- Email: [email protected]
Existence of solutions for a two-point boundary value problem on time scales
Applied Mathematics and Computation 150 (2004) 139–147 www.elsevier.com/locate/amc
Existence of solutions for a two-point boundary value problem on time scales Ruyun Ma *, Hua Luo Department of Mathematics, Northwest Normal University, Lanzhou 730070, Gansu, PR China
Abstract Let T be a time scale. Let f : ½0; rð1Þ R2 ! R be continuous. We study the existence of solutions for the nonlinear two-point boundary value problems on time scale xMM ðtÞ ¼ f ðt; xðtÞ; xM ðtÞÞ; xð0Þ ¼ 0;
t 2 ½0; 1
xM ðrð1ÞÞ ¼ 0:
We do not need any growth restrictions on f . Ó 2003 Elsevier Inc. All rights reserved. Keywords: Existence theorems; Boundary value problem; Time scales; Barrier strips
1. Introduction In 1994, Kelevedjiev [1] studied the second order boundary value problems of the form x00 ¼ f ðt; x; x0 Þ; xð0Þ ¼ A;
t 2 ½0; 1
0
x ð1Þ ¼ B;
ð1:1Þ ð1:2Þ
where A, B are two constants and f : ½0; 1 R2 ! R is continuous. A barrier strip P , which defined below.
*
Corresponding author. E-mail address: [email protected] (R. Ma).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00204-2
140
R. Ma, H. Luo / Appl. Math. Comput. 150 (2004) 139–147
(There are pairs (two or four) of suitable constants such that f ðt; x; pÞ does not change its sign on sets of the form ½0; 1 ½ M; M P , where M is a nonnegative constant, and P is a closed interval bounded by some pairs of constants, mentioned above.), play an important role in [1] to achieve existence results. This paper extends some results in [1] to the boundary value problem on time scales xMM ðtÞ ¼ f ðt; xðtÞ; xM ðtÞÞ; xð0Þ ¼ 0;
t 2 ½0; 1
M
x ðrð1ÞÞ ¼ 0;
ð1:3Þ ð1:4Þ
where f : ½0; rð1Þ R2 ! R is continuous. We obtain the existence of at least one solution to this problem without any growth restrictions on f . Our main tools are the Leray–Shauder principle and an existence assumption of barrier strips which arise from [1]. Moreover, the proofs in this paper are different from [1] to a great extent since time scales unify continuous and discrete analysis, and the continuous time orbits and the discrete time orbits are topologically different as well. See [2–4] for some pioneering work on the theory of time scales. In addition, we refer the reader to [5–7] for more results and references of Eq. (1.1) with diverse boundary conditions.
2. Some definitions on time scales In order to state our results we need some notation. A time scale T is a nonempty closed subset of R, assume that T has the topology that it inherits from the standards topology on R. We define the forward and backward jump operators r; q : T ! T by rðtÞ ¼ inffs > tjs 2 T g and
qðtÞ ¼ supfs < tjs 2 T g:
The point t 2 T is called right-scattered, right-dense, left-scattered, left-dense if rðtÞ > t, rðtÞ ¼ t, qðtÞ < t, qðtÞ ¼ t holds, respectively. The set T k is derived from the time scale T as follows: If T has a left-scattered maximum t0 , then T k ¼ T ft0 g. Otherwise, T k ¼ T . For a; b 2 T with a 6 b, define the closed interval ½a; b in T by ½a; b ¼ ft 2 T ja 6 t 6 bg; other open, half-open intervals in T can be similarly defined. Definition 2.1. If f : T ! R is a function and t 2 T k , then the delta derivative of f at the point t is defined to be the number f M ðtÞ (provided it exists) with the property that for each e > 0, there is a neighborhood U of t such that
