Abstract
In this paper, we study the multiplicity of positive doubly periodic solutions for a singular semipositone telegraph equation. The proof is based on a well-known fixed point theorem in a cone.MSC:34B15, 34B18.
Highlights
The existence and multiplicity of positive periodic solutions for a scalar singular equation or singular systems have been studied by using some fixed point theorems; see [ – ]
In [ ], the authors show that the method of lower and upper solutions is one of common techniques to study the singular problem
By a doubly periodic solution of Eq ( ) we mean that a u ∈ L ( ) satisfies Eq ( ) in the distribution sense, i.e., u φtt – φxx – cφt + a(t, x)φ dt dx = λ f (t, x, u)φ dt dx
Summary
The existence and multiplicity of positive periodic solutions for a scalar singular equation or singular systems have been studied by using some fixed point theorems; see [ – ]. For other results concerning the existence and multiplicity of positive doubly periodic solutions for a single regular telegraph equation or regular telegraph system, see, for example, the papers [ – ] and the references therein. In these references, the nonlinearities are nonnegative. The authors [ ] study the semipositone telegraph system ⎧ ⎨utt – uxx + c ut + a (t, x)u = b (t, x)f (t, x, u, v), ⎩vtt – vxx + c vt + a (t, x)v = b (t, x)g(t, x, u, v), where the nonlinearities f , g may change sign.
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