Abstract
The notion of complex valued metric spaces proved the common fixed point theorem that satisfies rational mapping of contraction. In the contraction mapping theory, several researchers demonstrated many fixed-point theorems, common fixed-point theorems and coupled fixed-point theorems by using complex valued metric spaces. The idea of b-metric spaces proved the fixed point theorem by the principle of contraction mapping. The notion of complex valued b-metric spaces, and this metric space was the generalization of complex valued metric spaces. They explained the fixed point theorem by using the rational contraction. In the metric spaces, we refer to this metric space as a quasi-metric space, the symmetric condition d(x, y) = d(y, x) is ignored. Metric space is a special kind of space that is quasi-metric. The Quasi metric spaces were discussed by many researchers. Banach introduced the theory of contraction mapping and proved the theorem of fixed points in metric spaces. We are now introducing the new notion of complex quasi b-metric spaces involving rational type contraction which proved the unique fixed point theorems with continuous as well as non-continuous functions. Illustrate this with example.
Highlights
The complex value metric spaces were first introduced by Azam, et al [1] and rational contraction mappings were used to prove the theorems
Uma Maheswari and A Anbarasan [8] were involved in generalizing the contraction mapping principle to prove very good results in the theory of fixed points
First we prove the unique fixed point theorem using complex quasi metric spaces
Summary
The complex value metric spaces were first introduced by Azam, et al [1] and rational contraction mappings were used to prove the theorems. In complex valued b-metric spaces, several fixed point theorems have been demonstrated by [11] A.k Dubey, Rita Shukla and R.P. Dubey. A new definition of complex valued quasi b-metric spaces satisfying rational form contraction is interested in proving fixed point theorems. A non- void set L is a function pcqb : L × L → C is known as complex valued quasi b metric spaces if for every a, b, e ∈ L :. A non-void set L is a mapping pcqb : L × L → C is known as complex valued quasi metric space and as well as complex valued quasi b-metric spaces. ( ) 3) Let L, pcqb is said to be complete if for all Cauchy sequence {bm} of L which converges to an element of L
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