Abstract
We consider a class of nonlinear discrete-time Volterra equations in Banach spaces. Estimates for the norm of operator-valued functions and the resolvents of quasi-nilpotent operators are used to find sufficient conditions that all solutions of such equations are elements of an appropriate Banach space. These estimates give us explicit boundedness conditions. The boundedness of solutions to Volterra equations with infinite delay is also investigated.
Highlights
In many phenomena of the real world, does their evolution prove to be dependent on the present state, but it is essentially specified by the entire previous history
To establish boundedness conditions of solutions, we will interpret the Volterra difference equations with nonlinear kernels as operator equations in appropriate spaces. Such an approach for linear Volterra difference equations has been used by Myshkis [5], Kolmanovskii et al [15], Kwapisz [19], and Medina [20, 21]
Which can be regarded as a retarded equation whose delay is infinite. In general this problem requires that one give an “initial function” on
Summary
In many phenomena of the real world, does their evolution prove to be dependent on the present state, but it is essentially specified by the entire previous history. In Song and Baker [12], several necessary and sufficient conditions for stability are obtained for solutions of the linear Volterra difference equations by considering the equations in various choices of Banach spaces. To establish boundedness conditions of solutions, we will interpret the Volterra difference equations with nonlinear kernels as operator equations in appropriate spaces. Such an approach for linear Volterra difference equations has been used by Myshkis [5], Kolmanovskii et al [15], Kwapisz [19], and Medina [20, 21].
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