Second-Order Differential Equation: Oscillation Theorems and Applications

<jats:p>Differential equations of second order appear in a wide variety of applications in physics, mathematics, and engineering. In this paper, necessary and sufficient conditions are established for oscillations of solutions to second-order half-linear delay differential equations of the form <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
                        <msup>
                           <mrow>
                              <mfenced open="(" close=")" separators="|">
                                 <mrow>
                                    <mi>ς</mi>
                                    <mrow>
                                       <mfenced open="(" close=")" separators="|">
                                          <mrow>
                                             <mi>y</mi>
                                          </mrow>
                                       </mfenced>
                                       <msup>
                                          <mrow>
                                             <mfenced open="(" close=")" separators="|">
                                                <mrow>
                                                   <msup>
                                                      <mrow>
                                                         <mi>u</mi>
                                                      </mrow>
                                                      <mrow>
                                                         <mo>′</mo>
                                                      </mrow>
                                                   </msup>
                                                   <mrow>
                                                      <mfenced open="(" close=")" separators="|">
                                                         <mrow>
                                                            <mi>y</mi>
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                                                      </mfenced>
                                                   </mrow>
                                                </mrow>
                                             </mfenced>
                                          </mrow>
                                          <mrow>
                                             <mi>a</mi>
                                          </mrow>
                                       </msup>
                                    </mrow>
                                 </mrow>
                              </mfenced>
                           </mrow>
                           <mrow>
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                        <mo>+</mo>
                        <mi>p</mi>
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                           <mrow>
                              <mi>y</mi>
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                        <msup>
                           <mrow>
                              <mi>u</mi>
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                           <mrow>
                              <mi>c</mi>
                           </mrow>
                        </msup>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>ϑ</mi>
                              <mrow>
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                                    <mrow>
                                       <mi>y</mi>
                                    </mrow>
                                 </mfenced>
                              </mrow>
                           </mrow>
                        </mfenced>
                        <mo>=</mo>
                        <mn>0</mn>
                        <mo>,</mo>
                        <mtext> for </mtext>
                        <mi>y</mi>
                        <mo>≥</mo>
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                           <mrow>
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                        <mo>,</mo>
                     </math>
                  </jats:inline-formula> under the assumption <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
                        <msup>
                           <mrow>
                              <mstyle displaystyle="true">
                                 <mo stretchy="false">∫</mo>
                              </mstyle>
                           </mrow>
                           <mrow>
                              <mi>∞</mi>
                           </mrow>
                        </msup>
                        <msup>
                           <mrow>
                              <mfenced open="(" close=")" separators="|">
                                 <mrow>
                                    <mi>ς</mi>
                                    <mrow>
                                       <mfenced open="(" close=")" separators="|">
                                          <mrow>
                                             <mi>η</mi>
                                          </mrow>
                                       </mfenced>
                                    </mrow>
                                 </mrow>
                              </mfenced>
                           </mrow>
                           <mrow>
                              <mo>−</mo>
                              <mfenced open="(" close=")" separators="|">
                                 <mrow>
                                    <mn>1</mn>
                                    <mo>/</mo>
                                    <mi>a</mi>
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                              </mfenced>
                           </mrow>
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                        <mo>=</mo>
                        <mi>∞</mi>
                     </math>
                  </jats:inline-formula>. Two cases are considered for <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <mi>a</mi>
                        <mo><</mo>
                        <mi>c</mi>
                     </math>
                  </jats:inline-formula> and <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
                        <mi>a</mi>
                        <mo>></mo>
                        <mi>c</mi>
                     </math>
                  </jats:inline-formula>, where <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
                        <mi>a</mi>
                     </math>
                  </jats:inline-formula> and <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
                        <mi>c</mi>
                     </math>
                  </jats:inline-formula> are the quotients of two positive odd integers. Two examples are given to show the effectiveness and applicability of the result.</jats:p>

For further work on the oscillation of this type of equations, we refer the readers to the references. Note that the majority of works consider only sufficient conditions, and merely a few consider both necessary and sufficient conditions. Hence, the objective of this work is to establish both necessary and sufficient conditions for the oscillation of solutions of (1) without using the comparison and the Riccati techniques. In this paper, we restrict our attention to the study (1), which includes the class of functional differential equations of neutral type.

Remark 1.
When the domain is not specified explicitly, all functional inequalities considered in this paper are assumed to hold eventually, i.e., they are satisfied for all y large enough.

