bN0 ; 2. maxt2½m2;mþbN Gðt; sÞ ¼ Gðs þ m 1; sÞ for each s 2 ½0; bN0 ; and m2 3. there exists a number c 2 (0, 1) such that
min
3ðbþmÞ bþm 6t6 4 4
Gðt; sÞ P c
for s 2 ½0; bN0 .
max
t2½m2;mþbN
m2
Gðt; sÞ ¼ cGðs þ m 1; sÞ;
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Now consider the operator S : X ! X defined by
Sðy1 ; y2 Þðt 1 ; t2 Þ :¼ ðS1 ðy1 ; y2 Þðt1 Þ; S2 ðy1 ; y2 Þðt2 ÞÞ;
ð2:1Þ
where we define S1 : X ! B1 by
S1 ðy1 ; y2 Þðt1 Þ :¼ a1 ðt1 Þw1 ðy1 Þ þ b1 ðt 1 Þ/1 ðy1 Þ þ k1
b X
G1 ðt 1 ; sÞa1 ðs þ m1 1Þf1 ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞ
ð2:2Þ
G2 ðt 2 ; sÞa2 ðs þ m2 1Þf2 ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞ;
ð2:3Þ
s¼0
and S2 : X ! B2 by
S2 ðy1 ; y2 Þðt2 Þ :¼ a2 ðt2 Þw2 ðy2 Þ þ b2 ðt 2 Þ/2 ðy2 Þ þ k2
b X s¼0
where, for each j = 1, 2, we define
aj ðtÞ :¼
bj ðtÞ :¼
1 1 mj 1 m 2 ; t tj bþ2 Cðmj 1Þ t
ð2:4Þ
mj 1
ð2:5Þ
mj 1 :
ðm þ bÞ
Moreover, Gj(t, s) is as given in Lemma 2.5 with m replaced by mj. We claim that whenever ðy1 ; y2 Þ 2 X is a fixed point of the operator S, it follows that the pair of functions y1(t) and y2(t) is a solution to problem (1.1)–(1.3). Theorem 2.7. Let fj : R2 ! ½0; þ1Þ and wj ; /j 2 Cð½mj 2; mj þ bNm 2 ; RÞ be given, for j = 1,2. If ðy1 ; y2 Þ 2 X is a fixed point of S, j then the pair of functions y1(t) and y2(t) is a solution to problem (1.1)–(1.3). Proof. Suppose that the operator S has a fixed point, say ðy1 ; y2 Þ 2 X . Let ðt1 ; t2 Þ 2 Nm1 2 Nm2 2 . Then, for example, we have that
y1 ðt 1 Þ ¼ S1 ðy1 ; y2 Þðt 1 Þ;
ð2:6Þ
where S1 is as in (2.2). It is easy to check that S1(y1, y2)(m1 2) = w(y1) and that S1(y1, y2)(m1 + b) = /1(y1), so that the boundary conditions in (1.2) and (1.3) are satisfied. Similarly, one can check that S1(y1, y2)(t1) satisfies the difference equation in (1.1). Since similar verifications may be made for S2(y1, y2)(t2), the claim follows. h The following lemma and its associated corollary are of particular importance in the sequel. Because the proofs of each of these is straightforward, we omit them. Lemma 2.8. For each admissible j (i.e., j = 1 or 2), the function aj(tj) is decreasing in tj, for tj 2 ½mj 2; mj þ bNm 2 . Also, j mintj 2½mj 2;mj þbN aj ðtj Þ ¼ 0 and maxtj 2½mj 2;mj þbN aj ðtj Þ ¼ 1. On the other hand, for each admissible j, the function bj(tj) is strictly mj 2 mj 2 increasing in tj, for tj 2 ½mj 2; mj þ bNm 2 . In addition, mintj 2½mj 2;mj þbN bj ðt j Þ ¼ 0 and maxtj 2½mj 2;mj þbN bj ðt j Þ ¼ 1. mj 2
j
Corollary 2.9. Let j be fixed and admissible (i.e., j = 1 or 2). Let Ij :¼ mintj 2Ij aj ðt j Þ ¼ M aj kaj k and mintj 2Ij bj ðtj Þ ¼ M bj kbj k.
h
bþmj 4
mj 2
;
i
3ðbþmj Þ 4
. There are constants M aj ; M bj 2 ð0; 1Þ such that
Let us conclude this section with a remark. Remark 2.10. Observe that unlike the integer-order problem considered in [2], in the fractional-order problem we encounter a significant problem with respect to the domains of the various operators. This has been clearly seen in the preceding discussion, wherein the functions y1(t) and y2(t) are defined on (possibly) different sets – namely Nm1 2 in the former case and Nm2 2 in the latter case. This complication arises in the discrete fractional calculus due to the domain shifting of the fractional difference and sum operators. We will continue to encounter this complication in Sections 3 and 4 of this paper.
3. Existence of a positive solution: Case I We now present the first of two theorems for the existence of at least one positive solution to problem (1.1)–(1.3). As was pointed out in Section 1, in this particular section we shall not assume that either wi(yi) or /i(yi), 1 6 i 6 2, is nonnegative for all yi P 0. Rather, we shall make some other assumptions about these functionals. Let us now present the conditions that we shall assume henceforth. We note that conditions (F1)–(F2) are the same conditions given by Henderson, et al. [2]. Moreover, condition (L1) is essentially the same condition (up to a constant multiple) as given in [2, Theorem 3.1].
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F1: There exist numbers f1 and f2 , with f1 ; f2 2 ð0; þ1Þ, such that
lim
ðy1 ;y2 Þ!ð0þ ;0þ Þ
f1 ðy1 ; y2 Þ ¼ f1 y1 þ y2
and
lim
ðy1 ;y2 Þ!ð0þ ;0þ Þ
f2 ðy1 ; y2 Þ ¼ f2 : y1 þ y2
F2: There exist numbers f1 and f2 , with f1 ; f2 2 ð0; þ1Þ, such that
lim
ðy1 ;y2 Þ!ð1;1Þ
f1 ðy1 ; y2 Þ ¼ f1 y1 þ y2
and
lim
ðy1 ;y2 Þ!ð1;1Þ
f2 ðy1 ; y2 Þ ¼ f2 : y1 þ y2
G1: For each admissible j, the functionals wj and /j are linear. In particular, we assume both that
wj ðyj Þ ¼
mj þb X i¼mj 2
cjimj þ2 yj ðiÞ
and that
/j ðyj Þ ¼
mj þb X
j
dkmj þ2 yj ðkÞ;
k¼mj 2 j
for constants cjimj þ2 ; dkmj þ2 2 R. G2: For each admissible j, we have both that mj þb X i¼mj 2
cjimj þ2 Gj ði; sÞ P 0
and that mj þb X
j
dkmj þ2 Gj ðk; sÞ P 0;
k¼mj 2
for each s 2 ½0; bN0 , and in addition that mj þb X i¼mj 2
cjimj þ2 þ
mj þb X k¼mj 2
j
dkmj þ2 6
1 : 4
G3: We have that each of wi(ai), wi(bi), /i(ai), and /i(bi) is nonnegative for each admissible i – that is, i = 1, 2. L1: The constants k1 and k2 satisfy
K1 < k i < K2 ; for each i, where
8 " #1 " #1 9 = <1 X b b X bþ1 1 b þ 1 þ m1 ; s a1 ðs þ m1 1Þf1 þ m2 ; s a2 ðs þ m2 1Þf2 K1 :¼ max cG1 ; cG2 ; :2 s¼0 2 2 s¼0 2 and
8 " #1 " #1 9 b b <1 X = X 1 K2 :¼ min G1 ðs þ m1 1; sÞa1 ðs þ m1 1Þf1 ; G2 ðs þ m2 1; sÞa2 ðs þ m2 1Þf2 ; :4 s¼0 ; 4 s¼0 where c 2 (0, 1) is a constant defined by
c :¼ minfMa1 ; Ma2 ; Mb1 ; Mb2 ; c1 ; c2 g; where M a1 ; M a2 ; M b1 , and M b2 each come from Corollary 2.9 and c1 and c2 are associated by Lemma 2.6 to G1(t, s) and G2(t, s), respecitively. (Recall that these are defined on possibly different time scales). In the sequel, we shall also make use of the cone
K :¼ fðy1 ; y2 Þ 2 X : y1 ; y2 P 0;
bþm
min bþm
4
4
ðt 1 ;t 2 Þ2
P 0; /j ðyj Þ P 0; for each j ¼ 1; 2g;
1 ;3ðbþm1 Þ 4
½y1 ðt 1 Þ þ y2 ðt 2 Þ P ckðy1 ; y2 Þk; wj ðyj Þ
2 ;3ðbþm2 Þ 4
ð3:1Þ
where c is defined exactly as in the statement of condition (L1) above. Clearly, we have that K # X . In order to show that S has a fixed point in K, we must first demonstrate that K is invariant under S – that is, S : K ! K. We show this now.
