Electron. J. Diff. Equ., Vol. 2010(2010), No. 149, pp. 1-13.

Monotone iterative method for semilinear impulsive evolution equations of mixed type in Banach spaces

Pengyu Chen, Jia Mu

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.

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Pengyu Chen
Department of Mathematics, Northwest Normal University
Lanzhou 730070, China
email: chpengyu123@163.com
Jia Mu
Department of Mathematics, Northwest Normal University
Lanzhou 730070, China
email: mujia05@lzu.cn

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