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Multiple positive solutions for p-Laplacian m-point boundary value problems on time scales
Applied Mathematics and Computation 182 (2006) 478–491 www.elsevier.com/locate/amc
Multiple positive solutions for p-Laplacian m-point boundary value problems on time scales q Hong-Rui Sun *, Wan-Tong Li School of Mathematics and Statistics, Lanzhou University, Tianshui Street 222, Lanzhou, Gansu 730000, People’s Republic of China
Abstract Let T be a time scale such that 0; T 2 T, ai P 0 for i = 1, . . ., m 2. Let ni satisfy 0 < n1 < n2 < < nm2 < q(T) and Pm2 i¼1 ai < 1. We consider the following p-Laplacian m-point boundary value problem on time scales ðup ðuD ðtÞÞÞr þ aðtÞf ðt; uðtÞÞ ¼ 0; t 2 ð0; T Þ; uð0Þ ¼ 0; up ðuD ðT ÞÞ ¼
m2 X
ai up ðuD ðni ÞÞ;
i¼1
where a 2 Cld ((0, T), [0, 1)) and f 2 Cld ((0, T) · [0, 1), [0, 1)). Some new results are obtained for the existence of at least twin or triple positive solutions of the above problem by applying Avery-Henderson and Leggett-Williams fixed point theorems respectively. In particular, our criteria extend and improve some known results. 2006 Elsevier Inc. All rights reserved. Keywords: Time scales; Positive solution; Cone; Fixed point
1. Introduction The study of dynamic equations on time scales goes back to its founder Stefan Hilger [14], and is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on time scales can build bridges between continuous and discrete equations. Further, the study of time scales has led to several important applications, e.g., in the study of insect population models, neural networks, heat transfer and epidemic models [1,10,11]. We begin by presenting some basic definitions which can be found in [1,10,14,17]. Another excellent source on dynamic equations on time scales is the book [11]. q Supported by the NNSF of China (10571078), the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China and the Fundamental Research Fund for Physics and Mathematic of Lanzhou University (Lzu05003). * Corresponding author. E-mail address: [email protected] (H.-R. Sun).
0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.04.009
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A time scale T is a nonempty closed subset of R. It follows that the jump operators r; q : T ! T: rðtÞ ¼ inffs 2 T : s > tg and qðtÞ ¼ supfs 2 T : s < tg; (supplemented by inf ; :¼ sup T and sup ; :¼ inf T) are well defined. The point t 2 T is left-dense, left-scattered, right-dense, right-scattered if q(t) = t, q(t) < t, r(t) = t, r(t) > t, respectively. If T has a right-scattered minimum m, define Tj ¼ T fmg; otherwise, set Tj ¼ T. If T has a left-scattered maximum M, define Tj ¼ T fMg; otherwise, set Tj ¼ T. The forward graininess is l(t) :¼ r(t) t. Similarly, the backward graininess is t(t) :¼ t q(t). We make the blanket assumption that 0, T are points in T. By an interval (0, T) we always mean the intersection of the real interval (0, T) with the given time scale, that is ð0; T Þ \ T. Other type of intervals are defined similarly. For f : T ! R and t 2 Tj , the delta derivative [10] of f at t, denoted by fD (t), is the number (provided it exists) with the property that given any > 0, there is a neighborhood U T of t such that: f ðrðtÞÞ f ðsÞ f D ðtÞ½rðtÞ sj 6 jrðtÞ s; for all s 2 U. For f : T ! R and t 2 Tj , the nabla derivative [6] of f at t, denoted by f$ (t), is the number (provided it exists) with the property that given any > 0, there is a neighborhood U of t such that: f ðqðtÞÞ f ðsÞ f r ðtÞ½qðtÞ sj 6 jqðtÞ s; for all s 2 U. A function f : T ! R is ld-continuous provided it is continuous at left dense points in T and its right sided limit exists (finite) at right dense points in T. If T ¼ R, then f is ld-continuous if and only if f is continuous. If T ¼ Z, then any function is ld-continuous. It is known [6] that if f is ld-continuous, then there is a function F(t) such that F$ (t) = f(t). In this case, we define Z b f ðsÞrs ¼ F ðbÞ F ðaÞ: a
Very recently, there is an increasing attention paid to question of positive solution for second order three point boundary value problems on time scales [3,4,12,15,16,19,21–23]. But very little work has been done to the existence of positive solutions for p-Laplacian boundary value problem on time scales [5,24,25]. In particular, we would like to mention some results of Anderson, Avery and Henderson [5], Gupta [13], Sun and Li [24,25], which motivate us to consider our problem. For convenience, throughout this paper we denote up(u) is p-Laplacian operator, i.e., up(u) = jujp2u, p > 1, (up)1 = uq, 1/p + 1/q = 1. In [5], Anderson, Avery and Henderson considered the following problem: r
ðup ðuD ðtÞÞÞ þ cðtÞf ðuÞ ¼ 0; D
uðaÞ B0 ðu ðmÞÞ ¼ 0;
t 2 ða; bÞ;
D
u ðbÞ ¼ 0;
where m 2 (a, b), f 2 Cld ([0, 1), [0, 1)), c 2 Cld ((a, b), [0, 1)) and Kmx 6 B0(x) 6 KMx for some positive constants Km, KM. They established the existence result of at least one positive solution by a fixed point theorem of cone expansion and compression of functional type. In [13], Gupta studied the multi-point boundary value problem: 0
ð/ðu0 ðtÞÞÞ ¼ f ðt; xðtÞ; x0 ðtÞÞ þ eðtÞ;
t 2 ð0; 1Þ;
and uð0Þ ¼ 0;
/ðu0 ð1ÞÞ ¼
m2 X
ai /ðu0 ðni ÞÞ;
i¼1
where / is an odd increasing homeomorphism form R on R with /(0) = 0. A new prior estimate of Poincare type and Leray-Schauder continuation theorem lead to the existence theorems of positive solutions.
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In [24], the authors considered the existence of at least single, twin or triple positive solutions to the following problem: D
ðup ðuD ðtÞÞÞ þ hðtÞf ður ðtÞÞ ¼ 0;
t 2 ½a; b;
with the boundary condition, uðaÞ B0 ðuD ðaÞÞ ¼ 0;
uD ðrðbÞÞ ¼ 0;
and in [25], we further considered the existence criteria of one or multiple positive solutions to the following p-Laplacian m-point boundary value problem: ðup ðuD ðtÞÞÞr þ aðtÞf ðt; uðtÞÞ ¼ 0; uD ð0Þ ¼ 0;
uðT Þ ¼
m2 X
t 2 ð0; T Þ;
ai uðni Þ:
i¼1
The main tools used in [24,25] are fixed point theorems in cones. In this paper, we shall study the existence of multiple positive solutions for the one-dimensional p-Laplacian m-point boundary value problem on time scales: r
ðup ðuD ðtÞÞÞ þ aðtÞf ðt; uðtÞÞ ¼ 0; uð0Þ ¼ 0;
up ðuD ðT ÞÞ ¼
m2 X
t 2 ð0; T Þ;
ð1:1Þ
ai up ðuD ðni ÞÞ:
ð1:2Þ
i¼1
Some new results are obtained for the existence of at least twin or triple positive solutions for the above problem by using the fixed point theorems due to Avery and Henderson [9] and Leggett-Williams [18]. The results are even new for the special cases of difference equations and differential equations, as well as in the general time scale setting. Pm2 It is also noted that when i¼1 ai ¼ 0, boundary value problem (1.1) and (1.2) reduce to one dimension pLaplacian two point boundary value problem. In the special case of p = 2, it has been extensively studied by many authors, see [10,11]. The rest of the paper is organized as follows. In Section 2, we first give four lemmas which are needed throughout this paper and then list two fixed point theorems due to Avery and Henderson and LeggettWilliams. In Section 3 we give two existence results and some corollaries. In particular, our Theorem 3.1 Pm2 includes and extends the main results of Avery, Chyan and Henderson [8] ðp ¼ 2; i¼1 ai ¼ 0; f ðt; uÞ ¼ f ðuÞÞ in the case of T ¼ R and T ¼ Z. Section 4 is to develop existence criteria for (at least) three positive and arbitrary odd positive solutions of problem (1.1) and (1.2). In particular, our results are new when T ¼ R (the continual case) and T ¼ Z (the discrete case). Finally, in Section 5, we give an example to illustrate our main results. For convenience, we list the following hypotheses: (A1) ai P 0 for i = 1, . . ., m 2, 0 < n1 < n2 < < nm2 < q(T) and d ¼ 1 (A2) a:(0, T) ! [0, 1) is ld-continuous such that a(t0) > 0 for at f:(0, T) · [0, 1) ! [0, 1) is ld-continuous.
Pm2
ai > 0; least one
i¼1
t0 2 [0, T)
and
2. Some lemmas To prove the main results in this paper, we will employ several lemmas, which are based on the linear boundary value problem r
ðup ðuD ðtÞÞÞ þ hðtÞ ¼ 0; uð0Þ ¼ 0;
up ðuD ðT ÞÞ ¼
t 2 ð0; T Þ; m2 X i¼1
ai up ðuD ðni ÞÞ:
ð2:1Þ ð2:2Þ
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481
Lemma 2.1. If d 5 0, then for h 2 Cld [0, T] the boundary value problem (2.1) and (2.2) has the unique solution: ! Z Z ni Z t Z s m2 1 T 1X uðtÞ ¼ uq hðsÞrs þ hðsÞrs ai hðsÞrs Ds: ð2:3Þ d 0 d i¼1 0 0 0 Proof. By the formula ð D
u ðtÞ ¼ uq
Z
t 0
and so up ðuD ðtÞÞ ¼
Z
Rt
f ðt; sÞDsÞD ¼ f ðrðtÞ; tÞ þ
a
1 hðsÞrs þ d
t
hðsÞrs þ
0
1 d
Observe that Z m2 m2 X X ai up ðuD ðni ÞÞ ¼ ai i¼1
¼
1 d
Pm2
T 0
Z
T
hðsÞrs 0
hðsÞrs þ ni
hðsÞrs þ
ai
i¼1
Z ni m2 1X ai hðsÞrs: d i¼1 0
Pm2
ni
Z
f D ðt; sÞDs, we have
a
! Z ni m2 1X hðsÞrs ai hðsÞrs ; d i¼1 0
0
i¼1 m2 X
Z
Rt
0
i¼1
Z
ai
d 1 d
Z ni m2 m2 X 1X hðsÞrs ai ai hðsÞrs d i¼1 0 i¼1
T
0 m2 X
Z ai
i¼1
!
