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
