The Asymptotic Behavior Of Solutions Of High Order Integral-differential Equations With Deviating Arguments

For the research of asymptotic behavior of solution of integro-differential equation has been the important question in the field of equation, because in certain conditions,using integral inequality, we obtain that nonlinear integro-differential equations can be obtained the asymptotic state with a homogeneous equation solution of asymptotic state.Therefore in the process of promoting ,we have a similar problem in the research of system of unified method. Gronwall-Bellman and Bihari integral inequality and their promotion has played an important role in the field of the asymptotic behavior of integro-differential equations. Many scholars and researchers in order to achieve different goals, in the past few years, they has already established some important Gronwall-Bellman and Bihari integral inequality, and by using this integral inequality, they have studied a few kinds of asymptotic behavior of integro-differential equations.In 2004. Fanwei Meng[6] studiecd the asymptotic behavior of second order integro-differential equations with deviating argument of the form:(a(t)x')' + b(t)x' + c(t)x=f[t,x(t),x'(t),x(?(t)),x'(?(t)),?0t(?)g(t,s,x(s),x'(s),x(?(s)),x'(?(s)))ds].In 2013, Meng Fanwei and Yao Jianli[7] studied the asymptotic behavior of higher order integro-differential equations with deviating arguments of the form:On this basis, this paper uses promotion of Gronwall-Bellman and Bihari integral inequality, and promote the above integro-differential equation, and studied the status of the asymptotic of solution, at the same time , some new results are obtained. Finally,through an extension of discrete type Bihari inequality, we can obtain the boundedness and the asymptotic behavior of the solutions of a class of third order nonlinear difference equation.According to the content , this article is divided into the following five chapters:Chapter 1 Preference, we introduce the main contents and background of this paper.Chapter 2 Using new Gronwall-Bellman and Bihari integral inequality, we will pro-mote the integro-differential equations , and we will obtain third order integro-differential equations with deviation argument, and studies its asymptotic behavior of solution:Herea a =a(t) is a continuously differentiable function on R+ = [0, ?) such that a(0) = 1;b = b(t), c = c(t), d = d(t) are continuous functions on R+; f ?C[R+ × R7, R] and g ? C[R+2 × R6, R] , respectively; ?(t), ?(t) are continuously differentiable satisfying that?(t) ? t,?(t) < t; ?'(t) > 0, ?'(t) > 0 and ?(t), ?(t) are eventually positive.Chapter 3 Using new Gronwall-Bellman and Bihari integral inequality, we will promote the integro-differential equations, we will obtain higher order integro-differential equations with deviation argument, and studies its asymptotic behavior of solution:where p(t) is a differentiable function defined on R+ = [0,?) with p(t) > 0, and p(0) =1; Ci(t)(i = 1,2, ...,n) are continuous functions on R+, ?? C[R+,R],?(t) < t , ?'(t) > 0, ?(t) < t , ?'(t) > 0, and a(t) , ?(t) is eventually positive ,f ? C[R+ x R2n+1,R], g ? C[R+2 × Rn,R].Chapter 4 Using new Gronwall-Bellman and Bihari integral inequality, we will promote the integro-differential equations, we will obtain higher order nonlinear integro-differential equations with deviation argument, and studies its asymptotic behavior of solution:Here p = p(t) is a positive and continuously differentiable function on R+ = [0,?)such that p(0) = 1; ci(t) (i = 1,2, ...,n) are continuous functions on R+; f ? C[R+ ×R2n+1, R] and g ? C[R+2×R2n, R] , respectively; ?(t), ?(t) are continuously differentiable satisfying that ?(t) ? t , ?(t) ? t; ?'(t) > 0, ?'(t) > 0 and ?(t), ?(t) are eventually positive.Chapter 5 Through an extension of discrete Bihari inequality, we study boundedness and asymptotic behavior of' solutions of a class of third order nonlinear difference equation:?(r2(n)?(r1(n)?(xp(n)))) + f(n,x(n)) = 0 There n ? N+(n0) = {n0,n0+1,...},n0?N+, and ? is the forward difference operator,r(n)is real sequence, f is a real function that defined in the interval N(no) × R × R.