Existence And Asymptotic Stability Of Solution For Nonlinear Functional Integral Equation

Consider the following nonlinear functional integral equations andThe conditions are list as below:(H1) Functionsα,β,γ:R+â†'R+ are continuous andα(t)â†'∞as tâ†'∞.(H2) The function f:R+×R×Râ†'R is continuous and there exist positive constants L,m1,m2, such that for all t∈R+,x1,x2,y1,y2∈R, where L>m1+m2.(H3) The function f(t,0,0) is bounded and denote(H4) The function g:R+×R+×R×Râ†'R is continuous and there exist functions a,b:R+â†'R+, such that |g(t,s,x,y)|≤a(t)b(s) for t,s∈R+.In addition,(H5) The function f:R+×R×R×Râ†'R is continuous and there exist positive constants k1,k2, a non-decreasing functionΦ:R+â†'R+ and continuous function m(t):R+â†'R+ such that 0≤k1+k2< 1,Φ(0)= 0, and where t∈R+,x1,x2,y1,y2,z1,z2∈R.(H6) The function f(t,0,0,0) is bounded and(H7) The function u:R+×E+×R×Râ†'R is continuous and there exist positive constant D, such thatThe main results obtained in this paper are in the following.Theoreml. Assume that conditions (H1)-(H4) hold. The integral equation (1) has at least one solution in the space BC(R+) which is global asymptotic stable.Theorem2. Assume that conditions (H1) and (H5)-(H7) hold. The integral equation (2) has at least one solution in the space BC(R+) which is asymptotic stable.The results in this paper generalize the corresponding those in the literature [Nonlinear Anal 2008; 69:949-952].