On the Jaggi-type Fixed Point Theorem in (\(α\), \(β\))-complex-valued b-metric Spaces

This study establishes existence and uniqueness results for Jaggi-type rational contractive mappings in complete (\(α\), \(β\))-complex-valued b-metric spaces. The analysis uses Picard iterative sequences, estimates of successive iterates, and the generalised (\(α\), \(β\))-triangle inequality. Two fixed-point criteria are developed. In the first, the successive-iterate estimate is governed by q = \(λ_1/(1− λ_2),\) together with the additional restriction \(βλ_1/(1−λ_2),\) < 1, which ensures convergence of the geometric series required in the Cauchy-sequence argument. The second theorem treats a Jaggi-type rational condition in which the rational termis weighted by \(λ_1\) and the direct distance termby \(λ_2\); its convergence requirement is \(βλ_2/(1−λ_1),\) <1. In both cases, completeness is used to obtain a limit of the Picard sequence, after which the contractive conditions establish that the limit is a fixed point and that the fixed point is unique. The results reduce to the corresponding complex-valued b-metric and complex-valued metric settings under symmetric parameter choices. The analysis also distinguishes the roles of the structural parameters: \(α\) enters as a multiplicative factor in the iterative estimates, whereas \(β\) directly affects the ratio of the geometric series used in the convergence argument. The numerical examples illustrate how the two structural parameters enter the contractive estimates and show the intended role of asymmetric parameter choices within the proposed framework.