Existence results for a class of abstract nonlocal Cauchy problems. (English) Zbl 0954.34055

Aizicovici, Sergiu; McKibben, Mark

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 39, No. 5, 649-668 (2000).

The authors study the global existence of a solution to nonlinear evolution equations with nonlocal conditions of the form \[ u'(t) + Au(t) \ni f(t,u(t)), \quad u(0)=g(u), \quad 0<t<T, \tag{*} \] in a real Banach space \(X\). Here, \(A\) is a nonlinear \(m\)-accretive (possibly multivalued) operator on \(X\), \(F: L^1(0,T;X) \to L^1(0,T;X)\) and \(g:L^1(0,T;X) \to \overline {D(A)}\). Using the Schauder fixed point theorem, the Fryszkowski selection theorem and some properties of compact semigroups, the authors prove the existence of integral solutions. This work is a continuation of the paper by S. Aizicovici and Y. Gao [J. Appl. Math. Stochastic Anal. 10, No. 2, 145-156 (1997; Zbl 0883.34065)].

Reviewer: J.Myjak (L’Aquila)

MSC:

34G25	Evolution inclusions
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
47H06	Nonlinear accretive operators, dissipative operators, etc.

Keywords:

nonlocal initial condition; \(m\)-accretive operator; compact semigroup; integral solution

Citations:

Zbl 0883.34065