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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
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