ANTIPERIODIC SOLUTIONS FOR nth-order FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

Elecronic Journal of Differenial Equaions, Vol. 216 (216), No. 44, pp. 1 8. ISSN: 172-6691. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu ANTIPERIODIC SOLUTIONS FOR nth-order FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY SUZETE M. AFONSO, ANDRÉ L. FURTADO Absrac. In his work, we esablish he exisence and uniqueness of aniperiodic soluion for a class of nh-order funcional differenial equaions wih infinie delay. The main ool in our sudy is he coincidence degree heory. An example is presened o illusrae he resuls obained. 1. Inroducion Given a posiive number T, we say ha a coninuous funcion x : R R is T -aniperiodic on R if x( + T ) = x() and x ( + T ) = x() for all R. 2 In his work, we consider he nh-order funcional differenial equaion wih infinie delay ( x (n) () = f, x (n 1), x (n 2),..., x, x ), R, (1.1) where f is coninuous and real defined on R C B ((, ], R) C B ((, ], R), }{{} n imes and T -periodic in he firs argumen, where C B ((, ], R) represens he space of he bounded coninuous funcions φ : (, ] R endowed wih he norm φ = sup τ (,] φ(τ) ; x denoes he mapping x : (, ] R defined by x (τ) = x( + τ) for τ (, ], where R; f ( + T 2, ϕ ) ( ) 1, ϕ 2,..., ϕ n = f, ϕ1, ϕ 2,..., ϕ n, for ϕi C B ((, ], R), i = 1,..., n; ] x is he derivaive of he unknown funcion x and x (j) is he derivaive of jh-order of x, j = 2,..., n 1. Funcional differenial equaions wih delay have an imporan role in modelling of naural processes in hose which is necessary o consider he influence of pas effecs for a beer undersanding of heir evoluion. For example, in populaion dynamics, he gesaion imes is a naural source of delays, since presen birh raes 21 Mahemaics Subjec Classificaion. 32A7, 34A5, 34K13, 47H11. Key words and phrases. Aniperiodic soluion; exisence; uniqueness; coincidence degree; funcional differenial equaion; infinie delay. c 216 Texas Sae Universiy. Submied November 14, 215. Published February 2, 216. 1

2 S. M. AFONSO, A. L. FURTADO EJDE-216/44 is srongly dependen on he number of individuals a fecundaion (see [18, 2, 25]). In classical physics, realisic models mus ake in accoun he ime-delays due o he finie propagaion speed of he classical fields (see [17, 26]). More examples of real phenomena wih delay effecs in physiology, epidemiology, engineering, economics, neural neworks, auomaic conrol, ec, can be found in [12, 16, 27]. Furhermore of imporance in applicaions, he funcional differenial equaions wih delay have several disinc mahemaical properies of ordinary and parial differenial equaions, which also provides hem wih a purely mahemaical ineres. Arising from problems in applied sciences, aniperiodic problems of nonlinear differenial equaions have been exensively sudied by many auhors during he pas weny years. We can cie [1]-[6], [8, 1, 15, 19] and references herein. To he bes of our knowledge here is no none work dedicaed o sudy of he exisence and uniqueness of aniperiodic soluion for nh-order funcional differenial equaions wih infinie delay wih level of generaliy of equaion (1.1). We will obain sufficien and necessary condiions for he exisence and uniqueness of an aniperiodic soluion on R of equaion (1.1) via he coincidence degree heory. This aricle is organized as follows. In Secion 2 we obain resuls ensuring exisence and uniqueness of a T -aniperiodic soluion for equaion (1.1). An illusraive example is given in Secion 3. 