On a two-point boundary value problem for the second order ordinary differential equations with singularities. (English) Zbl 1027.34022

Lomtatidze, A.; Malaguti, L.

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 52, No. 6, 1553-1567 (2003).

The authors consider the boundary value problem \[ u''= f(t, u,u'),\tag{1} \]
\[ u(a+)= 0,\qquad u(b-)= 0,\tag{2} \] with \(f=\in K_{\text{loc}}(]a,b[\times\mathbb{R}^2;\mathbb{R})\).
By a solution to problem (1), (2), the authors understand a function \(u\in\widetilde C_{\text{loc}}'(]a,b[; \mathbb{R})\) satisfying (1) almost everywhere in \(]a,b[\) and also conditions (2).
By using a lower solution \(\alpha(t)\) to (1) and an upper solution \(\beta(t)\) to (1), the existence of at least one solution \(u(t)\) to problem (1), (2), is proved, where \(\alpha(t)\leq u(t)\leq \beta(t)\) for \(a\leq t\leq b\).

Reviewer: Anatolij Ivan Kolosov (Khar’kov)

MSC:

34B16

Singular nonlinear boundary value problems for ordinary differential equations

Keywords:

second-order singular differential equation; two-point boundary value problem; lower and upper solutions; one-sided restriction

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