T
hðsÞrs ¼ up ðuD ðT ÞÞ:
0
(2.3) still holds. i¼1 ai 0, then (2.1) and (2.2) reduce to the two-point BVP. It is easy to see that P D We note that the boundary value problem (up(xD(t)))$ = 0, x(0) = 0, up ðxD ðT ÞÞ ¼ m2 i¼1 ai up ðx ðni ÞÞ has only the trivial solution if d 5 0. Thus u in (2.3) is the unique solution of (2.1) and (2.2). The proof is complete. h If
Lemma 2.2. Let d > 0. If h 2 Cld[0, T] and h P 0, then the unique solution u of (2.1) and (2.2) satisfies uðtÞ P 0;
t 2 ½0; T :
Proof. Since (up(uD(t)))$ = h(t) 6 0, we obtain that: up ðuD ðT ÞÞ ¼ In view of
Pm2 i¼1
m2 X
ai up ðuD ðni ÞÞ P
i¼1
m2 X
ai up ðuD ðT ÞÞ:
i¼1
D
ai < 1, we know u (T) P 0. By u(0) = 0, the result is obvious. The proof is complete.
h
Now we give a lemma which is needed in Lemma 2.4. Lemma 2.3. Assume g : R ! R is continuous, g : T ! R is delta differentiable on Tj , and f : R ! R is continuous differentiable. Then there exists c in the real interval [q(t), t] with ðf gÞr ðtÞ ¼ f 0 ðgðcÞÞgr ðtÞ:
ð2:4Þ
Proof. The proof is similar to that of [10, Theorem 1.87, p. 31]. For convenience, we list it here. Fix t 2 Tj . First we consider the case where t is left-scattered. In this case r
ðf gÞ ðtÞ ¼
f ðgðtÞÞ f ðgðqðtÞÞÞ : qðtÞ
If g(q(t)) = g(t), then we get (f g)$(t) = 0 and g$(t) = 0 and so (2.4) holds for any c in the real interval [q(t), t]. Hence we can assume g(q(t)) 5 g(t). Then ðf gÞr ðtÞ ¼
f ðgðtÞÞ f ðgðqðtÞÞÞ gðtÞ gðqðtÞÞ ¼ f 0 ðnÞgr ðtÞ; gðtÞ gðqðtÞÞ qðtÞ
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by the mean value theorem, where n is between g(q(t)) and g(t). Since g : R ! R is continuous, there is c 2 [q(t), t] such that g(c) = n, which gives us the desired result. It remains to consider the case when t is left-dense. In this case r
ðf gÞ ðtÞ ¼ lim s!t
f ðgðtÞÞ f ðgðsÞÞ gðtÞ gðsÞ ¼ lim f 0 ðns Þ s!t ts ts
by the mean value theorem in calculus, where ns is between g(s) and g(t). By the continuity of g we get that lims!tns = g(t) which give us the desired result. The proof is complete. h Lemma 2.4. Let d < 0. If h 2 Cld[0, T] and h P 0, then the boundary value problem (2.1) and (2.2) has no nonnegative solution. Proof. By Lemma 2.1 and d < 0, it is easy to see that u(T) < 0. The proof is complete.
h
Now, let the Banach space B = Cld[0, T] (see [3]) be endowed with the norm kuk = supt2[0,T]ju(t)j, and choose the cone P B defined by; P¼
u 2 B : uðtÞ P 0
for t 2 ½0; T
uDr ðtÞ 6 0; uD ðtÞ P 0
and
for t 2 ð0; T Þ; uð0Þ ¼ 0
:
Clearly, kuk = u(T) for u 2 P. Define the operator A : P ! B by ! Z Z ni Z s Z t m2 1 T 1X uq aðsÞf ðs; uðsÞÞrs þ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs Ds: AuðtÞ ¼ d 0 d i¼1 0 0 0 ð2:5Þ Obviously, Au(t) P 0 for t 2 [0, T]. From the definition of A, we claim that for each u 2 P, Au 2 P and satisfies (1.2) and Au(T) is the maximum value of Au(t) on [0, T]. In fact, ! Z t Z Z ni m2 1 T 1X D ðAuÞ ðtÞ ¼ uq aðsÞf ðs; uðsÞÞrsþ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs d 0 d i¼1 0 0 ! Z t Z Z T m2 1 T 1X P uq aðsÞf ðs; uðsÞÞrsþ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs d 0 d i¼1 0 0 Z T ¼ uq aðsÞf ðs; uðsÞÞrs P 0; t 2 ð0; T Þ: t
Moreover, uq(x) is a monotone increasing and continuously differentiable function and Z T r aðsÞf ðs; uðsÞÞrs ¼ aðtÞf ðt; uðtÞÞ 6 0; t
then by Lemma 2.3, we obtain ðAuÞDr ðtÞ 6 0; so, A : P ! P. Lemma 2.5. A : P ! P is completely continuous. Proof. First, we show that A maps bounded set into itself. Assume c > 0 is a constant and u 2 P c ¼ fx 2 P : kxk 6 cg. Note that the continuity of f(t, u) guarantees that there is a C > 0 such that f(t, u) 6 up(C) for t 2 [0, T]. So
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kAuk ¼ AuðT Þ ! Z T Z s Z Z ni m2 1 T 1X ¼ uq aðsÞf ðs; uðsÞÞrs þ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs Ds d 0 d i¼1 0 0 0 Z T Z T Z T Z T 1 1 6 uq aðsÞf ðs; uðsÞÞrs Ds 6 C uq aðsÞrs Ds: d 0 d 0 0 0 That is, AP c is uniformly bounded. In addition, notice that: ! Z t2 Z s Z T Z ni m2 X 1 1 jAuðt1 Þ Auðt2 Þj ¼ uq aðsÞf ðs; uðsÞÞrs þ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs Ds d 0 d i¼1 t1 0 0 Z t 2 Z T Z T 1 1 uq aðsÞf ðs; uðsÞÞrs Ds 6 Cjt1 t2 juq aðsÞrs : 6 d d t1 0 0 So, by applying Arzela–Ascoli theorem on time scales [2] we obtain that AP c is relatively compact. In view of Lebesgue’s dominated convergence theorem on time scales [7], it is easy to prove that A is continuous. Hence, A is completely continuous. The proof is complete. h Lemma 2.6. If u 2 P, then uðtÞ P Tt kuk for t 2 [0, T]. Proof. Since uD$(t) 6 0, it follows that uD(t) is nonincreasing. Thus, for 0 < t < T, Z t uD ðsÞDs P tuD ðtÞ uðtÞ uð0Þ ¼ 0
and uðT Þ uðtÞ ¼
Z
T
uD ðsÞDs 6 ðT tÞuD ðtÞ;
t
from which we have tuðT Þ þ ðT tÞuð0Þ t t P uðT Þ ¼ kuk: T T T The proof is complete. h uðtÞ P
In the rest of this section, we provide some background material from the theory of cones in Banach spaces, and we then state several fixed point theorems which we needed later. Let E be a Banach space and P be a cone in E. A map w : P ! [0, +1) is said to be a nonnegative, continuous and increasing functional provided w is nonnegative, continuous and satisfies w(x) 6 w(y) for all x, y 2 P and x 6 y. Given a nonnegative continuous functional w on a cone P of a real Banach space E, we define, for each d > 0, the set P ðw; dÞ ¼ fx 2 P : wðxÞ < dg: Lemma 2.7 [9]. Let P be a cone in a real Banach space E. Let a and c be increasing, nonnegative continuous functional on P, and let h be a nonnegative continuous functional on P with h(0) = 0 such that, for some c > 0 and H > 0, cðxÞ 6 hðxÞ 6 aðxÞ
and
kxk 6 H cðxÞ
for all x 2 P ðc; cÞ. Suppose there exist a completely continuous operator A : P ðc; cÞ ! P and 0 < a < b < c such that: hðkxÞ 6 khðxÞ
for 0 6 k 6 1 and x 2 @P ðh; bÞ
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and (i) c(Ax) > c for all x 2 oP(c, c); (ii) h(Ax) < b for all x 2 oP(h, b); (iii) P(a, a) 5 ; and a(Ax) > a for x 2 oP(a, a). Then, A has at least two fixed points, x1 and x2 belonging to P ðc; cÞ satisfying a < aðx1 Þ with hðx1 Þ < b;
and
b < hðx2 Þ with cðx2 Þ < c:
The following lemma is similar to Lemma 2.7. Lemma 2.8 [20]. Let P be a cone in a real Banach space E. Let a and c be increasing, nonnegative continuous functional on P, and let h be a nonnegative continuous functional on P with h(0) = 0 such that, for some c > 0 and H > 0, cðxÞ 6 hðxÞ 6 aðxÞ and
kxk 6 H cðxÞ;
for all x 2 P ðc; cÞ. Suppose there exist a completely continuous operator A : P ðc; cÞ ! P and 0 < a < b < c such that hðkxÞ 6 khðxÞ for 0 6 k 6 1
and
x 2 @P ðh; bÞ
