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Lyapunov type inequalities for even order differential equations with mixed nonlinearities
Journal of Inequalities and Applications volume 2015, Article number: 142 (2015)
Abstract
In the case of oscillatory potentials, we present Lyapunov and Hartman type inequalities for even order differential equations with mixed nonlinearities: \(x^{(2n)}(t)+(-1)^{n-1}\sum_{i=1}^{m}q_{i}(t)|x(t)|^{\alpha_{i}-1}x(t)=0\), where \(n,m\in{\mathbb{N}}\) and the nonlinearities satisfy \(0<\alpha_{1}<\cdots<\alpha_{j}<1<\alpha_{j+1}<\cdots<\alpha_{m}<2\).
1 Introduction
Consider the Hill equation
where \(q(t)\in\mathrm{L}^{1}[a,b]\) is a real-valued function. If there exists a nontrivial solution \(x(t)\) of (1.1) satisfying the Dirichlet boundary conditions
where \(a, b\in\mathbb{R}\) with \(a< b\) and \(x(t)\ne0\) for \(t\in(a,b)\), then the inequality
holds. This striking inequality was first proved by Lyapunov [1] and it is known as the Lyapunov inequality. Later Wintner [2] and thereafter some more authors achieved the replacement of the function \(|q(t)|\) in (1.3) by the function \(q^{+}(t)\), i.e. they obtained the following inequality:
where \(q^{+}(t)=\max\{q(t),0\}\), and the constant 4 in the right hand side of inequalities (1.3) and (1.4) is the best possible largest number (see [1] and [3], Theorem 5.1).
In [3], Hartman obtained an inequality sharper than both (1.3) and (1.4):
Clearly, (1.5) implies (1.4), since
for all \(t\in(a,b)\), and equality holds when \(t=(a+b)/2\).
It appears that the first generalization of Hartman’s result was obtained by Das and Vatsala [4], Theorem 3.1.
Theorem 1.1
(Hartman type inequality)
If \(x(t)\) is a nontrivial solution of the equation
satisfying the 2-point boundary conditions
where \(a, b\in\mathbb{R}\) with \(a< b\) are consecutive zeros, then the inequality
holds.
Note that (1.9) generalizes the Lyapunov type inequality
by (1.6) (see also [5], Corollary 3.3).
The proof of Theorem 1.1 is based on the Green’s function \({\mathcal{G}}_{n}(t,s)\) of the 2-point boundary value problem
satisfying (1.8), obtained in [4] as follows:
and
Note that \((-1)^{n-1}{\mathcal{G}}_{n}(t,s)\geq0\) and
for all \(t\in[a,b]\) (see [5]). In fact, in view of the symmetry of \({\mathcal{G}}_{n}(t,s)\), (1.14) also implies that
In view of the alternating term \((-1)^{n-1}\) in the Green’s function \({\mathcal{G}}_{n}(t,s)\), Hartman and Lyapunov type inequalities for the 2-point boundary value problem
satisfying the boundary conditions (1.8) can be obtained by replacing the function \(q^{+}(t)\) by \(|q(t)|\) in (1.9) and (1.10), respectively.
The Lyapunov inequality and its generalizations have been used successfully in connection with oscillation and Sturmian theory, asymptotic theory, disconjugacy, eigenvalue problems, and various properties of the solutions of (1.1) and related equations; see for instance [2, 3, 6–23] and the references cited therein. For some of its extensions to Hamiltonian systems, higher order differential equations, nonlinear and half-linear differential equations, difference and dynamic equations, and functional and impulsive differential equations, we refer in particular to [10, 11, 24–43].
The aim of our paper is to extend the well-known Lyapunov and Hartman type inequalities for even order nonlinear equations of the form
where \(n,m\in{\mathbb{N}}\), the potentials \(q_{i}(t)\), \(i=1,\ldots, m\), are real-valued functions and no sign restrictions are imposed on them. Further, the exponents in (1.17) satisfy
It is clear that the two special cases of (1.17) are the even order sub-linear equation
and the even order super-linear equation
Further, we note that letting \(\alpha_{i}\to1^{-}\), \(i=1,\ldots,j\), and \(\alpha_{i}\to1^{+}\), \(i=j+1,\ldots,m\), in (1.17) results in (1.7) with \(q(t)=\sum_{i=1}^{m}q_{i}(t)\), i.e.,
and as a consequence, our results extend and improve the main results of Das and Vatsala [4], i.e. Theorem 1.1, and in particular the classical Lyapunov [1] and Hartman’s [3] results.
