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Coincidence points of compatible multivalued mappings. (English) Zbl 0862.54039

Let \(CB(X)\) be the space of nonempty bounded closed subsets of a metric space \((X,d)\) with the Hausdorff metric. Mappings \(T:X\to CB(X)\), \(f:X\to X\) are said to be compatible if, for any sequence \(\{x_n\}\subset X\) satisfying \(\lim_{n\to\infty} fx_n\in \lim_{n\to\infty} Tx_n\) we have \(\lim_{n\to\infty} H(fTx_n,Tfx_n)=0\). A point \(z\) is called a coincidence point of \(f\) and \(T\) if and only if \(f(z)\in T(z)\). The main result in the paper is connected with coincidence points for pair mappings.
Theorem. Let \(X\) be a complete metric space. Let \(f,g: X\to X\) and \(S,T: X\to CB(X)\) be continuous mappings such that \(f\) is compatible with \(S\), and \(g\) is compatible with \(T\). Assume that \(SX\subseteq g(X)\), \(TX\subseteq f(X)\) and that for all \(x,y\in X\) \(H(Sx,Ty)\leq \lambda d(x,y)\), \(0<\lambda<1\). Then there is a common coincidence point for \(f\) and \(S\), as well as for \(g\) and \(T\).
In particular, if the following condition is satisfied: \[ \Omega\in Sz,\quad gz\in Tz \quad\text{ implies }\quad \lim_{n\to\infty} f^n z=\lim g^n z=t, \] then \(t\) is a common fixed point of \(f,g,S\) and \(T\).

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
47J05 Equations involving nonlinear operators (general)
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