Existence Results for a Nonlinear Semipositone Telegraph System with Repulsive Weak Singular Forces

Using the fixed point theorem of cone expansion/compression, we consider the existence results of positive solutions for a nonlinear semipositone telegraph system with repulsive weak singular forces.


Introduction
In this paper, we are concerned with the existence of positive solutions for the nonlinear telegraph system: x v g t, x, u , 1.1 with doubly periodic boundary conditions In particular, the function f t, x, v may be singular at v 0 or superlinear at v ∞, and g t, x, u may be singular at u 0 or superlinear at u ∞.

Mathematical Problems in Engineering
In the latter years, the periodic problem for the semilinear singular equation with a, b, c ∈ L 1 0, T and λ > 0, has received the attention of many specialists in differential equations.The main methods to study 1.3 are the following three common techniques: i the obtainment of a priori bounds for the possible solutions and then the applications of topological degree arguments; ii the theory of upper and lower solutions; iii some fixed point theorems in a cone.
We refer the readers to see 1-7 and the references therein.Equation 1.3 is related to the stationary version of the telegraph equation where c > 0 is a constant and λ ∈ R. Because of its important physical background, the existence of periodic solutions for a single telegraph equation or telegraph system has been studied by many authors; see 8-16 .Recently, Wang utilize a weak force condition to enable the achievement of new existence criteria for positive doubly periodic solutions of nonlinear telegraph system through a basic application of Schauder's fixed point theorem in 17 .Inspired by these papers, here our interest is in studying the existence of positive doubly periodic solutions for a semipositone nonlinear telegraph system with repulsive weak singular forces by using the fixed point theorem of cone expansion/compression.
Then, T has a fixed point in This paper is organized as follows: in Section 2, some preliminaries are given; in Section 3, we give the main results.

Preliminaries
Let 2 be the torus defined as 2.1 Doubly 2π-periodic functions will be identified to be functions defined on 2 .We use the notations to denote the spaces of doubly periodic functions with the indicated degree of regularity.The space D 2 denotes the space of distributions on 2 .By a doubly periodic solution of 1.1 -1.2 we mean that a u, v ∈ L 1 2 × L 1 2 satisfies 1.1 -1.2 in the distribution sense; that is, First, we consider the linear equation where c i > 0, λ i ∈ R, and Let £ λ i be the differential operator acting on functions on 2 .Following the discussion in 14 , we know that if λ i < 0, then £ λ i has the resolvent R λ i : where u i is the unique solution of 2.4 , and the restriction of For λ i −c 2 i /4, the Green function G i t, x of the differential operator £ λ i is explicitly expressed; see lemma 5.2 in 14 .From the definition of G i t, x , we have

2.7
Let E denote the Banach space C 2 with the norm u max t,x ∈ 2 |u t, x |, then E is an ordered Banach space with cone For convenience, we assume that the following condition holds throughout this paper: and 2 a i t, x dt dx > 0.
Next, we consider 2.4 when −λ i is replaced by a i t, x .In 10 , Li has proved the following unique existence and positive estimate result.Lemma 2.1.Let h i t, x ∈ L 1 2 ; E is the Banach space C 2 .Then; 2.4 has a unique solution u i P i h i ; P i : L 1 2 → C 2 is a linear bounded operator with the following properties; ii if h i t, x > 0, then a.e.t, x ∈ 2 , P i h i t, x has the positive estimate 2.9

Main Result
In this section, we establish the existence of positive solutions for the telegraph system

3.1
where a i ∈ C R 2 , R and f t, x, v may be singular at v 0. In particular, f t, x, v may be negative or superlinear at v ∞. g t, x, u has the similar assumptions.Our interest is in working out what weak force conditions of f t, x, v at v 0, g t, x, u at u 0 and what superlinear growth conditions of f t, x, v at v ∞, g t, x, u at u ∞ are needed to obtain the existence of positive solutions for problem 1.1 -1.2 .
We assume the following conditions throughout.
Proof.To show that 1.1 -1.2 has a positive solution, we will proof that In addition, by Lemma 2.1, it is clear to see that u, v ∈ C 2 2 × C 2 2 is a solution of 3.9 if and only if u, v ∈ C 2 × C 2 is a solution of the following system: 3.10 Evidently, 3.10 can be rewritten as the following equation:

3.11
Define a cone K ⊂ E as

3.12
We define an operator T : E → K by for u ∈ E and t, x ∈ 2 .We have the conclusion that T : E → E is completely continuous and T K ⊆ K.The complete continuity is obvious by Lemma 2.1.Now, we show that T K ⊆ K.
For any u ∈ K, we have

3.15
So, we get

Mathematical Problems in Engineering 7
First, we show Tu ≤ u , for u ∈ K ∩ ∂Ω r .

3.20
In addition, we also have 3.21 by H5 , H6 , and 3.20 .So, we have This implies that Tu ≤ u ; that is, 3.18 holds.Next, we show From H4 and H6 , we have This implies that Tu ≥ u ; that is, 3.23 holds.Finally, 3.18 , 3.23 , and Lemma 1.1 guarantee that T has a fixed point u ∈ K ∩Ω R \Ω r with r ≤ u ≤ R. Clearly, u > Mω 1 . Since 3.25 then we have a doubly periodic solution u, v of 3.9 with u > Mω 1 , v > Mω 2 , namely, u − Mω 1 , v − Mω 2 > 0, 0 is a positive solution of 1.1 with 1.2 .
Similarly, we also obtain the following result.
H7 There exists

3.30
Then, problem 1.1 -1.2 has a positive periodic solution.

An Example
Consider the following system: here we denote

4.3
where H max{ k 1 , k 2 } and the Green function G 1 G 2 G.Then, problem 4.1 has a positive solution.

4.6
This implies that there exists So, H5 is satisfied.Finally, since 4.9   this implies that there exists R. In addition, for fixed r, R, choosing λ sufficiently large, we have Thus, H6 is satisfied.So, all the conditions of Theorem 3.1 are satisfied.