Asymptotic behavior of even-order damped differential equations with p-Laplacian like operators and deviating arguments

Philos [14]

Let \(u\in\mathrm{ C}^{n}(\mathbb{I},(0,\infty))\). If \(u^{(n-1)}(t)u^{(n)}(t)\leq0\) for \(t\geq t_{u}\). Then, for every \(\lambda\in(0,1)\), there exists a constant \(M>0\) such that, for all sufficiently large t,

Agarwal et al. [5]

Let \(u\in\mathrm{ C}^{n}(\mathbb{I},(0,\infty))\) and \(u^{(n)}(t)\leq0\). If \(\lim_{t\rightarrow\infty}u(t)\neq0\), then, for every \(\lambda\in(0,1)\), there exists a \(t_{\lambda}\in\mathbb{I}\) such that, for all \(t\in[t_{\lambda},\infty)\),

Let conditions (1.2) and (1.6) be satisfied and Assume that there exists a function \(\rho\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\) such that, for all constants \(M>0\), If, for some constant \(\lambda_{0}\in(0,1)\), then every solution x of (1.1) is either oscillatory or satisfies condition \(\lim_{t\rightarrow\infty}x(t)=0\).

Assume that (1.1) has a nonoscillatory solution x which is eventually positive and such that Modifying the proof in Zhang et al. ([3], Lemma 2.3), we can show that, for all \(t\geq t_{1}\), there exist two possible cases: where \(t_{1}\geq t_{0}\) is sufficiently large. We consider each of the two cases separately.

Case I. Assume first that case (1) holds. For \(t\geq t_{1}\), we define the function ω by Then \(\omega(t)>0\) for all \(t\geq t_{1}\) and Let \(u:=x'\). It follows from Lemma 2.1 that, for some constant \(M>0\) and for all sufficiently large t, Thus, we deduce that From (1.1) and (3.5), we obtain Hence, we have Let Using the inequality where \(C>0\), \(y\geq0\), and \(D_{+}:=\max(0,D)\) (see Fišnarová and Mařík ([8], Lemma 1) for details), we get Integrating this inequality from \(t_{1}\) to t, we obtain which contradicts (3.2).

Case II. Assume now that case (2) is satisfied. For \(t\geq t_{1}\), we define another function v as follows: Then \(v(t)<0\) for all \(t\geq t_{1}\). Since we conclude that \(-a(t)(-x^{(n-1)}(t))^{p-1}E(t_{0},t)\) is decreasing. Thus, for all \(s\geq t\geq t_{1}\), Hence, for all \(s\geq t\geq t_{1}\), Integrating this inequality from t to ι, we obtain Taking \(\iota\rightarrow\infty\) and using the fact that \(\lim_{\iota\rightarrow\infty}x^{(n-2)}(\iota)\geq0\) and the definition of δ, we have Inequality (3.10) implies that Hence, by (3.8) and (3.11), we get Differentiation of (3.8) yields From (1.1) and (3.8), it follows that On the other hand, by Lemma 2.2, we have, for every \(\lambda\in(0,1)\) and for all sufficiently large t, Using (3.12) in (3.13), we have It follows from (3.14) and (3.15) that Multiplying (3.16) by \(\delta^{p-1}(t)E(t_{0},t)\) and integrating the resulting inequality from \(t_{1}\) to t, we obtain Let and Using inequalities (3.7), (3.12), and the definition of φ, we have which contradicts (3.3). This completes the proof. □

Let conditions (1.2), (1.6), and (3.1) be satisfied and \(n=2\). Suppose that there exists a function \(\rho\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\) such that If then (1.1) is oscillatory.

For \(t\geq1\), consider a second-order delay differential equation with damping where \(q_{0}>0\) is a constant. Let \(t_{0}=1\), \(p=2\), \(a(t)=t^{2}\), \(r(t)=t/2\), \(q(t)=q_{0}\), \(g(t)=t/2\), and \(\rho(t)=1\). Then \(h_{+}(t)=0\) and thus condition (3.17) is satisfied. It is easy to see that \(E(t_{0},t)=t^{1/2}\), \(\delta(t)=2t^{-3/2}/3\), \(\phi(t)=2t^{-1/2}/3\), and \(\varphi (t)=2t^{-1}/3\). Then condition (3.18) holds for \(q_{0}>1\). Therefore, by Theorem 3.2, equation (3.19) is oscillatory provided that \(q_{0}>1\). Observe, however, that if \(\gamma=1\), then and which mean that the results obtained in [7, 20] fail to apply in equation (3.19).

