Abstract
The aim of this study is to investigate the existence of solutions for a non-linear neutral differential equation with an unbounded delay. To achieve our goals, we take advantage of fixed point theorems for self-mappings satisfying a generalized ( α , φ ) rational contraction, as well as cyclic contractions in the context of F -metric spaces. We also supply an example to support the new theorem.
Highlights
The concept of a metric space was initiated by Frechet [1] in the following way: A metric on a non-empty set S is a mapping d : S × S → [0, +∞) satisfying the following properties: (i) d(u, v) = 0 ⇐⇒ u = v, (ii) d(u, v) = d(v, u), and (iii) d(u, w) ≤ d(u, v) + d(v, w), for all u, v, w ∈S
Jleli et al [5] introduced an attractive generalization of a metric space, as follows
In 2012, Samet et al [8] initiated the notions of α-admissible mappings and (α,φ) contractive mappings and proved various fixed point theorems for such mappings
Summary
The concept of a metric space was initiated by Frechet [1] in the following way: A metric on a non-empty set S is a mapping d : S × S → [0, +∞) satisfying the following properties:. (ii) d(u, v) = d(v, u), and (iii) d(u, w) ≤ d(u, v) + d(v, w), for all u, v, w ∈S. The pair (S , d) is called a metric space. Many interesting generalizations (or extensions) of the metric space have recently appeared. Initiated the notions of b-metric spaces, generalized metric spaces, and partial metric spaces resepctively. Jleli et al [5] introduced an attractive generalization of a metric space, as follows. Suppose that F is a set of functions f : (0, +∞) → R satisfying the assertions: Czerwik [2], Branciari [3], and Matthews [4]
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