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Hindawi Publishing Corporation Boundary Value Problems Volume 2008, Article ID 389028, 18 pages doi:10.1155/2008/389028 Research Article On Periodic Solutions of Higher-Order Functional Differential Equations I. Kiguradze, 1 N. Partsvania, 1 and B. P ˚ u ˇ za 2 1 Andrea Razmadze Mathematical Institute, 1 Aleksidze Street, 0193 Tbilisi, Georgia 2 Department of Mathematics and Statistics, Masaryk University, Jan ´ a ˇ ckovo n ´ am. 2a, 66295 Brno, Czech Republic Correspondence should be addressed to I. Kiguradze, kig@rmi.acnet.ge Received 8 September 2007; Accepted 23 January 2008 Recommended by Donal O’Regan For higher-order functional differential equations and, particularly, for nonautonomous differential equations with deviated arguments, new sufficient conditions for the existence and uniqueness of a periodic solution are established. Copyright q 2008 I. Kiguradze et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Statement of the main results 1.1. Statement of the problem Let n ≥ 2 be a natural number, ω>0, L ω the space of ω-periodic and Lebesgue integrable on 0,ω functions u : R → R with the norm u L ω   ω 0   us   ds. 1.1 Let C ω and C n−1 ω be, respectively, the spaces of continuous and n − 1-times continuously dif- ferentiable ω-periodic functions with the norms u C ω  max    ut   : t ∈ R  , u C n−1 ω  n  k1   u k−1   C ω , 1.2 and let  C n−1 ω be the space of functions u ∈ C n−1 ω for which u n−1 is absolutely continuous. 2 Boundary Value Problems We consider the functional differential equation u n tfut, 1.3 whose important particular case is the differential equation with deviated arguments u n tg  t, u  τ 1 t  , ,u n−1  τ n t  . 1.4 Throughout the paper, it is assumed that f : C n−1 ω → L ω is a continuous operator satisfying the condition f ∗ r ·sup    fu·   : u≤r  ∈ L ω for any r>0, 1.5 and g : R × R n → R is a function from the Carath ´ eodory class, satisfying the equality g  t  ω, x 1 , ,x n   g  t, x 1 , ,x n  1.6 for almost all t ∈ R and all x 1 , ,x n  ∈ R n . As for the functions τ k : R → R k  1, ,n,they are measurable on each finite interval and  τ k t  ω − τ k t  ω is an integer  k  1, ,n  1.7 foralmostallt ∈ R. A function u ∈  C n−1 ω is said to be an ω-periodic solution of 1.3 or 1.4 if it satisfies this equation almost everywhere on R. For the case τ k t ≡ tk  1, ,n, the problem on the existence and uniqueness of an ω-periodic solution of 1.4 has been investigated in detail see, e.g., 1–18 and the references therein.For1.3 and 1.4,whereτ k t / ≡ t k  1, ,n, the mentioned problem is studied mainly in the cases n ∈{1, 2} see 19–31, and for the case n>2, the problem remains so far unstudied. The present paper is devoted exactly to this case. Everywhere below the following notation will be used: ν k  ω 2  ω 2π  n−k−2  k  0, ,n− 2  ,ν n−1  1, 1.8 x −   |x|−x  /2forx ∈ R, 1.9 μumin    ut   :0≤ t ≤ ω} for u ∈ C ω . 1.10 1.2. Existence theorems The existence of an ω-periodic solution of 1.3 is proved in the cases where the operator f in the space C n−1 ω satisfies the conditions   ω 0 fusds  sgn  σu0  ≥ h  μu  − n−1  k1  1k   u k   C ω − c for μu > 0, 1.11      x t fusds     ≤ h  μu   n−1  k1  2k   u k   C ω  c for 0 ≤ t ≤ x ≤ ω, 1.12 I. Kiguradze et al. 3 or the conditions   ω 0 fusds  sgn  σu0  ≥ 0forμu >c 0 , 1.13      x t fusds     ≤ c 0  n−1  k0  k   u k   C ω for 0 ≤ t ≤ x ≤ ω. 1.14 Theorem 1.1. Let there exist an increasing function h : 0, ∞→ 0, ∞ and