## 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)

### Keywords:

pair mappings; common coincidence point
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