Positive Solutions to Singular and Delay Higher-Order Differential Equations on Time Scales
© Liang-Gen Hu et al. 2009
Received: 21 March 2009
Accepted: 1 July 2009
Published: 16 August 2009
We are concerned with singular three-point boundary value problems for delay higher-order dynamic equations on time scales. Theorems on the existence of positive solutions are obtained by utilizing the fixed point theorem of cone expansion and compression type. An example is given to illustrate our main result.
where , , , , , and . The functional is continuous and is continuous. Our nonlinearity may have singularity at and/or and may have singularity at .
(supplemented by and ) are well defined. The point is left-dense, left-scattered, right-dense, right-scattered if , , , , respectively. If has a left-scattered maximum (right-scattered minimum ), define ( ); otherwise, set ( ). By an interval we always mean the intersection of the real interval with the given time scale, that is, . Other types of intervals are defined similarly.
Theoretically, dynamic equations on time scales can build bridges between continuous and discrete mathematics. Practically, dynamic equations have been proposed as models in the study of insect population models, neural networks, and many physical phenomena which include gas diffusion through porous media, nonlinear diffusion generated by nonlinear sources, chemically reacting systems as well as concentration in chemical of biological problems . Hence, two-point and multipoint boundary value problems for dynamic equations on time scales have attracted many researchers' attention (see, e.g., [1–19] and references therein). Moreover, singular boundary value problems have also been treated in many papers (see, e.g., [4, 5, 12–14, 18] and references therein).
and obtained the existence of positive solutions by using a fixed point theorem due to Gatica et al. , where is decreasing in for every and may have singularity at .
where are fixed integers satisfying , . They obtained some existence theorems of positive solutions by using Krasnosel'skii fixed point theorem.
Recently, Anderson and Karaca  studied higher-order three-point boundary value problems on time scales and obtained criteria for the existence of positive solutions.
The purpose of this paper is to investigate further the singular BVP for delay higher-order dynamic equation (1.1). By the use of the fixed point theorem of cone expansion and compression type, results on the existence of positive solutions to the BVP (1.1) are established.
The paper is organized as follows. In Section 2, we give some lemmas, which will be required in the proof of our main theorem. In Section 3, we prove some theorems on the existence of positive solutions for BVP (1.1). Moreover, we give an example to illustrate our main result.
Throughout the paper, we assume that .
The following four lemmas can be found in .
So is true.
So holds. Thus is true by induction.
Lemma 2.9 (see ).
Let be a real Banach space and a cone. Assume that is completely continuous operator such that
(i) for and for ,
(ii) for and for .
Then has a fixed point with .
3. Main Results
(C1) is continuous;
for constants and with ;
We seek positive solutions , satisfying (1.1). For this end, we transform (1.1) into an integral equation involving the appropriate Green function and seek fixed points of the following integral operator.
Let (C1), (C2), and (C3) hold, and let , be fixed constants with . Then is completely continuous.
We separate the proof into four steps.
For each , is bounded.
Consequently, is bounded and well defined.
This leads to .
From (C1), (C2), and the Lebesgue dominated convergence theorem , we see that the right-hand side (3.19) can be sufficiently small for beingbig enough. Hence the sequence of compact operators converges uniformly to on so that operator is compact. Consequently, is completely continuous by using the Arzela-Ascoli theorem .
It holds that is a solution of (1.1) if and only if .
From [8, Lemma 3.1], we know that on . So we conclude that is the solution of BVP (1.1).
From condition (C2) and (3.12), we have .
Assume that there exist positive constants with , and such that
(i) and ;
(ii) , for all and .
If it is false, then there exists some with , that is, which implies that for .
Clearly, (3.31) contradicts (3.29). This means that (3.28) holds.
Suppose on the contrary that there exists some with for all .
yielding a contradiction with for all . This means that (3.32) holds. Therefore, from (3.28), (3.32) and Lemma 2.9, we conclude that the operator has at least one fixed point . From the definition of the cone and (2.18), we see that for all . Thus, Proposition 3.2 implies that is a solution of BVP (1.1). So we obtain the desired result.
Adopting the same argument as in Theorem 3.3, we obtain the following results.
Assume that there exist positive constants with , and , such that
(iv) , for all and .
The authors would like to thank the referees for helpful comments and suggestions. The work was supported partly by the NSF of China (10771202), the Research Fund for Shanghai Key Laboratory of Modern Applied Mathematics (08DZ2271900), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805).
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