Research Article | Open Access
Fanglei Wang, Yunhai Wang, Yukun An, "Existence Results for a Nonlinear Semipositone Telegraph System with Repulsive Weak Singular Forces", Mathematical Problems in Engineering, vol. 2011, Article ID 610812, 12 pages, 2011. https://doi.org/10.1155/2011/610812
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.
In this paper, we are concerned with the existence of positive solutions for the nonlinear telegraph system: with doubly periodic boundary conditions In particular, the function may be singular at or superlinear at , and may be singular at or superlinear at .
In the latter years, the periodic problem for the semilinear singular equation with , , and , 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.
Equation (1.3) is related to the stationary version of the telegraph equation where is a constant and . 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 . 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.
Lemma 1.1 (see ). Let be a Banach space, and let be a cone in . Assume that , are open subsets of with , , and let be a completely continuous operator such that either(i) and ; or(ii) and .Then, has a fixed point in .
Let be the torus defined as Doubly -periodic functions will be identified to be functions defined on . We use the notations to denote the spaces of doubly periodic functions with the indicated degree of regularity. The space denotes the space of distributions on .
Let be the differential operator acting on functions on . Following the discussion in , we know that if , then has the resolvent : where is the unique solution of (2.4), and the restriction of on or is compact. In particular, is a completely continuous operator.
For , the Green function of the differential operator is explicitly expressed; see lemma 5.2 in . From the definition of , we have Let denote the Banach space with the norm , then is an ordered Banach space with cone For convenience, we assume that the following condition holds throughout this paper:(H1), for , and .
Lemma 2.1. Let is the Banach space . Then; (2.4) has a unique solution is a linear bounded operator with the following properties;(i) is a completely continuous operator;(ii)if has the positive estimate
3. Main Result
In this section, we establish the existence of positive solutions for the telegraph system where and may be singular at . In particular, may be negative or superlinear at . has the similar assumptions. Our interest is in working out what weak force conditions of at , at and what superlinear growth conditions of at , at are needed to obtain the existence of positive solutions for problem (1.1)-(1.2).
We assume the following conditions throughout.(H2) is continuous, and there exists a constant such that (H3) for with continuous and nonincreasing on , continuous on and nondecreasing on . for with continuous and nonincreasing on , continuous on and nondecreasing on .(H4) for all with continuous and nonincreasing on , continuous on with nondecreasing on ; for all with continuous and nonincreasing on , continuous on with nondecreasing on .(H5) There exists such that here where , and is the unique solution to problem: (H6) There exists , such that where
Proof. To show that (1.1)-(1.2) has a positive solution, we will proof that
has a solution with , for . In addition, by Lemma 2.1, it is clear to see that is a solution of (3.9) if and only if is a solution of the following system:
Evidently, (3.10) can be rewritten as the following equation:
Define a cone as We define an operator by for and . We have the conclusion that is completely continuous and . The complete continuity is obvious by Lemma 2.1. Now, we show that .
For any , we have From (H1)–(H3) and Lemma 2.1, we have So, we get namely, .
Let Since for any , we have .
First, we show In fact, if , then and for. By (H3) and (H4), we have In addition, we also have by (H5), (H6), and (3.20).
So, we have for , since .
This implies that ; that is, (3.18) holds.
Next, we show If , then and for . From (H4) and (H6), we have for , since .
This implies that ; that is, (3.23) holds.
Finally, (3.18), (3.23), and Lemma 1.1 guarantee that has a fixed point with . Clearly, .
Since then we have a doubly periodic solution of (3.9) with , , namely, is a positive solution of (1.1) with (1.2).
Similarly, we also obtain the following result.
Theorem 3.2. Assume that (H1)–(H4) hold. In addition, we assume the following.(H7)There exists such that here (H8) There exists , such that where Then, problem (1.1)-(1.2) has a positive periodic solution.
4. An Example
Consider the following system: where , , , , , is continuous. When is chosen such that here we denote where and the Green function . Then, problem (4.1) has a positive solution.
To verify the result, we will apply Theorem 3.1 with and Clearly, (H1)–(H4) are satisfied.
Set Obviously, , , then there exists such that This implies that there exists such that So, (H5) is satisfied.
Finally, since this implies that there exists . In addition, for fixed , choosing sufficiently large, we have Thus, (H6) is satisfied. So, all the conditions of Theorem 3.1 are satisfied.
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