AK-iterative scheme for fixed point approximation in hyperbolic metric spaces and applications in Volterra integral equation

Abstract In this paper, we utilize the AK-iteration procedure for hyperbolic metric spaces, ensuring the symmetry condition is met and establish the weak $w^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>w</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -stability and data dependence results for contraction mappings. Additionally, we establish several Δ-convergence and strong convergence theorems for the generalized $(\alpha ,\beta )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> <mml:mo>)</mml:mo> </mml:math> -nonexpansive (GABNE) mappings. To demonstrate it, we provide a numerical example of the GABNE-mappings, emphasizing the enhanced efficiency of the AK-iteration method relative to other iterative approaches. Additionally, the applications of the main results are demonstrated by solving 2 D and 3 D Volterra integral equations. These findings extend and improve numerous related results found in the current literature.