New Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations

International Journal of Mathematical Analysis Vol. 11, 2017, no. 19, 945-954 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2017.79127 New Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations M. M. A. El-sheikh Department of Mathematics, Faculty of Science Minoufiya University, Shebeen EL-Koom, Egypt A. A. Soliman Department of Mathematics, Faculty of Science Benha University, Benha-Kalubia, Egypt M. H. Abdalla Department of Mathematics, Faculty of Science Benha University, Benha-Kalubia, Egypt A. M. Hassan Department of Mathematics, Faculty of Science Benha University, Benha-Kalubia, Egypt Copyright c 2017 M. M. A. El-sheikh, A. A. Soliman,M. H. Abdalla and A. M. Hassan. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We establish some new oscillation criteria for a class of second-order nonlinear neutral delay dynamic equations using a couple of Riccati substitutions. Our main results not only complement those related results in the literature, but also improve some known results for second-order delay dynamic equations without neutral terms. Mathematics Subject Classification: 34C10, 34K11

946 M. M. A. El-sheikh, A. A. Soliman, M. H. Abdalla and A. M. Hassan Keywords: Oscillation, second order, Nonlinear dynamic equations, Riccati technique 1 Introduction In this paper, we introduce new sufficient conditions for the oscillation of solutions of the neutral dynamic equation rt)ϕ z t)) ] + qt)fϕ xδt))) = 0 for t, ) T 1) where zt) := xt) + pt)xτt)), ϕ γ λ) := sgnλ) λ γ for λ R and γ R +,, R +. We assume the following conditions. H 1 ) r C 1 rd, ) T, R), r 1/ s =. H 2 ) τ, δ C 1 rd, ) T, T), τt) t, δt) t, τ δ = δ τ, lim t τt) =, lim t δt) =, τ t), and δ t) > 0 where is a constant. H 3 ) p, q C rd, ) T, R), 0 pt) p 0 <, qt) 0 and qt) is not identically zero for large t. H 4 ) f CT, T), and there exists a positive constant k such that fx) K x for all x 0. The theory of time scales was introduced by Hilger see 5]) in 1988 in order to unify continuous and discrete analysis. A time scale, which inherits the standard topology on R, is a nonempty closed subset of reals. Here, and later throughout this paper, a time scale will be denoted by the symbol T, and the intervals with a subscript T are used to denote the intersection of the usual interval with T. For t T, the forward jump operator σ : T T is defined by σt) := inft, ) T, while the backward jump operator ρ : T T is defined by ρt) := sup, t) T, and the graininess function µ : T R + is defined to be µt) := σt) t. A point t T is called right-dense if σt) = t and/or equivalently µt) = 0 holds; otherwise, it is called right-scattered, and similarly left-dense and left-scattered points are defined with respect to the backward jump operator. The set of all such rd-continuous functions is denoted by C rd T, R). The set of functions f : T R which are differentiable and whose derivative is an rd-continuous function is denoted by Crd 1 T, R). For some concepts related to the notion of time scales, see 4]. By a solution of 1) we mean a nontrivial function x C rd T x, ) T, R), where T x, ) T, which has the property that rt)ϕ x+p x τ) t)) ] C 1 rd T x, ) T, R) and satisfies 1) identically on T x, ) T. A solution x of 1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory. Equation 1) is called oscillatory if all its solutions oscillate.

