Research Article On Stability of Vector Nonlinear Integrodifferential Equations

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1 International Engineering Mathematics Volume 216, Article ID , 5 pages Research Article On Stability of Vector Nonlinear Integroifferential Equations Michael Gil Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, 8415 Beersheba, Israel Corresponence shoul be aresse to Michael Gil ; gilmi@bezeqint.net Receive 15 March 216; Accepte 5 May 216 Acaemic Eitor: Josè A. Tenereiro Machao Copyright 216 Michael Gil. This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. Let be a boune omain in a real Eucliean space. We consier the equation u(t, x)/ t = C(x)u(t, x) + K(x, s)u(t, s)s + [F(u)](t,x) (t > ; x ),where C( ) an K(, ) are matrix-value functions an F( ) is a nonlinear mapping. Conitions for the exponential stability of the steay state are establishe. Our approach is base on a norm estimate for operator commutators. 1. Introuction an Statement of the Main Result Throughout this paper, C n is the complex n-imensional Eucliean space with a scalar prouct (, ) n an norm n = (, ) n ; C n n is the set of n n-matrices; I is the unit operator in corresponing space; is a boune omain with a smooth bounary in a real Eucliean space; L 2 (C n,)=l 2 isthehilbertspaceoffunctionsefineon with values in C n,thescalarprouct (V,w) L 2 = (V (x),w(x)) n x (w, V L 2 ), (1) an the norm L 2 = (, ) L 2. Our main object in this paper is the equation u (t, x) t =C(x) u (t, x) + K (x, s) u (t, s) s + [F (u)] (t, x) (t>; x ), where C( ) an K(, ) are matrix-value functions efine on an, respectively, with values in C n n,anf( ) : L 2 L 2 satisfy conitions pointe out below, an u(, ) is unknown. Traitionally, (2) is calle the Barbashin type integroifferential equation or simply the Barbashin equation. It plays an essential role in numerous applications, in particular, in kinetic theory [1], transport theory [2], continuous mechanics [3], control theory [4], raiation theory [5, 6], an the (2) ynamics of populations [7]. Regaring other applications, see [8]. The classical results on the Barbashin equation are represente in the well-known book [9]. The recent results about various aspects of the theory of the Barbashin equation canbefoun,forinstance,in[1 14]anthereferences given therein. In particular, in [11], the author investigates the solvability conitions for the Cauchy problem for a Barbashin equation in the space of boune continuous functions an in the space of continuous vector-value functions with the values in an ieal Banach space. The stability an bouneness of solutions to a linear scalar nonautonomous Barbashinequationhavebeeninvestigatein[15]. The literature on the asymptotic properties of integroifferential equations is rather rich (cf. [16 22] an the references given therein), but the stability of nonlinear vector integroifferential equations is almost not investigate. It is at an early stage of evelopment. A solution of (2) is a function u(t, ) : [, ) L 2 having a measurable erivative boune on each finite interval. It is assume that uner consieration F provies the existence an uniqueness of solutions (e.g., it is Lipschitz continuous). The zero solution of (2) is sai to be exponentially stable, if there are constants m 1, δ >,anα>,such that u(t) L 2 m u() L 2e αt (t ), provie u() L 2 δ. It is globally exponentially stable if δ =. Suppose that, for a positive r, F (h) L 2 q h L 2 (h L 2 ; h L 2 r). (3)

