Multiple Positive Solutions to Multipoint Boundary Value Problem for a System of Second-Order Nonlinear Semipositone Differential Equations on Time Scales
We study a system of second-order dynamic equations on time scales , , satisfying four kinds of different multipoint boundary value conditions, is continuous and semipositone. We derive an interval of such that any lying in this interval, the semipositone coupled boundary value problem has multiple positive solutions. The arguments are based upon fixed-point theorems in a cone.
In this paper, we consider the following dynamic equations on time scales: satisfying one of the boundary value conditions where and is continuously and nonegative functionsquad, for ; the points for with .
In the past few years, the boundary value problems of dynamic equations on time scales have been studied by many authors (see [1–19] and references). Recently, multipoint boundary value problems on time scale have been studied, for instance, see [1–12].
In 2006, Anderson and Ma  studied the second-order multiple time-scale eigenvalue problem: where the functions and are continuous. The authors discuss conditions for the existence of at least one positive solution to the second-order Sturm-Liouville-type multiple eigenvalue problem on time scales.
In 2009, Feng et al.  studied where the functions , , , . This paper shows the existence of multiple positive solutions for the boundary value problem on time scales.
In 2009, Topal and Yantir  studied the second-order nonlinear -point boundary value problems where ; for all ; , , are continuously and nonegative functions. The authors deal with determining the value of , and the existences of multiple positive solutions of the equation are obtained. In 2010, Yuan and Liu  also study the second-order -point boundary value problems; Yuan and Liu shows the existence of multiple positive solutions if is semipositone and superlinear.
Motivated by the above results mentioned, we study the second-order nonlinear -point boundary value problem (1) with boundary condition (), and nonlinear term may be singularity and semipositone.
In this paper, the nonlinear term of (1) is suit to and semipositone and the superlinear case, we shall prove our two existence results for the problem (1) with () by using a nonlinear alternative of Leray-Schauder type and Krasnosel'skii fixed-point theorem. This paper is organized as follows. In Section 2, we start with some preliminary lemmas. In Section 3, we give the main result which state the sufficient conditions for (1) with -point boundary value () to have existence of positive solutions ().
In this section, we state the preliminary information that we need to prove the main results.
In this paper, for our constructions, we shall consider the Banach space equipped with standard norm ; for each , we write . Clearly, is a Banach space. Denote by and (), the solutions of the equation under the initial conditions respectively. So that and () satisfy respectively. For , set , the Green's function of the corresponding homogeneous boundary value problem is defined by From Lemmas and 3.3 in , we have the following lemma.
Lemma 1. If , is a solution of (1) with boundary value condition () if only and if where , and
For the rest of the paper, we need the following assumption:
From is nondecreasing on , is nonincreasing on (see [2, Proposition 2.3]), it is easy to verify the following inequalities:
Lemma 2. The Green's function has properties
Lemma 3. For , and , one has the conclusions and where and
Proof. From Lemma 2 and
For or 3, we have So, we have
Since , then we also have The proof is complete.
The following theorems will play a major role in our next analysis.
Theorem 4 (see ). Let be a Banach space, and closed and convex. Assume is a relatively open subset of with , and let be a compact, continuous map. Then either (1)has a fixed point in , or (2)there exists and , with .
Theorem 5 (see ). Let be a Banach space, and let be a cone in . Let be bounded open subsets of with , and let be a completely continuous operator such that, either (1), , , or (2), , . Then has a fixed point in .
3. Main Results
We make the following assumptions:, moreover there exists a function such that , for any , , .( may be singular at ; moreover, there exists a function such that , for any , .(, for .( There exists such that .( and for any , is any constant .In fact, we only consider the system with one of the boundary value conditions where and . For , from Lemma 1, () is the solution of the equation respectively, satisfying the following boundary value conditions:
We will show that there exists a solution to the boundary value problem () of the system (26) with . If this is true, then is a nonnegative solution (positive on ) of the system (1) with the boundary value problem , . Since for any , from we have As a result, we will concentrate our study on (26) with the boundary value problem ().
We define a cone by It is clearly that is a cone of , . Define the integral operator , , , , by where operators are defined by where . Clearly, if is a fixed point of , then is a solution of system (26) with .
On the other hand, when , we have Thus, . Hence .
When , we have Thus, . Hence .
Similarly discussion, we also have , . In addition, standard arguments show that is a completely continuous operator.
For simplicity, we adopt the notation: and , then, we can write , that is, , ().
Theorem 6. Suppose that - hold. Then there exists a constant such that, for any , (1) with boundary value condition () has at least one positive solution ().
Proof. Fix and (). From () let be such that
Let , and . We have
Set , since for any , fix the , we always have Then there exists a such that
Let , and be such that , that is, . We claim that . In fact for and , we have It follows that that is, which implies that . By the nonlinear alternative of Leray-Schauder type, has a fixed point . Moreover combining (40) and the fact that , we obtain
Then has a positive fixed point and ; that is, is a positive solution of the boundary value problem (26) with for .