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

We introduce a new unconditional two-parameter class of mappings, called unconditional \((α,β)\)-nonexpansive mappings, satisfying \(\|Tx-Ty\| ≤ α\|Tx-y\| +α\|Ty-x\| +β\|x-y\|\) for all \(x,y∈ C\), where \(α,β≥ 0\) and \(2α+β≤ 1\). This class contains the classical nonexpansive class as the special case \((α,β)=(0,1)\), 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 \((1/2)\|x-Tx\|≤\|x-y\|\) before the contraction estimate is invoked. We analyse quantitative stability of the fixed-point set \(F(T)\) under operator convergent perturbations \(T_n→ T\) (i.e.,\(η_n:=x∈ C\|T_nx-Tx\|→ 0\)) in two distinct regimes. In the interior regime \(2α+β<1\), the fixed-point set is a singleton \(\{x^*\}\) and we establish the explicit Hausdorff bound \(d_H(F(T_n),F(T)) ≤ 1-α/1-2α-βη_n,\) and prove that the constant \(1-α/1-2α-β\) is sharp for every admissible \((α,β)\); as a consequence, the fixed-point map \(T↦ x^*(T)\) is Lipschitz on the operator space. In the boundary regime \(2α+β=1\), the set of fixed points is closed and convex and is a subset of \(C\); we prove the Kuratowski upper limit inclusion \({w-Ls}(F(T_n))⊆ F(T)\) under the Opial property, and the full Kuratowski convergence \(F(T_n){w}F(T)\) 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.