A Suzuki type fixed point theorem for a generalized multivalued mapping on partial Hausdorff metric spaces
Let \((X,p)\) be a complete partial metric space. Denote by \(CB^p(X)\) the class of all nonempty closed bounded subsets of \((X,p)\). For the nonempty subsets \(A,B\) in \(X\), set \(H_p(A,B)=\max\{\delta_p(A,B), \delta_p(B,A)\}\), where \(\delta_p(A,B)=\sup\{p(a,B): a\in A\}\), \(\delta_p(B,A)=\sup\{p(b,A): b\in B\}\); it will be referred to as the {Hausdorff metric} induced by \(p\). Then, let \(\psi:[0,1)\to (0,1]\) be the function \[ \psi(t)=\begin{cases}1&\text{ if }0\leq t< 1/2;\\ \psi(t)=1-t&\text{ if }1/2\leq t< 1.\end{cases} \] Finally, given the multivalued map \(T:X\to CB^p(X)\), denote for \(x,y\in X\) \[ M_p(x,y)=\max\{p(x,y),p(x,Tx),p(y,Ty), (1/2([p(x,Ty)+p(y,Tx)]\} \] The main result of the paper is the following. Theorem. With the notations above, suppose that there exists \(r\in [0,1)\) such that [\(\psi(r)p(x,Tx)\leq p(x,y)\) implies \(H_p(Tx,Ty)\leq r M_p(x,y)\)]. Then \(T\) has at least one fixed point in \(X\). As a by-product, an existence and uniqueness result is obatined for the bounded solutions of a functional equation arising in dynamic programming.
scientific article; zbMATH DE number 6139373