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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 properties of the solution of the differential equation u ' ' + 2 u ' r = u - u 3

Valter Šeda, Ján Pekár (1990)

Aplikace matematiky

In the paper it is shown that each solution u ( r , α ) ot the initial value problem (2), (3) has a finite limit for r , and an asymptotic formula for the nontrivial solution u ( r , α ) tending to 0 is given. Further, the existence of such a solutions is established by examining the number of zeros of two different solutions u ( r , α ¯ ) , u ( r , α ^ ) .

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