Abstract
This article constitutes the new fixed point results of dynamic process D(ϒ, μ0) through FIC-integral contractions of the Ciric kind and investigates the said contraction to iterate a fixed point of set-valued mappings in the module of metric space. To do so, we use the dynamic process instead of the conventional Picard sequence. The main results are examined by tangible nontrivial examples which display the motivation for such investigation. The work is completed by giving an application to Liouville‐Caputo fractional differential equations.
Highlights
Introduction and PreliminariesIn the recent past, the study of metric fixed point theory untied a portal to a new area of pure and applied mathematics, the fixed point theory and its application. is concept was prolonged by either extending metric space into its extensions or by modifying the structure of the contractions.e most classical structure known as Banach contraction principle theorem was introduced by a Polish mathematician Banach in 1922 [8]. e applications of fixed points of Banach contraction mappings defined for different kinds of spaces is the guarantee of the existence and uniqueness of solutions of differential and integral type equations. e variety of these nonlinear problems imposes the search for more and better tools, which are recently very remarkable in the literature
The study of metric fixed point theory untied a portal to a new area of pure and applied mathematics, the fixed point theory and its application. is concept was prolonged by either extending metric space into its extensions or by modifying the structure of the contractions
E most classical structure known as Banach contraction principle theorem was introduced by a Polish mathematician Banach in 1922 [8]. e applications of fixed points of Banach contraction mappings defined for different kinds of spaces is the guarantee of the existence and uniqueness of solutions of differential and integral type equations. e variety of these nonlinear problems imposes the search for more and better tools, which are recently very remarkable in the literature
Summary
The study of metric fixed point theory untied a portal to a new area of pure and applied mathematics, the fixed point theory and its application. is concept was prolonged by either extending metric space into its extensions or by modifying the structure of the contractions (see [1,2,3,4,5,6,7]). E main purpose of this manuscript is to introduce the new concept of dynamic iterative process D (Υ, μ0) based on FCI -integral contractions and prove some related multivalued fixed point results in the class of metric space. En, we get μi0− μi0 ∈ Υμi0− which implies the existence of fixed point due to this consideration of dynamic process that satisfying (15) does not depreciate a generality of our approach. Each set-valued FCI -integral contraction Υ on a metric space (Δ, δ) with respect to dynamic process D (Υ, μ0) assures that τ U μi− 1, μi + F⎛⎝H(Υμi,Υμi+1) φ(s)ds⎞⎠. Let (Δ, δ) be a complete metric space, μ0 ∈ Δ and Υ: Δ ⟶ K(Δ) be a set-valued FCI -integral contraction with respect to the dynamic process (μi) ∈ D (Υ, μ0). All the required hypotheses of eorem 1 are satisfied and 0 is a fixed point of Υ
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