Estimates and asymptotic properties of solutions and their derivatives for a weakly nonlinear third order ordinary differential equation (Q2399408)

The author considers the following weakly linear ordinary differential equation of third order \[ {x}''' + a_2 \left( t \right){x}'' + a_1 \left( t \right){x}' + a_0 \left( t \right)x = f\left( t \right) + F\left( {t,x,{x}',{x}''} \right),\tag{1} \] where the function \(F(t,x,{x}',{x}'')\) satisfies the condition of weak nonlinearity, \[ \left| {F(t,x,{x}',{x}'')} \right| \leq F_0 \left( t \right) + g_0 \left( t \right)\left| x \right| + g_1 \left( t \right)\left| {x}' \right| + g_2 \left( t \right)\left| {x}'' \right| \] with nonnegative free term \(F_0 \) and Lipschitz coefficients, \(g_0 ,g_1 ,g_2 \). The author establishes sufficient conditions for estimate, boundedness, power absolute integrability on the half-interval \(J = \left[ {t_0 ,\infty } \right),\) tending to zero as \(t \to \infty ,\) including the exponential and power laws, for solutions and their first and second derivatives to equation (1). A theorem on that subject is proved. To reach the goal of the paper, the author proposes a method based on a nonstandard method of reduction to a system, weight functions and integral inequalities, where the Gronwall-Belman lemma is used. Finally, the author gives an example to clarify the proposed problem.