Convergence of an iterative scheme for multifunctions
Let \((X,d)\) be a metric space, \(\mathcal{C}(X)\) the set of all nonempty closed subsets of \(X\) and \(H\) the Pompeiu-Hausdorff metric. Let \(T: X\rightarrow \mathcal{C}(X)\) be a multivalued operator satisfying a Suzuki type contractive condition, i.e., there exist \(\alpha,\beta \in (0,1)\) such that \(\alpha (3-2\alpha=\beta)\leq 1\) and \[ \alpha d(x,Tx)\leq d(x,y) \Rightarrow H(Tx,Ty)\leq \beta d(x,y). \] Let \(\{T_i\}_{i=1}^{\infty}\) be a sequence of approximants of \(T\), that is, \[ H(Tx,T_i x)\leq \epsilon_i,\text{ for all }x\in X, \] where \(\sum\limits_{i=0}^{\infty}\epsilon_i<\infty\). The main result of the paper (Theorem 2.1) is a stability theorem that can be briefly stated as follows: if \[ \frac{\epsilon_i}{1-\alpha}\leq d(x_i,x_{i+1})\leq d(x_i,T_ix_i)+\delta_i,\quad x_{i+1}\in T_i x_i, \] where \(\sum\limits_{i=0}^{\infty}\delta_i<\infty\), then the ``perturbed'' sequence \(\{x_i\}\) still converges to a fixed point of \(T\).
scientific article; zbMATH DE number 6162219