A general fixed point theorem for multi-valued mapping in uniform space
Let \((X,\mathcal{U})\) be a complete Hausdorff uniform space and let the uniformity \(\mathcal{U}\) be generated by a family \((d_i)_{i\in I}\) of pseudometrics. Denote the Hausdorff metric corresponding to \(d_i\) by \(H_i\). For \(i\in I\) and \(r>0\) define \(V^*(i,r)\) as the set of all pairs of subsets \((A,B)\) of \(X\) such that \(H_i(A,B)<r\) and call \(\mathcal{U}^*\) the uniformity on the space \(2^X\) of nonempty bounded subsets of \(X\) generated by the family \(\{V^*(i,r)|\;i\in I, r>0\}\). Let \(F\) be a continuous map from \(X\) to \(2^X\) such that each \(F(x)\) is compact. Assume that there is a nondecreasing function \(K\) from \([0,\infty)\) to itself such that \(K(0)=0\) and \(K(t)<t\) whenever \(t>0\) and that \(H_i(Fx,Fy)\leq K(\max\{d_i(x,y),d_i(x,Fx),d_i(y,Fy),d_i(x,Fy),d_i(y,Fx)\})\) for all \(i\in I\) and \(x,y\in X\). Finally, assume that there is an \(x_0\) such that \(\sum_{n=1}^\infty K^n(d_i(x_0,Fx_0))<\infty\). The authors prove that in this situation \(F\) has a fixed point. \{There is a major typo in this result: The authors state the convergence of the infinite sum as the conclusion of the theorem, and do not mention the fixed point.\}
scientific article; zbMATH DE number 5541238
scientific article