Oscillatory Behaviour of a First-Order Neutral Differential Equation in relation to an Old Open Problem

<jats:p>In this paper, we obtain sufficient conditions for oscillation and nonoscillation of the solutions of the neutral delay differential equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msup><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>y</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mo>−</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mrow><mml:mo stretchy="false">∑</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>k</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mi>y</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced></mml:mrow></mml:mstyle></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi>q</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mi>G</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>y</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>g</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced><mml:mo>−</mml:mo><mml:mi>u</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mi>H</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>y</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>h</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mtext> and </mml:mtext><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math> for each <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>j</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>q</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>H</mml:mi><mml:mo>,</mml:mo><mml:mi>g</mml:mi><mml:mo>,</mml:mo><mml:mi>h</mml:mi><mml:mo>,</mml:mo><mml:mtext> and </mml:mtext><mml:mi>f</mml:mi></mml:math> are all continuous functions and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi>q</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>u</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>h</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mo><</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>g</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mo><</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mtext> and </mml:mtext><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mo><</mml:mo><mml:mi>t</mml:mi></mml:math> for each <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mi>j</mml:mi></mml:math>. Further, each <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mi>g</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:mi>h</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mo>⟶</mml:mo><mml:mi>∞</mml:mi></mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M10"><mml:mi>t</mml:mi><mml:mo>⟶</mml:mo><mml:mi>∞</mml:mi></mml:math>. This paper improves and generalizes some known results.</jats:p>


Introduction
Consider the neutral delay differential equation (NDDE in short) of the first order: y(t) − k j�1 p j (t)y r j (t) ′ + q(t)G(y(g(t)))

− u(t)H(y(h(t))) � f(t),
where p j and r j for each j and q, u, g, h, G, H, and f are in C(R, R) and q ≥ 0, u ≥ 0, g(t) < t, h(t) < t, and r j (t) < t for each j. Further, each r j (t), g(t), and h(t) ⟶ ∞ as t ⟶ ∞. We study the behavior of oscillation and nonoscillation of solutions of neutral differential equation (1) under the following assumptions: xG(x) > 0, for x ≠ 0. (2) ere exists a real-valued bounded function F(t) such that e function F(t) in (3) satisfies H is bounded and uH(u) > 0, for u ≠ 0. (

y(t) − p(t)y(t − τ)) ′ + q(t)G(y(t − σ))
− u(t)H(y(t − α)) � f(t). (9) e research papers [2-5, 7, 8] and many others while studying nonlinear NDDEs assume the condition "G is nondecreasing." It is found that the authors in [3-5, 7, 8, 10] use the lemmas [1] (Lemma 1.5.1 and 1.5.2) as the main tool to study NDDEs (8) or (9). If lim inf t⟶∞ y(t) � 0 and lim t⟶∞ (y(t) − p(t)y(t − τ)) exists finitely then, these lemmas help us to evaluate lim t⟶∞ y(t). Hence, the lemmas could only be applied to study (8) or (9) where there is only one functional delay term under the derivative but could not be applied to study (1) because of the presence of more than one functional delay term under the derivative. Further, "the note [1] (notes 1.8, page 31) suggests to extend these lemmas meant for the function with one delay to multiple delays and apply it to the study of NDDE (1) with several delays." But, it seems difficult to extend these lemmas for the said purpose. e motivation behind this work is that no result in the literature appears to have an answer to the qualitative behaviour of solutions to the NDDE: In this paper, we remove the condition "G is nondecreasing" that is assumed in [2-5, 7, 8] and study the oscillatory behaviour of solutions of the NDDE (1) and then apply the results to study the NDDE: where v(t) changes sign. e oscillatory behaviour of solutions of the discrete analogue of (1) with u � 0 is obtained in [11] (1) in some results is proved to be different from that in [11]. "Let t 1 be a fixed positive real number, and By a solution of (1), we mean a function y ∈ C([t 0 , ∞), R) such that y(t) − k i�1 p i (t)y(r i (t)) is differentiable on [t 0 , ∞) and the neutral equation (1) is satisfied by y(t) for all t ≥ t 1 . It is known that (1) has a unique solution provided that an initial function ϕ ∈ C([t 0 , t 1 ], R) is given to satisfy y(t) � ϕ(t) for all t ∈ [t 0 , t 1 ]. Such a solution is said to be nonoscillatory if it is eventually positive or eventually negative, otherwise it is called oscillatory." In this work, we assume the existence of solutions of (1) and study only their qualitative behaviour. In the sequel, unless otherwise specified, when we write a functional inequality, it will be assumed to hold for all sufficiently large values of t.

