Positive Solutions for the Initial Value Problems of Fractional Evolution Equation

This paper discusses the existence of positive solutions for the initial value problem of fractional evolution equation with noncompact semigroup , ; in a Banach space , where denotes the Caputo fractional derivative of order , is a closed linear operator, generates an equicontinuous semigroup, and is continuous. In the case where satisfies a weaker measure of noncompactness condition and a weaker boundedness condition, the existence results of positive and saturated mild solutions are obtained. Particularly, an existence result without using measure of noncompactness condition is presented in ordered and weakly sequentially complete Banach spaces. These results are very convenient for application. As an example, we study the partial differential equation of parabolic type of fractional order. 1. Introduction The theory of fractional differential equations is a new and important branch of differential equation theory, which has an extensive physical background and realistic mathematical model; see [1–6]. Correspondingly, the existence of solutions to fractional evolution equations in Banach space has also been studied by several authors; see [7–17]. In [7, 8], El-Borai first constructed the type of mild solutions to fractional evolution equations in terms of a probability density. And then they investigated the existence, uniqueness, and regularity of solutions to fractional integrodifferential equations in [9, 10]. Recently, this theory was developed by Zhou et al. [11–14]. In [15–17], the authors studied the existence of mild solutions to fractional impulsive evolutions equations. But as far as we know, there are seldom results on the existence of positive solutions to the fractional evolution equations; see [18–20]. In this paper, we use the Sadovskii’s fixed point theorem and monotone iterative technique to discuss the existence of positive and saturated mild solutions for the initial value problem (IVP) of fractional evolution equations: in Banach space , where denotes the Caputo fractional derivative of order , is a closed linear operator, generates a -semigroup ( ) in , and is continuous and will be specified later, . In some existing articles, the fractional evolution equations were treated under the hypothesis that (I) generates a compact semigroup or (II) the nonlinearity is Lipschitz continuous in on a bounded set. For the case (I), the continuity of nonlinearity can guarantee the local existence of solutions. Hence it is convenient to apply to partial differential equations with compact resolvent. But for the case of noncompact semigroup,