Nonnegative Solutions Of Nonlinear Discrete Volterra Equations

For nonlinear discrete Volterra equations with or without delay, we obtain several results concerning asymptotic behavior under certain conditions. We discuss the solution of some nonlinear discrete Volterra equations. we are interested in an equation of the formwhere n≥0 and j ≥ 0 are integers, vectors x(j) ∈ R~d, R~d is defined in chapt 2, a(j) are prescribed d x d matrices, and finally, f(j) ∈ R~d are a given sequence of perturbations. In particular we will look at asymptotic behavior and asymptotically constant solutions.In this paper we define the resolvent of Nα = {Na(j)}j≥0, say it rN = {rN(j)}j≥0, then we let rN= {rN(j)}j≥0 satisfies: for all positive number N and that rN is nonnegative for all positive N, and ∑_j~∞=0 rN(j) ≤ K(0 ≤ K < 1), rN is very useful in this paper. First we we give some assumptions to assure that Equation (1) has an unqiune nonnegative solution. To prove the Equation (1) has an unique solution, we use the theorem of contraction mapping. Then we give some basic results, we give a comparison theorem and some other theorems. At last we give a remark to rN = {rN(j)}j≥0, which assure that there exist a rN = {rN(j)}j≥0 satisfies the above condition.