New oscillation criteria for second-order neutral delay dynamic equations 947 In recent years there has been much research activities concerning the oscillation of solutions of several classes of neutral dynamic equations, see 7,9,10,12] Several papers are devoted to study the cases in which 0 < pt) < 1 and 0 < pt) p 0 <, for instance, in case of T = R Baculíková and Džurina 2] studied the second order neutral differential equation ] rt)xt) + pt)xτt))) + qt)fxδt))) = 0. They presented new oscillation criteria, where they replaced the traditional restriction 0 pt) < 1 by 0 < pt) p 0 < and δt) τt) < t. They use new comparison theorems, that enable them to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation. In 13] Zhang et al. introduce new oscillation criteria for the class of second order dynamic equations of the type, rt)xt) + pt)xτt))) ] + qt)fxδt))) = 0. s <. In rs) 3] Baculíková and Džurina studied the oscillation of the second-order neutral differential equations of the form )] rt) xt) + pt)xτt))) + qt)x δt)) = 0. Under the conditions 0 < pt) p 0 <, s = and rs) where, are the ratios of two positive odd integers, 0 pt) <, τt) t, δt) τt) t, but they did not consider the case τt) δt) t. Our aim in this paper is to obtain some new sufficient conditions for 1) and improve the results of 1, 3, 13, 14]. Now, we present some known results, which needed in the proof of our main results. Theorem 1.1. 4] Assume that v : T R is strictly increasing and T := vt) is a time scale. Let y : T R. If y vt)] and v t) exist for t T k, then yvt)]) = y vt)]v t). Lemma 1.2. 11] If X 0, Y 0, and 0 < λ 1, then X λ + Y λ X + Y ) λ. Lemma 1.3. 3] If X 0, Y 0, and λ 1, then X λ + Y λ 2 1 λ X + Y ) λ. Lemma 1.4. 6] If B > 0, A > 0, and > 0, then Au Bu +1 + 1) +1 A +1 B.

948 M. M. A. El-sheikh, A. A. Soliman, M. H. Abdalla and A. M. Hassan 2 Main results For convenience, we define K 0 = { K 0 < 1, 2 1 K, > 1., ρ +t) = max{0, ρ t)}. The following theorem introduces a new oscillation criterion when δt) τt) Theorem 2.1. Assume that H 1 )-H 4 ) and δt) τt) are satisfied. If there exists a function ρ Crd 1 T, R) such that for all constants λ 1, λ 2 > 0,we have t ) ) lim sup K 0 ρζ)qζ) 1+ p 0 /) ρ +ζ)) +1 rτζ)) ζ t + 1) +1 τ0 λ 1 ρ 1 ζ) 2) or t ) lim sup K 0 ρζ)qζ) 1 + p 0 ρ + ζ)) +1 ) λ 2 r / τζ)) ζ >, t + 1) +1 τ 0 ρ ζ) 3) where Qt) = min{qt), qτt))}, then Eq. 1) is oscillatory. Proof. Let xt) be a nonoscillatory solution of 1) with xt) > 0 on, ) T, then there exists t 1 such that xt) > 0, xτt)) > 0, xδt)) > 0, for all t, ) T. By the definition of zt), we have z > 0 and zt) xt), t t 1. From 1), we have rt)ϕ z t))] Kqt)x δt)) 0. 4) From H 1 ) and 4), one can easily obtain z t) > 0. then 4) becomes rt)z t)) ] Kqt)x δt)) 0. 5) It follows from Theorem 1.1 that rτt))z τt))) ] = rt)z t)) ] τ t), that there exists a t 2 T such that rτt))z τt))) ] τ t) Kqτt))x δτt))). But since τ t) > 0, we get for, t t 2, 1 rτt))z n 1 τt))) ] Kqτt))x δτt))). 6)

