Abstract
We investigate the solutions of functional-integral equation of fractional order in the setting of a measure of noncompactness on real-valued bounded and continuous Banach space. We introduce a new μ-set contraction operator and derive generalized Darbo fixed point results using an arbitrary measure of noncompactness in Banach spaces. An illustration is given in support of the solution of a functional-integral equation of fractional order.
Highlights
Theorem 1.2 ([9]) Let X be a closed, convex subset of a Banach space E
We will discuss the solutions u ∈ C(I, X) of functional-integral equation of fractional orderHu(t) t g (s) u(t) = f t, u(t) + Γ (γ )(g(t) – g(s))1–γ k t, s, u(s) ds, t ∈ I = [0, 1], 0 < γ < 1, in the setting of measure of noncompactness (MNC) on real-valued bounded and continuous Banach space
We introduce a μ-set contraction operator using new control functions and establish some new fixed point result, a Krasnoselskii fixed point result, that generalizes the results in [1,2,3, 10, 12, 13]
Summary
Theorem 1.2 ([9]) Let X be a closed, convex subset of a Banach space E. Every compact, continuous map T : X → X has at least one fixed point. Theorem 1.3 ([10]) Let T : Ω → Ω be a continuous and μ-set contraction operator, that is, there exists a constant k ∈ [0, 1) with μ(TM) ≤ kμ(M)
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