Fixed point theorems for nonlinear contractive mappings in ordered b -metric space with auxiliary function
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Fixed point theorems for nonlinear contractive mappings in ordered b‑metric space with auxiliary function N. Seshagiri Rao1*, K. Kalyani2 and Belay Mitiku1
Abstract Objectives: In this paper we present some fixed point theorems for self mappings satisfying generalized (φ, ψ)-weak contraction condition in partially ordered complete b-metric spaces. The results presented over here generalize and extend some existing results in the literature. Finally, we illustrate two examples to support our results. Result: We obtained a unique fixed point of a self mapping satisfying certain contraction condition which is involving an auxiliary function. Also, the results are presented for the existence of a common fixed point and a coincidence point for generalized (φ, ψ)-weak contraction mappings in partially ordered complete b-metric space. Keywords: Ordered b-metric space, Rational type generalized contraction mappings, Fixed points, Complete metric space Mathematics Subject Classification: 46T99, 41A50, 54H25 Introduction Fixed points of mappings satisfying contractive conditions in generalized metric spaces are highly useful in large number of mathematical problems of pure and applied mathematics. First, Ran and Reuings [1] have extended the result in this direction, discussed the existence of fixed points for certain maps in ordered metric space and also presented some applications to matrix linear equations. Afterwords, the result of [1] has been extended by Nieto et al. [2, 3] involving nondecreasing mappings and used their results in obtaining an unique solution of a first order differential equation. At the same time, the results regarded to generalized contractions in ordered spaces were studied by Agarwal et al. [4] and, O’Regan et al. [5]. The concept of coupled fixed points for certain mappings was first introduced by Bhaskar *Correspondence: [email protected] 1 Department of Applied Mathematics, School of Applied Natural Sciences, Adama Science and Technology University, Post Box No.1888, Adama, Ethiopia Full list of author information is available at the end of the article
and Lakshmikantham [6] and then applied their results to a periodic boundary value problem in acquiring an unique solution. Thereafter, the concept of coupled coincidence and common fixed point results was first initiated by Lakshmikantham and Ćirić [7], which were the extensions of Bhaskar and Lakshmikantham [6] involving monotone property for a function in ordered metric spaces. More work relevant to coupled fixed point results under different contractive conditions in various spaces can be found from [8–15]. Later Singh et al. [16] obtained a coincidence and common fixed point theorems for Suzuki type hybrid contractions in ordered metric spaces and presented the corresponding applications to the results in their work. Mean while a new type of coincidence and common fixed point theorems with applications was investigated by Singh et al. [17]. b-metric space is one of many generalizati
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