and (i) c(Ax) < c for all x 2 oP(c, c); (ii) h(Ax) > b for all x 2 oP(h, b); (iii) P(a, a) 5 ; and a(Ax) < a for x 2 o P(a, a). Then, A has at least two fixed points, x1 and x2 belonging to P ðc; cÞ satisfying a < aðx1 Þ with hðx1 Þ < b;
and
b < hðx2 Þ with cðx2 Þ < c:
Let 0 < a < b be given and let a be a nonnegative continuous concave functional on the cone P. Define the convex sets Pa, P(a, a, b) by P a ¼ fx 2 P : kxk < ag; P ða; a; bÞ ¼ fx 2 P : a 6 aðxÞ; kxk 6 bg: Finally we state the Leggett-Williams fixed point theorem [18]. Lemma 2.9. Let P be a cone in a real Banach space E, A : P c ! P c be completely continuous and a be a nonnegative continuous concave functional on P with a(x) 6 kxk for all x 2 P c . Suppose there exists 0 < d < a < b 6 c such that (i) {x 2 P(a, a, b) : a(x) > a} 5 ; and a(Ax) > a for x 2 P(a, a, b); (ii) kAxk < d for kxk 6 d; (iii) a(Ax) > a for x 2 P(a, a, c) with kAxk > b. Then A has at least three fixed points x1, x2, x3 satisfying kx1 k < d; a < aðx2 Þ; kx3 k > d
and
aðx3 Þ < a:
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485
3. Twin solutions In this section, let l ¼ maxft 2 T : 0 6 t 6 T =2g and fix c 2 T such that l < c < T, denote M¼
l T
Z
Z
T
T
uq l
Z aðsÞrs Ds; N ¼
T
0
s
uq
Z T 1 aðsÞrs Ds d 0
and L¼
c T
Z
Z
T
uq c
T
aðsÞrs Ds:
s
Define the nonnegative, increasing and continuous functionals c, h, and a on P by cðuÞ ¼ min uðtÞ ¼ uðlÞ; hðuÞ ¼ max uðtÞ ¼ uðlÞ t2½l;c
t2½0;l
and aðuÞ ¼ max uðtÞ ¼ uðcÞ: t2½0;c
We observe that, for each u 2 P, cðuÞ ¼ hðuÞ 6 aðuÞ:
ð3:1Þ
In addition, for each u 2 P ; cðuÞ ¼ uðlÞ P Tl kuk. Thus kuk 6
T cðuÞ; l
u 2 P:
ð3:2Þ
Finally, we also note that hðkuÞ ¼ khðuÞ;
06k61
and
u 2 @P ðh; b0 Þ:
We now present the results in this section. Theorem 3.1. Assume that there are positive numbers a 0 < b 0 < c 0 such that 0 < a0 <
L 0 lL 0 b < c: N TN
Assume further that f(t, u) satisfies the following conditions: 0
(i) f ðt; uÞ > up ðMc Þ, ðt; uÞ 2 ½l; T ½c0 ; Tl c0 , 0 (ii) f ðt; uÞ < up ðbN Þ, ðt; uÞ 2 ½0; T ½0; Tl b0 , 0 (iii) f ðt; uÞ > up ðaL Þ, ðt; uÞ 2 ½c; T ½a0 ; Tc a0 . Then (1.1) and (1.2) has at least two positive solutions u1 and u2 such that a0 < max u1 ðtÞ with max u1 ðtÞ < b0 t2½0;c
t2½0;l
and
b0 < max u2 ðtÞ with min u2 ðtÞ < c0 : t2½0;l
t2½l;c
Proof. By the definition of operator A and its properties, it suffices to show that the conditions of Lemma 2.7 hold with respect to A. We first show that if u 2 oP(c, c 0 ), then c(Au) > c 0 . Indeed, if u 2 oP(c, c 0 ), then c(u) = mint2[l, c]u(t) = u(l) = c 0 . Since u 2 P, kuk 6 Tl cðuÞ ¼ Tl c0 , we have 0 c0 c 6 uðtÞ 6 Tl c0 ; t 2 ½l; T . As a consequence of (i), f ðt; uðtÞÞ > up ðM Þ; t 2 ½l; T . Also, Au 2 P implies that
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l AuðT Þ T ! Z T Z s Z Z ni m2 l 1 T 1X ¼ u aðsÞf ðs; uðsÞÞrsþ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs Ds T 0 q d 0 d i¼1 0 0 ! Z T Z s Z Z T m2 l 1 T 1X P u aðsÞf ðs; uðsÞÞrsþ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs Ds T 0 q d 0 d i¼1 0 0 Z T Z T Z T Z T l l ¼ u aðsÞf ðs; uðsÞÞrs Ds P uq aðsÞf ðs; uðsÞÞrs Ds T 0 q s T l s Z T Z T lc0 uq aðsÞrs Ds ¼ c0 : > TM l s
cðAuÞ ¼ AuðlÞ P
Next, we verify that h(Au) < b 0 for u 2 oP(h, b 0 ). Let us choose u 2 oP(h, b 0 ), then h(u) = maxt2[0,l]u(t) = u(l) = b 0 . This implies 0 6 u(t) 6 b 0 , t 2 [0, l]. Since u 2 P, we also have b 0 6 u(t) 6 kuk 6 Tl uðlÞ ¼ Tl b0 for t 2 [l, T]. So 0 6 uðtÞ 6
T 0 b; l
t 2 ½0; T :
Using (ii), 0 b f ðt; uðtÞÞ < up ; N
t 2 ½0; T :
Also, Au 2 P implies that hðAuÞ ¼ AuðlÞ 6 AuðT Þ ! Z Z ni Z T Z s m2 1 T 1X ¼ uq aðsÞf ðs; uðsÞÞrsþ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs Ds d 0 d i¼1 0 0 0 Z T Z Z T Z T 1 b0 T 1 uq aðsÞf ðs; uðsÞÞrs Ds < uq aðsÞrs Ds ¼ b0 : 6 d d N 0 0 0 0 Finally, we prove that P(a, a 0 ) 5 ; and a(Au) > a 0 for all u 2 oP(a, a 0 ). 0 In fact, the constant function a2 2 P ða; a0 Þ. Moreover, for u 2 oP(a, a 0 ), we have a(u) = maxt2[0,c]u(t) = 0 u(c) = a 0 . This implies a0 6 uðtÞ 6 Tc a0 ; t 2 ½c; T . Using assumption (iii), f ðt; uðtÞÞ > up ðaL Þ; t 2 ½c; T . As before Au 2 P, we obtain Z T Z c c T u aðsÞf ðs; uðsÞÞrs Ds aðAuÞ ¼ ðAuÞðcÞ P AuðT Þ P T T 0 q s Z T Z Z Z T c T a0 c T u aðsÞf ðs; uðsÞÞrs Ds > u aðsÞrs Ds ¼ a0 : P T c q s L T c q s Thus, by Lemma 2.7, there exists at least two fixed points of A which are positive solutions u1 and u2, belonging to P ðc; c0 Þ, of the BVP (1.1) and (1.2) such that: a0 < aðu1 Þ with hðu1 Þ < b0 ; The proof is complete.
and
b0 < hðu2 Þ with cðu2 Þ < c0 :
h
Theorem 3.2. Assume that there are positive numbers a 0 < b 0 < c 0 such that 0 < a0 <
c 0 cN 0 b < c: T TM
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Assume further that f(t, u) satisfies the following conditions: 0
(i) f ðt; uÞ < up ðNc Þ for ðt; uÞ 2 ½0; T ½0; Tl c0 , 0 (ii) f ðt; uÞ > up ðMb Þ for ðt; uÞ 2 ½l; T ½b0 ; Tl b0 , 0 (iii) f ðt; uÞ < up ðaN Þ for ðt; uÞ 2 ½c; T ½0; Tc a0 . Then (1.1) and (1.2) has at least two positive solutions u1 and u2 such that a0 < max u1 ðtÞ with max u1 ðtÞ < b0 t2½0;c
and
t2½0;l
b0 < max u2 ðtÞ with max u2 ðtÞ < c0 : t2½0;l
t2½l;c
Using Lemma 2.8, the proof is similar to that of Theorem 3.1 and we omit it here. Corollary 3.1. Assume that f satisfies conditions (i) f0 ðtÞ ¼ limu!0 fuðt;uÞ > up ðM1 Þ, t 2 [l, T] and f1 ðtÞ ¼ limu!1 fuðt;uÞ > up ðL1Þ, t 2 [c, T]; p ðuÞ p ðuÞ 0 (ii) there exists a > 0 such that 0 a T f ðt; uÞ < up ; ðt; uÞ 2 ½0; T 0; a0 : L N Then (1.1) and (1.2) has at least two positive solutions. Corollary 3.2. Assume that f satisfies conditions (i) f0 ðtÞ < up ðTNl Þ, t 2 [0, T] and f1 ðtÞ < up ðTNc Þ, t 2 [c, T]; (ii) there exists b 0 > 0 such that 0 b T f ðt; uÞ > up ; ðt; uÞ 2 ½l; T b0 ; b0 : l M Then (1.1) and (1.2) has at least two positive solutions. By applying Theorems 3.1 and 3.2, it is easy to prove that Corollaries 3.1 and 3.2 hold, respectively. 4. Triple solutions Let the nonnegative continuous concave functional W: P ! [0, 1) be defined by WðuÞ ¼ min uðtÞ ¼ uðlÞ; u 2 P : t2½l;T
Note that for u 2 P, W(u) 6 kuk. Theorem 4.1. Suppose that there exist constants 0 < d 0 < a 0 such that (i) (ii) (iii) (D1) (D2)
0
f ðt; uÞ < up ðdN Þ, (t, u) 2 [0, T] · [0, d 0 ]; 0 f ðt; uÞ P up ðMa Þ, ðt; uÞ 2 ½l; T ½a0 ; Tl a0 ; one of the following conditions holds: limu!1 maxt2½0;T fuðt;uÞ < up ðN1 Þ; p ðuÞ 0 there exists a number c0 > Tl a0 such that f ðt; uÞ < up ðNc Þ for (t, u) 2 [0, T] · [0, c 0 ].
Then (1.1) and (1.2) has at least three positive solutions. Proof. By the definition of operator A and its properties, it suffices to show that the conditions of Lemma 2.9 hold with respect to A.