R. Ma, H. Luo / Appl. Math. Comput. 150 (2004) 139–147
141
jf ðrðtÞÞ f ðsÞ f M ðtÞðrðtÞ sÞj 6 ejrðtÞ sj for all s 2 U . The function f is called M-differentiable on T if f M ðtÞ exists for all t 2 T k. Definition 2.2. If F M ¼ f holds on T k . we define the integral of f by Z t f ðsÞMs ¼ F ðtÞ F ðsÞ; s; t 2 T k : s
3. Preliminaries Lemma 3.1 [8]. For f : T ! R and t 2 T k the following hold: (1) If f is M-differentiable at t, then f is continuous at t. (2) If f is continuous at t and t is right-scattered, then f is M-differentiable at t and f M ðtÞ ¼
f ðrðtÞÞ f ðtÞ : rðtÞ t
(3) If t is right-dense, then f is M-differentiable at t if and only if the limit lim s!t
f ðtÞ f ðsÞ t s
exists as a finite number. In this case f M ðtÞ is equal to this limit. (4) If f is M-differentiable at t, then f ðrðtÞÞ ¼ f ðtÞ þ ðrðtÞ tÞf M ðtÞ:
Lemma 3.2. Suppose that f : ½a; b ! R is M-differentiable on ½a; b, then f is nondecreasing (nonincreasing) on ½a; b if and only if f M ðtÞ P 0 ðf M ðtÞ 6 0Þ;
k
t 2 ½a; b :
Proof. we only present the proof in the case of nondecreasing, another case can be proved in a similar way. k For each t 2 ½a; b , we have 8 f ðtÞ f ðsÞ > > < lim ; rðtÞ ¼ t s!t
s M f ðtÞ ¼ f ðrðtÞÞt
f ðtÞ > > ; rðtÞ > t : rðtÞ t
142
R. Ma, H. Luo / Appl. Math. Comput. 150 (2004) 139–147
Since f is nondecreasing on ½a; b, in the cases of rðtÞ ¼ t and rðtÞ > t, f ðtÞ f ðsÞ P 0; t s f ðrðtÞÞ P f ðtÞ:
8s 2 U
holds, respectively, where U is an arbitrary neighborhood around t. it implies that f M ðtÞ P 0;
k
8t 2 ½a; b :
In reverse order, if r; s 2 ½a; bk , r < s, from Z s f ðsÞ f ðrÞ ¼ f M ðsÞMs P 0; r
we have f ðrÞ 6 f ðsÞ: In order to prove f is nondecreasing on ½a; b, we also need consider the point b. If qðbÞ ¼ b, then ½a; b ¼ ½a; bk , if qðbÞ < b, then rðqðbÞÞ ¼ b. It follows that f ðbÞ ¼ f ðqðbÞÞ þ f M ðqðbÞÞðb qðbÞÞ P f ðqðbÞÞ; as required. Hence the proof is completed. Lemma 3.3 [9]. Let X ,Z are real vector normed spaces, L : domL X ! Z a linear Fredholm mapping of index zero, X X is an open bounded subset, N : X ! Z a L-compact mapping. If ker L ¼ f0g, 0 2 X, and Lu kNu 6¼ 0; for every ðu; kÞ 2 ðdomL \ @XÞ ð0; 1Þ, then equation Lu ¼ Nu has at least one solution in domL \ X.
4. Existence Let T be a time scale with 0; 1 2 T . By C 2 ½0; r2 ð1Þ, we mean the Banach space of second-order continuous M-differentiable functions x : ½0; r2 ð1Þ ! R equipped with the norm kxk ¼ maxfjxj0 ; jxM j0 ; jxMM j0 g; where k k0 is the maximum norm. In this section, we state and prove our main result (see Theorem 4.2). Let D ¼ fx 2 C 2 ½0; r2 ð1Þjx satisfies (1.4)}.
R. Ma, H. Luo / Appl. Math. Comput. 150 (2004) 139–147
143
Theorem 4.1. Suppose f : ½0; rð1Þ R2 ! R is continuous, L : D \ C 2 ½0; r2 ð1Þ ! C½0; 1, Lx ¼ xMM and there is a constant c < 1 such that kxk < c for each solution xðtÞ to the boundary value problem ðLxÞðtÞ ¼ kf ðt; xðtÞ; xM ðtÞÞ;
t 2 ½0; 1;
x2D
ð4:1Þ