constants c ≥ 0,  ik ≥ 0 i  1, 2; k  1, ,n− 1,  ≥ 1,andσ ∈{−1, 1} such that hx → ∞ as x → ∞, n−1  k1   1k   2k  ν k < 1, 1.15 and inequalities 1.11 and 1.12 are satisfied in the space C n−1 ω .Then1.3 has at least one ω-periodic solution. Theorem 1.2. Let there exist constants c 0 ≥ 0,  k ≥ 0 k  0, ,n− 1,andσ ∈{−1, 1} such that n−1  k0  k ν k < 1, 1.16 and inequalities 1.13 and 1.14 are satisfied in the space C n−1 ω .Then1.3 has at least one ω-periodic solution. Theorems 1.1 and 1.2 imply the following propositions. Corollary 1.3. Let there exist constants λ>0, σ ∈{−1, 1}, and functions p ik ∈ L ω i, k  1, ,n, q ∈ L ω such that the inequalities g  t, x 1 , ,x n  sgn  σx 1  ≥ p 11 t   x 1   λ − n  k2 p 1k t   x k   − qt,   g  t, x 1 , ,x n    ≤ p 21 t   x 1   λ  n  k2 p 2k t   x k    qt 1.17 hold on the set R × R n . Let, moreover,  ω 0 p 11 tdt > 0, 1.18 and either λ<1 and n  k2 ν k−1  ω 0  p 1k sp 2k s  ds < 1, 1.19 or λ  1 and ν 0  ω 0    p 11 s  −  p 21 s  ds  n  k2 ν k−1  ω 0  p 1k sp 2k s  ds < 1, 1.20 where    ω 0 p 21 tdt/  ω 0 p 11 tdt.Then1.4 has at least one ω-periodic solution. 4 Boundary Value Problems Corollary 1.4. Let there exist constants c 0 ≥ 0, σ ∈{−1, 1}, and functions g 0 ∈ L ω , p k ∈ L ω k  1, ,n, q ∈ L ω such that  ω 0 g 0 sds  0 1.21 and the inequalities  g  t, x 1 , ,x n  − g 0 t  sgn  σx 1  ≥ 0 for   x 1   >c 0 ,   g  t, x 1 , ,x n    ≤ n  k1 p k t   x k    qt 1.22 hold on the set R × R n . If, moreover, n  k1 ν k−1  ω 0 p k sds < 1, 1.23 then 1.4 has at least one ω-periodic solution. 1.3. Uniqueness theorems The unique solvability of a periodic problem for 1.3 is proved in the cases where the operator f, for any u and v ∈ C n−1 ω , satisfies the conditions:   ω 0  fu  vs − fvs  ds  sgn  σu0  ≥  10 μu − n−1  k1  1k   u k   C ω for μu > 0, 1.24      x t  fu  vs − fvs  ds     ≤  20 μu n−1  k1  2k   u k   C ω for 0 ≤ t ≤ x ≤ ω, 1.25 or the conditions   ω 0  fu  vs − fvs  ds  sgn  σu0  > 0forμu > 0, 1.26      x t  fu  vs − fvs  ds     ≤  0 u C ω for 0 ≤ t ≤ x ≤ ω. 1.27 Theorem 1.5. Let there exist constants  20 ≥  10 > 0,  ik ≥ 0 i  1, 2; k  1, ,n − 1,and σ ∈{−1, 1} such that for arbitrary u, v ∈ C n−1 ω the operator f satisfies inequalities 1.24 and 1.25. If, moreover, inequality 1.15 holds, where    20 / 10 ,then1.3 has one and only one ω-periodic solution. Theorem 1.6. Let there exist constants  0 > 0 and σ ∈{−1, 1} such that for arbitrary u, v ∈ C n−1 ω an operator f satisfies conditions 1.26 and 1.27. If, moreover,  ω 0 f0sds  0 ,  0 ν 0 < 1, 1.28 then 1.3 has one and only one ω-periodic solution. I. Kiguradze et al. 5 From Theorem 1.5, the following corollary holds. Corollary 1.7. Let there exist a constant σ ∈{−1, 1} and functions p ik ∈ L ω i  1, 2; k  1, ,n such that for almost all t ∈ R and all x 1 , ,x n  and y 1 , ,y n  ∈ R n the conditions  g  t, x 1 , ,x n  − g  t, y 1 , ,y n  sgn  σ  x 1 − y 1  ≥ p 11 t   x 1 − y 1   − n  k2 p 1k t   x k − y k   ,   g  t, x 1 , ,x n  − g  t, y 1 , ,y n    ≤ n  k1 p 2k t   x k − y k   1.29 are satisfied. If, moreover, inequalities 1.18 and 1.20 hold, where    ω 0 p 21 sds/  ω 0 p 11 sds,then 1.4 has one and only one ω-periodic solution. Note that the functions p 1k k  2, ,n and p 2k k  1, ,n in this corollary as in Corollary 1.3 are nonnegative, and p 11 may change its sign. Consider now the equation u n tg  t, u  τt  , 1.30 which is d erived from 1.4 inthecasewheregt, x 1 , ,x n  ≡ gt, x 1  and τ 1 t ≡ τt.As above, we will assume that the function g : R × R → R belongs