2. Exisence and uniqueness of aniperiodic soluion We adop he following noaion: CT k = {x C k (R, R); x is T -periodic}, k {, 1, 2,... }, ( 1/2, x 2 = x() d) 2 x = max x(), for x [,T ] C T, x (k) = max [,T ] x(k) (), for x C k T. Definiion 2.1. A funcion x : R R is said o be a T -aniperiodic soluion of equaion (1.1) if he following condiions are fulfilled: ( ) (i) x (n) () = f, x (n 1), x (n 2),..., x, x for each R; (ii) x is T -aniperiodic on R. We use he following assumpion: (H1) There are posiive consans a 1,..., a n such ha ( ) i n i=1 a T i 2π < 1, and n f(, ϕ 1,..., ϕ n ) f(, ψ 1,..., ψ n ) a i ϕ i () ψ i (), for each R and ϕ i, ψ i C B ((, ], R), i = 1,..., n. In he nex lines, our goal is o prove he following resul. Theorem 2.2. If (H1) holds, hen (1.1) has a leas one T -aniperiodic soluion. To prove Theorem 2.2, we sar by recalling some conceps in he nex lemma, which is crucial in he argumens of his secion. Le X and Y be real normed vecor spaces. A linear operaor L : Dom L X Y is a Fredholm operaor if ker L and Y \ Img L are finie-dimensional and Img L is closed in Y. The index of L is defined by dim ker L codim Img L. If L is a Fredholm operaor of index zero, i=1

EJDE-216/44 ANTIPERIODIC SOLUTIONS FOR nth-order FDES 3 i is possible o prove (see [28]) ha if L is a Fredholm operaor of index zero, hen here exis coninuous linear and idempoen operaors P : X X and Q : Y Y such ha ker L = Img P and Img L = ker Q. (2.1) The firs equaliy in (2.1) implies ha he resricion of L o Dom L ker P, which we will denoe by L P, is an isomorphism ono is image. Indeed, by supposing ker L = Img P and aking x Dom L ker P such ha L P (x) =, we have ha x Img P, ha is, here exiss y X such ha P y = x. Since P is idempoen and x ker P, he las equaliy implies x = P y = P x =. By assuming ha L : Dom L X Y is a Fredholm operaor of index zero and P and Q are he aforemenioned operaors, we say ha a coninuous operaor N : X Y is L-compac on Ω, where Ω X is open and bounded, if QN(Ω) is bounded and he operaor (L P ) 1 (I Q)N : Ω X is compac. To prove Theorem 2.2 we need he following resul, whose proof can be found in [23]. Lemma 2.3. Le X, Y be Banach spaces, Ω X a bounded open se symmeric wih Ω. Suppose L : Dom L X Y is a Fredholm operaor of index zero wih Dom L Ω and N : X Y is a L-compac operaor on Ω. Assume, moreover, ha Lx Nx λ(lx + N( x)), for all x Dom L Ω and all λ (, 1], where Ω is he boundary of Ω wih respec o X. Under hese condiions, he equaion Lx = Nx has a leas one soluion on Dom L Ω. Nex, we consruc an equaion Lx = N x ha appropriaely mirrors problem (1.1) and so ha all he condiions of Lemma 2.3 are fulfilled. Define he ses X = { x CT n ; x ( + T ) } = x(), R, 2 By equipping X and Y wih he norms Y = { x C n 1 T ; x ( + T ) } = x(), R. 2 x X = max{ x, x,..., x (n) }, x Y = max{ x, x,..., x (n 1) }, respecively, we obain wo Banach spaces. Define he operaors L : X Y and N : X Y by Lx() = x (n) (), R, (2.2) ( Nx() = f, x (n 1), x (n 2),..., x, x ), R. (2.3) To prove Theorem 2.2, i is sufficien o show ha condiion (H1) implies ha he assumpions of Lemma 2.3 are saisfied when L and N are defined as in (2.2) and (2.3). I is easy o verify ha ker L = and Img L = {x Y ; x(s)ds = } = Y. Then dim ker L = = codim Img L and L is a linear Fredholm operaor of index zero.

