Jaroslav Jaroš

NONOSCILLATION THEORY FOR SECOND ORDER HALF-LINEAR DIFFERENTIAL EQUATIONS IN THE FRAMEWORK OF REGULAR VARIATION

Criteria are established for nonoscillation of all solutions of the second order half-linear differential equation A% MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

Forced oscillation of second order linear and half-linear difference equations

(\mid y^\prime \mid^{\alpha-1}y^\prime)^\prime + q(t)\mid y \mid^{\alpha -1}y = 0,\ \ \ t \geq 0,

PICONE-TYPE INEQUALITIES FOR NONLINEAR ELLIPTIC EQUATIONS WITH FIRST-ORDER TERMS AND THEIR APPLICATIONS

where α > 0 is a constant and q: [0, ∞) → ℝ is continuous. The criteria are designed to exhibit the role played by the integral of q(t) in guaranteeing the existence of nonoscillatory solutions of (A) in specific classes of regularly varying functions in the sense of Karamata.

Generalized Prüfer angle and oscillation of half-linear differential equations

Oscillation properties of solutions of the forced linear and half-linear second order difference equations are invesitgated. It is shown that if the forcing term does not oscillate, in some sense, too rapidly, then the oscillation of unforced equation implies oscillation of the forced equation.

On strongly decreasing solutions of cyclic systems of second-order nonlinear differential equations

Picone-type inequalities are established for nonlinear elliptic equations which are generalizations of nonself-adjoint linear elliptic equations, and Sturmian comparison theorems are derived as applications. Oscillation results are also obtained for forced superlinear elliptic equations and superlinear-sublinear elliptic equations.

Oscillation properties of solutions of a class of nonlinear parabolic equations

Abstract In this paper, we introduce a new modification of the half-linear Prufer angle. Applying this modification, we investigate the conditional oscillation of the half-linear second order differential equation ( ∗ ) t α − 1 r ( t ) Φ ( x ′ ) ′ + t α − 1 − p s ( t ) Φ ( x ) = 0 , Φ ( x ) = | x | p − 1 sgn x , where p > 1 , α ≠ p , and r , s are continuous functions such that r ( t ) > 0 for large t . We present conditions on the functions r , s which guarantee that Eq. ( ∗ ) behaves like the Euler type equation [ t α − 1 Φ ( x ′ ) ] ′ + λ t α − 1 − p Φ ( x ) = 0 , which is conditionally oscillatory with the oscillation constant λ 0 = | p − α | p ∕ p p .

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

The n -dimensional cyclic system of second-order nonlinear differential equations is analysed in the framework of regular variation. Under the assumption that α i and β i are positive constants such that α 1 … α n > β 1 … β n and p i and q i are regularly varying functions, it is shown that the situation in which the system possesses decreasing regularly varying solutions of negative indices can be completely characterized, and moreover that the asymptotic behaviour of such solutions is governed by a unique formula describing their order of decay precisely. Examples are presented to demonstrate that the main results for the system can be applied effectively to some classes of partial differential equations with radial symmetry to provide new accurate information about the existence and the asymptotic behaviour of their radial positive strongly decreasing solutions.

On an integral inequality of the Wirtinger type

Oscillations of solutions of a class of nonlinear parabolic equations are investigated, and the unboundedness of solutions is also studied as corollaries. Our approach is to employ the modifications of Picone-type identities for half-linear elliptic operators.

An Oscillation Test for a Class of Linear Neutral Differential Equations

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.

Existence and precise asymptotics of positive solutions for a class of nonlinear differential equations of the third order

In this paper, a new Picone-type identity for the quasilinear differential operator of the second order is used to establish an integral inequality involving functions and their derivatives which generalizes the classical Wirtinger inequality.