Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 149, pp. 1-13.
Title: Monotone iterative method for semilinear impulsive
evolution equations of mixed type in Banach spaces
Authors: Pengyu Chen (Northwest Normal Univ., Lanzhou, China)
Jia Mu (Northwest Normal Univ., Lanzhou, China)
Abstract:
We use a monotone iterative method in the presence of lower
and upper solutions to discuss the existence and uniqueness
of mild solutions for the initial value problem
$$\displaylines{
u'(t)+Au(t)= f(t,u(t),Tu(t)),\quad t\in J,\; t\neq t_k,\cr
\Delta u |_{t=t_k}=I_k(u(t_k)) ,\quad k=1,2,\dots ,m,\cr
u(0)=x_0,
}$$
where $A:D(A)\subset E\to E$ is a closed linear operator
and $-A$ generates a strongly continuous semigroup
$T(t)(t\geq 0)$ in $E$.
Under wide monotonicity conditions and the non-compactness measure
condition of the nonlinearity f, we obtain the
existence of extremal mild solutions and a unique mild solution
between lower and upper solutions requiring only that $-A$
generate a strongly continuous semigroup.
Submitted August 4, 2010. Published October 21, 2010.
Math Subject Classifications: 34K30, 34K45, 35F25.
Key Words: Initial value problem; lower and upper solution;
impulsive integro-differential evolution equation;
C0-semigroup; cone.