Lyapunov Stability of Quasilinear Implicit Dynamic Equations on Time Scales

Abstract

This paper studies the stability of the solution for a class of quasilinear implicit dynamic equations on time scales of the form . We deal with an index concept to study the solvability and use Lyapunov functions as a tool to approach the stability problem.

1. Introduction

The stability theory of quasilinear differential-algebraic equations (DAEs for short)

(1.1)

with being a given -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. März in [1] has dealt with the question whether the zero-solution of (1.1) is asymptotically stable in the Lyapunov sense with , with being constant and small perturbation .

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)

(1.2)

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 , in [5], 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 [6] where the nonlinear perturbation is small and does not depend on .

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 analysis on time scales. The most popular examples of time scales are and . Using "language" of time scales, we rewrite (1.1) and (1.2) under a unified form

(1.3)

with in time scale and being the derivative operator on . When , (1.3) is (1.1); if , we have a similar equation to (1.2) if it is rewritten under the form .

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

(1.4)

with small perturbation and for quasilinear implicit dynamic equations of the style

(1.5)

with the assumption of differentiability for . The main results of this paper are established in Section 3 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.

2. 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 , and we usually denote it by the symbol . We assume throughout that a time scale is endowed with the topology inherited from the real numbers with the standard topology. We define the forward jump operator and the backward jump operator by (supplemented by ) and (supplemented by ). The graininess is given by . A point is said to be right-dense if , right-scattered if , left-dense if , left-scattered if , and isolated if is right-scattered and left-scattered. For every , by , we mean the set . The set is defined to be if does not have a left-scattered maximum; otherwise, it is without this left-scattered maximum. Let be a function defined on , valued in . We say that is delta differentiable (or simply: differentiable) at provided there exists a vector , called the derivative of , such that for all there is a neighborhood around with for all . If is differentiable for every , then is said to be differentiable on. If , then delta derivative is from continuous calculus; if , the delta derivative is the forward difference, , from discrete calculus. A function defined on 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 to a Banach space is denoted by . A matrix function from to is said to be regressive if for all , and denote the set of regressive functions from to . Moreover, denote the set of positively regressive functions from to , that is, the set .

Theorem 2.1 (see [9–11]).

Let and let be a rd-continuous -matrix function and rd-continuous function. Then, for any , the initial value problem (IVP)

(2.1)

has a unique solution defined on . Further, if is regressive, this solution exists on .

The solution of the corresponding matrix-valued IVP , always exists for , even is not regressive. In this case, is defined only with (see [12, 13]) and is called the Cauchy operator of the dynamic equation (2.1). If we suppose further that is regressive, the Cauchy operator is defined for all .

We now recall the chain rule for multivariable functions on time scales, this result has been proved in [14]. Let and be continuously differentiable. Then is delta differentiable and there holds

(2.2)

where is the derivative (in the second variable of the function ) in normal meaning and is the scalar product.

We refer to [12, 15] for more information on the analysis on time scales.

2.2. Linear Equations with Small Nonlinear Perturbation

Let be a time scale. We consider a class of nonlinear equations of the form

(2.3)

The homogeneous linear implicit dynamic equations (LIDEs) associated to (2.3) are

(2.4)

where and is rd-continuous in . In the case where the matrices are invertible for every , we can multiply both sides of (2.3) by to obtain an ordinary dynamic equation

(2.5)

which has been well studied. If there is at least a such that is singular, we cannot solve explicitly the leading term . 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 do 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.

We introduce the so-called index-1 of (2.4). Suppose that rank for all and let such that is an isomorphism between and ; . Let be a projector onto satisfying . We can find such operators and by the following way: let matrix possess a singular value decomposition

(2.6)

where , are orthogonal matrices and is a diagonal matrix with singular values on its main diagonal. Since , on the above decomposition of , we can choose the matrix to be in (see [16]). Hence, by putting and , we obtain and as the requirement.

Let

(2.7)

and .

Under these notations, we have the following Lemma.

