Abstract
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 secondorder mpoint 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 pLaplacian operator. An example which supports our theoretical results is also indicated.
MSC: 34B10, 39A10.
Keywords:
symmetric positive solution; fixedpoint theorem; time scales; mpoint boundary value problem; increasing homeomorphism and homomorphism1 Introduction
The theory of time scales was introduced by Stefan Hilger [1] 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.[25]. 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 signchanging nonlinearity on time scales by using the fixed point theorem on cones. See [6,7] and references therein. In [8], 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 [9], Ji, Bai and Ge studied the following singular multipoint boundary value problem:
where
Sun, Wang and Fan [12] studied the nonlocal boundary value problem with pLaplacian of the form
where
Inspired by the mentioned works, in this paper we consider the following mpoint boundary value problem with an increasing homeomorphism and homomorphism:
where
(i) If
(ii) ϕ is a continuous bijection and its inverse mapping is also continuous;
(iii)
We assume that the following conditions are satisfied:
(H1)
(H2)
(H3)
By using four functionals fixed point theorem [5], we establish the existence of at least one symmetric positive solution for BVP (1.1)(1.2).
In particular, the nonlinear term
In this paper, a symmetric positive solution x of (1.1) and (1.2) means a solution of (1.1) and (1.2) satisfying
2 Preliminaries
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
or
where
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
or
where
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
Proof For any
So,
Therefore there exists a unique
Then
implies that there exists a unique
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
(i)
(ii)
(iii) there exists a unique
(iv)
Proof Suppose that
(i)
(ii) We have
If we continue like this, we have
Using (H1), we obtain
which implies that
If we continue in this way, we attain that
Using (H1), we have
(iii)
If there exist
which is a contradiction.
(iv) From Lemmas 2.1 and 2.2, we have
The lemma is proved. □
Lemma 2.4If
Proof Clearly,
Again, let
So,
Let
We define two cones by
Define the operator
and
where
Lemma 2.5If (H1) holds, then
Proof For
i.e.,
By
Hence
i.e.,
The proof is finalized. □
From Lemma 2.5, we obtain
Lemma 2.6Suppose that (H1)(H3) hold, then
Proof Let
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:
Theorem 3.1[5]
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
(i)
(ii)
(iii)
(iv)
(v)
ThenAhas a fixed pointxin
Suppose
and define the maps
and let
Theorem 3.2Assume (H1)(H3) hold. If there exist constantsr, j, n, Rwith
(C1)
(C2)
(C3)
Then BVP (1.1)(1.2) has at least one symmetric positive solution
Proof Boundary value problem (1.1)(1.2) has a solution
We first show that
which means that
Let
Clearly,
So,
For all
So,
For all
and for all
Thus (iii) and (v) in Theorem 3.1 hold true. We finally prove that (iv) in Theorem 3.1 holds.
For all
Thus, all conditions of Theorem 3.1 are satisfied. T has a fixed point x in
4 An example
Example 4.1 Let
where
Set
Choose
(1)
(2)
(3)
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
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
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