and for each k 2 ð0; 1Þ. Then the boundary value problem (1.3) and (1.4) has at least one solution in D. Proof. First, we show that L is one-to-one mapping. From Definition 2.2, integrate Lx ¼ 0 twice, we can get xðtÞ ¼ xð0Þ þ xM ðrð1ÞÞt and note that x 2 D, we have xðtÞ 0. i.e. ker L ¼ f0g. Hence, L is one-to-one mapping. Let N : C 2 ½0; r2 ð1Þ ! C½0; 1 and G : C½0; 1 ! C 2 ½0; r2 ð1Þ defined by ðNxÞðtÞ ¼ f ðt; xðtÞ; xM ðtÞÞ;
t 2 ½0; rð1Þ
and ðGyÞðtÞ ¼
Z
rð1Þ
H ðt; sÞyðsÞ Ds; 0
respectively, where H ðt; sÞ is GreenÕs function of homogeneous equation of (1.3) and (1.4). Obviously, LGy ¼ y, y 2 C½0; 1, GLx ¼ x, x 2 D, then GL is a Fredholm mapping of index zero, and it follows easily that GN : C 2 ½0; r2 ð1Þ ! C 2 ½0; r2 ð1Þ is GL-compact mapping. Eq. (4.1) is equivalent to the operator equation Lx ¼ kNx, and BVP (1.3) and (1.4) is equivalent to Lx ¼ Nx; which is equivalent to x ¼ GNx: Let X ¼ fx 2 C 2 ½0; r2 ð1Þjkxk < cg. Then GLx 6¼ kGNx: for every x 2 D \ @X, 8k 2 ð0; 1Þ. By Lemma 3.3, Lx ¼ Nx has at least one solution in D \ X. Hence boundary value problem (1.3) and (1.4) has at least one solution x with kxk 6 c. Theorem 4.2. Let f : ½0; rð1Þ R2 ! R be continuous. Suppose there are constants Li ; i ¼ 1; 2; 3; 4; such that L2 > L1 P 0, L3 < L4 6 0, f ðt; u; pÞ P 0;
for ðt; u; pÞ 2 ½0; rð1Þ R ½L1 ; L2
ð4:2Þ
144
R. Ma, H. Luo / Appl. Math. Comput. 150 (2004) 139–147
and for ðt; u; pÞ 2 ½0; rð1Þ R ½L3 ; L4
f ðt; u; pÞ 6 0;
ð4:3Þ
Then problem (1.3) and (1.4) has at least one solution in C 2 ½0; r2 ð1Þ. Remark 1. Theorem 4.2 generates [1, Theorem 3.1] when T ¼ R. Remark 2. We can find some elementary functions satisfying the conditions of Theorem 4.2. Consider the boundary value problem on time scales 4
2
xMM ðtÞ ¼ ðxM ðtÞÞ 5ðxM ðtÞÞ þ 4; xð0Þ ¼ 0;
t 2 ½0; 1
xM ðrð1ÞÞ ¼ 0;
Let f ðt; u; pÞ ¼ p4 5p2 þ 4. Obviously, f is continuous on ½0; rð1Þ, Choose L1 ¼ 0, L2 ¼ 1, L3 ¼ 2, L4 ¼ 1, it can be seen that all conditions of Theorem 4.2 are satisfied. Thus, the considered problem has at least one solution in C 2 ½0; r2 ð1Þ. Proof of Theorem 4.2. Define / : R ! R as follows: 8 < L2 ; v > L2 /ðvÞ ¼ v; L3 6 v 6 L2 : L3 ; v < L3 Let us consider the boundary value problem xMM ðtÞ ¼ f ðt; xðtÞ; /ðxM ðtÞÞÞ; xð0Þ ¼ 0;
t 2 ½0; 1
M
x ðrð1ÞÞ ¼ 0:
ð4:4Þ ð4:5Þ
Let x 2 D be any solution of equations xMM ðtÞ ¼ kf ðt; xðtÞ; /ðxM ðtÞÞÞ;
t 2 ½0; 1
ð4:6Þ
for some k 2 ð0; 1Þ. By Theorem 4.1, a priori bounds on x is needed. Let S1 ¼ ft 2 ½0; rð1ÞjxM ðtÞ > L1 g, we want to show that S1 ¼ £. Suppose to the contrary that there exists t0 2 S1 , then xM ðt0 Þ > L1 . Set A ¼ ft 2 ðt0 ; rð1ÞjxM ðtÞ 6 L1 g, then A is not empty, since rð1Þ 2 A. Choose t1 ¼ inf A, we claim that t1 2 A. In fact, if t1 62 A, then xM ðt1 Þ > L1 , we give an investigation to the right side of t1 : Case 1. rðt1 Þ ¼ t1 . Since xM ðtÞ is continuous on ½0; rð1Þ, set e ¼ xM ðt1 Þ L1 , then there exists a neighborhood U of t1 , such that
R. Ma, H. Luo / Appl. Math. Comput. 150 (2004) 139–147
145
jxM ðsÞ xM ðt1 Þj < e; for all s 2 U , s > t1 , followed by xM ðsÞ > xM ðt1 Þ e ¼ L1 , contrary to t1 ¼ inf A. Case 2. rðt1 Þ > t1 . In this case, either rðt1 Þ 2 A or rðt1 Þ 62 A holds, but both are contrary to the definition of t1 . Hence, t1 2 A, moreover, t1 > t0 . Next we consider the left side of t1 . Case 1. qðt1 Þ ¼ t1 . An inevitable outcome is xM ðt1 Þ ¼ L1 , we also have xM ðtÞ P L1 , t 2 ½t0 ; t1 , so L1 6 /ðxM ðtÞÞ 6 L2 , t 2 ½t0 ; t1 . This implies that Z t1 xMM ðsÞMs P 0; 0 > xM ðt1 Þ xM ðt0 Þ ¼ t0
