Nonnegative doubly periodic solutions for nonlinear telegraph system with twin-parameters

摘要

In this paper, by using Krasnosel’skii fixed-point theorem and under suitable conditions, we present the existence and multiplicity of nonnegative doubly periodic solutions for the following system:utt-uxx+c1ut+a11(t,x)u+a12(t,x)v=b1(t,x)f(t,x,u,v),vtt-vxx+c2vt+a21(t,x)u+a22(t,x)v=b2(t,x)g(t,x,u,v),where ci>0 is a constant, a11(t,x),a22(t,x),b1(t,x),b2(t,x)∈C(R2,R+),a12(t,x),a21(t,x)∈C(R2,R-),f(t,x,u,v),g(t,x,u,v)∈C(R2×R+×R+,R+), and aij,bi,f,g are 2π-periodic in t and x. We derive two explicit intervals of b1(t,x) and b2(t,x) such that for any b1(t,x) and b2(t,x) in the two intervals respectively, the existence of at least one solution for the system is guaranteed, and the existence of at least two solutions for b1(t,x) and b2(t,x) in appropriate intervals is also discussed.