Lemma 2.2.

The following assertions are equivalent

(i);

(ii)the matrix is nonsingular;

(iii), for all .

Proof.

(i)(ii) Let and such that . This equation implies . Since and , it follows that . Hence, which implies . This means that . Thus, , that is, the matrix is nonsingular.

(ii)(iii) It is obvious that . We see that and . Thus, and we have .

Let , that is, and . Since , there is a such that and since , . Therefore, . Hence, which follows that . Thus, and then . So, we have that (iii). (iii)(i) is obvious.

Lemma 2.2 is proved.

Lemma 2.3.

Suppose that the matrix is nonsingular. Then, there hold the following assertions:

(2.8)

(2.9)

(2.10)

(2.11)

(2.12)

(2.13)

Proof.

• (1) Noting that , we get (2.8).

• (2) From , it follows . Thus, we have (2.9).

• (3) and . This means that is a projector onto . From the proof of (iii), Lemma 2.2, we see that is the projector onto along .

• (4) Since for any ,

  

  (2.14)

  Therefore, so we have (2.11). Finally,

  

  (2.15)

  Thus, we get (2.12).

• (5) Let be another linear transformation from onto satisfying to be an isomorphism from onto and a projector onto . Denote . It is easy to see that

  

  (2.16)

Therefore, . The proof of Lemma 2.3 is complete.

Definition 2.4.

The LIDE (2.4) is said to be index-1 if for all , the following conditions hold:

(i)rank ,

(ii) ker.

Now, we add the following assumptions.

Hypothesis 2.5.

• (1) The homogeneous LIDE (2.4) is of index-1.

• (2) is rd-continuous and satisfies the Lipschitz condition,

  

  (2.17)

where

(2.18)

Remark 2.6.

By the item (2.13) of Lemma 2.3, the condition (2.18) is independent from the choice of and .

We assume further that we can choose the projector function onto such that for all right-dense and left-scattered ; is differentiable at every and is rd-continuous. For each , we have . Therefore,

(2.19)

and the implicit equation (2.3) can be rewritten as

(2.20)

Thus, we should look for solutions of (2.3) from the space :

(2.21)

Note that does not depend on the choice of the projector function since the relations and are true for each two projectors and along the space .

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 are nonsingular for all . Multiplying (2.3) by and , respectively, it yields

(2.22)

Therefore, by using the results of Lemma 2.3, we get

(2.23)

By denoting , (2.23) becomes a dynamic equation on time scale

(2.24)

and an algebraic relation

(2.25)

For fixed and , we consider a mapping given by

(2.26)

We see that

(2.27)

for any . Since , is a contractive mapping. Hence, by the fixed point theorem, there exists a mapping satisfying

(2.28)

and it is easy to see that is rd-continuous in .

Moreover,

(2.29)

This deduces

(2.30)

Thus, is Lipschitz continuous with the Lipschitz constant . Substituting into (2.24), we obtain

(2.31)

It is easy to see that the right-hand side of (2.31) satisfies the Lipschitz condition with the Lipschitz constant

(2.32)

Applying the global existence theorem (see [12]), we see that (2.31), with the initial condition has a unique solution .

Thus, we get the following theorem.

Theorem 2.7.

Let Hypothesis 2.5 and the assumptions on the projector be satisfied. Then, (2.3) with the initial condition

(2.33)

has a unique solution. This solution is expressed by

(2.34)

where is the solution of (2.31) with .

We now describe the solution space of the implicit dynamic equation (2.3). Denote

(2.35)

Lemma 2.8.

There hold the following statements:

(i),

(ii)If for all then .

Proof.

(i) Let , that is, . We have

(2.36)

Hence,

(2.37)

From

(2.38)

it yields

(2.39)

Conversely, suppose that , that is, there exists such that . We have to prove

(2.40)

or equivalently,

(2.41)

Indeed,

(2.42)

where we have already used a result of Lemma 2.3 that is a projector onto . So .