which is a contradiction. Case 2. qðt1 Þ < t1 . It is clear that rðqðt1 ÞÞ ¼ t1 . Since xM ðqðt1 ÞÞ > L1 and it follows that L1 < /ðxM ðqðt1 ÞÞÞ 6 L2 and so 0>
xM ðt1 Þ xM ðqðt1 ÞÞ ¼ xMM ðqðt1 ÞÞ P 0; t1 qðt1 Þ
which is a contradiction. All the discussion above imply that S1 ¼ £. In the same way, we can set S2 ¼ ft 2 ½0; rð1ÞjxM ðtÞ < L4 g, it will be followed that S2 ¼ £. Thus, L4 6 xM ðtÞ 6 L1 ;
t 2 ½0; rð1Þ
ð4:7Þ
Let c1 ¼ maxf L4 ; L1 g. Then jxM j0 6 c1 :
ð4:8Þ
From (4.7) we have L4 rð1Þ 6 xðtÞ ¼
Z
t
xM ðsÞMs 6 L1 rð1Þ;
t 2 ½0; rð1Þ
0
and for t ¼ r2 ð1Þ, xðr2 ð1ÞÞ ¼ xðrð1ÞÞ þ xM ðrð1ÞÞðr2 ð1Þ rð1ÞÞ 6 L1 rð1Þ þ L1 ðr2 ð1Þ rð1ÞÞ ¼ L1 r2 ð1Þ: Similarly xðr2 ð1ÞÞ P L4 r2 ð1Þ:
146
R. Ma, H. Luo / Appl. Math. Comput. 150 (2004) 139–147
To sum up, it is L4 r2 ð1Þ 6 xðtÞ 6 L1 r2 ð1Þ;
t 2 ½0; r2 ð1Þ
ð4:9Þ
it concludes that jxj0 6 c1 r2 ð1Þ:
ð4:10Þ
By the definition of / and (4.8), (4.10), together with the fact that x is a solution of (4.6) and (4.5), it is showed that jxMM j0 6 c2 ;
ð4:11Þ
for a constant c2 < 1 independent of k. Now choose c ¼ maxfc1 r2 ð1Þ; c2 g þ 1. From (4.8), (4.10) and (4.11), we get kxk < c: Since c is just the priori bound for which we look, from Theorem 4.1, boundary value problem (4.4) and (4.5) has at least one solution x 2 D C 2 ½0; r2 ð1Þ. Note that x obtained above satisfies (4.7), it follows by /ðxM ðtÞÞ ¼ xM ðtÞ;
t 2 ½0; rð1Þ
Thus, x is also a solution of boundary value problem (1.3) and (1.4), as required.
Acknowledgements Supported by the NSFC (No. 10271095), GG-110-10736-1003, NWNUKJCXGC-212 and the Foundation of Major Projects in Science and Technology of the Chinese Education Ministry.
References [1] P. Kelevedjiev, Existence of solutions for two-point boundary value problems, Nonlinear Anal. 22 (1994) 217–224. [2] B. Aulbach, S. Hilger, Linear dynamic processes with inhomogeneous time scale, in: Nonlinear dynamics and quantum dynamical systems, Akademic Verlag, Berlin, 1990. [3] B. Kaymakcalan, V. Lakshmikantham, S. Sivasundaram, Dynamic systems on measure chains, Kluwer Academic Publishers, Boston, MA, 1996. [4] R.P. Agarwal, M. Bohner, Basic calculus on time scales and some of its applications, Results Math. 35 (1999) 3–22. [5] P.B. Bailey, L.F. Shampine, P.E. Waltman, Nonlinear two-point boundary value problems, Academic Press, New York, 1968. [6] L.H. Erbe, Existence of solutions to boundary value problems for second order differential equations, Nonlinear Anal. 6 (1982) 1155–1162.
R. Ma, H. Luo / Appl. Math. Comput. 150 (2004) 139–147
147
[7] A. Granas, R.B. Guenther, J.W. Lee, Nonlinear boundary value problems for some classes of ordinary differential equations, Rocky Mount. J. Math. 10 (1980) 35–38. [8] F. Merdivenci Atici, G.Sh. Guseinov, On GreenÕs functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math. 141 (2002) 75–99. [9] J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, in: NSFCB Regional Conference Series in Mathematics, vol. 40, Amer. Math. Soc., Providence, RI, 1979.
