EXISTENCE OF PERIODIC SOLUTIONS FOR 2nTH-ORDER NONLINEAR p-laplacian DIFFERENCE EQUATIONS

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1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 46, Number 5, 2016 EXISTENCE OF PERIODIC SOLUTIONS FOR 2nTH-ORDER NONLINEAR p-laplacian DIFFERENCE EQUATIONS HAIPING SHI, XIA LIU AND YUANBIAO ZHANG ABSTRACT. By using the critical point theory, the existence of periodic solutions for 2nth-order nonlinear p- Laplacian difference equations is obtained. The main approaches used in our paper are variational techniques and the Saddle Point theorem. The problem is to solve the existence of periodic solutions for 2nth-order p-laplacian difference equations. The results obtained successfully generalize and complement the existing ones. 1. Introduction. In this paper, we consider the following 2nthorder p-laplacian difference equation (1.1) n (r n φ p ( n u 1 )) + ( 1) n q φ p ( u 1 ) = ( 1) n f(, u ), n Z(1), Z, where is the forward difference operator u = u +1 u, n u = ( n 1 u ), φ p (s) is the p-laplacian operator φ p (s) = s p 2 s, 1 < p <, {r } and {q } are real sequences, f C(Z R, R), T is a given positive integer, r +T = r > 0, q +T = q 0, f( + T, v) = f(, v). We may thin of equation (1.1) as a discrete analogue of the following 2nth-order differential equation: 2010 AMS Mathematics subject classification. Primary 39A11. Keywords and phrases. Existence, periodic solutions, nonlinear difference equation, discrete variational theory, p-laplacian. This project is supported by the Specialized Research Fund for Doctoral Program of Higher Eduction of China (No ), National Natural Science Foundation of China (No ), Natural Science Foundation of Guangdong Province (No. S ) and Project for High Level Talents of Guangdong Higher Education Institutes. The first author is the corresponding author. Received by the editors on August 21, 2013, and in revised form on November 3, DOI: /RMJ Copyright c 2016 Rocy Mountain Mathematics Consortium 1679

2 1680 HAIPING SHI, XIA LIU AND YUANBIAO ZHANG (1.2) d n [ ( d n )] u(t) dt n r(t)φ p dt n + ( 1) n q(t)(φ p (u (t))) = ( 1) n f(t, u(t)), t R. Existence of periodic solutions of higher-order differential equations has been the subject of many investigations [8, 15, 16, 17, 32, 36, 37]. By using various methods and techniques, such as fixed point theory, the Kaplan-Yore method, critical point theory, coincidence degree theory, bifurcation theory and dynamical system theory, etc., a series of existence results for periodic solutions have been obtained in the literature. Recently, the difference equations have widely occurred as the mathematical models describing real life situations in probability theory, matrix theory, electrical circuit analysis, combinatorial analysis, queuing theory, number theory, psychology and sociology, etc. For the general bacground of difference equations, one can refer to the monographs [1, 2, 3, 28]. Since the last decade, there has been much progress on the qualitative properties of difference equations, which included results on stability and attractivity [18, 28, 31, 46] and results on oscillation and other topics [1, 2, 3, 23, 24, 25, 31, 42, 43, 44, 45]. Only a few papers discuss the periodic solutions of higherorder difference equations. Therefore, it is worthwhile to explore this topic. The widely used tools for the existence of periodic solutions of difference equations are the various fixed point theorems in cones [1, 2, 3, 28]. It is well nown that critical point theory is a very powerful tool that deals with the problems of differential equations [8, 11, 14, 21, 22, 32, 38, 41]. Only since 2003, critical point theory has been employed to establish sufficient conditions on the existence of periodic solutions of difference equations. By using the critical point theory, Guo and Yu [23, 24, 25] and Shi et al. [39] established sufficient conditions on the existence of periodic solutions of second-order nonlinear difference equations. Compared to firstorder or second-order difference equations, the study of higher-order equations has received considerably less attention (see, for example, [4, 5, 8, 12, 13, 19, 28, 33, 35] and the references contained therein). Ahlbrandt and Peterson [4] in 1994 studied the 2nth-order difference