4 S. M. AFONSO, A. L. FURTADO EJDE-216/44 Proposiion 2.4. The operaor N is L-compac on any bounded open se Ω X. Proof. Le us consider he operaors P and Q given by P x = 1 T x()d, x X and Qy = 1 T y()d, y Y. Thus Img P = ker L and ker Q = Img L. Denoing by L 1 P : Img L X ker P he inverse of L X ker P, one can observe ha L 1 P is a compac operaor. Besides, i is no difficul o show ha, for any open bounded se Ω X, he se QN(Ω) is bounded and, using he Arzelà-Ascoli s Theorem, he operaor L 1 P (I Q)N : Ω X is compac. Therefore, N is L-compac on Ω. The nex lemma will be used laer. Is proof can be found in [14]. Lemma 2.5. If v : R R is a T -periodic absoluely coninuous funcion such ha v()d = and v () 2 d R, hen v() 2 d T 2 4π 2 v () 2 d. Proposiion 2.6. If condiion (H1) holds, hen here exiss a posiive number D, which does no depend on λ such ha, if hen x X D. Lx Nx = λ[lx + N( x)], λ (, 1], (2.4) Proof. Assume (H1) and ha x X saisfies (2.4). Then, by using he definiions of operaors L and N, given in (2.2) and (2.3), respecively, we obain x (n) () = 1 ( ) 1 + λ f, x (n 1), x (n 2),..., x, x λ ( 1 + λ f, x (n 1), x (n 2),..., x, x ). Thereby, considering F (, x) = f (, x (n 1), x (n 2) ),..., x, x, we have x (n) () = 1 λ F (, x) F (, x). 1 + λ 1 + λ Muliplying boh sides of his equaliy by x (n) () and subsequenly inegraing i from o T and using he riangle inequaliy, we obain x (n) 2 2 1 F (, x) x (n) () d + λ F (, x) x (n) () d 1 + λ 1 + λ 1 [ ] F (, x) F (, ) x (n) () d + F (, ) x (n) () d 1 + λ + λ [ ] F (, x) F (, ) x (n) () d + F (, ) x (n) () d. 1 + λ Therefore, x (n) 2 2 max{ F (, x) F (, ), F (, x) F (, ) } x (n) () d

EJDE-216/44 ANTIPERIODIC SOLUTIONS FOR nth-order FDES 5 + F (, ) x (n) () d. This, assumpion (H1), Lemma 2.5 and Hölder inequaliy, imply x (n) 2 2 a 1 x (n) 2 x (n 1) 2 + a 2 x (n) 2 x (n 2) 2 +... + a n x (n) 2 x 2 + R T x (n) 2, T ( T ) 2 x ( T ) n x a 1 2π x(n) 2 2 + a (n) 2 2 2 + + a (n) n 2 2 2π 2π + R T x (n) 2, where R = max [,T ] F (, ). Thus we obain x (n) 2 K, where K = R T 1 ( n i=1 a i T 2π ) i, (2.5) since, by hypohesis (H1), ( ) i n i=1 a T i 2π < 1. Then he inequaliies ( T ) n j, x (j) 2 K 2π j = 1,..., n, (2.6) follow from (2.5) and Lemma 2.5. On he oher hand, by mean value heorem for inegrals we conclude ha, for each j =,..., n 1, here exiss τ j [, T ] such ha x (j) (τ j ) =, because x(j) ()d =. Hence, by Hölder inequaliy, for each j =,..., n 1, we have x (j) () = x (j+1) (s)ds τ j x (j+1) (s)ds T x (j+1) 2, [, T ]. Consequenly, x (j) T x (j+1) 2 for j =,..., n 1. Now inequaliies (2.6) imply x X = max j n 1 x(j) D, (2.7) where D = K ( ) n j T T max 1 j n 2π and he saemen follows. Proposiion 2.7. If condiion (H1) is saisfied, hen here is a bounded open se Ω X such ha Lx Nx λ(lx + N( x)), (2.8) for all x Ω and all λ (, 1]. Proof. By (H1) and Proposiion 2.6 here exiss a posiive consan D, which does no depend on λ such ha, if x saisfies he equaliy Lx Nx = λ(lx + N( x)), λ (, 1], hen x X D. Thus, if where M > D, we conclude ha Ω = {x X; x X < M}, (2.9) Lx Nx λ(lx N( x)), for every x Ω = {x X; x X = M} and λ (, 1].

