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:
where is a time scale, is an increasing homeomorphism and homomorphism with . A projection is called an increasing homeomorphism and homomorphism if the following conditions are satisfied:
(i) If , then for all ;
(ii) ϕ is a continuous bijection and its inverse mapping is also continuous;
(iii) for all , where .
We assume that the following conditions are satisfied:
(H1) , , , , , ;
(H2) is symmetric on (i.e., for );
(H3) 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 , .
To prove the main result in this paper, we will employ several lemmas. These lemmas are based on the BVP
Lemma 2.1If condition (H1) holds, then for , boundary value problem (2.1) and (2.2) has a unique solution
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
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 2.2If is nonnegative on and on any subinterval of , then there exists a unique satisfying (2.5). Moreover, there is a unique such that .
Proof For any , define
So, is continuous and strictly increasing. It is easy to see that
Therefore there exists a unique satisfying (2.5). Furthermore, let
Then is continuous and strictly increasing on and , . Thus
implies that there exists a unique such that . Lemma is proved. □
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
Lemma 2.3Let (H1) hold. If is nonnegative on and on any subinterval of , then the unique solution of BVP (2.1)-(2.2) has the following properties:
(i) is concave on ,
(iii) there exists a unique such that ,
Proof Suppose that is a solution of BVP (2.1)-(2.2), then
(i) , is nonincreasing so is nonincreasing. This implies that is concave.
(ii) We have and . Furthermore, we get
If we continue like this, we have
Using (H1), we obtain
which implies that . Similarly,
If we continue in this way, we attain that
Using (H1), we have , . Therefore, we get , .
(iii) , imply that there is a such that .
If there exist , , such that , then
which is a contradiction.
(iv) From Lemmas 2.1 and 2.2, we have . Hence we obtain that . This implies .
The lemma is proved. □
Lemma 2.4If is symmetric nonnegative on and on any subinterval of , then the unique solution of (2.1), (2.2) is concave and symmetric with on .
Proof Clearly, is concave and from Lemma 2.3. We show that is symmetric on . For the symmetry of , it is easy to see that , i.e., . Therefore, from (2.9) and for , by the transformation , we have
Again, let . Then
So, is symmetric on . The proof is accomplished. □
Let . Then E is a Banach space with the norm
We define two cones by
Define the operator by
and as follows:
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.
Lemma 2.5If (H1) holds, then for , where
Proof For , one arrives at
i.e., . Hence,
By , we get
The proof is finalized. □
From Lemma 2.5, we obtain
Lemma 2.6Suppose 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
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
is a completely continuous operator, and is a bounded set. If
(ii) for all , with and ;
(iii) for all , with ;
(iv) for all , with and ;
(v) for all , with .
ThenAhas a fixed pointxin .
Suppose with . For the convenience, we take the notations
and define the maps
and let , and be defined by (3.1).
Theorem 3.2Assume (H1)-(H3) hold. If there exist constantsr, j, n, Rwith , and suppose thatfsatisfies the following conditions:
(C1) for ;
(C2) for ;
(C3) for .
Then BVP (1.1)-(1.2) has at least one symmetric positive solution such that
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.
We first show that is bounded and is completely continuous. For all with Lemma 2.5, we have
which means that is a bounded set. According to Lemma 2.6, it is clear that is completely continuous.
Clearly, . By direct calculation,
So, , which means that (i) in Theorem 3.1 is satisfied.
For all , with and , we have from concavity
So, . Hence (ii) in Theorem 3.1 is fulfilled.
For all , with ,
and for all , with ,
Thus (iii) and (v) in Theorem 3.1 hold true. We finally prove that (iv) in Theorem 3.1 holds.
For all , with and , we have
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
Example 4.1 Let . If we choose , , , , in boundary value problem (1.1)-(1.2), then we have the following BVP on time scale :
Set , . By simple calculation, we get
Choose , , and . It is easy to check that , .
(1) for ;
(2) for ;
(3) for .
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.
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