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C.S. Goodrich / Applied Mathematics and Computation 217 (2011) 4740–4753
Lemma 3.1. Let S : X ! X be the operator defined as in (2.1). Then S : K ! K. Proof. Suppose that ðy1 ; y2 Þ 2 K. We show first that ðt1 ;t 2 Þ2
bþm
min bþm
4
4
1 ;3ðbþm1 Þ 4
2 ;3ðbþm2 Þ 4
½S1 ðy1 ; y2 Þðt 1 Þ þ S2 ðy1 ; y2 Þðt 2 Þ P ckSðy1 ; y2 Þk;
whenever ðy1 ; y2 Þ 2 K. So note that
bþmin
S1 ðy1 ; y2 Þðt 1 Þ P m 3ðbþm Þ
t1 2
4
1;
1
4
þ t1 2
bþmin
m 3ðbþm Þ 4
1;
4
1
b X
t1 2
bþmin
½a1 ðt 1 Þ/1 ðy1 Þ þ b1 ðt1 Þw1 ðy1 Þ m 3ðbþm Þ 4
1;
4
1
G1 ðt1 ; sÞa1 ðs þ m1 1Þf1 ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞ
s¼0
P Ma1 ka1 k/1 ðy1 Þ þ M b1 kb1 kw1 ðy1 Þ þ k1 max
t 1 2½m1 2;m1 þb
c1 G1 ðs þ m1 1; sÞa1 ðs þ m1 1Þf1 ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞ
s¼0
" ~1 Pc
b X
a1 ðt1 Þ/1 ðy1 Þ þ b1 ðt1 Þw1 ðy1 Þ þ k1
b X
# G1 ðt 1 ; sÞa1 ðs þ m1 1Þf1 ðy1 ðs þ m1 1Þ; y2 ðs þ m1 1ÞÞ
s¼0
~1 kS1 ðy1 ; y2 Þk; ¼c
ð3:2Þ
where c~1 :¼ minfM a1 ; M b1 ; c1 g, whence t1 2
c 1 kS1 ðy1 ; y2 Þk; bþmin
S1 ðy1 ; y2 Þðt1 Þ P e m 3ðbþm Þ 4
1;
4
ð3:3Þ
1
as desired. In an entirely similar manner to (3.2), we deduce that t2 2
~2 kS2 ðy1 ; y2 Þk; bþmin
S2 ðy1 ; y2 Þðt2 Þ P c m 3ðbþm Þ 4
2;
4
ð3:4Þ
2
~2 :¼ minfM a2 ; M b2 ; c2 g. where c ~1 ; c ~2 g. Consequently, from (3.3) and (3.4) it follows that Now, put c :¼ minfc ðt1 ;t 2 Þ2
bþm
min bþm
4
4
1 ;3ðbþm1 Þ 4
2 ;3ðbþm2 Þ 4
½S1 ðy1 ; y2 Þðt 1 Þ þ S2 ðy1 ; y2 Þðt 2 Þ P
ðt 1 ;t 2 Þ2
bþm
min bþm
4
4
þ ðt 1 ;t 2 Þ2
1 ;3ðbþm1 Þ 4
S1 ðy1 ; y2 Þðt1 Þ
2 ;3ðbþm2 Þ 4
bþm
min bþm
4
4
1 ;3ðbþm1 Þ 4
S2 ðy1 ; y2 Þðt2 Þ
2 ;3ðbþm2 Þ 4
~2 kS2 ðy1 ; y2 ÞkÞ ~1 kS1 ðy1 ; y2 Þk þ c P ðc P ðckS1 ðy1 ; y2 Þk þ ckS2 ðy1 ; y2 ÞkÞ ¼ ckðS1 ðy1 ; y2 Þ; S2 ðy1 ; y2 ÞÞk ¼ ckSðy1 ; y2 Þk:
ð3:5Þ
So, from (3.5) we conclude that whenever ðy1 ; y2 Þ 2 X , we find that ðt1 ;t 2 Þ2
bþm
min bþm
4
4
1 ;3ðbþm1 Þ 4
2 ;3ðbþm2 Þ 4
½S1 ðy1 ; y2 Þðt 1 Þ þ S2 ðy1 ; y2 Þðt 2 Þ P ckSðy1 ; y2 Þk;
ð3:6Þ
as desired. We next show that for each admissible j we have wj(Sj(y1, y2)) P 0 whenever ðy1 ; y2 Þ 2 K. To see that this is true, note that
wj ðSj ðy1 ; y2 ÞÞ ¼
mj þb X i¼mj 2
¼ kj
cjimj þ2 Sj ðy1 ; y2 Þ ðiÞ
mj þb X i¼mj 2
þ
cjimj þ2
i¼mj 2 l¼mj 2
¼ kj
mj þb X
s¼0 i¼mj 2
Gj ði; sÞaj ðs þ mj 1Þfj ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞ
s¼0
mj þb X mj þb X
b X
b X
cjimj þ2 cjlmj þ2 yj ðlÞaj ðiÞ þ
mj þb X
mj þb X
i¼mj 2 k¼mj 2
j
cjimj þ2 dkmj þ2 yj ðkÞbj ðiÞ
cjimj þ2 Gj ði; sÞaj ðs þ mj 1Þfj ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞ þ wj ðaj Þwj ðyj Þ þ wj ðbj Þ/j ðyj Þ:
ð3:7Þ
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C.S. Goodrich / Applied Mathematics and Computation 217 (2011) 4740–4753
But by assumptions (G2) and (G3) together with the nonnegativity of fj(y1, y2) and the fact that ðy1 ; y2 Þ 2 K, we find from (3.7) that wj(Sj(y1, y2)) P 0, for each admissible j. An entirely dual argument, which we omit, shows that
/j ðSj ðy1 ; y2 ÞÞ P 0;
ð3:8Þ
too, whenever ðy1 ; y2 Þ 2 K and j is admissible. Finally, it is clear from the definitions of both S1 and S2 that
S1 ðy1 ; y2 Þðt 1 Þ P 0
ð3:9Þ
S2 ðy1 ; y2 Þðt 2 Þ P 0
ð3:10Þ
and
whenever ðy1 ; y2 Þ 2 K. Therefore, we conclude from (3.6)–(3.9) and (3.10) that whenever ðy1 ; y2 Þ 2 K, it follows that Sðy1 ; y2 Þ 2 K. Thus, S : K ! K, as desired. And this completes the proof. h We now prove the first of our two main theorems, Theorem 3.3. We first recall the Krasnosel’skii˘ fixed point theorem (see [40]), which we shall use to prove Theorem 3.3. We shall also require Lemma 3.2 in Section 4 of this paper. Lemma 3.2. Let B be a Banach space and let K # B be a cone. Assume that X1 and X2 are open sets contained in B such that 0 2 X1 and X1 # X2 . Assume, further, that T : K \ ðX2 n X1 Þ ! K is a completely continuous operator. If either 1. kTyk 6 kyk for y 2 K \ @ X1 and kTyk P kyk for y 2 K \ @ X2 ; or 2. kTyk P kyk for y 2 K \ @ X1 and kTyk 6 kyk for y 2 K \ @ X2 ; then T has at least one fixed point in K \ ðX2 n X1 Þ. Theorem 3.3. Suppose that conditions (F1)–(F2), (G1)–(G3), and (L1) hold. Then problem (1.1)–(1.3) has at least one positive solution. Proof. We have already shown in Lemma 3.1 that S : K ! K. Furthermore, it is evident that S is a completely continuous operator. Note first by condition (L1) that there is > 0 such that
8 " #1 " #1 9 b b <1 X = bþ1 1 X bþ1 þ m1 ; s a1 ðs þ m1 1Þ f1 þ m2 ; s a2 ðs þ m2 1Þ f2 max cG ; cG2 :2 s¼0 1 ; 2 2 s¼0 2 6 k1 ; k2 ð3:11Þ and
k1 ; k2 6 min
8 " b <1 X :4
G1 ðs þ m1 1; sÞa1 ðs þ m1 1Þ f1 þ
s¼0
#1
" #1 9 b = 1 X : ; G2 ðs þ m2 1; sÞa2 ðs þ m2 1Þ f2 þ ; 4 s¼0 ð3:12Þ
Now, given this , by condition (F1) it follows that there exists some number
r 1
> 0 such that
f1 ðy1 ; y2 Þ 6 f1 þ ðy1 þ y2 Þ;
ð3:13Þ
whenever k(y1, y2)k < r1. Similarly, by condition (F2), for the same , there exists a number r 1 > 0 such that
f2 ðy1 ; y2 Þ 6 f2 þ ðy1 þ y2 Þ;
ð3:14Þ
whenever k(y1, y2)k < r2. In particular, then, by putting r1 :¼ min r1 ; r 1 , we find that both (3.13) and (3.14) hold whenever k(y1, y2)k < r1. This suggests putting
X1 :¼ fðy1 ; y2 Þ 2 X : kðy1 ; y2 Þk < r 1 g; which we shall use in the sequel.
ð3:15Þ
C.S. Goodrich / Applied Mathematics and Computation 217 (2011) 4740–4753
4747
Now, let X1 be as in (3.15) above. Then for ðy1 ; y2 Þ 2 K \ @ X1 we find that
kS1 ðy1 ; y2 Þk ¼
ja1 ðt 1 Þ/1 ðy1 Þ þ b1 ðt 1 Þw1 ðy1 Þ þ k1
max
t 1 2½m1 2;m1 þbN
mX 1 þb
mX 1 þb
c1im1 þ2 y1 ðkÞ þ
i¼m1 2
G1 ðt 1 ; sÞa1 ðs þ m1 1Þf1 ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞj
s¼0
m1 2
6
b X
1
dkm1 þ2 y1 ðkÞ þ k1
k¼m1 2
max
b X
t 1 2½m1 2;m1 þbN
G1 ðt 1 ; sÞa1 ðs þ m1 1Þf1 ðy1 ðs þ m1 1Þ;
m1 2 s¼0
y2 ðs þ m2 1ÞÞ " # mX mX b 1 þb 1 þb X 1 1 cim1 þ2 þ dkm1 þ2 þ k1 G1 ðs þ m1 1; sÞa1 ðs þ m1 1Þ f1 þ ðy1 ðs þ m1 1Þ 6 r1 i¼m1 2
k¼m1 2
s¼0
" # b X 1 þ y2 ðs þ m2 1ÞÞ 6 kðy1 ; y2 Þk þ k1 G1 ðs þ m1 1; sÞa1 ðs þ m1 1Þ f1 þ ; 4 s¼0
ð3:16Þ
where we have used the fact that S1(y1, y2) is nonnegative whenever ðy1 ; y2 Þ 2 K. However, by the choice of k1 as given in (3.11) and (3.12), we deduce from (3.16) that
kS1 ðy1 ; y2 Þk 6
1 kðy1 ; y2 Þk: 2
ð3:17Þ
We may deduce by an entirely dual argument that
kS2 ðy1 ; y2 Þk 6
1 kðy1 ; y2 Þk: 2
ð3:18Þ
Thus, by putting (3.11)–(3.18) together we find that for ðy1 ; y2 Þ 2 K \ @ X1 we have
kSðy1 ; y2 Þk 6
1 1 kðy1 ; y2 Þk þ kðy1 ; y2 Þk ¼ kðy1 ; y2 Þk: 2 2
ð3:19Þ
Now, let be the same positive number selected at the beginning of this proof. For this same , by virtue of condition (F2) we can find a number ~r 2 > 0 such that
f1 ðy1 ; y2 Þ P f1 ðy1 þ y2 Þ
ð3:20Þ
f2 ðy1 ; y2 Þ P f2 ðy1 þ y2 Þ;
ð3:21Þ
and
whenever y1 þ y2 P ~r2 . Put
~r 2 ; r2 :¼ max 2r 1 ;
ð3:22Þ
c
where, as before, we take
c :¼ minfc~1 ; c~2 g: Moreover, put
X2 :¼ fðy1 ; y2 Þ 2 X : kðy1 ; y2 Þk < r 2 g:
ð3:23Þ
Note that if ðy1 ; y2 Þ 2 K \ @ X2 , then it follows that
y1 ðt 1 Þ þ y2 ðt2 Þ P
bþm
ðt 1 ;t 2 Þ2
4
min bþm
1 ;3ðbþm1 Þ 4
4
½y1 ðt 1 Þ þ y2 ðt 2 Þ P ckðy1 ; y2 Þk P ~r 2 :
2 ;3ðbþm2 Þ 4
ð3:24Þ
Now, define the numbers r1 and r2, with r1 < r2, by
r1 :¼ max
m1 þ b 4
m2 þ b m1 þ 1 ; m2 þ 1 4
and
r2 :¼ min
3ðm1 þ bÞ 3ðm2 þ bÞ m1 þ 1 ; m2 þ 1 ; 4 4
we assume in the sequel that b is sufficiently large so that ½r1 ; r2 \ N0 – £. Then for each ðy1 ; y2 Þ 2 K \ @ X2 we find that
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C.S. Goodrich / Applied Mathematics and Computation 217 (2011) 4740–4753
S1 ðy1 ; y2 Þ
mX mX b 1 þb 1 þb X bþ1 bþ1 1 þ m1 ¼ þ m1 ; s a1 ðs þ m1 c1im1 þ2 y1 ðkÞ þ dkm1 þ2 y1 ðkÞ þ k1 G1 2 2 s¼0 i¼m 2 k¼m 2 1
1
1Þf1 ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞ r2 X bþ1 1 P k1 þ m1 ; s a1 ðs þ m1 1Þ f1 c½ky1 k þ ky2 k P kðy1 ; y2 Þk; G1 2 2 s¼r
ð3:25Þ
1
where to arrive at the first inequality in (3.25) we have used the positivity assumption about w1(y1) and /1(y1) whenever ðy1 ; y2 Þ 2 K. Thus, we conclude from (3.25) that
kS1 ðy1 ; y2 Þk P
1 kðy1 ; y2 Þk: 2
ð3:26Þ
In a completely similar way, it can be shown that
kS2 ðy1 ; y2 Þk P
1 kðy1 ; y2 Þk: 2
ð3:27Þ
Consequently, (3.20)–(3.27) imply that
kSðy1 ; y2 Þk P kðy1 ; y2 Þk;
ð3:28Þ
whenever ðy1 ; y2 Þ 2 K \ @ X2 . Finally, notice that (3.19) implies that the operator S is a cone compression on K \ @ X1 , whereas (3.28) implies that S is a cone expansion on K \ @ X2 . Thus, all of the hypothesis of Lemma 3.2 are satisfied. Consequently we conclude that S has a fixed point, say ðy1 ; y2 Þ 2 K. As ðy1 ; y2 Þ is a positive solution of (1.1)–(1.3), the theorem is proved. h Remark 3.4. Observe that we may still be able to apply Theorem 3.3 to problem (1.1)–(1.3) even if none of the functionals
w1(y1), w2(y2), /1(y1), and /2(y2) is nonnegative for all yi P 0. This is the advantage of using the cone introduced by Infante and Webb. So, in particular, Theorem 3.3 provides a generalization and extension of [2, Theorem 3.3]. Importantly, Theorem 3.3 provides new results even in case m1 = m2 = 2 – that is, the integer-order case. Remark 3.5. Note that in the preceding arguments it is important that each of c1 and c2 and thus c is a constant. That c is constant here is a reflection of the fact that the Green’s function G satisfies a sort of Harnack-like inequality. Interestingly, however, in the continuous fractional setting, this may (see [20]) or may not (see [17]) be true. Remark 3.6. We point out that our results could be extended to the case of n equations and 2n boundary conditions. Since this is just a straightforward extension of the results presented in this section, we omit the details.
