Method of Lyapunov functionals construction in stability of delay evolution equations

Transcription

1 J. Math. Anal. Appl ) Metho of Lyapunov functionals construction in stability of elay evolution equations T. Caraballo a,1, J. Real a,1, L. Shaikhet b, a Departamento e Ecuaciones Diferenciales y Análisis Numérico, Universia e Sevilla, Apo. e Correos 1160, Sevilla, Spain b Department of Higher Mathematics, Donetsk State University of Management, Chelyuskintsev str., 163-a, Donetsk 83015, Ukraine Receive 1 August 005 Available online 3 January 007 Submitte by V. Raulescu Abstract The investigation of stability for hereitary systems is often relate to the construction of Lyapunov functionals. The general metho of Lyapunov functionals construction which was propose by V. Kolmanovskii an L. Shaikhet an successfully use alreay for functional ifferential equations, for ifference equations with iscrete time, for ifference equations with continuous time, is use here to investigate the stability of elay evolution equations, in particular, partial ifferential equations. 007 Elsevier Inc. All rights reserve. Keywors: Metho of Lyapunov functionals construction; Evolution equations; Stability; Partial ifferential equations; D Navier Stokes moel with elays 1. Introuction The stuy of functional ifferential equations is motivate by the fact that when one wants to moel some evolution phenomena arising in Physics, Biology, Engineering, etc., some hereitary characteristics such as aftereffect, time lag an time elay can appear in the variables. * Corresponing author. aresses: caraball@us.es T. Caraballo), jreal@us.es J. Real), leoni@sum.eu.ua, leoni.shaikhet@usa.net L. Shaikhet). 1 T. Caraballo an J. Real have been partly supporte by Ministerio e Eucación y Ciencia Spain) an FEDER European Community) grant MTM X/$ see front matter 007 Elsevier Inc. All rights reserve. oi: /j.jmaa

2 T. Caraballo et al. / J. Math. Anal. Appl ) Typical examples arise from the researches of materials with thermal memory, biochemical reactions, population moels, etc. see, for instance, 1 5,9,11 15,18,3 6,3 37] an references therein). On the other han, one important an interesting problem in the analysis of functional ifferential equations is the stability, the theory of which has been greatly evelope over the last years. There exist many works ealing with the construction of Lyapunov functionals for a wie range of equations containing some kin of hereitary properties. As it is well known, in the case without any hereitary features, Lyapunov s technique is available to obtain sufficient conitions for the stability of solutions of partial) ifferential equations. However, in the case of ifferential equations with hereitary properties, for instance, even in the case of constant time elays, Lyapunov s metho becomes ifficult to apply effectively as N.N. Krasovskii 1] pointe out. The main reason is that it is much more ifficult or even impossible in some cases) to construct proper Lyapunov functions or functionals) for functional ifferential equations than for those without any hereitary characteristics. Our interest in this paper is to investigate the stability of ynamical systems moelle by elay evolution equations, in particular, by partial ifferential equations with elays, using the general metho of Lyapunov functionals construction that was propose by V. Kolmanovskii an L. Shaikhet an successfully use alreay for functional ifferential equations, for ifference equations with iscrete time, an for ifference equations with continuous time 16,17,19,0, 7 31]. Taking into account that many interesting problems from applications have main operators which satisfy some kin of coercivity assumption, we will exploit this iea here an will be intereste in this class of operators Notations an efinitions Let U an H be two real separable Hilbert spaces such that U H H U, where the injections are continuous an ense. Let, an be the norms in U, H an U respectively,, )) an, ) be the scalar proucts in U an H respectively, an, the uality prouct between U an U. We assume that u β u, u U. 1.1) Let C h, 0,H) be the Banach space of all continuous functions from h, 0] to H, x t C h, 0,H) for each t 0, ), be the function efine by x t s) = xt + s) for all s h, 0]. The space C h, 0,U)is similarly efine. Let At, ) : U U, f 1 t, ) : C h, 0,H) U an f t, ) : C h, 0,U) U be three families of nonlinear operators efine for t>0, At, 0) = 0, f 1 t, 0) = 0, f t, 0) = 0. Consier the equation ut) = A t,ut) ) + f 1 t, u t ) + f t, u t ), t > 0, t us) = ψs), s h, 0]. 1.) Let us enote by u ; ψ) the solution of Eq. 1.) corresponing to the initial conition ψ. Definition 1.1. The trivial solution of Eq. 1.) is sai to be stable if for any ɛ>0 there exists δ>0 such that ut; ψ) <ɛfor all t 0, if ψ CH = sup s h,0] ψs) <δ.