We further remark that the Lyapunov type inequalities have been studied by many authors, see for instance the survey paper [44] and the references therein, but to the best of our knowledge there are no results in the literature for (1.17), and in particular for (1.19) and (1.20).
2 Main results
Throughout this paper we shall assume that \(q_{i}(t)\in\mathrm{L}^{1}[a,b]\), \(i=1,\ldots,m\).
We will need the following lemma.
Lemma 2.1
If A is positive, and B, z are nonnegative, then
for any \(\mu\in(0,2)\) with equality holding if and only if \(B=z=0\).
Proof
Let
where \(A>0\) and \(B\geq0\). Clearly, when \(z=0\) or \(B=0\), (2.1) is obvious. On the other hand, if \(B>0\), then it is easy to see that \({\mathcal{H}}\) attains its minimum at \(z_{0}=(\mu A^{-1}B/2)^{1/(2-\mu)}\) and
Thus, (2.1) holds. Note that if \(B>0\), then (2.1) is strict. □
Now we state and prove our first result.
Theorem 2.1
(Hartman type inequality)
If \(x(t)\) is a nontrivial solution of (1.17) satisfying the 2-point boundary conditions (1.8), where \(a,b\in\mathbb{R}\) with \(a< b\) are consecutive zeros, then the inequality
holds, where
with
Proof
Let \(x(t)\) be a nontrivial solution of (1.17) satisfying the boundary conditions (1.8), where \(a,b\in\mathbb{R}\) with \(a< b\) are consecutive zeros. Without loss of generality, we may assume that \(x(t)>0\) for \(t\in(a,b)\). In fact, if \(x(t)<0\) for \(t\in(a,b)\), then we can consider \(-x(t)\), which is also a solution. Then, by using the Green’s function of (1.11)-(1.8), \(x(t)\) can be expressed as
Let \(x(c)=\max_{t\in(a,b)}x(t)\). Then by (2.1) in Lemma 2.1 with \(A=B=1\), we have
Using this in (2.5), we obtain
which implies the quadratic inequality
where
and
But inequality (2.7) is possible if and only if \(\Theta_{1}\Theta _{2}>1/4\). Finally, we note that
This completes the proof of Theorem 2.1. □
Next, we prove the following result.
Theorem 2.2
(Lyapunov type inequality)
If \(x(t)\) is a nontrivial solution of (1.17) satisfying the 2-point boundary conditions (1.8) where \(a,b\in\mathbb{R}\) with \(a< b\) are consecutive zeros, then the inequality
holds, where the functions \(\widehat{Q}_{m}\) and \(\widetilde{Q}_{m}\) are defined in Theorem 2.1.
Proof
In the view of (1.6), (2.3) immediately implies (2.8). □
Remark 1
Since
where \(\theta_{i}\) is defined in (2.4), it is easy to see that inequalities (2.3) and (2.8) reduce to inequalities (1.9) and (1.10), respectively, with \(q^{+}(t)=\sum_{i=1}^{m}q^{+}_{i}(t)\). Thus, Theorems 2.1 and 2.2 reduce to Theorem 3.1 of Das and Vatsala [4], and Corollary 3.3 of Yang [5], respectively. Moreover, when \(n=1\), they reduce to the classical Lyapunov (1.4) and Hartman (1.5) inequalities with \(q^{+}(t)=\sum_{i=1}^{m}q^{+}_{i}(t)\).
Remark 2
It is of interest to find analogs of Theorems 2.1 and 2.2 for (1.17)-(1.8) without the term \((-1)^{n-1}\), i.e., for the equation
satisfying the 2-point boundary conditions (1.8). We state these results in the following.
Proposition 1
If \(x(t)\) is a nontrivial solution of (2.9) satisfying the 2-point boundary conditions (1.8) where \(a, b\in\mathbb{R}\) with \(a< b\) are consecutive zeros, then the following hold:
-
(i)
Hartman type inequality;
$$\begin{aligned} &\biggl(\int_{a}^{b}(b-t)^{2n-1}(t-a)^{2n-1} \widehat{P}_{m}(t)\,\mathrm{d}t \biggr) \biggl(\int_{a}^{b}(b-t)^{2n-1}(t-a)^{2n-1} \widetilde{P}_{m}(t)\,\mathrm{d}t \biggr) \\ &\quad >(2n-1)^{2}(n-1)!^{4}(b-a)^{4n-2}/4. \end{aligned}$$ -
(ii)
Lyapunov type inequality;
$$ \biggl(\int_{a}^{b}\widehat{P}_{m}(t)\, \mathrm{d}t \biggr) \biggl(\int_{a}^{b}\widetilde {P}_{m}(t)\,\mathrm{d}t \biggr)>\frac{4^{4n-3}(2n-1)^{2}(n-1)!^{4}}{(b-a)^{4n-2}}, $$
where
and \(\theta_{i}\) is defined in (2.4).
