Fixed points of asymptotically regular multivalued mappings. (English) Zbl 0765.54036

Beg, Ismat; Azam, Akbar

J. Aust. Math. Soc., Ser. A 53, No. 3, 313-326 (1992).

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

Keywords:

common fixed point; multivalued maps of Kannan type; coincidence points

Citations:

Zbl 0613.54029; Zbl 0611.47044; Zbl 0669.54023