(ii) Let . Then and . Since , we have . This means that . From the assumption , it follows that . The fact implies that . Thus . The lemma is proved.

Remark 2.9.

• (1) By virtue of Lemma 2.8, we find out that the solution space is independent from the choice of projector and operator .

• (2) Since and , the initial condition (2.33) is equivalent to the condition. This implies that the initial condition is not also dependent on choice of projectors.

• (3) Noting that if is a solution of (2.3) with the initial condition (2.33), then for all . Conversely, let and let , , be the solution of (2.3) satisfying the initial condition . We see that . This means that there exists a solution of (2.3) passing .

2.3. Quasilinear Implicit Dynamic Equations

Now we consider a quasilinear implicit dynamic equation of the form

(2.43)

with and assumed to be continuously differentiable in the variable and continuous in .

Suppose that rank for all . We keep all assumptions on the projector and operator stated in Section 2.2.

Equation (2.43) is said to be of index-1 if the matrix

(2.44)

is invertible for every and .

Denote

(2.45)

Further introduce the set

(2.46)

containing all solutions of (2.43). The subspace manifests its geometrical meaning

(2.47)

where is the tangent space of at the point .

Suppose that (2.43) is of index-1. Then, by Lemma 2.2, this condition is equivalent to one of the following conditions:

(1),

(2).

(3)Let be a matrix such that the matrix is invertible (we can choose , e.g.). From the relation

(2.48)

it follows that

(2.49)

is invertible.

Lemma 2.10.

Suppose that the bounded linear operator triplet: is given, where are Banach spaces. Then the operator is invertible if and only if is invertible.

Proof .

See [17, Lemma 1].

By virtue of (2.49) and Lemma 2.10, we get that

(2.50)

Now we come to split (2.43). Multiplying both sides of (2.43) by and , respectively, and putting , we obtain

(2.51)

Consider the function

(2.52)

We see that

(2.53)

where .

Let be a vector satisfying . Paying attention to , we have

(2.54)

Therefore, by (2.50) we get . This means that is an isomorphism of . By the implicit function theorem, equation has a unique solution . Moreover, the function is continuous in and continuously differentiable in . Its derivative is

(2.55)

Then, by substituting into the first equation of (2.51) we come to

(2.56)

It is obvious that the ordinary dynamic equation (2.56) with the initial condition

(2.57)

is locally uniquely solvable and the solution of (2.43) with the initial condition (2.33) can be expressed by .

Now suppose further that satisfies the Lipschitz condition in and we can find a matrix such that

(2.58)

is bounded for all and . Then, the right-hand side of (2.56) also satisfies the Lipschitz condition. Thus, from the global existence theorem (see [12]), (2.56) with the initial condition (2.57) has a unique solution defined on .

Therefore, we have the following theorem.

Theorem 2.11.

Given an index-1 quasilinear implicit dynamic equation (2.43), then there holds the following.

• (1) Equation (2.43) is locally solvable, that is, for any , , there exists a unique solution of (2.43), defined on with some , , satisfying the initial condition (2.33).

• (2) Moreover, if satisfies the Lipschitz condition in and we can find a matrix such that

  

  (2.59)

is bounded, then this solution is defined on and we have the expression

(2.60)

where is the solution of (2.56) with .

Remark 2.12.

• (1) We note that the expression depends only on choosing the matrix .

• (2) The assumption that is bounded for a matrix function seems to be too strong. In Section 3, we show a condition for the global solvability via Lyapunov functions.

• (3) If , there exists satisfying . Hence, . Therefore, by the same argument as in Section 2.2, we can prove that for every , there is a unique solution passing through .

3. Stability Theorems of Implicit Dynamic Equations

For the reason of our purpose, in this section we suppose that is an upper unbounded time scale, that is, . For a fixed , denote .

Consider an implicit dynamic equation of the form

(3.1)

where and .

First, we suppose that for each , (3.1) with the initial condition

(3.2)

has a unique solution defined on . The condition ensuring the existence of a unique solution can be refered to Section 2. We denote the solution with the initial condition (3.2) by . Remember that we look for the solution of (3.1) in the space . Let for all , which follows that (3.1) has the trivial solution .

