tailieunhanh - Existence of mild solutions for abstract mixed type semilinear evolution equations

This paper is concerned with the existence of global mild solutions and positive mild solutions to initial value problem for a class of mixed type semilinear evolution equations with noncompact semigroup in Banach spaces. The main method is based on a new fixed point theorem with respect to convex-power condensing operator. | Turk J Math 35 (2011) , 457 – 472. ¨ ITAK ˙ c TUB doi: Existence of mild solutions for abstract mixed type semilinear evolution equations ∗ Hong-Bo Shi, Wan-Tong Li and Hong-Rui Sun Abstract This paper is concerned with the existence of global mild solutions and positive mild solutions to initial value problem for a class of mixed type semilinear evolution equations with noncompact semigroup in Banach spaces. The main method is based on a new fixed point theorem with respect to convex-power condensing operator. Key Words: Semilinear evolution equation; convex-power condensing operator; fixed point theorem; C0 semigroup; measure of noncompactness. 1. Introduction In this paper, we are interested in the following initial value problem (IVP) of mixed type semilinear evolution equation in Banach space E : ⎧ t ⎪ ⎨ u (t) + Au(t) = f t, u(t), k(t, s)u(s)ds, ⎪ ⎩ 0 a h(t, s)u(s)ds , 0 t ∈ J, () u(0) = x0 , where A : D(A) → E is a dense and closed linear operator, −A is the infinitesimal generator of a C0 -semigroup T (t)(t ≥ 0) in E , and J = [0, a] , x0 ∈ E . For convenience, we denote (Ku)(t) = t k(t, s)u(s)ds, 0 (Su)(t) = a h(t, s)u(s)ds. 0 Then IVP () can be rewritten as u (t) + Au(t) = f(t, u(t), (Ku)(t), (Su)(t)), t ∈ J, u(0) = x0 . 2000 AMS Mathematics Subject Classification: 34G20; 47J35. by NSF of China (No. 10801065) ∗Supported 457 SHI, LI, SUN This kind of equation () and other special forms serve as models for various partial differential equations or partial integro-differential equations arising in heat flow in material with memory, viscoelasticity and reaction diffusion problems (see [16, 20]). In recent years, the existence, uniqueness and some other properties of solutions to semilinear evolution equations similar to () have been extensively studied. We can refer to [1, 2, 6, 7, 8, 9, 10, 11, 12, 13, 18, 20] and references cited therein. In particular, we would like to mention the results