Abstract
By means of the fixed point theory for a strict set contraction operator, this paper investigates the existence, nonexistence, and multiplicity of positive solutions for a nonlinear higher order boundary value problem with four point fractional integral boundary conditions in Banach spaces. In addition, an example is worked out to illustrate the main results.
Highlights
Let (E, · ) be a real Banach space and P ⊂ E be a cone of E
The goal of this paper is to study the existence, nonexistence, and multiplicity of positive solutions for the following higher order boundary value problem with fractional integral boundary conditions: cDαx(t) = λ a(t)f t, x, x, . . . , x(n– ) + λ b(t)g t, x, x, . . . , x(n– ), Tx, Sx, t ∈ (, ), x(i)( ) = θ, ≤ i ≤ n, x(n– )( ) + x(n– )( ) = Iδx(n– )(η), x(n– )( ) + x(n– )( ) + Iδx(n– )(μ) = θ, ( . ) ( . )
Intensive study has been done to investigate the positive solutions for the nonlinear boundary value problems of fractional differential equations
Summary
Let (E, · ) be a real Banach space and P ⊂ E be a cone of E. The goal of this paper is to study the existence, nonexistence, and multiplicity of positive solutions for the following higher order boundary value problem with fractional integral boundary conditions: cDαx(t) = λ a(t)f t, x, x , . Existence results of at least one or two positive solutions are established to the fractional functional differential equation by constructing a special cone and using the Krasnoselskii fixed point theorem. The existence results of positive solutions for integer order differential equations have been studied extensively by several researchers (see [ – ] and the references therein), but, as far as we know, only a few papers consider the BVP for higher order fractional differential equations in Banach spaces. A has a fixed point x ∈ Kr,R such that r ≤ x ≤ R
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