On the Boundedness of Solution of the Second Order Ordinary Differential Equation with Involution

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In the present paper, the initial value problem for the second order ordinary differential equation with involution is investigated. We obtain equivalent initial value problem for the fourth order ordinary differential equations to the initial value problem for second order linear differential equations with involution. Theorem on stability estimates for the solution of the initial value problem for the second order ordinary linear differential equation with involution is proved. Theorem on existence and uniqueness of bounded solution of initial value problem for second order ordinary nonlinear differential equation with involution is established.