Some fixed point results of generalized $$(\phi , \psi )$$-contractive mappings in ordered b-metric spaces

Abstract

Objectives The aim of this paper is to establish some fixed point, coincidence point and, coupled coincidence and coupled common fixed point results for generalized $$(\phi , \psi )$$ ( ϕ , ψ ) -contractive mappings in partially ordered b-metric spaces. Our results generalize, extend and unify most of the fundamental metrical fixed point theorems in the existing literature. Few examples are illustrated to justify our results. Result The existence and uniqueness theorems for a fixed point and coincidence point, coupled coincidence point and coupled common fixed points for two mappings satisfying generalized $$(\phi , \psi )$$ ( ϕ , ψ ) -contractive conditions in complete partially ordered b-metric spaces are proved. These results generalize several comparable results in the existing literature.

Fixed point results for weak contractions in partially ordered b-metric space

“…We demonstrate some fixed points results for mappings in ordered b -metric space that satisfy a generalized weak contraction in this paper. The results from [ 10 , 11 , 19 , 22 , 33 ] are expanded here as well as some examples noted to support the findings at the end of our work.…”

“…[ 10 , 15 ] In a b -metric space, if as then is said to be convergent to . if as then is a Cauchy sequence.…”

“…A b -metric space is one of several generalizations of a standard metric space proposed by Bakhtin in his work [ 9 ], and widely used by Czerwik in his work [ 14 , 15 ]. Following that, a lot of progress was made in acquiring the results of fixed points to single valued as well as multi-valued operators in the space, as evidenced by [ 1 , 2 , 4 – 8 , 10 , 13 , 16 – 18 , 20 , 21 , 23 , 24 , 26 , 27 , 31 , 38 – 41 ].…”

Generalized fixed point results of rational type contractions in partially ordered metric spaces

“…rkA lot of work on the results of fixed points, common fixed points, coupled fixed points in partly ordered metric spaces with different topological properties involved can be found from [ 29 – 41 ]. Some generalized fixed points results of monotone mappings in partially ordered b -metric spaces have been investigated by Seshagiri Rao et al [ 43 , 46 , 47 ], Kalyani et al [ 42 , 45 , 48 ] and Belay Mitiku et al [ 44 ]. Acar [ 49 ] explored some fixed point results of F -contraction for multivalued integral type mapping on a complete metric space.…”

Common fixed points for $$(\psi -\varphi )$$-weak contractions type in b-metric spaces

“…We would like to point out that many authors have extended and generalized the notion of (ψ − ϕ)-weak contraction mappings and analyzed the existence of (common) fixed points for this class of mappings in different metric-like spaces. To mention some, we recommend the recent works [21,32,42].…”