Abstract
The main objective of this paper is to introduce the ( α , β )-type ϑ -contraction, ( α , β )-type rational ϑ -contraction, and cyclic ( α - ϑ ) contraction. Based on these definitions we prove fixed point theorems in the complete metric spaces. These results extend and improve some known results in the literature. As an application of the proved fixed point Theorems, we study the existence of solutions of an integral boundary value problem for scalar nonlinear Caputo fractional differential equations with a fractional order in (1,2).
Highlights
Fixed point theorems are useful tools in nonlinear analysis, the theory of differential equations, and many other related areas of mathematics
In this paper, based on the proved fixed point theorems, we provide some new sufficient conditions for the existence of the solutions of an integral boundary value problem for a scalar nonlinear Caputo fractional differential equations with fractional order in
We introduced two new types of contractions: (α,β)-type θ-contraction and (α,β)-type rational θ-contraction
Summary
Fixed point theorems are useful tools in nonlinear analysis, the theory of differential equations, and many other related areas of mathematics. One of the most applicable method for various investigations is Banach’s contraction principle [1]. Many researchers generalized and extended this theorem to different directions. Samet et al in [3] defined α-admissible and α-ψ-contractive type mappings and studied some of their properties in the framework of complete metric spaces. Salimi et al in [4] introduced and investigated the twisted (α,β)-admissible mappings. Many extensions of the notion of α-ψ-contractive type mappings have been developed, see, for example, [5,6,7,8,9] and the references therein
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