Abstract
In this paper, we obtain the existence of multiple positive solutions of a boundary value problem for α-order nonlinear integro-differential equations in a Banach space by means of fixed point index theory of completely continuous operators. MSC:26A33, 34B15.
Highlights
1 Introduction Fractional differential equations (FDEs) have been of great interest for the last three decades [ – ]. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in the modeling of many phenomena in various fields of science and engineering
Proof If u ∈ BC[J, E] is a solution of boundary value problem (BVP) ( ), by applying Lemma . we reduce Dα u(t) + f (t, u(t), (Tu)(t), (Su)(t)) = θ to an equivalent integral equation u(t) = –I α+f t, u(t), (Tu)(t), (Su)(t) + c tα– + c tα– + c tα
The issue on the existence of multiple positive solutions of a boundary value problem for α-order nonlinear integro-differential equations in a Banach space has been addressed for the first time
Summary
Fractional differential equations (FDEs) have been of great interest for the last three decades [ – ]. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in the modeling of many phenomena in various fields of science and engineering. We shall use the fixed point index theory of completely continuous operators to investigate the multiple positive solutions of a boundary value problem for a class of α order nonlinear integro-differential equations in a Banach space. Let E be a real Banach space, P be a cone in E and P denote the interior points of P.
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