Abstract
We compare the newly defined bv(s)-metric spaces with several other abstract spaces like metric spaces, b-metric spaces and show that some well-known results, which hold in the latter class of spaces, may not hold in bv(s)-metric spaces. Besides, we introduce the notions of sequential compactness and bounded compactness in the framework of bv(s)-metric spaces. Using these notions, we prove some fixed point results involving Nemytzki–Edelstein type mappings in this setting, from which several comparable fixed point results can be deduced. In addition to these, we find some existence and uniqueness criteria for the solution to a certain type of mixed Fredholm–Volterra integral equations.
Highlights
There are many interesting extensions of the notion of metric spaces available in the literature where several classical fixed point results have been studied. One of such extensions is the concept of b-metric spaces, which was introduced by Bakhtin [3] in 1989, and later on, in 1993, it had been further investigated by Czerwik [5]
In the year of 2000, Branciari [4] coined the notion of rectangular metric spaces or generalized metric spaces by modifying the triangle inequality of the usual metric spaces
Some remarks on bv(s)-metric spaces and fixed point results we try to find a additional criteria on the underlying bv(s)-metric space X, which confirms the existence of a fixed point for a contractive mapping. To proceed in this direction, we introduce the notions of sequential compactness and bounded compactness of bv(s)-metric spaces and establish correlations between them
Summary
There are many interesting extensions of the notion of metric spaces available in the literature where several classical fixed point results have been studied. To proceed in this direction, we introduce the notions of sequential compactness and bounded compactness of bv(s)-metric spaces and establish correlations between them In such spaces, we establish some fixed point theorems related to contractive mapping, which improve and generalize some standard fixed point results due to Nemytzki [17], Edelstein [8] and Suzuki [19]. We establish some fixed point theorems related to contractive mapping, which improve and generalize some standard fixed point results due to Nemytzki [17], Edelstein [8] and Suzuki [19] Another importance of the (metric) fixed point theory is that it is an invaluable tool for finding existence and/or uniqueness criteria of solution(s) of several types of differential equations, integral equations, fractional integral equations, matrix equations, etc. At the end of this paper, we utilize one of our obtained results to find some criteria for the existence and uniqueness of solution of a special type of integral equation
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