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Zbl 1206.54064
Sintunavarat, Wutiphol; Kumam, Poom
Weak condition for generalized multi-valued $(f,\alpha ,\beta )$-weak contraction mappings. (English)
[J] Appl. Math. Lett. 24, No. 4, 460-465 (2011). ISSN 0893-9659

Summary: In 2007, {\it 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 $(\alpha ,L)$-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 [loc. cit.]. 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.

MSC 2000:: *54H25 Fixed-point theorems in topological spaces
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: generalized multi-valued $f$-weak contraction; generalized multi-valued $(f, \alpha , \beta )$-weak contraction; $T$-weakly commuting; coincidence points; common fixed points

Citations: Zbl 1128.54024

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