Abstract—Pantograph-type equations arise in diverse areas of applied mathematics, yet their stability and solvability properties remain difficult to analyze. This work addresses this problem by studying the initial value problem for pantograph-type equations on a fixed interval. The objectives are to establish Ulam–Hyers stability conditions and to derive sufficient conditions for unique solvability. The analysis employs the Banach fixed-point theorem, which provides a rigorous basis for proving existence, uniqueness, and stability. An illustrative example is included to demonstrate the applicability of the theoretical results. The findings indicate that, although exact solutions are often challenging to obtain in practice, the Ulam–Hyers theory offers an effective approach for approximating and estimating solutions of pantograph-type equations. The results contribute to the advancement of stability theory for functional differential equations and highlight the significance of Ulam–Hyers methods in the study of equations with proportional arguments.

Keywords—Banach fixed-point theorem, pantograph-type equation, Ulam-Hyers stability

Cite: Natalia Dilna, Svitlana Leshchuk, "Ulam-Hyers Stability of the Initial Value Problem for the Pantograph-Type Equation," International Journal of Applied Physics and Mathematics, vol. 15, no. 2, pp. 102-107, 2025.

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