WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 24, 2025
Fixed Point Theorem in MR-metric Spaces VIA Integral Type Contraction
Author:
Abstract: We present new fixed point theorems in MR-metric spaces using integral-type contractions. These
results build upon and extend previous research. The concept of an MR-metric space was originally introduced
by Malkawi, who also outlined the idea of sequence convergence in such spaces. Additionally, methods for
constructing MR-metrics from certain real-valued partial functions in three-dimensional Euclidean space were
proposed, along with a study of various convergence types in MR-metric spaces, analyzing the implications and
non-implications among them. In 2002, Branciari introduced a new generalization of the contractive condition
of the integral type. His work focused on the existence of fixed points for mappings defined over complete
metric spaces (X, d), subject to a broad integral-type contractive inequality. This condition is reminiscent of
the Banach-Caccioppoli criterion. Specifically, the study involves mappings f : X → X for which there exists a
constant c ∈ (0, 1) such that for any x, y ∈ X: $$\int_{0}^{d(f(x),f (y))}Ψ(t) dt ≤ c \int_{0}^{d(x,y)}Ψ(t) dt$$ Ψ(t) : [0,+∞) → [0,+∞] is a Lebesgue-integrable function. It is nonnegative, summable on every compact
subset of [0,+∞), and satisfies $$\int_{0}^{ϵ}$$ Ψ(t) dt > 0 for each ϵ > 0.
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Keywords: MR − metric space, M-Convergent, M-Cauchy, fixed point theorems, Integral Type Contraction
Pages: 295-299
DOI: 10.37394/23206.2025.24.28