3 113 T. Caraballo et al. / J. Math. Anal. Appl ) Definition 1.. The trivial solution of Eq. 1.) is sai to be exponentially stable if it is stable an there exists a positive constant λ such that for any ψ C h, 0,U) there exists C which may epen on ψ) such that ut; ψ) Ce λt for t> Lyapunov type stability theorem Let us now prove a theorem which will be crucial in our stability investigation. Theorem 1.1. Assume that there exists a functional Vt,u t ) such that the following conitions hol for some positive numbers c 1, c an λ: Vt,u t ) c 1 e λt ut), t 0, 1.3) V0,u 0 ) c ψ C H, t Vt,u t) 0, t 0. Then the trivial solution of Eq. 1.) is exponentially stable. 1.4) 1.5) Proof. Integrating 1.5) we obtain Vt,u t ) V0,u 0 ). From here an 1.3), 1.4) it follows c 1 ut) e λt V0,u 0 ) c ψ C H. The inequality c 1 ut) c ψ C H means that the trivial solution of Eq. 1.) is stable. Besies, from the inequality c 1 ut) e λt V0,u 0 ), it follows that the trivial solution of Eq. 1.) is exponentially stable. Note that Theorem 1.1 implies that the stability investigation of Eq. 1.) can be reuce to the construction of appropriate Lyapunov functionals. A formal proceure to construct Lyapunov functionals is escribe below Proceure of Lyapunov functionals construction The proceure consists of four steps. Step 1. To transform Eq. 1.) into the form zt,u t ) ) = A 1 t,ut) + A t, u t ) 1.6) t where zt, ) an A t, ) are families of nonlinear operators, zt, 0) = 0, A t, 0) = 0, operator A 1 t, ) only epens on t an ut), but oes not epen on the previous values ut + s), s<0. Step. Assume that the trivial solution of the auxiliary equation without memory yt) ) = A 1 t,yt) t 1.7) t is exponentially stable an therefore there exists a Lyapunov function vt,yt)), which satisfies the conitions of Theorem 1.1.

4 T. Caraballo et al. / J. Math. Anal. Appl ) Step 3. A Lyapunov functional Vt,u t ) for Eq. 1.6) is constructe in the form V = V 1 + V, where V 1 t, u t ) = vt,zt,u t )). Here the argument y of the function vt,y) is replace on the functional zt, x t ) from the left-han part of Eq. 1.6). Step 4. Usually, the functional V 1 t, u t ) almost satisfies the conitions of Theorem 1.1. In orer to fully satisfy these conitions, it is necessary to calculate t V 1t, u t ) an estimate it. Then, the aitional functional V t, u t ) can be chosen in a stanar way. Note that the representation 1.6) is not unique. This fact allows, using ifferent representations type of 1.6) or ifferent ways of estimating t V 1t, u t ), to construct ifferent Lyapunov functionals an, as a result, to get ifferent sufficient conitions of exponential stability.. Construction of Lyapunov functionals for equations with time-varying elay Consier the following evolution equation ut) = A t,ut) ) + F u t ht) )),.1) t where At, ), F : U U are appropriate partial ifferential operators see conitions below), which is a particular case of Eq. 1.). We will apply the metho escribe above to construct Lyapunov functionals for Eq..1), an, as a consequence, to obtain sufficient conitions ensuring the stability of the trivial solution. We will use two ifferent constructions which will provie ifferent stability regions for the parameters involve in the problem..1. The first way of Lyapunov functionals construction First we consier a quite general situation for the operators involve in Eq..1). Theorem.1. Assume that operators in Eq..1) satisfy the conitions: At, u), u γ u, γ >0, F : U U, Fu) α u, u U,.) an If ht) 0,h 0 ], ḣt) h 1 < 1..3) γ> α 1 h1, then the trivial solution of Eq..1) is exponentially stable. Proof. Owing to the proceure of Lyapunov functionals construction, let us consier the auxiliary equation without memory t yt) = A t,yt) )..5).4)

5 1134 T. Caraballo et al. / J. Math. Anal. Appl ) The function vt,y) = e λt y, λ>0, is a Lyapunov function for Eq..5), i.e. it satisfies the conitions of Theorem 1.1. Actually, it is easy to see that conitions 1.3), 1.4) hol for the function vt,yt)). Besies, since γ>0, there exists λ>0 such that γ >λβ. Using.5), 1.1) an.), we obtain t v t,yt) ) = λe λt yt) + e λt A t,yt) ),yt) e λt γ λβ ) yt) 0. Accoring to the proceure, we now construct a Lyapunov functional V for Eq..1) in the form V = V 1 + V, where V 1 t, u t ) = e λt ut). For Eq..1) we obtain t V 1t, u t ) = λv 1 t, u t ) + e λt A t,ut) ) + F u t ht) )),ut) Set now V t, u t ) = ɛα e λt λ ut) + γ ut) + α u t ht) ) ut) )] e λβ λt ut) γ ut) + α ɛ u t ht) ) + 1 ut) )] ɛ = e λt λβ γ + α ) ut) ) + ɛαu t ht) ]. ɛ e λs+h 0) us) s. Then t V t, u t ) = ɛα e λt+h 0 ) ut) 1 ḣt) ) e λ+h 0) u t ht) ) ) Thus, for V = V 1 + V we have t Vt,u t) ɛαeλt e λh 0 ut) )e λh 0 ht)) u t ht) ) ) ɛαe λt e λh 0 ut) u t ht) ) λβ γ + α 1 ). )] ɛ + ɛeλh0 e λt ut). Rewrite the expression in square brackets as 1 γ + α ɛ + ɛ ) + λβ + ɛα eλh 0 1. To minimize this expression in the brackets, choose ɛ =. As a consequence we obtain ) ] t Vt,u α t) γ ρλ) e λt ut).6) 1 h1 with ρλ) = λβ + α eλh h1.

6 T. Caraballo et al. / J. Math. Anal. Appl ) Since ρ0) = 0, then by conition.4) there exists λ>0 small enough such that ) α γ ρλ). 1 h1 From here an.6) it follows that t Vt,u t) 0. So, the functional Vt,u t ) constructe above satisfies the conitions in Theorem 1.1. This means that the trivial solution of Eq..1) is exponentially stable. Note, in particular, if ht) h 0 then h 1 = 0 an conition.4) takes the form γ>α... The secon way of Lyapunov functionals construction We now establish a secon result which implies that the operator F must be less general than in Theorem.1. However, as we will show later in the applications section, the stability regions provie by this theorem will be better than the ones given by Theorem.1. Theorem.. Suppose that operators in Eq..1) satisfy the following conitions: At, u) + Fu),u γ u, γ >0, an If At, u) + Fu) α 1 u, F : U U, ht) 0,h 0 ], ḣt) h 1 < 1, Fu) α u, u U,.7) ḣt) h..8) α h γ>α 1 α h α h 0 ),.9) 1 h1 then the trivial solution of Eq..1) is exponentially stable. Proof. To use the proceure of Lyapunov functionals construction, let us first transform Eq..1) as t zt, u t) = A t,ut) ) + F ut) ) + ḣt)f u t ht) )),.10) where zt, u t ) = ut) + F us) ) s..11) Consier the auxiliary equation without memory in the form t yt) = A t,yt) ) + F yt) )..1) The function vt,y) = e λt y is a Lyapunov function for Eq..1). Actually, since γ>0then there exists λ>0 such that γ >λβ. Using.1), 1.1),.7), we obtain

7 1136 T. Caraballo et al. / J. Math. Anal. Appl ) t v t,yt) ) = λe λt yt) + e λt A t,yt) ) + F yt) ),yt) e λt γ λβ ) yt). Next, we construct a Lyapunov functional V for Eqs..10),.11) in the form V = V 1 + V, where V 1 t, u t ) = e λt zt, ut ),.13) an zt, u t ) is efine by.11). Using.7) for Eqs..10) an.11) we have t V 1t, u t ) = λv 1 t, u t ) + e λt A t,ut) ) + F ut) ) + ḣt)f u t ht) )),zt,u t ) = λv 1 t, u t ) + e λt A t,ut) ) + F ut) ) + ḣt)f u t ht) )), ut) + F us) ) s = λv 1 t, u t ) + e λt A t,ut) ) + F ut) ),ut)+ + e λt ḣt) F u t ht) )),ut)+ λv 1 t, u t ) + e λt γ ut) + α 1 α + e λt ) ḣt) α u t ht) ut) + α λv 1 t, u t ) + e λt γ ut) + α 1 α F us) ) s F us) ) s ) ut) us) ] s + e λt ) ) ḣt) α ɛ u t ht) 1 + ut) ɛ + α ) ] ) ɛ 3 u t ht) 1 + us) s ɛ 3 ) ) u t ht) us) s 1 ut) + ɛ 1 us) ) ] s ɛ 1 = λv 1 t, u t ) + e λt γ + 1 ɛ 1 α 1 α ht) + 1 ɛ α ḣt) ) ut) + α ɛ + ɛ 3 α ht) ) ḣt) u t ht) )

8 From.13) an.10) it follows e λt V 1 t, u t ) = ut) + T. Caraballo et al. / J. Math. Anal. Appl ) α ɛ 1 α ɛ 3 α ḣt) ) ut) + t ut) + α β + α ht)β t us) s ]. )) t ut), F us) s + ut) F us) ) s + ht) F us) ) s ) ɛ 4 ut) 1 + us) s ɛ 4 us) s 1 + ɛ 4 α ht) ) β ut) + α β 1 ɛ 4 + α ht) F us) ) s ) t us) s. Therefore t V 1t, u t ) e λt λβ 1 + ɛ 4 α ht) ) γ + 1 α 1 α ht) + 1 ] ut) α ḣt) ɛ 1 ɛ Put now + e λt α ɛ + ɛ 3 α ht) ) ḣt) u t ht) ) + e λt α ɛ 1 α 1 + α )] t ḣt) + λβ 1 + α ht) ɛ 3 ɛ 4 e λt λβ 1 + ɛ 4 α h 0 ) γ + 1 ɛ 1 α 1 α h ɛ α h ] ut) + e λt ɛ + ɛ 3 α h 0 )α h u t ht) ) + e λt α ɛ 1 α 1 + α )] t 1 h + λβ + α h 0 ɛ 3 ɛ 4 V t, u t ) = ɛ + ɛ 3 α h 0 )α h + α ɛ 1 α 1 + α 1 h + λβ ɛ 3 e λs+h 0) us) s ɛ 4 + α h 0 )] t t h 0 us) s us) s. t h 0 e λs+h0) s t + h 0 ) us) s.