We mention again that . Noting that if is the solution of (3.1) and (3.2) then for all .

Definition 3.1.

The trivial solution of (3.1) is said to be

(1)-stable (resp., -stable) if, for each and , there exists a positive such that (resp., ) implies for all ,

(2)-uniformly (resp., -uniformly) stable if it is -stable (resp., -stable) and the number mentioned in the part (1). of this definition is independent of ,

(3)-asymptotically (resp., -asymptotically) stable if it is stable and for each , there exist positive such that the inequality (resp., ) implies . If is independent of , then the corresponding stability is -uniformly asymptotically (-uniformly asymptotically) stable,

(4)-uniformly globally asymptotically (resp., -uniformly globally asymptotically) stable if for any there exist functions , such that (resp., ) implies for all and if (resp., ) then for all ,

(5)P-exponentially stable if there is positive constant with such that for every there exists an , the solution of (3.1) with the initial condition satisfies . If the constant can be chosen independent of , then this solution is called -uniformly exponentially stable.

Remark 3.2.

From and , the notions of -stable and -stable as well as -asymptotically stable and -asymptotically stable are equivalent. Therefore, in the following theorems we will omit the prefixes and when talking about stability and asymptotical stability. However, the concept of -uniform stability implies -uniform stability if the matrices are uniformly bounded and -uniform stability implies -uniform stability if the matrices are uniformly bounded.

Denote

(3.3)

and is the domain of definition of .

Proposition 3.3.

The trivial solution of (3.1) is -uniformly (resp., -uniformly) stable if and only if there exists a function such that for each and any solution of (3.1) the inequality

(3.4)

holds, provided (resp., ).

Proof.

We only need to prove the proposition for the -uniformly stable case.

Sufficiency. Suppose there exists a function satisfying (3.4) for each ; we take such that , that is, . If is an arbitrary solution of (3.1) and , then , for all .

Necessity. Suppose that the trivial solution of (3.1) is -uniformly stable, that is, for each there exists such that for each the inequality implies , for all . For the sake of simplicity in computation, we choose . Denote

(3.5)

It is clear that is an increasing positive function in . Further, and by definition, there holds

(3.6)

By putting

(3.7)

it is seen that

(3.8)

Let be the inverse function of . It is clear that also belongs to .

For , we denote . If , then by (remember that does not imply that ). Consider the case where . If , then by the relations (3.6) and (3.8) we have . In particular, which is a contradiction. Thus , this implies , provided .

The proposition is proved.

Similarly, we have the following proposition.

Proposition 3.4.

The trivial solution of (3.1) is -stable (resp., -stable) if and only if for each and any solution of (3.1) there exists a function such that there holds the following:

(3.9)

provided (resp., ).

In order to use the Lyapunov function technique related to (3.1), we suppose that . By using (2.3), we can define the derivative of the function along every solution curve as follows:

(3.10)

Remark 3.5.

Note that when the function is independent of and even if the vector field associated with the implicit dynamic equation (3.1) is autonomous, the derivative may depend on .

Theorem 3.6.

Assume that there exist a constant and a function being rd-continuous and a function , defined on satisfying

(1) for all and ,

(2), for any and .

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 .

Proof.

Denote

(3.11)

By the condition (2), we have

(3.12)

Therefore, for all

(3.13)

From the condition (1), it follows that

(3.14)

or

(3.15)

The last inequality says that the solution can be lengthened on , that is, (3.1) is globally solvable.

Theorem 3.7.

Assume that there exist a function being rd-continuous and a function , defined on satisfying the conditions

(1) for all ,

(2) for all and ,

(3) for any and .

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 (2) and (3), it follows that (3.1) is globally solvable. Suppose on the contrary that the trivial solution of (3.1) is not stable. Then, there exists an such that for all there exists a solution of (3.1) satisfying and for some . Put .

