Nonlocal problems for integrodifferential equations. (English) Zbl 1163.45010

The paper deals with the nonlocal Cauchy problem for the nonlinear integrodifferential equation
\[ u'(t)=Au(t)+\int_0^tB(t-s)u(s)\,ds+f(t,u(t)),\quad 0\leq t\leq T,\tag{1} \]
\[ u(0)=u_0+g(u),\tag{2} \]
in a Banach space \(X\), where \(A:D(A)\subset X\to X\) is a densely defined, closed linear operator that generates a \(C_0\)-semigroup \(\{T(t),\;t\in [0,T]\}\), \(\{B(t),\;t\in [0,T]\}\) is a family of continuous linear operators from \(D(A)\) into \(X\), the function \(f:[0,T]\times X\to X\) is continuous and the operator \(g:C([0,T]\times X)\to X\) is compact, which satisfy some additional assumptions. The authors prove that the resolvent operator \(R(t)\) of equation \((1)\) is continuous in the uniform operator topology, for \(t>0\), and then they establish the existence of mild solutions of the problem (1)–(2), by using Schaefer’s fixed point theorem.


45N05 Abstract integral equations, integral equations in abstract spaces
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations