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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 979705, 27 pages doi:10.1155/2011/979705 Research Article Lyapunov Stability of Quasilinear Implicit Dynamic Equations on Time Scales N H Du,1 N C Liem,1 C J Chyan,2 and S W Lin2 Department of Mathematics, Mechanics and Informatics, Vietnam National University, 334 Nguyen Trai, Hanoi, Vietnam Department of Mathematics, Tamkang University, 151 Ying Chuang Road, Tamsui, Taipei County 25317, Taiwan Correspondence should be addressed to N H Du, dunh@vnu.edu.vn Received 29 September 2010; Accepted February 2011 Academic Editor: Stevo Stevic Copyright q 2011 N H Du et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited This paper studies the stability of the solution x ≡ for a class of quasilinear implicit dynamic equations on time scales of the form At xΔ f t, x We deal with an index concept to study the solvability and use Lyapunov functions as a tool to approach the stability problem Introduction The stability theory of quasilinear differential-algebraic equations DAEs for short At x t f t, x t , x t , f t, 0, 0 ∀t ∈ Ê, 1.1 with A being a given m × m-matrix function, has been an intensively discussed field in both theory and practice This problem can be seen in many real problems, such as in electric circuits, chemical reactions, and vehicle systems Mă rz in has dealt with the a question whether the zero-solution of 1.1 is asymptotically stable in the Lyapunov sense Bx t g t, x t , x t , with A being constant and small perturbation with f t, x t , x t g Together with the theory of DAEs, there has been a great interest in singular difference equation SDE also referred to as descriptor systems, implicit difference equations An x n f n, x n ,x n , n∈ 1.2 Journal of Inequalities and Applications This model appears in many practical areas, such as the Leontiev dynamic model of multisector economy, the Leslie population growth model, and singular discrete optimal control problems On the other hand, SDEs occur in a natural way of using discretization techniques for solving DAEs and partial differential-algebraic equations, and so forth, which have already attracted much attention from researchers cf 2–4 When f n, x n , x n Bn x n g n, x n , x n , in , the authors considered the solvability of Cauchy problem for 1.2 ; the question of stability of the zero-solution of 1.2 has been considered in where the nonlinear perturbation g n, x n , x n is small and does not depend on x n Further, in recent years, to unify the presentation of continuous and discrete analysis, a new theory was born and is more and more extensively concerned, that is, the theory of the Using analysis on time scales The most popular examples of time scales are Ì Ê and Ì “language” of time scales, we rewrite 1.1 and 1.2 under a unified form At x Δ t f t, xΔ t , x t , with t in time scale Ì and Δ being the derivative operator on Ì When Ì Ê, Ì Ỉ , we have a similar equation to 1.2 if it is rewritten under the form An x −An x n f n, x n , x n ; n ∈ Ỉ 1.3 1.3 is 1.1 ; if n −x n The purpose of this paper is to answer the question whether results of stability for 1.1 and 1.2 can be extended and unified for the implicit dynamic equations of the form 1.3 The main tool to study the stability of this implicit dynamic equation is a generalized direct Lyapunov method, and the results of this paper can be considered as a generalization of 1.1 and 1.2 The organization of this paper is as follows In Section 2, we present shortly some basic notions of the analysis on time scales and give the solvability of Cauchy problem for quasilinear implicit dynamic equations At x Δ Bt x f t, x , 1.4 with small perturbation f t, x and for quasilinear implicit dynamic equations of the style At x Δ f t, x , 1.5 with the assumption of differentiability for f t, x The main results of this paper are established in Section where we deal with the stability of 1.5 The technique we use in this section is somewhat similar to the one in 6–8 However, we need some improvements because of the complicated structure of every time scale Nonlinear Implicit Dynamic Equations on Time Scales 2.1 Some Basic Notations of the Theory of the Analysis on Time Scales A time scale is a nonempty closed subset of the real numbers Ê, and we usually denote it by the symbol Ì We assume throughout that a time scale Ì is endowed with the topology inherited from the real numbers with the standard topology We define the