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We first show that if (D1) holds, then there exists a number l0 > Tl a0 such that A : P l0 ! P l0 . Suppose that f ðt; uÞ 1 < up lim max u!1 t2½0;T up ðuÞ N holds, then there are s > 0 and d < N1 such that if u > s, then max
t2½0;T
f ðt; uÞ 6 up ðdÞ: up ðuÞ
That is to say, f ðt; uÞ 6 up ðduÞ; ðt; uÞ 2 ½0; T ½s; 1Þ: Set k = max{f(t, u) : (t, u) 2 [0, T] · [0, s]}, then f ðt; uÞ 6 k þ up ðduÞ; ðt; uÞ 2 ½0; T ½0; 1Þ: Taking
!) kup ðN Þ T 0 a ; uq l > max : l 1 up ðdN Þ
ð4:1Þ
(
0
ð4:2Þ
If u 2 P l0 , then by (2.5), (4.1), (4.2), we obtain kAuk ¼ AuðT Þ ! Z Z T Z s Z ni m2 1 T 1X ¼ uq aðsÞf ðs; uðsÞÞrsþ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs Ds d 0 d i¼1 0 0 0 Z T Z T Z T Z T 1 1 uq aðsÞf ðs; uðsÞÞrs Ds 6 uq aðsÞðk þ up ðduÞÞrs Ds 6 d 0 d 0 0 0 Z Z T 1 T uq aðsÞrs Ds ¼ uq ðk þ up ðdl0 ÞÞN < l0 : 6 uq ðk þ up ðdl0 ÞÞ d 0 0 Next we verify that if there is a positive number r 0 such that if f(t, u) < up(r 0 /N) for (t, u) 2 [0, T] · [0, r 0 ], then A : P r0 ! P r0 . Indeed, if u 2 P r0 , then kAuk ¼ AuðT Þ 6
Z
T
uq 0
Z T Z T Z 1 r0 T 1 aðsÞf ðs; uðsÞÞrs Ds < uq aðsÞrs Ds ¼ r0 ; d 0 d 0 N 0
thus, Au 2 P . Hence, we have shown that either (D1) or (D2) holds, then there exists a number c 0 with c0 > Tl a0 such that A : P c0 ! P c0 . It is also note from (i) that A : P d 0 ! P d 0 . Now, we show that fu 2 P ðW; a0 ; Tl a0 Þ : WðuÞ > a0 g 6¼ ; and W(Au) > a 0 for all u 2 P ðW; a0 ; Tl a0 Þ. In fact, ðl þ T Þa0 0 T 0 0 2 u 2 P W; a ; a : WðuÞ > a : u¼ l 2l r0
For u 2 P ðW; a0 ; Tl a0 Þ, we have a0 6 min uðtÞ ¼ uðlÞ 6 uðtÞ 6 t2½l;T
T 0 a; l
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for all t 2 [l, T]. Then, in view of (ii), we know that WðAuÞ ¼ min AuðtÞ ¼ AuðlÞ P t2½l;T
l AuðT Þ T
! Z s Z Z ni m2 1 T 1X uq aðsÞf ðs; uðsÞÞrsþ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs Ds d 0 d i¼1 0 0 0 Z T Z T Z T Z T l l u aðsÞf ðs; uðsÞÞrs Ds ¼ uq aðsÞf ðs; uðsÞÞrs Ds P T 0 q s T l s Z T Z T la0 uq aðsÞrs Ds ¼ a0 : > TM l s
l ¼ T
Z
T
Finally, we assert that if u 2 P(W, a 0 , c 0 ) and kAuk > Tl a0 , then W(Au) > a 0 . Suppose u 2 P(W, a 0 , c 0 ) and kAuk > Tl a0 , then WðAuÞ ¼ min AuðtÞ ¼ AuðlÞ P t2½l;T
l l AuðT Þ ¼ kAuk > a0 : T T
To sum up, the hypotheses of Lemma 2.9 are satisfied, hence (1.1) and (1.2) has at least three positive solutions u1, u2, u3 such that ku1 k < d 0 ; a0 < min u2 ðtÞ and
ku3 k > d 0 with min u3 ðtÞ < a0 :
t2½l;T
The proof is complete.
t2½l;T
h
From Theorem 4.1, we see that, when assumption like (i), (ii), and (iii) are imposed appropriately on f, we can establish the existence of an arbitrary odd number of positive solutions of (1.1) and (1.2). Theorem 4.2. Suppose that there exist constants 0 < d 01 < a01 <
T 0 T a < d 02 < a02 < a02 < d 03 < < d 0n ; n 2 N; l 1 l
such that the following conditions are satisfied: d0
(i) f ðt; uÞ < up ðNi Þ, ðt; uÞ 2 ½0; T ½0; d 0i ; a0 (ii) f ðt; uÞ P up ðMi Þ, ðt; uÞ 2 ½l; T ½a0i ; Tl a0i . Then (1.1) and (1.2) has at least 2n 1 positive solutions. Proof. When n = 1, it is immediate from condition (i) that A : P d 01 ! P d 01 P d 01 , which means that A has at least one fixed point u1 2 P d 01 by the Schauder fixed point theorem. When n = 2, it is clear that Theorem 4.1 holds (with c1 ¼ d 02 ). Then we can obtain at least three positive solutions u1, u2, and u3 satisfying ku1 k < d 01 ; min u2 ðtÞ > a01 t2½l;T
and
ku3 k > d 01 with min u3 ðtÞ < a01 : t2½l;T
Following this way, we finish the proof by induction. The proof is complete.
h
5. Examples N
Let T ¼ f1 ð1=2Þ 0 g [ f1g. Taking T = 1, p = 2, m = 4, a1 ¼ 13 ; n1 ¼ 12 ; a2 ¼ 12, and n2 ¼ 15 , we have d ¼ 16. 16 1 If we let a(s) 1, then M ¼ 8, N = 6. Suppose f ðt; uÞ ¼ f ðuÞ :¼
2000u2 ; u2 þ 5000
t 2 ½0; 1;
u P 0:
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Clearly f is always increasing. If we take d 0 ¼ 25, a 0 = 23, and c 0 = 12006, then 0 < d 0 < a0 < 2a0 < c0 : Now we check that (i), (ii) and (iii) of Theorem 4.1 are satisfied. In view of f(2/5) = 2000/31251 0.0639, we get f ðuÞ <
2 1 ¼ 0:0667; 5N 15
u 2 ½0; d 0 ;
so that (i) of Theorem 4.1 is met. To verify (ii), note that f(23) 191.355, so that f ðuÞ P 23=M ¼ 184;
u 2 ½a0 ; 2a0 :
Lastly, as limu!1f(u) = 2000, f ðuÞ 6 2000 < c0 =N ¼ 2001;
u 2 ½0; c0 ;
and (D2) of (iii) holds. Thus by Theorem 4.1, the boundary value problem 1 D 1 1 D 15 Dr D u ðtÞ þ f ðuðtÞÞ ¼ 0; uð0Þ ¼ 0; u ð1Þ ¼ u þ u ; 3 2 2 16 has at least three positive solutions u1, u2, u3 satisfying 2 ku1 k < ; 5
min u2 ðtÞ > 23
t2½12;1
and
ku3 k >
2 with min u3 ðtÞ < 23: t2½1=2;1 5
References [1] R.P. Agarwal, M. Bohner, W.T. Li, Nonoscillation and Oscillation Theory for Functional Differential Equations, Pure and Applied Mathematics Series, 267, Marcel Dekker, 2004. [2] R.P. Agarwal, M. Bohner, P. Rehak, Half-linear dynamic equations, Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday, 1, Kluwer Academic Publishers, Dordrecht, 2003, p. 1–57. [3] D.R. Anderson, Solutions to second-order three-point problems on time scales, J. Differen. Equ. Appl. 8 (2002) 673–688. [4] D.R. Anderson, Nonlinear triple point problems on time scales, Electron. J. Differen. Equ. 47 (2004) 1–12. [5] D.R. Anderson, R. Avery, J. Henderson, Existence of solutions for a one-dimensional p-Laplacian on time scales, J. Differen. Equ. Appl. 10 (2004) 889–896. [6] F.M. Atici, G.Sh. Guseinov, On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math. 141 (2002) 75–99. [7] B. Aulbach, L. Neidhart, Integration on measure chains, in: Proc. of the Sixth Int. Conf. on Difference Equations, CRC, Boca Raton, FL, 2004, pp. 239–252. [8] R.I. Avery, C.J. Chyan, J. Henderson, Twin solutions of boundary value problems for ordinary differential equations and finite difference equations, Comput. Math. Appl. 42 (2001) 695–704. [9] R.I. Avery, J. Henderson, Two positive fixed points of nonlinear operator on ordered Banach spaces, Commun. Appl. Nonlinear Anal. 8 (2001) 27–36. [10] M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001. [11] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003. [12] J.J. DaCunha, J.M. Davis, P.K. Singh, Existence results for singular three point boundary value problems on time scales, J. Math. Anal. Appl. 295 (2004) 378–391. [13] C.P. Gupta, A non-resonant multi-point boundary value problem for a p-Laplacian type operator, Electron. J. Differen. Equ. Conf. 10 (2003) 143–152. [14] S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math. 18 (1990) 18–56. [15] E.R. Kaufmann, Positive solutions of a three-point boundary value problem on a time scale, Electron. J. Differen. Equ. 2003 (82) (2003) 1–11. [16] E.R. Kaufmann, Y.N. Raffoul, Eigenvalue problems for a three-point boundary-value problem on a time scale, Electron. J. Qualitative Theory Differen. Equ. 2 (2004) 1–10. [17] V. Lakshmikantham, S. Sivasundaram, B. Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Boston, 1996. [18] R. Leggett, L. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979) 673–688. [19] W.T. Li, H.R. Sun, Multiple positive solutions for nonlinear dynamic systems on a measure chain, J. Comput. Appl. Math. 162 (2004) 421–430.
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[20] Y. Liu, W. Ge, Twin positive solutions of boundary value problems for finite difference equations with p-Laplacian operator, J. Math. Anal. Appl. 278 (2003) 551–561. [21] A.C. Peterson, Y.N. Raffoul, C.C. Tisdell, Three point boundary value problems on time scales, J. Differen. Equ. Appl. 10 (2004) 843– 849. [22] H.R. Sun, Existence of positive solutions to second-order time scale systems, Comput. Math. Appl. 49 (2005) 131–145. [23] H.R. Sun, W.T. Li, Existence of positive solutions for nonlinear three-point boundary value problems on time scales, J. Math. Anal. Appl. 299 (2004) 508–524. [24] H.R. Sun, W.T. Li, Existence theory for positive solutions to one-dimensional p-Laplacian boundary value problems on time scales, submitted for publication. [25] H.R. Sun, W.T. Li, Positive solutions of p-Laplacian m-point boundary value problems on time scales, Taiwan. J. Math., submitted for publication.
