On a general class of multi-valued weakly Picard mappings (Q865298)

Let \((X,d)\) be a metric space and \(T:X\rightarrow\mathcal{P}(X)\) be a multi-valued operator. \(T\) is said to be a generalized \((\alpha ,L)\)-weak contraction if there exist a constant \(L\geq 0\) and a function \(\alpha:[0,\infty)\rightarrow[0,1)\), with \(\lim \sup_{r\rightarrow t^+} \alpha (r)<1\) for every \(t\in [ 0,\infty )\), such that \[ H(Tx, Ty)\leq \alpha (d(x,y)) d(x,y)+LD(y,Tx) \] for all \(x,y\in X\). When \(\alpha (t)=\theta\in[0,1)\) for all \(t\in[0,\infty)\), we say that \(T\) is a \((\theta ,L)\)-weak contraction. The authors prove that a generalized \((\alpha ,L)\)-weak contraction \(T\) has a fixed point whenever \(X\) is complete and \(T\) has nonempty bounded and closed values. Moreover, if \(T\) is a \((\theta,L)\)-weak contraction, then for any \(x_0\in X\) there exists an orbit \(\{x_n\}_{n\geq 0}\) converging to a fixed point \(u\) of \(T\) for which the following estimate holds: \[ d(x_n,u)\leq\min\left\{\frac{h^n}{1-h}d(x_1,x_0),\frac{h}{1-h}d(x_{n-1},x_n)\right\} \] for a certain constant \(h<1\).