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Conjugacy criteria and principal solutions of self-adjoint differential equations

Ondřej Došlý, Jan Komenda (1995)

Archivum Mathematicum

Oscillation properties of the self-adjoint, two term, differential equation ( - 1 ) n ( p ( x ) y ( n ) ) ( n ) + q ( x ) y = 0 ( * ) are investigated. Using the variational method and the concept of the principal system of solutions it is proved that (*) is conjugate on R = ( - , ) if there exist an integer m { 0 , 1 , , n - 1 } and c 0 , , c m R such that 0 x 2 ( n - m - 1 ) p - 1 ( x ) d x = = 0 x 2 ( n - m - 1 ) p - 1 ( x ) d x and lim sup x 1 - , x 2 x 1 x 2 q ( x ) ( c 0 + c 1 x + + c m x m ) 2 d x 0 , q ( x ) ¬ 0 . Some extensions of this criterion are suggested.

Conjugacy criteria for half-linear differential equations

Simón Peňa (1999)

Archivum Mathematicum

Sufficient conditions on the function c ( t ) ensuring that the half-linear second order differential equation ( | u ' | p - 2 u ' ) ' + c ( t ) | u ( t ) | p - 2 u ( t ) = 0 , p > 1 possesses a nontrivial solution having at least two zeros in a given interval are obtained. These conditions extend some recently proved conjugacy criteria for linear equations which correspond to the case p = 2 .

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