The Class of JS-Contractions in Branciari Metric Spaces

Banach contraction principleBanach contraction principle has been generalized in many ways over the years. In some generalizations, the contraction is weakened; see [3, 6, 12, 16, 20, 21, 24, 30] and others. In other generalizations, the topology is weakened; see [1, 4, 5, 8, 9, 11, 13, 14, 22, 23, 27–29] and others. In [18], Nadler extended Banach fixed point theorem from single-valued maps to set-valued maps. Other fixed point results for set-valued maps can be found in [2, 7, 15, 17, 19] and references therein. In 2000, Branciari [4] introduced the concept of generalized metric spaces, where the triangle inequality is replaced by the inequality $$d(x,y)\le d(x,u)+d(u,v)+d(v,y)$$ for all pairwise distinct points $$x,y,u,v\in X$$ . Various fixed point results were established on such spaces; see, e.g., [1, 8, 13, 14, 22, 23, 28] and references therein. In this chapter, we present a recent generalization of Banach contraction principleBanach contraction principle on the setting of Branciari metric spacesBranciari metric, which is due to Jleli and Samet [10].