3 EXISTENCE OF PERIODIC SOLUTIONS 1681 equation of the form (1.3) n i i ( i) i u( i) ) = 0 i=0 in the context of the discrete calculus of variations, and Peil and Peterson [35] studied the asymptotic behavior of solutions of (1.3) with r i () 0 for 1 i n 1. In 1998, Anderson [5] considered (1.3) for Z(a) and obtained a formulation of generalized zeros and (n, n)- disconjugacy for (1.3). Migda [33] in 2004 studied an mth-order linear difference equation. If q 0, p = 2 and n = 2, and replacing f(, u ) with f(, u ), (1.1) reduces to the following equation: (1.4) u 2 ) + f(, u ) = 0, Z. In 2005, Cai, Yu and Guo [10] obtained some criteria for the existence of periodic solutions of the fourth-order difference equation for (1.4). Recently, Cai and Yu [9] obtained some criteria for the existence of periodic solutions of the following 2nth-order difference equation (1.5) n (r n n u n ) + f(, u ) = 0, n Z(3), Z, for the case where f grows superlinearly at both 0 and. A great deal of wor has also been done on the study of the existence of solutions to discrete boundary value problems with the p-laplacian operator. Because of potential applications in many fields, we refer the reader to the monograph by Agarwal, et al., and some recent contributions [6, 7, 8, 26, 27, 29, 30, 40]. However, to the best of our nowledge, the results on periodic solutions of higher-order nonlinear difference equations with p-laplacian are very scarce in the literature. The main intention of this paper is to give some sufficient conditions for the existence and multiplicity of periodic and subharmonic solutions for 2nth-order nonlinear p-laplacian difference equations. The proof is based on the Saddle Point theorem in combination with the variational technique. In particular, our results generalize and complement the results in the literature [9, 10]. In fact, one can see the following Remar 1.4 for details. The motivation for the present wor stems from the recent papers [13, 20].

4 1682 HAIPING SHI, XIA LIU AND YUANBIAO ZHANG Let r = q = min {r }, r = max {r }, Z(1,T ) Z(1,T ) min {q }, q = max {q }. Z(1,T ) Z(1,T ) Now we state the main results of this paper. Theorem 1.1. Assume that the following hypotheses are satisfied: (F 1 ) there exists a functional F (, v) C 1 (Z R, R), such that F (, v) v = f(, v) F ( + T, v) = F (, v); (F 2 ) there exists a constant M 0 > 0 for all (, v) Z R such that f(, v) M 0 ; (F 3 ) F (, v) + uniformly for Z as v +. Then, for any given positive integer m > 0, equation (1.1) has at least one mt -periodic solution. Remar 1.2. Assumption (F 2 ) implies that there exists a constant M 1 > 0 such that (F 2) F (, v) M 1 + M 0 v, for all (, v) Z R. Theorem 1.3. Assume that (F 1 ) holds; further, (F 4 ) there exist constants R 1 > 0 and α, 1 < α < 2, such that, for Z and v R 1, 0 < f(, v)v α pf (, v); 2 (F 5 ) there exist constants a 1 > 0, a 2 > 0 and γ, 1 < γ α such that F (, v) a 1 v (γ/2)p a 2, for all (, v) Z R. Then, for any given positive integer m > 0, equation (1.1) has at least one mt -periodic solution.

