Reich-Type and (α, F)-Contractions in Partially Ordered Double-Controlled Metric-Type Spaces with Applications to Non-Linear Fractional Differential Equations and Monotonic Iterative Method

Abstract

In this manuscript, we defined contractions in the context of double-controlled metric spaces and partially ordered double-controlled metric spaces. We established new fixed-point results and defined the notion of double-controlled metric space on a Reich-type contraction. Our findings are generalizations of a few well-known findings in the literature. Some non-trivial examples and certain consequences are also provided to illustrate the significance of the presented results. The existence and uniqueness of the solution of non-linear fractional differential equations and the monotone iterative method are also determined using the fixed-point method.

Fixed-Point Results of Generalized (ϕ,Ψ)-Contractive Mappings in Partially Ordered Controlled Metric Spaces with an Application to a System of Integral Equations

“…where ϕ ∈ Φ, ϕ ∈ Ψ, φ : [0, +∞) −→ [0, +∞) is continuous, and φ(t) < t ∀ t > 0 and φ(t) = 0 iff t = 0. As a result, we generalize and expand the findings of the study by [12][13][14][15][16] and several other comparable results.…”

“…Later, in 2010, Amini-Harandi and Emami [11] investigated the existence and uniqueness of solutions for periodic and boundary-value problems using partially ordered complete metric spaces and the Banach contraction principle (BCP), showcasing the applicability of the FP theory in addressing real-world problems in various domains. In 2022, Farhan et al [12] discussed Reich-type and (α, 𭟋)-contractions in partially ordered double controlled metric-type spaces (PODCMSs), illuminating the solution of nonlinear fractional differential equations through a monotonic iterative approach.…”

Common Fixed Point Results on a Double-Controlled Metric Space for Generalized Rational-Type Contractions with Application

“…The given results are improved and generalized to the existing ones in [11,18,20]. These results can be generalized by utilizing the notions in [19,[21][22][23][24][25][26].…”

“…Lateef [18] proved Fisher type fixed point results in controlled metric spaces. Farhan et al [19] proved numerous fixed point results for (α, F)-contraction and Reich-type contraction in the setting of DCMSs and partially ordered DCMSs. The authors in [20][21][22][23][24] generalized the notion of DCMSs by utilizing intuitionistic fuzzy sets and neutrosophic sets, and proved fixed point theorems with several applications.…”