6 S. M. AFONSO, A. L. FURTADO EJDE-216/44 Proof of Theorem 2.2. By (H1), clearly, he se Ω defined in (2.9) is symmeric, Ω and X Ω = Ω. Furhermore, i follows from Proposiion 2.7 ha if condiion (H1) is fulfilled hen Lx Nx λ[lx N( x)], for all x X Ω = Ω and all λ (, 1]. This ogeher wih Lemma 2.3 imply ha equaion (1.1) has a leas one T -aniperiodic soluion. Our purpose now is o show he following resul. Theorem 2.8. If (H1) holds, hen (1.1) has a mos one T -aniperiodic soluion. Proof. Assume (H1) and ha x and y are T -aniperiodic soluions of (1.1). To obain he resul, we show ha he funcion z = x y is idenically zero. Then, whereas x and y are T -periodic, i is sufficien o prove ha z() = for all [, T ]. Since x and y are soluions of equaion (1.1), z (n) () = f(, x (n 1), x (n 2),..., x, x ) f(, y (n 1), y (n 2),..., y, y ). (2.1) Muliplying boh sides of (2.1) by z (n) (), inegraing i from o T, using (H1) and Hölder inequaliy, we obain z (n) 2 2 = z (n) () f(, x (n 1), x (n 2),..., x, x ) f(, y (n 1), y (n 2),..., y, y ) d a 1 z (n) () z (n 1) () d + + a n z (n) () z() d (2.11) a 1 z (n) 2 z (n 1) 2 + + a n z (n) 2 z 2 = z (n) 2 (a 1 z (n 1) 2 + + a n z 2 ). On he oher hand, by Lemma 2.5, ( T ) j z z (n j) 2 (n) 2, j = 1,..., n. 2π From his and (2.11), we obain [ z (n) 2 2 z (n) 2 ( T ) ( T ) n ] 2 a 1 + + an. 2π 2π Then, since n i=1 a T ) i i( 2π < 1, we conclude ha z (n) and consequenly z (n 1) is a consan funcion. Le us see now ha z (n 1) is idenically zero. Indeed, since z (n 2) () = z (n 2) (T ) hen, by Mean Value Theorem, i follows ha here is τ [, T ] such ha z (n 1) (τ) =. Repeaing his argumen n 1 imes, we can conclude ha z and he proof is complee. Finally, we presen he main resul of his work. Theorem 2.9. If (H1) holds, hen (1.1) has a unique T -aniperiodic soluion. The above heorem follows immediaely from Theorems 2.2 and 2.8.

EJDE-216/44 ANTIPERIODIC SOLUTIONS FOR nth-order FDES 7 3. Example Se T = 2π. Le us consider he equaion x () = f(, x, x, x ), R, (3.1) where f : R C B ((, ], R) C B ((, ], R) C B ((, ], R) R is given by f(, ϕ 1, ϕ 2, ϕ 3 ) = sin4 28 ϕ 1() + cos2 1 ϕ 2() + 1 4 ϕ 3(), for [, + ) and ϕ 1, ϕ 2, ϕ 3 C B ((, ], R). Clearly, f is coninuous and f( + π, ϕ 1, ϕ 2, ϕ 3 ) = f(, ϕ 1, ϕ 2, ϕ 3 ). Furhermore, condiion (H1) is saisfied wih a 1 = 1 28, a 2 = 1 1 and a 3 = 1 4. Indeed, if R and ϕ 1, ϕ 2, ϕ 3, ψ 1, ψ 2, ψ 3 C B ((, ], R), we have f(, ϕ 1, ϕ 2, ϕ 3 ) f(, ψ 1, ψ 2, ψ 3 ) = sin4 28 (ϕ 1() ψ 1 ()) + cos2 1 (ϕ 2() ψ 2 ()) + 1 4 (ϕ 3() ψ 3 ()) sin4 28 ϕ 1() ψ 1 () + cos2 1 ϕ 2() ψ 2 () + 1 4 ϕ 3() ψ 3 () 1 28 ϕ 1() ψ 1 () + 1 1 ϕ 2() ψ 2 () + 1 4 ϕ 3() ψ 3 (). Then, by Theorem 2.9, equaion (3.1) has precisely one T -aniperiodic soluion on R. Acknowledgmens. This research was suppored by gran 214/1467-7, São Paulo Research Foundaion (FAPESP) (S. M. Afonso). References [1] A. R. Afabizadeh; S. Aizicovici; N. H. Pavel; On a class of second-order ani-periodic boundary value problems, J. Mah. Anal. Appl. 171 (1992), 31-32. [2] A. R. Afabizadeh; N. H. Pavel; Y. K. Huang; Ani-periodic oscillaions of some second-order differenial equaions and opimal conrol problems, J. Compu. Appl. Mah. 52 (1994), 3-21. [3] S. Aizicovici; N. H. Pavel; Ani-periodic soluions o a class of nonlinear differenial equaions in Hilber space, J. Func. Anal. 99 (1991), 387-48. [4] Y. Q. Chen; Ani-periodic soluions for semilinear evoluion equaions, J. Mah. Anal. Appl. 315 (26), 337-348. [5] Y. Q. Chen, J. J. Nieo, D. O Regan; Ani-periodic soluions for evoluion equaions associaed wih maximal monoone mappings, Applied Mahemaics Leers, 24 (211) 32-37. [6] Y. Q. Chen, J. J. Nieo, D. O Regan; Ani-periodic soluions for fully nonlinear firs-order differenial equaions, Mah. Compu. Modelling, 46 (27) 1183-119. [7] C. Chicone, S. Kopeikin, B. Mashhoon, D. G. Rezloff; Delay equaions and radiaion damping, Physics Leers, A 285 (21) 17-26. [8] W. Ding, Y.P. Xing, M.A. Han; Ani-periodic boundary value problems for firs order impulsive funcional differenial equaions, Appl. Mah. Compu, 186 (27), 45-53. [9] R. D. Driver; Ordinary and Delay Differenial Equaions, Spriger Verlag, New York, 1977. [1] Q. Y. Fan, W. T. Wang, X. J. Yi; Ani-periodic soluions for a class of nonlinear nh-order differenial equaions wih delays, Journal of Compuaional and Applied Mahemaics, 23 (29) 762-769. [11] K. Gopalsamy; Sabiliy and Oscillaions in Delay is Applicaions, 74. Kluwer Academic Publishers Group, Dordrech, 1992. [12] S. A. Gourley, Y. Kuang; A sage srucured predaor-prey model and is dependence on mau-raion delay and deah rae, J. Mah. Biol. 49 (24), 188-2.