9 1138 T. Caraballo et al. / J. Math. Anal. Appl ) Then e λt+h 0 ) t V t, u t ) = ɛ + ɛ 3 α h 0 )α h ut) e λ+h 0) u t ht) ) + α ɛ 1 α 1 + α )] 1 h + λβ + α h 0 ɛ 3 ɛ 4 t e λt+h0) h 0 ut) t h 0 e λs+h0) us) s ]. Since e λt e λs+h0) for s t h 0, then for V = V 1 + V, we obtain t Vt,u t) e λt λβ 1 + ɛ 4 α h 0 ) γ + 1 α 1 α h 0 ɛ ] α h + α h ɛ + ɛ 3 α h 0 ) eλh 0 ut) ɛ + e λt+h0) α h 0 ɛ 1 α )] 1 α h + λβ + α h 0 ut) ɛ 3 ɛ 4 ] = e λt λβ 1 + ɛ 4 α h 0 ) γ + 1 ɛ 1 α 1 α h ɛ α h + α h ɛ + ɛ 3 α h 0 ) eλh 0 1 h 1 + e λh 0 α h 0 ɛ 1 α )]] 1 α h + λβ + α h 0 ut) ɛ 3 ɛ 4 ) 1 = γ + α 1 α h 0 + ɛ 1 + α h + 1ɛ1 ɛ ) ɛ 1 + α h 0h + ɛ ) ] 3 + ρλ) e λt ut),.14) ɛ 3 where )] ρλ) = λ β 1 + ɛ 4 α h 0 ) + e λh 0 α h 0 β + α h 0 1ɛ4 + e λh 0 1 ) α h 0 ɛ 1 α 1 + α ) h + ɛ ] + ɛ 3 α h 0 )α h..15) ɛ 3 To minimize the right-han sie of inequality.14) we choose ɛ 1 = 1, ɛ = ɛ 3 =. Then, inequality.14) takes the form ) ] t Vt,u α h t) γ α 1 α h α h 0 ) ρλ) e λt ut)..16) 1 h1 From.15) it follows that ρ0) = 0. Thus, there exists λ>0 small enough such that from conition.7) we euce that ) α h γ α 1 α h α h 0 ) ρλ)..17) 1 h1

10 T. Caraballo et al. / J. Math. Anal. Appl ) This an.16) imply that t Vt,u t) 0 an, as a consequence, the functional Vt,u t ) constructe above satisfies conitions 1.4), 1.5). However, we cannot ensure that Theorem 1.1 hols true since the functional Vt,u t ) oes not satisfy the conition 1.3). Then, we will procee in a ifferent way. From.16),.17) it follows that there exists c>0 such that Therefore, Vt,u t ) V0,u 0 ) c 0 0 e λs us) s. e λs us) V0,u 0 ) s, Vt,u t ) V0,u 0 )..18) c Note also that zt, ut ) = ut) + ut) F us) ) s ut) α β From.7) it follows that ut) F us) ) s ut) us) s ut) α ut) ht) + β 1 α h 0 ) ut) α β t h 0 us) ) s us) s..19) γ u At, u) + Fu),u At, u) + Fu) u α 1 u, i.e. γ α 1.Using.9)wehaveα h 0 <γα So, from.19) we obtain ut) ut) + Fus))s + α β t h 0 us) s..0) 1 α h 0 Since V0,u 0 ) Vt,u t ) V 1 t, u t ) = e λt ut) + F us) ) s,

11 1140 T. Caraballo et al. / J. Math. Anal. Appl ) then ut) + F us) ) s e λt V0,u 0 )..1) It is easy to see that there exists C>0 such that V0,u 0 ) C u 0. Now, from.19).1) it follows that ut) K u 0, K = C + α h 0 + C c ). 1 α h 0 Therefore, the trivial solution of Eq..1) is stable. Thanks to.18) we have that there exists C 1 > 0 such that Hence, e λt h 0) t h 0 t h 0 us) s t h 0 e λs us) s h 0 e λs us) s C 1. us) s C 1 e λh 0 e λt.) an from.0).) it follows that, by conitions.7).9), the trivial solution of Eq..1) is exponentially stable. Note that if, in particular, ht) = h 0, then h 1 = h = 0 an conition.9) takes the form γ>α 1 α h Some applications In this section we will show some interesting applications to illustrate how our results work Application to a D Navier Stokes moel We first consier a D Navier Stokes moel with elays. Although this moel has alreay been analyse in etails in 6 8], there are some situations which still have not been consiere in those works. We aim to provie some aitional results on this moel as well as to improve some sufficient conitions establishe in 7] by applying Theorem.1. Let Ω R be an open an boune set with regular bounary Γ, T>0 given, an consier the following functional Navier Stokes problem: u t ν u + u u i = p + gt,u t ) x i i=1 in 0,T) Ω, iv u = 0 in0,t) Ω, u = 0 on0,t) Γ, u0,x)= u 0 x), x Ω, ut, x) = ψt,x), t h, 0), x Ω, 3.1)