2 2 International Engineering Mathematics For example, for an integer p>1,let[f(h)](x) = (Th(x)) p. Here, Th (x) = b (x, s) h (s) s (4) with a matrix kernel b(x, s) satisfying J p =( ( b (x, s) 2 p 1/2 n s) x) <. (5) Then, by the Schwarz inequality, Thus, Th (x) 2 n b (x, s) 2 n s h 2 L. (6) 2 [F (h)] (x) 2 x ( b (x, s) 2 p n s) x h 2p L 2 Hence, for any r<,wehaveconition(3)withq=r p 1 J p. The following notations are introuce: for a linear operator A, A is the ajoint operator, A is the operator norm, an σ(a) is the spectrum. For n n-matrix C,put g (C) =[N 2 2 (C) n k=1 λ k (C) 2 ] 1/2 (7), (8) where λ k (C), k = 1,...,n, are the eigenvalues of C, counte with their multiplicities; N 2 (C) = (Trace CC ) 1/2 is the Frobenius (Hilbert-Schmit) norm of C. Thefollowing relations are checke in [23, Section 2.1]: g 2 (C) N 2 2 (C) Trace C 2, g(e iτ C+zI)=g(C) (τ R, z C), g 2 (C) N2 2 (C C ). 2 If C is a normal matrix, CC = C C,theng(C) =. Furthermore, enote ξ fl 1 2 sup h L 2, h L 2 =1 h (x)) n s x, ((K (x, s) +K (s, x))h(s), γ=( C (x) K (x, s) K(x, s) C (s) 2 1/2 n s x) an assume that α fl ξ+sup x <. Rσ (C (x)) =ξ+sup x max Rλ k (C (x)) k (9) (1) (11) In aition, with the notation g = sup x g(c(x)),put χ= n 1 j,k= n 1 t k g k p (t) = (k!), 3/2 k= ζ =2 g j+k (k+j)! 2 j+k α j+k+1 (j!k!), 3/2 e 2αt t p (t) p (t s) p (s) s t. (12) Thisintegralissimplycalculate.IfC(x) is a normal matrix for all x,then g (C (x)) =, p (t) 1, χ= 1 α, ζ = 1 2 α 2. Now,weareinapositiontoformulateourmainresult. Theorem 1. Let conitions (3), (11), an (13) γζ +χq<1 (14) hol.then,thezerosolutionto(2)isexponentiallystable.if, in aition, r = in (3), then the zero solution is globally exponentially stable. This theorem is prove in the next 3 sections. It gives us goo results when γ is small, that is, if matrices K(x, s) an C(x) almost commute an sup x,s C(x) C(s) n is small. If (2) is scalar, then g =, χ= 1 α, γ=, (15) p (t) 1, ζ = 1 2 α 2. So, in the scalar case, conition (14) takes the form q< α. (16) This conition is similar to the stability test erive in [24] for scalar integroifferential equations. 2. Auxiliary Results Let H be a Hilbert space with a scalar prouct (, ) H an the norm H = (, ) H ; B(H) enotes the set of boune linear operators in H an [A 1,A 2 ]=A 1 A 2 A 2 A 1 is the commutator of A 1,A 2 B(H).

3 International Engineering Mathematics 3 Lemma 2. Let A, B B(H) an C=[A,B].Then, [e At t,b]= e As Ce A(t s) s. (17) Proof. Put J(t) = t eas Ce A(t s) s. Then,(/t)(J(t)e ta )= e At Ce At. On the other han, t ([eat,b] e ta ) = t (eat Be At B) =e At Ce At. (18) So, [e At,B]=J,asclaime. Let Then, the Lyapunov equation α (A) fl sup Rσ (A) <. (19) WA + (WA) = 2I (2) has a unique solution W B(H) an it can be represente as W=2 e A t e At t (21) (cf. [25]). Denote Λ B = (1/2) sup σ(b + B ), ζ (A) =2 eat t ψ (W, B) = { Λ B W if Λ B >, { Λ { B λ W if Λ B, where λ W = inf σ(w). Lemma 3. Uner conition (19), one has eas ea(t s) s t, Re (WB) = 1 2 ((WB) + (WB) ) (ψ(w, B) + C ζ (A))I. Proof. Making use of (21), we can write (22) (23) Re (WB) = (e A t e At B+B e A t e At )t. (24) But e At B=Be At +[e At,B], B e A t =e A t B +[B,e A t ]= e A t B +[e At,B].So2Re(WB) = J 1 +J 2,where We have J 1 = e A t (B + B )e At t, J 2 = (e A t [e At,B]+(e A t [e At,B]) )t. (25) J 1 2Λ B e A t e At t = Λ B W. (26) If Λ B >,thenj 1 Λ B W I.IfΛ B <,thenj 1 Λ B λ W I. So J 1 2ψ(W,B)I. In aition, by Lemma 2, J = C ζ (A). This proves the lemma. eat [eat,b] t eat C t 3. Equations in a Hilbert Space eas ea(t s) s t (27) In this section, for simplicity, we put H =.Putω(r) = {V H, V r}(<r ). Consier in H the equation u=(a+b) u+f(u) (t ), (28) t where A, B B(H) an F continuously maps ω(r) into H an satisfies FV q V (V ω(r)). (29) The solution an stability are efine as in Section 1. The existence an uniqueness of solutions are assume. Recall that W is a solution of (2). Lemma 4. Let conitions (19) an (29) with r=hol. Then, any solution of (28) satisfies the inequality u (t) ( W 1/2 ) u () e ]t, t, λ W (3) where ] fl 1 ψ(w, B) ζ(a) C q W. Proof. For brevity,we write [Fu](t) = Fu(t). Multiplying (28) by W anoingthescalarprouct,weget (Wu (t),u(t)) =(W (A+B) u (t),u(t)) + (WFu (t),u(t)). (31) Since (/t)(wu(t), u(t)) = (Wu (t), u(t)) + (u(t), Wu (t)), ueto(2)anlemma3,itcanbewrittenthat (Wu (t),u(t)) t =2Re (W (A+B) u (t),u(t)) +2Re (WFu (t),u(t)) 2( 1+ψ(W, B) +ζ(a) C ) (u (t),u(t)) +2Re (WFu (t),u(t)). Taking into account the fact that ue to (29) (WFu (t),u(t)) W Fu (t) u (t) W q u (t) 2, (32) (33)

4 4 International Engineering Mathematics we get (Wu (t),u(t)) 2] (u (t),u(t)). t (34) From this inequality, we have (Wu(t), u(t)) (Wu(), u())e 2]t.Hence, as claime. λ W (u (t),u(t)) W (u (),u()) e 2]t, (35) Lemma 5. Let conitions (29) an ] <hol.then,thezero solution to (28) is exponentially stable. If r=in (29), then the zero solution to (28) is globally exponentially stable. Proof. If r=, then the require result is ue to the previous lemma. If r<,then,taking u() < r(λ W / W ) 1/2 ue to the previous lemma, u(t) < r. Hence,weeasilyobtainthe require result. 4. Proof of Theorem 1 Take Ah (x) = (C (x) +ξi) h (x), Bh (x) = K (x, s) h (s) s ξh (x) (h L 2 ). Then, Λ(B) an [A, B] h (x) = [C (x) K (x, s) K(x, s) C (s)] h (s) s. So [A, B] L 2 γ.due to [23, Example 1.7.3], ec(x)t n eα(c(x))t n 1 k= where α(c(x)) = sup Re σ(c(x)).hence, eat L 2 (36) (37) t k g k (C (x)) (k!) 3/2 (t ), (38) sup x e(c(x)+ξ)t n eα t p (t) (t ), (39) since g(c(x) + ξi) = g(c(x)). Consequently,ζ(A) ζ.in aition, W L eat L 2 t 2 Now,therequireresultisuetoLemma5. e 2α t p 2 (t) t = χ. (4) Competing Interests The author eclares that there are no competing interests regaring the publication of this paper. References [1] C. Cercignani, Mathematical Methos in Kinetic Theory, Macmillian, New York, NY, USA, [2]H.G.Kaper,C.G.Lekkerkerker,anJ.Hejtmanek,Spectral Methos in Linear Transport Theory, vol.5ofoperator Theory: Avances an Applications, Birkhäuser, Basel, Switzerlan, [3] V.M.AleksanrovanE.V.Kovalenko,Problems in Continuous Mechanics with Mixe Bounary Conitions, Nauka,Moscow, Russia, 1986 (Russian). [4] A. L. Khoteev, An optimal control problem for integroifferential equations of Barbashin type, in Problemy Optimizacii Upravlenija, pp , Minsk, Russia, 1976 (Russian). [5] K. M. Case an P. F. Zweifel, Linear Transport Theory, Aison- Wesley,Reaing,Mass,USA,1967. [6] G. I. Marchuk, The Methos of Calculation for Nuclear Reactors, Atomizat, Moscow, Russia, 1961 (Russian). [7] H. R. Thieme, A Differential-Integral Equation Moelling the Dynamics of Populations with a Rank Structure, vol.68of Lecture Notes in Biomathematics, [8] A.W.Englan, Thermalmicrowaveemissionfromahalfspace containing scatterers, Raio Science,vol.9,no.4,pp , [9] J.M.Appell,A.S.Kalitvin,anP.P.Zabrejko,Partial Integral Operators an Integro-Differential Equations, MarcelDekker, New York, NY, USA, 2. [1] A. S. Kalitvin, On two problems for the Barbashin integroifferential equation, Mathematical Sciences,vol.126, no. 6, pp , 25. [11] A. S. Kalitvin, Some aspects of the theory of integro-ifferential Barbashin equations in function spaces, Mathematical Sciences,vol.188,no.3,pp ,213. [12] B. G. Pachpatte, On a parabolic type Freholm integroifferential equation, Numerical Functional Analysis an Optimization, vol.3,no.1-2,pp ,29. [13] B. G. Pachpatte, On a nonlinear Volterra integral equation in two variables, Sarajevo Mathematics, vol.6,no.19, pp.59 73,21. [14] B. G. Pachpatte, On a parabolic integroifferential equation of Barbashin type, Commentationes Mathematicae Universitatis Carolinae, vol. 52, no. 3, pp , 211. [15] M. Gil, On stability of linear Barbashin type integroifferential equations, Mathematical Problems in Engineering, vol.215, Article ID , 5 pages, 215. [16] R. P. Agarwal, A. Domoshnitsky, an Y. Goltser, Stability of partial functional integro-ifferential equations, Dynamical an Control Systems,vol.12,no.1,pp.1 31,26. [17] N.M.Chuong,T.D.Ke,anN.N.Quan, Stabilityforaclass of fractional partial integro-ifferential equations, Integral Equations an Applications, vol.26,no.2,pp , 214. [18] A. Domoshnitsky an Y. M. Goltser, Approach to stuy of bifurcations an stability of integro-ifferential equations, Mathematical an Computer Moelling,vol.36,no.6,pp , 22. [19] A. D. Drozov an M. I. Gil, Stability of linear integroifferential equations with perioic coefficients, Quarterly of Applie Mathematics, vol. 54, no. 4, pp , [2] Ya. Goltser an A. Domoshnitsky, Bifurcations an stability of integro-ifferential equations, Nonlinear Analysis: Theory, Methos & Applications,vol.47,no.2,pp ,21.

5 International Engineering Mathematics 5 [21] J. Cao an Z. Huang, Existence an exponential stability of weighte pseuo almost perioic classical solutions of integroifferential equations with analytic semigroups, Differential Equations an Dynamical Systems, vol.23,no.3,pp , 215. [22] N. T. Dung, On exponential stability of linear Levin-Nohel integro-ifferential equations, JournalofMathematicalPhysics, vol. 56, Article ID 2272, 215. [23] M. I. Gil, Operator Functions an Localization of Spectra, vol. 183 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 23. [24] M. I. Gil, Stability of Freholm type integro-parabolic equations, Mathematical Analysis an Applications, vol. 244, no. 2, pp , 2. [25] L. Daleckii an M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, American Mathematical Society, Provience, RI, USA, 1974.

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