Existence of Exponential p-Stability Nonconstant Equilibrium of Markovian Jumping Nonlinear Diffusion Equations via Ekeland Variational Principle

The authors obtained a delay-dependent exponential p-stability criterion for a class of Markovian jumping nonlinear diffusion equations by employing the Lyapunov stability theory and some variational methods. As far as we know, it is the first time to apply Ekeland variational principle to obtain the existence of exponential stability equilibrium of p-Laplacian dynamic system so that some methods used in this paper are different from those methods of many previous related literatures. In addition, the obtained existence criterion is only involved in the activation functions so that the criterion is simpler and easier than other existence criteria to be verified in practical application. Moreover, a numerical example shows the effectiveness of the proposed methods owing to the large allowable variation range of time-delay.


Introduction
Nonlinear diffusion equations have been investigated extensively by many authors owing to their physics and biological engineering backgrounds, population dynamics, and so on (see [1][2][3][4][5][6][7][8][9] and references therein).In addition, Markovian jumping systems have attracted rapidly growing interest due to the fact that Markovian jumping parameters are useful in modeling abrupt phenomena, such as random failures, operating in different points of a nonlinear plant, and changing in the interconnections of subsystems (see [8,[10][11][12] and references therein).On the other hand, fuzzy logic theory has shown to be an appealing and efficient approach to deal with the analysis and synthesis problems for complex nonlinear systems.Among various kinds of fuzzy methods, Takagi-Sugeno (T-S) fuzzy models provide a successful method to describe certain complex nonlinear system using some local linear subsystems (see [13][14][15] and references therein).As pointed out in the above related literature, the Markovian jumping T-S fuzzy mathematical models have always found their extensive applications in the real world.However, almost all the applications are greatly dependent on the stability of systems, which can often come down to the stability of the equilibrium solution for the corresponding mathematical models.So in this paper, we may consider the stability of the nonlinear -Laplace ( > 1) diffusion fuzzy equations with Markovian jumping.Note that when  = 2, the so-called reaction-diffusion equations have been widely investigated (see [16][17][18][19][20] and references therein).For example, under Dirichlet boundary conditions, the existence result of -stability equilibrium solution in the sense of  2 norm for a class of time-delay reaction-diffusion equations was obtained in [16].Motivated by the abovementioned literature, in this paper, we will synthetically employ Ekeland variational principle, Sobolev imbedding theorem, and the Lyapunov functional method to study the existence of exponential -stability nonconstant equilibrium solution in the sense of   norm ( > 1) for delayed Markovian jumping fuzzy equations with nonlinear -Laplace diffusion ( > 1).
Remark 1.Our methods employed in this paper are different from those of previous related literature.For example, homomorphic mapping theory was employed to obtain the existence of equilibrium of ordinary differential equations in [22]; topological degree theory was used to obtain the existence of equilibrium for fuzzy ordinary differential equations in [23] and of equilibrium for reaction-diffusion partial differential equations in [17].In this paper, Ekeland variational principle is originally proposed to solve the existence of nonconstant equilibrium for nonlinear diffusion equations.Note that the abovementioned constant equilibrium point  * = ( * 1 , . . .,  *  )  can actually be regarded as the special case of our nonconstant equilibrium point  =  * () = ( * 1 (), . . .,  *  ())  with  *  () ≡ constant for  = 1, 2, . . ., .In addition, our criterion about existence is only involved in the activation functions while other more parameters need be considered in the proof of the existence of constant equilibrium point in those previous literatures (see Remark 8 below for details).

Preparation
Throughout this paper, we assume the following: (A1) There exists a positive definition matrix  = diag( has a convergent subsequence.By the way, the above sequence {  } with (  ) →  and   (  ) → 0 is called the (PS)  sequence of  for a given  ∈ .
The following lemma originated from the famous Sobolev imbedding theorem.

Lemma 4.
Let Ω be a bounded subset in   with smooth boundary Ω.For 1 <  < , there exist the corresponding positive constants  1 and   such that, for any  ∈ where the Sobolev space In 1979, Ekeland proposed the following famous Ekeland variational principle and its proof in [24].As is well known, Ekeland variational principle has been the most important result in nonlinear analysis and has been applied to optimization theory, control theory, economic equilibrium theory, critical point theory, dynamic systems, and so forth.In this paper, we also need the following Ekeland variational principle.

Lemma 5 (Ekeland variational principle [24, Theorem 1]).
Let  be a complete metric space and let  :  → (−∞, +∞] be a lower semicontinuous function, bounded from below and not identical to +∞.Let  > 0 be given and let  ∈  be such that Then there exists V ∈  such that and, for each

Main Result
Theorem 6.Let  > 1. Assume that there exists a positive scalar  =  1 / 2 with 1 <  <  such that where  1 is an odd number and so is  2 .Assume, in addition, If there exist a sequence of positive scalars ℎ  (()) ( ∈ ,  ∈ N) such that where then there exists a nonconstant equilibrium solution for PDEs (4), which is stochastically global exponential -stability about   norm.
Proof.The following proof may be divided into two big steps.
Step 1. Firstly, we need prove that there exists a nonconstant equilibrium solution for (4).Consider the functional where It is obvious that   ∈  1 ( Similarly, we can also deduce that where Thereby, we have Since    () > 0, we know that   () >   (0) = 0 if  > 0 and   () <   (0) = 0 if  < 0. And hence B  () = ∫  0   () ⩾ 0 for all  ∈ .Besides, we know from the above analysis and the Sobolev imbedding theorem (Lemma 4) that there exist positive constants  1 = Denote () = (1/)D    − 2   − 1 .Owing to  >  > 1, there exists a large enough constant  0 > 0 such that () > 0 for all || >  0 .And hence which implies that   is bounded below.And the infimum may be defined as   = inf    , where we denote  =  1, 0 (Ω) for convenience and denote by  * its dual space.Define the operators A  , B  :  →  * as follows: where where   is a constant.we can conclude from (13) Owing to the boundedness of {  (  )}, it is not difficult to prove by the application of reduction to absurdity that {  } must be bounded in Then  is a complete metric space with the above metric.
From the continuity of   and the above analysis, we know that  :  Step 2. Below, we will prove the exponential -stability for the equilibrium point  * .
Consider the Lyapunov-Krasovskii functional as where Let L be the weak infinitesimal operator; then for any given mode () =  ∈ , taking the derivative of  1 (, ) with respect to  along the trajectory of (4) yields Next, we claim that To verify (40), we have to prove firstly the following proposition by the Yang inequality.Proposition 7.For ,  ∈ which proves (40).
In addition, we get by (A1) Further, we can derive by (A2) ).Now, we can conclude from Definition 2 that the nonconstant equilibrium solution of ( 4) is stochastically exponentially -stable about   norm.And that completes the proof of Theorem 6.
Remark 8.In [27], existence theorems of stochastic differential equations on  ∈ [ 0 , ) were given under some conditions on activation functions, where  > 0 is a constant.And in [28,29], existence theorems of stochastic differential equations were presented under some conditions on function  ∈  1,2 ([ 0 − , ) ×   ;  + ).Motivated by [27], we proposed some conditions on activation functions to set up existence criterion for the equilibrium solution of system (4).In [22,23], the constant equilibrium solution  =  * for all  ∈ [ 0 , +∞) was obtained by homomorphic mapping theory and matrix theory, or matrix theory and homotopy invariance theorem, where  * = ( * 1 ,  * 2 , . . .,  *  , . . .,  *  ), and each  *  is a constant.In this paper, we also need to consider the equilibrium solution of (4) defined on [ 0 , +∞).Different from [22,23], we consider the nonconstant equilibrium solution  =  * () = ( * 1 (),  * 2 (), . . .,  *  (), . . .,  *  ()) for all  ∈ [ 0 , +∞).This equilibrium solution is a solution for a nonlinear -Laplacian elliptic partial differential equation whose space frame may be considered as infinite dimension function space  1, 0 (Ω).And variational method is always a powerful tool to solve the problem.Although the variational method is more complicated than homomorphic mapping method, -matrix method, or homotopy invariance theorem, our criterion about existence is only involved in the activation functions (remark: condition ( 14) is not used in the proof of existence) and hence is simpler and more effective than other criteria, such as -matrix criteria and LMI-based criteria, because LMI-based criteria or -matrix criteria always involve the computer MATLAB programming in practical application while our condition ( 13) is easy to verify.So our existence criterion is actually simpler and more effective than LMI-based criteria and other criteria, which is the main contribution in this paper.Remark 9. LMI-based stability criteria or -matrix stability criteria are always proposed in many literatures related to the mean square stability (see, e.g., [30][31][32][33] and references therein).However, when  > 1 and  ̸ = 2, -stability criteria always involve more complicated mathematical method and mathematical deduction.For example, the stability criteria in [34] are not simpler than our stability criterion in Theorem 6.Similar phenomena exposed in many literatures related to stability (see [15,[34][35][36][37][38]). Besides, the nonlinear -Laplacian ( > 1) operator produces great difficulties in -stability proof.However, our condition ( 14) is still a LMI condition, which can be computed and verified by computer MATLAB LMI Toolbox in practical application.
This work was supported by the National Basic Research Program of China (2010CB732501), by the Scientific Research Fund of Science Technology Department of Sichuan Province (2012JY010), and by Sichuan Educational Committee Science Foundation (08ZB002, 12ZB349, and 14ZA0274).