The existence of symmetric positive solutions for a seconder-order difference equation with sum form boundary conditions
© Guo et al.; licensee Springer. 2014
Received: 23 March 2014
Accepted: 25 August 2014
Published: 9 September 2014
In this paper, we consider the existence of positive solutions for a second-order discrete boundary value problem subject to the boundary conditions: , , where , for , is symmetric on , is symmetric on , is continuous, for all , and is nonnegative and symmetric on . By the fixed point theorem and the Hölder inequality, we study the existence of symmetric positive solutions for the above difference equation with sum form boundary conditions.
Applying the fixed point index theorem and the Hölder inequality, the author studied the existence of symmetric positive solutions for BVP (1.1)-(1.3).
Throughout this paper, the following conditions are assumed:
(A1) , is symmetric on , and there exists such that on , for , and is symmetric on , h is nonnegative, symmetric on , and , where , is continuous and is symmetric on for all .
Remark 1 The conditions that g and h are symmetric on the different sets, which can guarantee the symmetry of associated kernel function for BVP (1.4)-(1.6). The kernel functions are then used to obtain the existence of symmetric positive solutions for BVP (1.4)-(1.6) by constructing a suitable operator.
In order to study the existence of symmetric positive solutions of problem (1.4)-(1.6), we need the following lemmas.
Lemma 1.1 
Let P be a cone of the real Banach space E and Ω be a bounded open subset of E and . Assume is a completely continuous operator and satisfies , , . Then .
Lemma 1.2 
, , .
Lemma 1.3 (Hölder)
In our main results, we will use the following lemmas.
and , .
where is defined by (2.4). The proof is complete. □
From the above work, we can prove that and have the following properties.
where , , .
Proof It is clear that (2.6) holds. Now we prove (2.7) holds.
The proof is completed. □
Remark 2 The symmetry of on can guarantee that is symmetric for , and the symmetry of on can guarantee that is symmetric for .
It can be observed that u is a solution of problem (1.4)-(1.6) if and only if u is a fixed point of operator T.
We can get the following lemma from Lemma 2.1.
then u is a solution of BVP (1.4)-(1.6).
Lemma 2.3 Assume (A1) holds. Then and is completely continuous.
Thus, and . It is clear that is completely continuous. □
Remark 3 The symmetry of the kernel function for can guarantee that Tu is symmetric on for .
3 Main results
Theorem 3.1 Assume that conditions (A1) hold. In addition, suppose that
(A2) , and , or
(A3) , and
are satisfied. Then problem (1.4)-(1.6) has at least one symmetric positive solution.
So, T has at least one fixed point on . Then it follows that problem (1.4)-(1.6) has a symmetric positive solution . The proof is complete. □
Remark 4 From the proof of Theorem 3.1, we can establish that problem (1.4)-(1.6) has another nonnegative solution , .
The following corollary deals with the case .
Corollary 3.1 Suppose that (A1), (A2) hold. Then problem (1.4)-(1.6) has at least one symmetric positive solution.
Proof It is similar to the proof of Theorem 3.1. Let replace and repeat the argument of Theorem 3.1. □
Finally, we consider the case of .
Corollary 3.2 Assume that (A1), (A2) hold. Then problem (1.4)-(1.6) has at least one symmetric positive solution.
So , , . By Lemma 1.1, we can get . This together with in the proof of Theorem 3.1 completes the proof. □
The authors express their sincere thanks to the referees for the careful and details reading of the manuscript and very helpful suggestions. The project was supported by the Natural Science Foundation of China (11371120), the Natural Science Foundation of Hebei Province (A2013208147) and the Education Department of Hebei Province Science and Technology Research Project (Z2014095).
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