On the Existence and Uniqueness of Two-Dimensional Nonlinear Fuzzy Difference Equations with Logarithmic Interactions

This paper investigates a new class of two-dimensional fuzzy difference equations that integrate logarithmic nonlinearities with interaction effects between system variables. Motivated by the need to model complex dynamical systems influenced by uncertainty and interdependencies, we propose a system that extends existing one-dimensional models to capture more realistic interactions within a discrete-time framework. Our approach employs the characterization theory to transform the fuzzy system into an equivalent family of classical difference equations, thereby facilitating a rigorous analysis of the existence, uniqueness, and boundedness of positive solutions. To support the theoretical findings, two numerical examples are provided, illustrating the model’s capacity to capture complex dynamical patterns under fuzzy conditions. An application to a fuzzy population growth model illustrates how the model captures both interaction effects and uncertainty while ensuring well-defined and stable solutions. Numerical simulations show that, for instance, with α=0.10, β=δ=1.0, γ=0.08, and ρx=ρy=0.10, the trajectories of (xt,yt) rapidly converge toward a stable fuzzy equilibrium, with uncertainty bands confirming the positivity and boundedness of the solutions.