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(α(t)),x’(α(t)),∫0t(?)g(t,s,x(s),x’(s),x(β(s)),x’(β(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; α(t), β(t) are continuously differentiable satisfying thatα(t) ≤ t,β(t) < t; α’(t) > 0, β’(t) > 0 and α(t), β(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+, φ∈ C[R+,R],α(t) < t , α’(t) > 0, β(t) < t , β’(t) > 0, and a(t) , β(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; α(t), β(t) are continuously differentiable satisfying that α(t) ≤ t , β(t) ≤ t; α’(t) > 0, β’(t) > 0 and α(t), β(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.