Fixed-Point Theorems for Rational Interpolative-Type Operators with Applications

Abstract

In the current manuscript, two fixed-point theorems for Dass-Gupta and Gupta-Saxena rational interpolative-type operators are studied in the setting of metric spaces. For the authenticity of the presented work, examples and applications to the existence of a solution to the Caputo-Fabrizio fractional derivative and Caputo-Fabrizio fractal-fractional derivative are also discussed.

Common fixed point theorems for interpolative rational-type mapping in complex-valued metric space

“…An application for an R-L-C differential equation is provided. The results obtained will be able to generalize several works from the literature such as [4,31,[41][42][43][44] and several other.…”

“…Definition 2.6. [31] Suppose (X, d) be a metric space. Furthermore, consider a continuous operator Υ : X → X.…”

“…Theorem 2.3. [31] Let (X, d) be a complete metric space and Υ be an interpolative rationaltype contraction. Then Υ possesses a fixed point.…”

Interpolative Prešić Type Contractions and Related Results

“…for all k, l ∈ L with k ≠ Nk and l ≠ Nl, where ζ ∈ [0, 1) and ϑ ∈ (0, 1). After that, many existing contraction-type conditions have been generalized in the sense of interpolative Kannan contraction, for example, Karapınar et al [9] studied interpolative Reich-Rus-Ćirić type contraction in partial metric spaces, Aydi et al [10] studied interpolative Ćirić-Reich-Rus type contractions in Branciari metric spaces, Mohammadi et al [11] extended the concept of F-contractions by interpolative Ćirić-Reich-Rus type F-contractions, Karapınar et al [12] studied interpolative Hardy-Rogers type contractions, Debnath and Sen [13] studied set-valued interpolative Hardy-Rogers and set-valued Reich-Rus-Ćirić-type contractions, Sarwar et al [14] presented rational type interpolative contractions, Khan et al [15] worked on interpolative (ϕ, ψ)-type Z-contractions, Altun and Tasdemir [16] presented interpolative proximal contractions for nonself mappings, Fulga and Yesilkaya [17] studied interpolative Suzuki-type contractions, Karapınar et al [18] defined (α, β, ψ, ϕ)-interpolative contractions, and Alansari and Ali [19] studied multivalued interpolative Reich-Rus-Ćirić-type contractions. Gaba and Karapınar [20] extended the notion of interpolative Kannan contraction through exponential powers, stated as, a map N:…”

Existence of Positive Solutions for a Higher-Order Fractional Differential Equation with Multi-Term Lower-Order Derivatives

“…Due to the inherent difficulties in the fractional calculus, to the best of our knowledge, if only the left (or right) Riemann-Liouville fractional derivatives are involved, the most feasible approach to study the existence of solutions of a boundary value problem is to convert it into a fixed point problem for an appropriate operator. This idea has been widely used by many researchers, for a small sample of such work, as can be seen in [25][26][27][28][29][30][31] and the references therein for more comments and citations.…”