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On Lyapunov-type inequality for quasilinear systems
Applied Mathematics and Computation 216 (2010) 3584–3591
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On Lyapunov-type inequality for quasilinear systems Devrim Çakmak a,*, Aydın Tiryaki b a b
Gazi University, Faculty of Education, Department of Mathematics Education, 06500 Teknikokullar, Ankara, Turkey Izmir University, Faculty of Arts and Sciences, Department of Mathematics and Computer Sciences, 35350 Uckuyular, Izmir, Turkey
a r t i c l e
i n f o
a b s t r a c t In this paper, we sketch some recent developments in the theory of Lyapunov-type inequalities and present some new results relating to a quasilinear system, special cases of which contain some well-known differential equations such as half-linear and linear equations. Our result generalize the Lyapunov-type inequality given in [18]. Ó 2010 Elsevier Inc. All rights reserved.
Keywords: Lyapunov-type inequality Quasilinear system Half-linear equation Linear equation
1. Introduction The Lyapunov inequality and many of its generalizations have proved to be useful tools in oscillation theory, disconjugacy, eigenvalue problems, and numerous other applications for the theories of differential and difference equations. Before we continue the description of the content of this paper, a few hints concerning the literature on the Lyapunov inequalities might be in order. Russian mathematician Lyapunov [17] proved the following remarkable result. Theorem A. If x(t) is a nontrivial solution of
x00 ðtÞ þ qðtÞxðtÞ ¼ 0;
ð1Þ
with x(a) = 0 = x(b) where a; b 2 R with a < b be consecutive zeros and x(t) – 0 for t 2 (a, b), then the so called Lyapunov inequality
Z
b
jqðsÞj ds >
a
4 ba
ð2Þ
holds. Since the appearance of Lyapunov’s fundamental paper, various proofs and generalizations or improvements have appeared in the literature. For example, Hartman [13] has generalized the classical Lyapunov inequality for the linear differential equation
ðrðtÞx0 ðtÞÞ0 þ qðtÞxðtÞ ¼ 0;
rðtÞ > 0;
ð3Þ
as follows. Theorem B. If a; b 2 R with a < b are consecutive zeros of nontrivial solution of Eq. (3), then
Z a
b
qþ ðsÞ ds > R b a
4 ðrðsÞÞ1 ds
;
* Corresponding author. E-mail addresses: [email protected] (D. Çakmak), [email protected] (A. Tiryaki). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.05.004
ð4Þ
D. Çakmak, A. Tiryaki / Applied Mathematics and Computation 216 (2010) 3584–3591
3585
where q+(t) = max{0, q(t)} is the nonnegative part of q(t). Thus, the inequality (2) is strengthened to
Z
b
qþ ðsÞ ds >
a
4 ; ba
ð5Þ
for the Eq. (1) by Theorem B. The inequality (5) is the best possible in the sense that if the constant 4 in (5) is replaced by any larger constant, then there exists an example of (1) for which (5) no longer holds (see [13, p. 345], [15]). However, stronger results were obtained in Brown and Hinton [2] and Kwong [15]. In [15], it is shown that
Z
c
qþ ðsÞ ds >
1 ; ca
qþ ðsÞ ds >
1 ; bc
a
ð6Þ
and
Z
b
c
ð7Þ 0
where c 2 (a, b) such that x (c) = 0. Hence
Z
b
qþ ðsÞ ds >
a
1 1 ba 4 þ ¼ P : c a b c ðc aÞðb cÞ ba
ð8Þ
In [2, Corollary 4.1], the authors obtained
Z b 4 ; qðsÞ ds > ba a
ð9Þ
from which (2) can be obtained. The Lyapunov inequality has been extended in many directions and its half-linear differential equation
0 rðtÞjx0 ðtÞjk2 x0 ðtÞ þ qðtÞjxðtÞjk2 xðtÞ ¼ 0;
rðtÞ > 0 and k > 1;
ð10Þ
˘ ehák’s recent book [7] as follows. extension found in Došly´ and R Theorem C. Let a; b 2 R with a < b be consecutive zeros of nontrivial solution of Eq. (10). Then
Z a
2k
b
qþ ðsÞ ds > R
ð11Þ
k1 ;
b ðrðsÞÞ1=ð1kÞ ds a
where q+(t) = max{0, q(t)} is the nonnegative part of q(t). A thorough literature review of continuous and discrete Lyapunov inequalities and their applications can be found in the survey paper [4] by Cheng and the references quoted therein. For authors who contributed the Lyapunov-type inequalities, we refer to Cheng [3], Eliason [9], Hartman [13], Kwong [15] and Reid [27]. We should also mention here that inequality (2) has been generalized to second order nonlinear differential equations by Eliason [10] and Pachpatte [20,21], to delay differential equations of the second order by Dahiya and Singh [6] and Eliason [11], to third order differential equations by Parhi and Panigrahi [23], and to certain higher order differential equations by Çakmak [5], Pachpatte [19], Panigrahi [22], Parhi and Panigrahi [24], Yang [31], and Yang and Lo [32]. Lyapunov-type inequalities can be found in Pachpatte’s paper [21] for the Emden–Fowler type equations, and were obtained for the first time by Elbert [8] for the half-linear equation, but the proof of ˘ ehák [7]. Lyapunov-type inequalities for the half-linear equation have its extension can be found in the book of Došly´ and R been rediscovered by Lee et al. [16] and Pinasco [25,26]. Although there is an extensive literature on the Lyapunov-type inequalities for various classes of differential equations, there is not much done for the linear Hamiltonian system, in the case of two scalar linear differential equations, has the form
y0 ðtÞ ¼ JHðtÞyðtÞ;
t 2 R;
ð12Þ
where
yðtÞ ¼ ðy1 ðtÞ; y2 ðtÞÞT ;
J¼
0
1
1 0
;
HðtÞ ¼
h11 ðtÞ h12 ðtÞ h21 ðtÞ h22 ðtÞ
;
with hjk(t), j, k = 1, 2 are real-valued continuous functions defined on R and h12(t) = h21(t). Setting y1(t) = x(t), y2(t) = u(t), h11(t) = b2(t), h12(t) = h21(t) = a1(t) and h22(t) = b1(t) in Eq. (12), one can easily obtains the following linear system
x0 ðtÞ ¼ a1 ðtÞxðtÞ þ b1 ðtÞuðtÞ; u0 ðtÞ ¼ b2 ðtÞxðtÞ a1 ðtÞuðtÞ;
ð13Þ
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D. Çakmak, A. Tiryaki / Applied Mathematics and Computation 216 (2010) 3584–3591
which is a special case of the linear counterpart of the nonlinear system
x0 ðtÞ ¼ a1 ðtÞxðtÞ þ b1 ðtÞjuðtÞjc2 uðtÞ;
ð14Þ
u0 ðtÞ ¼ b2 ðtÞjxðtÞjb2 xðtÞ a1 ðtÞuðtÞ;