Necessary and Sufficient Conditions
Lemma 1. Let (A1)-(A3) hold and that u is an eventually positive solution of (1). en, there exist y 1 ≥ y 0 and d > 0 such that for y ≥ y 1 .
Proof. Let u be an eventually positive solution of (1). en, by (A1), there exists a y * such that u(y) > 0 and u(ϑ(y)) > 0 for all y ≥ y * . From (1) it follows that erefore, ς(y)(u ′ (y)) a is nonincreasing for y ≥ y * . Next, we show that ς(y)(u ′ (y)) a is positive. By contradiction, assume that ς(y)(u ′ (y)) a ≤ 0 at a certain time y ≥ y * . Using that p is not identically zero on any interval [b, ∞) and by (6), there exists y 1 ≥ y * such that Recall that a is the quotient of two positive odd integers. en, Integrating from y 2 to y, we have By (A3), the right-hand side approaches − ∞; then, Integrating this inequality from y 1 to y and using that u is continuous, Since lim y⟶∞ Υ(y) � ∞, there exists a positive constant d such that (4) holds.
Since ς(y)(u ′ (y)) a is positive and nonincreasing, lim y⟶∞ ς(y)(u ′ (y)) a exists and is nonnegative. Integrating (1) from y to b, we have Letting limit as b ⟶ ∞, we get en, Since u(y 1 ) > 0, integrating the above inequality yields Since the integrand is positive, we can increase the lower limit of integration from η to y and then use the definition of Υ(y) to obtain which yields (5).
□ Theorem 1. Assume that there exists a constant b 1 , the quotient of two positive odd integers, such that 0 < c < b 1 < a.

If (A1)-(A3) hold, then each solution of (1) is oscillatory if and only if
Proof. On the contrary, we assume that u is eventually positive solution. So, Lemma 1 holds, and then there exists where Computing the derivative of w, we have us, w is nonnegative and nonincreasing. Since u > 0, by (A2), it follows that p(y)u c (ϑ(y)) cannot be identically zero in any interval [b, ∞); thus, w ′ cannot be identically zero, and w cannot be constant on any interval [b, ∞).
erefore, w(y) > 0 for y ≥ y 1 . Computing the derivative, we have Integrating (21) from y 2 to y and using that w > 0, we have Next, we find a lower bound for the right-hand side of (25), independent of the solution u. By (4) and (19), we have Since w is nonincreasing, b 1 /a > 0, and ϑ(η) < η, it follows that Going back to (22), we have Since (1 − b 1 /a) > 0, by (17) the right-hand side approaches +∞ as y ⟶ ∞.
is contradicts (25) and completes the proof of sufficiency for eventually positive solutions.
e eventually negative solution can be dealt similarly by introducing the variables v � − u.
Next, we show the necessity part by a contrapositive argument. If (17) does not hold, then for each κ > 0 there exists for all η ≥ y 1 . We define the set of continuous functions We define an operator Ω on S by Note that when u is continuous, Ωu is also continuous on [0, ∞). If u is a fixed point of Ω, i.e., Ωu � u, then u is a solution of (1).

(31)
Note that for each fixed y, we have v 1 (y) ≥ v 0 (y). Using mathematical induction, we can show that v n+1 (y) ≥ v n (y). erefore, the sequence v n converges pointwise to a function v. Using the Lebesgue dominated convergence theorem, we can show that v is a fixed point of Ω in S. is shows under assumption (26), there is a nonoscillatory solution that does not converge to zero. is completes the proof. □ Theorem 2. Assume that there exists a constant b 2 , the quotient of two positive odd integers such that 0 < a < b 2 < c.

If (A1)-(A4) hold and ς(y) is nondecreasing, then each solution of (1) is oscillatory if and only if
Proof. On the contrary, we assume that u is an eventually positive solution that does not converge to zero. Using the same argument as in Lemma 1, there exists y 1 ≥ y 0 such that u(ϑ(y)) > 0 and ς(y)(u ′ (y)) a is positive and nonincreasing.
For eventually negative solutions, we use the same change of variables as in eorem 1 and proceed as above.
To prove the necessity part, we assume that (32) does not hold and obtain an eventually positive solution that does not converge to zero. If (32) does not hold, then for each κ > 0 there exists y 1 ≥ y 0 such that We define the set of continuous function en, we define the operator Note that if u is continuous, Ωu is also continuous at y � y 1 . Also, note that if Ωu � u, then u is solution of (1).

(44)
Note that for each fixed y, we have v 1 (y) ≥ v 0 (y). Using mathematical induction, we can prove that v n+1 (y) ≥ v n (y). erefore, v n converges pointwise to a function v in S. en, v is a fixed point of Ω and a positive solution of (1). e proof is completed. □ Example 1. Consider the differential equations Here, a � 11/3, ς(y) � e − y , ϑ 1 (y) � y − 2, Υ(y) � So, every conditions of eorem 1 hold true, and therefore, all solutions of (45) are oscillatory or converge to zero.

Conclusion
e aim of this work is to establish necessary and sufficient conditions for the oscillation of solution to second-order half-linear differential equation. e obtained oscillation theorems complement the well-known oscillation results present in the literature. is work, as well as [31][32][33][34][35][36][37][38][39][40][41], leads us to pose an open problem: Can we find necessary and sufficient conditions for the oscillation of solutions to second-order differential equation r(t) (y(t) + p(t)y(τ(t))) ′ c ′ + m i�1 q i (t)y α i τ i (t) ) � 0, for p ∈ C R + , R ?. (49)

Data Availability
No data were used to support the findings of this study.

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