to the Carath ´ eodory class and gt  ω, xgt, x1.31 for almost all t ∈ R and all x ∈ R. As for the function τ : R → R, it is measurable on each finite interval and  τt  ω − τt  ω is an integer 1.32 foralmostallt ∈ R. Theorem 1.6 yields the following corollary. Corollary 1.8. Let there exist a constant σ ∈{−1, 1} and a function p ∈ L ω such that the condition 0 <  gt, x − gt, y  sgn  σx − y  ≤ pt|x − y| 1.33 holds for almost all t ∈ R and all x /  y. If, moreover,  ω 0 gs, 0ds  0,ν 0  ω 0 psds < 1, 1.34 then 1.30 has one and only one ω-periodic solution. 6 Boundary Value Problems 2. Auxiliary propositions 2.1. Lemmas on a priori estimates Everywhere in this section, we will assume that ν k k  0, ,n− 1 are numbers given by 1.13. Lemma 2.1. If u ∈ C n−1 ω ,then u C ω ≤ μuν 0   u n−1   C ω , 2.1   u k   C ω ≤ ν k   u n−1   C ω  k  1, ,n− 1  . 2.2 Proof. We choose t 0 ∈ 0,ω so that u  t 0   μu, 2.3 and suppose vtut − u  t 0  . 2.4 Then vt 0 vt 0  ω0. Thus   vt         t t 0 v  sds     ≤  t t 0   v  s   ds,   vt         t 0 ω t v  sds     ≤  t 0 ω t   v  s   ds for 0 ≤ t ≤ ω. 2.5 If we sum up these two inequalities, we obtain 2   vt   ≤  t 0 ω t 0   v  s   ds for 0 ≤ t ≤ ω. 2.6 Consequently, v C ω ≤ 1 2  t 0 ω t 0   v  s   ds. 2.7 However, u C ω ≤ μuv C ω ,  t 0 ω t 0   v  s   ds   ω 0   u  s   ds, 2.8 which together with the previous inequality yields u C ω ≤ μu 1 2  ω 0   u  s   ds ≤ μu 1 2 ω 1/2   ω 0   u  s   2 ds  1/2 . 2.9 On the other hand, by the Wirtinger inequality see 32, Theorem 258 and 13, Lemma 1.1, we have  ω 0   u  s   2 ds ≤  ω 2π  2n−4  ω 0   u n−1 s   2 ds ≤ ω  ω 2π  2n−4   u n−1   2 C ω . 2.10 Consequently, estimate 2.1 is valid. I. Kiguradze et al. 7 If now we take into account that u k ∈ C n−1−k ω and μu k 0 k  1, ,m, then the validity of estimates 2.2 becomes evident. Lemma 2.2. Let u ∈ C n−1 ω and   u n−1   C ω ≤ c 0  n−1  k0  k   u k   C ω , 2.11 where c 0 and  k k  0, ,n− 1 are nonnegative constants. If, moreover, δ  n−1  k0  k ν k < 1, 2.12 then   u n−1   C ω ≤ 1 − δ −1  c 0   0 μu  , 2.13 u C n−1 ω ≤ μu1 − δ −1  c 0   0 μu  n−1  k0 ν k . 2.14 Proof. By Lemma 2.1, the function u satisfies inequalities 2.1 and 2.2. In view of these in- equalities from 2.11 we find   u n−1   C ω ≤ c 0   0 μu  n−1  k0  k ν k    u n−1   C ω . 2.15 Hence, by virtue of condition 2.12, we have estimate 2.13. On the other hand, according to 2.13, inequalities 2.1 and 2.2 result in 2.14. Lemma 2.3. Let u ∈ C n−1 ω and μu ≤ ϕ    u n−1   C ω  ,   u n−1   C ω ≤ c 0  n−1  k1  k   u k   C ω , 2.16 where ϕ : 0, ∞→ 0, ∞ is a nondecreasing function, c 0 ≥ 0,  k ≥ 0 k  1, ,n− 1,and δ  n−1  k1  k ν k < 1. 2.17 Then u C n−1 ω ≤ r 0 , 2.18 where r 0  ϕ  1 − δ −1 c 0  1 − δ −1 c 0 n−1  k0 ν k . 2.19 Proof. Inequalities 2.16 and 2.17 imply inequalities 2.11 and 2.12,where 0  0. However, by Lemma 2.2, these inequalities guarantee the validity of the estimates   u n−1   C ω ≤ 1 − δ −1 c 0 , u C n−1 ω ≤ μu1 − δ −1 c 0 n−1  k0 ν k . 2.20 8 Boundary Value Problems On the other hand, according to the first inequality in 2.16,wehave μu ≤ ϕ  1 − δ −1 c 0  . 2.21 Consequently, estimate 2.18 is valid, where r 0 is a number given by equality 2.19. Analogously, from Lemma 2.2, the following hold. Lemma 2.4. Let u ∈ C n−1 ω and μu ≤ c 0 ,   u n−1   C ω ≤ c 0  n−1  k0  k   u k   C ω , 2.22 where c 0 ≥ 0,  k ≥ 0 k  0, ,n− 1. If, moreover, inequality 2.12 holds, then estimate 2.18 is valid, where r 0   1 1 − δ −1  1   0  n−1  k0 ν k  c 0 . 