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Fixed point theorems for a broad class of multimaps. (English) Zbl 1029.54049
We say that $$\varphi: \mathbb{R}^+\to [0,1)$$ has property $$A$$ if there exists an $$M\in (0,\infty)$$ such that $$\varphi$$ is monotonically increasing on $$[M,\infty)$$, $$\int^\infty_M(1-\varphi(t)) dt=+\infty$$ and $$\sup\{\varphi(t): r\leq t\leq M\}< 1$$, $$\forall r\in (0,M)$$, $$\varphi$$ has property $$B$$ if $$\sup\{\varphi(t): r\leq t<\infty\}<1$$, $$\forall r\in (0,\infty)$$. Let $$(X,d)$$ be a metric space, $$N(X)$$ the collection of all nonempty subsets of $$X$$ and $$F: X\to N(X)$$. We say that $$\varphi$$ has property $$C$$ w.r.t. $$(F,x_0)$$ if it is monotonically increasing in $$\mathbb{R}^+$$ and $$\int^\infty_0 (1-\varphi(t)) dt> d(x_0, Fx_0)$$.
The main result of this paper is the following theorem: Let $$F: X\to C(X)$$, where $$C(X)$$ is the collection of all nonempty closed subsets of $$X$$. Suppose that $$(X,d)$$ is $$F$$-orbitally complete, the function $$f$$ defined on $$X$$ as $$f(x)= d(x,Fx)$$ is $$F$$-orbitally semicontinuous at any cluster point of any orbit of $$F$$ w.r.t. $$x_0$$, the function $$\varphi$$ has property $$A$$ or $$B$$ or property $$C$$ w.r.t. $$(F,x_0)$$ and that $$d(y, Fy)\leq \varphi(d(x_0, x)) d(x,y)$$ whenever $$x\in 0(F,x_0)$$, $$y\in Fx$$ and $$x\not\in Fx$$. Then there exists an orbit $$\{x_n\}^\infty_0$$ of $$F$$ w.r.t. $$x_0$$ which converges to a fixed point of $$F$$.
This theorem is a proper generalization of Theorem 2.1 of [C.-K. Zhong, J. Zhu and P.-H. Zhao, Proc. Am. Math. Soc. 128, No. 8, 2439-2444 (2000; Zbl 0948.47058)].
Reviewer: V.Popa (Bacau)

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems 54C60 Set-valued maps in general topology 54H10 Topological representations of algebraic systems
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##### References:
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