Multiple positive solutions for p-Laplacian m-point boundary value problems on time scales q Hong-Rui Sun *, Wan-Tong Li School of Mathematics and Statistics, Lanzhou University, Tianshui Street 222, Lanzhou, Gansu 730000, People’s Republic of China
Abstract Let T be a time scale such that 0; T 2 T, ai P 0 for i = 1, . . ., m 2. Let ni satisfy 0 < n1 < n2 < < nm2 < q(T) and Pm2 i¼1 ai < 1. We consider the following p-Laplacian m-point boundary value problem on time scales ðup ðuD ðtÞÞÞr þ aðtÞf ðt; uðtÞÞ ¼ 0; t 2 ð0; T Þ; uð0Þ ¼ 0; up ðuD ðT ÞÞ ¼
m2 X
ai up ðuD ðni ÞÞ;
i¼1
where a 2 Cld ((0, T), [0, 1)) and f 2 Cld ((0, T) · [0, 1), [0, 1)). Some new results are obtained for the existence of at least twin or triple positive solutions of the above problem by applying Avery-Henderson and Leggett-Williams fixed point theorems respectively. In particular, our criteria extend and improve some known results. 2006 Elsevier Inc. All rights reserved. Keywords: Time scales; Positive solution; Cone; Fixed point
1. Introduction The study of dynamic equations on time scales goes back to its founder Stefan Hilger [14], and is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on time scales can build bridges between continuous and discrete equations. Further, the study of time scales has led to several important applications, e.g., in the study of insect population models, neural networks, heat transfer and epidemic models [1,10,11]. We begin by presenting some basic definitions which can be found in [1,10,14,17]. Another excellent source on dynamic equations on time scales is the book [11]. q Supported by the NNSF of China (10571078), the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China and the Fundamental Research Fund for Physics and Mathematic of Lanzhou University (Lzu05003). * Corresponding author. E-mail address: [email protected] (H.-R. Sun).
0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.04.009
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A time scale T is a nonempty closed subset of R. It follows that the jump operators r; q : T ! T: rðtÞ ¼ inffs 2 T : s > tg and qðtÞ ¼ supfs 2 T : s < tg; (supplemented by inf ; :¼ sup T and sup ; :¼ inf T) are well defined. The point t 2 T is left-dense, left-scattered, right-dense, right-scattered if q(t) = t, q(t) < t, r(t) = t, r(t) > t, respectively. If T has a right-scattered minimum m, define Tj ¼ T fmg; otherwise, set Tj ¼ T. If T has a left-scattered maximum M, define Tj ¼ T fMg; otherwise, set Tj ¼ T. The forward graininess is l(t) :¼ r(t) t. Similarly, the backward graininess is t(t) :¼ t q(t). We make the blanket assumption that 0, T are points in T. By an interval (0, T) we always mean the intersection of the real interval (0, T) with the given time scale, that is ð0; T Þ \ T. Other type of intervals are defined similarly. For f : T ! R and t 2 Tj , the delta derivative [10] of f at t, denoted by fD (t), is the number (provided it exists) with the property that given any > 0, there is a neighborhood U T of t such that: f ðrðtÞÞ f ðsÞ f D ðtÞ½rðtÞ sj 6 jrðtÞ s; for all s 2 U. For f : T ! R and t 2 Tj , the nabla derivative [6] of f at t, denoted by f$ (t), is the number (provided it exists) with the property that given any > 0, there is a neighborhood U of t such that: f ðqðtÞÞ f ðsÞ f r ðtÞ½qðtÞ sj 6 jqðtÞ s; for all s 2 U. A function f : T ! R is ld-continuous provided it is continuous at left dense points in T and its right sided limit exists (finite) at right dense points in T. If T ¼ R, then f is ld-continuous if and only if f is continuous. If T ¼ Z, then any function is ld-continuous. It is known [6] that if f is ld-continuous, then there is a function F(t) such that F$ (t) = f(t). In this case, we define Z b f ðsÞrs ¼ F ðbÞ F ðaÞ: a
Very recently, there is an increasing attention paid to question of positive solution for second order three point boundary value problems on time scales [3,4,12,15,16,19,21–23]. But very little work has been done to the existence of positive solutions for p-Laplacian boundary value problem on time scales [5,24,25]. In particular, we would like to mention some results of Anderson, Avery and Henderson [5], Gupta [13], Sun and Li [24,25], which motivate us to consider our problem. For convenience, throughout this paper we denote up(u) is p-Laplacian operator, i.e., up(u) = jujp2u, p > 1, (up)1 = uq, 1/p + 1/q = 1. In [5], Anderson, Avery and Henderson considered the following problem: r
ðup ðuD ðtÞÞÞ þ cðtÞf ðuÞ ¼ 0; D
uðaÞ B0 ðu ðmÞÞ ¼ 0;
t 2 ða; bÞ;
D
u ðbÞ ¼ 0;
where m 2 (a, b), f 2 Cld ([0, 1), [0, 1)), c 2 Cld ((a, b), [0, 1)) and Kmx 6 B0(x) 6 KMx for some positive constants Km, KM. They established the existence result of at least one positive solution by a fixed point theorem of cone expansion and compression of functional type. In [13], Gupta studied the multi-point boundary value problem: 0
ð/ðu0 ðtÞÞÞ ¼ f ðt; xðtÞ; x0 ðtÞÞ þ eðtÞ;
t 2 ð0; 1Þ;
and uð0Þ ¼ 0;
/ðu0 ð1ÞÞ ¼
m2 X
ai /ðu0 ðni ÞÞ;
i¼1
where / is an odd increasing homeomorphism form R on R with /(0) = 0. A new prior estimate of Poincare type and Leray-Schauder continuation theorem lead to the existence theorems of positive solutions.
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In [24], the authors considered the existence of at least single, twin or triple positive solutions to the following problem: D
ðup ðuD ðtÞÞÞ þ hðtÞf ður ðtÞÞ ¼ 0;
t 2 ½a; b;
with the boundary condition, uðaÞ B0 ðuD ðaÞÞ ¼ 0;
uD ðrðbÞÞ ¼ 0;
and in [25], we further considered the existence criteria of one or multiple positive solutions to the following p-Laplacian m-point boundary value problem: ðup ðuD ðtÞÞÞr þ aðtÞf ðt; uðtÞÞ ¼ 0; uD ð0Þ ¼ 0;
uðT Þ ¼
m2 X
t 2 ð0; T Þ;
ai uðni Þ:
i¼1
The main tools used in [24,25] are fixed point theorems in cones. In this paper, we shall study the existence of multiple positive solutions for the one-dimensional p-Laplacian m-point boundary value problem on time scales: r
ðup ðuD ðtÞÞÞ þ aðtÞf ðt; uðtÞÞ ¼ 0; uð0Þ ¼ 0;
up ðuD ðT ÞÞ ¼
m2 X
t 2 ð0; T Þ;
ð1:1Þ
ai up ðuD ðni ÞÞ:
ð1:2Þ
i¼1
Some new results are obtained for the existence of at least twin or triple positive solutions for the above problem by using the fixed point theorems due to Avery and Henderson [9] and Leggett-Williams [18]. The results are even new for the special cases of difference equations and differential equations, as well as in the general time scale setting. Pm2 It is also noted that when i¼1 ai ¼ 0, boundary value problem (1.1) and (1.2) reduce to one dimension pLaplacian two point boundary value problem. In the special case of p = 2, it has been extensively studied by many authors, see [10,11]. The rest of the paper is organized as follows. In Section 2, we first give four lemmas which are needed throughout this paper and then list two fixed point theorems due to Avery and Henderson and LeggettWilliams. In Section 3 we give two existence results and some corollaries. In particular, our Theorem 3.1 Pm2 includes and extends the main results of Avery, Chyan and Henderson [8] ðp ¼ 2; i¼1 ai ¼ 0; f ðt; uÞ ¼ f ðuÞÞ in the case of T ¼ R and T ¼ Z. Section 4 is to develop existence criteria for (at least) three positive and arbitrary odd positive solutions of problem (1.1) and (1.2). In particular, our results are new when T ¼ R (the continual case) and T ¼ Z (the discrete case). Finally, in Section 5, we give an example to illustrate our main results. For convenience, we list the following hypotheses: (A1) ai P 0 for i = 1, . . ., m 2, 0 < n1 < n2 < < nm2 < q(T) and d ¼ 1 (A2) a:(0, T) ! [0, 1) is ld-continuous such that a(t0) > 0 for at f:(0, T) · [0, 1) ! [0, 1) is ld-continuous.
Pm2
ai > 0; least one
i¼1
t0 2 [0, T)
and
2. Some lemmas To prove the main results in this paper, we will employ several lemmas, which are based on the linear boundary value problem r
ðup ðuD ðtÞÞÞ þ hðtÞ ¼ 0; uð0Þ ¼ 0;
up ðuD ðT ÞÞ ¼
t 2 ð0; T Þ; m2 X i¼1
ai up ðuD ðni ÞÞ:
ð2:1Þ ð2:2Þ
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Lemma 2.1. If d 5 0, then for h 2 Cld [0, T] the boundary value problem (2.1) and (2.2) has the unique solution: ! Z Z ni Z t Z s m2 1 T 1X uðtÞ ¼ uq hðsÞrs þ hðsÞrs ai hðsÞrs Ds: ð2:3Þ d 0 d i¼1 0 0 0 Proof. By the formula ð D
u ðtÞ ¼ uq
Z
t 0
and so up ðuD ðtÞÞ ¼
Z
Rt
f ðt; sÞDsÞD ¼ f ðrðtÞ; tÞ þ
a
1 hðsÞrs þ d
t
hðsÞrs þ
0
1 d
Observe that Z m2 m2 X X ai up ðuD ðni ÞÞ ¼ ai i¼1
¼
1 d
Pm2
T 0
Z
T
hðsÞrs 0
hðsÞrs þ ni
hðsÞrs þ
ai
i¼1
Z ni m2 1X ai hðsÞrs: d i¼1 0
Pm2
ni
Z
f D ðt; sÞDs, we have
a
! Z ni m2 1X hðsÞrs ai hðsÞrs ; d i¼1 0
0
i¼1 m2 X
Z
Rt
0
i¼1
Z
ai
d 1 d
Z ni m2 m2 X 1X hðsÞrs ai ai hðsÞrs d i¼1 0 i¼1
T
0 m2 X
Z ai
i¼1
!