5 EXISTENCE OF PERIODIC SOLUTIONS 1683 Remar 1.4. Assumption (F 4 ) implies that, for each Z, there exist constants a 3 > 0 and a 4 > 0 such that (F 4) F (, v) a 3 v (α/2)p + a 4, for all (, v) Z R. Remar 1.5. The results of Theorems 1.1 and 1.3 ensure that equation (1.1) has at least one mt -periodic solution. However, in some cases, we are interested in the existence of nontrivial periodic solutions for equation (1.1). In this case, we have Theorem 1.6. Assume that (F 1 ) holds; further, (F 6 ) F (, 0) = 0, f(, v) = 0 if and only if v = 0, for all Z; (F 7 ) there exists a constant α, 1 < α < 2, such that, for Z, 0 < f(, v)v α pf (, v), for all v 0; 2 (F 8 ) there exist constants a 5 > 0 and γ, 1 < γ α, such that F (, v) a 5 v (γ/2)p, for all (, v) Z R. Then, for any given positive integer m > 0, equation (1.1) has at least one nontrivial mt -periodic solution. Theorem 1.7. Assume that (F 1 ) (F 3 ) and (F 6 ) hold; further, (F 9 ) there exist constants a 6 > 0 and θ, 0 < θ < 2, such that F (, v) a 6 v (θ/2)p, for all (, v) Z R. Then, for any given positive integer m > 0, equation (1.1) has at least one nontrivial mt -periodic solution. If q 0, p = 2 and n = 2, replacing f(, u ) with ( 1) n+1 f(, u ), equation (1.1) reduces to equation (1.5). Then, we have the following results. Theorem 1.8. Assume that (F 1 ) holds; further, (F 10 ) F (, 0) = 0, for all Z;

6 1684 HAIPING SHI, XIA LIU AND YUANBIAO ZHANG (F 11 ) there exists a constant α, 1 < α < 2, such that, for Z, αf (, v) vf(, v) < 0, for all v = 0; (F 12 ) there exist constants a 7 > 0 and γ, 1 < γ α, such that F (, v) a 7 v γ, for all (, v) Z R. Then, for any given positive integer m > 0, equation (1.5) has at least one nontrivial mt -periodic solution. Theorem 1.9. Assume that (F 1 ), (F 2 ) and (F 6 ) hold; further, (F 13 ) F (, v) uniformly for Z as v + ; (F 14 ) there exist constants a 8 > 0 and θ, 0 < θ < 2, such that F (, v) a 8 v θ, for all (, v) Z R. Then, for any given positive integer m > 0, equation (1.5) has at least one nontrivial mt -periodic solution. Remar When β > 2, Cai, Yu and Guo [10, Theorem 1.1] have obtained some criteria for the existence of periodic solutions of (1.4) and Cai and Yu [9, Theorem 1.1] have obtained some criteria for the existence of periodic solutions of (1.5). When β < 2, we can still find the periodic solutions of equations (1.4) and (1.5). Hence, Theorems generalize and complement the existing ones. The rest of the paper is organized as follows. First, in Section 2, we shall establish the variational framewor associated with (1.1) and transfer the problem of the existence of periodic solutions of (1.1) into that of the existence of critical points of the corresponding functional. Some related fundamental results will also be recalled. Then, in Section 3, we shall complete the proof of the results by using the critical point method. Finally, in Section 4, we shall give two examples to illustrate the main results. Regarding the basis for variational methods, we refer the reader to [32, 34, 38]. 2. Variational structure and some lemmas. In order to apply the critical point theory, we shall establish the corresponding varia-

7 EXISTENCE OF PERIODIC SOLUTIONS 1685 tional framewor for (1.1) and give some lemmas which will be of fundamental importance in proving our main results. We start with some basic notation. Let S be the set of sequences u = (..., u,..., u 1, u 0, u 1,..., u,...) = {u } + =, that is, S = {{u } u R, Z}. For any u, v S, a, b R, au + bv is defined by au + bv = {au + bv } + =. Then S is a vector space. For any given positive integers m and T, E mt subspace of S by is defined as a E mt = {u S u +mt = u, for all Z}. Clearly, E mt is isomorphic to R mt. E mt can be equipped with the inner product (2.1) u, v = u j v j, for all u, v E mt, j=1 by which the norm can be induced by (2.2) u = ( mt 1/2 uj) 2, for all u E mt. j=1 It is obvious that E mt with the inner product (2.1) is a finitedimensional Hilbert space and linearly homeomorphic to R mt. On the other hand, we define the norm s on E mt as follows: (2.3) u s = for all u E mt and s > 1. ( mt ) 1/s u j s, j=1

