Asymptotic behavior of third order functional dynamic equations with γ-Laplacian and nonlinearities given by Riemann – Stieltjes integrals

In this paper, we study the third-order functional dynamic equations with γ-Laplacian and nonlinearities given by Riemann–Stieltjes integrals { r2 (t) φγ2 ([ r1 (t) φγ1 ( x∆ (t) )]∆)}∆ + ∫ b a q (t, s) φα(s) (x(g (t, s))) dζ (s) = 0, on an above-unbounded time scale T, where φγ(u) := |u|γ−1 u and ∫ b a f (s) dζ (s) denotes the Riemann–Stieltjes integral of the function f on [a, b] with respect to ζ. Results are obtained for the asymptotic and oscillatory behavior of the solutions. This work extends and improves some known results in the literature on third order nonlinear dynamic equations.


Introduction
We are concerned with the asymptotic and oscillatory behavior of the third order nonlinear functional dynamic equation q(t, s)φ α(s) (x(g(t, s))) dζ(s) = 0 (1.1) on an above-unbounded time scale T, where φ γ (u) := |u| γ−1 u, γ 1 , γ 2 > 0; α ∈ C[a, b] with −∞ < a < b < ∞ such that α(s) > 0 is strictly increasing, r i , i = 1, 2, are positive rdcontinuous functions on T; q is a positive rd-continuous function on T × [a, b]; and g : T × [a, b] → T is a rd-continuous function such that lim t→∞ g(t, s) = ∞ for s ∈ [a, b].Without loss of generality we assume 0 ∈ T. Hence we may discuss the solutions of Eq. (1.1) on [0, ∞) T .Here b a f (s) dζ(s) denotes the Riemann-Stieltjes integral of the function f on [a, b] with respect to ζ.We note that as special cases, the integral term in the equation becomes a finite sum when ζ(s) is a step function and a Riemann integral when ζ(s) = s.Throughout this paper, we let It is easy to see that all solutions of Eq. (1.1) can be extended to ∞ if either g (t, s) ≤ t − τ for some τ > 0 and all t ∈ T and s ∈ [a, b] or T is a discrete time scale and g (t, s) ≤ t for all t ∈ T and s ∈ [a, b].However, Eq. (1.1) may have both extendable solutions and nonextendable solutions in general.For the asymptotic and oscillation purposes, we are only interested in the solutions that are extendable to ∞.Thus, we use the following definition of solutions.Definition 1.1.By a solution of Eq. (1.1) we mean a nontrivial real-valued function x ∈ C 1 rd [T x , ∞) T for some T x ≥ t 0 such that x [1] , x [2] ∈ C 1 rd [T x , ∞) T , and x(t) satisfies Eq. (1.1) on [T x , ∞) T , where C rd is the space of right-dense continuous functions, and C 1 rd is the space of functions whose ∆-derivatives are right-dense on [T x , ∞) T .
In the last few years, there has been an increasing interest in obtaining sufficient conditions for the oscillation/nonoscillation of solutions of different classes of dynamic equations, we refer the reader to the papers [1,2,6,7,9,15,17,19,20,21,24,26,28] and the references cited therein.Regarding third order dynamic equations, Erbe, Peterson, and Saker [10,11] and Yu and Wang [29] obtained sufficient conditions for oscillation for the third order dynamic equations r 2 (t) r 1 (t)x ∆ (t) ∆ ∆ + p(t)x(t) = 0, r 2 (t) r 1 (t)x ∆ (t) ∆ γ ∆ + p(t)x γ (t) = 0, and r 2 (t) r 1 (t) x ∆ (t) where γ ≥ 1 is the quotient of odd positive integers and r 1 , r 2 , p ∈ C rd (T) are positive.Hassan [16] and Erbe, Hassan, and Peterson [12] extended their work to the dynamic equation with delay for the case that γ ≥ 1 and γ > 0, respectively, where h(t) is a monotone delay function on T. A number of sufficient conditions for oscillation were obtained for the cases when Asymptotic behavior of third order functional dynamic equations 3 respectively.Also, Han, Li, Sun, and Zhang [18] discussed the third order delay dynamic equation where g(t) ≤ t and Recently, Erbe, Hassan, and Peterson [13] extended these results to third-order dynamic equations of a more general form where certain restrictions on the delay terms were imposed.
In this paper, we study the asymptotic and oscillatory behavior of the third-order functional dynamic equation (1.1) with γ-Laplacian and nonlinearities given by Riemann-Stieltjes integrals for both the cases and The results improve and extend the oscillation criteria established in [8,10,11,12,13,16,18,24,25,26].