T
hðsÞrs ¼ up ðuD ðT ÞÞ:
0
(2.3) still holds. i¼1 ai 0, then (2.1) and (2.2) reduce to the two-point BVP. It is easy to see that P D We note that the boundary value problem (up(xD(t)))$ = 0, x(0) = 0, up ðxD ðT ÞÞ ¼ m2 i¼1 ai up ðx ðni ÞÞ has only the trivial solution if d 5 0. Thus u in (2.3) is the unique solution of (2.1) and (2.2). The proof is complete. h If
Lemma 2.2. Let d > 0. If h 2 Cld[0, T] and h P 0, then the unique solution u of (2.1) and (2.2) satisfies uðtÞ P 0;
t 2 ½0; T :
Proof. Since (up(uD(t)))$ = h(t) 6 0, we obtain that: up ðuD ðT ÞÞ ¼ In view of
Pm2 i¼1
m2 X
ai up ðuD ðni ÞÞ P
i¼1
m2 X
ai up ðuD ðT ÞÞ:
i¼1
D
ai < 1, we know u (T) P 0. By u(0) = 0, the result is obvious. The proof is complete.
h
Now we give a lemma which is needed in Lemma 2.4. Lemma 2.3. Assume g : R ! R is continuous, g : T ! R is delta differentiable on Tj , and f : R ! R is continuous differentiable. Then there exists c in the real interval [q(t), t] with ðf gÞr ðtÞ ¼ f 0 ðgðcÞÞgr ðtÞ:
ð2:4Þ
Proof. The proof is similar to that of [10, Theorem 1.87, p. 31]. For convenience, we list it here. Fix t 2 Tj . First we consider the case where t is left-scattered. In this case r
ðf gÞ ðtÞ ¼
f ðgðtÞÞ f ðgðqðtÞÞÞ : qðtÞ
If g(q(t)) = g(t), then we get (f g)$(t) = 0 and g$(t) = 0 and so (2.4) holds for any c in the real interval [q(t), t]. Hence we can assume g(q(t)) 5 g(t). Then ðf gÞr ðtÞ ¼
f ðgðtÞÞ f ðgðqðtÞÞÞ gðtÞ gðqðtÞÞ ¼ f 0 ðnÞgr ðtÞ; gðtÞ gðqðtÞÞ qðtÞ
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by the mean value theorem, where n is between g(q(t)) and g(t). Since g : R ! R is continuous, there is c 2 [q(t), t] such that g(c) = n, which gives us the desired result. It remains to consider the case when t is left-dense. In this case r
ðf gÞ ðtÞ ¼ lim s!t
f ðgðtÞÞ f ðgðsÞÞ gðtÞ gðsÞ ¼ lim f 0 ðns Þ s!t ts ts
by the mean value theorem in calculus, where ns is between g(s) and g(t). By the continuity of g we get that lims!tns = g(t) which give us the desired result. The proof is complete. h Lemma 2.4. Let d < 0. If h 2 Cld[0, T] and h P 0, then the boundary value problem (2.1) and (2.2) has no nonnegative solution. Proof. By Lemma 2.1 and d < 0, it is easy to see that u(T) < 0. The proof is complete.
h
Now, let the Banach space B = Cld[0, T] (see [3]) be endowed with the norm kuk = supt2[0,T]ju(t)j, and choose the cone P B defined by; P¼
u 2 B : uðtÞ P 0
for t 2 ½0; T
uDr ðtÞ 6 0; uD ðtÞ P 0
and
for t 2 ð0; T Þ; uð0Þ ¼ 0
:
Clearly, kuk = u(T) for u 2 P. Define the operator A : P ! B by ! Z Z ni Z s Z t m2 1 T 1X uq aðsÞf ðs; uðsÞÞrs þ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs Ds: AuðtÞ ¼ d 0 d i¼1 0 0 0 ð2:5Þ Obviously, Au(t) P 0 for t 2 [0, T]. From the definition of A, we claim that for each u 2 P, Au 2 P and satisfies (1.2) and Au(T) is the maximum value of Au(t) on [0, T]. In fact, ! Z t Z Z ni m2 1 T 1X D ðAuÞ ðtÞ ¼ uq aðsÞf ðs; uðsÞÞrsþ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs d 0 d i¼1 0 0 ! Z t Z Z T m2 1 T 1X P uq aðsÞf ðs; uðsÞÞrsþ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs d 0 d i¼1 0 0 Z T ¼ uq aðsÞf ðs; uðsÞÞrs P 0; t 2 ð0; T Þ: t
Moreover, uq(x) is a monotone increasing and continuously differentiable function and Z T r aðsÞf ðs; uðsÞÞrs ¼ aðtÞf ðt; uðtÞÞ 6 0; t
then by Lemma 2.3, we obtain ðAuÞDr ðtÞ 6 0; so, A : P ! P. Lemma 2.5. A : P ! P is completely continuous. Proof. First, we show that A maps bounded set into itself. Assume c > 0 is a constant and u 2 P c ¼ fx 2 P : kxk 6 cg. Note that the continuity of f(t, u) guarantees that there is a C > 0 such that f(t, u) 6 up(C) for t 2 [0, T]. So
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483
kAuk ¼ AuðT Þ ! Z T Z s Z Z ni m2 1 T 1X ¼ uq aðsÞf ðs; uðsÞÞrs þ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs Ds d 0 d i¼1 0 0 0 Z T Z T Z T Z T 1 1 6 uq aðsÞf ðs; uðsÞÞrs Ds 6 C uq aðsÞrs Ds: d 0 d 0 0 0 That is, AP c is uniformly bounded. In addition, notice that: ! Z t2 Z s Z T Z ni m2 X 1 1 jAuðt1 Þ Auðt2 Þj ¼ uq aðsÞf ðs; uðsÞÞrs þ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs Ds d 0 d i¼1 t1 0 0 Z t 2 Z T Z T 1 1 uq aðsÞf ðs; uðsÞÞrs Ds 6 Cjt1 t2 juq aðsÞrs : 6 d d t1 0 0 So, by applying Arzela–Ascoli theorem on time scales [2] we obtain that AP c is relatively compact. In view of Lebesgue’s dominated convergence theorem on time scales [7], it is easy to prove that A is continuous. Hence, A is completely continuous. The proof is complete. h Lemma 2.6. If u 2 P, then uðtÞ P Tt kuk for t 2 [0, T]. Proof. Since uD$(t) 6 0, it follows that uD(t) is nonincreasing. Thus, for 0 < t < T, Z t uD ðsÞDs P tuD ðtÞ uðtÞ uð0Þ ¼ 0
and uðT Þ uðtÞ ¼
Z
T
uD ðsÞDs 6 ðT tÞuD ðtÞ;
t
from which we have tuðT Þ þ ðT tÞuð0Þ t t P uðT Þ ¼ kuk: T T T The proof is complete. h uðtÞ P
In the rest of this section, we provide some background material from the theory of cones in Banach spaces, and we then state several fixed point theorems which we needed later. Let E be a Banach space and P be a cone in E. A map w : P ! [0, +1) is said to be a nonnegative, continuous and increasing functional provided w is nonnegative, continuous and satisfies w(x) 6 w(y) for all x, y 2 P and x 6 y. Given a nonnegative continuous functional w on a cone P of a real Banach space E, we define, for each d > 0, the set P ðw; dÞ ¼ fx 2 P : wðxÞ < dg: Lemma 2.7 [9]. Let P be a cone in a real Banach space E. Let a and c be increasing, nonnegative continuous functional on P, and let h be a nonnegative continuous functional on P with h(0) = 0 such that, for some c > 0 and H > 0, cðxÞ 6 hðxÞ 6 aðxÞ
and
kxk 6 H cðxÞ
for all x 2 P ðc; cÞ. Suppose there exist a completely continuous operator A : P ðc; cÞ ! P and 0 < a < b < c such that: hðkxÞ 6 khðxÞ
for 0 6 k 6 1 and x 2 @P ðh; bÞ
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and (i) c(Ax) > c for all x 2 oP(c, c); (ii) h(Ax) < b for all x 2 oP(h, b); (iii) P(a, a) 5 ; and a(Ax) > a for x 2 oP(a, a). Then, A has at least two fixed points, x1 and x2 belonging to P ðc; cÞ satisfying a < aðx1 Þ with hðx1 Þ < b;
and
b < hðx2 Þ with cðx2 Þ < c:
The following lemma is similar to Lemma 2.7. Lemma 2.8 [20]. Let P be a cone in a real Banach space E. Let a and c be increasing, nonnegative continuous functional on P, and let h be a nonnegative continuous functional on P with h(0) = 0 such that, for some c > 0 and H > 0, cðxÞ 6 hðxÞ 6 aðxÞ and
kxk 6 H cðxÞ;
for all x 2 P ðc; cÞ. Suppose there exist a completely continuous operator A : P ðc; cÞ ! P and 0 < a < b < c such that hðkxÞ 6 khðxÞ for 0 6 k 6 1
and
x 2 @P ðh; bÞ
