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

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:u"t"t-u"x"x+c"1u"t+a"1"1(t,x)u+a"1"2(t,x)v=b"1(t,x)f(t,x,u,v),v"t"t-v"x"x+c"2v"t+a"2"1(t,x)u+a"2"2(t,x)v=b"2(t,x)g(t,x,u,v),where c"i>0 is a constant, a"1"1(t,x),a"2"2(t,x),b"1(t,x),b"2(t,x)@?C(R^2,R^+),a"1"2(t,x),a"2"1(t,x)@?C(R^2,R^-),f(t,x,u,v),g(t,x,u,v)@?C(R^2xR^+xR^+,R^+), and a"i"j,b"i,f,g are [email protected] in t and x. We derive two explicit intervals of b"1(t,x) and b"2(t,x) such that for any b"1(t,x) and b"2(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 b"1(t,x) and b"2(t,x) in appropriate intervals is also discussed.

[1]  Maximum principles and the method of upper and lower solutions for time-periodic problems of the telegraph equations , 2007 .

[2]  J. Berkovits.  On the Leray–Schauder Formula and Bifurcation , 2001 .

[3]  Rafael Ortega,et al.  A Maximum Principle for Bounded Solutions of the Telegraph Equations and Applications to Nonlinear Forcings , 2000 .

[4]  Wan-Se Kim.  Double-periodic boundary value problem for non-linear dissipative hyperbolic equations , 1990 .

[5]  Jean Mawhin,et al.  Generalized periodic solutions of nonlinear telegraph equations , 1978 .

[6]  W. Ames,et al.  Nonlinear problems in abstract cones , 1988 .

[7]  Rafael Ortega,et al.  Maximum principles for bounded solutions of the telegraph equation in space dimensions two and three and applications , 2005 .

[8]  Yukun An,et al.  Positive solutions for 2p-order and 2q-order nonlinear ordinary differential systems , 2006 .

[9]  Haiyan Wang,et al.  Existence and multiplicity of positive solutions for elliptic systems , 1997 .

[10]  Xiyou Cheng,et al.  Existence of positive solutions for a second-order ordinary differential system☆ , 2005 .

[11]  Xiaojing Yang.  Existence of positive solutions for 2m-order nonlinear differential systems , 2005 .

[12]  Youming Zhou,et al.  Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations☆ , 2006 .

[13]  J. Berkovits,et al.  On nonresonance for systems of semilinear wave equations , 1997 .

[14]  Yongxiang Li.  Positive doubly periodic solutions of nonlinear telegraph equations , 2003 .

[15]  Rafael Ortega,et al.  A Maximum Principle for Periodic Solutions of the Telegraph Equation , 1998 .

[16]  Wan-Se Kim.  Multiple Doubly Periodic Solutions of Semilinear Dissipative Hyperbolic Equations , 1996 .