When \(q_{i}(t)=0\), for all \(i=2,3,\ldots,m-1\), then (1.17) and (2.9) reduce to the equations
and
respectively, where \(p(t)=q_{m}(t)\), \(q(t)=q_{1}(t)\), \(\gamma=\alpha_{1}\in (0,1)\), and \(\beta=\alpha_{m}\in(1,2)\).
For these equations we have the following corollaries.
Corollary 2.3
If \(x(t)\) is a nontrivial solution of (2.10) satisfying the 2-point boundary conditions (1.8) where \(a, b\in\mathbb{R}\) with \(a< b\) are consecutive zeros, then the following hold:
-
(i)
Hartman type inequality;
$$\begin{aligned} & \biggl(\int_{a}^{b}(b-t)^{2n-1}(t-a)^{2n-1} \bigl[p^{+}(t)+q^{+}(t)\bigr]\,\mathrm{d}t \biggr)\\ &\qquad{} \times \biggl(\int_{a}^{b}(b-t)^{2n-1}(t-a)^{2n-1} \bigl[\beta_{0}p^{+}(t)+\gamma _{0}q^{+}(t)\bigr]\,\mathrm{d}t \biggr)\\ &\quad>(2n-1)^{2}(n-1)!^{4}(b-a)^{4n-2}/4. \end{aligned}$$ -
(ii)
Lyapunov type inequality;
$$\biggl(\int_{a}^{b}\bigl[p^{+}(t)+q^{+}(t)\bigr]\, \mathrm{d}t \biggr) \biggl(\int_{a}^{b}\bigl[\beta _{0}p^{+}(t)+\gamma_{0}q^{+}(t)\bigr]\,\mathrm{d}t \biggr) > \frac{4^{4n-3}(2n-1)^{2}(n-1)!^{4}}{(b-a)^{4n-2}}, $$
where
Corollary 2.4
If \(x(t)\) is a nontrivial solution of (2.11) satisfying the 2-point boundary conditions (1.8) where \(a, b\in\mathbb{R}\) with \(a< b\) are consecutive zeros, then the following hold:
-
(i)
Hartman type inequality;
$$\begin{aligned} & \biggl(\int_{a}^{b}(b-t)^{2n-1}(t-a)^{2n-1} \bigl[\bigl|p(t)\bigr|+\bigl|q(t)\bigr|\bigr]\,\mathrm{d}t \biggr) \\ &\qquad{} \times \biggl(\int_{a}^{b}(b-t)^{2n-1}(t-a)^{2n-1} \bigl[\beta_{0}\bigl|p(t)\bigr|+\gamma _{0}\bigl|q(t)\bigr|\bigr]\,\mathrm{d}t \biggr)\\ &\quad>(2n-1)^{2}(n-1)!^{4}(b-a)^{4n-2}/4. \end{aligned}$$ -
(ii)
Lyapunov type inequality;
$$\biggl(\int_{a}^{b}\bigl[\bigl|p(t)\bigr|+\bigl|q(t)\bigr|\bigr]\, \mathrm{d}t \biggr) \biggl(\int_{a}^{b}\bigl[\beta _{0}\bigl|p(t)\bigr|+\gamma_{0}\bigl|q(t)\bigr|\bigr]\,\mathrm{d}t \biggr) > \frac{4^{4n-3}(2n-1)^{2}(n-1)!^{4}}{(b-a)^{4n-2}}, $$
where the constants \(\beta_{0}\) and \(\gamma_{0}\) are defined in Corollary 2.3.
Remark 3
Corollary 2.3 is of particular interest since it gives two new results for the even order sub-linear equation (when \(p(t)=0\)) and super-linear equation (when \(q(t)=0\)), i.e., (1.19) and (1.20). Moreover, classical results can also be obtained by the limiting process \(\gamma\to1^{-}\) and \(\beta\to1^{+}\) in inequalities (i) and (ii) given in Corollary 2.3.
3 Some special cases
In this section we consider the situations when the potentials \(q_{i}(t)\), \(i=1,\ldots,m\), are either linear, convex, or concave functions.
Corollary 3.1
Let \(q_{i}(t)=c_{i}t+d_{i}\), \(i=1,\ldots,m\), in (1.17) be positive on \([a,b]\). If \(x(t)\) is a nontrivial solution of (1.17) satisfying the 2-point boundary conditions (1.8), where \(a,b\in\mathbb{R}\) with \(a< b\) are consecutive zeros, then the inequality
holds, where
and \(\theta_{i}\) is the same as in (2.4).
