Periodic Solutions for A Class of Integro-Differential Equations with Vriable Delay and Exponential Nonlinearity

This paper establishes the existence and uniqueness of a periodic solution for a nonlinear integro-differential equation with variable delay, employing Kwasniewski’s Fixed-point Theorem. We consider the equation where a are continuous, periodic functions , with satisfying appropriate regularity and boundedness conditions . The nonlinear term exhibits a natural boundedness property: the function is globally bounded on R, satisfying . This crucial feature ensures the integral operator is well-behaved and facilitates the application of Kwasniewski’s theorem within the Banach space of continuous periodic functions. To apply the theorem, we decompose the associated integral operator into the sum of a compact operator and a contraction mapping, thereby verifying the requisite conditions. Uniqueness of the periodic solution is further established through refined differential estimates and a careful analysis of the Lipschitz properties induced by the exponential nonlinearity. The results presented herein constitute a significant generalization of prior work in the literature, as they address the challenging combination of variable delay and a highly nonlinear, non-Lipschitz kernel that cannot be handled by classical methods. This work contributes not only to the theoretical framework of functional differential equations but also provides a robust analytical tool applicable to models in population dynamics, neural networks, and control systems with memory effects.