Abstract
The aim of this paper is to discuss the existence of weak solutions for a nonlinear two-point boundary value problem of integrodifferential equations of fractional order $\alpha\in(1,2]$ . Our analysis relies on the Krasnoselskii fixed point theorem combined with the technique of measure of weak noncompactness.
Highlights
1 Introduction Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and they have been emerging as an important area of investigation in the last few decades; see [ – ]
In [ ], Zhou discusses the existence of solutions for a nonlinear multi-point boundary value problem of integrodifferential equations of fractional order as follows: cDα +x(t) = f (t, x(t), (Hx)(t), (Kx)(t)), t ∈ [, ], α ∈ (, ], a x( ) – b x ( ) = d x(ξ ), a x( ) + b x ( ) = d x(ξ ), where cDα + denotes the fractional Caputo derivative and t (Hx)(s) = g(t, s)u(s) ds
In [ ], Bouffak investigates the existence of weak solutions for a class of boundary value problem of fractional differential equations involving nonlinear integral conditions of the form cDα +x(t) = f (t, x(t)), t ∈ [, T], α ∈ (, ], x( ) + μ
Summary
Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and they have been emerging as an important area of investigation in the last few decades; see [ – ]. Is uniformly HK-integrable over I for every x ∈ Dr ( ) For each bounded set X, Y , Z ⊂ Dr, and each for each closed interval J ⊂ I, t ∈ I, there exists a positive constant L , L ∈ ( , ) such that β k (J, J)g(J, Y ) ≤ L β Y (J) , β k (J, J)h(J, Z) ≤ L β Z(J) , β f (t, X, Y , Z) ≤ Mr(t) max β(X), β(Y ), β(Z) .
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