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Boundedness of solutions of the fourth order differential equation with oscillatory restoring and forcing terms. (English) Zbl 1174.34035
The author investigates the boundedness of solutions of the fourth-order nonlinear differential equation $$ x^{(4)} + ax''' + bx'' + cx' + h(x) = p(t),$$ where $a=\xi_1+\xi_2+\xi_3$, $b=\xi_1\xi_2+\xi_1\xi_3+\xi_2\xi_3$, $c=\xi_1\xi_2\xi_3$, $\xi_1$, $\xi_2$, $\xi_3>0$ with $-\xi_i$ $(i=1, 2, 3)$ being roots of the cubic equation $$ \eta^3 + a\eta^2 + b\eta + c =0.$$ By employing the Cauchy formula for the particular solution of non-homogeneous linear differential equations with constant coefficients, it is proved that each solution of the differential equation above and its derivatives up to order three are bounded.
Reviewer: Qiru Wang (Guangzhou)
34C11Qualitative theory of solutions of ODE: growth, boundedness