We establish some new Lyapunov-type inequalities for one-dimensional p-Laplacian systems with antiperiodic boundary conditions. The lower bounds of eigenvalues are presented.Встановлено дєякі нові нєрівності типу Ляпунова для одновимірних p-лапласових систем з антиперіодичними граничними умовами. Наведено нижні межі для власних значень

Yannan Li

Yongzhen Bai

Youyu Wang

Ukrains’kyi Matematychnyi Zhurnal

English

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

UDC 517.9Youyu Wang, Yannan Li, Yongzhen Bai (Tianjin Univ. Finance and Economics, China)LYAPUNOV-TYPE INEQUALITIES FOR QUASILINEAR SYSTEMSWITH ANTIPERIODIC BOUNDARY CONDITIONSНЕРIВНОСТI ТИПУ ЛЯПУНОВА ДЛЯ КВАЗIЛIНIЙНИХ СИСТЕМЗ АНТИПЕРIОДИЧНИМИ ГРАНИЧНИМИ УМОВАМИWe establish some new Lyapunov-type inequalities for one-dimensional p-Laplacian systems with antiperiodic boundaryconditions. The lower bounds of eigenvalues are presented.Встановлено деякi новi нерiвностi типу Ляпунова для одновимiрних p-лапласових систем з антиперiодичнимиграничними умовами. Наведено нижнi межi для власних значень.1. Introduction. The Lyapunov inequality and many of its generalizations have proved to be usefultools in oscillation theory, disconjugacy, eigenvalue problems, and numerous other applications forthe theories of differential and difference equations. A classical result of Lyapunov [12] states that ifu(t) is a nontrivial solution of the differential systemu′′(t) + r(t)u(t) = 0, t ∈ (a, b),u(a) = 0 = u(b),where r(t) is a continuous and nonnegative function defined in [a, b], thenb∫ar(t)dt >4b− a,and the constant 4 cannot be replaced by a larger number.Since the appearance of Lyapunov’s fundamental paper, various proofs and generalizations orimprovements have appeared in the literature. For authors, who contributed to the Lyapunov-typeinequalities, we refer to [1, 3 – 11, 14 – 16, 20 – 23] and the references quoted therein, especially tothe survey papers [1, 3, 22]. In recent years, the Lyapunov inequality has been extended in manydirections.For example, in 2004, Pinasco [18] has generalized the classical Lyapunov inequality for thehalf-linear differential equation(|u′(t)|p−2u′(t))′ + r(t)|u(t)|p−2u(t) = 0, t ∈ (a, b),u(a) = u(b) = 0.(1.1)He obtained Lyapunov inequality for system (1.1) as follows:b∫ar(t)dt ≥ 2p(b− a)p/q ,c© YOUYU WANG, YANNAN LI, YONGZHEN BAI, 20131646 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12LYAPUNOV-TYPE INEQUALITIES FOR QUASILINEAR SYSTEMS WITH ANTIPERIODIC . . . 1647where p > 1, 1/p+ 1/q = 1 and r ∈ C([a, b], (0,+∞)).In 2006, Pinasco [17] studied the following Dirichlet – Neumann problem:(|u′(t)|p−2u′(t))′ + r(t)|u(t)|p−2u(t) = 0, t ∈ (a, b),u′(a) = u(b) = 0.He gave the following Lyapunov-type inequality:b∫ar(t)dt >1(b− a)p−1 .Although there is an extensive literature on the Lyapunov-type inequalities for various classes ofdifferential equations, there is not much done for the linear Hamiltonian systems.In 2006, Napoli and Pinasco [13] have interested in the