The solvability results for the third-order singular non-linear differential equation

Zh. B. Yeskabylova ; K. N. Ospanov ; T. N. Bekjan

Eurasian mathematical journal, Tome 10 (2019) no. 4, pp. 85-91

Résumé

We give some conditions for solvability in $L_2(\mathbb{R})$ ($\mathbb{R}=(-\infty,+\infty)$) of the following singular non-linear differential equation: $$ ly\equiv-y'''(x)+q(x,y,y')y'+s(x,y,y')y=h(x). $$ We assume that $q$ and $s$ are real-valued unbounded functions and $q$ does not obey the “potential” $s$. For the solution $y$ we prove that $$ ||y'''||_2+||q(\cdot,y,y')y'||_2+||s(\cdot,y,y')y||_2\infty, $$ where $||\cdot||_2$ is the norm in $L_2$. To establish these facts, we use coercive solvability results for the corresponding linear third-order differential equation obtained by us earlier.