Existence of solutions for a two-point boundary value problem on time scales Ruyun Ma *, Hua Luo Department of Mathematics, Northwest Normal University, Lanzhou 730070, Gansu, PR China
Abstract Let T be a time scale. Let f : ½0; rð1Þ R2 ! R be continuous. We study the existence of solutions for the nonlinear two-point boundary value problems on time scale xMM ðtÞ ¼ f ðt; xðtÞ; xM ðtÞÞ; xð0Þ ¼ 0;
t 2 ½0; 1
xM ðrð1ÞÞ ¼ 0:
We do not need any growth restrictions on f . Ó 2003 Elsevier Inc. All rights reserved. Keywords: Existence theorems; Boundary value problem; Time scales; Barrier strips
1. Introduction In 1994, Kelevedjiev [1] studied the second order boundary value problems of the form x00 ¼ f ðt; x; x0 Þ; xð0Þ ¼ A;
t 2 ½0; 1
0
x ð1Þ ¼ B;
ð1:1Þ ð1:2Þ
where A, B are two constants and f : ½0; 1 R2 ! R is continuous. A barrier strip P , which defined below.
*
Corresponding author. E-mail address: [email protected] (R. Ma).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00204-2
140
R. Ma, H. Luo / Appl. Math. Comput. 150 (2004) 139–147
(There are pairs (two or four) of suitable constants such that f ðt; x; pÞ does not change its sign on sets of the form ½0; 1 ½ M; M P , where M is a nonnegative constant, and P is a closed interval bounded by some pairs of constants, mentioned above.), play an important role in [1] to achieve existence results. This paper extends some results in [1] to the boundary value problem on time scales xMM ðtÞ ¼ f ðt; xðtÞ; xM ðtÞÞ; xð0Þ ¼ 0;
t 2 ½0; 1
M
x ðrð1ÞÞ ¼ 0;
ð1:3Þ ð1:4Þ
where f : ½0; rð1Þ R2 ! R is continuous. We obtain the existence of at least one solution to this problem without any growth restrictions on f . Our main tools are the Leray–Shauder principle and an existence assumption of barrier strips which arise from [1]. Moreover, the proofs in this paper are different from [1] to a great extent since time scales unify continuous and discrete analysis, and the continuous time orbits and the discrete time orbits are topologically different as well. See [2–4] for some pioneering work on the theory of time scales. In addition, we refer the reader to [5–7] for more results and references of Eq. (1.1) with diverse boundary conditions.
2. Some definitions on time scales In order to state our results we need some notation. A time scale T is a nonempty closed subset of R, assume that T has the topology that it inherits from the standards topology on R. We define the forward and backward jump operators r; q : T ! T by rðtÞ ¼ inffs > tjs 2 T g and
qðtÞ ¼ supfs < tjs 2 T g:
The point t 2 T is called right-scattered, right-dense, left-scattered, left-dense if rðtÞ > t, rðtÞ ¼ t, qðtÞ < t, qðtÞ ¼ t holds, respectively. The set T k is derived from the time scale T as follows: If T has a left-scattered maximum t0 , then T k ¼ T ft0 g. Otherwise, T k ¼ T . For a; b 2 T with a 6 b, define the closed interval ½a; b in T by ½a; b ¼ ft 2 T ja 6 t 6 bg; other open, half-open intervals in T can be similarly defined. Definition 2.1. If f : T ! R is a function and t 2 T k , then the delta derivative of f at the point t is defined to be the number f M ðtÞ (provided it exists) with the property that for each e > 0, there is a neighborhood U of t such that
