Some fixed point theorems on dislocated $b$-metric and dislocated quasi $b$-metric spaces -by P.G. Golhare and C. T. Aage

# Some fixed point theorems on dislocated $b$-metric and dislocated quasi $b$-metric spaces-by P.G. Golhare and C. T. Aage

Abstract: A. Branciari in his paper entitled(\cite{ref13}) \lq\lq A fixed point theorem for mappings satisfying a general contractive condition of integral type\rq\rq ~ has proved following theorem:
Let $(X,d)$ be a complete metric space, $c\in ]0,1[,$ and let $f:X\to X$ be a mapping such that for each $x,y\in X,$ $\int_{0}^{d(fx,fy)}\varphi(t)dt\leq c\int_{0}^{d(x,y)}\varphi(t)dt,$ where $\varphi:[0,+\infty)\to [0,+\infty]$ is Lebesgue-integrable mapping which is summable (i.e. with finite integral) on each compact subset of $[0,\infty[,$ non-negative and such that for each $\epsilon>0,\int_{0}^{\epsilon}\varphi(t)>0;$ then $f$ has a unique fixed point $a\in X$ such that for each $x\in X,\lim_{n\to\infty}f^n(x)=a.$
We extend this theorem to the setup of dislocated quasi $b$-metric spaces. In this continuation, we establish a well known fixed point theorems in dislocated quasi $b$-metric spaces.

Keywords: complete dqb-metric space; contraction mapping; self-mapping; Cauchy sequence; fixed point.

P.G. Golhare and C. T. Aage, Some fixed point theorems on dislocated $b$-metric and dislocated quasi $b$-metric spaces, International Journal of Advances in Mathematics, Volume 2019, Number 1, Pages 27-43, 2019.