Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals

Lemma 1 (see [15].)Suppose that (H₄) holds. Let x : 𝕋 → ℝ. If x^Δ exists for all sufficiently large t ∈ 𝕋, then (x(δ(t))) ^Δ = x^Δ(δ(t))δ^Δ(t) for all sufficiently large t ∈ 𝕋.

Lemma 2 (see [12].)Assume that x(t) is Δ-differentiable and eventually positive or eventually negative, then

Lemma 3 (see [16].)Suppose that X and Y are nonnegative, then

where equality holds if and only if X = Y.

Lemma 4 (see [17].)Let $$ and η(t) ∈ L_ξ[a, b] satisfy u(t)⩾0 (≢0), η(t) > 0 on $$, and

Then,

Lemma 5 (see [17].)Let τ(t) ∈ C_rd([t₀, ∞) _𝕋, [t₀, ∞) _𝕋) satisfying 0 ⩽ τ(t) ⩽ t, and $$. Assume $$ such that r(t) | x^Δ(t)|^α−1x^Δ(t) is nonincreasing on [t₀, ∞) _𝕋, where r(t) ∈ C_rd([t₀, ∞) _𝕋, (0, ∞)), α > 0 is a constant. Then,

Theorem 6. Assume that (H₁)–(H₆) hold. If there exist a function $$ and a function η(s) ∈ L_ξ[a, b] such that η(s) > 0 on $$,

where then (1) is oscillatory.

Proof. Suppose that (1) has a nonoscillatory solution x(t), then there exists T₀ (∈𝕋)⩾t₀ such that x(t) ≠ 0 for all t ∈ [T₀, ∞) _𝕋. Without loss of generality, we assume that x(t) > 0, x(τ(t)) > 0, x(δ(t)) > 0, and x(g(t, s)) > 0 for t ∈ [T₀, ∞) _𝕋, $$, because a similar analysis holds for x(t) < 0, x(τ(t)) < 0, x(δ(t)) < 0, and x(g(t, s)) < 0. Then, from (1), (H₂), and (H₅), we get

Therefore, r(t) | Z^Δ(t)|^α−1Z^Δ(t) is a nonincreasing function, and Z^Δ(t) is eventually of one sign.

We claim that

Otherwise, if there exists a t₁ (∈𝕋)⩾T₀ such that Z^Δ(t) < 0 for t ∈ [t₁, ∞) _𝕋, then, from (14), there exists some positive constant K such that that is, and integrating the above inequality from t₁ to t, we have Letting t → ∞, from (H₂), we get lim _t→∞Z(t) = −∞, which contradicts (14). Thus, we have proved (15).

We choose some T₁ (∈𝕋)⩾T₀ such that δ(t)⩾T₀ for t ∈ [T₁, ∞) _𝕋. Therefore, from (14), (15), and the fact δ(t) ⩽ σ(t), we have that

which follows that On the other hand, from (1), (H₅), and (15), we obtain Notice (15) and the fact Z(t)⩾x(t), we get where $$.

Define

Obviously, w(t) > 0. From (22), (23), it follows that Now, we consider the following two cases.

In the first case, α⩾1. By (15), (H₄), and Lemmas 1 and 2, we have

From (H₄), (20), (23)–(25), and the fact that Z(t) is nondecreasing, we obtain

In the second case, 0 < α < 1. By (15), (H₄), and Lemmas 1 and 2, we get

From (H₄), (20), (23)-(24), (27), and the fact that Z(t) is nondecreasing, we have Therefore, for α > 0, from (26) and (28), we get On the other hand, it is obvious that the conditions in Lemma 5 are satisfied with x(t), τ(t), p(t) replaced by Z(t), g(t, s), and r(t), respectively. So, we have in the view of x(t)⩾(1 − p(t))Z(t), we get From (29) and (31), we obtain By (10) and (11), we have Therefore, by Lemma 4 and (33), we have that for t ∈ [T₁, ∞) _𝕋 Substituting (34) into (32), we obtain where M(t) and (ϕ^Δ(t)) ₊ are defined by (13).

Taking

by Lemma 3 and (35), we obtain Integrating above inequality (37) from T₁ to t, we have Since w(t) > 0 for t > T₁, we have which contradicts (12). This completes the proof of Theorem 6.

Remark 7. If we take r(x, s) = 0 and use the convention that ln  0 = −∞, e^−∞ = 0, then Theorems 6 reduces to [15, Theorems 3.1]. If furthermore α⩾1 is a quotient of odd positive integer, then Theorem 6 reduces to [14, Theorem 3.1].

