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On some boundary value problems for second order nonlinear differential equations

Zuzana Došlá, Mauro Marini, Serena Matucci (2012)

Mathematica Bohemica

We investigate two boundary value problems for the second order differential equation with p -Laplacian ( a ( t ) Φ p ( x ' ) ) ' = b ( t ) F ( x ) , t I = [ 0 , ) , where a , b are continuous positive functions on I . We give necessary and sufficient conditions which guarantee the existence of a unique (or at least one) positive solution, satisfying one of the following two boundary conditions: i ) x ( 0 ) = c > 0 , lim t x ( t ) = 0 ; ii ) x ' ( 0 ) = d < 0 , lim t x ( t ) = 0 .

On some equations y'(x) = f(x,y(h(x)+g(y(x))))

Zbigniew Grande (2011)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In [4] W. Li and S.S. Cheng prove a Picard type existence and uniqueness theorem for iterative differential equations of the form y'(x) = f(x,y(h(x)+g(y(x)))). In this article I show some analogue of this result for a larger class of functions f (also discontinuous), in which a unique differentiable solution of considered Cauchy's problem is obtained.

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