Abstract
In this paper, we have established and proved fixed point theorems for the Boyd-Wong-type contraction in metric spaces. In particular, we have generalized the existing results for a pair of mappings that possess a fixed point but not continuous at the fixed point. We can apply this result for both continuous and discontinuous mappings. We have concluded our results by providing an illustrative example for each case and an application to the existence and uniqueness of a solution of nonlinear Volterra integral equations.
Highlights
Introduction and PreliminariesContinuity is an ideal property which is sometimes difficult to be fulfilled especially in some daily life applications
We transform different kinds of day to day real-world phenomena into threshold functions which satisfies our desirable continuity of the weaker form and a new type of contraction to provide a solution to some daily life applications
Pant and Pant [9] used ε − δ and k-continuity property to prove the above fixed point theorem for one self map
Summary
Continuity is an ideal property which is sometimes difficult to be fulfilled especially in some daily life applications. Most neural network systems like bar code scanning, speech recognition, and handwritten digit recognition lack the continuity property These neural network systems are some excellent prototypes for learning discontinuity phenomena. In 1969, Kannan [6] proved the following fixed point theorem for discontinuous mapping: Theorem 1 [6]. In the Kannan contractive condition, continuity of mapping T was not required for the existence of a fixed point. K-continuity of T does not imply continuity of Tn. Example 4 [9]. Xn, xmÞ exists and it is finite (iii) A metric space ðX, dÞ is complete if every Cauchy sequence fxng converges to a point x ∈ X such that dðx, lim n,m⟶+∞. If f is k-continuous or f k is continuous for some k ≥ 1 or f is orbitally continuous, f possesses a unique fixed point
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