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Abstract

We obtain an analogue of the integral Hille–Wintner comparison theorem for the half-linear differential equations of third order. We also give an example involving a differential equation of Euler type, which gives a condition under which half-linear differential equations have weak property B.

nonoscillatory solution in the form of a power function. The coefficient γ is considered as a function of λ 1 for b 0 = 0.5, λ 3 = 0.75, p = 3, α = 0. Conclusion In the paper we have established new oscillation criteria for second order neutral half-linear differential equations. We followed the works of previous authors and suggested a modification of the classical Riccati substitution method and comparison method which allows us to remove the restrictive condition on commutativity of the deviating arguments and also gives better oscillation constants

1 Introduction The second-order half-linear differential equation (HL) ( p ⁢ ( t ) ⁢ | x ′ | α ⁢ sgn ⁡ x ′ ) ′ + q ⁢ ( t ) ⁢ | x | α ⁢ sgn ⁡ x = 0 (p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0{} is considered under the assumption that the following conditions hold: (a) α > 0 {\alpha>0} is a constant. (b) p : [ a , ∞ ) → ( 0 , ∞ ) {p:[a,\infty)\to(0,\infty)} , q : [ a , ∞ ) → ℝ {q:[a,\infty)\to\mathbb{R}} , a > 0 {a>0} , are continuous functions and ∫ a ∞ q ⁢ ( s ) ⁢ 𝑑 s = lim t

DEMONSTRATIO MATHEMATICA Vol. XXXII No 2 1999 Guang Zhang, Sui Sun Cheng ON CONNECTED HALF-LINEAR DIFFERENTIAL EQUATIONS Abstract . In this paper connections between several classes of "half-linear" differ- ential equations with or without delays are established. By means of these connections, existence of eventually positive solutions can be inferred from the properties of either one of these families of equations. 1. Introduction To motivate our concerns in this paper, let us first look at the following four differential equations in a formal manner

Acknowledgement Second author was supported by Czech Science Foundation under Grant GA17-03224S. References [1] Hasil P., Mařík R., Veselý M., Conditional oscillation of half-linear differential equations with coefficients having mean values, Abstract Appl. Anal ., 2014 , article ID 258159, 1–14. 10.1155/2014/258159 Hasil P. Mařík R. Veselý M. Conditional oscillation of half-linear differential equations with coefficients having mean values Abstract Appl. Anal. 2014 article ID 258159 1 14 10.1155/2014/258159 [2] Agarwal R.P., Grace A.R., O’Regan D

Georgian Mathematical Journal Volume 10 (2003), Number 4, 785–797 OSCILLATION OF SECOND ORDER HALF-LINEAR DIFFERENTIAL EQUATIONS WITH DAMPING QIGUI YANG AND SUI SUN CHENG Abstract. This paper is concerned with a class of second order half-linear damped differential equations. Using the generalized Riccati transformation and the averaging technique, new oscillation criteria are obtained which are either extensions of or complementary to a number of the existing results. 2000 Mathematics Subject Classification: 34A30, 34C10. Key words and phrases: Half

[1] DOŠLÝ, O.— PÁTÍKOVÁ, Z.: Hille-Wintner type comparison criteria for half-linear second order differential equations, Arch. Math. (Brno) 42 (2006), 185–194. [2] DOŠLÝ, O.— ŘEHÁK, P.: Half-Linear Differential Equations. North-Holland Math. Stud. 202, Elsevier, Amsterdam, 2005. [3] DOŠLÝ, O.— ŰNAL, M.: Conditionally oscillatory half-linear differential equations, Acta Math. Hungar. (To appear). [4] HOWARD, H. C.— MARIĆ, V.: Regularity and nonoscillation of solutions of second order linear differential equations, Bull. Cl. Sci. Math. Nat. Sci. Math. 20 (1990

References [1] DOˇSL´Y, O.-HASIL, P.: Critical oscillation constant for half-linear differential equations with periodic coefficients, Annal. Mat. Pura Appl. (4) (to appear). [2] DOˇ SL´Y, O.-FIˇ SNAROV´A, S.: Half-linear oscillation criteria: perturbation in the term involving derivative, Nonlinear Anal. 73 (2010), 3756-3766. [3] DOˇ SL´Y, O.-ˇREH´AK, P.: Half-Linear Differential Equations. North-Holland Math. Stud., Vol. 202, Elsevier, Amsterdam, 2005. [4] DOˇ SL´Y, O.-¨UNAL, M.: Conditionally oscillatory half-linear differential equations, Acta Math. Hungar

References [1] BINGHAM, N. H.-GOLDIE, C. M.-TEUGELS, J. L.: Regular Variation. Encyclopedia Math. Appl. 27, Cambridge Univ. Press, Cambridge, 1987. [2] DOŠLÁ, Z.-VRKOČ, I.: On an extension of the Fubini theorem and its applications in ODEs, Nonlinear Anal. 57 (2004), 531-548. [3] DOŠLÝ, O.-ŘEHÁK, P.: Half-linear Differential Equations, North-Holland Math. Stud. 202, Elsevier, Amsterdam, 2005. [4] HOSHINO, H.-IMABAYASHI, R.-KUSANO, T.-TANIGAWA, T.: On second-order half-linear oscillations, Adv. Math. Sci. Appl. 8 (1998), 199-216. [5] HOWARD, H. C.-MARIĆ, V

{array}{} \displaystyle y'(t)+q(t)y(\sigma(t))\leq 0 \end{array}$$ (2.1) has no eventually positive solution . 3 Main results The following theorem presents the bridge between the second order neutral half-linear differential equation (1.1) and the first order delay differential inequality (2.1) . It introduces a bunch of parameters (both numbers and functions) and as we show hereinafter, a careful choice of the parameters allow us to cover the cases which have not been treated in the literature yet. A nontrivial example with noncommutative deviating arguments is