4. Existence of a positive solution: Case II We now wish to present an alternative method for deducing the existence of at least one positive solution to problem (1.1)–(1.3). In particular, in stead of using the cone given in (3.1), we shall now revert to a more traditional cone, whose use can be found in innumerable papers, including [2]. An advantage of this approach, however, is that it shall allow us to weaken hypothesis (G1). However, we do so at the expense of having to assume a priori the positivity of each of these functionals for all y P 0. Let us specifically state some new assumptions that we shall require in the sequel. G4: For i = 1, 2 we have that
lim
kyi k!0þ
wi ðyi Þ ¼ 0: kyi k
G5: For each i = 1, 2 we have that
lim
kyi k!0þ
/i ðyi Þ ¼ 0: kyi k
G6: For each i = 1, 2 we have that wi(yi) and /i(yi) are nonnegative for all yi P 0. L2: The constants k1 and k2 satisfy
K1 < k i < K2 ; for each admissible i, where
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C.S. Goodrich / Applied Mathematics and Computation 217 (2011) 4740–4753
8 " #1 " #1 9 b b <1 X = bþ1 1 X bþ1 þ m1 ; s a1 ðs þ m1 1Þf1 þ m2 ; s a2 ðs þ m2 1Þf2 K1 :¼ max cG ; cG2 :2 s¼0 1 ; 2 2 s¼0 2 and
8 " #1 " #1 9 <1 X = b b 1 X K2 :¼ min G1 ðs þ m1 1; sÞa1 ðs þ m1 1Þf1 ; G2 ðs þ m2 1; sÞa2 ðs þ m2 1Þf2 ; :3 s¼0 ; 3 s¼0 where fi and fi retain their earlier meaning from conditions (F1)–(F2), for each admissible i. Moreover, c is defined just as in Section 3. Before proceeding any further, let us point out explicitly that there do exist nontrivial functionals satisfying conditions (G4) and (G5). For example, consider the functional given by
/ðyÞ :¼ ½yðt 0 Þ6 ; where t0 is some number in the domain of y. Then it is clear that
0 6 limþ kyk!0
½yðt0 Þ6 ½yðt 0 Þ6 6 limþ ¼ limþ ½yðt 0 Þ5 ¼ 0; kyk yðt0 Þ kyk!0 kyk!0
from which it follows that / satisfies conditions (G4) and (G5). (Note that this naturally relies crucially upon the fact that / (y) is nonnegative for all y P 0.) Of course, not every possible functional will satisfy these conditions as is easy to see. We now present our existence theorem. This result complements Theorem 3.3 given in Section 3. Theorem 4.1. Suppose that conditions (F1)–(F2), (G4)–(G6), and (L2) hold. Then problem (1.1)–(1.3) has at least one positive solution. Proof. Begin by noting that by condition (L2) that there is
> 0 such that
8 " #1 " #1 9 b b <1 X = bþ1 1 X bþ1 þ m1 ; s a1 ðs þ m1 1Þ f1 þ m2 ; s a2 ðs þ m2 1Þ f2 max G ; G2 :2 s¼0 1 ; 2 2 s¼0 2 6 k1 ; k2 ð4:1Þ and
8 " #1 " #1 9 <1 X b b = 1 X k1 ; k2 6 min G ðs þ m1 1; sÞa1 ðs þ m1 1Þ f1 þ ; G2 ðs þ m2 1; sÞa2 ðs þ m2 1Þ f2 þ : :3 s¼0 1 ; 3 s¼0 ð4:2Þ Now, for the
determined by (4.1) and (4.2) we have by conditions (G4) and (G5) there exists a number g1 > 0 such that
/1 ðy1 Þ 6 ky1 k;
ð4:3Þ
whenever ky1k 6 g1, and a number g2 > 0 such that
w1 ðy2 Þ 6 ky1 k;
ð4:4Þ
whenever ky1k 6 g2. Put
g :¼ minfg1 ; g2 g:
ð4:5Þ
Then whenever k(y1, y2)k < g*, we have that both (4.3) and (4.4) hold. Now, for the same > 0 given in the first paragraph of this proof, we find that there exists a number g3 such that
f1 ðy1 ; y2 Þ 6 f1 þ ðy1 þ y2 Þ;
ð4:6Þ
whenever k(y1, y2)k < g3. Thus, by putting
g :¼ minfg ; g3 g;
ð4:7Þ
we get that (4.3), (4.4), and (4.6) are collectively true. So, define X1 by
X1 :¼ fðy1 ; y2 Þ 2 X : kðy1 ; y2 Þk < g g:
ð4:8Þ
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C.S. Goodrich / Applied Mathematics and Computation 217 (2011) 4740–4753
Then whenever ðy1 ; y2 Þ 2 K \ @ X1 we have, for any t1 2 ½m1 2; m1 þ bNm
1 2
S1 ðy1 ; y2 Þðt 1 Þ 6 ky1 k þ ky1 k þ k1
b X
,
G1 ðt 1 ; sÞa1 ðs þ m1 1Þf1 ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞ
s¼0
6 2ky1 k þ k1
b X s¼0
1 G1 ðs þ m1 1; sÞa1 ðs þ m1 1Þ f1 þ kðy1 ; y2 Þk 6 2 þ kðy1 ; y2 Þk; 3
ð4:9Þ
where we have used condition (L2) together with (4.2). An entirely dual argument reveals that
1 S2 ðy1 ; y2 ÞðtÞ 6 2 þ kðy1 ; y2 Þk: 3
ð4:10Þ
Therefore, putting (4.9) and (4.10) together we conclude that
kSðy1 ; y2 Þk 6 kðy1 ; y2 Þk;
ð4:11Þ
whenever ðy1 ; y2 Þ 2 K \ @ X1 and is chosen sufficiently small, which may be assumed without loss of generality. To complete the proof, we may give an argument nearly identical to the second half of the proof of Theorem 3.3. We omit this, and so, the proof is complete. h Remark 4.2. Observe that Theorem 4.1 provides a further generalization of the results in [2]. In particular, it allows for a somewhat greater range of nonlocal conditions and thus provides new results in the integer-order case. Moreover, it does not presuppose that the functionals are linear in y. Thus, as far as the author is aware, this provides new results in case m1 = m2 = 2, namely the classical integer-order case. Of course, it also provides, at least so far as the author is aware, new results in the fractional-order case. Remark 4.3. As in Section 3, it is clear how the results of this section can be extended to the case in which (1.1) is replaced with n equations and boundary conditions (1.2) and (1.3) are extended to 2n boundary conditions in the obvious way. As in Section 3, we omit the details of this extension.
5. An example In this section, we wish to provide an explicit numerical example of our results. In particular, it is worth noting that in [2] no numerical examples were provided for the reader. So, in that work it was not explicitly shown what sorts of nontrivial functions f(y1,y2), with f : [0, +1) [0, +1) ? [0, +1), satisfy conditions (F1)–(F2). Our Example 5.1 below will provide an explicit illustration. Example 5.1. Consider the boundary value problem
(
y1
3 3 7 D1:3 y1 ðtÞ ¼ k1 a1 t þ 10 f1 y1 t þ 10 ; y2 t þ 10 ; 7 3 7 f2 y1 t þ 10 ; y2 t þ 10 ; D1:7 y2 ðtÞ ¼ k2 a2 t þ 10
ð5:1Þ
7 1 13 1 53 ¼ ; y1 y1 10 12 10 25 10
ð5:2Þ
y1
213 1 83 1 73 ¼ ; y1 y1 10 30 10 100 10
3 1 17 1 77 y2 ¼ y2 y2 ; 10 40 10 150 10
y2
217 1 47 1 107 ¼ y2 y2 ; 10 17 20 30 20
ð5:3Þ
where we take
a1 ðtÞ :¼ et4 ;
ð5:4Þ
a2 ðtÞ :¼ et4 ;
ð5:5Þ
f1 ðy1 ; y2 Þ :¼ 5000ey2 ðy1 þ y2 Þ þ ðy1 þ y2 Þ
ð5:6Þ
f2 ðy1 ; y2 Þ :¼ 7500ey1 ðy1 þ y2 Þ þ ðy1 þ y2 Þ;
ð5:7Þ
and
with f1, f2 : [0, +1) [0, +1) ? [0, +1). It is clear from (5.2) and (5.3) that in problem (5.1)–(5.3) we have taken
w1 ðy1 Þ :¼
1 13 1 53 ; y1 y1 12 10 25 10
ð5:8Þ
C.S. Goodrich / Applied Mathematics and Computation 217 (2011) 4740–4753
4751
/1 ðy1 Þ :¼
1 83 1 73 ; y1 y1 30 10 100 10
ð5:9Þ
w2 ðy2 Þ :¼
1 17 1 77 y2 y2 ; 40 10 150 10
ð5:10Þ
/2 ðy2 Þ :¼
1 47 1 107 y2 y2 ; 17 20 30 20
ð5:11Þ
Note, in addition, that y1 is defined on the time scale
7 3 213 ; ; ;...; 10 10 10
whereas y2 is defined on the time scale
3 7 217 ; ;...; : 10 10 10
In particular, we have chosen m1 ¼ 13 ; m2 ¼ 17 , and b = 20 here. We shall select k1 and k2 below. 10 10 We now endeavor to check that each of conditions (F1)–(F2), (G1)–(G3), and (L1) holds. It is easy to check that (F1)–(F2) hold. On the other hand, since (5.8)–(5.11) reveals the functionals to be linear in y1 and y2, respectively, it is immediate that (G1) holds. On the other hand, to see that conditions (G2)–(G3) hold, let us note first both that
1 1 1 1 1 1 þ þ þ ¼ 6 12 25 30 100 6 4
ð5:12Þ
and that
1 1 1 1 421 1 þ þ þ ¼ 6 : 40 150 17 30 3400 4
ð5:13Þ
Furthermore, numerical calculations show that mj þb X i¼mj 2
cjimj þ2 Gj ði; sÞ P 0
ð5:14Þ
and that mj þb X
j
dkmj þ2 Gj ðk; sÞ P 0;
ð5:15Þ
k¼mj 2
for each j. So, it follows that condition (G2) holds. Finally, routine numerical calculations reveal that
w1 ða1 Þ 0:012; w1 ðb1 Þ 0:012; w2 ða2 Þ 0:012; w2 ðb2 Þ 0:0012
ð5:16Þ
and that
/1 ða1 Þ 0:00091; /1 ðb1 Þ 0:018; /2 ða2 Þ 0:015; /2 ðb2 Þ 0:000099:
ð5:17Þ
Hence, condition (G3) is satisfied. Finally, we check condition (L1) to determine the admissible range of eigenvalues, ki. To this end, recall from [26, Theorem 3.2] that the constant c in Lemma 2.6 is
8 9 2 3 m1 3ðbþmÞ > m1 < = 2 ðm þ b þ 1Þm1 7 bþm m1 > 4 1 6 3ðb þ mÞ c :¼ min 4 5; 4 m1 : m 1 m 1 > 4 ðm þ b 1Þ ðb þ mÞ > : 3ðbþmÞ ;
ð5:18Þ
4
Thus, using the definition of c provided by (5.18), we numerically estimate that
n o K1 max f1 3:288 107 ; f2 1:0322 107 ¼ maxf3:288 107 ; 1:0322 107 g ¼ 3:337 107 ;
ð5:19Þ
whereas
n o K2 min f1 1:871 109 ; f2 1:363 109 ¼ minf5001 1:871 109 ; 7501 1:363 109 g ¼ minf9:357 106 ; 1:022 105 g ¼ 9:357 106 : In summary, then, so long as k1, k2 are chosen so that
ð5:20Þ
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C.S. Goodrich / Applied Mathematics and Computation 217 (2011) 4740–4753
3:337 107 6 ki 6 9:357 106 ; for each admissible i, then condition (L1) is satisfied. So, suppose that
k1 ; k2 2 ½3:337 107 ; 9:357 106 :
ð5:21Þ
Then (5.4)–(5.20) together with condition (5.21) imposed on k1, k2 imply that the hypotheses of Theorem 3.3 are satisfied. Therefore, we conclude that problem (5.1)–(5.3) has at least one positive solution under these conditions, as desired. A similar example could be provided for Theorem 4.1. Remark 5.2. We note in passing that a fairly broad class of (nontrivial) functions satisfying conditions (F1)–(F2) are given by f(x) :¼ C1eg(x)rH(x), where g : Rnþ ! ½0; þ1Þ; f : Rnþ ! ½0; þ1Þ; C 1 > 0 is a constant, H : Rnþ ! Rnþ is the vector field defined P by HðxÞ :¼ ni¼1 12 x2i ei , where ei is the ith standard basis vector in Rn , and by Rnþ we mean the closure of the open positive cone in Rn . Obviously the class of functions L(y1,y2) = ay1 + ay2 trivially satisfies (F1)–(F2), for a > 0.