Asymptotic behavior
In this section, we discuss the asymptotic behavior of the solutions of (1.1) when (1.5) and (1.6) hold, respectively.The first theorem is under the assumption that (1.5) holds, the second is under the assumption that (1.6) holds, and the last one is for the general case.

If
∞ t b a q(w, s) dζ(s)∆w = ∞, we have reached a contradiction.Otherwise, Again, integrating this inequality from t to ∞ and noting that x [1] (t) ≤ 0 eventually, we get −x [1] , where L := l 1 γ 1 γ 2 > 0. Finally, integrating the last inequality from T 2 to t, we get Hence by (2.1), we have lim t→∞ x(t) = −∞, which contradicts the fact that x(t) is a positive solution of Eq. (1.1).This shows that lim t→∞ x(t) = 0 and hence completes the proof.

Remark 2.2. The conclusion of Theorem 2.1 remains intact if assumption (2.1) is replaced by the condition
Now we consider the case when (1.6) holds.We will use the following notations:

1) holds, and for any t
If Eq. (1.1) has eventually positive solution x(t), then x [2] (t) eventually, and either x ∆ (t) is eventually positive or x(t) tends to zero eventually.
Proof.Since x(t) is eventually positive solution of Eq. (1.1), then there is a 2), x [2] (t) is strictly decreasing on [T, ∞) T .This implies that x [1] (t) ∆ and x ∆ (t) are eventually of one sign.
(I) We show that x [1] (t) ∆ is eventually positive.Otherwise, it is eventually negative.We consider the following two cases: (a) x ∆ (t) < 0 and x [1] (t) ∆ < 0 eventually.In this case, there exists T 1 ≥ T such that x ∆ (t) < 0 and where where From (1.1) and (2.5) we find that Integrating this last inequality from T 2 to t, we see that which implies that Again, integrating the above inequality from T 2 to t, we get which yields From (2.3), we have lim t→∞ x(t) = −∞, which contradicts the fact that x is a positive solution of Eq. (1.1).(b) x ∆ (t) > 0 and x [1] (t) ∆ < 0 eventually.In this case, there exists T 1 ≥ T such that x ∆ (t) > 0 and Again, we let and where where > 0. By (1.1) and (2.8), Integrating both sides from T 2 to t, we have which implies that Again, integrating both sides from T 2 to t, we get −x [1] (T 2 ) < x [1] (t) − x [1] (T 2 ) which contradicts (2.4).
(II) With essentially the same proof as in Part (II) of the proof of Theorem 2.1, we can show that if x ∆ (t) is not eventually positive, then x(t) tends to zero eventually.We omit the details.