and (i) c(Ax) < c for all x 2 oP(c, c); (ii) h(Ax) > b for all x 2 oP(h, b); (iii) P(a, a) 5 ; and a(Ax) < a for x 2 o P(a, a). Then, A has at least two fixed points, x1 and x2 belonging to P ðc; cÞ satisfying a < aðx1 Þ with hðx1 Þ < b;
and
b < hðx2 Þ with cðx2 Þ < c:
Let 0 < a < b be given and let a be a nonnegative continuous concave functional on the cone P. Define the convex sets Pa, P(a, a, b) by P a ¼ fx 2 P : kxk < ag; P ða; a; bÞ ¼ fx 2 P : a 6 aðxÞ; kxk 6 bg: Finally we state the Leggett-Williams fixed point theorem [18]. Lemma 2.9. Let P be a cone in a real Banach space E, A : P c ! P c be completely continuous and a be a nonnegative continuous concave functional on P with a(x) 6 kxk for all x 2 P c . Suppose there exists 0 < d < a < b 6 c such that (i) {x 2 P(a, a, b) : a(x) > a} 5 ; and a(Ax) > a for x 2 P(a, a, b); (ii) kAxk < d for kxk 6 d; (iii) a(Ax) > a for x 2 P(a, a, c) with kAxk > b. Then A has at least three fixed points x1, x2, x3 satisfying kx1 k < d; a < aðx2 Þ; kx3 k > d
and
aðx3 Þ < a:
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3. Twin solutions In this section, let l ¼ maxft 2 T : 0 6 t 6 T =2g and fix c 2 T such that l < c < T, denote M¼
l T
Z
Z
T
T
uq l
Z aðsÞrs Ds; N ¼
T
0
s
uq
Z T 1 aðsÞrs Ds d 0
and L¼
c T
Z
Z
T
uq c
T
aðsÞrs Ds:
s
Define the nonnegative, increasing and continuous functionals c, h, and a on P by cðuÞ ¼ min uðtÞ ¼ uðlÞ; hðuÞ ¼ max uðtÞ ¼ uðlÞ t2½l;c
t2½0;l
and aðuÞ ¼ max uðtÞ ¼ uðcÞ: t2½0;c
We observe that, for each u 2 P, cðuÞ ¼ hðuÞ 6 aðuÞ:
ð3:1Þ
In addition, for each u 2 P ; cðuÞ ¼ uðlÞ P Tl kuk. Thus kuk 6
T cðuÞ; l
u 2 P:
ð3:2Þ
Finally, we also note that hðkuÞ ¼ khðuÞ;
06k61
and
u 2 @P ðh; b0 Þ:
We now present the results in this section. Theorem 3.1. Assume that there are positive numbers a 0 < b 0 < c 0 such that 0 < a0 <
L 0 lL 0 b < c: N TN
Assume further that f(t, u) satisfies the following conditions: 0
(i) f ðt; uÞ > up ðMc Þ, ðt; uÞ 2 ½l; T ½c0 ; Tl c0 , 0 (ii) f ðt; uÞ < up ðbN Þ, ðt; uÞ 2 ½0; T ½0; Tl b0 , 0 (iii) f ðt; uÞ > up ðaL Þ, ðt; uÞ 2 ½c; T ½a0 ; Tc a0 . Then (1.1) and (1.2) has at least two positive solutions u1 and u2 such that a0 < max u1 ðtÞ with max u1 ðtÞ < b0 t2½0;c
t2½0;l
and
b0 < max u2 ðtÞ with min u2 ðtÞ < c0 : t2½0;l
t2½l;c
Proof. By the definition of operator A and its properties, it suffices to show that the conditions of Lemma 2.7 hold with respect to A. We first show that if u 2 oP(c, c 0 ), then c(Au) > c 0 . Indeed, if u 2 oP(c, c 0 ), then c(u) = mint2[l, c]u(t) = u(l) = c 0 . Since u 2 P, kuk 6 Tl cðuÞ ¼ Tl c0 , we have 0 c0 c 6 uðtÞ 6 Tl c0 ; t 2 ½l; T . As a consequence of (i), f ðt; uðtÞÞ > up ðM Þ; t 2 ½l; T . Also, Au 2 P implies that
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l AuðT Þ T ! Z T Z s Z Z ni m2 l 1 T 1X ¼ u aðsÞf ðs; uðsÞÞrsþ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs Ds T 0 q d 0 d i¼1 0 0 ! Z T Z s Z Z T m2 l 1 T 1X P u aðsÞf ðs; uðsÞÞrsþ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs Ds T 0 q d 0 d i¼1 0 0 Z T Z T Z T Z T l l ¼ u aðsÞf ðs; uðsÞÞrs Ds P uq aðsÞf ðs; uðsÞÞrs Ds T 0 q s T l s Z T Z T lc0 uq aðsÞrs Ds ¼ c0 : > TM l s
cðAuÞ ¼ AuðlÞ P
Next, we verify that h(Au) < b 0 for u 2 oP(h, b 0 ). Let us choose u 2 oP(h, b 0 ), then h(u) = maxt2[0,l]u(t) = u(l) = b 0 . This implies 0 6 u(t) 6 b 0 , t 2 [0, l]. Since u 2 P, we also have b 0 6 u(t) 6 kuk 6 Tl uðlÞ ¼ Tl b0 for t 2 [l, T]. So 0 6 uðtÞ 6
T 0 b; l
t 2 ½0; T :
Using (ii), 0 b f ðt; uðtÞÞ < up ; N
t 2 ½0; T :
Also, Au 2 P implies that hðAuÞ ¼ AuðlÞ 6 AuðT Þ ! Z Z ni Z T Z s m2 1 T 1X ¼ uq aðsÞf ðs; uðsÞÞrsþ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs Ds d 0 d i¼1 0 0 0 Z T Z Z T Z T 1 b0 T 1 uq aðsÞf ðs; uðsÞÞrs Ds < uq aðsÞrs Ds ¼ b0 : 6 d d N 0 0 0 0 Finally, we prove that P(a, a 0 ) 5 ; and a(Au) > a 0 for all u 2 oP(a, a 0 ). 0 In fact, the constant function a2 2 P ða; a0 Þ. Moreover, for u 2 oP(a, a 0 ), we have a(u) = maxt2[0,c]u(t) = 0 u(c) = a 0 . This implies a0 6 uðtÞ 6 Tc a0 ; t 2 ½c; T . Using assumption (iii), f ðt; uðtÞÞ > up ðaL Þ; t 2 ½c; T . As before Au 2 P, we obtain Z T Z c c T u aðsÞf ðs; uðsÞÞrs Ds aðAuÞ ¼ ðAuÞðcÞ P AuðT Þ P T T 0 q s Z T Z Z Z T c T a0 c T u aðsÞf ðs; uðsÞÞrs Ds > u aðsÞrs Ds ¼ a0 : P T c q s L T c q s Thus, by Lemma 2.7, there exists at least two fixed points of A which are positive solutions u1 and u2, belonging to P ðc; c0 Þ, of the BVP (1.1) and (1.2) such that: a0 < aðu1 Þ with hðu1 Þ < b0 ; The proof is complete.
and
b0 < hðu2 Þ with cðu2 Þ < c0 :
h
Theorem 3.2. Assume that there are positive numbers a 0 < b 0 < c 0 such that 0 < a0 <
c 0 cN 0 b < c: T TM
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Assume further that f(t, u) satisfies the following conditions: 0
(i) f ðt; uÞ < up ðNc Þ for ðt; uÞ 2 ½0; T ½0; Tl c0 , 0 (ii) f ðt; uÞ > up ðMb Þ for ðt; uÞ 2 ½l; T ½b0 ; Tl b0 , 0 (iii) f ðt; uÞ < up ðaN Þ for ðt; uÞ 2 ½c; T ½0; Tc a0 . Then (1.1) and (1.2) has at least two positive solutions u1 and u2 such that a0 < max u1 ðtÞ with max u1 ðtÞ < b0 t2½0;c
and
t2½0;l
b0 < max u2 ðtÞ with max u2 ðtÞ < c0 : t2½0;l
t2½l;c
Using Lemma 2.8, the proof is similar to that of Theorem 3.1 and we omit it here. Corollary 3.1. Assume that f satisfies conditions (i) f0 ðtÞ ¼ limu!0 fuðt;uÞ > up ðM1 Þ, t 2 [l, T] and f1 ðtÞ ¼ limu!1 fuðt;uÞ > up ðL1Þ, t 2 [c, T]; p ðuÞ p ðuÞ 0 (ii) there exists a > 0 such that 0 a T f ðt; uÞ < up ; ðt; uÞ 2 ½0; T 0; a0 : L N Then (1.1) and (1.2) has at least two positive solutions. Corollary 3.2. Assume that f satisfies conditions (i) f0 ðtÞ < up ðTNl Þ, t 2 [0, T] and f1 ðtÞ < up ðTNc Þ, t 2 [c, T]; (ii) there exists b 0 > 0 such that 0 b T f ðt; uÞ > up ; ðt; uÞ 2 ½l; T b0 ; b0 : l M Then (1.1) and (1.2) has at least two positive solutions. By applying Theorems 3.1 and 3.2, it is easy to prove that Corollaries 3.1 and 3.2 hold, respectively. 4. Triple solutions Let the nonnegative continuous concave functional W: P ! [0, 1) be defined by WðuÞ ¼ min uðtÞ ¼ uðlÞ; u 2 P : t2½l;T
Note that for u 2 P, W(u) 6 kuk. Theorem 4.1. Suppose that there exist constants 0 < d 0 < a 0 such that (i) (ii) (iii) (D1) (D2)
0
f ðt; uÞ < up ðdN Þ, (t, u) 2 [0, T] · [0, d 0 ]; 0 f ðt; uÞ P up ðMa Þ, ðt; uÞ 2 ½l; T ½a0 ; Tl a0 ; one of the following conditions holds: limu!1 maxt2½0;T fuðt;uÞ < up ðN1 Þ; p ðuÞ 0 there exists a number c0 > Tl a0 such that f ðt; uÞ < up ðNc Þ for (t, u) 2 [0, T] · [0, c 0 ].
Then (1.1) and (1.2) has at least three positive solutions. Proof. By the definition of operator A and its properties, it suffices to show that the conditions of Lemma 2.9 hold with respect to A.
