Power comparison theorems for oscillation problems for second order differential equations with p(t) -Laplacian
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POWER COMPARISON THEOREMS FOR OSCILLATION PROBLEMS FOR SECOND ORDER DIFFERENTIAL EQUATIONS WITH p(t)-LAPLACIAN K. FUJIMOTO Department of Mathematics and Statistics, Masaryk University, Kotl´ aˇsk´ a 2, CZ-61137 Brno, Czech Republic e-mail: [email protected] (Received October 17, 2019; revised January 12, 2020; accepted January 13, 2020)
Abstract. This paper deals with the nonlinear differential equation (r(t)|x′ |p(t)−2 x′ )′ + c(t)|x|p(t)−2 x = 0, where r(t) > 0 and c(t) are continuous functions, and p(t) > 1 is a smooth function. We establish a comparison theorem for the oscillation problem for this equation with respect to the power p(t). Using our result, we can utilize oscillation criteria given for half-linear differential equations to equations with p(t)-Laplacian.
1. Introduction We consider the second order nonlinear differential equation (1.1)
(r(t)|x′ |p(t)−2 x′ )′ + c(t)|x|p(t)−2 x = 0,
where r(t) and c(t) are positive continuous functions satisfying (1.2)
0 < lim inf r(t) and t→∞
lim sup r(t) < ∞, t→∞
and p(t) > 1 is a smooth function defined on (0, ∞). A nontrivial solution x(t) of (1.1) is said to be oscillatory if there exists a sequence {tn } tending to ∞ such that x(tn ) = 0. Otherwise, it is said to be nonoscillatory, that is, it is eventually positive (or eventually negative). For simplicity, we call it a positive solution (or negative solution). This work was supported by JSPS KAKENHI Grant Number JP17J00259. Key words and phrases: oscillation, comparison theorem, p(t)-Laplacian, half-linear differential equation, Riccati technique. Mathematics Subject Classification: primary 34C10, secondary 34C15. c 2020 0236-5294/$ 20.00 © � 0 Akad´ emiai Kiad´ o, ´ Budapest 0236-5294/$20.00 Akade ´miai Kiado , Budapest, Hungary
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K. K. FUJIMOTO FUJIMOTO
The differential operator in (1.1) is called p(t)-Laplacian. Such operator appears in mathematical models in the study of image processing and electrorheological fluids (see [1,10,11]). In recent years, increasing attention has been paid to the study of oscillation problems for nonlinear differential equations with p(t)-Laplacian. For example, those results can be found in [9,13–15,18–20]. If p(t) ≡ p > 1, then p(t)-Laplacian is the well-known p-Laplacian, and (1.1) becomes the half-linear differential equation
(1.3)
r(t)|x′ |p−2 x′
′
+ c(t)|x|p−2 x = 0,
whose solution space has just one half of the properties which characterize linearity, namely homogeneity. Numerous papers have been devoted to the study of oscillation problems for half-linear differential equations; we can refer to [2–8,12,16,17] and the references cited therein. For example, the following Leighton–Wintner type oscillation criterion is well-known (see [7]). Theorem A. All nontrivial solutions of (1.3) are oscillatory provided ∞ ∞ 1−˜ p (r(t)) dt = ∞ and c(t) dt = ∞, 1
1
where p˜ = p/(p − 1).
Moreover, according to [7], we see that the classical linear Sturmian comparison theorem extends verbatim to (1.3). From the Sturmian comparison theorem, the com
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