2.23 2.2. Lemma on the solvability of a periodic problem Below, by C n−1 0,ω we denote the space of n − 1-times continuously differentiable func- tions u : 0,ω → R with the norm u C n−1 0,ω  n  k1 max    u k−1 t   :0≤ t ≤ ω  , 2.24 and by L0,ω we denote the space of Lebesgue integrable functions u : 0,ω → R with the norm u L0,ω   ω 0   ut   dt. 2.25 Consider the differential equation u n tfut2.26 with the periodic boundary conditions u i−1 0u i−1 ωi  1, ,n, 2.27 where f : C n−1 0,ω → L0,ω is a continuous operator such that f r ·sup    fu·   : u C n−1 0,ω ≤ r  ∈ L  0,ω  2.28 for any r>0. The following lemma is valid. I. Kiguradze et al. 9 Lemma 2.5. Let there exist a linear, bounded operator p : C n−1 0,ω → L0,ω and a positive constant r 0 such that the linear differential equation u n tput2.29 with the periodic conditions 2.27 has only a trivial solution and for an arbitrary λ ∈0, 1 every solution of the differential equation u n tλput1 − λfut, 2.30 satisfying condition 2.27, admits the estimate u C n−1 0,ω ≤ r 0 . 2.31 Then problem 2.26, 2.27 has at least one solution. For the proof of this lemma see 33, Corollary 2. Lemma 2.6. Let f : C n−1 ω → L ω be a continuous operator satisfying condition 1.5 for any r>0.Let, moreover, there exist constants a /  0 and r 0 > 0 such that for an arbitrary λ ∈0, 1, every ω-periodic solution of the functional differential equation u n tλau01 − λfut2.32 admits estimate 2.18.Then1.3 has at least one ω-periodic solution. Proof. Let c 1 , ,c n be arbitrary constants. Then the problem y 2n t0,y i−1 00,y i−1 ωc i i  1, ,n2.33 has a unique solution. Let us denote by yt; c 1 , ,c n  the solution of that problem. For any u ∈ C n−1 0,ω, we set zutut − y  t; uω − u0, ,u n−1 ω − u n−1 0  for 0 ≤ t ≤ ω, 2.34 and extend zu· to R periodically with a period ω. Then, it is obvious that z : C n−1 0,ω → C n−1 ω is a linear, bounded operator. Suppose futf  zu  t. 2.35 Consider the boundary value problem 2.26, 2.27. If the function u is an ω-periodic solution of 1.3, then its restriction to 0,ω is a solution of problem 2.26, 2.27, and vice versa, if u is a solution of problem 2.26, 2.27, then its periodic extension to R with a period ω is an ω-periodic solution of 1.3. Thus to prove the lemma, it suffices to state that problem 2.26, 2.27 has at least one solution. By virtue of equalities 2.34, 2.35 and condition 1.5, f : C n−1 0,ω → L0,ω is a continuous operator, satisfying condition 2.28 for any r>0. On the other hand, it is evident that if put ≡ αu0, then problem 2.29, 2.27 has only a trivial solution. By these conditions and Lemma 2.5, problem 2.26, 2.27 is solvable if for any λ ∈0, 1 every solution u of problem 2.30, 2.27,whereput ≡ αu0, admits estimate 2.31. Let u be a solution of problem 2.30, 2.27 for some λ ∈0, 1 . Then its periodic extension to R with a period ω is a solution of 2.32, and according to one of the conditions of the lemma, admits estimate 2.18. Therefore, estimate 2.31 is valid. 10 Boundary Value Problems 3. Proof of the main results Proof of Theorem 1.1. Without loss of generality, it can be assumed that h00. On the other hand, according to condition 1.15, we can choose a constant a so that σa > 0 and the numbers  k   1k   2k  k  1, ,n− 2  , n−1   1n−1   2n−1  ων 0 |a| 3.1 satisfy inequality 2.17. Let h 0 xmin  |a|ωx, hx  , 3.2 let h −1 0 be a function, inverse to h 0 , ϕxh −1 0  n−1  k1  1k ν k  x  c  ,c 0  2c, 3.3 and let r 0 be a number given by equality 2.19. By virtue of Lemma 2.6, to prove the theorem, it suffices to state that