Weak condition for generalized multi-valued $(f,\alpha ,\beta )$-weak contraction mappings

Abstract

In 2007, Kamran [T. Kamran, Multivalued $f$-weakly Picard mappings, Nonlinear Anal. 67 (2007) 2289–2296] extended the notion of multi-valued mapping from weak contraction and generalized $(\alpha ,L)$-weak contraction to $f$-weak contraction and generalized multi-valued $f$-weak contraction. He also obtained some common fixed point theorems with the notion of $T$-weakly commuting at a coincidence point of a hybrid pair. In this paper, we can drop the condition of $T$-weakly commuting in Theorems 2.9 and 3.5 in [5]. We further extend the notion of generalized multi-valued $f$-weak contraction and introduce the notion of generalized multi-valued $(f,\alpha ,\beta )$-weak contraction. We also establish some coincidence and common fixed point theorems with generalized multi-valued $(f,\alpha ,\beta )$-weak contraction mappings. Our results extend and generalize several common fixed point theorems of many authors.