Multi-valued fixed point theorems by altering distance between the points.

*(English)*Zbl 1027.47060Fixed point results involving altering distances were introduced by M. S. Khan, M. Swaleh and S. Sessa [Bull. Aust. Math. Soc. 30, 1-9 (1984; Zbl 0553.54023)]. There exists a vast literature which generalizes the results of this paper. An altering distance is a mapping \(\phi: [0,\infty)\to [0,\infty)\) which satisfies: (a) \(\phi\) is increasing continuous and (b) \(\phi(t)= 0\) if and only if \(t= 0\). In the present paper, the authors extend the results from [M. El Amrani and A. B. Mbarki, Southwest J. Pure Appl. Math. 1, 16-21 (2000; Zbl 0966.47038)] and M. A. Krasnoselski and P. P. Zabreiko [“Geometrical methods of nonlinear analysis”, Grundlehren 263, Springer-Verlag, Berlin (1984; Zbl 0546.47030), p. 206] to multivalued maps.

Let \((X,d)\) be a metric space. We denote by \(CB(X)\) the set of all nonempty closed bounded subsets of \((X,d)\) and by \(H\) the Hausdorff-Pompeiu metric on \(CB(X)\), i.e., \[ H(A,B)= \max\{\sup_{x\in A} d(x,B), \sup_{x\in B} d(x,A)\}, \] where \(A,B\in CB(X)\) and \(d(x,A)= \inf_{y\in A} \{d(x,y)\}\). We define also \(\delta(A,B)\) by \(\delta(A,B)= \sup\{d(a, b): a\in A,b\in B\}\). Let \(F: X\to 2^X\) be a multi-valued map, then a point \(x\in X\) is called a fixed point of \(F\) if \(x\in Fx\); it is called absolutely fixed if \(Fx= \{x\}\).

The authors prove the following Theorem: Let \((X,d)\) be a complete metric space, \(T: X\to CB(X)\) a multi-valued map satisfying the following condition: \(\phi(\delta(Tx,Ty))\leq a\phi(dx,y))+ b[\phi(\delta(x, Tx))+ \phi(\delta(y, Ty))]+ c\min\{\phi(d(x, Ty)),\phi(d(y, Tx))\}\), \(\forall x,y\in X\), where \(\phi: \mathbb{R}^+\to \mathbb{R}^+\) is continuous and strictly increasing such that \(\phi(0)= 0\) and \(a\), \(b\), \(c\) are three positive constants such that \(a+ 2b< 1\) and \(a+ c< 1\). Then \(T\) has a unique absolutely fixed point.

A similar theorem for multi-valued maps with compact values is proved. In the last part, a generalization of Theorem 34.5 of Krasnoselski and Zabreiko [loc. cit.] is proved.

Let \((X,d)\) be a metric space. We denote by \(CB(X)\) the set of all nonempty closed bounded subsets of \((X,d)\) and by \(H\) the Hausdorff-Pompeiu metric on \(CB(X)\), i.e., \[ H(A,B)= \max\{\sup_{x\in A} d(x,B), \sup_{x\in B} d(x,A)\}, \] where \(A,B\in CB(X)\) and \(d(x,A)= \inf_{y\in A} \{d(x,y)\}\). We define also \(\delta(A,B)\) by \(\delta(A,B)= \sup\{d(a, b): a\in A,b\in B\}\). Let \(F: X\to 2^X\) be a multi-valued map, then a point \(x\in X\) is called a fixed point of \(F\) if \(x\in Fx\); it is called absolutely fixed if \(Fx= \{x\}\).

The authors prove the following Theorem: Let \((X,d)\) be a complete metric space, \(T: X\to CB(X)\) a multi-valued map satisfying the following condition: \(\phi(\delta(Tx,Ty))\leq a\phi(dx,y))+ b[\phi(\delta(x, Tx))+ \phi(\delta(y, Ty))]+ c\min\{\phi(d(x, Ty)),\phi(d(y, Tx))\}\), \(\forall x,y\in X\), where \(\phi: \mathbb{R}^+\to \mathbb{R}^+\) is continuous and strictly increasing such that \(\phi(0)= 0\) and \(a\), \(b\), \(c\) are three positive constants such that \(a+ 2b< 1\) and \(a+ c< 1\). Then \(T\) has a unique absolutely fixed point.

A similar theorem for multi-valued maps with compact values is proved. In the last part, a generalization of Theorem 34.5 of Krasnoselski and Zabreiko [loc. cit.] is proved.

Reviewer: Valeriu Popa (Bacau)