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

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 \(a\leq t\leq b\).


34B16 Singular nonlinear boundary value problems for ordinary differential equations
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