forward jump inf{s ∈ Ì : s > t} operator and the backward jump operator σ, ρ : Ì → Ì by σ t Journal of Inequalities and Applications sup{s ∈ Ì : s < t} supplemented by supplemented by inf ∅ sup Ì and ρ t σ t − t A point t ∈ Ì sup ∅ inf Ì The graininess μ : Ì → Ê ∪ {0} is given by μ t is said to be right-dense if σ t t, right-scattered if σ t > t, left-dense if ρ t t, left-scattered if ρ t < t, and isolated if t is right-scattered and left-scattered For every a, b ∈ Ì, by a, b , we mean the set {t ∈ Ì : a t b} The set Ìk is defined to be Ì if Ì does not have a left-scattered maximum; otherwise, it is Ì without this left-scattered maximum Let f be a function defined on Ì, valued in Êm We say that f is delta differentiable or simply: differentiable at t ∈ Ìk provided there exists a vector f Δ t ∈ Êm , called the derivative of f, such that for all > there is a neighborhood V around t with f σ t − f s − f Δ t σ t − s |σ t − s| for all s ∈ V If f is differentiable for every t ∈ Ìk , then f is said to be differentiable on Ì If Ì Ê, then delta derivative is f t from continuous calculus; if Ì , the delta derivative is the forward difference, Δf, from discrete calculus A function f defined on Ì is rd-continuous if it is continuous at every right-dense point and if the left-sided limit exists at every left-dense point The set of all rd-continuous functions from Ì to a Banach space X is denoted by Crd Ì, X A matrix function f from Ì to Êm×m is said to be regressive if det I μ t f t / for all t ∈ Ìk , and denote R the set of regressive functions from Ì to Êm×m Moreover, denote R the set of positively regressive functions from Ì to Ê, that is, the set {f : Ì → Ê : μ t f t > ∀t ∈ Ì} Theorem 2.1 see 9–11 Let t ∈ Ì and let At be a rd-continuous m × m-matrix function and qt rd-continuous function Then, for any t0 ∈ Ìk , the initial value problem (IVP) xΔ has a unique solution x · defined on t At x qt , x t0 2.1 x0 t0 Further, if At is regressive, this solution exists on t ∈ Ì At X, X s I always The solution of the corresponding matrix-valued IVP X Δ s, even At is not regressive In this case, ΦA t, s is defined only with t s exists for t see 12, 13 and is called the Cauchy operator of the dynamic equation 2.1 If we suppose further that At is regressive, the Cauchy operator ΦA t, s is defined for all s, t ∈ Ì We now recall the chain rule for multivariable functions on time scales, this result has been proved in 14 Let V : Ì × Êm → Ê and g : Ì → Êm be continuously differentiable Then V ·, g · : Ì → Ê is delta differentiable and there holds V Δ t, g t VtΔ t, g t VtΔ Vx σ t , g t hμ t g Δ t , g Δ t 2.2 t, g σ t dh Vx t, g t hμ t g Δ t ,g Δ t dh, V t, x where Vx is the derivative in the second variable of the function V meaning and ·, · is the scalar product We refer to 12, 15 for more information on the analysis on time scales in normal Journal of Inequalities and Applications 2.2 Linear Equations with Small Nonlinear Perturbation Let Ì be a time scale We consider a class of nonlinear equations of the form At x Δ Bt x 2.3 f t, x The homogeneous linear implicit dynamic equations LIDEs associated to 2.3 are At x Δ 2.4 Bt x, where A , B ∈ Crd Ìk , Êm×m and f t, x is rd-continuous in t, x ∈ Ì × Êm In the case where the matrices At are invertible for every t ∈ Ì, we can multiply both sides of 2.3 by A−1 to t obtain an ordinary dynamic equation xΔ A−1 Bt x t A−1 f t, x , t t ∈ Ì, 2.5 which has been well studied If there is at least a t such that At is singular, we cannot solve explicitly the leading term xΔ In fact, we are concerned with a so-called ill-posed problem where the solutions of Cauchy problem may exist only on a submanifold or even they not exist One of the ways to solve this equation is to impose some further assumptions stated under the form of indices of the equation r for all t ∈ Ì We introduce the so-called index-1 of 2.4 Suppose that rank At and let Tt ∈ GL Êm such that Tt |ker At is an isomorphism between ker At and ker Aρ t ; T ∈ Crd Ìk , Êm×m Let Qt be a projector onto ker At satisfying Q ∈ Crd Ìk , Êm×m We can find such operators Tt and Qt by the following way: let matrix At possess a singular value decomposition At Ut Σt Vt , 2.6 where Ut , Vt are orthogonal matrices and Σt is a diagonal matrix with singular values σt1 σt2 · · · σtr > on its main diagonal Since A ∈ Crd Ìk , Êm×m , on the above decomposition of At , we can choose the matrix Vt to be in Crd Ìk , Êm×m see 16 Hence, by putting Qt Vt diag O, Im−r Vt and Tt Vρ t Vt−1 , we obtain Qt and Vt as the requirement Let St {x ∈ Êm , Bt x ∈ imAt }, and Pt : I − Qt Under these notations, we have the following Lemma Lemma 2.2 The following assertions are equivalent i kerAρ t ∩ St {0}; ii the matrix Gt At − Bt Tt Qt is nonsingular; iii Ê m kerAρ t ⊕ St , for all t ∈ Ì 2.7 Journal of Inequalities and Applications Ax This Proof i ⇒ ii Let t ∈ Ì and x ∈ Êm such that At − Bt Tt Qt x ⇔ Bt Tt Qt x equation implies Tt Qt x ∈ St Since ker Aρ t ∩ St {0} and Tt Qt x ∈ ker Aρ t , it follows that Tt Qt x Hence, Qt x which implies At x This means that x ∈ ker At Thus, x Qt x 0, that is, the matrix Gt At − Bt Tt Qt is nonsingular ii ⇒ iii It is obvious that x I Tt Qt G−1 Bt x−Tt Qt G−1 Bt x We see that Tt Qt G−1 Bt x ∈ t t t ker Aρ t and Bt I Tt Qt G−1 Bt x Bt x − At − Bt Tt Qt G−1 Bt x At G−1 Bt x At G−1 Bt x ∈ imAt t t t t Thus, I Tt Qt G−1 Bt x ∈ St and we have Êm St ker Aρ t t Let x ∈ ker Aρ t ∩ St , that is, x ∈ St and x ∈ ker Aρ t Since x ∈ St , there is a z ∈ Êm such that Bt x At z At Pt z and since x ∈ ker Aρ t , Tt−1 x ∈ ker At Therefore, Tt−1 x Qt Tt−1 x Hence, At − Bt Tt Qt Tt−1 x − At − Bt Tt Qt Pt z which follows that Tt−1 x −Pt z Thus, Tt−1 x and then x So, we have that iii iii ⇒ i is obvious Lemma 2.2 is proved Lemma 2.3 Suppose that the matrix Gt is nonsingular Then, there hold the following assertions: Pt G−1 At , t 2.8 Qt −G−1 Bt Tt Qt , t 2.9 Qt : −Tt Qt G−1 Bt is the projector onto ker Aρ t along St , t 2.10 a Pt Gt −1 Bt Pt G−1 Bt Pρ t , t 2.11 b Qt Gt −1 Bt Qt G−1 Bt Pρ t − Tt −1 Qρ t , t 2.12 Tt Qt G−1 does not depend on the choice of Tt and Qt t 2.13 At − Bt Tt Qt Pt At Pt At , we get 2.8 Proof Noting that Gt Pt From Bt Tt Qt At − Gt , it follows G−1 Bt Tt Qt Pt − I −Qt Thus, we have 2.9 t 2.9 −Aρ t Qt Tt Qt G−1 Bt Tt Qt G−1 Bt −Tt Qt Qt G−1 Bt −Tt Qt G−1 Bt Qt and Aρ t Qt t t t t −1 Tt Qt Gt Bt This means that Qt is a projector onto ker Aρ t From the proof of iii , Lemma 2.2, we see that Qt is the projector onto ker Aρ t along St Since Tt−1 Qρ t x ∈ ker At for any x, Pt G−1 Bt Qρ t t Therefore, Pt G−1 Bt t Pt G−1 Bt Tt Tt−1 Qρ t t −Pt G−1 At − Bt Tt Qt Qt Tt−1 Qρ t t Pt G−1 Bt Pρ t so we have 2.11 Finally, t Qt G−1 Bt t Qt G−1 Bt Pρ t t Qt G−1 Bt Tt Qt Tt−1 Qρ t t Qt G−1 Bt Pρ t − Qt G−1 At − Bt Tt Qt Qt Tt−1 Qρ t t t Qt G−1 Bt Pρ t − Qt Tt−1 Qρ t t Thus, we get 2.12 2.14 Qt G−1 Bt Pρ t − Tt−1 Qρ t t 2.15 Journal of Inequalities and Applications Let Tt be another linear transformation from Êm onto Êm satisfying Tt |ker At to be an isomorphism from ker At onto ker Aρ t and Qt a projector onto ker At Denote Gt At − Bt Tt Qt It is easy to see that Tt Qt G−1 Gt t Tt Qt G−1 At − Bt Tt Qt t Therefore, Tt Qt G−1 t Tt Qt Pt − Tt Qt G−1 Bt Tt Qt t Tt Qt 2.16 Tt Qt Gt−1 The proof of Lemma 2.3 is complete Definition 2.4 The LIDE 2.4 is said to be index-1 if for all t ∈ hold: i rank At Tt Qt Tt Qt r ii kerAρ t ∩ St constant r Ì, the following conditions m−1 , {0} Now, we add the following assumptions Hypothesis 2.5 The homogeneous LIDE 2.4 is of index-1 f t, x is rd-continuous and satisfies the Lipschitz condition, f t, w − f t, w ∀w, w ∈ Êm , Lt w − w , 2.17 where γt : Lt Tt Qt G−1 < ∀t ∈ Ìk t 2.18 Remark 2.6 By the item 2.13 of Lemma 2.3, the condition 2.18 is independent from the choice of Tt and Qt We assume further that we can choose the projector function Qt onto ker At such that Qt for all right-dense and left-scattered t; Qρ t is differentiable at every t ∈ Ìk and Qρ t Δ Qρ t is rd-continuous For each t ∈ Ìk , we have Pρ t x t Δ Pρ σ t xΔ t Pρ t Δ x t Therefore, At x Δ t A t Pt x Δ t Pρ t x t At Δ − Pρ t Δ x t , 2.19 and the implicit equation 2.3 can be rewritten as A t Pρ t x Δ A t Pρ t Δ Bt x f t, x , t ∈ Ìk 2.20 Thus, we should look for solutions of 2.3 from the space CN : CN Ìk , Êm x · ∈ Crd Ìk , Êm : Pρ t x t is differentiable at every t ∈ Ìk 2.21 Note that CN does not depend on the choice of the projector function since the relations Pt P t P t and P t Pt Pt are true for each two projectors Pt and P t along the space ker At Journal of Inequalities and Applications We now describe shortly the decomposition technique for 2.3 as follows Since 2.3 has index-1 and by virtue of Lemma 2.2, we see that the matrices Gt are nonsingular for all t ∈ Ìk Multiplying 2.3 by Pt G−1 and Qt G−1 , respectively, it yields t t Pt x Δ Pt G−1 Bt x t Qt G−1 Bt x t Pt G−1 f t, x , t 2.22 Qt G−1 f t, x t Therefore, by using the results of Lemma 2.3, we get Pρ t x Δ Pρ t Δ Pρ t uΔ Pρ t Pρ t x, v Δ I Δ Tt Qt G−1 t Tt Qt G−1 Bt Pρ t x t Qρ t x By denoting u Tt Qt G−1 Bt Pρ t x t I Pt G−1 Bt Pρ t x t Pt G−1 f t, x , t 2.23 Tt Qt G−1 f t, x t Qρ t x, 2.23 becomes a dynamic equation on time scale Tt Qt G−1 Bt u t Pt G−1 Bt u t Pρ t Δ Tt Qt G−1 t Pt G−1 f t, u t v , 2.24 and an algebraic relation v Tt Qt G−1 Bt u t Tt Qt G−1 f t, u t 2.25 v For fixed u ∈ Êm and t ∈ Ìk , we consider a mapping Ct : im Qρ t → im Qρ t given by Ct v : Tt Qt G−1 Bt u t Tt Qt G−1 f t, u t 2.26 v We see that Tt Qt G−1 t Ct v − Ct v f t, u v − f t, u γt v − v , v 2.27 for any v, v ∈ im Qρ t Since γt < 1, Ct is a contractive mapping Hence, by the fixed point theorem, there exists a mapping gt : im Pρ t → im Qρ t satisfying gt u Tt Qt G−1 Bt u t Tt Qt G−1 f t, u t 2.28 gt u , and it is easy to see that gt u is rd-continuous in t Moreover, gt u − gt u Tt Qt G−1 Bt t u−u Tt Qt G−1 t Tt Qt G−1 Bt t u−u Lt Tt Qt G−1 t f t, u gt u u−u − f t, u gt u gt u − gt u 2.29 Journal of Inequalities and Applications This deduces γt − γt gt u − gt u −1 −1 Lt u−u Bt Lt Thus, gt is Lipschitz continuous with the Lipschitz constant δt : Substituting gt into 2.24 , we obtain uΔ Δ Pρ t I Tt Qt G−1 Bt u t Pt G−1 Bt u t Δ Pρ t Tt Qt G−1 t 2.30 γt − γt −1 −1 Lt Pt G−1 f t, u t Lt Bt gt u 2.31 It is easy to see that the right-hand side of 2.31 satisfies the Lipschitz condition with the Lipschitz constant ωt Pρ t Δ I Tt Qt G−1 Bt t Pt G−1 Bt t Lt Pρ t δt Δ Tt Qt G−1 t Pt G−1 t 2.32 Applying the global existence theorem see 12 , we see that 2.31 , with the initial condition u t0 Pρ t0 x0 has a unique solution u t u t; t0 , x0 , t t0 Thus, we get the following theorem Theorem 2.7 Let Hypothesis 2.5 and the assumptions on the projector Qt be satisfied Then, 2.3 with the initial condition Pρ t0 x t0 − x0 2.33 has a unique solution This solution is expressed by x t where u t x t; t0 , x0 u t; t0 , x0 gt u t; t0 , x0 , u t; t0 , x0 is the solution of 2.31 with u t0 t t0 , t ∈ Ìk , 2.34 Pρ t0 x0 We now describe the solution space of the implicit dynamic equation 2.3 Denote Łt x ∈ Êm : Qρ t x Ωt Tt Qt G−1 Bt Pρ t x t x ∈ Êm : Bt x Tt Qt G−1 f t, x t , 2.35 f t, x ∈ im At Lemma 2.8 There hold the following statements: i Łt Ωt , ii If f t, 0 for all t ∈ Ì then Ωt ∩ ker Aρ t Proof i Let y ∈ Łt , that is, Qρ t y y Pρ t y Qρ t y Tt Qt G−1 Bt Pρ t y t I {0} Tt Qt G−1 f t, y We have t Tt Qt G−1 Bt Pρ t y t Tt Qt G−1 f t, y t 2.36 Journal of Inequalities and Applications Hence, Tt Qt G−1 Bt Pρ t y t I Bt Tt Qt G−1 f t, y t I Bt Tt Qt G−1 Bt Pρ t y t I Bt Tt Qt G−1 f t, y t I Bt y Bt Tt Qt G−1 t f t, y Bt I f t, y Bt Pρ t y 2.37 From Bt Tt Qt G−1 t I I At − Gt G−1 t At G−1 , t 2.38 it yields Bt y At G−1 Bt Pρ t y t f t, y ∈ im At ⇒ y ∈ Ωt f t, y Conversely, suppose that y ∈ Ωt , that is, there exists z ∈ Êm such that Bt y have to prove Qρ t y Tt Qt G−1 Bt Pρ t y t f t, y Tt Qt G−1 f t, y , t 2.39 At z We 2.40 or equivalently, y Tt Qt G−1 f t, y t Tt Qt G−1 Bt Pρ t y t 2.41 Pρ t y Indeed, Tt Qt G−1 f t, y t Tt Qt G−1 Bt Pρ t y t Tt Qt G−1 f t, y t Pρ t y Tt Qt G−1 Bt y − Tt Qt G−1 Bt Qρ t y t t Tt Qt G−1 f t, y t Bt y − Tt Qt G−1 Bt Qρ t y t Tt Qt G−1 At z − Tt Qt G−1 Bt Qρ t y t t Pρ t y Pρ t y 2.42 Pρ t y Tt Qt Pt z − Tt Qt G−1 Bt Qρ t y t Pρ t y −Tt Qt G−1 Bt Qρ t y t Qρ t y Pρ t y Pρ t y y, where we have already used a result of Lemma 2.3 that Q −Tt Qt G−1 Bt is a projector onto t ker Aρ t So Łt Ωt ii Let y ∈ Ωt ∩ ker Aρ t Then y ∈ Ωt and Pρ t y Since Ωt Łt , we have y ∈ Łt This means that Qρ t y Tt Qt G−1 Bt Pρ t y Tt Qt G−1 f t, y Tt Qt G−1 f t, Qρ t y From the t t t −1 assumption f t, 0, it follows that Qρ t y Lt Tt Qt Gt Qρ t y γt Qρ t y The fact γt < implies that Qρ t y Thus y Pρ t y Qρ t y The lemma is proved 10 Journal of Inequalities and Applications Remark 2.9 By virtue of Lemma 2.8, we find out that the solution space Łt is independent from the choice of projector Qt and operator Tt Pρ t0 and Aρ t0 Pρ t0 Aρ t0 , the initial condition 2.33 is Since G−1t0 Aρ t0 ρ Aρ t0 x0 This implies that the initial condition is equivalent to the condition Aρ t0 x t0 not also dependent on choice of projectors Noting that if x t is a solution of 2.3 with the initial condition 2.33 , then x t ∈ t0 Conversely, let x0 ∈ Łt Ωt and let x s; t, x0 , s t, be the solution of Łt for all t 2.3 satisfying the initial condition Pρ t x t; t, x0 − x0 We see that x t; t, x0 Pρ t x Pρ t x0 gt Pρ t x0 x0 This means that there exists a solution of 2.3 passing g t Pρ t x x0 ∈ Łt 2.3 Quasilinear Implicit Dynamic Equations Now we consider a quasilinear implicit dynamic equation of the form At x Δ 2.43 f t, x , with A ∈ Crd Ìk , Êm×m and f : Ì × Êm → Êm assumed to be continuously differentiable in the variable x and continuous in t, x Suppose that rank At r for all t ∈ Ì We keep all assumptions on the projector Qt and operator Tt stated in Section 2.2 Equation 2.43 is said to be of index-1 if the matrix Gt : At − fx t, x Tt Qt 2.44 is invertible for every t ∈ Ì and x ∈ Êm Denote S t, x z ∈ Êm , fx t, x z ∈ imAt ; ker At Nt 2.45 Further introduce the set Ωt x ∈ Êm , f t, x ∈ imAt , 2.46 containing all solutions of 2.43 The subspace S t, x manifests its geometrical meaning S t, x Tx Ωt for x ∈ Ωt , 2.47 where Tx is the tangent space of Ωt at the point x Suppose that 2.43 is of index-1 Then, by Lemma 2.2, this condition is equivalent to one of the following conditions: Journal of Inequalities and Applications 13 is bounded, then this solution is defined on t0 , sup Ì and we have the expression x t; t0 , x0 u t; t0 , x0 gt u t; t0 , x0 , where u t; t0 , x0 is the solution of 2.56 with u t0 t t0 , 2.60 Pρ t0 x0 Remark 2.12 We note that the expression Tt Qt G−1 Bt depends only on choosing the t matrix Bt The assumption that Tt Qt G−1 fx t, x |Qρ t Êm −1 Tt Qt G−1 fx t, x |Pρ t Êm is bounded for t t a matrix function Bt seems to be too strong In Section 3, we show a condition for the global solvability via Lyapunov functions If x0 ∈ Ωt , there exists z ∈ Êm satisfying At z f t, x0 Hence, Tt Qt G−1 f t, x0 t Therefore, by the same argument as in Section 2.2, we can prove that for every x0 ∈ Ωt , there is a unique solution passing through x0 Stability Theorems of Implicit Dynamic Equations For the reason of our purpose, in this section we suppose that Ì is an upper unbounded time scale, that is, sup Ì ∞ For a fixed τ ∈ Ì, denote Ìτ {t ∈ Ì, t τ} Consider an implicit dynamic equation of the form At x Δ f t, x , t ∈ Ìτ , 3.1 where A ∈ Crd Ìk , Êm×m and f ·, · ∈ Crd Ìτ × Êm , Êm τ First, we suppose that for each t0 ∈ Ìk , 3.1 with the initial condition τ Aρ t0 x t0 − x0 3.2 has a unique solution defined on Ìt0 The condition ensuring the existence of a unique solution can be refered to Section We denote the solution with the initial condition 3.2 by x t x t; t0 , x0 Remember that we look for the solution of 3.1 in the space CN Ìk , Êm τ Let f t, 0 for all t ∈ Ìτ , which follows that 3.1 has the trivial solution x ≡ We mention again that Ωt {x ∈ Êm , f t, x ∈ im At } Noting that if x t x t; t0 , x0 is the solution of 3.1 and 3.2 then x t ∈ Ωt for all t ∈ Ìt0 Definition 3.1 The trivial solution x ≡ of 3.1 is said to be A-stable resp., P -stable if, for each > and t0 ∈ Ìk , there exists a positive δ τ δ t0 , such that Aρ t0 x0 < δ resp., Pρ t0 x0 < δ implies x t; t0 , x0 < for all t t0 , A-uniformly resp., P -uniformly stable if it is A-stable resp., P -stable and the number δ mentioned in the part of this definition is independent of t0 , A-asymptotically resp., P -asymptotically stable if it is stable and for each t0 ∈ Ìk , there exist positive δ δ t0 such that the inequality Aρ t0 x0 < δ τ If δ is independent of resp., Pρ t0 x0 < δ implies limt → ∞ x t; t0 , x0 t0 , then the corresponding stability is A-uniformly asymptotically P -uniformly asymptotically stable, 14 Journal of Inequalities and Applications A-uniformly globally asymptotically resp., P -uniformly globally asymptotically stable if for any δ0 > there exist functions δ · , T · such that Aρ t0 x0 < δ implies x t; t0 , x0 < for all t t0 and if Aρ t0 x0 < δ0 resp., Pρ t0 x0 < δ resp., Pρ t0 x0 < δ0 then x t; t0 , x0 < for all t t0 T , P-exponentially stable if there is positive constant α with −α ∈ R such that for 1, the solution of 3.1 with the initial every t0 ∈ Ìk there exists an N N t0 τ satisfies x t; t0 , x0 N Pρ t0 x0 e−α t, t0 , t condition Pρ t0 x t0 − x0 t0 , t ∈ Ìτ If the constant N can be chosen independent of t0 , then this solution is called P -uniformly exponentially stable Remark 3.2 From G−1 At Pt and At At Pt , the notions of A-stable and P -stable as well as At asymptotically stable and P -asymptotically stable are equivalent Therefore, in the following theorems we will omit the prefixes A and P when talking about stability and asymptotical stability However, the concept of A-uniform stability implies P -uniform stability if the matrices At are uniformly bounded and P -uniform stability implies A-uniform stability if the matrices Gt are uniformly bounded Denote Þ: and φ ∈ C 0, a , Ê , φ 0, φ is