An existence theorem for a difference inclusion in general Banach spaces
In the present paper the authors have studied the problem of existence of solution of the difference inclusion \[ u_{i+ 1}- 2u_ i+ u_{i- 1}\in C_ i Au_ i+ f_ i,\quad i= 1,2,\dots,\;u_ 0= x,\quad \sup\{| u_ i|: i\geq 0\}< \infty,\tag{1} \] where \(A\) is a nonlinear \(m\)- accretive operator in a Banach space \((X,|.|)\), \(\{u_ i\}\) is a given sequence of positive numbers, and \(\{f_ i\}\) is a given sequence in \(X\). The main result proved in this paper is as follows: Theorem. Let \(X\) be a Banach space and \(A\subset X\times X\) an \(m\)- accretive operator. If problem (1) has a solution for some \(x\) in \(X\), then it has a unique solution for all \(x\) in \(X\). An existence theorem for the quasi-autonomous incomplete Cauchy problem is also given.
existence of solution of the difference inclusion
quasi-autonomous incomplete Cauchy problem
scientific article; zbMATH DE number 60987