DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS
DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS
DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS
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EXISTENCE RESULTS FOR EVOLUTION INCLUSIONS<br />
IN BANACH SPACES<br />
VASILE STAICU<br />
We survey some new results obtained recently in joint papers with S. Aizicovici and N. S.<br />
Papageorgiou, concerning the existence of integral solutions for nonlocal Cauchy problem<br />
and for the periodic problem to evolution inclusions in Banach spaces.<br />
Copyright © 2006 Vasile Staicu. This is an open access article distributed under the Creative<br />
Commons Attribution License, which permits unrestricted use, distribution, and<br />
reproduction in any medium, provided the original work is properly cited.<br />
1. Introduction<br />
In this paper we study the existence of integral solutions for the nonlocal Cauchy problem<br />
and for the periodic problem<br />
u ′ (t) ∈−Au(t)+F ( t,u(t) ) , t ∈ [0,b]; u(0) = g(u), (1.1)<br />
−u ′ (t) ∈ Au(t)+F ( t,u(t) ) , t ∈ [0,b]; u(0) = u(b), (1.2)<br />
where A : D(A) ⊂ X → X is an m-accretive operator, F : T × X → 2 X is a multivalued map,<br />
and g : C(I;D(A)) → D(A).<br />
The study of nonlocal initial value problems in Banach spaces was initiated by<br />
Byszewski [14], who considered an equation of the form (1.1) withA linear, F single<br />
valued, and g of a special structure. Results on fully nonlinear abstract nonlocal Cauchy<br />
problemhavebeenobtainedin[2–4] and very recently in [24]. These papers are primarily<br />
concerned with equations governed by accretive operators and single-valued perturbations.<br />
To our knowledge, the only existing result for (1.1) withA nonlinear and<br />
F multivalued was obtained in [4], where F is supposed to be closed-valued and lower<br />
semicontinuous in its second variable. On the other hand, finite-dimensional versions of<br />
(1.1)(withA = 0) appear in [12, 18], while abstract semilinear evolution inclusions with<br />
nonlocal initial conditions have been considered in [1, 9–11]. In particular, in [1], the<br />
problem (1.1) is analyzed under the assumption that −A is the infinitesimal generator<br />
Hindawi Publishing Corporation<br />
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 1019–1027
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