Initial value problems for a nonlinear integro-differential equation of mixed type in Banach spaces (Q2792555)

The authors consider an initial value problem (IVP) of the first-order nonlinear integro-differential equations of mixed type in a Banach space of the form NEWLINE\[NEWLINEu=f(t,u,Tu,Su), \,u(t_0)=u_0,NEWLINE\]NEWLINE where \(f \in C[J \times E \times E \times E, E]\), \(J=[t_0,t_0+a]\), where \(a>0\), \(u_0 \in E\), and NEWLINE\[NEWLINE(Tu)(t)= \int_{t_0}^{t} K(t,s)u(s)ds,\,\, Su(t)=\int_{t_0}^{t_0+a} K(t,s)u(s)ds.NEWLINE\]NEWLINE The important fact in this problem is that the kernal \(k(t,s)\) in the integral operator T is unbounded.NEWLINENEWLINEThe main result of this paper is an existence result guaranteeing the existence of a maximal and minimal solution in a sector for the above IVP.NEWLINENEWLINEUsing the Kuratowski measure of compactness and the Hausdorff measure of compactness, the solutions of the integro-differential equation in a Banach space \(E\) are shifted to the real line \(\mathbb R\) and the lemmas in \(\mathbb R\) are applied.NEWLINENEWLINEThe existence of the solution of the linear integro-differential equation is obtained by using the contraction theorem and that of the uniform convergence of the sequence of iterates is obtained by Ascoli's theorem in the abstract set up.