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Journal of Vibration Testing and System Dynamics 10(2) (2026) 119--134 | DOI:10.5890/JVTSD.2026.06.002

Tharmalingam Gunasekar$^{1,2}$, Srinivasan Madhumitha$^{1}$, Prabakaran Raghavendran$^{1}$,\\ Kamalendra Kumar$^{3}$

$^{1}$ Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, India

$^{2}$ Department of Mathematics, Srinivas University, Mukka, Mangaluru - 574146, Karnataka, India

$^{3}$ Shri Ram Murti Smarak College of Engineering & Technology, Bareilly (U.P.), Uttar Pradesh, India

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Abstract

This study explores the second-order functional differential and integro-differential equations incorporating delays and stochastic influences. We employ the Kakutani fixed point theorem to establish the existence results. The investigation focuses on the existence results for equations with infinite delay and random disturbances, which emphasize the importance of stochastic processes in the analysis. An illustrative example, along with a graph, is provided to demonstrate the theoretical findings and highlight the practical applications of the conceptual framework. Analytical results are validated through rigorous mathematical examination, demonstrating the applicability of the Kakutani fixed point theorem in stochastic differential equations with functional interrelationships. This study contributes to understanding the interaction between delays, randomness, and second-order dynamics in mathematical modeling. \textcolor{black}{Such understanding ensures more accurate modeling of complex systems where memory and uncertainty play a significant role. The main objective is to provide a solid theoretical basis for analyzing stochastic systems influenced by past states and probabilistic effects.}

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