strictly increasing; a > , 3.3 φ is the domain of definition of φ Proposition 3.3 The trivial solution x ≡ of 3.1 is A-uniformly (resp., P -uniformly) stable if and only if there exists a function ϕ ∈ Þ such that for each t0 ∈ Ìk and any solution x t; t0 , x0 of 3.1 τ the inequality x t; t0 , x0 ϕ Aρ t0 x0 holds, provided Aρ t0 x0 ∈ , resp., x t; t0 , x0 ϕ (resp., Pρ t0 x0 ∈ ϕ Pρ t0 x0 ∀t t0 , 3.4 ϕ ) Proof We only need to prove the proposition for the A-uniformly stable case Sufficiency Suppose there exists a function ϕ ∈ Þ satisfying 3.4 for each > 0; we take δ δ > such that ϕ δ < , that is, ϕ−1 > δ If x t; t0 , x0 is an arbitrary solution of ϕ Aρ t0 x0 < ϕ δ < , for all t t0 3.1 and Aρ t0 x0 < δ, then x t; t0 , x0 Necessity Suppose that the trivial solution x ≡ of 3.1 is A-uniformly stable, that is, for each > there exists δ δ > such that for each t0 ∈ Ìk the inequality Aρ t0 x0 < δ τ implies x t; t0 , x0 < , for all t t0 For the sake of simplicity in computation, we choose δ < Denote γ It is clear that γ there holds sup δ :δ has such a property is an increasing positive function in Further, γ Aρ t0 x0 < γ then x t; t0 , x0 < ∀t 3.5 and by definition, t0 3.6 Journal of Inequalities and Applications 15 By putting β : γ t dt, 3.7 it is seen that β ∈ Þ, 0 Aρ t0 x0 The proposition is proved Similarly, we have the following proposition Proposition 3.4 The trivial solution x ≡ of 3.1 is A-stable (resp., P -stable) if and only if for each t0 ∈ Ìk and any solution x t; t0 , x0 of 3.1 there exists a function ϕt0 ∈ Þ such that there holds the τ following: x t; t0 , x0 Aρ t0 x0 ϕt0 provided Aρ t0 x0 ∈ resp., x t; t0 , x0 ϕt0 (resp., Pρ t0 x0 ∈ ϕt0 ∀t Pρ t0 x0 t0 , 3.9 ϕt0 ) In order to use the Lyapunov function technique related to 3.1 , we suppose that Aρ t ∈ Crd Ìk , Êm×m By using 2.3 , we can define the derivative of the function V : Ìτ × τ m Ê → Ê along every solution curve as follows: VΔ 3.10 t, Aρ t x VtΔ t, Aρ t x Vx σ t , A ρ t x hμ t Aρ t x Δ , Aρ t x Δ 3.10 dh Remark 3.5 Note that when the function V is independent of t and even if the vector field associated with the implicit dynamic equation 3.1 is autonomous, the derivative V Δ may 3.10 depend on t Theorem 3.6 Assume that there exist a constant c > 0, −c ∈ R and a function V : Ìτ × Êm → being rd-continuous and a function ψ ∈ Þ, ψ defined on 0, ∞ satisfying ψ x VΔ 3.10 V t, Aρ t x for all x ∈ Ωt and t ∈ Ìτ , t, Aρ t x c/ − cμ t V t, Aρ t x , for any x ∈ Ωt and t ∈ Ìk τ Ê 16 Journal of Inequalities and Applications Assume further that 3.1 is locally solvable Then, 3.1 is globally solvable, that is, every solution with the initial condition 3.2 is defined on Ìt0 Proof Denote W t, x V t, x e−c t, t0 3.11 By the condition , we have WΔ 3.10 t, Aρ t x VΔ 3.10 t, Aρ t x e−c σ t , t0 − cV t, Aρ t x e−c t, t0 c V t, Aρ t x − cμ t e−c t, t0 − cV t, Aρ t x e−c t, t0 − cμ t 3.12 t0 Therefore, for all t W t, Aρ t x t − W t0 , A ρ t x t0 t t0 WΔ 3.10 τ, Aρ τ x τ Δτ 3.13 From the condition , it follows that e−c t, t0 ψ x t W t, Aρ t x t W t0 , A ρ t x t0 V t0 , A ρ t x t0 3.14 or x t ψ −1 V t0 , Aρ t0 x t0 e −c t, t0 ψ −1 V t0 , Aρ t0 x t0 e c/ 1−cμ t t, t0 3.15 The last inequality says that the solution x t can be lengthened on Ìt0 , that is, 3.1 is globally solvable Theorem 3.7 Assume that there exist a function V : Ìτ × Êm → a function ψ ∈ Þ, ψ defined on 0, ∞ satisfying the conditions Ê being rd-continuous and V t, ≡ for all t ∈ Ìτ , ψ x V t, Aρ t x for all x ∈ Ωt and t ∈ Ìτ , VΔ t, Aρ t x 3.10 for any x ∈ Ωt and t ∈ Ìk τ Assume further that 3.1 is locally solvable Then the trivial solution of 3.1 is stable Proof By virtue of Theorem 3.6 and the conditions and , it follows that 3.1 is globally solvable Suppose on the contrary that the trivial solution x ≡ of 3.1 is not stable Then, there exists an > such that for all δ > there exists a solution x t of 3.1 satisfying t0 Put ψ Aρ t0 x t0 < δ and x t1 ; t0 , x t0 for some t1 Journal of Inequalities and Applications 17 and V t, x is rd-continuous, we can find δ0 > By the assumption that V t0 , such that if y < δ0 then V t0 , y < With given δ0 > 0, let x t be a solution of 3.1 such t0 that Aρ t0 x t0 < δ0 and x t1 ; t0 , x t0 for some t1 Since x t ∈ Ωt and by the condition , t1 t0 VΔ 3.10 t, Aρ t x t Δt V t1 , A ρ t x t1 − V t0 , A ρ t x t0 3.16 V t0 , Aρ t0 x t0 < Further, x t1 ∈ Ωt1 and by the condition ψ x t1 ψ This is a contradiction The theorem Therefore, V t1 , Aρ t1 x t1 we have V t1 , Aρ t1 x t1 is proved Theorem 3.8 Assume