problem of finding the Lyapunov-typeinequality for the following quasilinear systems:− (|u′1(t)|p1−2u′1(t))′ = r1(t)|u1(t)|α1−2|u2(t)|α2u1(t),− (|u′2(t)|p2−2u′2(t))′ = r2(t)|u1(t)|α1 |u2(t)|α2−2u2(t)(1.2)with the Dirichlet boundary conditionsu1(a) = u1(b) = 0 = u2(a) = u2(b), u1(t) > 0, u2(t) > 0 ∀t ∈ (a, b),where r1, r2 are real-valued positive continuous functions for all x ∈ R, the exponents satisfy1 < p1, p2 <∞, and the positive parameters α1, α2 satisfy α1p1+α2p2= 1. They gave the followingLyapunov-type inequality:(b− a)α1+α2−1 b∫ar1(t)dtα1p1 b∫ar2(t)dtα2p2≥ 2α1+α2 . (1.3)More recently, by adopting the method used in Napoli and Pinasco [13], Cakmak and Tiryaki [2]generalized Lyapunov-type inequality (1.3) to the following more general quasilinear systems:− (|u′1(t)|p1−2u′1(t))′ = r1(t)|u1(t)|α1−2|u2(t)|α2 . . . |un(t)|αnu1(t),− (|u′2(t)|p2−2u′2(t))′ = r2(t)|u1(t)|α1 |u2(t)|α2−2 . . . |un(t)|αnu2(t),· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·− (|u′n(t)|pn−2u′n(t))′ = rn(t)|u1(t)|α1 |u2(t)|α2 . . . |un(t)|αn−2un(t)(1.4)with Dirichlet boundary conditionsui(a) = 0 = ui(b), ui(t) 6≡ 0 ∀t ∈ (a, b), i = 1, 2, . . . , n.They established the Lyapunov-type inequalityISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 121648 YOUYU WANG, YANNAN LI, YONGZHEN BAIn∏i=1 b∫ar+i (t)dtαipi≥ 2m(b− a)1−m,where m =∑ni=1αi and r+i (t) = max{ri(t), 0} for i = 1, 2, · · · , n.Motivated by the paper [2] and [19], the purpose of this paper is to get three types of Lyapunovinequalities for one-dimensional p-Laplacian system. In Section 2, we show Lyapunov inequality forone-dimensional p-Laplacian problem− (|u′(t)|p−2u′(t))′ = r(t)|u(t)|p−2u(t), t ∈ (a, b),u(a) + u(b) = 0, u′(a) + u′(b) = 0,(1.5)where r : [a, b] → (0,∞) is a continuous function, p > 1, q be a conjugate exponent of p, i.e.,1/p+ 1/q = 1.In Section 3, the more general system than (1.2)− (|u′1(t)|p1−2u′1(t))′ = r1(t)|u1(t)|α1−2|u2(t)|α2u1(t),− (|u′2(t)|p2−2u′2(t))′ = r2(t)|u1(t)|β1 |u2(t)|β2−2u2(t),ui(a) + ui(b) = 0, u′i(a) + u′i(b) = 0, i = 1, 2,(1.6)will be studied, we establish new Lyapunov inequality for this system. Where r1, r2 are real-valuedpositive continuous functions for all t ∈ R, the exponents satisfy 1 < p1, p2 <∞, α1, α2, β1, β2 > 0satisfyα1p1+α2p2= 1 andβ1p1+β2p2= 1.In Section 4, we consider the Lyapunov inequality for system (1.4) with antiperiodic conditions,i.e.,− (|u′1(t)|p1−2u′1(t))′ = r1(t)|u1(t)|α1−2|u2(t)|α2 . . . |un(t)|αnu1(t),− (|u′2(t)|p2−2u′2(t))′ = r2(t)|u1(t)|α1 |u2(t)|α2−2 . . . |un(t)|αnu2(t),· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·− (|u′n(t)|pn−2u′n(t))′ = rn(t)|u1(t)|α1 |u2(t)|α2 . . . |un(t)|αn−2un(t),ui(a) + ui(b) = 0, u′i(a) + u′i(b) = 0, i = 1, 2, . . . , n.(1.7)As an application of our Lyapunov-type inequalities, in Section 5, we focus on the estimate oflower bound for eigenvalues.2. Lyapunov-type inequality for system (1.5). In the proof of our results, the following lemmais very important.Lemma 2.1. If u′ ∈ Lp(a, b) and u(t) satisfying the condition u(a) + u(b) = 0, then|u(t)| ≤ 12(b− a)1/q b∫a|u′(x)|pdx1/p .ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12LYAPUNOV-TYPE INEQUALITIES FOR QUASILINEAR SYSTEMS WITH ANTIPERIODIC . . . 1649Proof. Let us define,H(x, y) =12, a ≤ x ≤ y,−12, y ≤ x ≤ b.Then, we haveu(y) =b∫au′(x)H(x, y)dx, a ≤ y ≤ b.So, by Ho¨lder’s inequality, we have|u(y)| ≤ b∫a|H(x, y)|qdx1/q b∫a|u′(x)|pdx1/p ==12(b− a)1/q b∫a|u′(x)|pdx1/p .The main result of this section is the following Lyapunov-type inequality.Theorem 2.1. If u(t) is a solution of system (1.5) and u(t) 6≡ 0, t ∈ [a, b], thenb∫ar(t)dt >2p(b− a)p−1 .Proof. Multiplying the equation in (1.5) by u and integrating over [a, b], yields−b∫a(|u′|p−2u′)′udt =b∫ar|u|pdt,|u′(a)|p−2u′(a)u(a)− |u′(b)|p−2u′(b)u(b) +b∫a|u′|pdt =b∫ar|u|pdt,by the antiperiodic boundary condition in (1.5), we get |u′(b)|p−2u′(b)u(b) = |u′(a)|p−2u′(a)u(a),so we haveb∫a|u′|pdt =b∫ar|u|pdt,by Lemma 2.1, we obtainb∫a|u′|pdt =b∫ar|u|pdt < ( maxa≤t≤b|u|)pb∫ar(t)dt ≤ (b− a)p−12pb∫a|u′(t)|pdtb∫ar(t)dt. (2.1)Now we claim thatISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 121650 YOUYU WANG, YANNAN LI, YONGZHEN BAIb∫a|u′(t)|pdt > 0. (2.2)If not, we have∫ ba|u′(t)|pdt = 0, then u′(t) = 0, t ∈ [a, b]. By the antiperiodic boundary conditionu(a) + u(b) = 0, we can obtain u(t) ≡ 0, t ∈ [a, b], which contradicts u(t) 6≡ 0, t ∈ [a, b]. Therefore(2.2) holds. Divided the inequality (2.1) by∫ ba|u′(t)|pdt, we haveb∫ar(t)dt >2p(b− a)p−1 .3. Lyapunov-type inequality for system (1.6). In this section, the main result is as follows.Theorem 3.1. If system (1.6) has a solution u1(t), u2(t) and ui(t) 6≡ 0 ∀t ∈ [a, b], i = 1, 2,then(b− a)β1+α2−β1p1−α2p2 b∫ar1(t)dtβ1p1 b∫ar2(t)dtα2p2≥ 2β1+α2 . (3.1)Proof. Multiplying the first equation in (1.6) by u1 and integrating over [a, b] together with theantiperiodic condition, yieldsb∫a|u′1(t)|p1dt =b∫ar1(t)|u1(t)|α1 |u2(t)|α2dt, (3.2)similarly,b∫a|u′2(t)|p2dt =b∫ar2(t)|u1(t)|β1 |u2(t)|β2dt, (3.3)by Lemma 2.1, we get|u1(t)|p1 ≤ (b− a)p1−12p1b∫a|u′1(x)|p1dx. (3.4)Now, it follows from (3.2), (3.4) and the Ho¨lder inequality thatb∫ar1(t)|u1(t)|p1dt ≤ (b− a)p1−12p1b∫ar1(t)dtb∫a|u′1(t)|p1dt ==(b− a)p1−12p1b∫ar1(t)dtb∫ar1(t)|u1(t)|α1 |u2(t)|α2dt ≤ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12LYAPUNOV-TYPE INEQUALITIES FOR QUASILINEAR SYSTEMS WITH ANTIPERIODIC . . . 1651≤ (b− a)p1−12p1b∫ar1(t)dt b∫ar1(t)|u1(t)|p1dtα1p1 b∫ar1(t)|u2(t)|p2dtα2p2==M1 b∫ar1(t)|u1(t)|p1dtα1p1 b∫ar1(t)|u2(t)|p2dtα2p2(3.5)andb∫ar2(t)|u1(t)|p1dt ≤ (b− a)p1−12p1b∫ar2(t)dtb∫a|u′1(t)|p1dt ==(b− a)p1−12p1b∫ar2(t)dtb∫ar1(t)|u1(t)|α1 |u2(t)|α2dt ≤≤ (b− a)p1−12p1b∫ar2(t)dt b∫ar1(t)|u1(t)|p1dtα1p1 b∫ar1(t)|u2(t)|p2dtα2p2==M2 b∫ar1(t)|u1(t)|p1dtα1p1 b∫ar1(t)|u2(t)|p2dtα2p2, (3.6)whereM1 =(b− a)p1−12p1b∫ar1(t)dt, M2 =(b− a)p1−12p1b∫ar2(t)dt. (3.7)Similarly, we also have|u2(t)|p2 ≤ (b− a)p2−12p2b∫a|u′2(x)|p2dx. (3.8)It follows from (3.3), (3.8) and the Ho¨lder inequality thatb∫ar1(t)|u2(t)|p2dt ≤ (b− a)p2−12p2b∫ar1(t)dtb∫a|u′2(t)|p2dt ==(b− a)p2−12p2b∫ar1(t)dtb∫ar2(t)|u1(t)|β1 |u2(t)|β2dt ≤ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 121652 YOUYU WANG, YANNAN LI, YONGZHEN BAI≤ (b− a)p2−12p2b∫ar1(t)dt b∫ar2(t)|u1(t)|p1dtβ1p1 b∫ar2(t)|u2(t)|p2dtβ2p2==M3 b∫ar2(t)|u1(t)|p1dtβ1p1 b∫ar2(t)|u2(t)|p2dtβ2p2(3.9)andb∫ar2(t)|u2(t)|p2dt ≤ (b− a)p2−12p2b∫ar2(t)dtb∫a|u′2(t)|p2dt ==(b− a)p2−12p2b∫ar2(t)dtb∫ar2(t)|u1(t)|β1 |u2(t)|β2dt ≤≤ (b− a)p2−12p2b∫ar2(t)dt b∫ar2(t)|u1(t)|p1dtβ1p1 b∫ar2(t)|u2(t)|p2dtβ2p2==M4 b∫ar2(t)|u1(t)|p1dtβ1p1 b∫ar2(t)|u2(t)|p2dtβ2p2, (3.10)whereM3 =(b− a)p2−12p2b∫ar1(t)dt, M4 =(b− a)p2−12p2b∫ar2(t)dt. (3.11)Next, we prove thatb∫ar1(t)|u1(t)|p1dt > 0. (3.12)In fact, if (3.12) is not true, thenb∫ar1(t)|u1(t)|p1dt = 0. (3.13)From (3.2) and (3.13), we have0 ≤b∫a|u′1(t)|p1dt =b∫ar1(t)|u1(t)|α1 |u2(t)|α2dt ≤ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12LYAPUNOV-TYPE INEQUALITIES FOR QUASILINEAR SYSTEMS WITH ANTIPERIODIC . . . 1653≤ b∫ar1(t)|u1(t)|p1dtα1p1 b∫ar1(t)|u2(t)|p2dtα2p2= 0.So u′1(t) ≡ 0, a ≤ t ≤ b, by the antiperiodic boundary condition, we obtain u1(t) ≡ 0 for a ≤ t ≤ b,which contradicts u1(t) 6≡ 0 ∀t ∈ [a, b]. Therefore, (3.12) holds. Similarly, we getb∫ar1(t)|u2(t)|p2dt > 0,b∫ar2(t)|u2(t)|p2dt > 0,b∫ar2(t)|u1(t)|p1dt > 0.From (3.5), (3.6), (3.9), (3.10), we obtainMα1β1p211 Mβ1α2p1p22 Mβ1α2p1p23 Mα2β2p224 ≥ 1.It follows from (3.7), (3.11) that (3.1) holds.4. Lyapunov-type inequality for system (1.7). In this section, we establish new Lyapunov-typeinequality for system (1.7). Assume that(H1) ri, i = 1, 2, . . . , n, are real-valued positive continuous functions for all t ∈ R,(H2) the exponents satisfy 1 < pi <∞ and the positive parameters αi satisfy∑ni=1αipi= 1.Theorem 4.1. If system (1.7) has a solution (u1(t), u2(t), . . . , un(t)) with ui(t) 6≡ 0, t ∈ [a, b],i = 1, 2, . . . , n, thenn∏i=1n∏j=1(b− a)pi−12pib∫arj(t)dtαiαjpipj≥ 1. (4.1)Proof. Multiplying the ith equation in (1.7) by ui and integrating over [a, b] together with theantiperiodic condition, yieldsb∫a|u′i(t)|pidt =b∫ari(t)n∏k=1|uk(t)|αkdt, i = 1, 2, . . . , n, (4.2)similarly, by Lemma 2.1, we get|ui(t)|pi ≤ (b− a)pi−12pib∫a|u′i(x)|pidx, i = 1, 2, . . . , n. (4.3)Now, it follows from (4.2), (4.3) and the generalized Ho¨lder inequality thatb∫arj(t)|ui(t)|pidt ≤ (b− a)pi−12pib∫arj(t)dtb∫a|u′i(t)|pidt ==(b− a)pi−12pib∫arj(t)dtb∫ari(t)n∏k=1|uk(t)|αkdt =ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 121654 YOUYU WANG, YANNAN LI, YONGZHEN BAI=Mijb∫ari(t)n∏k=1|uk(t)|αkdt ≤Mijn∏k=1 b∫ari(t)|uk(t)|pkdtαkpk, i, j = 1, 2, . . . , n, (4.4)whereMij =(b− a)pi−12pib∫arj(t)dt, i, j = 1, 2, . . . , n. (4.5)Next, we prove thatb∫ari(t)|uj(t)|pjdt > 0, i, j = 1, 2, . . . , n. (4.6)If (4.6) is not true, then there exists i0, j0 ∈ {1, 2, . . . , n} such thatb∫ari0(t)|uj0(t)|pj0dt = 0. (4.7)From (4.2) and (4.7), we have0 ≤b∫a|u′i0(t)|pi0dt =b∫ari0(t)n∏k=1|uk(t)|αkdt ≤n∏k=1 b∫ari0(t)|uk(t)|pkdtαkpk= 0.So thatu′i0(t) = 0, a ≤ t ≤ b. (4.8)Combining the antiperiodic boundary and (4.8), we obtain that