Existence of symmetric positive solutions for a multipoint boundary value problem with sign-changing nonlinearity on time scales
© Tokmak and Karaca; licensee Springer 2013
Received: 30 September 2012
Accepted: 12 February 2013
Published: 14 March 2013
In this paper, we make use of the four functionals fixed point theorem to verify the existence of at least one symmetric positive solution of a second-order m-point boundary value problem on time scales such that the considered equation admits a nonlinear term f whose sign is allowed to change. The discussed problem involves both an increasing homeomorphism and homomorphism, which generalizes the p-Laplacian operator. An example which supports our theoretical results is also indicated.
The theory of time scales was introduced by Stefan Hilger  in his PhD thesis in 1988 in order to unify continuous and discrete analysis. This theory was developed by Agarwal, Bohner, Peterson, Henderson, Avery, etc. [2–5]. Some preliminary definitions and theorems on time scales can be found in books [3, 4] which are excellent references for calculus of time scales.
There have been extensive studies on a boundary value problem (BVP) with sign-changing nonlinearity on time scales by using the fixed point theorem on cones. See [6, 7] and references therein. In , Feng, Pang and Ge discussed the existence of triple symmetric positive solutions by applying the fixed point theorem of functional type in a cone.
where , , , for . By using fixed point index theory  and the Legget-Williams fixed point theorem , sufficient conditions for the existence of countably many positive solutions are established.
where and and , for and . By using the four functionals fixed point theorem and five functionals fixed point theorem, they obtained the existence criteria of at least one positive solution and three positive solutions.
If , then for all ;
ϕ is a continuous bijection and its inverse mapping is also continuous;
for all , where .
We assume that the following conditions are satisfied:
, , , , , ;
is symmetric on (i.e., for );
symmetric on (i.e., for ) and on any subinterval of .
By using four functionals fixed point theorem , we establish the existence of at least one symmetric positive solution for BVP (1.1)-(1.2). In particular, the nonlinear term is allowed to change sign. The remainder of this paper is organized as follows. Section 2 is devoted to some preliminary lemmas. We give and prove our main result in Section 3. Section 4 contains an illustrative example. To the best of our knowledge, symmetric positive solutions for multipoint BVP for an increasing homeomorphism and homomorphism with sign-changing nonlinearity on time scales by using four functionals fixed point theorem  have not been considered till now. In this paper, we intend to fill in such gaps in the literature.
In this paper, a symmetric positive solution x of (1.1) and (1.2) means a solution of (1.1) and (1.2) satisfying and , .
where satisfies (2.5).
On the other hand, it is easy to verify that if x is the solution of (2.3) or (2.4), then x is a solution of BVP (2.1), (2.2). The proof is accomplished. □
Lemma 2.2 If is nonnegative on and on any subinterval of , then there exists a unique satisfying (2.5). Moreover, there is a unique such that .
implies that there exists a unique such that . Lemma is proved. □
is concave on ,
there exists a unique such that ,
, is nonincreasing so is nonincreasing. This implies that is concave.
, imply that there is a such that .
From Lemmas 2.1 and 2.2, we have . Hence we obtain that . This implies .
The lemma is proved. □
Lemma 2.4 If is symmetric nonnegative on and on any subinterval of , then the unique solution of (2.1), (2.2) is concave and symmetric with on .
So, is symmetric on . The proof is accomplished. □
where . Obviously, x is a solution of BVP (2.1)-(2.2) if and only if x is a fixed point of the operator F.
The proof is finalized. □
Lemma 2.6 Suppose that (H1)-(H3) hold, then is completely continuous.
Proof Let . According to the definition of T and Lemma 2.3, it follows that , which implies the concavity of on . On the other hand, from the definition of f and h, holds for , i.e., Tx is symmetric on . So, . By applying the Arzela-Ascoli theorem on time scales, we can obtain that is relatively compact. In view of the Lebesgue convergence theorem on time scales, it is obvious that T is continuous. Hence, is a completely continuous operator. The proof is completed. □
3 Existence of one symmetric positive solution
Theorem 3.1 
for all , with and ;
for all , with ;
for all , with and ;
for all , with .
Then A has a fixed point x in .
and let , and be defined by (3.1).
Theorem 3.2 Assume (H1)-(H3) hold. If there exist constants r, j, n, R with , and suppose that f satisfies the following conditions:
Proof Boundary value problem (1.1)-(1.2) has a solution if and only if x solves the operator equation . Thus we set out to verify that the operator T satisfies four functionals fixed point theorem, which will prove the existence of a fixed point of T.
which means that is a bounded set. According to Lemma 2.6, it is clear that is completely continuous.
So, , which means that (i) in Theorem 3.1 is satisfied.
So, . Hence (ii) in Theorem 3.1 is fulfilled.
Thus (iii) and (v) in Theorem 3.1 hold true. We finally prove that (iv) in Theorem 3.1 holds.
Thus, all conditions of Theorem 3.1 are satisfied. T has a fixed point x in . Clearly, , . By condition (C3), we have , , that is, . Hence, . This means that x is a fixed point of the operator F. Therefore, BVP (1.1)-(1.2) has at least one symmetric positive solution. The proof is completed. □
4 An example
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