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.
MSC: 34B10, 39A10.
Keywords:symmetric positive solution; fixed-point theorem; time scales; m-point boundary value problem; increasing homeomorphism and homomorphism
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.
In , Ji, Bai and Ge studied the following singular multipoint boundary value problem:
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.
Sun, Wang and Fan  studied the nonlocal boundary value problem with p-Laplacian of the form
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.
Inspired by the mentioned works, in this paper we consider the following m-point boundary value problem with an increasing homeomorphism and homomorphism:
(ii) ϕ is a continuous bijection and its inverse mapping is also continuous;
We assume that the following conditions are satisfied:
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.
To prove the main result in this paper, we will employ several lemmas. These lemmas are based on the BVP
Proof Suppose x is a solution of BVP (2.1), (2.2). Integrating (2.1) from 0 to t, we have
Integrating (2.6) from 0 to t, we get
or integrating the same equation from t to 1, we achieve
Using boundary condition (2.2), we get
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. □
Remark 2.1 By Lemmas 2.1 and 2.2, the unique solution of BVP (2.1), (2.2) can be rewritten in the form
If we continue like this, we have
Using (H1), we obtain
If we continue in this way, we attain that
which is a contradiction.
The lemma is proved. □
We define two cones by
The proof is finalized. □
From Lemma 2.5, we obtain
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
Let α and Ψ be nonnegative continuous concave functionals on P, and let β and θ be nonnegative continuous convex functionals on P, then for positive numbers r, j, n and R, we define the sets:
IfPis a cone in a real Banach spaceE, αand Ψ are nonnegative continuous concave functionals onP, βandθare nonnegative continuous convex functionals onPand there exist positive numbersr, j, nandRsuch that
and define the maps
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.
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
So, all conditions of Theorem 3.2 hold. Thus, by Theorem 3.2, BVP (4.1) has at least one symmetric positive solution x such that
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
Hilger, S: Analysis on measure chains - a unified approach to continuous and discrete calculus. Results Math.. 18, 18–56 (1990). Publisher Full Text
Agarwal, RP, Bohner, M: Basic calculus on time scales and some of its applications. Results Math.. 35, 3–22 (1999). Publisher Full Text
Avery, R, Henderson, J, O’Regan, D: Four functionals fixed point theorem. Math. Comput. Model.. 48, 1081–1089 (2008). Publisher Full Text
Hamal, NA, Yoruk, F: Symmetric positive solutions of fourth order integral BVP for an increasing homeomorphism and homomorphism with sign-changing nonlinearity on time scales. Comput. Math. Appl.. 59, 3603–3611 (2010). Publisher Full Text
Xu, F, Meng, Z: The existence of positive solutions for third-order p-Laplacian m-point boundary value problems with sign changing nonlinearity on time scales. Adv. Differ. Equ.. 2009, Article ID 169321 (2009)
Feng, H, Pang, H, Ge, W: Multiplicity of symmetric positive solutions for a multipoint boundary value problem with a one-dimensional p-Laplacian. Nonlinear Anal.. 69, 3050–3059 (2008). Publisher Full Text
Ji, D, Bai, Z, Ge, W: The existence of countably many positive solutions for singular multipoint boundary value problems. Nonlinear Anal.. 72, 955–964 (2010). Publisher Full Text
Leggett, RW, Williams, LR: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J.. 28, 673–688 (1979). Publisher Full Text