New oscillation criteria for second-order neutral delay dynamic equations 949 Combining 5) and 6), we obtain rt)z t)) ] + p 0 rτt))z n 1 τt))) ] + Kqt)x δt)) + p 0Kqτt))x δτt))) 0 7) Assume tha < 1. Since δ τ = τ δ and Lemma 1.2, we get rt)z t)) ] + p 0 rτt))z τt))) ] Kqt)x δt)) p 0 Kqτt))x δτt))) Now, if > 1. Similarly, in view of Lemma 1.3, we have KQt)x δt)) + x δτt)))] KQt)xδt)) + xδτt)))] KQt)z δt)). 8) rt)z t)) ] + p 0 rτt))z τt))) ] KQt)x δt)) + x δτt)))] It follows from 8) and 9) that 2 1 KQt)xδt)) + xδτt)))] 2 1 KQt)z δt)). 9) rt)z t)) ] + p 0 rτt))z τt))) ] K0 Qt)z δt)) 10) Now, we define a Riccati substitution ωt) := ρt) rt)z t)) z τt)) It is clear that ω > 0 for all t t 1, and ω t) = ρt) ] ρt) z τt)) rt)z t)) ] + rt)z t)) ] σ z τt)) =ρt) rt)z t)) ] z τt)) + ρ t) ρσt)) ωσt)) τ ρt) t) ρσt)) for allt t 1, ) T 11) z τt)) ωσt)). 12) zτt)) Since rt)z t)) decreasing, then rt)z t)) rτt))z τt))), i.e., ) rt) 1/ z τt)) z t). 13) rτt)) This with 12) leads to ω t) ρt) rt)z t)) ] z τt)) + ρ t) ρσt)) ωσt)) ρt) ρσt)) rt) rτt)) ) 1/ z t) zτt)) ωσt)) 14)

950 M. M. A. El-sheikh, A. A. Soliman, M. H. Abdalla and A. M. Hassan Since, and z t) > 0, then there exists a constant λ 1 > 0 such that zt) zτt)) λ 1. Using 13) and 14), we get ω t) ρt) rt)z t)) ] z + ρ t) τt)) Applying Lemma 1.4, we obtain ρσt)) ωσt)) ρ +1 ρt)λ 1 σt))r 1 τt)) ω+1)/ σt)) 15) ω t) ρt) rt)z t)) ] z τt)) Similarly, define another Riccati substitution + + 1) +1 ρ t)) +1 rτt)) τ 0 λ 1 ρ 1 t) 16) νt) := ρt) rτt))z τt))) z τt)) for allt t 1, ) T. 17) Then we have νt) > 0. Differentiating 17), by rτt))z τt))) ] rτt))z τt))) ] σ > 0 and zτ σ t)) λ 1 ν t) = ρt) ] ρt) z τt)) rτt))z τt))) ] + rτt))z τt))) ] σ z τt)) =ρt) rτt))z τt))) ] z τt)) Applying Lemma 1.4, we get ν t) ρt) rτt))z τt))) ] z τt)) Combining 16) and 18), we conclude that ω t) + p 0 ν t) Recalling 10), implies ρt) z τt)) ) + 1 + p 0 + ρ t) ρσt)) νσt)) ρt)λ1 ρ +1 σt))r 1 τt)) ν +1 + rt)z t)) ] + p 0 + 1) +1 ρ t)) +1 rτt)) τ 0 λ 1 ρ 1 t) + 1) +1 ρ t)) +1 rτt)) τ 0 λ 1 ρ 1 t) ) rτt))z τt))) ] σt)) ) ω t) + p 0 ν t) K 0 ρt)qt) z δt)) z τt)) + 1 + p 0 ρ t)) +1 rτt)) + 1) +1 τ0 λ 1 ρ 1 t). 20) Since δt) τt) and z t) > 0, then zδt)) zτt)). This leads to ) ω t) + p 0 ν t) K 0 ρt)qt) + 1 + p 0 ρ t)) +1 rτt)) + 1) +1 τ0 λ 1 ρ 1 t). 21) 18) 19)

