Abstract
In this paper, we establish some new fixed point theorems for generalized ϕ–ψ-contractive mappings satisfying an admissibility-type condition in a Hausdorff rectangular metric space with the help of C-functions. In this process, we rectify the proof of Theorem 3.2 due to Budhia et al. [New fixed point results in rectangular metric space and application to fractional calculus, Tbil. Math. J., 10(1):91–104, 2017]. Some examples are given to illustrate the theorems. Finally, we apply our result (Corollary 3.6) to establish the existence of a solution for an initial value problem of a fractional-order functional differential equation with infinite delay.
Highlights
One of the fundamental results in the evolution of the field of fixed point theory is the Banach contraction principle [8]
Metric space in which the triangle inequality is replaced with a weaker assumption called quadrilateral inequality, and an analogue of the Banach contraction principle is proved
Rectangular metric spaces can lack the Hausdorffness separation, it is not useful for our theory as Hausdorffness separation plays an important role in Theorem 1 and its corollaries
Summary
One of the fundamental results in the evolution of the field of fixed point theory is the Banach contraction principle [8] It has been generalized and extended in various directions. In 2012, Samet et al [34] introduced the concept of α−ψ-contractive mapping, which is interesting since it does not require the contractive conditions to hold for every pair of points in the domain unlike Banach contraction principle. It includes the case of discontinuous mappings. Following the ideas from [11] and [2], we provide new fixed point results in generalized metric spaces, which are utilized to establish the existence of a solution for an initial value problem of a fractional-order functional differential equation with infinite delay
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