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Existence of solutions to boundary value problems for a nonlinear second order equation with weak Carathéodory functions. (English) Zbl 0891.34022
The authors consider the following boundary value problems $[\phi(y')]'= f(x,y,y'),\quad a<x<b,\tag{1}$ $y(a)= A,\quad y(b)= B,\tag{2}$ $y'(a)= A,\quad y(b)= B,\tag{3}$ $[\phi(y')]'= f(x,y,y'),\quad a<x<+\infty,\quad y(a)= A,\tag{4}$ where $$A$$ and $$B$$ are prescribed real numbers.
The authors prove the existence of solutions of problems (1), (2) and (1), (3) when $$\phi:\mathbb{R}\to\mathbb{R}$$ is continuous and strictly increasing; $$f:I\times \mathbb{R}^2\to\mathbb{R}$$ is a weak Carathéodory function and satisfies a Nagumo condition.

MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations