The generalized Banach contraction conjecture in b-metric spaces

Abstract

Using some combinatorial arguments and, in particular, the pigeonhole principle, we prove that the generalized Banach contraction conjecture in b-metric spaces is true if every b-metric space (X, D, K) is a metric-type space and $$M \in (0,\frac{1}{K}) $$ . Within $$M \in (0,\frac{1}{K}) $$ , we also prove that the generalized Banach contraction conjecture in b-metric space (X, D, K) is true for the case $$J = 2$$ ; and for the case $$J = 3$$ if the map $$T: X \longrightarrow X$$ is continuous. These results are generalizations of corresponding results in Jachymski et al. (J Combin Theory Ser A 87:273–286, 1999), Jachymski and Stein (J Aust Math Soc 66:224–243, 1999) and Merryfield and Stein (J Math Anal Appl 273:112–120, 2002) to the setting of b-metric spaces.