(C , Ψ * , G ) Class of Contractions and Fixed Points in a Metric Space Endowed with a Graph

Abstract

In this paper, we introduce the ( C , Ψ * , G ) class of contraction mappings using C-class functions and some improved control functions for a pair of set valued mappings as well as a pair of single-valued mappings, and prove common fixed point theorems for such mappings in a metric space endowed with a graph. Our results unify and generalize many important fixed point results existing in literature. As an application of our main result, we have derived fixed point theorems for a pair of α -admissible set valued mappings in a metric space.

Some Results on (s − q)-Graphic Contraction Mappings in b-Metric-Like Spaces

“…For more details on other generalized metric spaces see [8][9][10][11][12][13][14]. The next proposition helps us to construct some more examples of b-metric (respectively partial b-metric, b-metric-like) spaces.…”

Some Generalized Contraction Classes and Common Fixed Points in b-Metric Space Endowed with a Graph

“…Khan et al [1] introduced the altering distance function ψ which is a control function where ψ : [0, ∞) → [0, ∞) is continuous, non decreasing and satisfies the condition ψ(t) = 0 if and only if t = 0. Recently in [2], the authors considered the condition ∑ ∞ n=1 ψ(t) < +∞ which in turn implies ψ(t) < t and ψ(0) = 0 where as George et al [3] considered the control function ψ which is continuous, non decreasing and satisfies the condition ψ(t) = 0 implies t = 0 which clearly implies that ψ(0) is not necessarily 0. Doric [4] considered contraction conditions involving two control functions for two single valued self mappings defined on a metric space.…”

“…Very recently the C-class functions were introduced in [19] which later proved to be a powerful tool in fixed point theory. In the present work using C-class functions and some modified versions of control functions as in George et al [3], we have introduced generalized classes of contractions and rational contractions for a pair of multivalued mappings and proved common fixed point theorems in a b-metric space endowed with a graph. In Section 3.1 we have introduced R-weakly graph preserving pair of mappings and R-weakly α-admissible pair of mappings which are proper extension and generalization of the class of graph preserving mappings, weak graph preserving mappings and α-admissible multivalued mappings of type S, α * -admissible multivalued mappings of type S and α * -orbital admissible multivalued mappings of type S respectively.…”