12 T. Caraballo et al. / J. Math. Anal. Appl ) where we assume that ν>0 is the kinematic viscosity, u is the velocity fiel of the flui, p the pressure, u 0 the initial velocity fiel, g is an external force containing some hereitary characteristic an ψ the initial atum in the interval of time h, 0), where h is a positive fixe number. To begin with we consier the following usual abstract spaces see 10,] for more etails): U = { u C 0 Ω)) :ivu = 0 }, H = the closure of U in L Ω)) with the norm, an inner prouct, ), where for u, v L Ω)), u, v) = u j x)v j x) x, j=1ω U = the closure of U in H 1 0 Ω)) with the norm, an associate scalar prouct, )), where for u, v H 1 0 Ω)), u, v)) = i,j=1ω u j x i v j x i x. It follows that U H H U, where the injections are ense an compact. Now we enote au,v) = u, v)), an efine the trilinear form b on U U U by bu,v,w) = i,j=1ω Assume that the elay term is given by gt,u t ) = Gu t ht) ), u i v j x i w j x, u, v, w U. where G LU, U ) is a self-ajoint linear operator, an the elay function ht) satisfies the assumptions in Theorem.1. Then problem 3.1) can be set in the abstract formulation see 6 8,10,6] for a etaile escription) To fin u L h, T ; U) L 0,T; H) such that for all v U, ) ) ) ) ) ut), v + νa ut), v + b ut), ut), v = Gu t ht),v, t u0) = u 0, ut)= ψt), t h, 0), 3.) where the equation in 3.) must be unerstoo in the sense of D 0,T). Observe that Eq. 3.) can be rewritten as Eq..1) by enoting At, ), F : U U the operators efine as At, u) = νau, ) bu, u, ), F u) = Gu, u U. By arguing as in case 3) from 6, p. 448], it is not ifficult to check that conitions in Theorem.1 hol provie ν> G LU,U ), an we can therefore ensure that there exists a unique solution to this problem 3.) which, in aition, satisfies u C 0 0,T; H) for any T>0. As G is linear, then we have that 0 is a stationary solution to our moel an we can analyse its stability. When G maps U or H into H in other wors, G is a first or zero-orer linear partial

13 114 T. Caraballo et al. / J. Math. Anal. Appl ) ifferential operator), Theorems 3.3 an 3.5 in 7] guarantee the exponential stability of the trivial solution provie the viscosity parameter ν is large enough. For instance, in the case that G maps H into H, the null solution of Eq. 3.) is exponentially stable if νλ 1 > h 1) G LH,H ), 3.3) where λ 1 is the first eigenvalue of the Stokes operator see also Corollary 3.7 in 7] for another sufficient conition when G maps U into H ). However, the results obtaine in 7] o not cover the more general situation in which G may contain secon-orer partial erivatives. This is why we consier this situation. Thus, in the present situation, i.e. for the operator G LU, U ) an the function gt,u t ) = Gut ht)) efine above, we have that γ = ν, α = G LU,U ), β = λ 1/ 1 an assumptions in Theorem.1 hol assuming that ν> G LU,U ) 1 h1. Remark 3.1. Observe that if G LH, H ), then G LU, U ) an, in aition, we have that G LU,U ) λ 1 1 G LH,H ), so, if we assume that νλ 1 > G LH,H ) 1 h1 3.4) it also follows that ν> G LU,U ) 1 h1 an, consequently, we have the exponential stability of the trivial solution. It is worth pointing out that conition 3.4) improves the conition establishe in 7], which is 3.3). 3.. Application to some reaction iffusion equations In this subsection we will consier three ifferent reaction iffusion equations to show how we can obtain ifferent stability regions for the parameters involve in the equation. Let us then consier the following three problems: ut,x) t ut,x) t ut,x) t with conitions = ν ut, x) x + μ ut ht), x) x, 3.5) = ν ut, x) x + μ ut ht), x), 3.6) x = ν ut, x) x + μu t ht), x ) 3.7) t 0, x a,b], ut,a)= ut, b) = 0, ht) 0,h 0 ], ḣt) h 1 < 1, ḣt) h,

