# Article

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Keywords:
regular half-linear equation; principal solution; Picone’s identity; Riccati-type equation
Summary:
We introduce the concept of the regular (nonoscillatory) half-linear second order differential equation $\left(r(t)\Phi (x^{\prime })\right)^{\prime }+c(t)\Phi (x)=0\,,\quad \Phi (x):=|x|^{p-2}x\,,\quad p>1 \qquad \mathrm {{(*)}}$ and we show that if (*) is regular, a solution $x$ of this equation such that $x^{\prime }(t)\ne 0$ for large $t$ is principal if and only if $\int ^\infty \frac{dt}{r(t)x^2(t)|x^{\prime }(t)|^{p-2}}=\infty \,.$ Conditions on the functions $r,c$ are given which guarantee that (*) is regular.
References:
[1] Allegretto W., Huang Y. X.: A Picone’s identity for the $p$-Laplacian and applications. Nonlin. Anal. 32 (1998), 819–830. MR 1618334 | Zbl 0930.35053
[2] Cecchi M., Došlá Z., Marini M.: Principal solutions and minimal set for quasilinear differential equations. to appear in Dynam. Syst. Appl. MR 2140874
[3] Došlý O., Elbert Á.: Integral characterization of the principal solution of half-linear differential equations. Studia Sci. Math. Hungar. 36 (2000), No. 3-4, 455–469. MR 1798750
[4] Došlý O., Lomtatidze A.: Oscillation and nonoscillation criteria for half-linear second order differential equations. submitted. Zbl 1123.34028
[5] Elbert Á.: A half-linear second order differential equation. Colloq. Math. Soc. János Bolyai 30 (1979), 158–180.
[6] Elbert Á.: Asymptotic behaviour of autonomous half-linear differential systems on the plane. Studia Sci. Math. Hungar. 19 (1984), 447–464. MR 0874513 | Zbl 0629.34066
[7] Elbert Á.: The Wronskian and the half-linear differential equations. Studia Sci. Math. Hungar. 15 (1980), 101–105. MR 0681431 | Zbl 0522.34034
[8] Elbert Á., Kusano T.: Principal solutions of nonoscillatory half-linear differential equations. Advances in Math. Sci. Appl. 18 (1998), 745–759. MR 1657164
[9] Elbert Á., Schneider A.: Perturbation of the half-linear Euler differential equations. Result. Math. 37 (2000), 56–83. MR 1742294
[10] Hartman P.: Ordinary Differential Equations. John Wiley, New York, 1964. MR 0171038 | Zbl 0125.32102
[11] Jaroš J., Kusano T.: A Picone type identity for half-linear differential equations. Acta Math. Univ. Comenianea 68 (1999), 137–151. MR 1711081
[12] Leighton W., Morse M.: Singular quadratic functionals. Trans. Amer. Math. Soc. 40 (1936) 252–286. MR 1501873 | Zbl 0015.02701
[13] Lorch L., Newman J. D.: A supplement to the Sturm separation theorem, with applications. Amer. Math. Monthly 72 (1965), 359–366, 390. MR 0176147 | Zbl 0135.29702
[14] Mirzov J. D.: On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems. J. Math. Anal. Appl. 53 (1976), 418–426. MR 0402184 | Zbl 0327.34027
[15] Mirzov J. D.: Principal and nonprincipal solutions of a nonoscillatory system. Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 31 (1988), 100–117. MR 1001343

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