6. Conclusions In this paper we have considered a coupled system of discrete FBVPs as given by (1.1)–(1.3). We have shown that under standard assumptions on the ai’s and the nonlinearities fi(y1, y2), where i = 1, 2, problem (1.1)–(1.3) may admit a positive solution even if some or all of the functionals wi(yi) and /i(yi), where i = 1, 2, are nonpositive for some yi P 0. This provides a generalization of the results in [2] as well as certain recent research on discrete FBVPs. While assuming that the wi’s and /i’s are nonnegative for all yi P 0 is a standard assumption to make, it worthwhile to see how the idea of Infante and Webb [1] allows us to weaken this assumption. Moreover, we have shown that under somewhat different conditions, a more traditional cone may be used to deduce the existence of at least one positive solution of problem (1.1)–(1.3). This also provides new results in case m1 = m2 = 2, which is the integer-order case, and thus improves, in this sense, the result given in [2], for instance. We would also like to point out that it might be interesting to generalize certain of the results here to delta-nabla problems – that is, where the ith equation in (1.1) is replaced by
Dli rmi yðt þ mÞ ¼ ki ai ðt þ li þ mi 1Þfi ðy1 ðt þ li þ mi 1Þ; . . . ; yn ðt þ li þ mi 1ÞÞ;
ð6:1Þ
with li, mi 2 (1, 2], for each 1 6 i 6 n. Given that the study of discrete fractional delta-nabla problems is nontrivial, in no small part due to the composition of two fractional derivatives (e.g., see [34]), studying this sort of generalization could be interesting. In addition, it might be interesting to generalize Eq. (1.1) by allowing f to depend upon one or both of the fractional differences Dm1 y and Dm2 y. Such results would build upon the results given in [41], for example, and could prove to be mathematically interesting. In any case, we hope that these ideas can be further extended both in the fractional and integer-order settings. References [1] G. Infante, J.R.L. Webb, Positive solutions of nonlocal boundary value problems: a unified approach, J. London Math. Soc. 74 (2) (2006) 673–693. [2] J. Henderson, S.K. Ntouyas, I.K. Purnaras, Positive solutions for systems of nonlinear discrete boundary value problems, J. Differ. Equat. Appl. 15 (2009) 895–912. [3] X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett. 22 (2009) 64–69. [4] D.R. Anderson, Solutions to second-order three-point problems on time scales, J. Differ. Equat. Appl. 8 (2002) 673–688. [5] E.R. Kaufmann, Y.N. Raffoul, Positive solutions for a nonlinear functional dynamic equation on a time scale, Nonlinear Anal. TMA 62 (2005) 1267–1276. [6] W. Cheung, J. Ren, P.J.Y. Wong, D. Zhao, Multiple positive solutions for discrete nonlocal boundary value problems, J. Math. Anal. Appl. 330 (2007) 900– 915. [7] R. Agarwal, D. O’Regan, A coupled system of difference equations, Appl. Math. Comput. 114 (2000) 39–49. [8] D.R. Anderson, Existence of solutions for first-order multi-point problems with changing-sign nonlinearity, J. Differ. Equat. Appl. 14 (2008) 657–666. [9] R.I. Avery, J. Henderson, Twin solutions of boundary value problems for ordinary differential equations and finite difference equations, Comput. Math. Appl. 42 (2001) 695–704. [10] J. Henderson, P.J.Y. Wong, Double symmetric solutions for discrete Lidstone boundary value problems, J. Differ. Equat. Appl. 7 (2001) 811–828. [11] R. Ma, Y.N. Raffoul, Positive solutions of three-point nonlinear discrete second order boundary value problem, J. Differ. Equat. Appl. 10 (2004) 129–138. [12] C. Wang, X. Han, C. Li, Positive solutions to nonlinear second-order three-point boundary-value problems for difference equation with change of sign, Electon. J. Differ. Eqs. (87) (2008) 10. [13] P. Wang, Y. Wang, Existence of positive solutions for second-order m-point boundary value problems on time scales, Acta Mathmeticae Applicatae Sinica, English Series 22 (2006) 457–468. [14] R. Almeida, D.F.M. Torres, Calculus of variations with fractional derivatives and fractional integrals, Appl. Math. Lett. 22 (2009) 1816–1820. [15] A. Arara et al, Fractional order differential equations on an unbounded domain, Nonlinear Anal. TMA 72 (2010) 580–586. [16] A. Babakhani, V. Daftardar-Gejji, Existence of positive solutions of nonlinear fractional differential equations, J. Math. Anal. Appl. 278 (2003) 434–442. [17] Z. Bai, H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005) 495–505. [18] J.V. Devi, Generalized monotone method for periodic boundary value problems of Caputo fractional differential equation, Commun. Appl. Anal. 12 (2008) 399–406. [19] K. Diethelm, N. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229–248. [20] C.S. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett. 23 (2010) 1050–1055. [21] A. Malinowska, D.F.M. Torres, Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative, Comput. Math. Appl. 59 (2010) 3110–3116.
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[22] X. Xu, D. Jiang, C. Yuan, Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Anal. TMA 71 (2009) 4676–4688. [23] F.M. Atici, P.W. Eloe, Fractional q-calculus on a time scale, J. Nonlinear Math. Phys. 14 (3) (2007) 333–344. [24] F.M. Atici, P.W. Eloe, A transform method in discrete fractional calculus, Int. J. Differ. Equat. 2 (2) (2007) 165–176. [25] F.M. Atici, P.W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137 (3) (2009) 981–989. [26] F.M. Atici, P.W. Eloe, Two-point boundary value problems for finite fractional difference equations, J. Differ. Equat. Appl., in press, doi:10.1080/ 10236190903029241. [27] F.M. Atici, S. Sßengül, Modeling with fractional difference equations, J. Math. Anal. Appl., 2010, doi:10.1016/j.jmaa.2010.02.009. [28] C.S. Goodrich, Continuity of solutions to discrete fractional initial value problems, Comput. Math. Appl. 59 (2010) 3489–3499. [29] C.S. Goodrich, Solutions to a discrete right-focal fractional boundary value problem, Int. J. Differ. Equat. 5 (2010), in press. [30] C.S. Goodrich, On a discrete fractional three-point boundary value problem, J. Differ. Equat. Appl., doi:10.1080/10236198.2010.503240. [31] C.S. Goodrich, Some new existence results for fractional difference equations, Int. J. Dyn. Syst. Differ. Equat., in press. [32] C.S. Goodrich, Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions, Comput. Math. Appl., doi:10.1016/ j.camwa.2010.10.041. [33] C.S. Goodrich, On positive solutions to nonlocal fractional and integer-order difference equations, submitted for publication. [34] C.S. Goodrich, On the operational properties of the fractional difference applied to a fractional delta-nabla difference equation with a nonlocal condition, submitted for publication. [35] N.R.O. Bastos, et al., Discrete-time fractional variational problems, Signal Process., 2010, doi:10.1016/j.sigpro.2010.05.001. [36] G.A. Anastassiou, Principles of delta fractional calculus on time scales and inequalities, Math. Comput. Model. 52 (2010) 556–566. [37] F.M. Atici, P.W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qualitative Theor. Differ. Equat. (spec. ed. I) 3 (2009) 1–12. [38] G.A. Anastassiou, Foundations of nabla fractional calculus on time scales and inequalities, Comput. Math. Appl. 59 (2010) 3750–3762. [39] D. Dunninger, H. Wang, Existence and multiplicity of positive solutions for elliptic systems, Nonlinear Anal. TMA 29 (1997) 1051–1060. [40] R. Agarwal, M. Meehan, D. O’Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001. [41] J. Henderson, H.B. Thompson, Existence of multiple solutions for second-order discrete boundary value problems, Comput. Math. Appl. 43 (2002) 1239–1248.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Existence of a positive solution to a system of discrete fractional boundary value problems Christopher S. Goodrich Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA
a r t i c l e
i n f o
a b s t r a c t
Keywords: Discrete fractional calculus Eigenvalues Positive solution Coupled system of boundary value problems Cone
We analyze a system of discrete fractional difference equations subject to nonlocal boundary conditions. We consider the system of equations given by Dmi yi ðtÞ ¼ ki ai ðtþ mi 1Þfi ðy1 ðt þ m1 1Þ; y2 ðt þ m2 1ÞÞ, for t 2 ½0; bN0 , subject to yi(mi 2) = wi(yi) and yi(mi + b) = /i(yi), for i = 1, 2, where wi ; /i : Rbþ3 ! R are given functionals. We also assume that mi 2 (1, 2], for each i. Although we assume that both ai and fi(y1, y2) are nonnegative for each i, we do not necessarily presume that each wi(yi) and /i(yi) is nonnegative for each i and each yi P 0. This generalizes some recent results both on discrete fractional boundary value problems and on discrete integer-order boundary value problems, and our techniques provide new results in each case. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction In this paper we are concerned with a pair of discrete fractional boundary value problems of the form
Dm1 y1 ðtÞ ¼ k1 a1 ðt þ m1 1Þf1 ðy1 ðt þ m1 1Þ; y2 ðt þ m2 1ÞÞ;
ð1:1Þ
Dm2 y2 ðtÞ ¼ k2 a2 ðt þ m2 1Þf2 ðy1 ðt þ m1 1Þ; y2 ðt þ m2 1ÞÞ; for t 2 ½0; bN0 , subject to the boundary conditions
y1 ðm1 2Þ ¼ w1 ðy1 Þ;
y2 ðm2 2Þ ¼ w2 ðy2 Þ;
y1 ðm1 þ bÞ ¼ /1 ðy1 Þ;
y2 ðm2 þ bÞ ¼ /2 ðy2 Þ;
ð1:2Þ ð1:3Þ bþ3
where ki > 0; ai : R ! ½0; þ1Þ; mi 2 ð1; 2 for each 1 6 i 6 2, and for each i we have that wi ; /i : R ! R are given functionals. We shall also assume that fi: [0, +1) [0, +1) ? [0, +1) and is continuous for each admissible i. We point out that we do not necessarily assume that either the wi’s or the /i’s are nonnegative for all y P 0 – i.e., for all nonnegative functions y(t). In spite of this, we shall still show that problem (1.1)–(1.3) can admit a positive solution for certain values of the ki’s (i.e., the eigenvalues) and some standard assumptions on the nonlinearities fi, for 1 6 i 6 2. We shall use some ideas introduced recently by Infante and Webb [1] to accomplish this. In addition to this, we shall show that by weakening the assumptions made on the functionals but at the same time imposing a nonnegativity condition on them, it remains possible to deduce the existence of at least one positive solution to (1.1)–(1.3). Thus, in the sequel we shall give two relatively distinct sets of conditions under which problem (1.1)–(1.3) will admit at least one positive solution. It should be noted, as will be described in the sequel, that our theorems provide completely new results and techniques even in the case when (1.1) is of integer-order – that is, m1 = m2 = 2. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.11.029
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To place problem (1.1)–(1.3) in an appropriate context, we first point out that a simpler, integer-order version of (1.1)– (1.3) was considered recently by Henderson, Ntouyas, and Purnaras [2]. In particular, they considered the problem
8 2 > < D uðn 1Þ þ kaðnÞf ðuðnÞ; v ðnÞÞ ¼ 0 D2 v ðn 1Þ þ lbðnÞgðuðnÞ; v ðnÞÞ ¼ 0 > :
ð1:4Þ
subject to the multipoint-type boundary conditions
uð0Þ ¼ buðgÞ;
uðNÞ ¼ auðgÞ;
v ð0Þ ¼ bv ðgÞ; v ðNÞ ¼ av ðgÞ;
ð1:5Þ