Oscillation Results
In this section, we present some results which prove that (5) is sufficient for any solution of (1) to be oscillatory or tending to zero as t ⟶ ∞. We need the following lemmas for our work.
Lemma 1 (see [12]). "Let u(t) and v(t) be two real-valued continuous functions defined for t ≥ t 0 ≥ 0. en, provided that no sum is of the form ∞ − ∞." Lemma 2 (see [12]). "Let u(t) and v(t) be two nonnegative real-valued continuous functions defined for t ≥ t 0 . en, provided that no product is of the form 0 × ∞." Theorem 1. Suppose that (2)- (7) hold, assume that there exists a positive constant p such that p j (t) for j � 1, 2, . . . ., k satisfies the following condition: en, every solution of (1) oscillates or tends to zero as t ⟶ ∞.
Proof. Let y(t) be any solution of (1) for t ≥ t 0 , where t 0 is a positive real number. If it oscillates, then there is nothing to prove; otherwise, it leads to two distinct possibilities, either y(t) > 0 or y(t) < 0 for t ≥ t 1 > t 0 . Consider the first one i.e., y(t) > 0 eventually. ere exists positive real t 2 ≥ t 1 such that for t ≥ t 2 , we have y(t) > 0, y(g(t)) > 0, y(h(t)) > 0, and y(r j (t)) > 0 for each j. Let us define for t ≥ t 1 : Clearly, due to the assumptions (6) and (7), c(t) is well defined and the improper integral c(t 1 ) is convergent at ∞, say to α.
en, there exists t 3 ≥ t 2 such that w(t) is monotonic and is of a constant sign for t ≥ t 3 . For the sake of a contradiction, assume that y(t) is not bounded. en, there exists a sequence y(a n ) such that a n ⟶ ∞, y a n ⟶ ∞ as n ⟶ ∞, y a n � max y(t): t 3 ≤ t ≤ a n .
From (4) and (17), it follows that for ϵ > 0, we can find a positive real t 4 > t 3 such that, for t ≥ t 4 , it implies |F(t)| < ϵ and |c(t)| < ϵ. Since each r j (t) ⟶ ∞ as t ⟶ ∞, we may choose n large enough so that r j (a n ) ≥ t 4 for each j. From (15) and using (19), (20), and (23), we obtain w a n � y a n − k j�1 p j a n y r j a n + c a n − F a n ≥ 1 − k j�1 p j a n ⎛ ⎝ ⎞ ⎠ y a n − 2ϵ >(1 − p)y a n − 2ϵ. (24) Taking n ⟶ ∞, we find lim t⟶∞ w(t) � ∞, a contradiction as w(t) is monotonic decreasing. Hence, y(t) is bounded which implies w(t) and z(t) are bounded and lim w(t) exists. Further, it follows that lim inf t⟶∞ y(t) and lim sup t⟶∞ y(t) exists. We claim lim inf t⟶∞ y(t) � 0. Otherwise, let y(t) ≥ α > 0. Next, since y(t) is bounded above, there exists β > 0 such that y(t) ≤ β. Hence, we have 0 < α ≤ y(t) ≤ β, which will be used for bounding the G term in (1) from below.
From the continuity of G and the assumption (2), it follows that there exists a positive lower bound m for G on [α, β]. Hence, there exists t 5 such that G(y(g(t))) > m > 0 for t > t 5 . en, integrating (21) from t � t 5 to s, we obtain In the above inequality, the left hand side is bounded, while the right hand side approaches +∞, as s ⟶ ∞. us, we have a contradiction. is yields lim inf t⟶∞ y(t) � 0. From (4), monotonic nature of w(t), and (20), it follows that lim t⟶∞ z(t) exists finitely. Let lim t⟶∞ z(t) � δ.
and lim inf t⟶∞ z(t) ≤ lim inf t⟶∞ y(t), it implies δ ≤ 0. en, using Lemmas 1 and 2 we obtain Hence, lim sup t⟶∞ y(t) ≤ 0, by (15), which implies that lim t⟶∞ y(t) � 0. If y(t) < 0 for t > t 1 , then we set x(t) � − y(t) to obtain x(t) > 0 and then (1) reduces to where (28) Further, In view of the above facts, it can be easily verified that G, H, and F satisfy the corresponding conditions satisfied by the functions G, H, and F in the theorem. Proceeding as in International Journal of Mathematics and Mathematical Sciences the proof for the case y(t) > 0, we may complete the proof of the theorem. □ Theorem 2. Suppose that (2)- (7) hold. Assume that there exists a positive constant p such that the functions p j (t) for j � 1, 2, . . . ., k satisfies the condition: en, every solution of (1) oscillates or tends to zero as t ⟶ ∞.