with c = 2 and b = 2. Recently, Guseinov and Kaymakçalan [12] and Tiryaki et al. [28] have obtained the Lyapunov-type inequalities for system (13) and (14), respectively. The discrete and time scale analogues of Lyapunov-type inequalities for systems have been also found in the papers Ünal et al. [29], Jiang and Zhou [14] and Ünal and Ç akmak [30]. More recently, Bonder and Pinasco [1] and Napoli and Pinasco [18] have interested in the problem of finding lower and upper bounds for the eigenvalues of a quasilinear elliptic system. In [18], the authors have also obtained the following Lyapunov-type inequality for quasilinear system
0 ju0 ðxÞjp2 u0 ðxÞ ¼ f1 ðxÞjuðxÞja2 uðxÞjv ðxÞjb ; 0 jv 0 ðxÞjq2 v 0 ðxÞ ¼ f2 ðxÞjuðxÞja jv ðxÞjb2 v ðxÞ;
ð15Þ
where 1 < p, q < 1, the positive parameters a and b satisfy ap þ bq ¼ 1, and f1, f2 are real-valued positive continuous functions for all x 2 R, as follows. Theorem D. If system (15) has a real nontrivial solution (u(x), v(x)) such that u(a) = u(b) = 0 = v(a) = v(b) where a; b 2 R with a < b be consecutive zeros and u and v are not identically zero on [a, b], then the following inequality
2aþb 6 ðb aÞaþb1
Z
b
!a=p Z
b
f1 ðxÞ dx
a
!b=q f2 ðxÞ dx
ð16Þ
a
holds. As an application of Theorem D, the authors have improved the lower bounds on the eigenvalues problem of the following one-dimensional system
0 ju0 ðxÞjp2 u0 ðxÞ ¼ karðxÞjuðxÞja2 uðxÞjv ðxÞjb ; 0 jv 0 ðxÞjq2 v 0 ðxÞ ¼ lbrðxÞjuðxÞja jv ðxÞjb2 v ðxÞ;
ð17Þ
as follows. Theorem E. There exist a function h(k) such that l P h (k) for every generalized eigenvalue (k, l) of problem (17), where h(k) is given by
hðkÞ ¼
!q=b 1 C ; R b ka=p b rðxÞ dx a
ð18Þ
and the constant C is given by
C¼
2aþb
aa=p ðb aÞaþb1
:
ð19Þ
The principal aim of this paper is to state and prove a Lyapunov-type inequality for quasilinear systems in the form
0 r 1 ðxÞju0 ðxÞjp2 u0 ðxÞ ¼ f1 ðxÞjuðxÞja2 uðxÞjv ðxÞjb ; 0 r 2 ðxÞjv 0 ðxÞjq2 v 0 ðxÞ ¼ f2 ðxÞjuðxÞjh jv ðxÞjc2 v ðxÞ;
ð20Þ
whose special cases contain the well-known equations such as half-linear and linear equations. Our result is an extension of result by Napoli and Pinasco [18]. By using this inequality, we prove better lower bound than Napoli and Pinasco for the first eigenvalue of problem (17). Our motivation comes from the recent papers of Lee et al. [16], Napoli and Pinasco [18], Pachpatte [21], and Tiryaki et al. [28]. In this paper, we derive a Lyapunov-type inequality for quasilinear system (20), where the both components of the solution (u(x), v(x)) have consecutive zeros at the points a; b 2 R with a < b in I ¼ ½t 0 ; 1Þ R. For the special cases of system (20), we also derive some Lyapunov-type inequalities which not only relates points a and b in I at which the both components of the solution (u(x), v(x)) have consecutive zeros but also any point in (a, b) where the both components of the solution (u(x), v(x)) are maximized.
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Since our attention is restricted to the Lyapunov-type inequality for the quasilinear system of differential equations, we shall assume the existence of nontrivial solution (u(x), v(x)) of system (20) and state our basic hypothesis with respect to the same system: (i) r1, r2, f1, f2 are real-valued continuous functions such that r1(x) > 0 and r2(x) > 0 for all x 2 R. (ii) The exponents satisfy 1 < p, q < 1, and the positive parameters a, b, h and c yield ap þ bq ¼ 1 and ph þ cq ¼ 1. For the sake of convenience, we define the following integral operator
Z
Mðt; s; lÞ ¼
t
!1l 1l Z b l0 l0 ðsðxÞÞ l dx þ ðsðxÞÞ l dx ;
a
ð21Þ
t 0
where t 2 (a, b), s is a real-valued continuous function such that s(x) > 0 for all x 2 R, and l is the conjugate exponent of l 2 (1, +1), that is, l1 þ l10 ¼ 1. For a given s and l, set F(t) = M(t, s, l) for t 2 (a, b). F(t) obtains its minimum at the point t 2 (a, b) such that
Z
t
l0
ðsðxÞÞ l dx ¼
a
Z
b
l0
ðsðxÞÞ l dx
ð22Þ
t
holds. Thus, we have
FðtÞ P F min ðtÞ ¼ 2
Z
t
1l l0 ðsðxÞÞ l dx ¼ Nðt; s; lÞ:
ð23Þ
a
2. Main results The main result of this paper is the following theorem which is an extension of Theorem D. Theorem 1. Let the hypothesis (i) and (ii) hold. If system (20) has a real nontrivial solution (u(x), v(x)) such that u(a) = u(b) = 0 = v(a) = v(b) where a; b 2 R with a < b be consecutive zeros, and u and v are not identically zero on [a, b], then the following inequality
fMðc; r1 ; pÞg
h=p
b=q
fMðd; r 2 ; qÞg
Z 6 a
!h=p Z
b
f1þ ðxÞ dx
a
!b=q
b
f2þ ðxÞ dx
ð24Þ
holds, where juðcÞj ¼ max juðxÞj; jv ðdÞj ¼ max jv ðxÞj such that M(c, r1, p) and M(d, r2, q) exist, and fiþ ðxÞ ¼ maxf0; fi ðxÞg for i = 1, 2. a
a
Proof. It follows from u(a) = u(b) = 0 = v(a) = v(b) where a; b 2 R with a < b be consecutive zeros, and u and v are not identically zero on [a, b], one can choose c, d 2 (a, b) such that juðcÞj ¼ max juðxÞj > 0 and jv ðdÞj ¼ max jv ðxÞj > 0. From the existence a
a
0
0
of M(c, r1, p) and M(d, r2, q), c and d cannot be too close to a or b. By using Rolle’s theorem, clearly u (c) = 0 and v (d) = 0. Therefore, for c 2 (a, b) and u(a) = 0, we have
Z juðcÞj ¼
c
a
Z c u0 ðxÞ dx 6 ju0 ðxÞj dx:
ð25Þ
a
0
By using Hölder inequality on the integral of the right-hand side of (25) with indices p and p , we obtain
juðcÞj 6
Z
c
ju0 ðxÞj dx ¼
Z
a
c
1
1
ðr 1 ðxÞÞp ðr 1 ðxÞÞp ju0 ðxÞj dx 6
a
Z
c
10 Z c 1p p p0 ðr1 ðxÞÞ p dx r 1 ðxÞju0 ðxÞjp dx ;
a
ð26Þ
a
where 1p þ p10 ¼ 1. On the other hand, multiplying the first equation of system (20) by u(x) and integrating from a to c and tak0 ing into account that u(a) = 0 and u (c) = 0, we get
Z
c
r 1 ðxÞju0 ðxÞjp dx ¼
a
Z
c
f1 ðxÞjuðxÞja jv ðxÞjb dx:
ð27Þ
a
Therefore, by using (27) in (26), we have
juðcÞj 6
Z
c a
10 Z p p0 ðr 1 ðxÞÞ p dx a
c
1p Z c 10 Z c 1p p p0 f1 ðxÞjuðxÞja jv ðxÞjb dx 6 ðr 1 ðxÞÞ p dx f1þ ðxÞjuðxÞja jv ðxÞjb dx : a
ð28Þ
a
If we take the pth power of both sides of the inequality (28), and ju(x)j is maximum at the point c, and jv(x)j is maximum at the point d, we obtain
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D. Çakmak, A. Tiryaki / Applied Mathematics and Computation 216 (2010) 3584–3591
1 6 juðcÞjap jv ðdÞjb
Z
c
p1 Z p0 ðr 1 ðxÞÞ p dx
a
f1þ ðxÞ dx :
c
a
ð29Þ
Now, since u(b) = 0, we get
Z Z b b juðcÞj ¼ j uðcÞj ¼ u0 ðxÞ dx 6 ju0 ðxÞj dx; c c
ð30Þ
and repeating the above procedure step by step, one can easily obtain
1 6 juðcÞj
ap
Z
b
jv ðdÞj
b
0
pp
ðr 1 ðxÞÞ
dx
!p1 Z
c
!
b
f1þ ðxÞ dx
c
ð31Þ
:
Thus we obtain from (29) and (31)
Z
b
a
f1þ ðxÞ dx P juðcÞjpa jv ðdÞjb Mðc; r 1 ; pÞ;
ð32Þ
where 1p þ p10 ¼ 1. By using similar manner, we get
Z
b
a
f2þ ðxÞ dx P juðcÞjh jv ðdÞjqc Mðd; r 2 ; qÞ;
ð33Þ
where 1q þ q10 ¼ 1. Raising inequality (32) to a power e1, inequality (33) to a power e2 and multiplying the resulting equations, we obtain !e !e
juðcÞjðpaÞe1 he2 jv ðdÞjbe1 þðqcÞe2 fMðc; r 1 ; pÞge1 fMðd; r2 ; qÞge2 6
Z
1
b
f1þ ðxÞ dx
a
Z
a
2
b
f2þ ðxÞ dx
:
ð34Þ
Now, we choose e1 and e2 such that ju(c)j and jv(d)j cancels out, i.e. solves the homogeneous linear system
ðp aÞe1 he2 ¼ 0;
ð35Þ
be1 þ ðq cÞe2 ¼ 0:
We observe that by hypothesis (ii), this system admits a nontrivial solution, indeed both equations are equivalent to
e1
b h ¼ e2 : q p
ð36Þ
Hence, we may take e1 ¼ ph ; e2 ¼ bq and we get inequality (24) which completes the proof. h By using (22) and (23), we have the following result from Theorem 1. Corollary 1. Let all assumptions of Theorem 1 satisfy. Then the following inequality h=p
fNðc; r 1 ; pÞg
b=q
fNðd; r2 ; qÞg
Z
!h=p Z
b
6 a
f1þ ðxÞ dx
a
!b=q
b
f2þ ðxÞ dx
ð37Þ
holds, where c; d; Nðc; r 1 ; pÞ; Nðd; r 2 ; qÞ; f1þ and f2þ are defined as before such that
Z
c
Z
1
ðr 1 ðxÞÞ1p dx ¼
a
and
Z
b
1
ðr1 ðxÞÞ1p dx;
ð38Þ
c d
1
ðr 2 ðxÞÞ1q dx ¼
a
Z
b
1
ðr 2 ðxÞÞ1q dx:
ð39Þ
d
Remark 1. Since r1(x) > 0 and r2(x) > 0 for all x 2 R, by using (38) and (39), it is easy to see that c and d cannot be too close to a or b. p0 1 Rc Since the function h(x) = x1 p is convex for x > 0, Jensen’s inequality h yþz 6 2 ½hðyÞ þ hðzÞ with y ¼ a ðr1 ðxÞÞ p dx and 2 p0 Rb z ¼ c ðr 1 ðxÞÞ p dx implies ! !
Mðc; r 1 ; pÞ ¼
Z
c
1p Z b p0 p0 ðr 1 ðxÞÞ p dx þ ðr 1 ðxÞÞ p dx
a
c
Similarly, we obtain
Mðd; r 2 ; qÞ P 2
q
Z
b
ðr2 ðxÞÞ
q0
q
1p
P 2p
Z
b
p0
ðr 1 ðxÞÞ p dx
1p
:
ð40Þ
a
!1q dx
:
a
Thus, by using the inequalities (40) and (41), we have the following result from Theorem 1.
ð41Þ
D. Çakmak, A. Tiryaki / Applied Mathematics and Computation 216 (2010) 3584–3591
3589
Corollary 2. Let all assumptions of Theorem 1 satisfy. Then the following inequality
2hþb 6
Z
b
!hðp1Þ=p Z
1
b
ðr 1 ðxÞÞ1p dx
a
1
!bðq1Þ=q Z
a
!h=p Z
b
f1þ ðxÞ dx
ðr 2 ðxÞÞ1q dx
a
a
!b=q
b
f2þ ðxÞ dx
ð42Þ
holds, where f1þ and f2þ are defined as before. Remark 2. It is easy to see that Corollaries 1 and 2 are equivalent for r1(x) = r2(x) = 1 in system (20). Remark 3. When r1(x) = r2(x) = 1 and h = a (or c = b) and fi(x) > 0 for i = 1, 2, Corollaries 1 or 2 reduces to Theorem D by Napoli and Pinasco [18]. Remark 4. It is easy to see that the following inequalities
f þ ðxÞ 6 jf ðxÞj; Z Z b f þ ðxÞ dx 6 a
ð43Þ b
jf ðxÞj dx
a
ð44Þ
Rb
hold. Hence, the integrals of a fiþ ðxÞ dx for i = 1, 2 in the above results may be replaced by Thus, we shall arrive to the following inequality
fMðc; r1 ; pÞg
h=p
b=q
fMðd; r 2 ; qÞg
Z 6 a
b
!h=p Z jf1 ðxÞj dx
Rb a
jfi ðxÞj dx for i = 1, 2, respectively.
!b=q
b
jf2 ðxÞj dx
;
ð45Þ
a
which is immediate consequence of the inequality (24) in Theorem 1. Remark 5. Let us observe from above results that for h = 0 (or b = 0), we obtain the result for the case of a single equation of system (20). In this case, the inequalities (2), (4), (5) and (11) are relaxed to the inequality (24) by Theorem 1. In addition to this, Corollary 2 reduces to Theorem C for half-linear differential equation (10), and to Theorem B for linear differential equation (3) if we take p = 2 (or q = 2). When r1(x) = 1 and p = 2 (or r2(x) = 1 and q = 2), Corollary 2 reduces to Theorem A for linear differential equation (1). Moreover, one can easily obtain from the proof of Theorem 1 that inequalities (29) and (31) in the specialZcase reduce to c
a
and
Z
f1þ ðxÞ dx P
1 ; ca
ð46Þ
f1þ ðxÞ dx P
1 ; bc
ð47Þ
1 1 ba 4 þ ¼ P : c a b c ðc aÞðb cÞ ba
ð48Þ
b
c
respectively. Hence
Z
a
b
f1þ ðxÞ dx P
Thus, the inequalities (6)–(8) are strengthened to the inequalities (46)–(48) by Theorem 1, respectively. Now, we present an application of the obtained Lyapunov-type inequality in Theorem 1. Let (k, l) be generalized eigenvalues of the system (20) with r1(x) = 1 = r2(x), and h = a (or c = b). The function r(x) also be positive for all x 2 R. Therefore, system (20) with f1(x) = kar(x) > 0 and f2(x) = lbr(x) > 0 for all x 2 R reduces to system (17). By using techniques similar to Napoli and Pinasco [18], we obtain the following result which gives better lower bounds for the first eigencurve in the generalized spectra. The proof of this result is rather elementary, based on above generalization of the Lyapunov-type inequality, exactly as in that of Theorem 1.4 of Napoli and Pinasco [18]. Theorem 2. There exists a function h1(k) such that l P h1(k) for every generalized eigenvalue (k, l) of problem (17), where h1(k) is given by
1 D h1 ðkÞ ¼ R b ka=p b rðxÞ dx a
!q=b
and the constant D is given by
n
D¼
ðc aÞ1p þ ðb cÞ1p
ð49Þ
;
oa=p n ob=q ðd aÞ1q þ ðb dÞ1q
where c and d are defined as before.
aa=p
;
ð50Þ
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D. Çakmak, A. Tiryaki / Applied Mathematics and Computation 216 (2010) 3584–3591
Proof. Let (k, l) be a generalized eigenpair, and (u(x), v(x)) be the corresponding nontrivial solution of system (17). For all x 2 R, by substituting f1(x) = kar(x) > 0 and f2(x) = lbr(x) > 0 in the Lyapunov inequality (24), we obtain
n oa=p n ob=q ðc aÞ1p þ ðb cÞ1p ðd aÞ1q þ ðb dÞ1q 6
Z
b
!a=p Z karðxÞ dx
a
!b=q
b
lbrðxÞ dx
:
ð51Þ
a
By rearranging terms in (51), and using the condition ap þ bq ¼ 1, we obtain
Z n oa=p n ob=q ðc aÞ1p þ ðb cÞ1p ðd aÞ1q þ ðb dÞ1q 6 ðkaÞa=p ðlbÞb=q
b
! rðxÞ dx ;
ð52Þ
a
which gives
!q=b
D
ka=p
Rb a
rðxÞ dx
6 lb;
and the proof is completed.
ð53Þ
h
Remark 6. Since h1 is a continuous function, and h1(k) ? +1 as k ? 0+. Therefore, there exists a ball centered in the origin such that the generalized spectrum is contained in its exterior. Also, by rearranging terms in (53), we obtain
lb=q ka=p P
bb=q
D : Rb rðxÞ dx a
ð54Þ
It is clear that when the interval collapses, right-hand side of (54) goes to infinity. Thus, we obtain the desired generalizations of Napoli and Pinasco [18]’s result for one-dimensional nonlinear systems. Remark 7. If we compare Theorem 2 with Theorem E, we obtain D P C since (40) and (41) hold. Thus, the inequality h1(k) P h(k) holds. Therefore, Theorem 2 presents better lower bound than Theorem E. Acknowledgement The authors would like to thank the referee for his valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
J.F. Bonder, J.P. Pinasco, Estimates for eigenvalues of quasilinear elliptic systems, Part II, J. Differ. Equ. 245 (2008) 875–891. R.C. Brown, D.B. Hinton, Opial’s inequality and oscillation of 2nd order equations, Proc. Amer. Math. Soc. 125 (1997) 1123–1129. S.S. Cheng, A discrete analogue of the inequality of Lyapunov, Hokkaido Math. J. 12 (1983) 105–112. S.S. Cheng, Lyapunov inequalities for differential and difference equations, Fasc. Math. 23 (1991) 25–41. D. Çakmak, Lyapunov-type integral inequalities for certain higher order differential equations, Appl. Math. Comput. 216 (2010) 368–373. R.S. Dahiya, B. Singh, A Liapunov inequality and nonoscillation theorem for a second order nonlinear differential–difference equations, J. Math. Phys. Sci. 7 (1973) 163–170. ˇ ehák, Half-linear differential equations, Mathematics Studies, vol. 202, North-Holland, 2005. O. Došly´, P. R Á. Elbert, A half-linear second order differential equation, Colloq. Math. Soc. János Bolyai 30 (1979) 158–180. S.B. Eliason, A Lyapunov inequality, J. Math. Anal. Appl. 32 (1970) 461–466. S.B. Eliason, A Lyapunov inequality for a certain nonlinear differential equation, J. London Math. Soc. 2 (1970) 461–466. S.B. Eliason, Lyapunov type inequalities for certain second order functional differential equations, SIAM J. Appl. Math. 27 (1) (1974) 180–199. G. Guseinov, B. Kaymakçalan, Lyapunov inequalities for discrete linear Hamiltonian system, Comput. Math. Appl. 45 (2003) 1399–1416. P. Hartman, Ordinary Differential Equations, second ed., Birkhäuser, Boston, 1982. L. Jiang, Z. Zhou, Lyapunov inequality for linear Hamiltonian systems on time scales, J. Math. Anal. Appl. 310 (2005) 579–593. M.K. Kwong, On Lyapunov’s inequality for disfocality, J. Math. Anal. Appl. 83 (1981) 486–494. C. Lee, C. Yeh, C. Hong, R.P. Agarwal, Lyapunov and Wirtinger inequalities, Appl. Math. Lett. 17 (2004) 847–853. A.M. Liapunov, Probleme général de la stabilité du mouvement (French Translation of a Russian paper dated 1893), Ann. Fac. Sci. Univ. Toulouse 2 (1907) 27–247. Reprinted as Ann. Math. Studies, vol. 17, Princeton, 1947. P.L. Napoli, J.P. Pinasco, Estimates for eigenvalues of quasilinear elliptic systems, J. Differ. Equ. 227 (2006) 102–115. B.G. Pachpatte, On Lyapunov-type inequalities for certain higher order differential equations, J. Math. Anal. Appl. 195 (1995) 527–536. B.G. Pachpatte, Lyapunov type integral inequalities for certain differential equations, Georgian Math. J. 4 (2) (1997) 139–148. B.G. Pachpatte, Inequalities related to the zeros of solutions of certain second order differential equations, Facta Universitatis, Ser. Math. Inform. 16 (2001) 35–44. S. Panigrahi, Lyapunov-type integral inequalities for certain higher order differential equations, Electron. J. Differ. Equ. 2009 (28) (2009) 1–14. N. Parhi, S. Panigrahi, On Liapunov-type inequality for third-order differential equations, J. Math. Anal. Appl. 233 (2) (1999) 445–460. N. Parhi, S. Panigrahi, Liapunov-type inequality for higher order differential equations, Math. Slovaca 52 (1) (2002) 31–46. J.P. Pinasco, Lower bounds for eigenvalues of the one-dimensional p-Laplacian, Abstr. Appl. Anal. 2004 (2004) 147–153. J.P. Pinasco, Comparison of eigenvalues for the p-Laplacian with integral inequalities, Appl. Math. Comput. 182 (2006) 1399–1404. T.W. Reid, A matrix Lyapunov inequality, J. Math. Anal. Appl. 32 (1970) 424–434. A. Tiryaki, M. Ünal, D. Çakmak, Lyapunov-type inequalities for nonlinear systems, J. Math. Anal. Appl. 332 (2007) 497–511. M. Ünal, D. Çakmak, A. Tiryaki, A discrete analogue of Lyapunov-type inequalities for nonlinear systems, Comput. Math. Appl. 55 (2008) 2631–2642.
D. Çakmak, A. Tiryaki / Applied Mathematics and Computation 216 (2010) 3584–3591 [30] M. Ünal, D. Çakmak, Lyapunov-type inequalities for certain nonlinear systems on time scales, Turk. J. Math. 32 (2008) 255–275. [31] X. Yang, On Liapunov-type inequality for certain higher-order differential equations, Appl. Math. Comput. 134 (2003) 307–317. [32] X. Yang, K. Lo, Lyapunov-type inequality for a class of even-order differential equations, Appl. Math. Comput. 215 (2010) 3884–3890.
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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On Lyapunov-type inequality for quasilinear systems Devrim Çakmak a,*, Aydın Tiryaki b a b
Gazi University, Faculty of Education, Department of Mathematics Education, 06500 Teknikokullar, Ankara, Turkey Izmir University, Faculty of Arts and Sciences, Department of Mathematics and Computer Sciences, 35350 Uckuyular, Izmir, Turkey
a r t i c l e
i n f o
a b s t r a c t In this paper, we sketch some recent developments in the theory of Lyapunov-type inequalities and present some new results relating to a quasilinear system, special cases of which contain some well-known differential equations such as half-linear and linear equations. Our result generalize the Lyapunov-type inequality given in [18]. Ó 2010 Elsevier Inc. All rights reserved.
Keywords: Lyapunov-type inequality Quasilinear system Half-linear equation Linear equation
1. Introduction The Lyapunov inequality and many of its generalizations have proved to be useful tools in oscillation theory, disconjugacy, eigenvalue problems, and numerous other applications for the theories of differential and difference equations. Before we continue the description of the content of this paper, a few hints concerning the literature on the Lyapunov inequalities might be in order. Russian mathematician Lyapunov [17] proved the following remarkable result. Theorem A. If x(t) is a nontrivial solution of
x00 ðtÞ þ qðtÞxðtÞ ¼ 0;
ð1Þ
with x(a) = 0 = x(b) where a; b 2 R with a < b be consecutive zeros and x(t) – 0 for t 2 (a, b), then the so called Lyapunov inequality
Z
b
jqðsÞj ds >
a
4 ba
ð2Þ
holds. Since the appearance of Lyapunov’s fundamental paper, various proofs and generalizations or improvements have appeared in the literature. For example, Hartman [13] has generalized the classical Lyapunov inequality for the linear differential equation
ðrðtÞx0 ðtÞÞ0 þ qðtÞxðtÞ ¼ 0;
rðtÞ > 0;
ð3Þ
as follows. Theorem B. If a; b 2 R with a < b are consecutive zeros of nontrivial solution of Eq. (3), then
Z a
b
qþ ðsÞ ds > R b a
4 ðrðsÞÞ1 ds
;
* Corresponding author. E-mail addresses: [email protected] (D. Çakmak), [email protected] (A. Tiryaki). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.05.004
ð4Þ
D. Çakmak, A. Tiryaki / Applied Mathematics and Computation 216 (2010) 3584–3591
3585
where q+(t) = max{0, q(t)} is the nonnegative part of q(t). Thus, the inequality (2) is strengthened to
Z
b
qþ ðsÞ ds >
a
4 ; ba
ð5Þ
for the Eq. (1) by Theorem B. The inequality (5) is the best possible in the sense that if the constant 4 in (5) is replaced by any larger constant, then there exists an example of (1) for which (5) no longer holds (see [13, p. 345], [15]). However, stronger results were obtained in Brown and Hinton [2] and Kwong [15]. In [15], it is shown that
Z
c
qþ ðsÞ ds >
1 ; ca
qþ ðsÞ ds >
1 ; bc
a
ð6Þ
and
Z
b
c
ð7Þ 0
where c 2 (a, b) such that x (c) = 0. Hence
Z
b
qþ ðsÞ ds >
a
1 1 ba 4 þ ¼ P : c a b c ðc aÞðb cÞ ba
ð8Þ
In [2, Corollary 4.1], the authors obtained
Z b 4 ; qðsÞ ds > ba a
ð9Þ
from which (2) can be obtained. The Lyapunov inequality has been extended in many directions and its half-linear differential equation
0 rðtÞjx0 ðtÞjk2 x0 ðtÞ þ qðtÞjxðtÞjk2 xðtÞ ¼ 0;
rðtÞ > 0 and k > 1;
ð10Þ
˘ ehák’s recent book [7] as follows. extension found in Došly´ and R Theorem C. Let a; b 2 R with a < b be consecutive zeros of nontrivial solution of Eq. (10). Then
Z a
2k
b
qþ ðsÞ ds > R
ð11Þ
k1 ;
b ðrðsÞÞ1=ð1kÞ ds a
where q+(t) = max{0, q(t)} is the nonnegative part of q(t). A thorough literature review of continuous and discrete Lyapunov inequalities and their applications can be found in the survey paper [4] by Cheng and the references quoted therein. For authors who contributed the Lyapunov-type inequalities, we refer to Cheng [3], Eliason [9], Hartman [13], Kwong [15] and Reid [27]. We should also mention here that inequality (2) has been generalized to second order nonlinear differential equations by Eliason [10] and Pachpatte [20,21], to delay differential equations of the second order by Dahiya and Singh [6] and Eliason [11], to third order differential equations by Parhi and Panigrahi [23], and to certain higher order differential equations by Çakmak [5], Pachpatte [19], Panigrahi [22], Parhi and Panigrahi [24], Yang [31], and Yang and Lo [32]. Lyapunov-type inequalities can be found in Pachpatte’s paper [21] for the Emden–Fowler type equations, and were obtained for the first time by Elbert [8] for the half-linear equation, but the proof of ˘ ehák [7]. Lyapunov-type inequalities for the half-linear equation have its extension can be found in the book of Došly´ and R been rediscovered by Lee et al. [16] and Pinasco [25,26]. Although there is an extensive literature on the Lyapunov-type inequalities for various classes of differential equations, there is not much done for the linear Hamiltonian system, in the case of two scalar linear differential equations, has the form
y0 ðtÞ ¼ JHðtÞyðtÞ;
t 2 R;
ð12Þ
where
yðtÞ ¼ ðy1 ðtÞ; y2 ðtÞÞT ;
J¼
0
1
1 0
;
HðtÞ ¼
h11 ðtÞ h12 ðtÞ h21 ðtÞ h22 ðtÞ
;
with hjk(t), j, k = 1, 2 are real-valued continuous functions defined on R and h12(t) = h21(t). Setting y1(t) = x(t), y2(t) = u(t), h11(t) = b2(t), h12(t) = h21(t) = a1(t) and h22(t) = b1(t) in Eq. (12), one can easily obtains the following linear system
x0 ðtÞ ¼ a1 ðtÞxðtÞ þ b1 ðtÞuðtÞ; u0 ðtÞ ¼ b2 ðtÞxðtÞ a1 ðtÞuðtÞ;
ð13Þ
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D. Çakmak, A. Tiryaki / Applied Mathematics and Computation 216 (2010) 3584–3591
which is a special case of the linear counterpart of the nonlinear system
x0 ðtÞ ¼ a1 ðtÞxðtÞ þ b1 ðtÞjuðtÞjc2 uðtÞ;
ð14Þ
u0 ðtÞ ¼ b2 ðtÞjxðtÞjb2 xðtÞ a1 ðtÞuðtÞ;
with c = 2 and b = 2. Recently, Guseinov and Kaymakçalan [12] and Tiryaki et al. [28] have obtained the Lyapunov-type inequalities for system (13) and (14), respectively. The discrete and time scale analogues of Lyapunov-type inequalities for systems have been also found in the papers Ünal et al. [29], Jiang and Zhou [14] and Ünal and Ç akmak [30]. More recently, Bonder and Pinasco [1] and Napoli and Pinasco [18] have interested in the problem of finding lower and upper bounds for the eigenvalues of a quasilinear elliptic system. In [18], the authors have also obtained the following Lyapunov-type inequality for quasilinear system
0 ju0 ðxÞjp2 u0 ðxÞ ¼ f1 ðxÞjuðxÞja2 uðxÞjv ðxÞjb ; 0 jv 0 ðxÞjq2 v 0 ðxÞ ¼ f2 ðxÞjuðxÞja jv ðxÞjb2 v ðxÞ;
ð15Þ
where 1 < p, q < 1, the positive parameters a and b satisfy ap þ bq ¼ 1, and f1, f2 are real-valued positive continuous functions for all x 2 R, as follows. Theorem D. If system (15) has a real nontrivial solution (u(x), v(x)) such that u(a) = u(b) = 0 = v(a) = v(b) where a; b 2 R with a < b be consecutive zeros and u and v are not identically zero on [a, b], then the following inequality
2aþb 6 ðb aÞaþb1
Z
b
!a=p Z
b
f1 ðxÞ dx
a
!b=q f2 ðxÞ dx
ð16Þ
a
holds. As an application of Theorem D, the authors have improved the lower bounds on the eigenvalues problem of the following one-dimensional system
0 ju0 ðxÞjp2 u0 ðxÞ ¼ karðxÞjuðxÞja2 uðxÞjv ðxÞjb ; 0 jv 0 ðxÞjq2 v 0 ðxÞ ¼ lbrðxÞjuðxÞja jv ðxÞjb2 v ðxÞ;
ð17Þ
as follows. Theorem E. There exist a function h(k) such that l P h (k) for every generalized eigenvalue (k, l) of problem (17), where h(k) is given by
hðkÞ ¼
!q=b 1 C ; R b ka=p b rðxÞ dx a
ð18Þ
and the constant C is given by
C¼
2aþb
aa=p ðb aÞaþb1
:
ð19Þ
The principal aim of this paper is to state and prove a Lyapunov-type inequality for quasilinear systems in the form
0 r 1 ðxÞju0 ðxÞjp2 u0 ðxÞ ¼ f1 ðxÞjuðxÞja2 uðxÞjv ðxÞjb ; 0 r 2 ðxÞjv 0 ðxÞjq2 v 0 ðxÞ ¼ f2 ðxÞjuðxÞjh jv ðxÞjc2 v ðxÞ;
ð20Þ
whose special cases contain the well-known equations such as half-linear and linear equations. Our result is an extension of result by Napoli and Pinasco [18]. By using this inequality, we prove better lower bound than Napoli and Pinasco for the first eigenvalue of problem (17). Our motivation comes from the recent papers of Lee et al. [16], Napoli and Pinasco [18], Pachpatte [21], and Tiryaki et al. [28]. In this paper, we derive a Lyapunov-type inequality for quasilinear system (20), where the both components of the solution (u(x), v(x)) have consecutive zeros at the points a; b 2 R with a < b in I ¼ ½t 0 ; 1Þ R. For the special cases of system (20), we also derive some Lyapunov-type inequalities which not only relates points a and b in I at which the both components of the solution (u(x), v(x)) have consecutive zeros but also any point in (a, b) where the both components of the solution (u(x), v(x)) are maximized.
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Since our attention is restricted to the Lyapunov-type inequality for the quasilinear system of differential equations, we shall assume the existence of nontrivial solution (u(x), v(x)) of system (20) and state our basic hypothesis with respect to the same system: (i) r1, r2, f1, f2 are real-valued continuous functions such that r1(x) > 0 and r2(x) > 0 for all x 2 R. (ii) The exponents satisfy 1 < p, q < 1, and the positive parameters a, b, h and c yield ap þ bq ¼ 1 and ph þ cq ¼ 1. For the sake of convenience, we define the following integral operator
Z
Mðt; s; lÞ ¼
t
!1l 1l Z b l0 l0 ðsðxÞÞ l dx þ ðsðxÞÞ l dx ;
a
ð21Þ
t 0
where t 2 (a, b), s is a real-valued continuous function such that s(x) > 0 for all x 2 R, and l is the conjugate exponent of l 2 (1, +1), that is, l1 þ l10 ¼ 1. For a given s and l, set F(t) = M(t, s, l) for t 2 (a, b). F(t) obtains its minimum at the point t 2 (a, b) such that
Z
t
l0
ðsðxÞÞ l dx ¼
a
Z
b
l0
ðsðxÞÞ l dx
ð22Þ
t
holds. Thus, we have
FðtÞ P F min ðtÞ ¼ 2
Z
t
1l l0 ðsðxÞÞ l dx ¼ Nðt; s; lÞ:
ð23Þ
a
2. Main results The main result of this paper is the following theorem which is an extension of Theorem D. Theorem 1. Let the hypothesis (i) and (ii) hold. If system (20) has a real nontrivial solution (u(x), v(x)) such that u(a) = u(b) = 0 = v(a) = v(b) where a; b 2 R with a < b be consecutive zeros, and u and v are not identically zero on [a, b], then the following inequality
fMðc; r1 ; pÞg
h=p
b=q
fMðd; r 2 ; qÞg
Z 6 a
!h=p Z
b
f1þ ðxÞ dx
a
!b=q
b
f2þ ðxÞ dx
ð24Þ
holds, where juðcÞj ¼ max juðxÞj; jv ðdÞj ¼ max jv ðxÞj such that M(c, r1, p) and M(d, r2, q) exist, and fiþ ðxÞ ¼ maxf0; fi ðxÞg for i = 1, 2. a
a
Proof. It follows from u(a) = u(b) = 0 = v(a) = v(b) where a; b 2 R with a < b be consecutive zeros, and u and v are not identically zero on [a, b], one can choose c, d 2 (a, b) such that juðcÞj ¼ max juðxÞj > 0 and jv ðdÞj ¼ max jv ðxÞj > 0. From the existence a
a
0
0
of M(c, r1, p) and M(d, r2, q), c and d cannot be too close to a or b. By using Rolle’s theorem, clearly u (c) = 0 and v (d) = 0. Therefore, for c 2 (a, b) and u(a) = 0, we have
Z juðcÞj ¼
c
a
Z c u0 ðxÞ dx 6 ju0 ðxÞj dx:
ð25Þ
a
0
By using Hölder inequality on the integral of the right-hand side of (25) with indices p and p , we obtain
juðcÞj 6
Z
c
ju0 ðxÞj dx ¼
Z
a
c
1
1
ðr 1 ðxÞÞp ðr 1 ðxÞÞp ju0 ðxÞj dx 6
a
Z
c
10 Z c 1p p p0 ðr1 ðxÞÞ p dx r 1 ðxÞju0 ðxÞjp dx ;
a
ð26Þ
a
where 1p þ p10 ¼ 1. On the other hand, multiplying the first equation of system (20) by u(x) and integrating from a to c and tak0 ing into account that u(a) = 0 and u (c) = 0, we get
Z
c
r 1 ðxÞju0 ðxÞjp dx ¼
a
Z
c
f1 ðxÞjuðxÞja jv ðxÞjb dx:
ð27Þ
a
Therefore, by using (27) in (26), we have
juðcÞj 6
Z
c a
10 Z p p0 ðr 1 ðxÞÞ p dx a
c
1p Z c 10 Z c 1p p p0 f1 ðxÞjuðxÞja jv ðxÞjb dx 6 ðr 1 ðxÞÞ p dx f1þ ðxÞjuðxÞja jv ðxÞjb dx : a
ð28Þ
a
If we take the pth power of both sides of the inequality (28), and ju(x)j is maximum at the point c, and jv(x)j is maximum at the point d, we obtain
3588
D. Çakmak, A. Tiryaki / Applied Mathematics and Computation 216 (2010) 3584–3591
1 6 juðcÞjap jv ðdÞjb
Z
c
p1 Z p0 ðr 1 ðxÞÞ p dx
a
f1þ ðxÞ dx :
c
a
ð29Þ
Now, since u(b) = 0, we get
Z Z b b juðcÞj ¼ j uðcÞj ¼ u0 ðxÞ dx 6 ju0 ðxÞj dx; c c
ð30Þ
and repeating the above procedure step by step, one can easily obtain
1 6 juðcÞj
ap
Z
b
jv ðdÞj
b
0
pp
ðr 1 ðxÞÞ
dx
!p1 Z
c
!
b
f1þ ðxÞ dx
c
ð31Þ
:
Thus we obtain from (29) and (31)
Z
b
a
f1þ ðxÞ dx P juðcÞjpa jv ðdÞjb Mðc; r 1 ; pÞ;
ð32Þ
where 1p þ p10 ¼ 1. By using similar manner, we get
Z
b
a
f2þ ðxÞ dx P juðcÞjh jv ðdÞjqc Mðd; r 2 ; qÞ;
ð33Þ
where 1q þ q10 ¼ 1. Raising inequality (32) to a power e1, inequality (33) to a power e2 and multiplying the resulting equations, we obtain !e !e
juðcÞjðpaÞe1 he2 jv ðdÞjbe1 þðqcÞe2 fMðc; r 1 ; pÞge1 fMðd; r2 ; qÞge2 6
Z
1
b
f1þ ðxÞ dx
a
Z
a
2
b
f2þ ðxÞ dx
:
ð34Þ
Now, we choose e1 and e2 such that ju(c)j and jv(d)j cancels out, i.e. solves the homogeneous linear system
ðp aÞe1 he2 ¼ 0;
ð35Þ
be1 þ ðq cÞe2 ¼ 0:
We observe that by hypothesis (ii), this system admits a nontrivial solution, indeed both equations are equivalent to
e1
b h ¼ e2 : q p
ð36Þ
Hence, we may take e1 ¼ ph ; e2 ¼ bq and we get inequality (24) which completes the proof. h By using (22) and (23), we have the following result from Theorem 1. Corollary 1. Let all assumptions of Theorem 1 satisfy. Then the following inequality h=p
fNðc; r 1 ; pÞg
b=q
fNðd; r2 ; qÞg
Z
!h=p Z
b
6 a
f1þ ðxÞ dx
a
!b=q
b
f2þ ðxÞ dx
ð37Þ
holds, where c; d; Nðc; r 1 ; pÞ; Nðd; r 2 ; qÞ; f1þ and f2þ are defined as before such that
Z
c
Z
1
ðr 1 ðxÞÞ1p dx ¼
a
and
Z
b
1
ðr1 ðxÞÞ1p dx;
ð38Þ
c d
1
ðr 2 ðxÞÞ1q dx ¼
a
Z
b
1
ðr 2 ðxÞÞ1q dx:
ð39Þ
d
Remark 1. Since r1(x) > 0 and r2(x) > 0 for all x 2 R, by using (38) and (39), it is easy to see that c and d cannot be too close to a or b. p0 1 Rc Since the function h(x) = x1 p is convex for x > 0, Jensen’s inequality h yþz 6 2 ½hðyÞ þ hðzÞ with y ¼ a ðr1 ðxÞÞ p dx and 2 p0 Rb z ¼ c ðr 1 ðxÞÞ p dx implies ! !
Mðc; r 1 ; pÞ ¼
Z
c
1p Z b p0 p0 ðr 1 ðxÞÞ p dx þ ðr 1 ðxÞÞ p dx
a
c
Similarly, we obtain
Mðd; r 2 ; qÞ P 2
q
Z
b
ðr2 ðxÞÞ
q0
q
1p
P 2p
Z
b
p0
ðr 1 ðxÞÞ p dx
1p
:
ð40Þ
a
!1q dx
:
a
Thus, by using the inequalities (40) and (41), we have the following result from Theorem 1.
ð41Þ
D. Çakmak, A. Tiryaki / Applied Mathematics and Computation 216 (2010) 3584–3591
3589
Corollary 2. Let all assumptions of Theorem 1 satisfy. Then the following inequality
2hþb 6
Z
b
!hðp1Þ=p Z
1
b
ðr 1 ðxÞÞ1p dx
a
1
!bðq1Þ=q Z
a
!h=p Z
b
f1þ ðxÞ dx
ðr 2 ðxÞÞ1q dx
a
a
!b=q
b
f2þ ðxÞ dx
ð42Þ
holds, where f1þ and f2þ are defined as before. Remark 2. It is easy to see that Corollaries 1 and 2 are equivalent for r1(x) = r2(x) = 1 in system (20). Remark 3. When r1(x) = r2(x) = 1 and h = a (or c = b) and fi(x) > 0 for i = 1, 2, Corollaries 1 or 2 reduces to Theorem D by Napoli and Pinasco [18]. Remark 4. It is easy to see that the following inequalities
f þ ðxÞ 6 jf ðxÞj; Z Z b f þ ðxÞ dx 6 a
ð43Þ b
jf ðxÞj dx
a
ð44Þ
Rb
hold. Hence, the integrals of a fiþ ðxÞ dx for i = 1, 2 in the above results may be replaced by Thus, we shall arrive to the following inequality
fMðc; r1 ; pÞg
h=p
b=q
fMðd; r 2 ; qÞg
Z 6 a
b
!h=p Z jf1 ðxÞj dx
Rb a
jfi ðxÞj dx for i = 1, 2, respectively.
!b=q
b
jf2 ðxÞj dx
;
ð45Þ
a
which is immediate consequence of the inequality (24) in Theorem 1. Remark 5. Let us observe from above results that for h = 0 (or b = 0), we obtain the result for the case of a single equation of system (20). In this case, the inequalities (2), (4), (5) and (11) are relaxed to the inequality (24) by Theorem 1. In addition to this, Corollary 2 reduces to Theorem C for half-linear differential equation (10), and to Theorem B for linear differential equation (3) if we take p = 2 (or q = 2). When r1(x) = 1 and p = 2 (or r2(x) = 1 and q = 2), Corollary 2 reduces to Theorem A for linear differential equation (1). Moreover, one can easily obtain from the proof of Theorem 1 that inequalities (29) and (31) in the specialZcase reduce to c
a
and
Z
f1þ ðxÞ dx P
1 ; ca
ð46Þ
f1þ ðxÞ dx P
1 ; bc
ð47Þ
1 1 ba 4 þ ¼ P : c a b c ðc aÞðb cÞ ba
ð48Þ
b
c
respectively. Hence
Z
a
b
f1þ ðxÞ dx P
Thus, the inequalities (6)–(8) are strengthened to the inequalities (46)–(48) by Theorem 1, respectively. Now, we present an application of the obtained Lyapunov-type inequality in Theorem 1. Let (k, l) be generalized eigenvalues of the system (20) with r1(x) = 1 = r2(x), and h = a (or c = b). The function r(x) also be positive for all x 2 R. Therefore, system (20) with f1(x) = kar(x) > 0 and f2(x) = lbr(x) > 0 for all x 2 R reduces to system (17). By using techniques similar to Napoli and Pinasco [18], we obtain the following result which gives better lower bounds for the first eigencurve in the generalized spectra. The proof of this result is rather elementary, based on above generalization of the Lyapunov-type inequality, exactly as in that of Theorem 1.4 of Napoli and Pinasco [18]. Theorem 2. There exists a function h1(k) such that l P h1(k) for every generalized eigenvalue (k, l) of problem (17), where h1(k) is given by
1 D h1 ðkÞ ¼ R b ka=p b rðxÞ dx a
!q=b
and the constant D is given by
n
D¼
ðc aÞ1p þ ðb cÞ1p
ð49Þ
;
oa=p n ob=q ðd aÞ1q þ ðb dÞ1q
where c and d are defined as before.
aa=p
;
ð50Þ
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Proof. Let (k, l) be a generalized eigenpair, and (u(x), v(x)) be the corresponding nontrivial solution of system (17). For all x 2 R, by substituting f1(x) = kar(x) > 0 and f2(x) = lbr(x) > 0 in the Lyapunov inequality (24), we obtain
n oa=p n ob=q ðc aÞ1p þ ðb cÞ1p ðd aÞ1q þ ðb dÞ1q 6
Z
b
!a=p Z karðxÞ dx
a
!b=q
b
lbrðxÞ dx
:
ð51Þ
a
By rearranging terms in (51), and using the condition ap þ bq ¼ 1, we obtain
Z n oa=p n ob=q ðc aÞ1p þ ðb cÞ1p ðd aÞ1q þ ðb dÞ1q 6 ðkaÞa=p ðlbÞb=q
b
! rðxÞ dx ;
ð52Þ
a
which gives
!q=b
D
ka=p
Rb a
rðxÞ dx
6 lb;
and the proof is completed.
ð53Þ
h
Remark 6. Since h1 is a continuous function, and h1(k) ? +1 as k ? 0+. Therefore, there exists a ball centered in the origin such that the generalized spectrum is contained in its exterior. Also, by rearranging terms in (53), we obtain
lb=q ka=p P
bb=q
D : Rb rðxÞ dx a
ð54Þ
It is clear that when the interval collapses, right-hand side of (54) goes to infinity. Thus, we obtain the desired generalizations of Napoli and Pinasco [18]’s result for one-dimensional nonlinear systems. Remark 7. If we compare Theorem 2 with Theorem E, we obtain D P C since (40) and (41) hold. Thus, the inequality h1(k) P h(k) holds. Therefore, Theorem 2 presents better lower bound than Theorem E. Acknowledgement The authors would like to thank the referee for his valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
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