R. Ma, H. Luo / Appl. Math. Comput. 150 (2004) 139–147
141
jf ðrðtÞÞ f ðsÞ f M ðtÞðrðtÞ sÞj 6 ejrðtÞ sj for all s 2 U . The function f is called M-differentiable on T if f M ðtÞ exists for all t 2 T k. Definition 2.2. If F M ¼ f holds on T k . we define the integral of f by Z t f ðsÞMs ¼ F ðtÞ F ðsÞ; s; t 2 T k : s
3. Preliminaries Lemma 3.1 [8]. For f : T ! R and t 2 T k the following hold: (1) If f is M-differentiable at t, then f is continuous at t. (2) If f is continuous at t and t is right-scattered, then f is M-differentiable at t and f M ðtÞ ¼
f ðrðtÞÞ f ðtÞ : rðtÞ t
(3) If t is right-dense, then f is M-differentiable at t if and only if the limit lim s!t
f ðtÞ f ðsÞ t s
exists as a finite number. In this case f M ðtÞ is equal to this limit. (4) If f is M-differentiable at t, then f ðrðtÞÞ ¼ f ðtÞ þ ðrðtÞ tÞf M ðtÞ:
Lemma 3.2. Suppose that f : ½a; b ! R is M-differentiable on ½a; b, then f is nondecreasing (nonincreasing) on ½a; b if and only if f M ðtÞ P 0 ðf M ðtÞ 6 0Þ;
k
t 2 ½a; b :
Proof. we only present the proof in the case of nondecreasing, another case can be proved in a similar way. k For each t 2 ½a; b , we have 8 f ðtÞ f ðsÞ > > < lim ; rðtÞ ¼ t s!t
s M f ðtÞ ¼ f ðrðtÞÞt
f ðtÞ > > ; rðtÞ > t : rðtÞ t
142
R. Ma, H. Luo / Appl. Math. Comput. 150 (2004) 139–147
Since f is nondecreasing on ½a; b, in the cases of rðtÞ ¼ t and rðtÞ > t, f ðtÞ f ðsÞ P 0; t s f ðrðtÞÞ P f ðtÞ:
8s 2 U
holds, respectively, where U is an arbitrary neighborhood around t. it implies that f M ðtÞ P 0;
k
8t 2 ½a; b :
In reverse order, if r; s 2 ½a; bk , r < s, from Z s f ðsÞ f ðrÞ ¼ f M ðsÞMs P 0; r
we have f ðrÞ 6 f ðsÞ: In order to prove f is nondecreasing on ½a; b, we also need consider the point b. If qðbÞ ¼ b, then ½a; b ¼ ½a; bk , if qðbÞ < b, then rðqðbÞÞ ¼ b. It follows that f ðbÞ ¼ f ðqðbÞÞ þ f M ðqðbÞÞðb qðbÞÞ P f ðqðbÞÞ; as required. Hence the proof is completed. Lemma 3.3 [9]. Let X ,Z are real vector normed spaces, L : domL X ! Z a linear Fredholm mapping of index zero, X X is an open bounded subset, N : X ! Z a L-compact mapping. If ker L ¼ f0g, 0 2 X, and Lu kNu 6¼ 0; for every ðu; kÞ 2 ðdomL \ @XÞ ð0; 1Þ, then equation Lu ¼ Nu has at least one solution in domL \ X.
4. Existence Let T be a time scale with 0; 1 2 T . By C 2 ½0; r2 ð1Þ, we mean the Banach space of second-order continuous M-differentiable functions x : ½0; r2 ð1Þ ! R equipped with the norm kxk ¼ maxfjxj0 ; jxM j0 ; jxMM j0 g; where k k0 is the maximum norm. In this section, we state and prove our main result (see Theorem 4.2). Let D ¼ fx 2 C 2 ½0; r2 ð1Þjx satisfies (1.4)}.
R. Ma, H. Luo / Appl. Math. Comput. 150 (2004) 139–147
143
Theorem 4.1. Suppose f : ½0; rð1Þ R2 ! R is continuous, L : D \ C 2 ½0; r2 ð1Þ ! C½0; 1, Lx ¼ xMM and there is a constant c < 1 such that kxk < c for each solution xðtÞ to the boundary value problem ðLxÞðtÞ ¼ kf ðt; xðtÞ; xM ðtÞÞ;
t 2 ½0; 1;
x2D
ð4:1Þ
