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In this paper, we modify the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mi>F</mi></mrow></semantics></math></inline-formula>-iteration process into hyperbolic metric spaces where the symmetry condition is satisfied and establish the weak <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>w</mi><mn>2</mn></msup></semantics></math></inline-formula>-stability and data dependence results for contraction mappings. We also prove some <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Δ</mo></semantics></math></inline-formula>-convergence and strong convergence theorems for generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></mrow></semantics></math></inline-formula>-nonexpansive type 1 mappings. Finally, we offer a numerical example of generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></mrow></semantics></math></inline-formula>-nonexpansive type 1 mappings and show that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mi>F</mi></mrow></semantics></math></inline-formula>-iteration process is more effective than some other iterations. Our results generalize and improve several relevant results in the literature.