Let conditions (1.6) and (1.7) hold. Assume that there exists a function \(\rho\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\) such that, for all constants \(M>0\), If there exists a function \(m\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\) such that and, for some constant \(\lambda_{0}\in(0,1)\), then the conclusion of Theorem  3.1 remains intact.

Assume that x is an eventually positive solution of (1.1) that satisfies (3.4). Similar analysis to that in Zhang et al. ([3], Lemma 2.3) leads to the conclusion that, for all \(t\geq t_{1}\), there exist two possible cases (1) and (2) (as those in the proof of Theorem 3.1), where \(t_{1}\geq t_{0}\) is sufficiently large. Assume first that case (1) holds. We define the function ω by With a similar proof as that of Case I in Theorem 3.1, one arrives at a contradiction with condition (3.20). Assume, instead, that case (2) holds. Define the function v as in (3.8). As in the proof of Theorem 3.1, we obtain (3.11), (3.12), (3.14), and (3.15). On the other hand, we derive from (3.11) that Using the latter inequality and (3.21), we have which implies that \(x^{(n-2)}/m\) is nondecreasing. Hence, it follows from (1.7) that Thus, by (3.14) and (3.15), we have The remaining proof is similar to that of Case II in Theorem 3.1, and hence is omitted. □

The optional function m satisfying condition (3.21) exists and can be constructed by taking \(m(t):=\delta(t)\).

Let conditions (1.6) and (1.7) be satisfied and \(n=2\). Assume that there exists a function \(\rho\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\) such that If there exists a function \(m\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\) such that (3.21) is satisfied and then the conclusion of Theorem  3.2 remains intact.

For \(t\geq1\), consider a second-order advanced differential equation with damping where \(q_{0}>0\) is a constant. Let \(t_{0}=1\), \(p=2\), \(a(t)=t^{2}\), \(r(t)=t/2\), \(q(t)=q_{0}\), \(g(t)=2t\), \(\rho(t)=1\), and \(m(t)=\delta(t)=2t^{-3/2}/3\). Similar analysis to that in Example 3.3 implies that condition (3.23) holds and condition (3.24) is satisfied for \(q_{0}>2\sqrt{2}\). Thus, by Theorem 3.6, equation (3.25) is oscillatory if \(q_{0}>2\sqrt{2}\). Observe that the results reported in [7, 20] cannot be applied to equation (3.25) since \(g(t)>t\).

Let conditions (1.7) and (3.26) hold. Assume that there exists a function \(\rho\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\) such that (3.20) holds for all constants \(M>0\). If there exists a function \(\xi\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\) such that and, for some constant \(\lambda_{0}\in(0,1)\), and then the conclusion of Theorem  3.1 remains intact.

Assuming again that x is an eventually positive solution of (1.1) that satisfies (3.4) and proceeding as in the proof of Theorem 3.4, we end up having to show case (2) (as the corresponding case in Theorem 3.1). As in the proof of Case II in Theorem 3.1, one arrives at the inequalities (3.9), (3.10), and (3.14) which holds for all \(\lambda\in(0,1)\). Inequalities (3.10) and (3.14) yield, for all \(\lambda_{0}\in(0,1)\) and for all sufficiently large t, From (1.1) and (3.30), we obtain Using (3.9) and condition (1.7), we have Thus, by (3.28), we get which implies that, for all \(s\geq t\geq t_{1}\), Integrating this inequality from t to ι, we obtain Letting \(\iota\rightarrow\infty\) and using the definition of A, we get which yields Now, we define the function v by (3.8). From (3.8) and (3.31), we see that Differentiating (3.8) and using (1.1), we have (3.13). On the other hand, by (3.27) and (3.31), we obtain which shows that \(x^{(n-2)}/\xi\) is nondecreasing. Hence, using condition (1.7), we get Thus, from (3.13), (3.14), (3.32), and (3.33), it follows that Multiplying (3.34) by \(A^{p-1}(t)\) and integrating the resulting inequality from \(t_{1}\) to t, we have Let Using inequality (3.7), we derive from (3.32) that which contradicts (3.29). This completes the proof. □

The optional function ξ satisfying assumption (3.27) can reasonably be constructed by taking \(\xi(t):=A(t)\).