that there exist a function V : Ìτ × Êm → Ê being rd-continuous and functions ψ, φ ∈ Þ, ψ defined on 0, ∞ , δ ∈ Crd t0 , ∞ , 0, ∞ such that t δ s Δs −→ ∞ as t −→ ∞, 3.17 t0 satisfying the conditions limx → V t, x ψ x VΔ 3.10 uniformly in t ∈ Ìτ , V t, Aρ t x for all x ∈ Ωt and t ∈ Ìτ , t, Aρ t x for any x ∈ Ωt and t ∈ Ìk τ −δ t φ Aρ t x Further, 3.1 is locally solvable Then the trivial solution of 3.1 is asymptotically stable Proof Also from Theorem 3.6 and the conditions and , it implies that 3.1 is globally solvable And since V Δ t, Aρ t x −δ t φ Aρ t x 0, the trivial solution of 3.1 is 3.10 stable by Theorem 3.7 Consider a bounded solution x t of 3.1 First, we show that lim inft → ∞ V t, Aρ t x t Assume on the contrary that inft∈Ìt0 V t, Aρ t x t > From the condition , it follows that inft∈Ìt0 Aρ t x t : r > By the condition , we have t V t0 , A ρ t x t0 V t, Aρ t x t t0 V t0 , A ρ t x t0 − VΔ 3.10 s, Aρ s x s Δs t δ s φ Aρ s x s Δs V t0 , x t0 3.18 t0 −φ r t δ s Δs −→ −∞, t0 as t → ∞, which gets a contradiction Further, from the condition for any s Thus, inft∈Ìt0 V t, Aρ t x t V t, Aρ t x t − V s, Aρ s x s t s VΔ 3.10 τ, Aρ τ x τ Δτ t we get 3.19 18 Journal of Inequalities and Applications This means that V t, Aρ t x t is a decreasing function Consequently, lim V t, Aρ t x t inf V t, Aρ t x t which follows that limt → ∞ x t t∈ t, Aρ t x VΔ 3.10 3.20 by the condition Þ, a defined on Theorem 3.9 Suppose that there exist a function a ∈ Crd Ìτ × Êm , Ê such that limx → V t, x t ∈ Ìτ , 0, Ìt0 t→∞ uniformly in t ∈ Ìτ and a x 0, ∞ , and a function V ∈ V t, Aρ t x for all x ∈ Ωt and 0, for any x ∈ Ωt and t ∈ Ìk τ Assume further that 3.1 is locally solvable Then, the trivial solution of 3.1 is A-uniformly stable Proof The proof is similar to the one of Theorem 3.7 with a remark that since limx → V t, x uniformly in t ∈ Ìτ , we can find δ0 > such that if y < δ0 then supt∈Ìτ V t, y < The proof is complete Remark 3.10 The conclusion of Theorem 3.9 is still true if the condition is replaced by “there exist two functions a, b ∈ Þ, a defined on 0, ∞ and a function V ∈ Crd Ìτ × Êm , Ê such V t, Aρ t x b Aρ t x for all x ∈ Ωt and t ∈ Ìτ ” that a x We present a theorem of uniform global asymptotical stability Theorem 3.11 If there exist functions a, b, c ∈ Þ, a defined on 0, ∞ , and a function V ∈ Crd satisfying Êm , Ê a x V t, Aρ t x t, Aρ t x VΔ 3.10 b Aρ t x −c Aρ t x Ìτ × for all x ∈ Ωt and t ∈ Ìτ , for any x ∈ Ωt and t ∈ Ìk τ Assume further that 3.1 is locally solvable Then, the trivial solution of 3.1 is A-uniformly globally asymptotically stable Proof Let δ0 > be given Define δ T T min{b−1 a max μ t Ì t∈ , δ0 } and 2b δ0 c δ is not necessary in Ì Let x t be a solution of 3.1 with Aρ t0 x t0 and a t0 ∈ Ìk τ such that ∀δ > 0; there exists a solution x t x t; t0 , x0 of 3.1 satisfying Pρ t0 x0 < δ t0 Let ψ Since V t0 , 0, it is possible to find and x t1 ; t0 , x0 , for some t1 aδ δ , t0 > satisfying V t0 , Pρ t0 z < when Pρ t0 z < δ, z ∈ Êm Consider the t0 solution x t satisfying Pρ t0 x0 < δ and x t1 ; t0 , x0 for a t1 From the assumption , it follows that t1 t0 VΔ 3.26 t, Pρ t x t Δt V t1 , P ρ t x t1 − V t0 , Pρ t0 x0 3.27 This implies V t0 , Pρ t0 x0 V t1 , P ρ t x t1 ψ x t1 ψ 3.28 We get a contradiction because > V t0 , Pρ t0 x0 when Pρ t0 x0 < δ The proof of the theorem is complete Theorem 3.14 Assume that 3.1 is locally solvable If there exist two functions a, b ∈ on 0, ∞ and a function V : Ìτ × Êm → Ê being rd-continuous such that a x VΔ 3.26 V t, Pρ t x t, Pρ t x b Pρ t x Þ, a defined for all x ∈ Ωt and t ∈ Ìτ , for all x ∈ Ωt and t ∈ Ìk , τ then the trivial solution of 3.1 is P -uniformly stable Proof The proof is similar to the one of Theorem 3.9 Theorem 3.15 If there exist functions a, b, c ∈ Þ, a defined on 0, ∞ and a function V ∈ Crd Êm , Ê satisfying a x VΔ 3.10 V t, Pρ t x t, Pρ t x b Pρ t x −c Pρ t x Ìτ × for all x ∈ Ωt and t ∈ Ìτ , for any x ∈ Ωt and t ∈ Ìk τ Assume further that 3.1 is locally solvable Then, the trivial solution of 3.1 is P -uniformly globally asymptotically stable Proof Similarly to the proof of Theorem 3.11 It is difficult to establish the inverse theorem for Theorems from 3.7 to 3.15, that is, if the trivial solution of 3.1 is stable, there exists a function V satisfying the assertions in the Journal of Inequalities and Applications 21 Ì is rather simple we have the above theorems However, if the structure of the time scale following theorem Theorem 3.16 Suppose that Ìτ contains no right-dense points