ui0(t) ≡ 0 for a ≤ t ≤ b, whichcontradicts ui(t) 6≡ 0, t ∈ [a, b], i = 1, 2, . . . , n. Therefore (4.6) holds. From (4.4), we haven∏i=1n∏j=1Mαiαjpipjij ≥ 1.It follows from (4.5) that (4.1) holds.5. Lower bounds for eigenvalues problem. In this section, we apply our Lyapunov-type in-equalities to obtain lower bounds for eigenvalues. Firstly, we investigate the problem− (|u′(t)|p−2u′(t))′ = λr(t)|u(t)|p−2u(t), t ∈ (a, b),u(a) + u(b) = 0, u′(a) + u′(b) = 0.(5.1)As a corollary of Theorem 2.1, we have the result for system (5.1).ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12LYAPUNOV-TYPE INEQUALITIES FOR QUASILINEAR SYSTEMS WITH ANTIPERIODIC . . . 1655Theorem 5.1. Assume that p > 1, r(t) is real-valued positive continuous function for all t ∈ Rand system (5.1) has a solution u(t) satisfying u(t) 6≡ 0, t ∈ [a, b]. Let λ be the eigenvalue of system(5.1). Thenλ >2p(b− a)p−1∫ bar(x)dx.Secondly, we consider the eigenvalues problem− (|u′1(t)|p1−2u′1(t))′ = λ1α1r(t)|u1(t)|α1−2|u2(t)|α2 . . . |un(t)|αnu1(t),− (|u′2(t)|p2−2u′2(t))′ = λ2α2r(t)|u1(t)|α1 |u2(t)|α2−2 . . . |un(t)|αnu2(t),· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·− (|u′n(t)|pn−2u′n(t))′ = λnαnr(t)|u1(t)|α1 |u2(t)|α2 . . . |un(t)|αn−2un(t),ui(a) + ui(b) = 0, u′i(a) + u′i(b) = 0, i = 1, 2, . . . , n.(5.2)Let (λ1, λ2, . . . , λn) be eigenvalue of problem (5.2) and (u1(t), u2(t), . . . , un(t)) be the eigen-functions associated with (λ1, λ2, . . . , λn). Then (u1(t), u2(t), . . . , un(t)) is a solution of system(1.7) with ri(t) = λiαir(t) > 0 for i = 1, 2, . . . , n.Theorem 5.2. Assume that 1 < pi < ∞, αi > 0 satisfy∑ni=1αipi= 1, r(t) is a real-valuedpositive continuous function defined on R, system (5.2) has a solution (u1(t), u2(t), . . . , un(t))with ui(t) 6≡ 0, t ∈ [a, b], i = 1, 2, . . . , n. Then there exists a function g(λ1, λ2, . . . , λn−1)such that λn > g(λ1, λ2, . . . , λn−1) for every eigenvalue (λ1, λ2, . . . , λn) of system (5.2), whereg(λ1, λ2, . . . , λn−1) is given byg(λ1, λ2, . . . , λn−1) =1αnn−1∏j=1(λjαj)αjpjn∏i=1(b− a)pi−12pib∫arj(t)dtαipi− pnαn.Proof. For system (1.7), ri(t) = λiαir(t) > 0 for i = 1, 2, . . . , n. Hence, it follows from (4.1)that1 ≤n∏i=1n∏j=1(b− a)pi−12pib∫arj(t)dtαiαjpipj==n∏j=1(λjαj)αjpjn∏i=1(b− a)pi−12pib∫arj(t)dtαipi== (λnαn)αnpnn−1∏j=1(λjαj)αjpjn∏i=1(b− a)pi−12pib∫arj(t)dtαipi.ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 121656 YOUYU WANG, YANNAN LI, YONGZHEN BAIHence, we haveλn >1αnn−1∏j=1(λjαj)αjpjn∏i=1(b− a)pi−12pib∫arj(t)dtαipi− pnαn.1. Brown R. C., Hinton D. B. Lyapunov inequalities and their applications // Survey Classical Inequalities / Ed.T. M. Rassias. – Dordrecht, The Netherlands: Kluwer Acad. Publ., 2000. – P. 1 – 25.2. Cakmak D., Tiryaki A. Lyapunov-type inequality for a class of Dirichlet quasilinear systems involving the(p1, p2, . . . , pn)-Laplacian // J. Math. Anal. and Appl. – 2010. – 369. – P. 76 – 81.3. Cheng S. Lyapunov inequalities for differential and difference equations // Fasc. Math. – 1991. – 23. – P. 25 – 41.4. Dos˘ly´ O., R˘eha´k P. Half-linear differential equations // North-Holland Math. Stud. – 2005. – 202.5. Eliason S. B. A Lyapunov type inequality for certain nonlinear differential equation // J. London Math. Soc. – 1970. –3. – P. 461 – 466.6. Guseinov G., Kaymakcalan B. Lyapunov inequalities for discrete linear Hamiltonian system // Comput. Math. andAppl. – 2003. – 45. – P. 1399 – 1416.7. Ha C. Eigenvalues of a Sturm – Liouville problem and inequalities of Lyapunov type // Proc. Amer. Math. Soc. –1998. – 126. – P. 3507 – 3511.8. Hartmann P. Ordinary differential equations. – Second ed. – Boston: Birkha¨user, 1982.9. Hochstadt H. On an inequality of Lyapunov // Proc. Amer. Math. Soc. – 1969. – 22. – P. 282 – 284.10. Lee C., Yeh C., Hong C., Agarwal R. P. Lyapunov and Wirtinger inequalities // Appl. Math. Lett. – 2004. – 17. –P. 847 – 853.11. Leighton W. On Lyapunov’s inequality // Proc. Amer. Math. Soc. – 1972. – 33. – P. 627 – 628.12. Lyapunov A. Probleme General de la Stabilite du Mouvement // Ann. Math. Stud. – Princeton, NJ, USA: PrincetonUniv. Press, 1949. – 17.13. Napoli P. L., Pinasco J. P. Estimates for eigenvalues of quasilinear elliptic systems // J. Differen. Equat. – 2006. –227. – P. 102 – 115.14. Nehari Z. On an inequality of Lyapunov // Stud. Math. Anal. and Relat. Top. – Stanford, CA: Stanford Univ. Press,1962. – P. 256 – 261.15. Pachpatte B. G. On Lyapunov-type inequalities for certain higher order differential equations // J. Math. Anal. andAppl. – 1995. – 195. – P. 527 – 536.16. Parhi N., Panigrahi S. On Liapunov-type inequality for third-order differential equations // J. Math. Anal. and Appl. –1999. – 233. – P. 445 – 464.17. Pinasco J. P. Comparison of eigenvalues for the p-Laplacian with integral inequalities // Appl. Math. and Comput. –2006. – 182. – P. 1399 – 1404.18. Pinasco J. P. Lower bounds for eigenvalues of the one-dimensional p-Laplacian // Abst. Appl. Anal. – 2004. – 2004. –P. 147 – 153.19. Tang X., He X. Lower bounds for generalized eigenvalues of the quasilinear systems // J. Math. Anal. and Appl. –2012. – 385. – P. 72 – 85.20. Tang X., Zhang M. Lyapunov inequalities and stability for linear Hamiltonian systems // J. Different. Equat. – 2012. –252. – P. 358 – 381.21. Tiryaki A., Unal M., Cakmak D. Lyapunov-type inequalities for nonlinear systems // J. Math. Anal. and Appl. – 2007. –332. – P. 497 – 511.22. Tiryaki A. Recent development of Lyapunov-type inequalities // Adv. Dynam. Syst. and Appl. – 2010. – 5. –P. 231 – 248.23. Yang X. On inequalities of Lyapunov type // Appl. Math. and Comput. – 2003. – 134. – P. 293 – 300.Received 18.05.12,after revision — 21.05.13ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12

Lyapunov-type inequalities for quasilinear systems with antiperiodic boundary conditions

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Lyapunov-type inequalities for quasilinear systems with antiperiodic boundary conditions

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