Unconditional Two-Parameter Nonexpansive Mappings: Hausdorff Stability of Fixed Point Sets under Operator-Convergent Perturbations

Abstract

We introduce a new unconditional two-parameter class of mappings, called unconditional \((\alpha,\beta)\)-nonexpansive mappings, satisfying \(\|Tx-Ty\| \leq \alpha\|Tx-y\| +\alpha\|Ty-x\| +\beta\|x-y\|\) for all \(x,y\in C\), where \(\alpha,\beta\geq 0\) and \(2\alpha+\beta\leq 1\). This class contains the classical nonexpansive class as the special case \((\alpha,\beta)=(0,1)\), and is not subsumed by the conditional frameworks of Pant and Shukla (2017) or Ullah et al. (2020), which impose a Suzuki-type pre-condition \((1/2)\|x-Tx\|\le\|x-y\|\) before the contraction estimate is invoked. We analyse quantitative stability of the fixed-point set \(F(T)\) under operator convergent perturbations \(T_n\to T\) (i.e.,\(\eta_n:=\sup_{x\in C}\|T_nx-Tx\|\to 0\)) in two distinct regimes. In the interior regime \(2\alpha+\beta<1\), the fixed-point set is a singleton \(\{x^*\}\) and we establish the explicit Hausdorff bound \(d_H(F(T_n),F(T)) \leq \frac{1-\alpha}{1-2\alpha-\beta}\eta_n,\) and prove that the constant \(\frac{1-\alpha}{1-2\alpha-\beta}\) is sharp for every admissible \((\alpha,\beta)\); as a consequence, the fixed-point map \(T\mapsto x^*(T)\) is Lipschitz on the operator space. In the boundary regime \(2\alpha+\beta=1\), the set of fixed points is closed and convex and is a subset of \(C\); we prove the Kuratowski upper limit inclusion \(\mathrm{w\text{-}Ls}(F(T_n))\subseteq F(T)\) under the Opial property, and the full Kuratowski convergence \(F(T_n)\xrightarrow{w}F(T)\) when \(F(T)\) is a singleton, which occurs in particular under a strict form of the defining inequality. As an application, we derive explicit stability estimates for the solution set of parametric Hammerstein integral equations on \(L^2([0,1])\) under \(L^2\)-convergent kernel perturbations.