Let (1.7) and (3.26) be satisfied and \(n=2\). Suppose that there exists a function \(\rho\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\) such that (3.23) holds. If and there exists a function \(\xi\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\) such that (3.27) holds and then the conclusion of Theorem  3.2 remains intact.

For \(t\geq1\) and \(q_{0}>0\), consider the second-order advanced differential equation (3.25). Let \(t_{0}=1\), \(p=2\), \(a(t)=t^{2}\), \(r(t)=t/2\), \(q(t)=q_{0}\), \(g(t)=2t\), \(\rho(t)=1\), and \(\xi(t)=A(t)=t^{-1}\). Then \(E(t_{0},t)=t^{1/2}\), \(\delta(t)=2t^{-3/2}/3\), and \(h_{+}(t)=0\). It is not difficult to verify that all conditions of Theorem 3.10 are satisfied for \(q_{0}\geq3\sqrt{2}/2\). Therefore, using Theorem 3.10, equation (3.25) is oscillatory provided that \(q_{0}\geq3\sqrt{2}/2\), whereas Theorem 3.6 implies that equation (3.25) is oscillatory if \(q_{0}>2\sqrt {2}\). Hence, Theorem 3.10 improves Theorem 3.6 in some cases. However, to achieve such improvement, an additional assumption (3.35) is required. Therefore, we observe that Theorems 3.4, 3.6, 3.8, and 3.10 are of independent interest.

For \(t\geq1\), consider a second-order differential equation with damping where \(r_{0}\geq0\) and \(q_{0}>0\) are constants. Let \(t_{0}=1\), \(p=2\), \(a(t)=t^{2}\), \(r(t)=r_{0}\), \(q(t)=q_{0}\), \(g(t)=t\), and \(\rho(t)=1\). Then \(h_{+}(t)=0\) and so condition (3.23) is satisfied. It is not hard to verify that \(1\leq E(t_{0},t)\leq\mathrm{ e}^{r_{0}}\), \(\mathrm{ e}^{-r_{0}}t^{-1}\leq\delta(t)\leq t^{-1}\), and \(A(t)=1/t\). Then condition (3.35) is satisfied for all sufficiently large t and, for \(q_{0}>1/4\), Hence, by Theorem 3.10, equation (3.37) is oscillatory provided that \(q_{0}>1/4\) (it is well known that \(q_{0}>1/4\) is the best possible for the oscillation of equation (3.37) when \(r_{0}=0\)). Observe, however, that if \(\gamma=1\), then and which mean that the results reported in [7, 20] cannot be applied to equation (3.37).

For \(t\geq1\), consider the second-order Euler differential equation where \(q_{0}>0\) is a constant. Let \(t_{0}=1\), \(p=2\), \(a(t)=t^{2}\), \(r(t)=t/2\), \(q(t)=q_{0}\), \(g(t)=t/2\), \(\rho(t)=1\), \(m(t)=\delta(t)=2t^{-3/2}/3\), and \(\xi (t)=A(t)=t^{-1}\). It is easy to see that \(h_{+}(t)=0\), \(E(t_{0},t)=t^{1/2}\), \(\phi(t)=2t^{-1/2}/3\), and \(\varphi(t)=2t^{-1}/3\). Applications of Theorems 3.2 and 3.6 imply that equation (3.38) is oscillatory if \(q_{0}>1\), whereas Theorem 3.10 yields oscillation of equation (3.38) for \(q_{0}>3/4\). Similar analysis to that in Example 3.3 shows that the results obtained in [7, 20] fail to apply in equation (3.38). However, it is well known that equation (3.38) is oscillatory if and only if \(q_{0}>9/16\). How to extend this sharp result to equation (1.1) remains open at the moment.

