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… Click to show full 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 $$M \in (0,\frac{1}{K}) $$M∈(0,1K). Within $$M \in (0,\frac{1}{K}) $$M∈(0,1K), we also prove that the generalized Banach contraction conjecture in b-metric space (X, D, K) is true for the case $$J = 2$$J=2; and for the case $$J = 3$$J=3 if the map $$T: X \longrightarrow X$$T:X⟶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.
               
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