for any λ ∈0, 1 every ω-periodic solution of 2.32 admits estimate 2.18. Due to condition 1.12,from2.32, we find   u n−1   C ω ≤ max       x t u n sds     :0≤ t ≤ x ≤ ω  ≤ λω|a|   u0   1 − λh  μu   n−1  k1  2k   u k   C ω  c. 3.4 On the other hand, if μu > 0, then by condition 1.11 we have 0    w 0 u n sds  sgn  σu0  ≥ λω|a|   u0   1 − λh  μu  − n−1  k1  1k   u k   C ω − c, 3.5 and consequently, λω|a|   u0   1 − λh  μu  ≤ n−1  k1  1k   u k   C ω  c. 3.6 If μu > 0, then by Lemma 2.1 and notations 3.1–3.3,from3.4 and 3.6, inequali- ties 2.16 hold. And if μu0, then by Lemma 2.1,   u0   ≤ ν 0   u n−1   C ω . 3.7 On the other hand, hμu  h00. Thus from 3.4 we obtain   u n−1   C ω ≤ ων 0 |a|   u n−1   C ω  n−1  k1  2k   u k   C ω  c. 3.8 [...]... a, On periodic solutions of nonlinear functional differential equations,” Geor˚z gian Mathematical Journal, vol 6, no 1, pp 47–66, 1999 20 I Kiguradze and B Puˇ a, On periodic solutions of systems of differential equations with deviating ˚z arguments,” Nonlinear Analysis: Theory, Methods & Applications, vol 42, no 2, pp 229–242, 2000 21 R Hakl, A Lomtatidze, and B Puˇ a, On periodic solutions of first... 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Proof of Theorem 1.6 For v t ≡ 0, 1.26 – 1.28 yield conditions 1.13 , 1.14 , and 1.16 , where w f 0 s ds, c0 0 k k 1, , n − 1 3.29 0 Consequently, all the conditions of Theorem 1.2 are satisfied which guarantee the existence of at least one ω -periodic solution of 1.3 Suppose now that u1 and u2 are arbitrary ω -periodic solutions of 1.3 and u t u2 t − u1 t If we assume that μ u > 0, then in view of. .. translation in Mathematical Notes, vol 37, no.1, pp 28–36, 1985, Russian 9 G T Gegelia, On bounded and periodic solutions of even-order nonlinear ordinary differential equations,” Differentsial’nye Uravneniya, vol 22, no 3, pp 390–396, 547, 1986, Russian 10 G T Gegelia, On periodic solutions of ordinary differential equations,” in Qualitative Theory of Differential Equations (Szeged, 1988), vol 53 of Colloquia... 1.34 imply conditions 1.26 – 1.28 , where quently, the operator f satisfies all the conditions of Theorem 1.6 3.32 0 ω p 0 s ds Conse- 4 Examples From the main results of the present paper new and optimal in some sense sufficient conditions for the existence of periodic solutions of linear and sublinear differential equations with I Kiguradze et al 15 deviated arguments and differential equations with bounded... inequalities 1.11 and 1.12 , where w hx 10 x, f 0 s ds c 3.22 0 Consequently, all the conditions of Theorem 1.1 are satisfied which guarantee the existence of at least one ω -periodic solution of 1.3 It remains to prove that 1.3 has no more than one ω -periodic solution Let u1 and u2 be arbitrary ω -periodic solutions of 1.3 and u2 t − u1 t ut 3.23 If we assume that μ u > 0, then from 1.24 we find 10 μ... 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Corporation Boundary Value Problems Volume 2008, Article ID 389028, 18 pages doi:10.1155/2008/389028 Research Article On Periodic Solutions of Higher-Order Functional Differential Equations I Lomtatidze, and B. P ˚ u ˇ za, On periodic solutions of first order nonlinear functional dif- ferential equations of non-Volterra’s type,” Memoirs on Differential Equations and Mathematical Physics, vol all the conditions of Theorem 1.1 are fulfilled which guarantee the existence of at least one ω -periodic solution of 1.4. I. Kiguradze et al. 13 Proof of Corollary 1.4. Without loss of generality,