Existence Results for Neutral and Second-Order Functional Differential Equations with Causal Operators in $L^p_{\mathrm{loc}}$ Spaces

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This paper studies the particular class of second order functional differential equations involving casual operators on a function space $L^p_{\mathrm{loc}}(\mathbb{R}_+, \mathbb{R}^n)$. Previous studies\cite{Corduneanu2008,Mahdavi2008} discussed this equations in the following different function spaces$C(\mathbb{R}_+,\mathbb{R}^n)$ and $L^2_{\mathrm{loc}}(\mathbb{R}_+,\mathbb{R}^n)$. We establish the existence and uniqueness of solutions for both linear and nonlinear cases. Our worked based on the resolvent kernel method, Hölder's inequality, and successive approximation techniques. Finally, We provide examples to illustrate our results.

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