Existence of solutions for \(n\)th-order integro-differential equations in Banach spaces (Q5948736)

The author considers the initial value problem for the following \(n\)th-order nonlinear integro-differential equation of mixed type in a Banach space \(E\): NEWLINE\[NEWLINEu^{(n)}(t) = f (t, u, u', \dots , u^{(n-1)}, (Tu)(t), (Su)(t)) (t\in [0, \infty)) NEWLINE\]NEWLINE NEWLINE\[NEWLINE u^{(k)}(0)=u_k\in E, k=0, \dots , n-1 NEWLINE\]NEWLINE where NEWLINE\[NEWLINE f: [0,\infty)\times E^{n+2}\rightarrow E NEWLINE\]NEWLINE is a continuous function, NEWLINE\[NEWLINE (Tu)(t)=\int_0^t k(t,s)u(s) ds NEWLINE\]NEWLINE and NEWLINE\[NEWLINE (Su)(t)=\int_0^\infty h(t,s)u(s) ds. NEWLINE\]NEWLINE Here \(k\) and \(h\) are continuous kernels. The existence of solutions is proved by using the Schauder fixed-point theorem.