and the trivial solution x ≡ of 3.1 is P -uniformly stable Then, there exists a function V : Ìτ × U → Ê being rd-continuous satisfying the conditions (1), (2), and (3) of Theorem 3.13, where U is an open neighborhood of in Êm Proof Suppose the trivial solution of 3.1 is P -uniformly stable Due to Proposition 3.3, there exist functions ϕ ∈ Þ such that for any solution x t; t0 , x0 of 3.1 , we have x t; t0 , x0 provided Pρ t0 x0 ∈ ϕ ϕ 0, a and U Let t ∈ Ìτ , we put ϕ Pρ t0 x0 ∀t t0 , {x : x < a} For any z ∈ Êm 3.29 satisfying Pρ t0 z < a and V t, z : sup x s; t, z , 3.30 s t where x s; t, z is the unique solution of 3.1 satisfying the initial condition Pρ t x t Pρ t z ϕ , V t, ≡ 0, and V t, x ∈ It is seen that V is defined for all z satisfying Pρ t0 z ∈ Crd Ìτ × Êm , Ê sups t x s; t, Pρ t y x t; t, Pρ t y Let y ∈ Ωt By the definition, V t, Pρ t y u s; t, Pρ t y g s, u s; t, Pρ t y for all s ∈ Ìt In particular, From 2.60 , x s; t, Pρ t y x t; t, Pρ t y Pρ t y g t, Pρ t y y Thus, V t, Pρ t y y ∀y ∈ Ωt , t ∈ Ìτ Hence, we have the assertion of the theorem x s; σ t , x σ t , t, Due to the unique solvability of 3.1 , we have x s; t, Pρ t y sups t x s; t, Pρ t y and Pρ t y with s σ t Therefore, V t, Pρ t y V σ t , Pρ σ t sup x s; σ t , x σ t , t, Pρ t y x σ t , t, Pρ t y s σ t 3.31 sup x s; t, Pρ t y V t, Pρ t y s σ t This implies VΔ 3.26 t, Pρ t y t V σ t , Pρ σ t x σ t , t, Pρ t y μ t − V t, Pρ t y 3.32 The proof is complete Now we give an example on using Lyapunov functions to test the stability of equations The following result finds out that the stability of a linear equation will be ensured if nonlinear perturbations are sufficiently small Lipschitz Consider a nonlinear equation of the form 2.3 AxΔ Bx f t, x , 3.33 22 Journal of Inequalities and Applications where A and B are constant matrices with ind A, B satisfing the Lipschitz condition 1, f t, 0 ∀t ∈ Ì, and f such that VΔ 3.26 P x −β P x By Theorem 3.15, 3.33 is P -uniformly globally asymptotically stable 3.46 Journal of Inequalities and Applications Example 3.18 Let Ì ∪∞ 2k, 2k k 25 and consider At x Δ Bt x 3.47 f t, x , with At t 1 , 0 −t − 0 Bt −t − , sin x1 0, t f t, x 3.48 0 is We have ker At span{ 0, }, rank At for all t ∈ Ì It is easy to verify that Qt 01 Let us choose T the canonical projector onto ker At , Pt I − Qt I We see that t 00 At − Bt Tt Qt Gt t 1 3.49 Since t 0, det Gt t / 0, 3.47 has index-1 It is obvious that f t, w1 − f t, w2 1/ t w1 − w2 , ∀w1 , w2 ∈ Ê2 Further, −1 γt Lt Tt Qt Gt 1/ t < 1, for all t ∈ Ì Thus, according to Theorem 2.7 for each t0 ∈ Ì, 3.47 with the initial condition Pρ t0 x t0 Pρ t0 x0 has the unique solution 1/ t 1 , Tt Qt G−1 Bt Pρ t x 0, , Tt Qt G−1 f t, x It is easy to compute, G−1 t t t 01 sin x1 / t x1 , x2 , Pt G−1 Bt t 0, , where x 0, Therefore, u t Pρ t x t satisfies uΔ Łt x x1 , x2 − 1/ t ∈ Ê2 , x2 −1/ t t 20 0 sin x1 t t 20 0 , and Pt G−1 f t, x t u Moreover, we have 3.50 Let the Lyapunov function be V t, x : x , t ∈ Ì, x ∈ Ê2 Put x x1 , x2 ∈ Łt , we have V t, Pρ t x Pρ t x 2|x1 | and x x1 x2 1/2 x1 sin2 x1 t 1/2 x1 sin2 x1 1/2 2|x1 | 3.51 Hence, x V t, Pρ t x Pρ t x , ∀x ∈ Łt , t ∈ Ì We have for any solution x t of 3.47 and t ∈ Ì noting that t , 3.52 26 Journal of Inequalities and Applications if t is right-scattered then V Δ t, Pρ t x t 3.26 u Δ t − ut u2 / t u2 1/2 − u Pρ t x u2 t, Pρ t x t if t is right-dense then V Δ 3.26 u2 / t u 0, where u ut u2 1/2 − ut ut 0, Vx t, Pρ t x , F t, Pρ t x u1 , u2 , F t, u −1/ t t 20 0 −2 t u In both two cases, we have V Δ t, Pρ t x t 0, so the trivial solution of 3.47 is 3.26 P -uniformly stable by Theorem 3.14 Note that if we let V t, x : x , t ∈ Ì, x ∈ Ê2 then the result is still true Indeed, by the simple calculations we obtain a a x ay V t, Pρ t x 1/2 y2 , b y b VΔ t, Pρ t x t 3.26 VΔ 3.26 t, Pρ t x t u u Pρ t x u1 ,u2 Ì, where a, b b Pρ t x , ∀x ∈ Łt , t ∈ y2 , y ∈ Ê , 2u, F t, u ⎧ ⎪ t u2 ⎪− ⎪ ⎪ ⎨ t ⎪ t t u2 ⎪ ⎪− ⎪ ⎩ t μ t F t, u ∈ Þ defined by Thus, if t is right-dense, 3.53 if t is right-scattered, Therefore, having the above result is obvious Conclusion We have studied some criteria ensuring the stability for a class of quasilinear dynamic equations on time scales So far, the inverse theorem of the theorems of the stability in 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Linear Equations with Small Nonlinear Perturbation Let Ì be a time scale We consider a class of nonlinear equations of the form At x Δ Bt x 2.3 f t, x The homogeneous linear implicit dynamic equations