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1. Introduction.
In this paper we study the asymptotic behaviour of solutions of a fourth order differential equation of the form $y^{(4)}=\alpha_0p(t)\varphi(y)\quad$(1). The purpose of this paper is to obtain the asymptotics $P_\omega(Y_0,\lambda_0)$ solutions of the differential equation (1) for the special case when $\lambda_0=1$.
2. Object of research.
Consider a differential equation of the form (1) where $\alpha_0\in\{-1,1\}$, $p:[a,\omega[\longrightarrow]0,+\infty[$ -is a continuous function, $-\infty
3. Basic definitions and notations.
The solution $y$ of the differential equation (1) is called $P_\omega(Y_0,\lambda_0)$-solution, where $-\infty\le \lambda_0\le +\infty,$ if it is defined on the segment $[t_0,\omega[\subset[a,\omega[$ and satisfies the following conditions $y(t)\in \Delta_{Y_0}\quad\mbox{at}\quad t\in [t_0,\omega[,\quad\lim\limits_{t\uparrow\omega}y(t)=Y_0,\quad$
$\lim\limits_{t\uparrow\omega}y^{(k)}(t)=\left[\begin{array}{l}
\mbox{or}\quad 0,\\
\mbox{or}\quad \pm\infty,
\end{array}
\right.\ (k=1,2,3),\quad
\lim\limits_{t\uparrow\omega}\frac{[y^{(3 )}(t)]^2}{y^{(2)}(t)y^{(4)}(t)}=\lambda_0.$
Let us introduce additional auxiliary notations
$J_0(t)=\int\limits_{A_0}^t p_0^\frac{1}{4}(\tau),\,
q(t)=\frac{(\Phi^{-1}(\alpha_0J_0(t)))'}{\alpha_0J_3(t)},\,
H(t)=\frac{\Phi^{-1}(\alpha_0J_0(t))\varphi'(\Phi^{-1}(\alpha_0J_0(t)))}{\varphi(\Phi^{-1}(\alpha_0J_0(t)))},\,
\\
J_1(t)=\int\limits_{A_1}^t p_0 (\tau)\varphi(\Phi^{-1}(\alpha_0J_0(\tau)))\, d\tau, \,
J_2(t)=\int\limits_{A_2}^t J_1(\tau)\, d\tau,\,
J_3(t)=\int\limits_{A_3}^t J_2(\tau)\, d\tau,$ where the integration boundary $A_i$ is either $\omega$ or constant and is defined so that the integral tends either to 0 or to $\pm\infty$.
4. Main results.
The following two theorems are valid for equation (1).
Theorem 1. For the existence $P\omega (Y_0, 1)$-solutions of differential equation (1) that the inequalities $\alpha_0\nu_2>0,\ \alpha_0\mu_0 J_0(t)<0,\mbox{at},\ t\in ]a,\omega[,\alpha_0\nu_0<0,\mbox{or},\
Y_0=0,\ \alpha_0\nu_0>0,\mbox{or},\ Y_0=\pm \infty$ (2),
and conditions $\frac{\alpha_0J_3(t)}{\Phi^{-1}(\alpha_0J_0(t))}\sim\frac{J'_1(t)}{J_1(t)}\sim\frac{J'_2(t)}{J_2(t)}\sim\frac{J'_3(t)}{J_3(t)}\sim\frac{(\Phi^{-1}(\alpha_0J_0(t)))'}{\Phi^{-1}(\alpha_0J_0(t))}\mbox{at}\ t\uparrow\omega,\alpha_0\lim_{t\uparrow\omega}J_0(t)=Z_0,$
$\lim_{t\uparrow\omega}\frac{\pi_\omega(t)(\Phi^{-1}(\alpha_0J_0(t)))'}{\Phi^{-1}(\alpha_0J_0(t)))}=\pm\infty,\quad \lim_{t\uparrow\omega}\frac{\pi_\omega(t) J_0'(t)}{J_0(t)}=\pm\infty$ (3). Moreover, for each such solution, the asymptotic representations at$\ y(t)=\Phi^{-1}(\alpha_0J_0(t))\left[1+\frac{o(1)}{H(t)}\right],\,y'(t)=\alpha_0 J_3(t)[1+o(1)],$
$y''(t)=\alpha_0 J_2(t)[1+o(1)],\, y'''(t)=\alpha_0 J_1(t)[1+o(1)]\, (4).$
Theorem 2. Let $p_0 : [a, \omega [ \rightarrow ]0, +\infty [$ - a continuously differentiable function and along with the
(2) - (3) conditions $\lim_{t\uparrow \omega}\frac{q'(t)J_2(t)|H(t)|^\frac14}{J_2'(t)}=0,\quad
\lim_{y\to Y_0\atop y\in\Delta_{Y_0}}\frac{\Bigr(\frac{\varphi'(y)}{\varphi(y)}\Bigl)'}{\Bigr(\frac{\varphi'(y)}{\varphi(y)}\Bigl)^2}\Biggr|\frac{y\varphi'(y)}{\varphi(y)}\Biggl|^\frac34=0$ then the differential equation (1) contains at $\alpha_0\mu_0=-1$ a two-parameter family of $P_\omega (Y_0, 1)$-solutions which admit at $t \uparrow \omega$ asymptotic representations (4) and furthermore such first, second and third order derivatives of which satisfy at $t \uparrow \omega$ the asymptotic relations $y'(t)=\alpha_0 J_3(t)\Biggr[1+\frac{o(1)}{|H(t)|^{\frac34}}\Biggl],\ y''(t)=\alpha_0 J_2(t)\Biggr[1+\frac{o(1)}{|H(t)|^{\frac12}}\Biggl],\ y'''(t)=\alpha_0 J_1(t)\Biggr[1+\frac{o(1)}{|H(t)|^{\frac14}}\Biggl].$The question of whether the differential equation (1) has $P_\omega(Y_0,\lambda_0)$- solutions admitting at $t\uparrow\omega$ asymptotic representations (4) in the case when $\alpha_0\mu_0=1$ is still open.
Bibliography: 1. V.M. Evtukhov, A.M. Samoilenko. \ Conditions of existence of vanishing solutions of real nonautonomous systems of quasilinear differential equations at a special point// Ukr. Mat. Mag. - 2010. - 62, № 1. - p. 52-80. 2. Maric V. Regular variation and differential equations. Lecture Notes in Math.,1726(2000).- 127p. 3. А. G. Chernikova, Asymptotic images of solutions of differential equations with rapidly changing nonlinearities, PhD thesis, Odesa (2019)
