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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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