Abstract
In this paper, we get a new form equivalent integral equation for a class of evolution equations of fractional order with nonlocal conditions on the half-line. With the aid of it, the uniqueness of the mild solution is obtained by the Banach contraction theorem. Also, we present the existence and uniqueness theorem of positive mild solutions by the monotone iterative method without assumption of lower and upper solutions.
Highlights
1 Introduction In this paper, we deal with the following functional differential abstract equation of fractional order with nonlocal conditions in the Banach space E:
Motivated by the papers [, ], in this paper, we study the fractional differential equations ( ) and ( ) with nonlocal conditions on the unbounded domains
Employing the monotone iterative method, without the assumption of lower and upper solutions, we present some new results on the existence of positive mild solutions for the abstract fractional differential equations ( )
Summary
We deal with the following functional differential abstract equation of fractional order with nonlocal conditions in the Banach space E:. One branch of the studies on fractional differential equations is devoted to investigating the fractional evolution equation with nonlocal conditions, which is a valuable tool to describe the physics phenomena. T ≥ , With the help of the monotone iterative method, the existence and uniqueness of the mild positive solutions were obtained in the paper. As far as we know, recently, evolution equations of fractional order have attracted increasing attention, and we refer to the papers [ – ] and the references therein. Employing the monotone iterative method, without the assumption of lower and upper solutions, we present some new results on the existence of positive mild solutions for the abstract fractional differential equations ( ). C -semigroup with the growth bound ω (ω < ), and A is the infinitesimal generator of
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