where g 2 {1, . . . , N 1}, n 2 {1, 2, . . . , N 1}, N P 4, and a, b, k, l > 0. They then deduced, under various assumptions on f, g, k, l, a and b, the existence of at least one positive solution to (1.4) and (1.5). Clearly, if we put m1 = m2 = 2, and appropriately choose wi(yi) and /i(yi), 1 6 i 6 2, then (1.1)–(1.3) reduces to problem (1.4) and (1.5). So, in particular, our results in this paper will provide a generalization of the results provided in [2]. Moreover, it should be noted that a similar problem in the continuous fractional calculus was recently considered by Su [3]. We next point out that discrete multipoint (or nonlocal) boundary value problems (BVPs) have been extensively studied in recent years. This is also true in the more general case of an arbitrary time scale T. As for but a few examples, we note that Anderson [4] considered the problem uDr(t) + f(t, u(t)) = 0, u(0) = 0, a u(g) = u(T), on a time scale T. Kaufmann and Raffoul [5] considered a closely related problem. Similarly, Cheung, et al. [6] considered a delta-nabla difference equation of the form rDu(k) + f(k, u(k)) = 0 together with a couple of a different specific nonlocal conditions. For some other papers on discrete nonlocal BVPs, we refer the reader to [7–13] and the references therein. On the other hand, the discrete fractional calculus is a relatively new and emerging area of mathematics. While the continuous fractional calculus has been studied for some time and put to good use in modeling and applied mathematics (see, for example, [14–22] and the references therein), the discrete fractional calculus has only recently begun to be developed seriously. In a series of very useful and interesting papers, namely [23–26], Atici and Eloe developed some of the basic theory of both discrete fractional IVPs and BVPs with delta derivative on the time scale Z. Moreover, a recent paper by Atici and Sßengül [27] has provided some analysis of discrete fractional variational problems; their paper also provided some initial attempts at using the discrete fractional calculus to model biological processes. Similarly, the present author has established in [28–34] some results on both discrete fractional IVPs and BVPs. Finally, recent papers by Bastos, et al. [35] and Anastassiou [36] have provided extensions of the discrete fractional calculus with delta derivative to more general time scales. Thus, the discrete fractional calculus has recently attracted increasing attention from a growing number of researchers. It is also worth noting that some progress made toward developing the discrete fractional calculus with the nabla derivative on the time scale Z. Three works come to mind here. Firstly, we are aware of a work by Atici and Eloe [37], in which certain of the basic theory of fractional nabla derivatives on the time scale Z was developed. Secondly, a recent paper by Anastassiou [38] has established some results regarding the fractional calculus on time scales with nabla derivative. Thirdly, in a recent work by the present author [34], a discrete fractional delta-nabla BVP of the form Dlrmy(t) = f(t + l + m 1, y(t + l + m 1)) subject to the boundary conditions y(l + m 2) = w(y), y(l + m + b) = 0 was considered for 0 < l 6 1, 0 < m 6 1, and l + m 2 (1, 2]. Since delta-nabla BVPs have proved to be interesting to study in the integer-order case, we feel that their study in the fractional case will continue to be worthwhile. Underlying many of these papers, at least the ones that consider FBVPs with delta derivative, is a consideration of the general discrete fractional boundary value problem (FBVP)
Dm yðtÞ þ f ðt þ m 1; yðt þ m 1ÞÞ ¼ 0; subject to the boundary conditions
yðm 2Þ ¼ wðyÞ;
yðb þ mÞ ¼ /ðyÞ;
for m 2 (1, 2], where w and / are given functionals. For example, in [26] the authors studied this problem in case w / 0 – that is, the so-called conjugate FBVP. Similarly, in [29], the author studied this problem in case w(y) 0 and /(y) = Dy(m + b) – that is, the so-called right-focal FBVP. With this context in mind, it is clear that problem (1.1)–(1.3), in addition to a generalization of [2], is a generalization of certain of the FBVPs considered in [26,29,30]. Indeed, our results in this work will generalize some of the previous results on discrete FBVPs as well as discrete integer-order BVPs. Specifically, we will analyze (1.1)–(1.3) by developing a representation for solutions of (1.1)–(1.3) in terms of the fixed point of an operator. Having fixed our framework, we shall then use some ideas presented by Infante and Webb [1] to deduce the existence of a positive solution (1.1)–(1.3) even if none of the wi’s and /i’s is nonnegative for all y P 0. In addition, we shall then present in Section 4 a different existence result that does not use the cone introduced by Infante and Webb but rather uses a more traditional cone together with some different assumptions about the each of the wi’s and /i’s. Finally, in Section 5 we shall give a specific numerical example of our result, something lacking in [2]. We conclude the paper in Section 6 with some thoughts for future research in the directions outlined in this paper.
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2. Preliminaries We first wish to collect some basic lemmas that will be important to us in the sequel. These and other related results and their proofs can be found, for example, in [23–26]. Definition 2.1. We define
Cðt þ 1Þ ; Cðt þ 1 mÞ for any t and m for which the right-hand side is defined. We also appeal to the convention that if t + 1 m is a pole of the tm :¼
Gamma function and t + 1 is not a pole, then tm = 0. Definition 2.2. The mth fractional sum of a function f defined on the set Na :¼ fa; a þ 1; . . .g, for m > 0, is defined to be
Dm f ðtÞ ¼ Dm f ðt; aÞ :¼
tm 1 X ðt s 1Þm1 f ðsÞ; CðmÞ s¼a
where t 2 fa þ m; a þ m þ 1; . . .g ¼: Naþm . We also define the mth fractional difference, where m > 0 and 0 6 N 1 < m 6 N with N 2 N, to be
Dm f ðtÞ :¼ DN DðNmÞ f ðtÞ; where t 2 NaþNm . Lemma 2.3. Let t and m be any numbers for which tm and tm1 are defined. Then
Dtm ¼ mt m1 : Lemma 2.4. Let 0 6 N 1 < m 6 N, where N 2 N and N 1P0. Then
Dm Dm yðtÞ ¼ yðtÞ þ C 1 tm1 þ C 2 t m2 þ þ C N t mN ; for some C i 2 R, with 1 6 i 6 N. We now wish to fix our framework for the study of problem (1.1)–(1.3). First of all, we let Bi represent the Banach space of all maps from ½mi 2; . . . ; mi þ bNm 2 into R when equipped with the usual maximum norm, k k. We shall then put i
X :¼ B1 B2 : By equipping X with the norm
kðy1 ; y2 Þk :¼ ky1 k þ ky2 k; it follows that ðX ; k kÞ is a Banach space, too – see, for example, [2,39]. Next we wish to develop a representation for a solution of (1.1)–(1.3) as the fixed point of an appropriate operator on X . To accomplish this, however, we present now some straightforward adaptations of results from [26] that will be of use here. Because the proofs of these adaptions are evident, we do not include them. Lemma 2.5 [26]. Let 1 < m 6 2 and h : ½m 1; m þ b 1Nm1 ! R be given. The unique solution of the FBVP Dmy(t) = h(t + m 1), P y(m 2) = 0 = y(m + b) is given by yðtÞ ¼ bs¼0 Gðt; sÞhðs þ m 1Þ, where G : ½m 2; m þ bNm2 ½0; bN0 ! R is defined by
8 m1 > t ðmþbs1Þm1 ðt s 1Þm1 ; 0 6 s < t m þ 1 6 b; 1 < ðmþbÞm1 Gðt; sÞ :¼ m1 ðmþbs1Þm1 t CðmÞ > : ; 0 6 t m þ 1 6 s 6 b: ðmþbÞm1
Lemma 2.6 [26]. The Green’s function G(t, s) given in Lemma 2.5 satisfies: 1. G(t, s) P 0 for each ðt; sÞ 2 ½m 2; m þ bNm2 ½0; bN0 ; 2. maxt2½m2;mþbN Gðt; sÞ ¼ Gðs þ m 1; sÞ for each s 2 ½0; bN0 ; and m2 3. there exists a number c 2 (0, 1) such that
min
3ðbþmÞ bþm 6t6 4 4
Gðt; sÞ P c
for s 2 ½0; bN0 .
max
t2½m2;mþbN
m2
Gðt; sÞ ¼ cGðs þ m 1; sÞ;
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Now consider the operator S : X ! X defined by
Sðy1 ; y2 Þðt 1 ; t2 Þ :¼ ðS1 ðy1 ; y2 Þðt1 Þ; S2 ðy1 ; y2 Þðt2 ÞÞ;
ð2:1Þ
where we define S1 : X ! B1 by
S1 ðy1 ; y2 Þðt1 Þ :¼ a1 ðt1 Þw1 ðy1 Þ þ b1 ðt 1 Þ/1 ðy1 Þ þ k1
b X
G1 ðt 1 ; sÞa1 ðs þ m1 1Þf1 ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞ
ð2:2Þ
G2 ðt 2 ; sÞa2 ðs þ m2 1Þf2 ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞ;
ð2:3Þ
s¼0
and S2 : X ! B2 by
S2 ðy1 ; y2 Þðt2 Þ :¼ a2 ðt2 Þw2 ðy2 Þ þ b2 ðt 2 Þ/2 ðy2 Þ þ k2
b X s¼0
where, for each j = 1, 2, we define
aj ðtÞ :¼
bj ðtÞ :¼
1 1 mj 1 m 2 ; t tj bþ2 Cðmj 1Þ t
ð2:4Þ
mj 1
ð2:5Þ
mj 1 :
ðm þ bÞ
Moreover, Gj(t, s) is as given in Lemma 2.5 with m replaced by mj. We claim that whenever ðy1 ; y2 Þ 2 X is a fixed point of the operator S, it follows that the pair of functions y1(t) and y2(t) is a solution to problem (1.1)–(1.3). Theorem 2.7. Let fj : R2 ! ½0; þ1Þ and wj ; /j 2 Cð½mj 2; mj þ bNm 2 ; RÞ be given, for j = 1,2. If ðy1 ; y2 Þ 2 X is a fixed point of S, j then the pair of functions y1(t) and y2(t) is a solution to problem (1.1)–(1.3). Proof. Suppose that the operator S has a fixed point, say ðy1 ; y2 Þ 2 X . Let ðt1 ; t2 Þ 2 Nm1 2 Nm2 2 . Then, for example, we have that
y1 ðt 1 Þ ¼ S1 ðy1 ; y2 Þðt 1 Þ;
ð2:6Þ
where S1 is as in (2.2). It is easy to check that S1(y1, y2)(m1 2) = w(y1) and that S1(y1, y2)(m1 + b) = /1(y1), so that the boundary conditions in (1.2) and (1.3) are satisfied. Similarly, one can check that S1(y1, y2)(t1) satisfies the difference equation in (1.1). Since similar verifications may be made for S2(y1, y2)(t2), the claim follows. h The following lemma and its associated corollary are of particular importance in the sequel. Because the proofs of each of these is straightforward, we omit them. Lemma 2.8. For each admissible j (i.e., j = 1 or 2), the function aj(tj) is decreasing in tj, for tj 2 ½mj 2; mj þ bNm 2 . Also, j mintj 2½mj 2;mj þbN aj ðtj Þ ¼ 0 and maxtj 2½mj 2;mj þbN aj ðtj Þ ¼ 1. On the other hand, for each admissible j, the function bj(tj) is strictly mj 2 mj 2 increasing in tj, for tj 2 ½mj 2; mj þ bNm 2 . In addition, mintj 2½mj 2;mj þbN bj ðt j Þ ¼ 0 and maxtj 2½mj 2;mj þbN bj ðt j Þ ¼ 1. mj 2
j
Corollary 2.9. Let j be fixed and admissible (i.e., j = 1 or 2). Let Ij :¼ mintj 2Ij aj ðt j Þ ¼ M aj kaj k and mintj 2Ij bj ðtj Þ ¼ M bj kbj k.
h
bþmj 4
mj 2
;
i
3ðbþmj Þ 4
. There are constants M aj ; M bj 2 ð0; 1Þ such that
Let us conclude this section with a remark. Remark 2.10. Observe that unlike the integer-order problem considered in [2], in the fractional-order problem we encounter a significant problem with respect to the domains of the various operators. This has been clearly seen in the preceding discussion, wherein the functions y1(t) and y2(t) are defined on (possibly) different sets – namely Nm1 2 in the former case and Nm2 2 in the latter case. This complication arises in the discrete fractional calculus due to the domain shifting of the fractional difference and sum operators. We will continue to encounter this complication in Sections 3 and 4 of this paper.
3. Existence of a positive solution: Case I We now present the first of two theorems for the existence of at least one positive solution to problem (1.1)–(1.3). As was pointed out in Section 1, in this particular section we shall not assume that either wi(yi) or /i(yi), 1 6 i 6 2, is nonnegative for all yi P 0. Rather, we shall make some other assumptions about these functionals. Let us now present the conditions that we shall assume henceforth. We note that conditions (F1)–(F2) are the same conditions given by Henderson, et al. [2]. Moreover, condition (L1) is essentially the same condition (up to a constant multiple) as given in [2, Theorem 3.1].
