The first boundary-value problem for a singular nonlinear ordinary differential equation of fourth order期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:

M N Yakovlev

Institution:

(1) St.Petersburg Department of the Steklov Mathematical Institute, St.Petersburg, Russia

Abstract:

The solvability of the boundary-value problem

$\begin{gathered} u^{(4)} - (p_1 (t)u')' - (p_2 (t)u']^{2k + 1} )' + p_0 (t)u + f_0 (t)\varphi (u) + f_1 (t)u^{2m + 1} = f(t), 0 < t < 1, \hfill \\ u(0) = u'(0) = u(1) = u'(1) = 0 \hfill \\ \end{gathered}$

in the space H ₀ ² (0, 1) is proved under the following assumptions: p₀(t)t³(1 − t)³ ∈ L(0, 1), p₁(t)t(1 − t) ∈ L(0, 1), f(t)t^3/2(1 − t)^3/2 ∈ L(0, 1), 0 ≤ p₂(t)t(1 − t)]^k+1 ∈ L(0, 1), 0 ≤ f₀(t)t(1 − t)]^3/2 ∈ L(0, 1), 0 ≤ f₁(t)t(1 − t)]^3m+3 ∈ L(0, 1), ϕ(u)u ≥ −c|u|, c > 0,

$1 - \int\limits_0^1 {p_1^ - (t)t(1 - t)dt - \frac{1}{3}\int\limits_0^1 {p_0^ - (t)t^3 (1 - t)^3 dt > 0} }$

. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 233–245.

Keywords:

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