8 1686 HAIPING SHI, XIA LIU AND YUANBIAO ZHANG Since u s and u 2 are equivalent, there exist constants c 1, c 2 such that c 2 c 1 > 0, and (2.4) c 1 u 2 u s c 2 u 2, for all u E mt. Clearly, u = u 2. For all u E mt, define the functional J on E mt as follows: (2.5) J(u) = 1 r 1 n u 1 p 1 q u p + F (, u ) p p where := H(u) + F (, u ), H(u) = 1 r 1 n u 1 p + 1 q u p, p p F (, v) v = f(, v). It is evident that J C 1 (E mt, R) and, for any u = {u } Z E mt, by using u 0 = u mt and u 1 = u mt +1, we can compute the partial derivative as J u = ( 1) n n (r n φ p ( n u 1 )) q φ p ( u 1 ) + f(, u ). Thus, u is a critical point of J on E mt if and only if n (r n φ p ( n u 1 )) + ( 1) n q φ p ( u 1 ) = ( 1) n f(, u ), for all Z(1, mt ). Due to the periodicity of u = {u } Z E mt and f(, v) in the first variable, we reduce the existence of periodic solutions of (1.1) to the existence of critical points of J on E mt, that is, the functional J is just the variational framewor of (1.1).

9 EXISTENCE OF PERIODIC SOLUTIONS 1687 Let P = be an mt mt matrix. By matrix theory, we see that the eigenvalues of P are ( (2.6) λ j = 2 1 cos 2j ) mt π, j = 0, 1, 2,..., mt 1. Thus, λ 0 = 0, λ 1 > 0, λ 2 > 0,..., λ mt 1 > 0. Therefore, (2.7) { λ min = min{λ 1, λ 2,..., λ mt 1 } = 2 ( 1 cos 2 mt π) λ max = max{λ 1, λ 2,..., λ mt 1 } { 4 when mt is even, = 2 ( 1 + cos 1 mt π) when mt is odd. Let Then W = er P = {u E mt P u = 0 R mt }. W = {u E mt u = {c}, c R}. Let V be the direct orthogonal complement of E mt to W, i.e., E mt = V W. For convenience, we identify u E mt with u = (u 1, u 2,, u mt ). Let E be a real Banach space, J C 1 (E, R), i.e., J is a continuously Fréchet-differentiable functional defined on E. J is said to satisfy the Palais-Smale condition (abbreviated PS condition) if any sequence {u (i) } E for which {J(u (i) )} is bounded and J (u (i) ) 0 (i ) possesses a convergent subsequence in E. Let B ρ denote the open ball in E about 0 of radius ρ, and let B ρ denote its boundary.

10 1688 HAIPING SHI, XIA LIU AND YUANBIAO ZHANG Lemma 2.1 (Saddle Point theorem [32, 38]). Let E be a real Banach space, E = E 1 E 2, where E 1 {0} and is finite-dimensional. Suppose that J C 1 (E, R) satisfies the PS condition and (J 1 ) there exist constants σ, ρ > 0 such that J Bρ E 1 σ; (J 2 ) there exists e B ρ E 1 and a constant ω σ such that J e+e2 ω. Then J possesses a critical value c ω, where c = inf max J(h(u)), Γ = {h C(B ρ E 1, E) h Bρ E 1 = id} u B ρ E 1 h Γ and id denotes the identity operator. Lemma 2.2. Assume that (F 1 ) (F 3 ) are satisfied. Then J satisfies the PS condition. Proof. Let {u (i) } E mt be such that {J(u (i) )} is bounded and J (u (i) ) 0 as i. Then there exists a positive constant M 2 such that J(u (i) ) M 2. Let u (i) = v (i) + w (i) V + W. For i large enough, since u 2 J (u (i) ), u = H (u (i) ), u + f(, u (i) )u, and combining with (F 2 ) and (F 3 ), we have H (u (i) ), v (i) f(, u (i) )v(i) + v(i) 2 M 0 v (i) + v(i) 2 On the other hand, we now that H (u (i) ), v (i) = r n v (i) (M 0 mt + 1 ) v (i) 2. mt p + q v (i) p = ph(v (i) ).