Proof
In this special case, we need to compute the integral
for real constants c and d. Writing \(ct+d=c(t-a)+ca+d\) and making the substitution \(t=(b-a)z+a\), we obtain
where \(B(\cdot,\cdot)\) is the Beta function. However, since
we have
Using this in (2.3) with \(q_{i}(t)=c_{i}t+d_{i}\) the result follows. □
Corollary 3.2
Let \(q_{i}(t)\), \(i=1,\ldots,m\), in (1.17) be continuous, positive, and convex on \([a,b]\). If \(x(t)\) is a nontrivial solution of (1.17) satisfying the 2-point boundary conditions (1.8), where \(a,b\in \mathbb{R}\) with \(a< b\) are consecutive zeros, then the inequality
holds.
Corollary 3.3
Let \(q_{i}(t)\), \(i=1,\ldots,m\), in (1.17) be continuous, positive, and concave on \([a,b]\). If \(x(t)\) is a nontrivial solution of (1.17) satisfying the 2-point boundary conditions (1.8), where \(a,b\in\mathbb{R}\) with \(a< b\) are consecutive zeros, then the inequality
holds.
The proofs of Corollaries 3.2 and 3.3 are similar to those of Propositions 4.2 and 4.3 of Das and Vatsala [4], and hence they are omitted.
Finally, we conclude this paper with the following remark. When \(n=1\), the results obtained in this paper for (1.17) (or (2.11)) can easily be extended to the second order equations
i.e., for Emden-Fowler sub-linear and Emden-Fowler super-linear equations with positive and negative coefficients. The formulations of these results are left to the reader.
It will be of interest to find similar results for the even order mixed nonlinear equations of the form (1.17) for some \(\alpha_{k}\geq2\), or the super-linear equation (1.20) for \(\beta\in [2,\infty)\). In fact, the case when \(n=1\) (Emden-Fowler super-linear) is of immense interest.
References
Lyapunov, AM: Probleme général de la stabilité du mouvement (French Translation of a Russian paper dated 1893). Ann. Fac. Sci. Univ. Toulouse 2, 27-247 (1907); Reprinted as Ann. Math. Studies, No. 17, Princeton (1947)
Wintner, A: On the nonexistence of conjugate points. Am. J. Math. 73, 368-380 (1951)
Hartman, P: Ordinary Differential Equations. Wiley, New York (1964); Birkhäuser, Boston (1982)
Das, AM, Vatsala, AS: Green function for n-n boundary value problem and an analogue of Hartman’s result. J. Math. Anal. Appl. 51, 670-677 (1975)
Yang, X: On inequalities of Lyapunov type. Appl. Math. Comput. 134, 293-300 (2003)
Agarwal, RP, Wong, PJY: Error Inequalities in Polynomial Interpolation and Their Applications. Kluwer Academic, Dordrecht (1993)
Beurling, A: Un théorème sur les fonctions bornées et uniformément continues sur l’axe réel. Acta Math. 77, 127-136 (1945)
Borg, G: On a Lyapunov criterion of stability. Am. J. Math. 71, 67-70 (1949)
Brown, RC, Hinton, DB: Opial’s inequality and oscillation of 2nd order equations. Proc. Am. Math. Soc. 125, 1123-1129 (1997)
Cheng, SS: Lyapunov inequalities for differential and difference equations. Fasc. Math. 23, 25-41 (1991)
Dahiya, RS, Singh, B: A Lyapunov inequality and nonoscillation theorem for a second order nonlinear differential-difference equations. J. Math. Phys. Sci. 7, 163-170 (1973)
Eliason, SB: Lyapunov inequalities and bounds on solutions of certain second order equations. Can. Math. Bull. 17(4), 499-504 (1974)
Hochstadt, H: A new proof of stability estimate of Lyapunov. Proc. Am. Math. Soc. 14, 525-526 (1963)
Kwong, MK: On Lyapunov’s inequality for disfocality. J. Math. Anal. Appl. 83, 486-494 (1981)
Lee, C, Yeh, C, Hong, C, Agarwal, RP: Lyapunov and Wirtinger inequalities. Appl. Math. Lett. 17, 847-853 (2004)
Mitrinović, DS, Pečarić, JE, Fink, AM: Inequalities Involving Functions and Their Integrals and Derivatives. Mathematics and Its Applications (East European Series), vol. 53. Kluwer Academic, Dordrecht (1991)
Napoli, PL, Pinasco, JP: Estimates for eigenvalues of quasilinear elliptic systems. J. Differ. Equ. 227, 102-115 (2006)
Nehari, Z: Some eigenvalue estimates. J. Anal. Math. 7, 79-88 (1959)
