The first boundary-value problem for a~singular nonlinear ordinary differential equation of fourth order

M. N. Yakovlev

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XIX, Tome 334 (2006), pp. 233-245

Résumé

The solvability of the boundary-value problem \begin{gather*} u^{(4)}-(p_1(t)u')'-(p_2(t)[u']^{2k+1})'+p_0(t)u+f_0(t)\varphi(u)+f_1(t)u^{2m+1}=f(t), \enskip 01, \\ u(0)=u'(0)=u(1)=u'(1)=0, \end{gather*} in the space $H^2_0(0,1)$ is proved under the following assumptions: $p_0(t)t^3(1-t)^3\in L(0,1)$, $p_1(t)t(1-t)\in L(0,1)$, $f(t)t^{3/2}(1-t)^{3/2}\in L(0,1)$, $0\le p_2(t)[t(1-t)]^{k+1}\in L(0,1)$, $0\le f_0(t)[t(1-t)]^{3/2}\in L(0,1)$, $0\le f_1(t)[t(1-t)]^{3m+3}\in L(0,1)$, $\varphi(u)u\ge-c|u|$, $c>0$, $$ 1-\int^1_0p^-_1(t)t(1-t)dt-\frac13\int^1_0p^-_0(t)t^3(1-t)^3\,dt>0. $$