Let conditions (1.2), (1.6), and (3.1) be satisfied. Assume that there exists a function \(H\in \mathfrak{H}\) such that, for all constants \(M>0\) and for all \(t_{1}\geq t_{0}\), If there exists a function \(K\in\mathfrak{K}\) such that, for some constant \(\lambda_{0}\in(0,1)\) and for all \(t_{1}\geq t_{0}\), then the conclusion of Theorem  3.1 remains intact.

Assume that x is an eventually positive solution of (1.1) that satisfies (3.4). Similar analysis to that in Zhang et al. ([3], Lemma 2.3) leads to the conclusion that, for all \(t\geq t_{1}\), there exist two possible cases (1) and (2) (as those in the proof of Theorem 3.1), where \(t_{1}\geq t_{0}\) is sufficiently large. First, assume that case (1) holds. Defining the function ω by (3.5) and proceeding as in the proof of Theorem 3.1, we arrive at inequality (3.6). Multiplying (3.6) by \(H(t,s)\) and integrating the resulting inequality from \(t_{1}\) to t, we obtain Let and Using the following inequality (a variation of the well-known Young inequality) where \(p>1\), \(C\geq0\), and \(D\geq0\), we conclude that Hence, we have which contradicts (4.1). Assume now that case (2) holds and define the function v as in (3.8). As in the proof of Theorem 3.1, we obtain (3.16). Multiplying (3.16) by \(K(t,s)\) and integrating the resulting inequality from \(t_{1}\) to t, we have Let Using inequality (4.3), we obtain which contradicts (4.2). This completes the proof. □

Let conditions (1.2), (1.6), and (3.1) be satisfied and \(n=2\). Assume that there exists a function \(H\in\mathfrak{H}\) such that, for all \(t_{1}\geq t_{0}\), If there exists a function \(K\in\mathfrak{K}\) such that, for all \(t_{1}\geq t_{0}\), then the conclusion of Theorem  3.2 remains intact.

Let conditions (1.6) and (1.7) hold. Assume that there exists a function \(H\in\mathfrak{H}\) such that, for all constants \(M>0\) and for all \(t_{1}\geq t_{0}\), Furthermore, suppose that there exists a function \(m\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\) such that (3.21) is satisfied. If there exists a function \(K\in\mathfrak{K}\) such that, for some constant \(\lambda_{0}\in(0,1)\) and for all \(t_{1}\geq t_{0}\), then the conclusion of Theorem  3.1 remains intact.

Let conditions (1.6) and (1.7) be satisfied and \(n=2\). Assume that there exists a function \(H\in\mathfrak{H}\) such that, for all \(t_{1}\geq t_{0}\), Furthermore, suppose that there exists a function \(m\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\) such that (3.21) is satisfied. If there exists a function \(K\in\mathfrak{K}\) such that, for all \(t_{1}\geq t_{0}\), then the conclusion of Theorem  3.2 remains intact.

Let conditions (1.7) and (3.26) hold and let condition (3.28) be satisfied for some constant \(\lambda_{0}\in(0,1)\). Assume that there exists a function \(H\in\mathfrak{H}\) such that (4.6) holds for all constants \(M>0\) and for all \(t_{1}\geq t_{0}\). Furthermore, suppose that there exists a function \(\xi\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\) such that (3.27) holds. If there exists a function \(K\in\mathfrak{K}\) such that, for all \(t_{1}\geq t_{0}\), then the conclusion of Theorem  3.1 remains intact.

Let conditions (1.7), (3.26), and (3.35) be satisfied and \(n=2\). Assume that there exists a function \(H\in\mathfrak{H}\) such that (4.8) holds for all \(t_{1}\geq t_{0}\). Furthermore, suppose that there exists a function \(\xi\in\mathrm{ C}^{1}(\mathbb{I},(0,\infty))\) such that (3.27) holds. If there exists a function \(K\in \mathfrak{K}\) such that, for all \(t_{1}\geq t_{0}\), then the conclusion of Theorem  3.2 remains intact.