New oscillation criteria for second-order neutral delay dynamic equations 951 Integrating 21) from t 1 to t, we see that t ) K 0 ρζ)qζ) 1 + p 0 /) ρ ζ)) +1 rτζ)) t 1 + 1) +1 τ0 λ 1 ρ 1 ζ) ) ζ ωt 1 ) + p 0 νt 1 ), 22) which contradicts 2). Caseii): >. Define the function ω by 11). Then 14) holds. Since z > 0, there exists a constant λ 2 > 0 such that rt)z t)) rτt))z τt))) λ 2, hence from 11), and 14), we obtain z t)) ω t) ρt) rt)z t)) ] z + ρ t) τt)) This with 23) leads to ω t) ρt) rt)z t)) ] z + ρ t) τt)) Applying Lemma 1.4, we conclude rt) λ 2 ) ρσt)) ωσt)) τ 0ρt)r ρσt)) ωσt)) ω t) ρt) rt)z t)) ] z τt)) ρ +1 t)r 1/ τt)) σt))z t)) 23) ω +1 σt)) 24) ρt)λ2 ω +1 ρ +1 σt)) 25) σt))r 1/ τt)) + ρ t)) +1 λ 2 r / τt)) + 1) +1 τ 0 ρ t) On the other hand, define ν as in 17). Similarly, we have ν t) ρt) rt)z t)) ] z τt)) + ρ t)) +1 By virtue of 10), 26), and 27), we deduce that rt)z t)) ] + p 0 ω t) + p 0 ν t) + ρt) z τt)) 1 + p 0 ) ρ t)) +1 λ 2 r / τt)) + 1) +1 τ 0 ρ t) ) rτt))z τt))) ] λ 2 r / τt)) + 1) +1 τ 0 ρ t) 1 + p 0 K 0 ρt)qt) z δt)) z τt)) + ) ρ t)) +1 26) 27) λ 2 r / τt)) + 1) +1 τ 0 ρ t) 28) Since δt) τt) and z t) > 0, then we obtain ω t) + p 0 ν t) K 0 ρt)qt) + 1 + p 0 ) ρ t)) +1 λ 2 r / τt)) + 1) +1 τ 0 ρ t) 29)

952 M. M. A. El-sheikh, A. A. Soliman, M. H. Abdalla and A. M. Hassan Consequently, t t 1 K 0 ρζ)qζ) 1 + p 0 λ ) 2 r / τζ)) + 1) +1 τ 0 ρ ζ) ) ρ ζ)) +1 this contradicts 3). The proof is complete. ζ ωt 1 ) + p 0 νt 1 ) Example 2.2. Consider the second-order neutral differential equation 30) 1 xt) + 2 xt 1)) γ + xt) = 0 for t 1. 31) t2 where γ > 0 is a constant, rt) = 1, p 0 = 1 2, τt) = t 1, = 1, Qt) = γ t 2 and δt) = t.clearly, = = 1 and τt) δt). Choose ρt) = t, then by 2) lim sup t t K 0 ρζ)qζ) = lim sup t t K 0 γ ζ 1 + p 0 3 8 ) 1 ζ ) ) /) ρ ζ)) +1 rτζ)) ζ + 1) +1 τ0 λ 1 ρ 1 ζ) ) dζ = 32) provided that γ > 3 8K 0. Hence, 31) is oscillatory if γ > 3 8K 0. For any K 0 > 1 our result is better than results obtained in 13]. Example 2.3. Consider the second order neutral differential equation t 1/2 xt) + p 0 xγt)) ] a + xµt) = 0 33) t3/2 where 0 < γ <, 0 < µ < 1 and a > 0. Here rt) = t 1/2, 0 < p 0 <, τt) = γt, = γ, qt) = Qt) = a and δt) = µt. Here = = 1. t 3/2 In 3], the authors studied this example in some cases for γ,, but they didn t get results in case of τt) < δt) < t, to obtain this case choose 0 < γ < µ < 1 and ρt) = t. Application of 2), then we get lim sup t t K 0 ρζ)qζ) = lim sup t t 1 + p 0 a ζ 3/2 ζ 1 4 Hence, 33) is oscillatory if a > 1 4 ) ) /) ρ ζ)) +1 rτζ)) ζ + 1) +1 τ0 λ 1 ρ 1 ζ) ) 1 )dζ = γ ζ 1 + p 0 γ ) 1 + p 0 γ 1 γ. 34)

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