Fixed point theorems for nonlinear contractive mappings in cone-valued $\theta$-type multiplicative metric spaces

Fixed point theory plays a fundamental role in nonlinear analysis and has significant applications in differential equations, integral equations, optimization, and applied mathematics. Inspired by developments in generalized metric structures, we introduced and studied the Geraghty-type contractive conditions in the setting of cone $\theta$-type multiplicative metric spaces defined on ordered Banach algebras. By extending the classical notions of $\theta$-metric and multiplicative metric spaces, we constructed a framework based on solid multiplicative cones and established their essential topological and convergence properties. Within this generalized structure, we formulated new fixed point theorems for self-mappings satisfying cone-valued $\theta$-type multiplicative Geraghty contractions, which weakened the traditional Lipschitz condition while preserving the existence and uniqueness of fixed points in complete spaces. The proposed results significantly extended and unified several well-known contraction principles, including those of Banach, Kannan, Chatterjea, and standard Geraghty contractions, under a broader cone-valued multiplicative setting. Furthermore, we developed extensions to higher-dimensional frameworks to enhance applicability in complex nonlinear systems. Illustrative examples are presented to substantiate the theoretical findings and to demonstrate the effectiveness of the introduced approach in generalized analytical environments.