# zbMATH — the first resource for mathematics

Fixed points of asymptotically regular multivalued mappings. (English) Zbl 0765.54036
In Section 3 the authors prove a common fixed point theorem for two multivalued maps satisfying a generalized contractive condition. In Section 4 multivalued maps of Kannan type are considered. Section 5 deals with coincidence points of a multivalued map $$T$$ and of a single-valued map $$f$$ defined on a metric space $$(X,d)$$ which are compatible in the following sense: let $$Tx$$ be a nonempty bounded closed subset of $$X$$ for any $$x$$ in $$X$$, $$T$$ and $$f$$ are compatible if whenever there exists a sequence $$\{x_ n\}$$ satisfying $$\lim fx_ n\in\lim Tx_ n$$ (if these exist), then $$H(fTx_ n,Tfx_ n)\to 0$$, where $$H$$ is the Hausdorff metric. The authors prove in Theorem 5.1 that if $$X$$ is complete, $$f$$ is continuous, $$T(X)\subseteq f(X)$$ and $$H(Tx,Ty)<\phi(d(x,y))\cdot d(fx,fy)$$ for all $$x$$, $$y$$ in $$X$$, where $$\phi: (0,+\infty)\to [0,1)$$ is a real function having suitable properties, then there exists a sequence $$\{x_ n\}$$ asymptotically $$T$$-regular with respect to $$f$$, i.e. $$d(fx_ n,Tx_ n)\to 0$$, and $$\{fx_ n\}$$ converges to a coincidence point of $$T$$ and $$f$$. Related results can be found in the papers of, e.g., G. Jungck [Int. J. Math. Math. Sci. 9, 771-779 (1986; Zbl 0613.54029)], H. Kaneko [Kobe J. Math. 3, 37-45 (1986; Zbl 0611.47044)], B. E. Rhoades, M. S. Khan and the reviewer [Int. J. Math. Math. Sci. 11, 375-392 (1988; Zbl 0669.54023)].
Reviewer: S.Sessa (Napoli)

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 54C60 Set-valued maps in general topology