14 T. Caraballo et al. / J. Math. Anal. Appl ) where ν>0 an μ is an arbitrary constant. Note that in all of these situations we can consier U = H0 1a,b]) an H = L a,b]). The constant β for the injection U H equals β = λ 1/ 1, where λ 1 = πb a) 1 is the first eigenvalue of the operator with Dirichlet bounary x conitions. We can therefore apply Theorem.1 to all these examples yieling the following sufficient stability conitions. For Eq. 3.5) ν> μ, 1 h1 for Eq. 3.6) ν> for Eq. 3.7) μ λ1 ), μ ν>. 3.8) λ 1 1 h1 Note that in the particular case a,b]=0,π] it hols λ 1 = 1 an these three conitions given by Theorem.1 are the same. Observe that Theorem. can be applie only to Eq. 3.7). For this equation the parameters of Theorem. are γ = α 1 = ν μλ 1 1, α = μ λ 1/ 1. It gives the following sufficient stability conition: ν> μ μ h λ1 + μ h 0 λ1 + ), μ <. 3.9) λ 1 λ1 ) λ1 μ h 0 h 0 Figure 1 shows the stability regions for Eq. 3.7) given by conitions 3.8) the boun 1)) an 3.9) the boun )) for the following values of the parameters h 0 = 1, h 1 = h = 0.1, Fig. 1.

15 1144 T. Caraballo et al. / J. Math. Anal. Appl ) Fig.. λ 1 = 1. One can see that for some negative μ conition 3.9) gives an aitional part of stability region, i.e. for some negative μ conition 3.9) is better than 3.8). It is easy to show also that if + h λ1 < 1 then conition 3.9) is better than 3.8) an for some positive μ. For example, Fig. shows the stability regions for h 0 = 1, h 1 = h = 0.95, λ 1 = 0.5. If μ = 0.06 then the right-han part of inequality 3.8) equals but the right-han part of inequality 3.9) equals 0.889, i.e. is less than References 1] S. Aizicovici, On a semilinear Volterra integroifferential equation, Israel J. Math ) ] M. Artola, Sur les perturbations es équations évolution, application à es problèmes e retar, Ann. Sci. École Norm. Sup. 4) 1969) ] M. Artola, Équations et inéquations variationnelles à retar, Ann. Sci. Math. Québec 1 ) 1977) ] A. Bensoussan, G. Da Prato, M.C. Delfour, S.K. Mitter, Representation an Control of Infinite Dimensional Systems, vol. I, Systems an Control: Founations an Applications, Birkhäuser, Boston, ] T. Caraballo, Nonlinear partial functional ifferential equations: Existence an stability, J. Math. Anal. Appl ) ] T. Caraballo, J. Real, Navier Stokes equations with elays, Proc. R. Soc. Lonon Ser. A ) 001) ] T. Caraballo, J. Real, Asymptotic behaviour of D-Navier Stokes equations with elays, Proc. R. Soc. Lonon Ser. A ) ] T. Caraballo, J. Real, Attractors for D-Navier Stokes moels with elays, J. Differential Equations ) ] M.G. Cranall, S.O. Lonen, J.A. Nohel, An abstract nonlinear Volterra integroifferential equation, J. Math. Anal. Appl ) ] R. Dautray, J.L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, Masson, Paris, ] W.E. Fitzgibbon, Asymptotic behavior of solutions to a class of Volterra integroifferential equations, J. Math. Anal. Appl ) ] J.K. Hale, S.M.V. Lunel, Introuction to Functional Differential Equations, Springer-Verlag, New York, 1993.