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We first show that if (D1) holds, then there exists a number l0 > Tl a0 such that A : P l0 ! P l0 . Suppose that f ðt; uÞ 1 < up lim max u!1 t2½0;T up ðuÞ N holds, then there are s > 0 and d < N1 such that if u > s, then max
t2½0;T
f ðt; uÞ 6 up ðdÞ: up ðuÞ
That is to say, f ðt; uÞ 6 up ðduÞ; ðt; uÞ 2 ½0; T ½s; 1Þ: Set k = max{f(t, u) : (t, u) 2 [0, T] · [0, s]}, then f ðt; uÞ 6 k þ up ðduÞ; ðt; uÞ 2 ½0; T ½0; 1Þ: Taking
!) kup ðN Þ T 0 a ; uq l > max : l 1 up ðdN Þ
ð4:1Þ
(
0
ð4:2Þ
If u 2 P l0 , then by (2.5), (4.1), (4.2), we obtain kAuk ¼ AuðT Þ ! Z Z T Z s Z ni m2 1 T 1X ¼ uq aðsÞf ðs; uðsÞÞrsþ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs Ds d 0 d i¼1 0 0 0 Z T Z T Z T Z T 1 1 uq aðsÞf ðs; uðsÞÞrs Ds 6 uq aðsÞðk þ up ðduÞÞrs Ds 6 d 0 d 0 0 0 Z Z T 1 T uq aðsÞrs Ds ¼ uq ðk þ up ðdl0 ÞÞN < l0 : 6 uq ðk þ up ðdl0 ÞÞ d 0 0 Next we verify that if there is a positive number r 0 such that if f(t, u) < up(r 0 /N) for (t, u) 2 [0, T] · [0, r 0 ], then A : P r0 ! P r0 . Indeed, if u 2 P r0 , then kAuk ¼ AuðT Þ 6
Z
T
uq 0
Z T Z T Z 1 r0 T 1 aðsÞf ðs; uðsÞÞrs Ds < uq aðsÞrs Ds ¼ r0 ; d 0 d 0 N 0
thus, Au 2 P . Hence, we have shown that either (D1) or (D2) holds, then there exists a number c 0 with c0 > Tl a0 such that A : P c0 ! P c0 . It is also note from (i) that A : P d 0 ! P d 0 . Now, we show that fu 2 P ðW; a0 ; Tl a0 Þ : WðuÞ > a0 g 6¼ ; and W(Au) > a 0 for all u 2 P ðW; a0 ; Tl a0 Þ. In fact, ðl þ T Þa0 0 T 0 0 2 u 2 P W; a ; a : WðuÞ > a : u¼ l 2l r0
For u 2 P ðW; a0 ; Tl a0 Þ, we have a0 6 min uðtÞ ¼ uðlÞ 6 uðtÞ 6 t2½l;T
T 0 a; l
H.-R. Sun, W.-T. Li / Applied Mathematics and Computation 182 (2006) 478–491
489
for all t 2 [l, T]. Then, in view of (ii), we know that WðAuÞ ¼ min AuðtÞ ¼ AuðlÞ P t2½l;T
l AuðT Þ T
! Z s Z Z ni m2 1 T 1X uq aðsÞf ðs; uðsÞÞrsþ aðsÞf ðs; uðsÞÞrs ai aðsÞf ðs; uðsÞÞrs Ds d 0 d i¼1 0 0 0 Z T Z T Z T Z T l l u aðsÞf ðs; uðsÞÞrs Ds ¼ uq aðsÞf ðs; uðsÞÞrs Ds P T 0 q s T l s Z T Z T la0 uq aðsÞrs Ds ¼ a0 : > TM l s
l ¼ T
Z
T
Finally, we assert that if u 2 P(W, a 0 , c 0 ) and kAuk > Tl a0 , then W(Au) > a 0 . Suppose u 2 P(W, a 0 , c 0 ) and kAuk > Tl a0 , then WðAuÞ ¼ min AuðtÞ ¼ AuðlÞ P t2½l;T
l l AuðT Þ ¼ kAuk > a0 : T T
To sum up, the hypotheses of Lemma 2.9 are satisfied, hence (1.1) and (1.2) has at least three positive solutions u1, u2, u3 such that ku1 k < d 0 ; a0 < min u2 ðtÞ and
ku3 k > d 0 with min u3 ðtÞ < a0 :
t2½l;T
The proof is complete.
t2½l;T
h
From Theorem 4.1, we see that, when assumption like (i), (ii), and (iii) are imposed appropriately on f, we can establish the existence of an arbitrary odd number of positive solutions of (1.1) and (1.2). Theorem 4.2. Suppose that there exist constants 0 < d 01 < a01 <
T 0 T a < d 02 < a02 < a02 < d 03 < < d 0n ; n 2 N; l 1 l
such that the following conditions are satisfied: d0
(i) f ðt; uÞ < up ðNi Þ, ðt; uÞ 2 ½0; T ½0; d 0i ; a0 (ii) f ðt; uÞ P up ðMi Þ, ðt; uÞ 2 ½l; T ½a0i ; Tl a0i . Then (1.1) and (1.2) has at least 2n 1 positive solutions. Proof. When n = 1, it is immediate from condition (i) that A : P d 01 ! P d 01 P d 01 , which means that A has at least one fixed point u1 2 P d 01 by the Schauder fixed point theorem. When n = 2, it is clear that Theorem 4.1 holds (with c1 ¼ d 02 ). Then we can obtain at least three positive solutions u1, u2, and u3 satisfying ku1 k < d 01 ; min u2 ðtÞ > a01 t2½l;T
and
ku3 k > d 01 with min u3 ðtÞ < a01 : t2½l;T
Following this way, we finish the proof by induction. The proof is complete.
h
5. Examples N
Let T ¼ f1 ð1=2Þ 0 g [ f1g. Taking T = 1, p = 2, m = 4, a1 ¼ 13 ; n1 ¼ 12 ; a2 ¼ 12, and n2 ¼ 15 , we have d ¼ 16. 16 1 If we let a(s) 1, then M ¼ 8, N = 6. Suppose f ðt; uÞ ¼ f ðuÞ :¼
2000u2 ; u2 þ 5000
t 2 ½0; 1;
u P 0:
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Clearly f is always increasing. If we take d 0 ¼ 25, a 0 = 23, and c 0 = 12006, then 0 < d 0 < a0 < 2a0 < c0 : Now we check that (i), (ii) and (iii) of Theorem 4.1 are satisfied. In view of f(2/5) = 2000/31251 0.0639, we get f ðuÞ <
2 1 ¼ 0:0667; 5N 15
u 2 ½0; d 0 ;
so that (i) of Theorem 4.1 is met. To verify (ii), note that f(23) 191.355, so that f ðuÞ P 23=M ¼ 184;
u 2 ½a0 ; 2a0 :
Lastly, as limu!1f(u) = 2000, f ðuÞ 6 2000 < c0 =N ¼ 2001;
u 2 ½0; c0 ;
and (D2) of (iii) holds. Thus by Theorem 4.1, the boundary value problem 1 D 1 1 D 15 Dr D u ðtÞ þ f ðuðtÞÞ ¼ 0; uð0Þ ¼ 0; u ð1Þ ¼ u þ u ; 3 2 2 16 has at least three positive solutions u1, u2, u3 satisfying 2 ku1 k < ; 5
min u2 ðtÞ > 23
t2½12;1
and
ku3 k >
2 with min u3 ðtÞ < 23: t2½1=2;1 5
References [1] R.P. Agarwal, M. Bohner, W.T. Li, Nonoscillation and Oscillation Theory for Functional Differential Equations, Pure and Applied Mathematics Series, 267, Marcel Dekker, 2004. [2] R.P. Agarwal, M. Bohner, P. Rehak, Half-linear dynamic equations, Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday, 1, Kluwer Academic Publishers, Dordrecht, 2003, p. 1–57. [3] D.R. Anderson, Solutions to second-order three-point problems on time scales, J. Differen. Equ. Appl. 8 (2002) 673–688. [4] D.R. Anderson, Nonlinear triple point problems on time scales, Electron. J. Differen. Equ. 47 (2004) 1–12. [5] D.R. Anderson, R. Avery, J. Henderson, Existence of solutions for a one-dimensional p-Laplacian on time scales, J. Differen. Equ. Appl. 10 (2004) 889–896. [6] F.M. Atici, G.Sh. Guseinov, On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math. 141 (2002) 75–99. [7] B. Aulbach, L. Neidhart, Integration on measure chains, in: Proc. of the Sixth Int. Conf. on Difference Equations, CRC, Boca Raton, FL, 2004, pp. 239–252. [8] R.I. Avery, C.J. Chyan, J. Henderson, Twin solutions of boundary value problems for ordinary differential equations and finite difference equations, Comput. Math. Appl. 42 (2001) 695–704. [9] R.I. Avery, J. Henderson, Two positive fixed points of nonlinear operator on ordered Banach spaces, Commun. Appl. Nonlinear Anal. 8 (2001) 27–36. [10] M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001. [11] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003. [12] J.J. DaCunha, J.M. Davis, P.K. Singh, Existence results for singular three point boundary value problems on time scales, J. Math. Anal. Appl. 295 (2004) 378–391. [13] C.P. Gupta, A non-resonant multi-point boundary value problem for a p-Laplacian type operator, Electron. J. Differen. Equ. Conf. 10 (2003) 143–152. [14] S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math. 18 (1990) 18–56. [15] E.R. Kaufmann, Positive solutions of a three-point boundary value problem on a time scale, Electron. J. Differen. Equ. 2003 (82) (2003) 1–11. [16] E.R. Kaufmann, Y.N. Raffoul, Eigenvalue problems for a three-point boundary-value problem on a time scale, Electron. J. Qualitative Theory Differen. Equ. 2 (2004) 1–10. [17] V. Lakshmikantham, S. Sivasundaram, B. Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Boston, 1996. [18] R. Leggett, L. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979) 673–688. [19] W.T. Li, H.R. Sun, Multiple positive solutions for nonlinear dynamic systems on a measure chain, J. Comput. Appl. Math. 162 (2004) 421–430.
H.-R. Sun, W.-T. Li / Applied Mathematics and Computation 182 (2006) 478–491
491
[20] Y. Liu, W. Ge, Twin positive solutions of boundary value problems for finite difference equations with p-Laplacian operator, J. Math. Anal. Appl. 278 (2003) 551–561. [21] A.C. Peterson, Y.N. Raffoul, C.C. Tisdell, Three point boundary value problems on time scales, J. Differen. Equ. Appl. 10 (2004) 843– 849. [22] H.R. Sun, Existence of positive solutions to second-order time scale systems, Comput. Math. Appl. 49 (2005) 131–145. [23] H.R. Sun, W.T. Li, Existence of positive solutions for nonlinear three-point boundary value problems on time scales, J. Math. Anal. Appl. 299 (2004) 508–524. [24] H.R. Sun, W.T. Li, Existence theory for positive solutions to one-dimensional p-Laplacian boundary value problems on time scales, submitted for publication. [25] H.R. Sun, W.T. Li, Positive solutions of p-Laplacian m-point boundary value problems on time scales, Taiwan. J. Math., submitted for publication.