Existence of solutions to nonlinear integrodifferential equations of Sobolev type with nonlocal condition in Banach spaces (Q1570009)

The authors deal with the nonlocal Cauchy problem \[ \bigl(Bu (t)\bigr)' +Au(t)=f \bigl(t,u(t) \bigr)+ \int^t_0g\bigl( t,s,u(s)\bigr) ds, \;0<t \leq a,\tag{1} \] \[ u(0)+\sum^p_{k=1}c_ku(t_k)=u_0,\tag{2} \] where \(A\) and \(B\) are closed linear operators in a Banach space \(X\) with \(D(B)\subset D(A)\) and the compact \(B^{-1}\), \(0\leq t_1<t_2<\cdots<t_p\leq a\), \(u_0\in X\), and \(f: [t_0, t_0+a]\times X\to X\), \(g:\{(s,t):0\leq s\leq t\leq a\}\times X\to X\) are given functions. The main results are the existence of mild (under assumptions about the boundedness of \(f\) and \(g)\) and unique strong (under assumptions about the boundedness of \(f,g\), Lipschitzian continuity of \(f(\cdot,u)\) with respect to \(u\) and Lipschitzian continuity of \(g(t,\cdot,\cdot)\) with respect to \(t)\) solutions to problem (1), (2) based on the Schauder fixed-point principle. As an example the following problem \[ {\partial\over \partial t}\bigl(z(t,x)-z_{xx} (t,x) \bigr)- z_{xx}(t,x)= \mu\bigl(t,z(t,x) \bigr)+\int^t_0 \eta \bigl(t,s,z (s,x)\bigr) ds,\;0\leq x\leq\pi,\;0<t\leq a, \] \[ z(t,0)= z(t,\pi)=0, \quad z(0,x)+ \sum^p_{k=1} z(t_k,x)=z_0(x), \] is considered.