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Weak condition for generalized multi-valued $\left(f,\alpha ,\beta \right)$-weak contraction mappings. (English) Zbl 1206.54064
Summary: In 2007, T. Kamran [Nonlinear Anal., Theory Methods Appl. 67, No. 7, A, 2289–2296 (2007; Zbl 1128.54024)] extended the notion of multi-valued mapping from weak contraction and generalized $\left(\alpha ,L\right)$-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 [loc. cit.]. We further extend the notion of generalized multi-valued $f$-weak contraction and introduce the notion of generalized multi-valued $\left(f,\alpha ,\beta \right)$-weak contraction. We also establish some coincidence and common fixed point theorems with generalized multi-valued $\left(f,\alpha ,\beta \right)$-weak contraction mappings. Our results extend and generalize several common fixed point theorems of many authors.
##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces