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A new Picone’s dynamic inequality on time scales with applications
Accepted Manuscript A new Picone’s dynamic inequality on time scales with applications S.H. Saker, R.R. Mahmoud, A. Peterson PII: DOI: Reference:
S0893-9659(15)00117-2 http://dx.doi.org/10.1016/j.aml.2015.03.015 AML 4760
To appear in:
Applied Mathematics Letters
Received date: 16 February 2015 Revised date: 25 March 2015 Accepted date: 26 March 2015 Please cite this article as: S.H. Saker, R.R. Mahmoud, A. Peterson, A new Picone’s dynamic inequality on time scales with applications, Appl. Math. Lett. (2015), http://dx.doi.org/10.1016/j.aml.2015.03.015 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A NEW PICONE’S DYNAMIC INEQUALITY ON TIME SCALES WITH APPLICATIONS S. H. SAKER1 , R. R. MAHMOUD2 AND A. PETERSON3
Abstract. In this paper, we will derive a new dynamic Picone-type inequality for half-linear dynamic equations and dynamic inequalities of second order on an arbitrary time scale T. As a consequence, we will apply this new Picone inequality to get a new Wirtinger-type inequality on time scales with two different weighted functions. The results contain the Wirtinger inequalities formulated by Beesack, Lee and Jaroˇs for the continuous case. For the discrete case our results are also new. 2010 Mathematics Subject Classification: 26D10, 26D15, 34A40, 34N05. Key words and phrases. Wirtinger’s inequality, Picone’s inequality, Chain rule, Time scales.
1. Introduction In 1910 Picone [12] derived the following identity i 0 u2 0 0 d hu 0 0 0 (1.1) vp1 u − up2 v = u p1 u − p2 v dx v v 0 2 0 u 0 2 + (p1 − p2 ) u + p2 u − v , v 0
0
where u, v, p1 u , p2 v are differentiable functions with respect to x and v(x) 6= 0 for x ∈ I ⊂ R. This identity was discovered by Picone during his attempt to introduce a generalization of the famous Sturm Comparison Theorem for the formally self-adjoint second order linear differential equations 0 0 0 0 (1.2) p1 (x)u (x) + q1 (x)u(x) = 0, p2 (x)v (x) + q2 (x)v(x) = 0.
The classical Picone identity was a very useful tool in the development of the classical Sturmian theory. Much work has been done and many papers which deal with various generalizations and extensions, even for second order half-linear differential equations with p−Laplacian, have appeared in the literature, we refer the reader to the book [8] and the papers [9, 14]. These various generalizations have also been extended to difference equations (see [5, 13] and the references cited therein). One of the main applications of identity (1.1) is the investigation of Wirtinger type inequalities which may be used and applied in obtaining lower bounds of the eigenvalues of the eigenvalue problems of second order ordinary differential equations (see [2, 7, 11]). In the following, we recall some of the related work that motivates and explains the aim of our paper. As a connection between Wirtinger type inequalities and 1
S. H. SAKER1 , R. R. MAHMOUD2 AND A. PETERSON3
2
solutions of differential equations, Beesack [3] considered a solution y1 (x) of the following second-order linear differential equation 00
y (x) + p(x)y(x) = 0 0
for x ∈ [0, a],
with y1 (0) = 0, y1 (0) ≥ 0, y1 (x) > 0, where p(x) is assumed to be continuous on 0 [0, a] and proved that if f ∈ L2 with f (0) = 0, then Z a Z a 2 0 (1.3) f (x) dx ≥ p(x) (f (x))2 dx. 0
0
Moreover, equality holds if and only if f (x) = Ay1 (x). Lee [11] considered a more general differential equation to establish a generalized Wirtinger-type inequality. In particular, he considered the following differential operator 0 0 α−1 0 + q |v|α−1 v, v Lv := p v for x in the interval I = [a, b] ⊆ R and v is a positive solution for the following differential inequality −Lv ≥ λ0 rv, x ∈ I, to obtain the validity of the following Wirtinger-type inequality Z Z 0 2 0 α−1 α−1 2 (1.4) (q |v| + λ0 r)u dx ≤ p u v dx + S1 (u, v) − S2 (u, v), I
I
for every function u ∈ AC(I), i.e., the set of all real-valued functions which are absolutely continuous on every closed subinterval of I, where 0 α−1 0 α−1 p(x)u2 (x) v (x) p(x)u2 (x) v (x) , S2 (u, v) := lim . S1 (u, v) := lim v(x) v(x) x→b− x→a+
Recently in 2011 Jaroˇs [10] extended his results in [9] and made use of the close connection between Wirtinger-type inequalities with Euler-Lagrange differential equations associated with variational problems to prove some new Wirtinger-type inequalities. In particular, Jaroˇs considered the following differential inequality 0 0 α−1 0 (1.5) p v v + q |v|α−1 v + λr |v|β−1 v ≤ 0, in I =[a, b],
and proved that Z Z 0 β+1 0 α−β α−β β+1 (1.6) (q |v| + λr) |u| dx ≤ p u dx + Sa (u, v) − Sb (u, v), v where
I
I
Sa (u, v) := limx→a+ Sb (u, v) := limx→b−
0 α−1 0 p(x)|u|β+1 (x) v (x) v (x)
|v|β−1 0 v α−1 0 p(x)|u|β+1 (x) v (x) v (x) |v|
β−1
v
,
,
holds for any positive solution v of (1.5) and u ∈ AC(I). To the best of the authors knowledge nothing is known regarding the discrete analogue of the inequality (1.6) for the difference inequalities. The natural question arises now: Is it possible to prove a new dynamic Piconetype inequality on an arbitrary time scale T, which contains the continuous and the discrete inequalities as special cases?
PICONE-TYPE INEQUALITY
3
The main aim of this paper, in Section 2, is to give an affirmative answer to this question. The main result as a special case on time scales contains the results formulated by Agarwal et al. [1]. On the other hand as special cases when T = R the results contain the Wirtinger-type inequality (1.6) due to Jaroˇs and the results due to Beesack [3] and Lee [11]. When T = N the results are essentially new for the corresponding difference inequalities. 2. Main Results In this section, before we prove the main results, we present some basic definitions and inequalities concerning the delta calculus on time scales that we will use in this article. A time scale T is an arbitrary nonempty closed subset of the real numbers. For t ∈ T, we define the forward jump operator σ : T → T by σ(t) := inf{s ∈ T : s > t}. The mapping ν : T → R+ 0 such that ν(t) = t − ρ(t) is called the backward graininess. If σ(t) > t, we say that t is right-scattered, while if ρ(t) < t, we say that t is left-scattered. Also, if t < sup T and σ(t) = t, then t is called right-dense, and if t > inf T and ρ(t) = t, then t is called left-dense. A function f : [a, b)T → R is said to be right–dense continuous (rd−continuous) if it is right continuous at each right–dense point and there exists a finite left limit at all left–dense points. The space of rd−continuous functions is denoted (n) by Crd (T, R). We also denote by Crd (T) the space of all functions f ∈ Crd (T) such that f ∆i ∈ Crd (T) for i = 0, 1, 2, ..., n for n ∈ N. The product and quotient rules for the derivative of the product f g and the quotient f /g (where gg σ 6= 0, where g σ is the composition function g ◦ σ) of two differentiable functions f and g are given by ∆ f ∆g − f g∆ f ∆ ∆ σ ∆ ∆ ∆ σ = . (2.1) (f g) = f g + f g = f g + f g , g gg σ The time scales chain rule that we will use in this paper is Z 1 [hxσ + (1 − h)x]γ−1 dhx∆ (t), γ > 0, (2.2) (xγ (t))∆ = γ 0
which is a simple consequence of Keller’s chain rule [4, Theorem 1.90]. Now, we are ready to state and prove the main results. We begin by deducing a Picone-type inequality which is also an interesting result in its own right. Consider the second order half-linear dynamic operator L defined on the time scale interval I = [a, b]T := [a, b] ∩ T with a < σ (a) and ρ (b) < b, as h i ∆ (2.3) Lv(t) = pϕα (v ∆ ) + qϕα (v σ ) (t), for t ∈ I, 1 (I, R) with p > 0 on I. The where ϕα (v) = |v|α−1 v, q ∈ Crd (I, R) and p ∈ Crd domain D of L is defined to be the set of all real-valued functions v on I such that all delta-derivatives in (2.3) exist and are rd-continuous at each point in I. We will consider here solutions v ∈ D of the dynamic inequality
(2.4)
[Lv + λrϕβ (v σ )] (t) ≤ 0, for t ∈ I,
where β > 0, λ is a real number and r is a positive rd-continuous function in I. We denote by ACloc (I) the space of all real-valued absolutely rd-continuous functions on every closed subinterval of I.
4
S. H. SAKER1 , R. R. MAHMOUD2 AND A. PETERSON3
Theorem 2.1 (Picone-type inequality). Let v ∈ D be a positive solution of (2.4) on I. If u ∈ ACloc (I), then !∆ |u|β+1 pϕα (v ∆ ) (2.5) ϕβ (v) ( ∆ β+1 ) σ v∆ uv ∆ α−β ∆ β+1 u ∆ u − (β + 1) u ϕβ ( σ ) + β σ ≤ −p v v v β+1 ∆ α−β |uσ |β+1 v + +p u∆ [Lv − qϕα (v σ )] . ϕβ (v σ )
Proof. Without loss of generality, we may assume that v ∆ (t) and u∆ (t) > 0. For convenience we omit the argument t sometimes in the computations. From (2.1) we get that !∆ !∆ |u|β+1 |uσ |β+1 |u|β+1 ∆ ∆ ∆ pϕα (v ) = pϕα (v ∆ ) + pϕ (v ) . (2.6) α ϕβ (v) ϕβ (v) ϕβ (v σ )
Clearly, the second term on the right-hand side is equal to |uσ |β+1 [Lv − qϕα (v σ )] . ϕβ (v σ )
(2.7)
Computing the delta-derivative of the quotient |u|β+1 /ϕβ (v) in the first term on the right-hand side of (2.6), we obtain that ∆ ! β+1 β+1 ∆ |u| ϕβ (v) − |u|β+1 (ϕβ (v))∆ |u| (2.8) = . ϕβ (v) ϕβ (v)ϕβ (v σ ) By (2.2), we get that Z 1 ∆ (2.9) |u|β+1 = (β + 1) [h |uσ | + (1 − h) |u|]β dhu∆ 0 Z 1 ≤ (β + 1) [h |uσ | + (1 − h) |uσ |]β dhu∆ 0
≤ (β + 1) |uσ |β u∆ ,
since v ∆ (t) > 0, we can similarly get that ∆ (2.10) (ϕβ (v))∆ = |v|β ≥ β |v|β−1 v ∆ .
Substituting (2.9) and (2.10) into (2.8), we have that !∆ (β + 1) |uσ |β u∆ ϕβ (v) − |u|β+1 β |v|β−1 v ∆ |u|β+1 ≤ ϕβ (v) ϕβ (v)ϕβ (v σ ) = (2.11)
(β + 1) |uσ |β u∆ ϕβ (v) β |u|β+1 |v|β−1 v ∆ − ϕβ (v σ ) ϕβ (v)ϕβ (v σ ) ∆ ∆ α−β
≤ (β + 1) u
v
∆ α−β uv ∆ β+1 uσ v ∆ ϕβ ( σ ) − β v . vσ v
PICONE-TYPE INEQUALITY
5
Substituting (2.7) and (2.11) into (2.6), adding and subtracting the term β+1 ∆ α−β v p u∆ , we get that !∆ |u|β+1 ∆ pϕα (v ) ϕβ (v) ( ∆ β+1 ) σ v∆ ∆ α−β ∆ β+1 uv u ∆ u − (β + 1) u ϕβ ( σ ) + β σ ≤ −p v v v ∆ β+1 ∆ α−β |uσ |β+1 v [Lv − qϕα (v σ )] . +p u + ϕβ (v σ )
which is the required inequality (2.5). This completes the proof.
Remark 2.1. As a special case when T = R, we get the following inequality !0 |u|β+1 pϕα (v 0 ) ϕβ (v) 0 β+1 0 uv 0 α−β 0 β+1 uv − (β + 1) u0 ϕβ ( ≤ −p v )+β u v v 0 β+1 0 α−β |u|β+1 [Lv − qϕα (v)] , +p u + v ϕβ (v)
due to Jaroˇs [10] where v is a solution of (1.5) and u is any absolutely continuous function. As a special case when T = Z, we get the following discrete Picone-type inequality which is essentially new. Corollary 2.1. Assume that vk is a solution of the following difference inequality ∆ (pk ϕα (∆vk )) + qk ϕα (vk+1 ) + λrk ϕβ (vk+1 ) ≤ 0,
where k ∈ [m, n + 1] ≡ {m, m + 1, . . . , n, n + 1} is a discrete interval with m, n ∈ Z, pk , qk and rk are real-valued sequences with pk 6= 0. Then ! |uk |β+1 pk ϕα (∆vk ) ∆ ϕβ (vk ) ( ) uk ∆vk β+1 uk+1 ∆vk α−β β+1 ≤ −p |∆vk | |∆uk | − (β + 1) ∆uk ϕβ + β vk+1 vk+1 + p |∆uk |β+1 |∆vk |α−β +
|uk+1 |β+1 [∆ (pk ϕα (∆vk )) + qk ϕα (vk+1 ) − qϕα (vk+1 )] , ϕβ (vk+1 )
for every real-valued sequence uk 6= 0 for k ∈ [m, n + 1]. In the following, we shall apply the Picone dynamic inequality to obtain new generalized dynamic Wirtinger-type inequalities on time scales. For a positive
S. H. SAKER1 , R. R. MAHMOUD2 AND A. PETERSON3
6
solution v for (2.4), we shall consider functions u ∈ ACloc (I) such that the limits p|u|β+1 ϕ (v∆ ) α (a), if ϕβ (v(a)) > 0, ϕβ (v) β+1 p|u| ϕα (v ∆ ) (σ(a)), if ϕβ (v(a)) = 0, (2.12) Sh (u, v) := ϕβ (v) β+1 ∆ (x)) p(x)|u(x)| ϕ (v α lim , if a = σ(a), + x→a x∈T
(2.13)
ϕβ (v(x))
p|u|β+1 ϕ (v∆ ) α (b), if ϕβ (v(b)) > 0, ϕβ (v) β+1 p|u| ϕα (v ∆ ) (ρ(b)), if ϕβ (v(b)) = 0, Sk (u, v) := ϕβ (v) β+1 ∆ (x)) p(x)|u(x)| ϕ (v α limx→b− , if b = ρ(b), ϕβ (v(x)) x∈T
exist and are finite.
Theorem 2.2. Let v ∈ D be a positive solution of (2.4) in I. If u ∈ ACloc (I) such that the limits in (2.12) and (2.13) exist and are finite, then Z (q |v σ |α−β + λr) |uσ |β+1 ∆x I Z β+1 ∆ α−β v ≤ p u∆ (2.14) ∆x + Sh (u, v) − Sk (u, v), I
holds if the integrals exist.
Proof. From the Picone-type inequality (2.5), we have that !∆ |u|β+1 ∆ pϕα (v ) ϕβ (v) ( ∆ β+1 ) σ v∆ ∆ α−β ∆ β+1 uv u u − (β + 1) u∆ ϕβ ( σ ) + β σ ≤ −p v v v β+1 ∆ α−β |uσ |β+1 v [Lv − qϕα (v σ )] . +p u∆ + ϕβ (v σ )
Integrating the above inequality over a subinterval [h, k]T of I with σ (a) ≤ h if ϕβ (v(a)) = 0 and k ≤ ρ(b) if ϕβ (v(b)) = 0, we obtain that " #k Z i kh β+1 ∆ α−β σ α−β |u|β+1 ∆ v pϕα (v ) ≤ p u∆ + λr |uσ |β+1 ∆x. − q |v | ϕβ (v) h h
Finally, then by taking limits as h → a and k → b (notice that since a is leftscattered and b is right-scattered, we need only to consider the limit as h → a from the right and the limit as k → b from the left, respectively), when possible, we get the required inequality (2.14). This completes the proof. We have the following special cases. Remark 2.2. If T = R in Theorem 2.2, then v σ (x) = v(x), uσ (x) = u(x) and the result (2.14) reduces to (1.6) due to Jaroˇs . Remark 2.3. If T = R and β = 1 in Theorem 2.2, we get the continuous result (1.4) due to Lee et al. [11].
PICONE-TYPE INEQUALITY
7
Remark 2.4. If T = Z and β = 1 in Theorem 2.2, we get the following discrete result. Corollary 2.2. Assume that vk is a positive solution of the following difference inequality − ∆ pk (∆vk )α−1 ∆vk + qk (vk+1 )α−1 vk+1 ≥ λ1 rk vk+1 , k ∈ [m, n + 1].
Then the following discrete Wirtinger-type inequality holds for every sequence uk of real numbers X X (qk |vk |α−1 + λ1 rk )u2k ≤ pk (∆uk )2 |∆vk |α−1 + S3 (uk , vk ) − S4 (uk , vk ), k
k
where
2 α−1 pk uk |∆vk | (a), 2 vk α−1 S3 (uk , vk ) := p u |∆v | k k k (a + 1), vk 2 α−1 pk uk |∆vk | (b), vk 2 S4 (uk , vk ) := α−1 pk uk |∆vk | (b − 1), vk
if v(a) > 0, if v(a) = 0, if v(b) > 0, if v(b) = 0.
Remark 2.5. Assume α = β = 1 in Theorem 2.2. Then the result (2.14) reduces to the following consequence of Agarwal et al. [1]. Corollary 2.3. Assume that v is a positive solution for the differential inequality ∆ − pv ∆ − qv σ ≥ λ0 rv σ , on I,
u ∈ AC(I). Then Z b Z b 2 σ 2 (q + λ0 r) (u ) ∆x ≤ p u∆ ∆x + S5 (u, v) − S6 (u, v), a
where
a
2 ∆ pu v (a), if v(a) > 0, 2v ∆ pu v (σ(a)), if v(a) = 0, S5 (u, v) := v 2 ∆ (x)v (x) limx→a+ p(x)u v(x) , if a = σ(a), x∈T 2 ∆ pu v (b), if v(b) > 0, 2v ∆ pu v (ρ(b)), if v(b) = 0, S6 (u, v) := v p(x)u2 (x)v ∆ (x) lim , if b = ρ(b). − x→b x∈T
v(x)
Remark 2.6. If α = β = λ = 1, q(x) = 0 and Sh (u, v) = Sk (u, v) = 0, then Theorem 2.2 reduces to the following inequality Z b Z b 2 2 σ (2.15) r(x) (u (x)) ∆x ≤ u∆ (x) ∆x, a
a
which can be considered as the time scale version of Beesack’s result (1.3).
S. H. SAKER1 , R. R. MAHMOUD2 AND A. PETERSON3
8
Remark 2.7. As a special case of (2.15) if we set a = 0, y(0) = 0 and r(x) = 1/4x2 , we get that Z b Z b σ 2 (y (x))2 ∆ y (x) ∆x < ∆x, 4x2 0 0 which can be considered as the time scale version of the inequality Z b 2 Z b 2 y (x) 0 dx, dx < y (x) 4x2 0 0 due to Hardy [6, Theorem 253]. References [1] R. P. Agarwal, V. Otero-Espinar, K. Perera, and D. R.Vivero, Wirtinger’s inequalities on time scales. Can. Math. Bull. 51 (2) (2008), 161-171. [2] R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications. Second edition. Monographs and Textbooks in Pure and Applied Mathematics 228. Marcel Dekker, New York, (2000). [3] P. R. Beesack, Integral inequalities of the Wirtinger type, Duke Math. J. 25 (1958), 477-498. [4] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkh¨ auser, Boston, Mass, USA, (2001). [5] O. Doˇsl´ y , J. R. Graef, and J. Jaroˇs, Forced oscillation of second order linear and half-linear difference equations, Proc. Amer. Math. Soc. 131 (9) (2003), 2859-2867. [6] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, 2nd edn. Cambridge University Press, Cambridge, (1952) [7] D. B. Hinton and R. T. Lewis, Discrete spectra criteria for singular differential operators with middle terms, Math. Proc. Cambridge Philos. Soc. 77 (1975), 337–347. [8] K. Kreith, Oscillation theory, Springer (1973). [9] J. Jaroˇs and T. Kusano, A Picone type identity for second order half-linear differential equations, Acta Math. Univ. Comenianae 68 (1999), 137-151. [10] J. Jaroˇs, On an integral inequality of the Wirtinger type. Appl. Math. Lett. 24 (8) (2011), 1389-1392. [11] C. F. Lee, C. C. Yeh, C. H. Hong and R. P. Agarwal, Lyapunov and Wirtinger inequalities. Appl. Math. lett. 17 (7) (2004), 847-853. [12] M. Picone, Sui valori eccezionali di un parametro da cui dipende un’equazione differenziale lineare ordinaria del secondo ordine, Ann. Scuola Norm. Pisa 11 (1910), 1-141. [13] P. Rehak, Oscillatory properties of second order half-linear difference equations, Czechoslovak Math. J. 51 (2) (2001), 303-321. [14] A. Tiryaki, Sturm-Picone type theorems for second-order nonlinear differential equations, Elec. J. Diff. Eqns. 2014 (146) (2014), 1-11. 1 Department of Mathematics, Faculty of Science,, Mansoura University, MansouraEgypt, E-mail: [email protected], 2
Department of Mathematics, Faculty of Science,, Fayoum University, FayoumEgypt, E-mail: [email protected], 3 Department of Mathematics,, University of Nebraska–Lincoln,, Lincoln, NE 68588-0130, U.S.A., E-mail: [email protected].
S0893-9659(15)00117-2 http://dx.doi.org/10.1016/j.aml.2015.03.015 AML 4760
To appear in:
Applied Mathematics Letters
Received date: 16 February 2015 Revised date: 25 March 2015 Accepted date: 26 March 2015 Please cite this article as: S.H. Saker, R.R. Mahmoud, A. Peterson, A new Picone’s dynamic inequality on time scales with applications, Appl. Math. Lett. (2015), http://dx.doi.org/10.1016/j.aml.2015.03.015 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A NEW PICONE’S DYNAMIC INEQUALITY ON TIME SCALES WITH APPLICATIONS S. H. SAKER1 , R. R. MAHMOUD2 AND A. PETERSON3
Abstract. In this paper, we will derive a new dynamic Picone-type inequality for half-linear dynamic equations and dynamic inequalities of second order on an arbitrary time scale T. As a consequence, we will apply this new Picone inequality to get a new Wirtinger-type inequality on time scales with two different weighted functions. The results contain the Wirtinger inequalities formulated by Beesack, Lee and Jaroˇs for the continuous case. For the discrete case our results are also new. 2010 Mathematics Subject Classification: 26D10, 26D15, 34A40, 34N05. Key words and phrases. Wirtinger’s inequality, Picone’s inequality, Chain rule, Time scales.
1. Introduction In 1910 Picone [12] derived the following identity i 0 u2 0 0 d hu 0 0 0 (1.1) vp1 u − up2 v = u p1 u − p2 v dx v v 0 2 0 u 0 2 + (p1 − p2 ) u + p2 u − v , v 0
0
where u, v, p1 u , p2 v are differentiable functions with respect to x and v(x) 6= 0 for x ∈ I ⊂ R. This identity was discovered by Picone during his attempt to introduce a generalization of the famous Sturm Comparison Theorem for the formally self-adjoint second order linear differential equations 0 0 0 0 (1.2) p1 (x)u (x) + q1 (x)u(x) = 0, p2 (x)v (x) + q2 (x)v(x) = 0.
The classical Picone identity was a very useful tool in the development of the classical Sturmian theory. Much work has been done and many papers which deal with various generalizations and extensions, even for second order half-linear differential equations with p−Laplacian, have appeared in the literature, we refer the reader to the book [8] and the papers [9, 14]. These various generalizations have also been extended to difference equations (see [5, 13] and the references cited therein). One of the main applications of identity (1.1) is the investigation of Wirtinger type inequalities which may be used and applied in obtaining lower bounds of the eigenvalues of the eigenvalue problems of second order ordinary differential equations (see [2, 7, 11]). In the following, we recall some of the related work that motivates and explains the aim of our paper. As a connection between Wirtinger type inequalities and 1
S. H. SAKER1 , R. R. MAHMOUD2 AND A. PETERSON3
2
solutions of differential equations, Beesack [3] considered a solution y1 (x) of the following second-order linear differential equation 00
y (x) + p(x)y(x) = 0 0
for x ∈ [0, a],
with y1 (0) = 0, y1 (0) ≥ 0, y1 (x) > 0, where p(x) is assumed to be continuous on 0 [0, a] and proved that if f ∈ L2 with f (0) = 0, then Z a Z a 2 0 (1.3) f (x) dx ≥ p(x) (f (x))2 dx. 0
0
Moreover, equality holds if and only if f (x) = Ay1 (x). Lee [11] considered a more general differential equation to establish a generalized Wirtinger-type inequality. In particular, he considered the following differential operator 0 0 α−1 0 + q |v|α−1 v, v Lv := p v for x in the interval I = [a, b] ⊆ R and v is a positive solution for the following differential inequality −Lv ≥ λ0 rv, x ∈ I, to obtain the validity of the following Wirtinger-type inequality Z Z 0 2 0 α−1 α−1 2 (1.4) (q |v| + λ0 r)u dx ≤ p u v dx + S1 (u, v) − S2 (u, v), I
I
for every function u ∈ AC(I), i.e., the set of all real-valued functions which are absolutely continuous on every closed subinterval of I, where 0 α−1 0 α−1 p(x)u2 (x) v (x) p(x)u2 (x) v (x) , S2 (u, v) := lim . S1 (u, v) := lim v(x) v(x) x→b− x→a+
Recently in 2011 Jaroˇs [10] extended his results in [9] and made use of the close connection between Wirtinger-type inequalities with Euler-Lagrange differential equations associated with variational problems to prove some new Wirtinger-type inequalities. In particular, Jaroˇs considered the following differential inequality 0 0 α−1 0 (1.5) p v v + q |v|α−1 v + λr |v|β−1 v ≤ 0, in I =[a, b],
and proved that Z Z 0 β+1 0 α−β α−β β+1 (1.6) (q |v| + λr) |u| dx ≤ p u dx + Sa (u, v) − Sb (u, v), v where
I
I
Sa (u, v) := limx→a+ Sb (u, v) := limx→b−
0 α−1 0 p(x)|u|β+1 (x) v (x) v (x)
|v|β−1 0 v α−1 0 p(x)|u|β+1 (x) v (x) v (x) |v|
β−1
v
,
,
holds for any positive solution v of (1.5) and u ∈ AC(I). To the best of the authors knowledge nothing is known regarding the discrete analogue of the inequality (1.6) for the difference inequalities. The natural question arises now: Is it possible to prove a new dynamic Piconetype inequality on an arbitrary time scale T, which contains the continuous and the discrete inequalities as special cases?
PICONE-TYPE INEQUALITY
3
The main aim of this paper, in Section 2, is to give an affirmative answer to this question. The main result as a special case on time scales contains the results formulated by Agarwal et al. [1]. On the other hand as special cases when T = R the results contain the Wirtinger-type inequality (1.6) due to Jaroˇs and the results due to Beesack [3] and Lee [11]. When T = N the results are essentially new for the corresponding difference inequalities. 2. Main Results In this section, before we prove the main results, we present some basic definitions and inequalities concerning the delta calculus on time scales that we will use in this article. A time scale T is an arbitrary nonempty closed subset of the real numbers. For t ∈ T, we define the forward jump operator σ : T → T by σ(t) := inf{s ∈ T : s > t}. The mapping ν : T → R+ 0 such that ν(t) = t − ρ(t) is called the backward graininess. If σ(t) > t, we say that t is right-scattered, while if ρ(t) < t, we say that t is left-scattered. Also, if t < sup T and σ(t) = t, then t is called right-dense, and if t > inf T and ρ(t) = t, then t is called left-dense. A function f : [a, b)T → R is said to be right–dense continuous (rd−continuous) if it is right continuous at each right–dense point and there exists a finite left limit at all left–dense points. The space of rd−continuous functions is denoted (n) by Crd (T, R). We also denote by Crd (T) the space of all functions f ∈ Crd (T) such that f ∆i ∈ Crd (T) for i = 0, 1, 2, ..., n for n ∈ N. The product and quotient rules for the derivative of the product f g and the quotient f /g (where gg σ 6= 0, where g σ is the composition function g ◦ σ) of two differentiable functions f and g are given by ∆ f ∆g − f g∆ f ∆ ∆ σ ∆ ∆ ∆ σ = . (2.1) (f g) = f g + f g = f g + f g , g gg σ The time scales chain rule that we will use in this paper is Z 1 [hxσ + (1 − h)x]γ−1 dhx∆ (t), γ > 0, (2.2) (xγ (t))∆ = γ 0
which is a simple consequence of Keller’s chain rule [4, Theorem 1.90]. Now, we are ready to state and prove the main results. We begin by deducing a Picone-type inequality which is also an interesting result in its own right. Consider the second order half-linear dynamic operator L defined on the time scale interval I = [a, b]T := [a, b] ∩ T with a < σ (a) and ρ (b) < b, as h i ∆ (2.3) Lv(t) = pϕα (v ∆ ) + qϕα (v σ ) (t), for t ∈ I, 1 (I, R) with p > 0 on I. The where ϕα (v) = |v|α−1 v, q ∈ Crd (I, R) and p ∈ Crd domain D of L is defined to be the set of all real-valued functions v on I such that all delta-derivatives in (2.3) exist and are rd-continuous at each point in I. We will consider here solutions v ∈ D of the dynamic inequality
(2.4)
[Lv + λrϕβ (v σ )] (t) ≤ 0, for t ∈ I,
where β > 0, λ is a real number and r is a positive rd-continuous function in I. We denote by ACloc (I) the space of all real-valued absolutely rd-continuous functions on every closed subinterval of I.
4
S. H. SAKER1 , R. R. MAHMOUD2 AND A. PETERSON3
Theorem 2.1 (Picone-type inequality). Let v ∈ D be a positive solution of (2.4) on I. If u ∈ ACloc (I), then !∆ |u|β+1 pϕα (v ∆ ) (2.5) ϕβ (v) ( ∆ β+1 ) σ v∆ uv ∆ α−β ∆ β+1 u ∆ u − (β + 1) u ϕβ ( σ ) + β σ ≤ −p v v v β+1 ∆ α−β |uσ |β+1 v + +p u∆ [Lv − qϕα (v σ )] . ϕβ (v σ )
Proof. Without loss of generality, we may assume that v ∆ (t) and u∆ (t) > 0. For convenience we omit the argument t sometimes in the computations. From (2.1) we get that !∆ !∆ |u|β+1 |uσ |β+1 |u|β+1 ∆ ∆ ∆ pϕα (v ) = pϕα (v ∆ ) + pϕ (v ) . (2.6) α ϕβ (v) ϕβ (v) ϕβ (v σ )
Clearly, the second term on the right-hand side is equal to |uσ |β+1 [Lv − qϕα (v σ )] . ϕβ (v σ )
(2.7)
Computing the delta-derivative of the quotient |u|β+1 /ϕβ (v) in the first term on the right-hand side of (2.6), we obtain that ∆ ! β+1 β+1 ∆ |u| ϕβ (v) − |u|β+1 (ϕβ (v))∆ |u| (2.8) = . ϕβ (v) ϕβ (v)ϕβ (v σ ) By (2.2), we get that Z 1 ∆ (2.9) |u|β+1 = (β + 1) [h |uσ | + (1 − h) |u|]β dhu∆ 0 Z 1 ≤ (β + 1) [h |uσ | + (1 − h) |uσ |]β dhu∆ 0
≤ (β + 1) |uσ |β u∆ ,
since v ∆ (t) > 0, we can similarly get that ∆ (2.10) (ϕβ (v))∆ = |v|β ≥ β |v|β−1 v ∆ .
Substituting (2.9) and (2.10) into (2.8), we have that !∆ (β + 1) |uσ |β u∆ ϕβ (v) − |u|β+1 β |v|β−1 v ∆ |u|β+1 ≤ ϕβ (v) ϕβ (v)ϕβ (v σ ) = (2.11)
(β + 1) |uσ |β u∆ ϕβ (v) β |u|β+1 |v|β−1 v ∆ − ϕβ (v σ ) ϕβ (v)ϕβ (v σ ) ∆ ∆ α−β
≤ (β + 1) u
v
∆ α−β uv ∆ β+1 uσ v ∆ ϕβ ( σ ) − β v . vσ v
PICONE-TYPE INEQUALITY
5
Substituting (2.7) and (2.11) into (2.6), adding and subtracting the term β+1 ∆ α−β v p u∆ , we get that !∆ |u|β+1 ∆ pϕα (v ) ϕβ (v) ( ∆ β+1 ) σ v∆ ∆ α−β ∆ β+1 uv u ∆ u − (β + 1) u ϕβ ( σ ) + β σ ≤ −p v v v ∆ β+1 ∆ α−β |uσ |β+1 v [Lv − qϕα (v σ )] . +p u + ϕβ (v σ )
which is the required inequality (2.5). This completes the proof.
Remark 2.1. As a special case when T = R, we get the following inequality !0 |u|β+1 pϕα (v 0 ) ϕβ (v) 0 β+1 0 uv 0 α−β 0 β+1 uv − (β + 1) u0 ϕβ ( ≤ −p v )+β u v v 0 β+1 0 α−β |u|β+1 [Lv − qϕα (v)] , +p u + v ϕβ (v)
due to Jaroˇs [10] where v is a solution of (1.5) and u is any absolutely continuous function. As a special case when T = Z, we get the following discrete Picone-type inequality which is essentially new. Corollary 2.1. Assume that vk is a solution of the following difference inequality ∆ (pk ϕα (∆vk )) + qk ϕα (vk+1 ) + λrk ϕβ (vk+1 ) ≤ 0,
where k ∈ [m, n + 1] ≡ {m, m + 1, . . . , n, n + 1} is a discrete interval with m, n ∈ Z, pk , qk and rk are real-valued sequences with pk 6= 0. Then ! |uk |β+1 pk ϕα (∆vk ) ∆ ϕβ (vk ) ( ) uk ∆vk β+1 uk+1 ∆vk α−β β+1 ≤ −p |∆vk | |∆uk | − (β + 1) ∆uk ϕβ + β vk+1 vk+1 + p |∆uk |β+1 |∆vk |α−β +
|uk+1 |β+1 [∆ (pk ϕα (∆vk )) + qk ϕα (vk+1 ) − qϕα (vk+1 )] , ϕβ (vk+1 )
for every real-valued sequence uk 6= 0 for k ∈ [m, n + 1]. In the following, we shall apply the Picone dynamic inequality to obtain new generalized dynamic Wirtinger-type inequalities on time scales. For a positive
S. H. SAKER1 , R. R. MAHMOUD2 AND A. PETERSON3
6
solution v for (2.4), we shall consider functions u ∈ ACloc (I) such that the limits p|u|β+1 ϕ (v∆ ) α (a), if ϕβ (v(a)) > 0, ϕβ (v) β+1 p|u| ϕα (v ∆ ) (σ(a)), if ϕβ (v(a)) = 0, (2.12) Sh (u, v) := ϕβ (v) β+1 ∆ (x)) p(x)|u(x)| ϕ (v α lim , if a = σ(a), + x→a x∈T
(2.13)
ϕβ (v(x))
p|u|β+1 ϕ (v∆ ) α (b), if ϕβ (v(b)) > 0, ϕβ (v) β+1 p|u| ϕα (v ∆ ) (ρ(b)), if ϕβ (v(b)) = 0, Sk (u, v) := ϕβ (v) β+1 ∆ (x)) p(x)|u(x)| ϕ (v α limx→b− , if b = ρ(b), ϕβ (v(x)) x∈T
exist and are finite.
Theorem 2.2. Let v ∈ D be a positive solution of (2.4) in I. If u ∈ ACloc (I) such that the limits in (2.12) and (2.13) exist and are finite, then Z (q |v σ |α−β + λr) |uσ |β+1 ∆x I Z β+1 ∆ α−β v ≤ p u∆ (2.14) ∆x + Sh (u, v) − Sk (u, v), I
holds if the integrals exist.
Proof. From the Picone-type inequality (2.5), we have that !∆ |u|β+1 ∆ pϕα (v ) ϕβ (v) ( ∆ β+1 ) σ v∆ ∆ α−β ∆ β+1 uv u u − (β + 1) u∆ ϕβ ( σ ) + β σ ≤ −p v v v β+1 ∆ α−β |uσ |β+1 v [Lv − qϕα (v σ )] . +p u∆ + ϕβ (v σ )
Integrating the above inequality over a subinterval [h, k]T of I with σ (a) ≤ h if ϕβ (v(a)) = 0 and k ≤ ρ(b) if ϕβ (v(b)) = 0, we obtain that " #k Z i kh β+1 ∆ α−β σ α−β |u|β+1 ∆ v pϕα (v ) ≤ p u∆ + λr |uσ |β+1 ∆x. − q |v | ϕβ (v) h h
Finally, then by taking limits as h → a and k → b (notice that since a is leftscattered and b is right-scattered, we need only to consider the limit as h → a from the right and the limit as k → b from the left, respectively), when possible, we get the required inequality (2.14). This completes the proof. We have the following special cases. Remark 2.2. If T = R in Theorem 2.2, then v σ (x) = v(x), uσ (x) = u(x) and the result (2.14) reduces to (1.6) due to Jaroˇs . Remark 2.3. If T = R and β = 1 in Theorem 2.2, we get the continuous result (1.4) due to Lee et al. [11].
PICONE-TYPE INEQUALITY
7
Remark 2.4. If T = Z and β = 1 in Theorem 2.2, we get the following discrete result. Corollary 2.2. Assume that vk is a positive solution of the following difference inequality − ∆ pk (∆vk )α−1 ∆vk + qk (vk+1 )α−1 vk+1 ≥ λ1 rk vk+1 , k ∈ [m, n + 1].
Then the following discrete Wirtinger-type inequality holds for every sequence uk of real numbers X X (qk |vk |α−1 + λ1 rk )u2k ≤ pk (∆uk )2 |∆vk |α−1 + S3 (uk , vk ) − S4 (uk , vk ), k
k
where
2 α−1 pk uk |∆vk | (a), 2 vk α−1 S3 (uk , vk ) := p u |∆v | k k k (a + 1), vk 2 α−1 pk uk |∆vk | (b), vk 2 S4 (uk , vk ) := α−1 pk uk |∆vk | (b − 1), vk
if v(a) > 0, if v(a) = 0, if v(b) > 0, if v(b) = 0.
Remark 2.5. Assume α = β = 1 in Theorem 2.2. Then the result (2.14) reduces to the following consequence of Agarwal et al. [1]. Corollary 2.3. Assume that v is a positive solution for the differential inequality ∆ − pv ∆ − qv σ ≥ λ0 rv σ , on I,
u ∈ AC(I). Then Z b Z b 2 σ 2 (q + λ0 r) (u ) ∆x ≤ p u∆ ∆x + S5 (u, v) − S6 (u, v), a
where
a
2 ∆ pu v (a), if v(a) > 0, 2v ∆ pu v (σ(a)), if v(a) = 0, S5 (u, v) := v 2 ∆ (x)v (x) limx→a+ p(x)u v(x) , if a = σ(a), x∈T 2 ∆ pu v (b), if v(b) > 0, 2v ∆ pu v (ρ(b)), if v(b) = 0, S6 (u, v) := v p(x)u2 (x)v ∆ (x) lim , if b = ρ(b). − x→b x∈T
v(x)
Remark 2.6. If α = β = λ = 1, q(x) = 0 and Sh (u, v) = Sk (u, v) = 0, then Theorem 2.2 reduces to the following inequality Z b Z b 2 2 σ (2.15) r(x) (u (x)) ∆x ≤ u∆ (x) ∆x, a
a
which can be considered as the time scale version of Beesack’s result (1.3).
S. H. SAKER1 , R. R. MAHMOUD2 AND A. PETERSON3
8
Remark 2.7. As a special case of (2.15) if we set a = 0, y(0) = 0 and r(x) = 1/4x2 , we get that Z b Z b σ 2 (y (x))2 ∆ y (x) ∆x < ∆x, 4x2 0 0 which can be considered as the time scale version of the inequality Z b 2 Z b 2 y (x) 0 dx, dx < y (x) 4x2 0 0 due to Hardy [6, Theorem 253]. References [1] R. P. Agarwal, V. Otero-Espinar, K. Perera, and D. R.Vivero, Wirtinger’s inequalities on time scales. Can. Math. Bull. 51 (2) (2008), 161-171. [2] R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications. Second edition. Monographs and Textbooks in Pure and Applied Mathematics 228. Marcel Dekker, New York, (2000). [3] P. R. Beesack, Integral inequalities of the Wirtinger type, Duke Math. J. 25 (1958), 477-498. [4] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkh¨ auser, Boston, Mass, USA, (2001). [5] O. Doˇsl´ y , J. R. Graef, and J. Jaroˇs, Forced oscillation of second order linear and half-linear difference equations, Proc. Amer. Math. Soc. 131 (9) (2003), 2859-2867. [6] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, 2nd edn. Cambridge University Press, Cambridge, (1952) [7] D. B. Hinton and R. T. Lewis, Discrete spectra criteria for singular differential operators with middle terms, Math. Proc. Cambridge Philos. Soc. 77 (1975), 337–347. [8] K. Kreith, Oscillation theory, Springer (1973). [9] J. Jaroˇs and T. Kusano, A Picone type identity for second order half-linear differential equations, Acta Math. Univ. Comenianae 68 (1999), 137-151. [10] J. Jaroˇs, On an integral inequality of the Wirtinger type. Appl. Math. Lett. 24 (8) (2011), 1389-1392. [11] C. F. Lee, C. C. Yeh, C. H. Hong and R. P. Agarwal, Lyapunov and Wirtinger inequalities. Appl. Math. lett. 17 (7) (2004), 847-853. [12] M. Picone, Sui valori eccezionali di un parametro da cui dipende un’equazione differenziale lineare ordinaria del secondo ordine, Ann. Scuola Norm. Pisa 11 (1910), 1-141. [13] P. Rehak, Oscillatory properties of second order half-linear difference equations, Czechoslovak Math. J. 51 (2) (2001), 303-321. [14] A. Tiryaki, Sturm-Picone type theorems for second-order nonlinear differential equations, Elec. J. Diff. Eqns. 2014 (146) (2014), 1-11. 1 Department of Mathematics, Faculty of Science,, Mansoura University, MansouraEgypt, E-mail: [email protected], 2
Department of Mathematics, Faculty of Science,, Fayoum University, FayoumEgypt, E-mail: [email protected], 3 Department of Mathematics,, University of Nebraska–Lincoln,, Lincoln, NE 68588-0130, U.S.A., E-mail: [email protected].