and for each k 2 ð0; 1Þ. Then the boundary value problem (1.3) and (1.4) has at least one solution in D. Proof. First, we show that L is one-to-one mapping. From Definition 2.2, integrate Lx ¼ 0 twice, we can get xðtÞ ¼ xð0Þ þ xM ðrð1ÞÞt and note that x 2 D, we have xðtÞ 0. i.e. ker L ¼ f0g. Hence, L is one-to-one mapping. Let N : C 2 ½0; r2 ð1Þ ! C½0; 1 and G : C½0; 1 ! C 2 ½0; r2 ð1Þ defined by ðNxÞðtÞ ¼ f ðt; xðtÞ; xM ðtÞÞ;
t 2 ½0; rð1Þ
and ðGyÞðtÞ ¼
Z
rð1Þ
H ðt; sÞyðsÞ Ds; 0
respectively, where H ðt; sÞ is GreenÕs function of homogeneous equation of (1.3) and (1.4). Obviously, LGy ¼ y, y 2 C½0; 1, GLx ¼ x, x 2 D, then GL is a Fredholm mapping of index zero, and it follows easily that GN : C 2 ½0; r2 ð1Þ ! C 2 ½0; r2 ð1Þ is GL-compact mapping. Eq. (4.1) is equivalent to the operator equation Lx ¼ kNx, and BVP (1.3) and (1.4) is equivalent to Lx ¼ Nx; which is equivalent to x ¼ GNx: Let X ¼ fx 2 C 2 ½0; r2 ð1Þjkxk < cg. Then GLx 6¼ kGNx: for every x 2 D \ @X, 8k 2 ð0; 1Þ. By Lemma 3.3, Lx ¼ Nx has at least one solution in D \ X. Hence boundary value problem (1.3) and (1.4) has at least one solution x with kxk 6 c. Theorem 4.2. Let f : ½0; rð1Þ R2 ! R be continuous. Suppose there are constants Li ; i ¼ 1; 2; 3; 4; such that L2 > L1 P 0, L3 < L4 6 0, f ðt; u; pÞ P 0;
for ðt; u; pÞ 2 ½0; rð1Þ R ½L1 ; L2
ð4:2Þ
144
R. Ma, H. Luo / Appl. Math. Comput. 150 (2004) 139–147
and for ðt; u; pÞ 2 ½0; rð1Þ R ½L3 ; L4
f ðt; u; pÞ 6 0;
ð4:3Þ
Then problem (1.3) and (1.4) has at least one solution in C 2 ½0; r2 ð1Þ. Remark 1. Theorem 4.2 generates [1, Theorem 3.1] when T ¼ R. Remark 2. We can find some elementary functions satisfying the conditions of Theorem 4.2. Consider the boundary value problem on time scales 4
2
xMM ðtÞ ¼ ðxM ðtÞÞ 5ðxM ðtÞÞ þ 4; xð0Þ ¼ 0;
t 2 ½0; 1
xM ðrð1ÞÞ ¼ 0;
Let f ðt; u; pÞ ¼ p4 5p2 þ 4. Obviously, f is continuous on ½0; rð1Þ, Choose L1 ¼ 0, L2 ¼ 1, L3 ¼ 2, L4 ¼ 1, it can be seen that all conditions of Theorem 4.2 are satisfied. Thus, the considered problem has at least one solution in C 2 ½0; r2 ð1Þ. Proof of Theorem 4.2. Define / : R ! R as follows: 8 < L2 ; v > L2 /ðvÞ ¼ v; L3 6 v 6 L2 : L3 ; v < L3 Let us consider the boundary value problem xMM ðtÞ ¼ f ðt; xðtÞ; /ðxM ðtÞÞÞ; xð0Þ ¼ 0;
t 2 ½0; 1
M
x ðrð1ÞÞ ¼ 0:
ð4:4Þ ð4:5Þ
Let x 2 D be any solution of equations xMM ðtÞ ¼ kf ðt; xðtÞ; /ðxM ðtÞÞÞ;
t 2 ½0; 1
ð4:6Þ
for some k 2 ð0; 1Þ. By Theorem 4.1, a priori bounds on x is needed. Let S1 ¼ ft 2 ½0; rð1ÞjxM ðtÞ > L1 g, we want to show that S1 ¼ £. Suppose to the contrary that there exists t0 2 S1 , then xM ðt0 Þ > L1 . Set A ¼ ft 2 ðt0 ; rð1ÞjxM ðtÞ 6 L1 g, then A is not empty, since rð1Þ 2 A. Choose t1 ¼ inf A, we claim that t1 2 A. In fact, if t1 62 A, then xM ðt1 Þ > L1 , we give an investigation to the right side of t1 : Case 1. rðt1 Þ ¼ t1 . Since xM ðtÞ is continuous on ½0; rð1Þ, set e ¼ xM ðt1 Þ L1 , then there exists a neighborhood U of t1 , such that
R. Ma, H. Luo / Appl. Math. Comput. 150 (2004) 139–147
145
jxM ðsÞ xM ðt1 Þj < e; for all s 2 U , s > t1 , followed by xM ðsÞ > xM ðt1 Þ e ¼ L1 , contrary to t1 ¼ inf A. Case 2. rðt1 Þ > t1 . In this case, either rðt1 Þ 2 A or rðt1 Þ 62 A holds, but both are contrary to the definition of t1 . Hence, t1 2 A, moreover, t1 > t0 . Next we consider the left side of t1 . Case 1. qðt1 Þ ¼ t1 . An inevitable outcome is xM ðt1 Þ ¼ L1 , we also have xM ðtÞ P L1 , t 2 ½t0 ; t1 , so L1 6 /ðxM ðtÞÞ 6 L2 , t 2 ½t0 ; t1 . This implies that Z t1 xMM ðsÞMs P 0; 0 > xM ðt1 Þ xM ðt0 Þ ¼ t0
