Math Sci (2015) 9:189–192 DOI 10.1007/s40096-015-0166-5 ORIGINAL RESEARCH Asymptotic equilibrium of integro-differential equations with infinite delay Le Anh Minh1 • Dang Dinh Chau2 Received: 25 June 2014 / Accepted: September 2015 / Published online: 21 September 2015 Ó The Author(s) 2015 This article is published with open access at Springerlink.com Abstract The asymptotic equilibrium problems of ordinary differential equations in a Banach space have been considered by several authors In this paper, we investigate the asymptotic equilibrium of the integro-differential equations with infinite delay in a Hilbert space Keywords Asymptotic equilibrium Á Integro-differential equations Á Infinite delay Zt > > < dxtị ẳ Atị@ xtị ỵ kt hịxhịdhA; dt > > : xtị ẳ uðtÞ; t > 0; t60 ð1Þ where AðtÞ : H ! H, u in the phase space B, and xt is defined as xt hị ẳ xt ỵ hị; 1\h 0: Introduction The asymptotic equilibrium problems of ordinary differential equations in a Banach space have been considered by several authors, Mitchell and Mitchell [3], Bay et al [1], but the results for the asymptotic equilibrium of integrodifferential equations with infinite delay still is not presented In this paper, we extend the results in [1] to a class of integro-differential equations with infinite delay in a Hilbert space H which has the following form: Preliminaries We assume that the phase space ðB; jj:jjB Þ is a seminormed linear space of functions mapping ðÀ1; 0 into H satisfying the following fundamental axioms (we refer reader to [2]) (A1 ) For a [ 0, if x is a function mapping ðÀ1; a into H, such that x B and x is continuous on [0, a], then for every t ½0; a the following conditions hold: (i) (ii) (iii) & Le Anh Minh leanhminh@hdu.edu.vn Dang Dinh Chau chaudida@gmail.com Department of Mathematical Analysis, Hong Duc University, Thanh Ho´a, Vietnam Department of Mathematics, Hanoi University of Science, VNU, Hanoi, Vietnam xt belongs to B; jjxðtÞjj Gjjxt jjB ; jjxt jjB Ktị sups2ẵ0;t jjxsịjj ỵ Mtịjjx0 jjB where G is a possitive constant, K; M : ẵ0; 1ị ! ẵ0; 1Þ, K is continuous, M is locally bounded, and they are independent of x (A2 ) (A3 ) For the function x in (A1 ), xt is a B-valued continuous function for t in [0, a] The space B is complete 123 190 Math Sci (2015) 9:189–192 Example (i) (M4 ) Let BC be the space of all bounded continuous functions from ðÀ1; 0 to H, we define C0 :ẳ fu BC : limh!1 uhị ẳ 0g and C1 :ẳ fu BC : limh!1 uhị exists in Hg endowed with the norm jjujjB ¼ (ii) sup & ' u CððÀ1; 0; XÞ : lim ech uðhÞ exists in H T herein S(0, 1) is a unit ball in H, j ẳ LK ỵ Mị ỵ 1; where K, M, L are given in (c1 ), (c2 ) and (M3 ) h!À1 Theorem If (M1 ), ðM2 Þ, ðM3 Þ and ðM4 Þ are satisfied, then Eq (1) has an asymptotic equilibrium Proof We shall begin with showing that all solutions of (1) has a finite limit at infinity Indeed, Eq (1) may be rewritten as Z0 dxtị ẳ Atị@ xtị ỵ khịxt hịdhA ; dt À1 endowed with the norm jjujjB ¼ sup ð2Þ jjuðhÞjj h2ðÀ1;0 then C0 ; C1 satisfies (A1 )–(A3 ) However, BC satisfies (A1 ) and (A3 ), but (A2 ) is not satisfied For any real constant c, we define the functional spaces Cc by Cc ¼ There exists a constant T [ such that Z1 jjAðtÞhjjdt\q\ ; sup j h2Sð0;1Þ ech jjuðhÞjj: h2ðÀ1;0 Then conditions (A1 )–(A3 ) are satisfied in Cc then for t > s > T we have Z0 Zt khịxs hịdhAds xtị ẳ xsị ỵ Asị@ xsị ỵ À1 s Remark In this paper, we use the following acceptable hypotheses on K(t), M(t) in (A1 )(iii) which were introduced by Hale and Kato [2] to estimate solutions as t ! 