By the assumption that and is rd-continuous, we can find such that if then . With given , let be a solution of (3.1) such that and for some .

Since and by the condition (3),

(3.16)

Therefore, . Further, and by the condition (2) we have . This is a contradiction. The theorem is proved.

Theorem 3.8.

Assume that there exist a function being rd-continuous and functions , defined on , such that

(3.17)

satisfying the conditions

(1) uniformly in ,

(2) for all and ,

(3) for any and .

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 (2) and (3), it implies that (3.1) is globally solvable.

And since , the trivial solution of (3.1) is stable by Theorem 3.7. Consider a bounded solution of (3.1). First, we show that Assume on the contrary that . From the condition (1), it follows that By the condition (3), we have

(3.18)

as , which gets a contradiction.

Thus, Further, from the condition (3) for any we get

(3.19)

This means that is a decreasing function. Consequently,

(3.20)

which follows that by the condition (2).

Theorem 3.9.

Suppose that there exist a function , defined on , and a function such that

(1) uniformly in and for all and ,

(2) for any and .

Assume further that (3.1) is locally solvable. Then, the trivial solution of (3.1) is -uniformly stable.

Proof.

The proof is similar to the one of Theorem 3.7 with a remark that since uniformly in , we can find such that if then

The proof is complete.

Remark 3.10.

The conclusion of Theorem 3.9 is still true if the condition (1) is replaced by "there exist two functions,defined onand a functionsuch thatfor alland".

We present a theorem of uniform global asymptotical stability.

Theorem 3.11.

If there exist functions , defined on , and a function satisfying

(1) for all and ,

(2) for any and .

Assume further that (3.1) is locally solvable. Then, the trivial solution of (3.1) is -uniformly globally asymptotically stable.

Proof.

Let be given. Define and

(3.21)

( is not necessary in ).

Let be a solution of (3.1) with . From the condition (2), we see that

(3.22)

Therefore,

(3.23)

Hence, for all .

Because the trivial solution of (3.1) is -uniformly stable, we only need to show that there exists a such that . Assume that such a does not exist, that is for all . From the condition (2), we get

(3.24)

Since ,

(3.25)

which contradicts the definition of in (3.21). The proof is complete.

When is not differentiable, one supposes that there exists a -differentiable projector onto and is rd-continuous on ; moreover, for all . Let .

We choose matrix functions such that is an isomorphism between and and the matrix is invertible. Define

(3.26)

where (see (2.51)).

From now on we remain following the above assumptions on the operators whenever is mentioned.

By the same argument as Theorem 3.6, we have the following theorem.

Theorem 3.12.

Assume that there exist a constant and a function being rd-continuous and a function , defined on satisfying

(1) for all and ,

(2), for any and .

Assume further that (3.1) is locally solvable. Then, (3.1) is globally solvable.

Theorem 3.13.

Assume that (3.1) is locally solvable. Then, the trivial solution of (3.1) is stable if there exist a function being rd-continuous and a function , defined on such that

(1) for all ,

(2) for all and ,

(3) for all and .

Proof.

Assume that there is a function satisfying the assertions (1), (2), and (3) but the trivial solution of (3.1) is not stable. Then, there exist a positive and a such that ; there exists a solution of (3.1) satisfying and , for some . Let . Since , it is possible to find a satisfying when . Consider the solution satisfying and for a .

From the assumption (3), it follows that

(3.27)

This implies

(3.28)

We get a contradiction because when .

The proof of the theorem is complete.

Theorem 3.14.

Assume that (3.1) is locally solvable. If there exist two functions , defined on and a function being rd-continuous such that

(1) for all and ,

(2) for all and ,

then the trivial solution of (3.1) is -uniformly stable.

Proof.

The proof is similar to the one of Theorem 3.9.

Theorem 3.15.

If there exist functions , defined on and a function satisfying

(1) for all and ,

(2) for any and .

