A Four-Parameter Dass-Gupta-Reich Fixed Point Theorem in (α, β)-Complex-Valued b-Metric Spaces
Abstract
In this paper, we establish a four-parameter fixed point theorem for self-mappings on complete (α, β)-complex-valued b-metric spaces. The proposed contractive condition combines a Dass–Gupta rational term with two Reich-type self-distance terms and allows the coefficients of d(x, Tx) and d(y, Ty) to be different. The principal parameter restriction is expressed in the compact form
β(λ + γ1) + μ + γ2 < 1, αγ1 < 1,
which simultaneously guarantees the geometric decay required in the Picard iteration and the final fixed-point verification. A detailed example on [0, 1] verifies all four contractive components directly, including the rational term in both real and imaginary components. An application to a nonlinear Fredholm integral equation is obtained on an invariant space of constant functions. Several classical contractions arise as parameter specializations, including Banach-, Kannan-, Reich-, asymmetric Reich-, and Dass–Gupta-type conditions.
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References
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