Abstract
In this paper, we investigate the Krasnoselskii-type fixed point results for the operator F of two variables by assuming that the family F x , . : x is equiexpansive. The results may be considered as variants of the Krasnoselskii fixed point theorem in a general setting. We use our main results to obtain the existence of solutions of a fractional evolution differential equation. An example of a controlled system is given to illustrate the application.
Highlights
Fractional evolution equations give a unique way to evaluate the well-posedness of many complicated systems
Most of the results involve contractive operators with more restrictive conditions. This is the reason that many existence results cover a restrictive class of physical problems
We present an existence result for controlled problem with less conditions
Summary
Fractional evolution equations give a unique way to evaluate the well-posedness of many complicated systems. Many fractional order controlled problems and fractional evolution differential equations are recently studied Their existence results can be seen in [1,2,3,4,5]. While studying the solutions of delay and neutral differential/integral equations, it has been noticed that the solution can be expressed as a sum of contractive and compact operators This theorem plays an important role in the existence of solutions of delay integral equations and neutral functional equations. There is a υ in Ω such that ξυ + ζυ = υ: In Krasnoselskii fixed point theorem, there are two operators in which ξ is compact and continuous and ζ is a contraction mapping. Let X be a Banach space and Γ : X × X ⟶ X: The family fΓðν,:Þ: ν ∈ Xg is called equiexpansive if there is h > 1 such that kΓðν, υ1Þ − Γðν, υ2Þk ≥ hkυ1 − υ2k, ð2Þ for all ðν, υ1Þ,ðν, υ2Þ in the domain of Γ:
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