1, and jjxðtÞjj * + Zt Z0 khịxs hịdhAds; h ẳ sup xsị ỵ Asị@xsị ỵ h2S0;1ị s * + Z0 Z t ds xsị ỵ jjxsịjj ỵ sup khịx hịdh;Asịh s h2Sð0;1Þ s À1 ! (c1 ) K ẳ Ktị is a constant for all t > 0; (c2 ) MðtÞ M for all t > and some M Example For the functional space Cc in Example 1, the hypotheses (c1 ) and (c2 ) are satisfied if c > Definition Equation (1) has an asymptotic equilibrium if every solution of it has a finite limit at infinity and, for every h0 H, there exists a solution x(t) of it such that xðtÞ ! h0 as t ! Main results jjxsịjj ỵ q LK ỵ 1ị sup jjxnịjj ỵ LMjjujjB n2ẵ0;t 3ị implies jjjxtịjjj jjxsịjj þ q ðLK þ 1ÞjjjxðtÞjjj þ LMjjujjB or Now, we consider the asymptotic equilibrium of Eq (1) which satisfies the following assumptions: (M1 ) (M2 ) (M3 ) A(t) is a strongly continuous bounded linear operator for each t Rỵ ; A(t) is a self-adjoint operator for each t Rỵ ; k satisfies Zỵ1 jkhịjdh ẳ L\ ỵ 1; and 123 jjjxtịjjj jjxsịjj ỵ qLMjjujjB qLK ỵ 1ị where jjjxtịjjj ẳ sup jjxnịjj: 06n6t ð4Þ Math Sci (2015) 9:189–192 191 Now, we conclude that x(t) is bounded since 0\q\ 1 ¼ \ ) qLK ỵ 1ị\1 j LK ỵ Mị ỵ LK ỵ and by (4) Putting and jjx1 tịjj jjh0 jj1 ỵ qjị: Now, we consider the functional g2 t; hị ẳ hh0 ; hi + Zỵ1* Zs Asị@x1 tị ỵ ks hịx1 hịdhA; h ds: M ẳ sup jjxtịjj; t2R t we have By an argument analogous to the previous one, we get jjxtị xsịjj ẳ sup j \xtị xðsÞ; h [ j h2Sð0;1Þ Zt Z0 ds \Asị@xsị ỵ A kðÀhÞx ðhÞdh ; h [ sup s ; h2S0;1ị s ẵM LK þ 1Þ þ LMjjujjB sup h2Sð0;1Þ Zt jg2 ðt; hịj jjh0 jjẵjjhjj ỵ qj ỵ qjị2 and there exists an element x2 ðtÞ in H, such that g2 t; hị ẳ hx2 tị; hi with jjAsịhjjds ! jjx2 tịjj jjh0 jj1 ỵ qj ỵ qjị2 ị: s as t > s ! ỵ1 That means all solutions of (1) have a finite limit at infinity To complete the proof, it remains to show that for any h0 H, there exists a solution x(t) of (1) such that lim xtị ẳ h0 : Continuing this process, we obtain the linear continuous functional gn ðt; hÞ ẳ hh0 ; hi + Zỵ1* Zs Asị@xn1 tị ỵ ks hịxn1 hịdhA; h ds t t!ỵ1 Indeed, let h0 be an arbitrary fixed element of H; we choose the initial function u belongs to B such that u0ị ẳ h0 and jjujjB jjh0 jj and consider the functional g1 t; hị ẳ hh0 ; hi + Z1* Zs Asị@h0 ỵ kðs À hÞx0 ðhÞdhA; h ds ð5Þ and xn ðtÞ H such that gn t; hị ẳ hxn tị; hi satisfies the following estimate jjxn tịjj ỵ qj ỵ qjị2 ỵ ỵ qjịn ịjjh0 jj À1 t jjh0 jj : À qj Futhermore, We have jg1 t; hịj jjh0 jjjjhjj ỵ Zỵ1 kxn tị xn1 tịk jjh0 jjqjịn : kx0 sị t ỵ Zs ks hịx0 hịdhkjjAsịhjjds:: Since x0 ðsÞ h0 ; then jg1 ðt; hÞj jjh0 jjjjhjj ỵ qjị: It follows from Riesz representation theorem that there exists an element x1 ðtÞ in H, such that g1 t; hị ẳ hx1 tị; hi This inequality shows that fxn tịg is uniformly convergent on ẵT; ỵ1ị since qj\1 Put xtị ẳ lim xn tị: n!ỵ1 In (5), let n ! ỵ1; we have hxtị; hi ẳ hh0 ; hi + Zỵ1* Zs Asị@xtị ỵ ks hịxhịdhA; h ds t 6ị and since 123 192 Math Sci (2015) 9:189–192 jhxn ðtÞ; h0 ij\ Zỵ1 kxn1 sị T ỵ Zs creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made kðs À hÞxnÀ1 ðhÞdhkk AðsÞhkds À1 or jjh0 jjq ; jhxn ðtÞ; h0 ij À qj we have xn ðtÞ ! h0 as q ! 0, which means that there exists a solution of (1) converging to h0 The theorem is proved Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// 123 References Bay, N.S., Hoan, N.T., Man, N.M.: On the asymptotic equilibrium and asymptotic equivalence of differential equations in Banach spaces Ukr Math J 60(5), 716–729 (2008) Hale, J.K., Kato, J.: Phase space for retarded equations with infinite delay Fukcialaj Ekvacioj 21, 11–41 (1978) Mitchell, A.R., Mitchell, R.W.: Asymptotic equilibrium of ordinary differential systems in a Banach space Theory Comput Syst 9(3), 308–314 (1975) ... the asymptotic equilibrium and asymptotic equivalence of differential equations in Banach spaces Ukr Math J 60(5), 716–729 (2008) Hale, J.K., Kato, J.: Phase space for retarded equations with infinite. .. satisfied, then Eq (1) has an asymptotic equilibrium Proof We shall begin with showing that all solutions of (1) has a finite limit at infinity Indeed, Eq (1) may be rewritten as Z0 dxtị ẳ At @... satisfied if c > Definition Equation (1) has an asymptotic equilibrium if every solution of it has a finite limit at infinity and, for every h0 H, there exists a solution x(t) of it such that