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F1: There exist numbers f1 and f2 , with f1 ; f2 2 ð0; þ1Þ, such that
lim
ðy1 ;y2 Þ!ð0þ ;0þ Þ
f1 ðy1 ; y2 Þ ¼ f1 y1 þ y2
and
lim
ðy1 ;y2 Þ!ð0þ ;0þ Þ
f2 ðy1 ; y2 Þ ¼ f2 : y1 þ y2
F2: There exist numbers f1 and f2 , with f1 ; f2 2 ð0; þ1Þ, such that
lim
ðy1 ;y2 Þ!ð1;1Þ
f1 ðy1 ; y2 Þ ¼ f1 y1 þ y2
and
lim
ðy1 ;y2 Þ!ð1;1Þ
f2 ðy1 ; y2 Þ ¼ f2 : y1 þ y2
G1: For each admissible j, the functionals wj and /j are linear. In particular, we assume both that
wj ðyj Þ ¼
mj þb X i¼mj 2
cjimj þ2 yj ðiÞ
and that
/j ðyj Þ ¼
mj þb X
j
dkmj þ2 yj ðkÞ;
k¼mj 2 j
for constants cjimj þ2 ; dkmj þ2 2 R. G2: For each admissible j, we have both that mj þb X i¼mj 2
cjimj þ2 Gj ði; sÞ P 0
and that mj þb X
j
dkmj þ2 Gj ðk; sÞ P 0;
k¼mj 2
for each s 2 ½0; bN0 , and in addition that mj þb X i¼mj 2
cjimj þ2 þ
mj þb X k¼mj 2
j
dkmj þ2 6
1 : 4
G3: We have that each of wi(ai), wi(bi), /i(ai), and /i(bi) is nonnegative for each admissible i – that is, i = 1, 2. L1: The constants k1 and k2 satisfy
K1 < k i < K2 ; for each i, where
8 " #1 " #1 9 = <1 X b b X bþ1 1 b þ 1 þ m1 ; s a1 ðs þ m1 1Þf1 þ m2 ; s a2 ðs þ m2 1Þf2 K1 :¼ max cG1 ; cG2 ; :2 s¼0 2 2 s¼0 2 and
8 " #1 " #1 9 b b <1 X = X 1 K2 :¼ min G1 ðs þ m1 1; sÞa1 ðs þ m1 1Þf1 ; G2 ðs þ m2 1; sÞa2 ðs þ m2 1Þf2 ; :4 s¼0 ; 4 s¼0 where c 2 (0, 1) is a constant defined by
c :¼ minfMa1 ; Ma2 ; Mb1 ; Mb2 ; c1 ; c2 g; where M a1 ; M a2 ; M b1 , and M b2 each come from Corollary 2.9 and c1 and c2 are associated by Lemma 2.6 to G1(t, s) and G2(t, s), respecitively. (Recall that these are defined on possibly different time scales). In the sequel, we shall also make use of the cone
K :¼ fðy1 ; y2 Þ 2 X : y1 ; y2 P 0;
bþm
min bþm
4
4
ðt 1 ;t 2 Þ2
P 0; /j ðyj Þ P 0; for each j ¼ 1; 2g;
1 ;3ðbþm1 Þ 4
½y1 ðt 1 Þ þ y2 ðt 2 Þ P ckðy1 ; y2 Þk; wj ðyj Þ
2 ;3ðbþm2 Þ 4
ð3:1Þ
where c is defined exactly as in the statement of condition (L1) above. Clearly, we have that K # X . In order to show that S has a fixed point in K, we must first demonstrate that K is invariant under S – that is, S : K ! K. We show this now.
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Lemma 3.1. Let S : X ! X be the operator defined as in (2.1). Then S : K ! K. Proof. Suppose that ðy1 ; y2 Þ 2 K. We show first that ðt1 ;t 2 Þ2
bþm
min bþm
4
4
1 ;3ðbþm1 Þ 4
2 ;3ðbþm2 Þ 4
½S1 ðy1 ; y2 Þðt 1 Þ þ S2 ðy1 ; y2 Þðt 2 Þ P ckSðy1 ; y2 Þk;
whenever ðy1 ; y2 Þ 2 K. So note that
bþmin
S1 ðy1 ; y2 Þðt 1 Þ P m 3ðbþm Þ
t1 2
4
1;
1
4
þ t1 2
bþmin
m 3ðbþm Þ 4
1;
4
1
b X
t1 2
bþmin
½a1 ðt 1 Þ/1 ðy1 Þ þ b1 ðt1 Þw1 ðy1 Þ m 3ðbþm Þ 4
1;
4
1
G1 ðt1 ; sÞa1 ðs þ m1 1Þf1 ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞ
s¼0
P Ma1 ka1 k/1 ðy1 Þ þ M b1 kb1 kw1 ðy1 Þ þ k1 max
t 1 2½m1 2;m1 þb
c1 G1 ðs þ m1 1; sÞa1 ðs þ m1 1Þf1 ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞ
s¼0
" ~1 Pc
b X
a1 ðt1 Þ/1 ðy1 Þ þ b1 ðt1 Þw1 ðy1 Þ þ k1
b X
# G1 ðt 1 ; sÞa1 ðs þ m1 1Þf1 ðy1 ðs þ m1 1Þ; y2 ðs þ m1 1ÞÞ
s¼0
~1 kS1 ðy1 ; y2 Þk; ¼c
ð3:2Þ
where c~1 :¼ minfM a1 ; M b1 ; c1 g, whence t1 2
c 1 kS1 ðy1 ; y2 Þk; bþmin
S1 ðy1 ; y2 Þðt1 Þ P e m 3ðbþm Þ 4
1;
4
ð3:3Þ
1
as desired. In an entirely similar manner to (3.2), we deduce that t2 2
~2 kS2 ðy1 ; y2 Þk; bþmin
S2 ðy1 ; y2 Þðt2 Þ P c m 3ðbþm Þ 4
2;
4
ð3:4Þ
2
~2 :¼ minfM a2 ; M b2 ; c2 g. where c ~1 ; c ~2 g. Consequently, from (3.3) and (3.4) it follows that Now, put c :¼ minfc ðt1 ;t 2 Þ2
bþm
min bþm
4
4
1 ;3ðbþm1 Þ 4
2 ;3ðbþm2 Þ 4
½S1 ðy1 ; y2 Þðt 1 Þ þ S2 ðy1 ; y2 Þðt 2 Þ P
ðt 1 ;t 2 Þ2
bþm
min bþm
4
4
þ ðt 1 ;t 2 Þ2
1 ;3ðbþm1 Þ 4
S1 ðy1 ; y2 Þðt1 Þ
2 ;3ðbþm2 Þ 4
bþm
min bþm
4
4
1 ;3ðbþm1 Þ 4
S2 ðy1 ; y2 Þðt2 Þ
2 ;3ðbþm2 Þ 4
~2 kS2 ðy1 ; y2 ÞkÞ ~1 kS1 ðy1 ; y2 Þk þ c P ðc P ðckS1 ðy1 ; y2 Þk þ ckS2 ðy1 ; y2 ÞkÞ ¼ ckðS1 ðy1 ; y2 Þ; S2 ðy1 ; y2 ÞÞk ¼ ckSðy1 ; y2 Þk:
ð3:5Þ
So, from (3.5) we conclude that whenever ðy1 ; y2 Þ 2 X , we find that ðt1 ;t 2 Þ2
bþm
min bþm
4
4
1 ;3ðbþm1 Þ 4
2 ;3ðbþm2 Þ 4
½S1 ðy1 ; y2 Þðt 1 Þ þ S2 ðy1 ; y2 Þðt 2 Þ P ckSðy1 ; y2 Þk;
ð3:6Þ
as desired. We next show that for each admissible j we have wj(Sj(y1, y2)) P 0 whenever ðy1 ; y2 Þ 2 K. To see that this is true, note that
wj ðSj ðy1 ; y2 ÞÞ ¼
mj þb X i¼mj 2
¼ kj
cjimj þ2 Sj ðy1 ; y2 Þ ðiÞ
mj þb X i¼mj 2
þ
cjimj þ2
i¼mj 2 l¼mj 2
¼ kj
mj þb X
s¼0 i¼mj 2
Gj ði; sÞaj ðs þ mj 1Þfj ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞ
s¼0
mj þb X mj þb X
b X
b X
cjimj þ2 cjlmj þ2 yj ðlÞaj ðiÞ þ
mj þb X
mj þb X
i¼mj 2 k¼mj 2
j
cjimj þ2 dkmj þ2 yj ðkÞbj ðiÞ
cjimj þ2 Gj ði; sÞaj ðs þ mj 1Þfj ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞ þ wj ðaj Þwj ðyj Þ þ wj ðbj Þ/j ðyj Þ:
ð3:7Þ
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C.S. Goodrich / Applied Mathematics and Computation 217 (2011) 4740–4753
But by assumptions (G2) and (G3) together with the nonnegativity of fj(y1, y2) and the fact that ðy1 ; y2 Þ 2 K, we find from (3.7) that wj(Sj(y1, y2)) P 0, for each admissible j. An entirely dual argument, which we omit, shows that
/j ðSj ðy1 ; y2 ÞÞ P 0;
ð3:8Þ
too, whenever ðy1 ; y2 Þ 2 K and j is admissible. Finally, it is clear from the definitions of both S1 and S2 that
S1 ðy1 ; y2 Þðt 1 Þ P 0
ð3:9Þ
S2 ðy1 ; y2 Þðt 2 Þ P 0
ð3:10Þ
and
whenever ðy1 ; y2 Þ 2 K. Therefore, we conclude from (3.6)–(3.9) and (3.10) that whenever ðy1 ; y2 Þ 2 K, it follows that Sðy1 ; y2 Þ 2 K. Thus, S : K ! K, as desired. And this completes the proof. h We now prove the first of our two main theorems, Theorem 3.3. We first recall the Krasnosel’skii˘ fixed point theorem (see [40]), which we shall use to prove Theorem 3.3. We shall also require Lemma 3.2 in Section 4 of this paper. Lemma 3.2. Let B be a Banach space and let K # B be a cone. Assume that X1 and X2 are open sets contained in B such that 0 2 X1 and X1 # X2 . Assume, further, that T : K \ ðX2 n X1 Þ ! K is a completely continuous operator. If either 1. kTyk 6 kyk for y 2 K \ @ X1 and kTyk P kyk for y 2 K \ @ X2 ; or 2. kTyk P kyk for y 2 K \ @ X1 and kTyk 6 kyk for y 2 K \ @ X2 ; then T has at least one fixed point in K \ ðX2 n X1 Þ. Theorem 3.3. Suppose that conditions (F1)–(F2), (G1)–(G3), and (L1) hold. Then problem (1.1)–(1.3) has at least one positive solution. Proof. We have already shown in Lemma 3.1 that S : K ! K. Furthermore, it is evident that S is a completely continuous operator. Note first by condition (L1) that there is > 0 such that
8 " #1 " #1 9 b b <1 X = bþ1 1 X bþ1 þ m1 ; s a1 ðs þ m1 1Þ f1 þ m2 ; s a2 ðs þ m2 1Þ f2 max cG ; cG2 :2 s¼0 1 ; 2 2 s¼0 2 6 k1 ; k2 ð3:11Þ and
k1 ; k2 6 min
8 " b <1 X :4
G1 ðs þ m1 1; sÞa1 ðs þ m1 1Þ f1 þ
s¼0
#1
" #1 9 b = 1 X : ; G2 ðs þ m2 1; sÞa2 ðs þ m2 1Þ f2 þ ; 4 s¼0 ð3:12Þ
Now, given this , by condition (F1) it follows that there exists some number
r 1
> 0 such that
f1 ðy1 ; y2 Þ 6 f1 þ ðy1 þ y2 Þ;
ð3:13Þ
whenever k(y1, y2)k < r1. Similarly, by condition (F2), for the same , there exists a number r 1 > 0 such that
f2 ðy1 ; y2 Þ 6 f2 þ ðy1 þ y2 Þ;
ð3:14Þ
whenever k(y1, y2)k < r2. In particular, then, by putting r1 :¼ min r1 ; r 1 , we find that both (3.13) and (3.14) hold whenever k(y1, y2)k < r1. This suggests putting
X1 :¼ fðy1 ; y2 Þ 2 X : kðy1 ; y2 Þk < r 1 g; which we shall use in the sequel.