Assume further that (3.1) is locally solvable. Then, the trivial solution of (3.1) is -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 satisfying the assertions in the above theorems. However, if the structure of the time scale is rather simple we have the following theorem.

Theorem 3.16.

Suppose that contains no right-dense points and the trivial solution of (3.1) is -uniformly stable. Then, there exists a function being rd-continuous satisfying the conditions (1), (2), and (3) of Theorem 3.13, where is an open neighborhood of 0 in .

Proof.

Suppose the trivial solution of (3.1) is -uniformly stable. Due to Proposition 3.3, there exist functions such that for any solution of (3.1), we have

(3.29)

provided .

Let and . For any satisfying and , we put

(3.30)

where is the unique solution of (3.1) satisfying the initial condition . It is seen that is defined for all satisfying , , and .

Let . By the definition, . From (2.60), for all . In particular, . Thus, . Hence, we have the assertion (2) of the theorem.

Due to the unique solvability of (3.1), we have with . Therefore, and

(3.31)

This implies

(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)

(3.33)

where and are constant matrices with ind , , and satisfing the Lipschitz condition

(3.34)

where is sufficiently small. Let be defined by (2.9) with and . By Theorem 2.7, we see that there exists a unique solution satisfying the condition for any .

Besides, also consider the homogeneous equation associated to (3.33)

(3.35)

and suppose this equation has index-1. As in Section 2, multiplying (3.33) by we get

(3.36)

where .

Note that the general solution of (3.35) is

(3.37)

in there is denoted the set of right-scattered points of the interval .

Denote . It is easy to show that the trivial solution of (3.35) is -uniformly exponentially stable if and only if , where is the domain of uniform exponential stability of . On the exponential stable domain of a time scale, we can refer to [10, 18, 19]. By the definition of exponential stability, it implies that the graininess function of the time scale is upper bounded. Let .

We denote the set

(3.38)

and suppose . Since , this condition implies that (3.35) is -uniformly exponentially stable.

If , define

(3.39)

where the matrix is supposed to be symmetric positive definite. It is clear that is symmetric positive definite.

Since , the above series is convergent. Further, for any we have

(3.40)

Thus,

(3.41)

Letting and paying attention to , we obtain

(3.42)

In the case where and is symmetric positive definite, by putting

(3.43)

we can examine easily that the matrix also satisfies (3.42), is symmetric and positive definite.

Theorem 3.17.

Suppose that and the homogeneous equation (3.35) is of index-1 and the constant is sufficiently small. Then, the trivial solution of (3.33) is -uniformly globally asymptotically stable.

Proof.

Let be a symmetric and positive definite (constant) matrix satisfying (3.42). Consider the Lyapunov function . The derivative of along the solution of (3.33) is

(3.44)

From the Lipschitz condition and (2.25), it is seen that where . Therefore,

(3.45)

Combining this inequality and the above appreciation, we see that when is sufficiently small there exists such that

(3.46)

By Theorem 3.15, (3.33) is -uniformly globally asymptotically stable.

Example 3.18.

Let and consider

(3.47)

with

(3.48)

We have span, rank for all . It is easy to verify that is the canonical projector onto , . Let us choose . We see that

(3.49)

Since , , (3.47) has index-1.

It is obvious that . Further, . Thus, according to Theorem 2.7 for each , (3.47) with the initial condition has the unique solution.

It is easy to compute, , , , where , and .

Therefore, satisfies . Moreover, we have

(3.50)

Let the Lyapunov function be .

Put , we have and

(3.51)

Hence,

(3.52)

We have for any solution of (3.47) and (noting that ),

(+) if is right-scattered then ,

(+)if is right-dense then , where .

In both two cases, we have , so the trivial solution of (3.47) is -uniformly stable by Theorem 3.14.

Note that if we let then the result is still true. Indeed, by the simple calculations we obtain

(a), where defined by ,

(b). Thus,

(3.53)

Therefore, having the above result is obvious.

4. 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 Section 3 of this paper is still an open problem for an arbitrary time scale meanwhile it is true for discrete and continuous time scales.