Fixed point results in $M_{\nu }$-metric spaces with an application

Abstract

In this paper, we introduce the concept of M ν-metric as a generalization of M-metric and ν-generalized metric and also prove an analogue of Banach contraction principle in an M ν-metric space. Also, we adopt an example to highlight the utility of our main result which extends and improves the corresponding relevant results of the existing literature. Finally, we use our main result to examine the existence and uniqueness of solution for a Fredholm integral equation.

On Unique and Nonunique Fixed Points and Fixed Circles in M v b

“…Theorem 25 is an extension, improvement, and generalization of Banach [21] (α = β = 0), Kannan [22] (α = β, γ = 0), Reich [23], and Asim et al [10] (α = β = 0) to an M b v -metric space. Now, we prove the result for a Hardy and Rogers type contraction [24], which includes all the results stated above as a special case.…”

“…Þ is an improvement and extension of an M v -metric space [10]. Inclusion of the terms containing nonzero self-distances demonstrate that it is a proper generalization of a notion of an M v -metric space and consequently partial metric space as well.…”

“…Theorem 26 is an extension, improvement, and generalization of Banach [21] (α = β = δ = κ = 0), Kannan [22] (α = β = γ = 0, δ = κ), Chatterjee [25] (γ = δ = κ = 0, α = β), Reich [23] (α = β = 0), Hardy and Rogers [24] (s = 1), Asim et al [10] (α = β = δ = κ = 0), and references therein to an M b v -metric space.…”

“…Hausdorff [2] reexamined it as a metric space in the setting of points which has been refined, discussed, and generalized in numerous ways. Bakhtin [3], Branciari [4], Asadi et al [5], George et al [6], Mitrović and Radenović [7], Özgür et al [8], Karahan and Isik [9], and Asim et al [10] introduced the notions of a b-metric, a rectangular metric and a v-generalized metric, an M-metric, a rectangular b-metric, a b v ðsÞ-metric, a rectangular M-metric, a generalized p b v -partial metric, and an M v -metric, respectively. Further, one may also refer to Fernandez et al [11] and Kanwal et al [12,13] for work in N b -cone metric spaces over Banach algebra, orthogonal F -metric spaces, and weak partial b-metric spaces, respec-tively.…”

Some New Aspects of Metric Fixed Point Theory

“…In 2014, Asadi et al [11] extended the concept of partial metric space and introduced the notion of m-metric spaces to investigate fixed point. This concept was extended in many different ways, such as m b -metric space [12], m v -metric space [13,14], and rectangular m-metric space [15]. Also in this sequel, Patle et al [16] extended the notion of m-metric by proving fixed point results for Nadler and Kannan type set valued mappings in m-metric spaces.…”