Conditional oscillation of Riemann-Weber half-linear differential equations with asymptotically almost periodic coefficients

2014 ◽  
Vol 51 (3) ◽  
pp. 303-321
Author(s):  
Petr Hasil ◽  
Michal Veselý
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Almost Periodic ◽  
Periodic Coefficients ◽  
Conditional Oscillation ◽  
Linear Differential ◽  
Asymptotically Almost Periodic ◽  
Oscillation Constant

We analyse the oscillation and non-oscillation of second-order half-linear differential equations with periodic and asymptotically almost periodic coefficients, where the equations have the so-called Riemann-Weber form. For these equations, we find an explicit oscillation constant. Corollaries and examples are mentioned as well.

Conditional oscillation of Euler type half-linear differential equations with unbounded coefficients

2016 ◽  
Vol 53 (1) ◽  
pp. 22-41
Author(s):  
Jaroslav Jaroš ◽  
Michal Veselý
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
Oscillation Criterion ◽  
Linear Differential Equations ◽  
Unbounded Coefficients ◽  
Conditional Oscillation ◽  
Linear Differential ◽  
Oscillation Constant ◽  
Oscillatory Properties

The oscillatory properties of half-linear second order Euler type differential equations are studied, where the coefficients of the considered equations can be unbounded. For these equations, we prove an oscillation criterion and a non-oscillation one. We also mention a corollary which shows how our criteria improve the known results. In the corollary, the criteria give an explicit oscillation constant.

scholarly journals Conditional Oscillation of Half-Linear Differential Equations with Coefficients Having Mean Values

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Petr Hasil ◽  
Robert Mařík ◽  
Michal Veselý
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Linear Differential Equations ◽  
Almost Periodic ◽  
Mean Values ◽  
Conditional Oscillation ◽  
The Mean ◽  
Linear Differential ◽  
Positive Coefficients ◽  
Oscillation Constant

We prove that the existence of the mean values of coefficients is sufficient for second-order half-linear Euler-type differential equations to be conditionally oscillatory. We explicitly find an oscillation constant even for the considered equations whose coefficients can change sign. Our results cover known results concerning periodic and almost periodic positive coefficients and extend them to larger classes of equations. We give examples and corollaries which illustrate cases that our results solve. We also mention an application of the presented results in the theory of partial differential equations.

Uniformly almost periodic solutions of non-linear differential equations of the second order I. General exposition

1955 ◽  
Vol 51 (4) ◽  
pp. 604-613
Author(s):  
Chike Obi
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Almost Periodic ◽  
Non Linear ◽  
Independent Variable ◽  
Linear Differential ◽  
The Given ◽  
Analytical Expressions

1·1. A general problem in the theory of non-linear differential equations of the second order is: Given a non-linear differential equation of the second order uniformly almost periodic (u.a.p.) in the independent variable and with certain disposable constants (parameters), to find: (i) the non-trivial relations between these parameters such that the given differential equation has a non-periodic u.a.p. solution; (ii) the number of periodic and non-periodic u.a.p. solutions which correspond to each such relation; and (iii) explicit analytical expressions for the u.a.p. solutions when they exist.

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