Abstract
In the present study, we work on the problem of the existence of positive solutions of fractional integral equations by means of measures of noncompactness in association with Darbo’s fixed point theorem. To achieve the goal, we first establish new fixed point theorems using a new contractive condition of the measure of noncompactness in Banach spaces. By doing this we generalize Darbo’s fixed point theorem along with some recent results of (Aghajani et al. (J. Comput. Appl. Math. 260:67-77, 2014)), (Aghajani et al. (Bull. Belg. Math. Soc. Simon Stevin 20(2):345-358, 2013)), (Arab (Mediterr. J. Math. 13(2):759-773, 2016)), (Banaś et al. (Dyn. Syst. Appl. 18:251-264, 2009)), and (Samadi et al. (Abstr. Appl. Anal. 2014:852324, 2014)). We also derive corresponding coupled fixed point results. Finally, we give an illustrative example to verify the effectiveness and applicability of our results.
Highlights
Fractional calculus is the study of integrals and derivatives of an arbitrary order. Fractional calculus seeks to find the integrals and derivatives of a real or even complex order using the Gamma function, Euler’s generalization of the factorials
1 Introduction Fractional calculus is the study of integrals and derivatives of an arbitrary order
In modern times differential/integral equations with nonintegral order have drawn the attention of numerous researchers due to their wide applications in several fields of science and engineering
Summary
Fractional calculus is the study of integrals and derivatives of an arbitrary order. Fractional calculus seeks to find the integrals and derivatives of a real or even complex order using the Gamma function, Euler’s generalization of the factorials. ([ ]) A mapping β : ME −→ R+ is said to be an MNC in E if it satisfies the following conditions: ( ◦) the family ker β = {X ∈ ME : β(X) = } is nonempty and ker β ⊂ NE, ( ◦) X ⊂ Y ⇒ β(X) ≤ β(Y ), ( ◦) β(X) = β(X), ( ◦) β(Conv X) = β(X), ( ◦) β(λX + ( – λ)Y ) ≤ λβ(X) + ( – λ)β(Y ) for λ ∈ [ , ], ( ◦) if {Xn} is a sequence of closed sets from ME such that Xn+ ⊂ Xn for n = , , .
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