which is a contradiction. Case 2. qðt1 Þ < t1 . It is clear that rðqðt1 ÞÞ ¼ t1 . Since xM ðqðt1 ÞÞ > L1 and it follows that L1 < /ðxM ðqðt1 ÞÞÞ 6 L2 and so 0>
xM ðt1 Þ xM ðqðt1 ÞÞ ¼ xMM ðqðt1 ÞÞ P 0; t1 qðt1 Þ
which is a contradiction. All the discussion above imply that S1 ¼ £. In the same way, we can set S2 ¼ ft 2 ½0; rð1ÞjxM ðtÞ < L4 g, it will be followed that S2 ¼ £. Thus, L4 6 xM ðtÞ 6 L1 ;
t 2 ½0; rð1Þ
ð4:7Þ
Let c1 ¼ maxf L4 ; L1 g. Then jxM j0 6 c1 :
ð4:8Þ
From (4.7) we have L4 rð1Þ 6 xðtÞ ¼
Z
t
xM ðsÞMs 6 L1 rð1Þ;
t 2 ½0; rð1Þ
0
and for t ¼ r2 ð1Þ, xðr2 ð1ÞÞ ¼ xðrð1ÞÞ þ xM ðrð1ÞÞðr2 ð1Þ rð1ÞÞ 6 L1 rð1Þ þ L1 ðr2 ð1Þ rð1ÞÞ ¼ L1 r2 ð1Þ: Similarly xðr2 ð1ÞÞ P L4 r2 ð1Þ:
146
R. Ma, H. Luo / Appl. Math. Comput. 150 (2004) 139–147
To sum up, it is L4 r2 ð1Þ 6 xðtÞ 6 L1 r2 ð1Þ;
t 2 ½0; r2 ð1Þ
ð4:9Þ
it concludes that jxj0 6 c1 r2 ð1Þ:
ð4:10Þ
By the definition of / and (4.8), (4.10), together with the fact that x is a solution of (4.6) and (4.5), it is showed that jxMM j0 6 c2 ;
ð4:11Þ
for a constant c2 < 1 independent of k. Now choose c ¼ maxfc1 r2 ð1Þ; c2 g þ 1. From (4.8), (4.10) and (4.11), we get kxk < c: Since c is just the priori bound for which we look, from Theorem 4.1, boundary value problem (4.4) and (4.5) has at least one solution x 2 D C 2 ½0; r2 ð1Þ. Note that x obtained above satisfies (4.7), it follows by /ðxM ðtÞÞ ¼ xM ðtÞ;
t 2 ½0; rð1Þ
Thus, x is also a solution of boundary value problem (1.3) and (1.4), as required.
Acknowledgements Supported by the NSFC (No. 10271095), GG-110-10736-1003, NWNUKJCXGC-212 and the Foundation of Major Projects in Science and Technology of the Chinese Education Ministry.
References [1] P. Kelevedjiev, Existence of solutions for two-point boundary value problems, Nonlinear Anal. 22 (1994) 217–224. [2] B. Aulbach, S. Hilger, Linear dynamic processes with inhomogeneous time scale, in: Nonlinear dynamics and quantum dynamical systems, Akademic Verlag, Berlin, 1990. [3] B. Kaymakcalan, V. Lakshmikantham, S. Sivasundaram, Dynamic systems on measure chains, Kluwer Academic Publishers, Boston, MA, 1996. [4] R.P. Agarwal, M. Bohner, Basic calculus on time scales and some of its applications, Results Math. 35 (1999) 3–22. [5] P.B. Bailey, L.F. Shampine, P.E. Waltman, Nonlinear two-point boundary value problems, Academic Press, New York, 1968. [6] L.H. Erbe, Existence of solutions to boundary value problems for second order differential equations, Nonlinear Anal. 6 (1982) 1155–1162.
R. Ma, H. Luo / Appl. Math. Comput. 150 (2004) 139–147
147
[7] A. Granas, R.B. Guenther, J.W. Lee, Nonlinear boundary value problems for some classes of ordinary differential equations, Rocky Mount. J. Math. 10 (1980) 35–38. [8] F. Merdivenci Atici, G.Sh. Guseinov, On GreenÕs functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math. 141 (2002) 75–99. [9] J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, in: NSFCB Regional Conference Series in Mathematics, vol. 40, Amer. Math. Soc., Providence, RI, 1979.