16 T. Caraballo et al. / J. Math. Anal. Appl ) ] A.G. Kartsatos, On the construction of methos of lines for functional evolutions in general Banach spaces, Nonlinear Anal ) ] A.G. Kartsatos, M.E. Parrott, Existence of solutions an Galerkin approximations for nonlinear functional evolution equations, Tôhoku Math. J. 34 ) 198) ] A.G. Kartsatos, M.E. Parrott, Convergence of the Kato approximants for evolution equations involving functional perturbations, J. Differential Equations ) ] V.B. Kolmanovskii, L.E. Shaikhet, New results in stability theory for stochastic functional ifferential equations SFDEs) an their applications, in: Proceeings of Dynamical Systems an Applications, vol. 1, Dynamic Publishers, 1994, pp ] V.B. Kolmanovskii, L.E. Shaikhet, General metho of Lyapunov functionals construction for stability investigations of stochastic ifference equations, in: Dynamical Systems an Applications, in: Worl Sci. Ser. Appl. Anal., vol. 4, Worl Scientific, 1995, pp ] V.B. Kolmanovskii, L.E. Shaikhet, Control of Systems with Aftereffect, Transl. Math. Monogr., vol. 157, Amer. Math. Soc., Provience, RI, ] V.B. Kolmanovskii, L.E. Shaikhet, A metho of Lyapunov functionals construction for stochastic ifferential equations of neutral type, Differ. Uravn ) 1995) in Russian), translation in: Differ. Equ ) 1996) ] V. Kolmanovskii, L. Shaikhet, Construction of Lyapunov functionals for stochastic hereitary systems: A survey of some recent results, Math. Comput. Moelling 36 6) 00) ] N.N. Krasovskii, Stability of Motions, Stanfor Univ. Press, Stanfor, ] J.L. Lions, Quelque méthoes e résolution es problèmes aux limites non linéaires, Duno, Gauthier Villars, Paris, ] X. Mao, Exponential stability of nonlinear ifferential elay equations, Syst. Control Lett ) ] R.H. Martin, H.L. Smith, Abstract functional ifferential equations an reaction iffusion systems, Trans. Amer. Math. Soc ) ] W. Ruess, Existence of solutions to partial functional ifferential equations with elay, in: A.G. Kartsatos E.), Theory an Applications of Nonlinear Operators of Accretive an Monotone Type, Marcel Dekker, New York, 1996, pp ] W. Ruess, Existence an stability of solutions to partial functional ifferential equations with elay, Av. Differential Equations 4 6) 1999) ] L. Shaikhet, Moern state an evelopment perspectives of Lyapunov functionals metho in the stability theory of stochastic hereitary systems, Theory Stoch. Process. 18) 1 ) 1996) ] L. Shaikhet, Necessary an sufficient conitions of asymptotic mean square stability for stochastic linear ifference equations, Appl. Math. Lett. 10 3) 1997) ] L. Shaikhet, Lyapunov functionals construction for stochastic ifference secon kin Volterra equations with continuous time, Av. Difference Equ ) 004) ] L. Shaikhet, Construction of Lyapunov functionals for stochastic ifference equations with continuous time, Math. Comput. Simulation 66 6) 004) ] L. Shaikhet, General metho of Lyapunov functionals construction in stability investigations of nonlinear stochastic ifference equations with continuous time, in: Proceeings of Workshop Stochastic Dynamics with Delay an Memory, Wittenberg, February 5, 004, Stoch. Dyn. 5 ) 005) ] C.C. Travis, G.F. Webb, Existence an stability for partial functional ifferential equations, Trans. Amer. Math. Soc ) ] C.C. Travis, G.F. Webb, Existence, stability, an compactness in the α-norm for partial functional ifferential equations, Trans. Amer. Math. Soc ) ] G.F. Webb, Autonomous nonlinear functional ifferential equations an nonlinear semigroups, J. Math. Anal. Appl ) ] G.F. Webb, Functional ifferential equations an nonlinear semigroups in L p -spaces, J. Differential Equations ) ] G.F. Webb, Asymptotic stability for abstract nonlinear functional ifferential equations, Proc. Amer. Math. Soc ) ] J. Wu, Theory an Applications of Partial Functional Differential Equation, Springer-Verlag, New York, 1996.