A double-inertial $ Z $-iteration for generalized $ (\alpha, \beta) $-nonexpansive maps in Banach spaces with applications

We propose a double-inertial $ Z $-iteration to approximate fixed points of generalized $ (\alpha, \beta) $-nonexpansive maps in Banach spaces. The scheme injects a two-term inertial extrapolation, depending on three consecutive iterates $ x_m, x_{m-1}, x_{m-2} $, into the first step of a three-layer $ Z $-iteration. Under convergence conditions on the inertial parameters and boundedness conditions on both convex-combination parameters, we prove that the generated sequence is bounded and that $ \lim_{m\to\infty}\left\lVert{x_m-q}\right\rVert $ exists for every fixed point (FP) $ q $. In a uniformly convex Banach space (UCBS) satisfying Opial's property, we further establish weak convergence to an FP, and we derive a non-asymptotic bound $ O(1/M) $. When the operator is in addition compact, we obtain strong convergence. As a first application, we use this strong convergence result to solve a nonlinear Hammerstein-type integral equation in $ L^{2}([a, b], \mathbb{R}) $, whose integral operator is shown to be nonexpansive and compact, and illustrate it with two examples. As a second application, we specialize the scheme to the proximal-gradient operator of the $ \ell_1 $-regularized least-squares model and show, on image deblurring, that it attains higher signal-to-noise ratio (SNR) and peak signal-to-noise ratio (PSNR) than the classical $ Z $-iteration, a single-inertial $ Z $-algorithm, and an alternative double-inertial scheme, and reaches prescribed quality targets in fewer iterations and less CPU time.