8 S. M. AFONSO, A. L. FURTADO EJDE-216/44 [13] J. K. Hale, S. M. Verduyn Lunel; Inroducion o Funcional Differenial Equaions, Springer Verlag, Berlin, 1993. [14] G. H. Hardy, J. E. Lilewood, G. Pólya; Inequaliies, Cambridge Univ. Press, London, 1964. [15] T. Jankowski; Aniperiodic Boundary Value Problems for Funcional Differenial Equaions, Applicable Analysis, 81(22) 341-356. [16] V. Kolmanovskii, A. Myshkis; Inroducion o he Theory and Applicaions of Funcional Differenial Equaions, Kluwer Academic Publishers, Dordrech, 1999. [17] V. Kolmanovskii, A. Myshkis; Applied Theory of Funcional Differenial Equaions, Kluwer Academic Press Publisher, Dordrec, 1992. [18] Y. Kuang; Delay Differenial Equaions wih Applicaions in Populaion Dynamics, Academic Press, New York, 1993. [19] L. Liu, Y. Li; Exisence and uniqueness of ani-periodic soluions for a class of nonlinear n-h order funcional differenial equaions, Opuscula Mahemaica, vol. 31, n.1, 211. [2] J. D. Murray; Mahemaical Biology: An Inroducion, Springer-Verlag, Third Ediion, New York, 22 [21] H. Okochi; On he exisence of ani-periodic soluions o nonlinear parabolic equaions in noncylindrical domains, Nonlinear Anal. 14 (199), 771-783. [22] H. Okochi; On he exisence of ani-periodic soluions o a nonlinear evoluion equaion associaed wih odd subdifferenial operaors, J. Func. Anal., 91 (199) 246-258. [23] D. O Regan, Y. J. Chao, Y. Q. Chen; Topological Degree Theory and Applicaion, Taylor and Francis Group, Boca Raon, London, New York, 26 [24] S. Pinsky, U. Trimann; Ani-periodic boundary condiions in supersymmeric discree ligh cone quanizaion, Physics Rev., 62 (2), 8771, 4 pp. [25] R. W. Shonkkwiler, J. Herod; Mahemaical Biology, an Inroducion whih Mapple and Malab, second ediion, Springer, San Francisco, 29. [26] S. P. Travis; A one-dimensional wo-body problem of classical elecrodinamics, J. Appl. Mah., 28 (1975) 611-632. [27] H. Smih; An Inroducion o Delay Differenial Equaions wih Applicaions o he Life Sciences, Springer, New York, 211. [28] R. E. Gaines, J. Mawhin; Coincidence Degree and Nonlinear Differenial Equaions, Lecure Noes in Mah., 568, Springer, 1977. Suzee M. Afonso Univ Esadual Paulisa, Insiuo de Geociências e Ciências Exaas, 1356-9, Rio Claro SP, Brasil E-mail address: smafonso@rc.unesp.br André L. Furado Deparameno de Análise Maemáica, Insiuo de Maemáica e Esaísica, Universidade do Esado do Rio de Janeiro, Rio de Janeiro, Brasil E-mail address: andre.furado@ime.uerj.br