ð3:15Þ
C.S. Goodrich / Applied Mathematics and Computation 217 (2011) 4740–4753
4747
Now, let X1 be as in (3.15) above. Then for ðy1 ; y2 Þ 2 K \ @ X1 we find that
kS1 ðy1 ; y2 Þk ¼
ja1 ðt 1 Þ/1 ðy1 Þ þ b1 ðt 1 Þw1 ðy1 Þ þ k1
max
t 1 2½m1 2;m1 þbN
mX 1 þb
mX 1 þb
c1im1 þ2 y1 ðkÞ þ
i¼m1 2
G1 ðt 1 ; sÞa1 ðs þ m1 1Þf1 ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞj
s¼0
m1 2
6
b X
1
dkm1 þ2 y1 ðkÞ þ k1
k¼m1 2
max
b X
t 1 2½m1 2;m1 þbN
G1 ðt 1 ; sÞa1 ðs þ m1 1Þf1 ðy1 ðs þ m1 1Þ;
m1 2 s¼0
y2 ðs þ m2 1ÞÞ " # mX mX b 1 þb 1 þb X 1 1 cim1 þ2 þ dkm1 þ2 þ k1 G1 ðs þ m1 1; sÞa1 ðs þ m1 1Þ f1 þ ðy1 ðs þ m1 1Þ 6 r1 i¼m1 2
k¼m1 2
s¼0
" # b X 1 þ y2 ðs þ m2 1ÞÞ 6 kðy1 ; y2 Þk þ k1 G1 ðs þ m1 1; sÞa1 ðs þ m1 1Þ f1 þ ; 4 s¼0
ð3:16Þ
where we have used the fact that S1(y1, y2) is nonnegative whenever ðy1 ; y2 Þ 2 K. However, by the choice of k1 as given in (3.11) and (3.12), we deduce from (3.16) that
kS1 ðy1 ; y2 Þk 6
1 kðy1 ; y2 Þk: 2
ð3:17Þ
We may deduce by an entirely dual argument that
kS2 ðy1 ; y2 Þk 6
1 kðy1 ; y2 Þk: 2
ð3:18Þ
Thus, by putting (3.11)–(3.18) together we find that for ðy1 ; y2 Þ 2 K \ @ X1 we have
kSðy1 ; y2 Þk 6
1 1 kðy1 ; y2 Þk þ kðy1 ; y2 Þk ¼ kðy1 ; y2 Þk: 2 2
ð3:19Þ
Now, let be the same positive number selected at the beginning of this proof. For this same , by virtue of condition (F2) we can find a number ~r 2 > 0 such that
f1 ðy1 ; y2 Þ P f1 ðy1 þ y2 Þ
ð3:20Þ
f2 ðy1 ; y2 Þ P f2 ðy1 þ y2 Þ;
ð3:21Þ
and
whenever y1 þ y2 P ~r2 . Put
~r 2 ; r2 :¼ max 2r 1 ;
ð3:22Þ
c
where, as before, we take
c :¼ minfc~1 ; c~2 g: Moreover, put
X2 :¼ fðy1 ; y2 Þ 2 X : kðy1 ; y2 Þk < r 2 g:
ð3:23Þ
Note that if ðy1 ; y2 Þ 2 K \ @ X2 , then it follows that
y1 ðt 1 Þ þ y2 ðt2 Þ P
bþm
ðt 1 ;t 2 Þ2
4
min bþm
1 ;3ðbþm1 Þ 4
4
½y1 ðt 1 Þ þ y2 ðt 2 Þ P ckðy1 ; y2 Þk P ~r 2 :
2 ;3ðbþm2 Þ 4
ð3:24Þ
Now, define the numbers r1 and r2, with r1 < r2, by
r1 :¼ max
m1 þ b 4
m2 þ b m1 þ 1 ; m2 þ 1 4
and
r2 :¼ min
3ðm1 þ bÞ 3ðm2 þ bÞ m1 þ 1 ; m2 þ 1 ; 4 4
we assume in the sequel that b is sufficiently large so that ½r1 ; r2 \ N0 – £. Then for each ðy1 ; y2 Þ 2 K \ @ X2 we find that
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C.S. Goodrich / Applied Mathematics and Computation 217 (2011) 4740–4753
S1 ðy1 ; y2 Þ
mX mX b 1 þb 1 þb X bþ1 bþ1 1 þ m1 ¼ þ m1 ; s a1 ðs þ m1 c1im1 þ2 y1 ðkÞ þ dkm1 þ2 y1 ðkÞ þ k1 G1 2 2 s¼0 i¼m 2 k¼m 2 1
1
1Þf1 ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞ r2 X bþ1 1 P k1 þ m1 ; s a1 ðs þ m1 1Þ f1 c½ky1 k þ ky2 k P kðy1 ; y2 Þk; G1 2 2 s¼r
ð3:25Þ
1
where to arrive at the first inequality in (3.25) we have used the positivity assumption about w1(y1) and /1(y1) whenever ðy1 ; y2 Þ 2 K. Thus, we conclude from (3.25) that
kS1 ðy1 ; y2 Þk P
1 kðy1 ; y2 Þk: 2
ð3:26Þ
In a completely similar way, it can be shown that
kS2 ðy1 ; y2 Þk P
1 kðy1 ; y2 Þk: 2
ð3:27Þ
Consequently, (3.20)–(3.27) imply that
kSðy1 ; y2 Þk P kðy1 ; y2 Þk;
ð3:28Þ
whenever ðy1 ; y2 Þ 2 K \ @ X2 . Finally, notice that (3.19) implies that the operator S is a cone compression on K \ @ X1 , whereas (3.28) implies that S is a cone expansion on K \ @ X2 . Thus, all of the hypothesis of Lemma 3.2 are satisfied. Consequently we conclude that S has a fixed point, say ðy1 ; y2 Þ 2 K. As ðy1 ; y2 Þ is a positive solution of (1.1)–(1.3), the theorem is proved. h Remark 3.4. Observe that we may still be able to apply Theorem 3.3 to problem (1.1)–(1.3) even if none of the functionals
w1(y1), w2(y2), /1(y1), and /2(y2) is nonnegative for all yi P 0. This is the advantage of using the cone introduced by Infante and Webb. So, in particular, Theorem 3.3 provides a generalization and extension of [2, Theorem 3.3]. Importantly, Theorem 3.3 provides new results even in case m1 = m2 = 2 – that is, the integer-order case. Remark 3.5. Note that in the preceding arguments it is important that each of c1 and c2 and thus c is a constant. That c is constant here is a reflection of the fact that the Green’s function G satisfies a sort of Harnack-like inequality. Interestingly, however, in the continuous fractional setting, this may (see [20]) or may not (see [17]) be true. Remark 3.6. We point out that our results could be extended to the case of n equations and 2n boundary conditions. Since this is just a straightforward extension of the results presented in this section, we omit the details.
4. Existence of a positive solution: Case II We now wish to present an alternative method for deducing the existence of at least one positive solution to problem (1.1)–(1.3). In particular, in stead of using the cone given in (3.1), we shall now revert to a more traditional cone, whose use can be found in innumerable papers, including [2]. An advantage of this approach, however, is that it shall allow us to weaken hypothesis (G1). However, we do so at the expense of having to assume a priori the positivity of each of these functionals for all y P 0. Let us specifically state some new assumptions that we shall require in the sequel. G4: For i = 1, 2 we have that
lim
kyi k!0þ
wi ðyi Þ ¼ 0: kyi k
G5: For each i = 1, 2 we have that
lim
kyi k!0þ
/i ðyi Þ ¼ 0: kyi k
G6: For each i = 1, 2 we have that wi(yi) and /i(yi) are nonnegative for all yi P 0. L2: The constants k1 and k2 satisfy
K1 < k i < K2 ; for each admissible i, where
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C.S. Goodrich / Applied Mathematics and Computation 217 (2011) 4740–4753
8 " #1 " #1 9 b b <1 X = bþ1 1 X bþ1 þ m1 ; s a1 ðs þ m1 1Þf1 þ m2 ; s a2 ðs þ m2 1Þf2 K1 :¼ max cG ; cG2 :2 s¼0 1 ; 2 2 s¼0 2 and
8 " #1 " #1 9 <1 X = b b 1 X K2 :¼ min G1 ðs þ m1 1; sÞa1 ðs þ m1 1Þf1 ; G2 ðs þ m2 1; sÞa2 ðs þ m2 1Þf2 ; :3 s¼0 ; 3 s¼0 where fi and fi retain their earlier meaning from conditions (F1)–(F2), for each admissible i. Moreover, c is defined just as in Section 3. Before proceeding any further, let us point out explicitly that there do exist nontrivial functionals satisfying conditions (G4) and (G5). For example, consider the functional given by
/ðyÞ :¼ ½yðt 0 Þ6 ; where t0 is some number in the domain of y. Then it is clear that
0 6 limþ kyk!0
½yðt0 Þ6 ½yðt 0 Þ6 6 limþ ¼ limþ ½yðt 0 Þ5 ¼ 0; kyk yðt0 Þ kyk!0 kyk!0
from which it follows that / satisfies conditions (G4) and (G5). (Note that this naturally relies crucially upon the fact that / (y) is nonnegative for all y P 0.) Of course, not every possible functional will satisfy these conditions as is easy to see. We now present our existence theorem. This result complements Theorem 3.3 given in Section 3. Theorem 4.1. Suppose that conditions (F1)–(F2), (G4)–(G6), and (L2) hold. Then problem (1.1)–(1.3) has at least one positive solution. Proof. Begin by noting that by condition (L2) that there is
> 0 such that
8 " #1 " #1 9 b b <1 X = bþ1 1 X bþ1 þ m1 ; s a1 ðs þ m1 1Þ f1 þ m2 ; s a2 ðs þ m2 1Þ f2 max G ; G2 :2 s¼0 1 ; 2 2 s¼0 2 6 k1 ; k2 ð4:1Þ and
8 " #1 " #1 9 <1 X b b = 1 X k1 ; k2 6 min G ðs þ m1 1; sÞa1 ðs þ m1 1Þ f1 þ ; G2 ðs þ m2 1; sÞa2 ðs þ m2 1Þ f2 þ : :3 s¼0 1 ; 3 s¼0 ð4:2Þ Now, for the
determined by (4.1) and (4.2) we have by conditions (G4) and (G5) there exists a number g1 > 0 such that
/1 ðy1 Þ 6 ky1 k;
ð4:3Þ
whenever ky1k 6 g1, and a number g2 > 0 such that
w1 ðy2 Þ 6 ky1 k;
ð4:4Þ
whenever ky1k 6 g2. Put
g :¼ minfg1 ; g2 g:
ð4:5Þ
Then whenever k(y1, y2)k < g*, we have that both (4.3) and (4.4) hold. Now, for the same > 0 given in the first paragraph of this proof, we find that there exists a number g3 such that
f1 ðy1 ; y2 Þ 6 f1 þ ðy1 þ y2 Þ;
ð4:6Þ
whenever k(y1, y2)k < g3. Thus, by putting
g :¼ minfg ; g3 g;
ð4:7Þ
we get that (4.3), (4.4), and (4.6) are collectively true. So, define X1 by
X1 :¼ fðy1 ; y2 Þ 2 X : kðy1 ; y2 Þk < g g:
ð4:8Þ
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C.S. Goodrich / Applied Mathematics and Computation 217 (2011) 4740–4753
Then whenever ðy1 ; y2 Þ 2 K \ @ X1 we have, for any t1 2 ½m1 2; m1 þ bNm
1 2
S1 ðy1 ; y2 Þðt 1 Þ 6 ky1 k þ ky1 k þ k1
b X
,
G1 ðt 1 ; sÞa1 ðs þ m1 1Þf1 ðy1 ðs þ m1 1Þ; y2 ðs þ m2 1ÞÞ
s¼0
6 2ky1 k þ k1
b X s¼0
1 G1 ðs þ m1 1; sÞa1 ðs þ m1 1Þ f1 þ kðy1 ; y2 Þk 6 2 þ kðy1 ; y2 Þk; 3
ð4:9Þ
where we have used condition (L2) together with (4.2). An entirely dual argument reveals that
1 S2 ðy1 ; y2 ÞðtÞ 6 2 þ kðy1 ; y2 Þk: 3
ð4:10Þ
Therefore, putting (4.9) and (4.10) together we conclude that
kSðy1 ; y2 Þk 6 kðy1 ; y2 Þk;
ð4:11Þ
whenever ðy1 ; y2 Þ 2 K \ @ X1 and is chosen sufficiently small, which may be assumed without loss of generality. To complete the proof, we may give an argument nearly identical to the second half of the proof of Theorem 3.3. We omit this, and so, the proof is complete. h Remark 4.2. Observe that Theorem 4.1 provides a further generalization of the results in [2]. In particular, it allows for a somewhat greater range of nonlocal conditions and thus provides new results in the integer-order case. Moreover, it does not presuppose that the functionals are linear in y. Thus, as far as the author is aware, this provides new results in case m1 = m2 = 2, namely the classical integer-order case. Of course, it also provides, at least so far as the author is aware, new results in the fractional-order case. Remark 4.3. As in Section 3, it is clear how the results of this section can be extended to the case in which (1.1) is replaced with n equations and boundary conditions (1.2) and (1.3) are extended to 2n boundary conditions in the obvious way. As in Section 3, we omit the details of this extension.
