Approximation and fixed-point theorems for condensing composites of multifunctions
Let \(B\) be a non-empty bounded subset of a Banach space \(X\) and \(\gamma(B)\) be the Kuratowski measure of noncompactness. Let \(S\) be a non-empty subset of \(X\) and \(f\) be a continuous map from \(S\) into \(X\). If for every non-empty bounded subset \(B\) of \(S\) with \(\gamma(B)> 0\), we have \(\gamma(f(B))< \gamma(B)\), then \(f\) is called condensing. The inward set of \(S\) at \(x\in S\) is defined by \(I_S= \{x+ c(z- x)\mid z\in S, c>0\}\). Let \(F: X\to Y\) be a multifunction. We denote \(F\in \mathbb{U}^K_C(X, Y)\), if for any compact subset \(K\) of \(X\), there is a multifunction \(\widetilde F: K\to Y\), u.s.c. and compact-valued such that \(\widetilde F(x)\subset F(x)\) for each \(x\in X\). In [Can. Math. Bull. 22, 513-515 (1979; Zbl 0429.47019)], \textit{T. C. Lin} proved a Fan's theorem for a continuous condensing map defined on a closed ball in a Banach space or on a closed convex subset of a Hilbert space. In the following theorems of this paper, the authors extend Lin's results to multifunctions. Theorem. Let \(B\) be a closed ball with center at origin and radius \(r\) in a Banach space \(X\). If \(F\in \mathbb{U}^K_C(B, X)\) is condensing, then there exist \(x_0\in B\) and \(y_0\in F(x_0)\) such that \(\| x_0- y_0\|= d(y_0, B)= d(y_0,\text{cl }I_B(x_0))\). More precisely, either \(F\) has a fixed point \(x_0\in B\); or there exist \(x_0\in \text{bdy }B\) and \(y_0\in F(x_0)\) such that \[ 0<\| x_0- y_0\|= d(y_0,B)= d(y_0, \text{cl }I_B(x_0)). \] Theorem. Let \(S\) be a closed convex subset in a Hilbert space \(X\). If \(F\in \mathbb{U}^K_C(S, X)\) is condensing, and \(F(S)\) is bounded, then there exist \(x_0\in S\) and \(y_0\in F(x_0)\) such that \(\| x- x_0\|= d(y_0, S)= d(y_0, \text{cl }I_S(x_0))\). More precisely, either \(F\) has a fixed point \(x_0\in S\), or there exist \(x_0\in \text{bdy }S\) and \(y_0\in F(x_0)\) such that \[ 0<\| x_0- y_0\|= d(y_0, S)= d(y_0,\text{cl }I_S(x_0)). \] {}.
scientific article; zbMATH DE number 1215657
scientific article