The aim of our paper is to present a new class of functions and to define some new contractive mappings in b-metric spaces. We establish some fixed point results for these new contractive mappings in b-metric spaces. Furthermore, we extend our main result in the framework of b-metric-like spaces. Some consequences of main results are also deduced. We present some examples to illustrate and support our results. We provide an application to solve simultaneous linear equations. In addition, we present some open problems.

In the present article, the notion of αH-ψH-multivalued contractive type mappings is introduced and some fixed point results in complete metric spaces are studied. These theorems generalize Nadler’s (Multivalued contraction mappings, Pac. J. Math., 30, 475–488, 1969) and Suzuki-Kikkawa's (Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal., 69, 2942–2949, 2008) results that exist in the literature. The effectiveness of the obtained results has been verified with the help of some comparative examples. Moreover, a homotopy result has been presented as an application.

The idea of b -metric was proposed from the works of Bourbaki and Bakhtin. Czerwik gave an axiom which was weaker than the triangular inequality and formally defined bmetric spaces with a view of generalizing the Banach contraction mapping theorem. Further, in 2006, Mustafa and Sims have introduced an alternative more robust generalization of metric spaces to overcome fundamental flaws in B.C. Dhage s theory of generalized metric spaces and named it as G -metric spaces. In this paper, inspired by the concept of b -metric spaces and G -metric spaces, a new generalization of G -metric spaces (named as G b -metric spaces ) are introduced that recovers a large class of topological spaces including standard metric spaces, b -metric spaces, G -metric spaces etc. In such spaces, a new version of known fixed point theorems in b -metric spaces as well as in G -metric spaces have been proved. As an application of our result, we establish an existence and uniqueness result for system of linear equations in G b -complete metric spaces.Mathematics subject classification (2010): 47H10, 54H25.