Half-linear Differential Equations Research Articles

A new application method for nonoscillation criteria of Hille-Wintner type

The present paper deals with nonoscillation problem for the Sturm–Liouville half-linear differential equation $$\begin{aligned} \big (r(t)\phi _p(x')\big )' + c(t)\phi _p(x) = 0, \end{aligned}$$ where r, \(c\!:[a,\infty ) \rightarrow \mathbb {R}\) are continuous functions, \(r(t) > 0\) for \(t \ge a\), and \(\phi _p(z) = |z|^{p-2}z\) with \(p > 1\). The purpose of this paper is to show that it is possible to broaden the application range of Hille-Wintner type nonoscillation criteria. To this end, we derive a comparison theorem by means of Riccati’s technique. Our result is new even in the linear case that \(p = 2\). By the obtained result, we can compare two differential equations having a different power p of the above-mentioned type. To illustrate our comparison theorem, we present two examples of which all non-trivial solutions of the Sturm-Liouville linear differential equation are nonoscillatory even if \(\int _a^t\!\frac{1}{r(s)}ds\int _t^\infty \!\!c(s)ds\) or \(\int _t^\infty \!\!\frac{1}{r(s)}ds\int _a^t\!c(s)ds\) is less than the lower bound \(-3/4\).

Generalized Prüfer angle and oscillation of half-linear differential equations

In this paper, we introduce a new modification of the half-linear Prüfer angle. Applying this modification, we investigate the conditional oscillation of the half-linear second order differential equation (∗)tα−1r(t)Φ(x′)′+tα−1−ps(t)Φ(x)=0,Φ(x)=|x|p−1sgnx,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 [tα−1Φ(x′)]′+λtα−1−pΦ(x)=0, which is conditionally oscillatory with the oscillation constant λ0=|p−α|p∕pp.

Asymptotic formulae for solutions of half-linear differential equations

We establish asymptotic formulae for regularly varying solutions of the half-linear differential equation(r(t)|y′|α−1sgny′)′=p(t)|y|α−1sgny,where r, p are positive continuous functions on [a, ∞) and α ∈ (1, ∞). The results can be understood in several ways: Some open problems posed in the literature are solved. Results for linear differential equations are generalized; some of the observations are new even in the linear case. A refinement on information about behavior of solutions in standard asymptotic classes is provided. A precise description of regularly varying solutions which are known to exist is given. Regular variation of all positive solutions is proved.

Solutions of half-linear differential equations in the classes Gamma and Pi

We study asymptotic behavior of (all) positive solutions of the non\-oscillatory half-linear differential equation of the form $$ (r(t)|y'|^ {\alpha-1}\text{sgn}\, y')'=p(t)|y|^{\alpha-1}\text{sgn}\, y , $$ where $\alpha\in(1,\infty)$ and $r,p$ are positive continuous functions on $[a,\infty)$, with the help of the Karamata theory of regularly varying functions and the de Haan theory. We show that increasing resp. decreasing solutions belong to the de Haan class $\Gamma$ resp. $\Gamma_-$ under suitable assumptions. Further we study behavior of slowly varying solutions for which asymptotic formulas are established. Some of our results are new even in the linear case $\alpha=2$.

Decaying solutions for discrete boundary value problems on the half line

Some nonlocal boundary value problems, associated to a class of functional difference equations on unbounded domains, are considered by means of a new approach. Their solvability is obtained by using properties of the recessive solution to suitable half-linear difference equations, a half-linearization technique and a fixed point theorem in Frechét spaces. The result is applied to derive the existence of nonoscillatory solutions with initial and final data. Examples and open problems complete the paper.

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

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.

Oscillation and non-oscillation criterion for Riemann–Weber type half-linear differential equations

Kombinaci modifikovane pololinearni Pruferovy metody a Riccatiho metody jsou studovany oscilacni vlastnosti pololinearnich diferencialnich rovnic. Užitim transformacni teorie pololinearnich rovnic a jistých znamých výsledků je ukazano, že analyzovane rovnice v Riemannově-Weberově tvaru s perturbacemi v obou clenech jsou podminěně oscilatoricke. V ramci metody jsou identifikovany kriticke oscilacni hodnoty jejich koeficientů a nasledně je rozhodnuto, kdy uvažovane rovnice jsou oscilatoricke a kdy jsou neoscilatoricke. Jako přimý důsledek hlavniho výsledku je řesen tzv. kritický připad pro jistý typ pololinearnich neperturbovaných rovnic.

