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One of the most intensively studied and generalized results in metric fixed point theory is Branciari’s fixed point theorem, which asserts that in a complete metric space ( M , ρ ) , a mapping T : M → M satisfying ∫ 0 ρ ( T x , T y ) ω ( t ) d t ≤ c ∫ 0 ρ ( x , y ) ω ( t ) d t for some ω : [ 0 , ∞ ) → [ 0 , ∞ ) , c ∈ ( 0 , 1 ) , and all x , y ∈ M , admits a unique fixed point x ∗ ∈ M . This reduces to Banach fixed theorem when ω ( t ) = 1 . We extend this to Riemann–Liouville fractional integral contractions, proving the existence of a fixed point for T under 1 Γ ( α ) ∫ 0 ρ ( T x , T y ) ( ρ ( T x , T y ) − t ) α − 1 φ ( t ) d t ≤ c ⋅ 1 Γ ( α ) ∫ 0 ρ ( x , y ) ( ρ ( x , y ) − t ) α − 1 φ ( t ) d t , which recovers Branciari’s integral condition for α = 1 .

In this article, we analyze the boundedness for the fractional bilinear Hardy operators on variable exponent weighted Morrey–Herz spaces M K ˙ q , p ( ⋅ ) α ( ⋅ ) , λ ( w ) . Similar estimates are obtained for their commutators when the symbol functions belong to BMO space with variable exponents.

In the present article, we have introduced the notions of γ-admissibility for the pair of q-ROF set-valued maps and admissible hybrid q-ROF Z-contraction. Notions introduced in the article generalizes the existing concepts in fuzzy literature. Common fixed point result for a pair of γ-admissible q-ROF mappings in b-metric spaces utilizing the introduced contraction is presented. A nontrivial example to support the obtained results is also included. As an application, we have discussed the existence of solution of system of non-linear n-th order differential inclusions with non-local and integral boundary conditions.

This paper introduces the concept of ( P , ψ ) -type almost contractive conditions for fuzzy mappings in b -metric spaces. This novel framework is employed to establish certain fuzzy fixed point results in complete b -metric spaces. An illustrative example is provided to validate the assumptions of the main theorem, ensuring the existence of fuzzy fixed points. Furthermore, the existence of a solution to a second-order nonlinear boundary value problem is demonstrated by transforming the problem into a fixed point equation and applying the proven results. Several corollaries are derived as consequences of the main findings. The results presented in this work extend and generalize numerous existing fixed point theorems in the literature.

One of the well-studied generalizations of a metric space is known as a partial metric space. The partial metric space was further generalized to the so-called M-metric space. In this paper, we introduce the Double-Controlled Quasi M-metric space as a new generalization of the M-metric space. In our new generalization of the M-metric space, the symmetry condition is not necessarily satisfied and the triangle inequality is controlled by two binary functions. We establish some fixed point results, along with the examples and applications to illustrate our results.

The double-controlled metric-type space (X,D) is a metric space in which the triangle inequality has the form D(η,μ)≤ζ1(η,θ)D(η,θ)+ζ2(θ,μ)D(θ,μ) for all η,θ,μ∈X. The maps ζ1,ζ2:X×X→[1,∞) are called control functions. In this paper, we introduce a novel generalization of a metric space called a double-composed metric space, where the triangle inequality has the form D(η,μ)≤αD(η,θ)+βD(θ,μ) for all η,θ,μ∈X. In our new space, the control functions α,β:[0,∞)→[0,∞) are composed of the metric D in the triangle inequality, where the control functions ζ1,ζ2:X×X→[1,∞) in a double-controlled metric-type space are multiplied with the metric D. We establish some fixed-point theorems along with the examples and applications.

Recently, Ji et al. established certain fixed-point results using Mann’s iterative scheme tailored to Gb-metric spaces. Stimulated by the notion of the F-contraction introduced by Wardoski, the contraction condition of Ji et al. was generalized in this research. Several fixed-point results with Mann’s iterative scheme endowed with F-contractions in Gb-metric spaces were proven. One non-trivial example was elaborated to support the main theorem. Moreover, for application purposes, the existence of the solution to an integral equation is provided by using the axioms of the proven result. The obtained results are generalizations of several existing results in the literature. Furthermore, the results of Ji. et al. are the special case of theorems provided in the present research.

One of the well-studied generalizations of a metric space is known as a partial metric space. The partial metric space was further generalized to the so-called M-metric space. In this paper, we introduce the Double-Controlled Quasi M-metric space as a new generalization of the M-metric space. In our new generalization of the M-metric space, the symmetry condition is not necessarily satisfied and the triangle inequality is controlled by two binary functions. We establish some fixed point results, along with the examples and applications to illustrate our results.

Recently, Ji et al. established certain fixed-point results using Mann’s iterative scheme tailored to G[sub.b] -metric spaces. Stimulated by the notion of the F-contraction introduced by Wardoski, the contraction condition of Ji et al. was generalized in this research. Several fixed-point results with Mann’s iterative scheme endowed with F-contractions in G[sub.b] -metric spaces were proven. One non-trivial example was elaborated to support the main theorem. Moreover, for application purposes, the existence of the solution to an integral equation is provided by using the axioms of the proven result. The obtained results are generalizations of several existing results in the literature. Furthermore, the results of Ji. et al. are the special case of theorems provided in the present research.

Recently, Ji et al. established certain fixed-point results using Mann’s iterative scheme tailored to G b -metric spaces. Stimulated by the notion of the F -contraction introduced by Wardoski, the contraction condition of Ji et al. was generalized in this research. Several fixed-point results with Mann’s iterative scheme endowed with F -contractions in G b -metric spaces were proven. One non-trivial example was elaborated to support the main theorem. Moreover, for application purposes, the existence of the solution to an integral equation is provided by using the axioms of the proven result. The obtained results are generalizations of several existing results in the literature. Furthermore, the results of Ji. et al. are the special case of theorems provided in the present research.