Abstract
In 2010, Chistyakov, V. V., defined the theory of modular metric spaces. After that, in 2011, Mongkolkeha, C. et. al. studied and proved the new existence theorems of a fixed point for contraction mapping in modular metric spaces for one single-valued map in modular metric space. Also, in 2012, Chaipunya, P. introduced some fixed point theorems for multivalued mapping under the setting of contraction type in modular metric space. In 2014, Abdou, A. A. N., Khamsi, M. A. studied the existence of fixed points for contractive-type multivalued maps in the setting of modular metric space. In 2016, Dilip Jain et. al. presented a multivalued F-contraction and F-contraction of Hardy-Rogers-type in the case of modular metric space with specific assumptions. In this work, we extended these results into the case of a pair of multivalued mappings on proximinal sets in a regular modular metric space. This was done by introducing the notions of best approximation, proximinal set in modular metric space, conjoint F-proximinal contraction, and conjoint F-proximinal contraction of Hardy-Rogers-type for two multivalued mappings. Furthermore, we give an example showing the conditions of the theory which found a common fixed point of a pair of multivalued mappings on proximinal sets in a regular modular metric space. Also, the applications of the obtained results can be used in multiple fields of science like Electrorheological fluids and FORTRAN computer programming as shown in this communication.
Highlights
The concept of weak contraction was defined by Alber and Guerre-Delabriere in 1997 [1]
In 2012, Wardowski characterized the idea of Fcontraction which generalized the Banach contraction principle in various manners, and he utilized the new concept of contraction to find the fixed point theorem [4]
In 2016, Dilip Jain et al presented a multivalued Fcontraction in the case of modular metric space with specific assumptions [13]
Summary
Throughout this paper, let XB D denote the set of all nonempty closed and bounded subsets of D, X D denotes the set of all nonempty closed subsets of D, and PR D denotes the set of all closed proximinal subsets of D. Definition 2.1 [13]: Let F: R → R be a function satisfying the following conditions: (F1) F is strictly increasing on R , (F2) For every sequence 'sG) in R , we have limG→( sG = 0 if and only if limG→( F sG = −∞, (F3) There exists a number k ∈ (0, 1) such that limg→%= shF s = 0. Definition 2.2 [13]: (F-contraction) Let D be a non-empty ω-bounded subset of a modular metric space ( X , ω ). Definition 2.3 [13]: (F-contraction of Hardy-Rogers-type) Let D be a non-empty ω-bounded subset of a modular metric space (X, ω). A multivalued mapping T: D → XB D is called F-contraction of Hardy-Rogers-type if there exists F ∈ F, and τ ∈ R such that.
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