Unified Feng-Liu type fixed point theorems solving control problems
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Unified Feng-Liu type fixed point theorems solving control problems Hemant Kumar Nashine1,2 Rajendra Pant2
· Rabha W. Ibrahim3,4 · B. E. Rhoades5 ·
Received: 7 September 2019 / Accepted: 23 September 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract The propose of this work is to study unified Feng-Liu type fixed point theorems using (α, η)muti-valued admissible mappings with more general contraction condition in complete metric spaces. The obtained results generalize and improve several existing theorems in the literature. We use these results in metric spaces endowed with binary relations and partially ordered sets. Some non-trivial example have been presented to illustrate facts and show genuineness of our work. At the end, the established results will be used to obtain existence solutions for a fractional-type integral inclusion. Keywords Fixed point · Muti-valued mapping · Fractional integral inclusion Mathematics Subject Classification Primary 47H10; Secondary 54H25
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Hemant Kumar Nashine [email protected]; [email protected] Rabha W. Ibrahim [email protected] B. E. Rhoades [email protected] Rajendra Pant [email protected]; [email protected]
1
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, TN 632014, India
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Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park 2006, South Africa
3
Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
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Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
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Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA 0123456789().: V,-vol
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H. K. Nashine et al.
1 Introduction Fixed point theory is one of the noteworthy and stimulating area of nonlinear functional analysis that blends topology, analysis and applied mathematics. In this theory, the controllability problem is converted to a fixed point problem for an applicable nonlinear operator in some space. A fixed point of this operator turns out to be a solution of the given problem. Using the classical Banach fixed point theorem, one can get a unique solution of a nonlinear operator equation (or system of equations) if we convert this operator to a contraction operator in a complete metric space. In 1969, Nadler [1] obtained the multi-valued version of Banach fixed point theorem. His theorem was further extended and generalized by many authors. For some recent results for multi-valued mappings, one may refer to [2,3]. Feng and Liu [4] explored Nadler’s theorem as follows: Theorem 1.1 Let (Z , d) be a metric space and Q : Z → Pcl (Z ), where Pcl (Z ) is the collection of all closed subset of Z . Let f : Z → R be the lower semi-continuous function defined by f () = d(, Q). If there exist b, c ∈ (0, 1) with b < c such that, for any ∈ Z there is a ϑ ∈ Q satisfying c d(, ϑ) ≤ f () and f (ϑ) ≤ b d(, ϑ). Then Q has a fixed point in Z . Nicolae [5] extended their results and unified
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