Abstract
In this article, we introduce a new type of generalized multivalued Hardy and Roger’s type proximal contractive and proximal cyclic contractive mappings of b‐metric spaces and develop some results for the existence of best proximity point(s). Moreover, we obtain some results for the existence and uniqueness of best proximity points for single‐valued mappings. Examples are given to explain the main results.
Highlights
A mapping that satisfies (1) is known as Banach contraction
Many mathematicians contributed for the development of fixed-point theory by producing many results with different generalized contractive mappings in complete metric spaces, for details one can see [2,3,4,5,6,7,8] and the references therein
The metric space has been generalized to b-metric space; the fixed point theory has been further generalized for single-valued and multivalued mappings in the context of b-metric space, for instance, Bakhtin [15] in 1989 and Czerwik [16] in 1993
Summary
A mapping that satisfies (1) is known as Banach contraction. After this remarkable result, many mathematicians contributed for the development of fixed-point theory by producing many results with different generalized contractive mappings in complete metric spaces, for details one can see [2,3,4,5,6,7,8] and the references therein. A mapping T: R ⟶ ℘(S) is said to be a new type of generalized multivalued Hardy and Roger’s proximal contractive mapping if ρb r1, Tr3 ρb(R, S) ⎫⎪⎪⎪⎪⎬ ρb r2, Tr4 ρb(R, S) ⎪⎪⎪⎪⎭, r3 ≠ r4 implies, ρb r1, r2 < αρb r1, r3 + βρb c r2, r4 + b2ρb r3, r4 ⎫⎪⎪⎪⎪⎪⎬
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