5. An example In this section, we wish to provide an explicit numerical example of our results. In particular, it is worth noting that in [2] no numerical examples were provided for the reader. So, in that work it was not explicitly shown what sorts of nontrivial functions f(y1,y2), with f : [0, +1) [0, +1) ? [0, +1), satisfy conditions (F1)–(F2). Our Example 5.1 below will provide an explicit illustration. Example 5.1. Consider the boundary value problem
(
y1
3 3 7 D1:3 y1 ðtÞ ¼ k1 a1 t þ 10 f1 y1 t þ 10 ; y2 t þ 10 ; 7 3 7 f2 y1 t þ 10 ; y2 t þ 10 ; D1:7 y2 ðtÞ ¼ k2 a2 t þ 10
ð5:1Þ
7 1 13 1 53 ¼ ; y1 y1 10 12 10 25 10
ð5:2Þ
y1
213 1 83 1 73 ¼ ; y1 y1 10 30 10 100 10
3 1 17 1 77 y2 ¼ y2 y2 ; 10 40 10 150 10
y2
217 1 47 1 107 ¼ y2 y2 ; 10 17 20 30 20
ð5:3Þ
where we take
a1 ðtÞ :¼ et4 ;
ð5:4Þ
a2 ðtÞ :¼ et4 ;
ð5:5Þ
f1 ðy1 ; y2 Þ :¼ 5000ey2 ðy1 þ y2 Þ þ ðy1 þ y2 Þ
ð5:6Þ
f2 ðy1 ; y2 Þ :¼ 7500ey1 ðy1 þ y2 Þ þ ðy1 þ y2 Þ;
ð5:7Þ
and
with f1, f2 : [0, +1) [0, +1) ? [0, +1). It is clear from (5.2) and (5.3) that in problem (5.1)–(5.3) we have taken
w1 ðy1 Þ :¼
1 13 1 53 ; y1 y1 12 10 25 10
ð5:8Þ
C.S. Goodrich / Applied Mathematics and Computation 217 (2011) 4740–4753
4751
/1 ðy1 Þ :¼
1 83 1 73 ; y1 y1 30 10 100 10
ð5:9Þ
w2 ðy2 Þ :¼
1 17 1 77 y2 y2 ; 40 10 150 10
ð5:10Þ
/2 ðy2 Þ :¼
1 47 1 107 y2 y2 ; 17 20 30 20
ð5:11Þ
Note, in addition, that y1 is defined on the time scale
7 3 213 ; ; ;...; 10 10 10
whereas y2 is defined on the time scale
3 7 217 ; ;...; : 10 10 10
In particular, we have chosen m1 ¼ 13 ; m2 ¼ 17 , and b = 20 here. We shall select k1 and k2 below. 10 10 We now endeavor to check that each of conditions (F1)–(F2), (G1)–(G3), and (L1) holds. It is easy to check that (F1)–(F2) hold. On the other hand, since (5.8)–(5.11) reveals the functionals to be linear in y1 and y2, respectively, it is immediate that (G1) holds. On the other hand, to see that conditions (G2)–(G3) hold, let us note first both that
1 1 1 1 1 1 þ þ þ ¼ 6 12 25 30 100 6 4
ð5:12Þ
and that
1 1 1 1 421 1 þ þ þ ¼ 6 : 40 150 17 30 3400 4
ð5:13Þ
Furthermore, numerical calculations show that mj þb X i¼mj 2
cjimj þ2 Gj ði; sÞ P 0
ð5:14Þ
and that mj þb X
j
dkmj þ2 Gj ðk; sÞ P 0;
ð5:15Þ
k¼mj 2
for each j. So, it follows that condition (G2) holds. Finally, routine numerical calculations reveal that
w1 ða1 Þ 0:012; w1 ðb1 Þ 0:012; w2 ða2 Þ 0:012; w2 ðb2 Þ 0:0012
ð5:16Þ
and that
/1 ða1 Þ 0:00091; /1 ðb1 Þ 0:018; /2 ða2 Þ 0:015; /2 ðb2 Þ 0:000099:
ð5:17Þ
Hence, condition (G3) is satisfied. Finally, we check condition (L1) to determine the admissible range of eigenvalues, ki. To this end, recall from [26, Theorem 3.2] that the constant c in Lemma 2.6 is
8 9 2 3 m1 3ðbþmÞ > m1 < = 2 ðm þ b þ 1Þm1 7 bþm m1 > 4 1 6 3ðb þ mÞ c :¼ min 4 5; 4 m1 : m 1 m 1 > 4 ðm þ b 1Þ ðb þ mÞ > : 3ðbþmÞ ;
ð5:18Þ
4
Thus, using the definition of c provided by (5.18), we numerically estimate that
n o K1 max f1 3:288 107 ; f2 1:0322 107 ¼ maxf3:288 107 ; 1:0322 107 g ¼ 3:337 107 ;
ð5:19Þ
whereas
n o K2 min f1 1:871 109 ; f2 1:363 109 ¼ minf5001 1:871 109 ; 7501 1:363 109 g ¼ minf9:357 106 ; 1:022 105 g ¼ 9:357 106 : In summary, then, so long as k1, k2 are chosen so that
ð5:20Þ
4752
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3:337 107 6 ki 6 9:357 106 ; for each admissible i, then condition (L1) is satisfied. So, suppose that
k1 ; k2 2 ½3:337 107 ; 9:357 106 :
ð5:21Þ
Then (5.4)–(5.20) together with condition (5.21) imposed on k1, k2 imply that the hypotheses of Theorem 3.3 are satisfied. Therefore, we conclude that problem (5.1)–(5.3) has at least one positive solution under these conditions, as desired. A similar example could be provided for Theorem 4.1. Remark 5.2. We note in passing that a fairly broad class of (nontrivial) functions satisfying conditions (F1)–(F2) are given by f(x) :¼ C1eg(x)rH(x), where g : Rnþ ! ½0; þ1Þ; f : Rnþ ! ½0; þ1Þ; C 1 > 0 is a constant, H : Rnþ ! Rnþ is the vector field defined P by HðxÞ :¼ ni¼1 12 x2i ei , where ei is the ith standard basis vector in Rn , and by Rnþ we mean the closure of the open positive cone in Rn . Obviously the class of functions L(y1,y2) = ay1 + ay2 trivially satisfies (F1)–(F2), for a > 0.
6. Conclusions In this paper we have considered a coupled system of discrete FBVPs as given by (1.1)–(1.3). We have shown that under standard assumptions on the ai’s and the nonlinearities fi(y1, y2), where i = 1, 2, problem (1.1)–(1.3) may admit a positive solution even if some or all of the functionals wi(yi) and /i(yi), where i = 1, 2, are nonpositive for some yi P 0. This provides a generalization of the results in [2] as well as certain recent research on discrete FBVPs. While assuming that the wi’s and /i’s are nonnegative for all yi P 0 is a standard assumption to make, it worthwhile to see how the idea of Infante and Webb [1] allows us to weaken this assumption. Moreover, we have shown that under somewhat different conditions, a more traditional cone may be used to deduce the existence of at least one positive solution of problem (1.1)–(1.3). This also provides new results in case m1 = m2 = 2, which is the integer-order case, and thus improves, in this sense, the result given in [2], for instance. We would also like to point out that it might be interesting to generalize certain of the results here to delta-nabla problems – that is, where the ith equation in (1.1) is replaced by
Dli rmi yðt þ mÞ ¼ ki ai ðt þ li þ mi 1Þfi ðy1 ðt þ li þ mi 1Þ; . . . ; yn ðt þ li þ mi 1ÞÞ;
ð6:1Þ
with li, mi 2 (1, 2], for each 1 6 i 6 n. Given that the study of discrete fractional delta-nabla problems is nontrivial, in no small part due to the composition of two fractional derivatives (e.g., see [34]), studying this sort of generalization could be interesting. In addition, it might be interesting to generalize Eq. (1.1) by allowing f to depend upon one or both of the fractional differences Dm1 y and Dm2 y. Such results would build upon the results given in [41], for example, and could prove to be mathematically interesting. In any case, we hope that these ideas can be further extended both in the fractional and integer-order settings. References [1] G. Infante, J.R.L. Webb, Positive solutions of nonlocal boundary value problems: a unified approach, J. London Math. Soc. 74 (2) (2006) 673–693. [2] J. Henderson, S.K. Ntouyas, I.K. Purnaras, Positive solutions for systems of nonlinear discrete boundary value problems, J. Differ. Equat. Appl. 15 (2009) 895–912. [3] X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett. 22 (2009) 64–69. [4] D.R. Anderson, Solutions to second-order three-point problems on time scales, J. Differ. Equat. Appl. 8 (2002) 673–688. [5] E.R. Kaufmann, Y.N. Raffoul, Positive solutions for a nonlinear functional dynamic equation on a time scale, Nonlinear Anal. TMA 62 (2005) 1267–1276. [6] W. Cheung, J. Ren, P.J.Y. Wong, D. Zhao, Multiple positive solutions for discrete nonlocal boundary value problems, J. Math. Anal. Appl. 330 (2007) 900– 915. [7] R. Agarwal, D. O’Regan, A coupled system of difference equations, Appl. Math. Comput. 114 (2000) 39–49. [8] D.R. Anderson, Existence of solutions for first-order multi-point problems with changing-sign nonlinearity, J. Differ. Equat. Appl. 14 (2008) 657–666. [9] R.I. Avery, J. Henderson, Twin solutions of boundary value problems for ordinary differential equations and finite difference equations, Comput. Math. Appl. 42 (2001) 695–704. [10] J. Henderson, P.J.Y. Wong, Double symmetric solutions for discrete Lidstone boundary value problems, J. Differ. Equat. Appl. 7 (2001) 811–828. [11] R. Ma, Y.N. Raffoul, Positive solutions of three-point nonlinear discrete second order boundary value problem, J. Differ. Equat. Appl. 10 (2004) 129–138. [12] C. Wang, X. Han, C. Li, Positive solutions to nonlinear second-order three-point boundary-value problems for difference equation with change of sign, Electon. J. Differ. Eqs. (87) (2008) 10. [13] P. Wang, Y. Wang, Existence of positive solutions for second-order m-point boundary value problems on time scales, Acta Mathmeticae Applicatae Sinica, English Series 22 (2006) 457–468. [14] R. Almeida, D.F.M. Torres, Calculus of variations with fractional derivatives and fractional integrals, Appl. Math. Lett. 22 (2009) 1816–1820. [15] A. Arara et al, Fractional order differential equations on an unbounded domain, Nonlinear Anal. TMA 72 (2010) 580–586. [16] A. Babakhani, V. Daftardar-Gejji, Existence of positive solutions of nonlinear fractional differential equations, J. Math. Anal. Appl. 278 (2003) 434–442. [17] Z. Bai, H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005) 495–505. [18] J.V. Devi, Generalized monotone method for periodic boundary value problems of Caputo fractional differential equation, Commun. Appl. Anal. 12 (2008) 399–406. [19] K. Diethelm, N. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229–248. [20] C.S. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett. 23 (2010) 1050–1055. [21] A. Malinowska, D.F.M. Torres, Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative, Comput. Math. Appl. 59 (2010) 3110–3116.
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