DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

EXISTENCE AND UNIQUENESS OF AN INTEGRAL
SOLUTION TO SOME CAUCHY PROBLEM
WITH NONLOCAL CONDITIONS
GASTON M. N’GUÉRÉKATA
Using the contraction mapping principle, we prove the existence and uniqueness of an
integral solution to a semilinear differential equation in a Banach space with a nondensely
defined operator and nonlocal conditions.
Copyright © 2006 Gaston M. N’Guérékata. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Preliminaries and notations
The aim of this short note is to prove the existence and uniqueness of an integral solution
to the nonlocal evolution equation in a Banach space E,
du(t)
dt
= Au(t)+F ( t,Bu(t) ) , t ∈ [0,T],
u(0) + g(u) = u 0 ,
(1.1)
where A : D(A) ⊂ E → E and B : D(B) → E are closed linear operators.
We assume that the domain D(A)ofA is not dense in E and
(i) F :[0,T] × E → E is continuous,
(ii) g : C → E is continuous, where C := C([0,T];E) is the Banach space of all continuous
functions [0,T] → E equipped with the uniform norm topology.
An example of such problem is the following.
Example 1.1. Consider the partial differential equation
∂ ∂2
u(t,x) = u(t,x)+ f (t,Bx),
∂t ∂x2 (t,x) ∈ [0,T] × (0,1),
u(t,0)= u(t,1)= 0, t ∈ [0,T],
u(0,x)+
n∑
λ i u ( t i ,x ) = Φ(x), x ∈ (0,1),
i
(1.2)
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 843–849