Nehari, Z: On an inequality of Lyapunov. In: Studies in Mathematical Analysis and Related Topics. Stanford University Press, Stanford (1962)
Pachpatte, BG: Inequalities related to the zeros of solutions of certain second order differential equations. Facta Univ., Ser. Math. Inform. 16, 35-44 (2001)
Reid, TW: A matrix equation related to an non-oscillation criterion and Lyapunov stability. Q. Appl. Math. Soc. 23, 83-87 (1965)
Reid, TW: A matrix Lyapunov inequality. J. Math. Anal. Appl. 32, 424-434 (1970)
Singh, B: Forced oscillation in general ordinary differential equations. Tamkang J. Math. 6, 5-11 (1975)
Cakmak, D: Lyapunov-type integral inequalities for certain higher order differential equations. Appl. Math. Comput. 216, 368-373 (2010)
Cheng, SS: A discrete analogue of the inequality of Lyapunov. Hokkaido Math. J. 12, 105-112 (1983)
Došlý, O, Řehák, P: Half-Linear Differential Equations. Elsevier, Heidelberg (2005)
Elbert, A: A half-linear second order differential equation. Colloq. Math. Soc. János Bolyai 30, 158-180 (1979)
Eliason, SB: A Lyapunov inequality for a certain non-linear differential equation. J. Lond. Math. Soc. 2, 461-466 (1970)
Eliason, SB: Lyapunov type inequalities for certain second order functional differential equations. SIAM J. Appl. Math. 27(1), 180-199 (1974)
Guseinov, GS, Kaymakcalan, B: Lyapunov inequalities for discrete linear Hamiltonian systems. Comput. Math. Appl. 45, 1399-1416 (2003)
Guseinov, GS, Zafer, A: Stability criteria for linear periodic impulsive Hamiltonian systems. J. Math. Anal. Appl. 35, 1195-1206 (2007)
Jiang, L, Zhou, Z: Lyapunov inequality for linear Hamiltonian systems on time scales. J. Math. Anal. Appl. 310, 579-593 (2005)
Kayar, Z, Zafer, A: Stability criteria for linear Hamiltonian systems under impulsive perturbations. Appl. Math. Comput. 230, 680-686 (2014)
Pachpatte, BG: On Lyapunov-type inequalities for certain higher order differential equations. J. Math. Anal. Appl. 195, 527-536 (1995)
Pachpatte, BG: Lyapunov type integral inequalities for certain differential equations. Georgian Math. J. 4(2), 139-148 (1997)
Panigrahi, S: Lyapunov-type integral inequalities for certain higher order differential equations. Electron. J. Differ. Equ. 2009, 28 (2009)
Parhi, N, Panigrahi, S: On Lyapunov-type inequality for third-order differential equations. J. Math. Anal. Appl. 233(2), 445-460 (1999)
Parhi, N, Panigrahi, S: Lyapunov-type inequality for higher order differential equations. Math. Slovaca 52(1), 31-46 (2002)
Tiryaki, A, Unal, M, Cakmak, D: Lyapunov-type inequalities for nonlinear systems. J. Math. Anal. Appl. 332, 497-511 (2007)
Unal, M, Cakmak, D, Tiryaki, A: A discrete analogue of Lyapunov-type inequalities for nonlinear systems. Comput. Math. Appl. 55, 2631-2642 (2008)
Unal, M, Cakmak, D: Lyapunov-type inequalities for certain nonlinear systems on time scales. Turk. J. Math. 32, 255-275 (2008)
Yang, X: On Lyapunov-type inequality for certain higher-order differential equations. Appl. Math. Comput. 134, 307-317 (2003)
Yang, X: Lyapunov-type inequality for a class of even-order differential equations. Appl. Math. Comput. 215, 3884-3890 (2010)
Tiryaki, A: Recent developments of Lyapunov-type inequalities. Adv. Dyn. Syst. Appl. 5(2), 231-248 (2010)
Acknowledgements
This work was carried out when the second author was on academic leave, visiting TAMUK (Texas A&M University-Kingsville) and he wishes to thank TAMUK. This work is partially supported by TUBITAK (The Scientific and Technological Research Council of Turkey).
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Agarwal, R.P., Özbekler, A. Lyapunov type inequalities for even order differential equations with mixed nonlinearities. J Inequal Appl 2015, 142 (2015). https://doi.org/10.1186/s13660-015-0633-4
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DOI: https://doi.org/10.1186/s13660-015-0633-4