Weak condition for generalized multi-valued (f,α,β)(f,α,β)-weak contraction mappings

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