Oscillation criteria
In this section, by using the results in Section 2, we study the oscillatory behavior of the solutions of Eq. (1.1) under the assumptions (1.5) and (1.6), respectively.First, we establish oscillation criteria for Eq.(1.1) under the assumption that (1.5) holds.
Theorem 3.2.Assume that (1.5) and (2.1) hold.Suppose that for any t Then every solution of Eq. (1.1) is either oscillatory or tends to zero eventually.
Proof.Assume Eq. (1.1) has a nonoscillatory solution x(t).Then without loss of generality, assume there is a T ∈ [0, ∞) T such that x(t) > 0 on [T, ∞) T and x(g(t, s)) eventually and either x ∆ (t) is eventually positive or x(t) tends to zero eventually.We suppose that x [2] (t) ∆ < 0, x [1] (t) ∆ > 0, and x ∆ (t) > 0 eventually.Then there exists where c 1 := min s∈[a,b] c α(s) > 0. The rest of the proof is similar to that of Theorem 3.1 and hence is omitted.
In the following, we let γ := γ We note from the definition of m and n that 0 < m < 1 < n.The next lemma is a generalized arithmetic-geometric mean inequality established in [27].where we use the convention that ln 0 = −∞ and e −∞ = 0.
In the following, we denote k + := max{k, 0} for any k ∈ R. The theorem below is derived from Theorem 2.4.Theorem 3.5.Assume that (1.5) and (2.1) hold.Furthermore, suppose that there exists a positive function ϕ ∈ C 1 rd [0, ∞) T and that, for all sufficiently large t 0 ∈ [0, ∞) T , there is a t 1 > t 0 such that g(t, s) > t 0 for t ≥ t 1 and s ∈ [a, b], and where with q(u, s, t 0 ) := q(u, s)G(u, s, t 0 ) and G(u, s, t 0 Then every solution of Eq. (1.1) is either oscillatory or tends to zero eventually.
Proof.Assume Eq. ( 1.1) has a nonoscillatory solution x(t).Then without loss of generality, assume there is a By Theorem 2.1, we have eventually and either x ∆ (t) is eventually positive or x(t) tends to zero.We suppose that ∆ > 0, and x ∆ (t) > 0 eventually.Then there exists T 1 ≥ T such that Consider the Riccati substitution where γ = γ 1 γ 2 .By the product rule and the quotient rule, we get From (1.1) and the definition of w(t) we have for t ≥ T 1 , x γ (t) dζ(s) Let t ∈ [T 1 , ∞) T and s ∈ [a, b] be fixed.If g(t, s) ≥ t, then x(g(t, s)) ≥ x(t) by the fact that x(t) is strictly increasing.Now we consider the case when g(t, s) ≤ t.In view of Theorem 2.
R(t,T 1 ) is decreasing on (T 1 , ∞) T , we see that there exists T 2 ≥ T 1 such that g(t, s) > T 1 for t ≥ T 2 and s ∈ [a, b], and so In both cases, from the definition of q(t, s, T 1 ) we have that for t ≥ T 2 and s ∈ [a, b], We let η ∈ L ζ (a, b) be defined as in Lemma 3.3.Then η satisfies (3.5).It follows that This together with (3.9) shows that Then by the Pötzsche chain rule we obtain that x ∆ (t) x(σ(t)) x(σ(t)) x(t) γ ; and if γ ≥ 1, then x ∆ (t) x (σ(t)) x (σ(t)) x(t) .
Note that as x(t) is strictly increasing on [T 2 , ∞) T , we see that for γ > 0, where β := γ+1 γ .Define and Then, using the inequality (see [14]) we get that From this and (3.12) we have Integrating both sides from T 2 to t we get which leads to a contradiction to (3.6).Theorem 3.6.Assume that (1.5) and (2.1) hold.Furthermore, suppose that there exists a positive function ρ ∈ C 1 rd [0, ∞) T and that for all sufficiently large t 0 ∈ [0, ∞) T , there is a t 1 > t 0 such that g(t, s) > t 0 for t ≥ t 1 and s ∈ [a, b], and where with q(u, s, t 0 ) := R γ (u, t 0 )G(u, s, t 0 )q(u, s) and G(u, s, t 0 ) is given by (3.7).Then every solution of Eq. (1.1) is either oscillatory or tends to zero eventually.
where g(t {t, g(t, s)} and with q(t, s) := ∞ t q(u, s)∆u.Then every solution of Eq. (1.1) is either oscillatory or tends to zero eventually.
At the end of this paper, we establish parallel results to Theorems 3.1-3.7 under the assumption that (1.6) holds.Theorem 3.12.Assume that g(t, s) be a nondecreasing function with respect to t. Assume that (2.1), (2.3), (2.4) and (3.18) hold.Then every solution of Eq. (1.1) is either oscillatory or tends to zero eventually.