Oscillation criteria for neutral half-linear differential equations without commutativity in deviating arguments

Oscillation criteria for neutral half-linear differential equations without commutativity in deviating arguments

On conjugacy of second-order half-linear differential equations on the real axis

On conjugacy of second-order half-linear differential equations on the real axis

Precise asymptotic behavior of regularly varying solutions of second order half-linear differential equations

Precise asymptotic behavior of regularly varying solutions of second order half-linear differential equations

Lyapunov-type inequalities for higher order half-linear differential equations

The Lyapunov inequality for second order linear differential equations has been extended to higher order linear differential equations in various forms. It has also been extended to second and third order half-linear equations. However, it has not yet been developed for higher order half-linear differential equations due to the nonlinear feature of the equation and the complexity caused by the multiple p-Laplacian operators in the equation. In this paper, by subtle applications of forward and backward inductions, we establish several Lyapunov-type inequalities for even and odd order half-linear differential equations. These inequalities are applied to determine the nonexistence of nontrivial solutions of higher order boundary value problems and to estimate eigenvalues for higher order eigenvalue problems. Our results cover many results in the literature when the equations become linear.

Oscillations of Third Order Half Linear Neutral Differential Equations

In this paper the oscillation criterion was investigated for all solutions of the third-order half linear neutral differential equations. Some necessary and sufficient conditions are established for every solution of (a(t)[(x(t)±p(t)x(?(t) ) )^'' ]^? )^'+q(t) x^? (?(t) )=0, t?t_0, to be oscillatory. Examples are given to illustrate our main results.

Oscillations of Third Order Half Linear Neutral Differential Equations

In this paper the oscillation criterion was investigated for all solutions of the third-order half linear neutral differential equations. Some necessary and sufficient conditions are established for every solution of (a(t)[(x(t)±p(t)x(?(t) ) )^'' ]^? )^'+q(t) x^? (?(t) )=0, t?t_0, to be oscillatory. Examples are given to illustrate our main results.

Oscillations of even order half-linear impulsive delay differential equations with damping

In this paper, a kind of half-linear impulsive delay differential equations with damping is studied. By employing a generalized Riccati technique and the impulsive differential inequality, we derive several oscillation criteria which are either new or improve several recent results in the literature. In addition, we provide several examples to illustrate the use of our results.

Oscillation constants for half-linear difference equations with coefficients having mean values

We investigate second order half-linear Euler type difference equations whose coefficients have mean values. We show that these equations are conditionally oscillatory and we explicitly identify the corresponding oscillation constants given by the coefficients. Our results generalize the known ones concerning equations with positive constant, periodic, or (asymptotically) almost periodic coefficients. We also demonstrate the obtained results on examples and we give corollaries. In particular, we get new results even for linear difference equations.

Non-oscillation of perturbed half-linear differential equations with sums of periodic coefficients

We investigate perturbed second order Euler type half-linear differential equations with periodic coefficients and with the perturbations given by the finite sums of periodic functions which do not need to have any common period. Our main interest is to study the oscillatory properties of the equations in the case when the coefficients give exactly the critical oscillation constant. We prove that any of the considered equations is non-oscillatory in this case.

Exponential estimates for solutions of half-linear differential equations

This paper is concerned with estimates, unimprovable in a certain sense, for positive solutions to the half-linear differential equation ( $${|y^{\prime}|^{\alpha - 1} {\rm sgn} y^{\prime})^{\prime} = p(t) |y|^{\alpha - 1} {\rm sgn} y}$$ , where p is a continuous nonnegative function on $${[0, \infty)}$$ and $${\alpha> 1}$$ . It is shown that any positive increasing solution y of the equation satisfies $${y(t) \geqq y(0) exp{h \int_{0}^{t} p^{\frac{1}{\alpha}}(s) {\rm d}s}}$$ , with $${h< (\alpha - 1)^{-\frac{1}{\alpha}}}$$ , for all t on the complement of a set of finite Lebesgue measure. Under an additional assumption, this estimate holds for all t. Further, a condition is established which guarantees that the equation has exponentially increasing solutions and exponentially decreasing solutions.

Oscillation and non-oscillation of Euler type half-linear differential equations

We investigate oscillatory properties of second order Euler type half-linear differential equations whose coefficients are given by periodic functions and functions having mean values. We prove the conditional oscillation of these equations. In addition, we prove that the known oscillation constants for the corresponding equations with only periodic coefficients do not change in the studied more general case. The presented results are new for linear equations as well.

Hyers-Ulam-Rassias Stability by Fixed Point Technique for Half-linear Differential Equations with Unbounded Delay

Hyers-Ulam-Rassias Stability by Fixed Point Technique for Half-linear Differential Equations with Unbounded Delay

Lyapunov-type inequalities for $(m+1)$th order half-linear differential equations with anti-periodic boundary conditions

In this work, we will establish several new Lyapunov-type inequalities for (m + 1)th order half-linear differential equations with anti-periodic boundary condi- tions, the results of this paper are new and generalize and improve some early results in the literature.