□
For the next two results we assume Note that (38) is less restrictive than (34).
Proof. Let y(t) be a nonoscillatory solution of (1) for t ≥ t 1 .

Theorem 5. Assume conditions
en, every solution of (1) oscillates or tends to zero as t ⟶ ∞.
Proof. Let y(t) be a nonoscillatory solution of (1) for t ≥ t 1 .

International Journal of Mathematics and Mathematical Sciences
From (38), we find b j > 0 such that 0 ≤ − p j (t) ≤ b j for each j. en, this implies due to continuity of G and (2) that there exists a real d > 0 such that G(− p j (t)) < d for each j.
Note that the above result holds for f(t) ≡ 0. Further note that condition (46) implies (5), but the converse is not necessarily true. However, if q(t) is monotonic, then both the conditions are equivalent.
Next, we intend to present a result where p j (t), j � 1, 2, 3, . . . , k, satisfies the following condition: p j (t) > 0, for every j � 1, 2, . . . k and there exists, For that purpose, we give an example which would lead us to our next result. □ Example 1. Consider the first-order NDDE with variable several delay: Note that, in the above NDDE with the several delay term under a derivative sign, p 1 (t) � (e − t + � e √ ) and p 2 (t) � (e − t + e 2 ) satisfy (53).
is NDDE has an unbounded solution y(t) � e t tending to ∞ as t ⟶ ∞, unlike other results presented so far. e above example is the motivating point to the statement of our next result. Theorem 6. Suppose that (2)- (7) hold. Assume that the functions p j (t) for j � 1, 2, . . . ., k satisfy condition (53). en, every bounded solution of (1) oscillates or tends to zero as t ⟶ ∞.