Remark 8. The function η(t) satisfying (10) and (11) in Theorem 6 can be constructed explicitly for any nondecreasing function ξ. In fact, if we assume that $$, and let $$,

It is easy to see that $$ and

Moreover,

Let Then, we obtain that By the continuous dependence of η(s, l) on l, there exists l^* ∈ (0,1) such that η(s)≔η(s, l^*) satisfies

Remark 9. Set 𝕋₁ = ℕ, a = 1, b = n for n ∈ ℕ, and

•  

  ξ(s) = s;

•  

  θ(s) = β_s, (s = 1,2, …, n) satisfying β₁ > β₂ > ⋯>β_m > α > β_m+1 > ⋯>β_n;

•  

  k(t, s) = q_s(t), s = 1,2, …, n;

•  

  g(t, s) = τ_s(t), s = 1,2, …, n;

Then, (1) reduces to So, if we take 𝕋 for some peculiar cases in Theorem 6, we can obtain various results. For example, if we take 𝕋 = ℝ, p(t) = 0, |f(t, u)| = q(t) | u^α|, and n = 2 in (46), then Theorem 6 generalizes the results by [18, Theorem 2].

Next, we use the general weighted functions from the class Ϝ which will be extensively used in the sequel.

Let 𝔻 ≡ {(t, s)∈[t₀, ∞) _𝕋 × [t₀, ∞) _𝕋 : t⩾s⩾t₀}, we say that a continuous function H(t, s) ∈ C_rd(𝔻, ℝ) belongs to the class Ϝ if

• (i)

  H(t, t) = 0 for t ∈ [t₀, ∞) _𝕋 and H(t, s) > 0 for t > s⩾t₀ where t, s ∈ [t₀, ∞) _𝕋;

• (ii)

  H(t, s) has a nonpositive right-dense continuous Δ-partial derivative $$ with respect to the second variable.

Theorem 10. Assume that (H₁)–(H₆) hold. If there exist functions H(t, s) ∈ Ϝ, $$, η(t) ∈ L_ξ[a, b] such that η(s) > 0 on $$, (10) and (11) hold, and

where M(t) and (ϕ^Δ(t)) ₊ are defined as in Theorem 6, then (1) is oscillatory.

Proof. We proceed as in the proof of Theorem 6 to have (35). From (35), we obtain

Multiplying (50) (with t replaced by s) by H(t, s), integrating it with respect to s from T₁ to t for t ∈ (T₁, ∞) _𝕋, and using integration by parts and (i)-(ii), we get where Φ₊(t, s) is defined as in (49).

Taking

by Lemma 3 and (51), we obtain where U(t, s) is defined as in (48).

Then, it follows that

Thus, from (54), we get

Therefore,

which contradicts (47). This completes the proof of Theorem 10.

Remark 11. In the literature, there are so many results for second-order nonlinear neutral functional dynamic equation; however, to the best of our knowledge, there is no work done attempting to study the neutral functional dynamic equation with an infinite number of nonlinear terms. Hence, our paper seems to be the first one dealing with this untouched problem. Our results not only unify the existing results in the literature, but also extend the existing results to a wider class of dynamic equations.

Example 1. Consider the following dynamic equation:

In (57), r(t) = 1, α = 3/2, p(t) = 1/(1 + t²), f(t, u) = 0, a = 1, b = 2, θ(s) = s, ξ(s) = s, and q₀ > 1 is a constant.

If $$ and $$, δ(t) = t/q₀, g(t, s) = q₀t, and $$, where k₀ is an arbitrary positive integer, then δ^Δ(t) = 1/q₀. We can choose η(t) = 1, ϕ(t) = t. Then, it is easy to get that M(t) = 1/t, and therefore,

Hence, by Theorem 6, (57) is oscillatory.

Example 2. Consider on 𝕋 = ℝ the following differential equation:

In (59), r(t) = t^1/2, α = 3/2, p(t) = 1 − 1/(1 + t²),  τ(t) = t − 1,  δ(t) = t/2,  β, λ > 0 are constants, f(t, x(δ(t)) = (β/t²)((4 + t²)/4) ^3/2 | x(t/2)|^1/2x(t/2), q(t) = (β/t²)((4 + t²)/4) ^3/2,  k(t, s) = λ(1 + 2t²) ^3s/t²,    g(t, s) = t,  a = 0,  b = 1,  θ(s) = 3s, and  ξ(s) = s.

We can choose η(t) = 1, ϕ(t) = t. Then, it is easy to get that M(t) = (β + λ)/t, δ^′(t) = 1/2, and therefore, from Theorem 6,

Hence, by Theorem 6, (59) is oscillatory if β + λ > 2^7/2/5^5/2.