International Journal of Mathematics and Mathematical Sciences
Proof. Let y(t) be an eventually positive solution of (56). en, setting z(t) as in (19) we obtain and it follows that z(t) is monotonic and of constant sign on some interval [t 1 , ∞). By this, we have two distinct possibilities, i.e., z(t) > 0 or z(t) < 0. As p j (t), j � 1, 2, . . . , k, satisfies (53), then let i � 1. Suppose that y(t) is bounded. is implies z(t) is bounded. As z(t) is monotonic also then λ : � lim t⟶∞ z(t) exists as a finite number. If z(t) > 0 or z(t) < 0, then lim t⟶∞ z(t) exists finitely. Integrating (57) from t = t 0 to s and taking limit s ⟶ ∞, we obtain ∞ t 0 q(t)G(y(g(t)))dt < ∞. (58) Now, we claim that lim infy(t) � 0. Taking integration on (57), Since z(t) is bounded, the above integral is convergent. is in turn, by (5), it implies that lim inf s⟶∞ G(y(g(s))) � 0. As G(x) ≠ 0 for x ≠ 0, lim inf s⟶∞ y(g(s)) � 0 and because lim t⟶∞ g(t) � ∞, lim inf t⟶∞ y(t) � 0. Following the line of proof as in eorem 6, we find that lim t⟶∞ z(t) � δ � 0, which implies that z(t) > 0 because z is decreasing. Since p i (t) ≥ 1, then y(t) > k j�1 p j (t)y(r j (t)) ≥ y(r i (t)). Consequently, lim inf t⟶∞ y(t) ≠ 0, a contradiction. Hence, the bounded solution y(t) cannot be eventually positive. e proof for the case when y(t) is bounded and eventually negative solution of (56) is similar. us, every bounded solution y(t) oscillates, and the theorem is proved. □ Example 2. Consider the first-order NDDE (10) with several delays under the derivative. It satisfies all the conditions of eorem 1. As such the equation has a solution y(t) � e − t which tends to zero as t ⟶ ∞. However, no result in the literature appears to have an answer to the qualitative behaviour of solutions to this NDDE.

Remark 1.
In this section, we have not assumed the condition that G is nondecreasing unlike the authors in [3,4,10].

Nonoscillation Results
In this section, we show that (5) is necessary for every solution of (1) to be oscillatory or tending to zero as t ⟶ ∞. Or equivalently ∞ t 0 q(t)dt < ∞ is sufficient for (1) to have a bounded positive solution which does not tend to zero as t ⟶ ∞, even if the limit exists. For this, we need the following lemma.
Lemma 3 (Krasnoselskii's fixed-point theorem [13]). Let X be a Banach space and S be a bounded closed convex subset of X. Let A and B be operators from S to X such that Ax + By ∈ S for every pair of x, y ∈ S. If A is a contraction and B is completely continuous, then the equation has a solution in S. (3) and (6) hold. Further, assume that one of the conditions of (15) or (30) hold. en, (5) is a necessary condition for all solution of (1) to be oscillatory or tending to zero as t ⟶ ∞.

Theorem 8. Assume that
Proof. Suppose condition (15) holds. e proof for the case when (30) holds, would follow on similar lines. Assume for the sake of contradiction, that (5) does not hold. is implies that there exists real t 0 such that us, all we need to show is the existence of a bounded solution y(t) of (1) with lim inf t⟶∞ y(t) > 0. From (3), we find a positive constant c and a positive real t 1 > t 0 > 0 such that Choose a positive constant L such that L ≥ 5c/1 − p.
From (6), we find t 2 > t 1 such that t > t 2 implies en by using (61), one can fix t 3 > t 2 such that for t ≥ t 3 it follows that Choose T 1 > t 3 such that Clearly S is a bounded closed and convex subset of X. Define two operators A and B: S ⟶ X as follows. For y ∈ S, define International Journal of Mathematics and Mathematical Sciences It may be verified that the NDDE (79) satisfies all the conditions of eorem 10. Hence, every solution of (79) oscillates or tends to zero as t ⟶ ∞. As such, it admits a positive solution y(t) � e − t which tends to zero as t ⟶ ∞.

Remark 6.
e results of this article seem to be significant as no result in literature can be applied to the NDDEs (79) and (84).

Conclusion
every solution of oscillates.

Corollary 1. Suppose that (76) and (77) hold. en, every bounded solution of
(with v(t) changing sign) oscillates or tends to zero as t ⟶ ∞.
Data Availability e data used are published research articles in different referred journals which can be assessed in their websites.

Conflicts of Interest
e authors declare that they have no conflicts of interest.