11 EXISTENCE OF PERIODIC SOLUTIONS 1689 Since H(v (i) ) r p cp 1 [(x(i) ) P (x (i) )] p/2 + q p cp 1 [(v(i) ) P (v (i) )] p/2 r p cp 1 λp/2 min x(i) p 2 + q p cp 1 λp/2 min v(i) p 2, and H(v (i) ) r p cp 2 [(x(i) ) P (x (i) )] p/2 + q p cp 2 [(v(i) ) P (v (i) )] p/2 λ (n 1)p/2 min r p cp 2 λp/2 max x (i) p 2 + q p cp 2 λp/2 max v (i) p 2, mt v (i) p 2 x(i) p 2 = ( n 2 v (i) +1 n 2 v (i) λ p/2 max )p ( n 2 v (i) )p λ (n 1)p/2 max v (i) p 2, where x (i) = ( n 1 v (i) 1, n 1 v (i) 2,..., n 1 v (i) mt ), we get (2.8) r p cp 1 λnp/2 min v(i) p 2 + q p cp 1 λp/2 min v(i) p 2 H(v(i) ) Thus, we have r p cp 2 λnp/2 max v (i) p 2 + q p cp 2 λp/2 max v (i) p 2. rc p 1 λ(np)/2 min v (i) p 2 + q p cp 1 λp/2 min v(i) p 2 (M 0 mt + 1) v (i) 2. The above inequality implies that {v (i) } is bounded. Next, we shall prove that {w (i) } is bounded. Since M 2 J(u (i) ) = H(u (i) ) + F (, u (i) ) mt = H(v (i) ) + [F (, u (i) ) F (, w(i) )] + F (, w (i) ),

12 1690 HAIPING SHI, XIA LIU AND YUANBIAO ZHANG combining with (2.8), we get F (, w (i) ) M 2 + H(v (i) ) + F (, u (i) ) F (, w(i) ) M 2 + p cp 2 λ(np)/2 max + q ) p cp 2 λp/2 max v (i) p 2 + f(, w (i) + θv(i) )v(i) M 2 + p cp 2 λ(np)/2 max + q ) p cp 2 λp/2 max v (i) p 2 +M 0 mt v (i) 2, where θ (0, 1). It is not difficult to see that is bounded. { mt F (, w (i) ) } By (F 3 ), {w (i) } is bounded. Otherwise, assume that w (i) 2 + as i. Since there exist z (i) R, i N, such that w (i) = (z (i), z (i),..., z (i) ) E mt, then w (i) 2 = ) 1/2 ( ) 1/2 w (i) 2 = z (i) 2 = mt z (i) + ( mt as i. Since F (, w (i) ) = F (, z(i) ), then F (, w (i) ) + as i. This contradicts the fact that { mt F (, w(i) )} is bounded. Thus the PS condition is verified. Lemma 2.3. Assume that (F 1 ), (F 4 ) and (F 5 ) are satisfied. Then J satisfies the PS condition. Proof. Let {u (i) } E mt be such that {J(u (i) )} is bounded and J (u (i) ) 0 as i. Then there exists a positive constant M 3 such that J(u (i) ) M 3. For i large enough, we have J (u (i) ), u (i) u (i) 2.

13 EXISTENCE OF PERIODIC SOLUTIONS 1691 So M p u(i) 2 J(u (i) ) 1 p J (u (i) ), u (i) Tae By (F 4 ), we have = F (, u (i) ) 1 p I 1 = { Z(1, mt ) u (i) R 1}, I 2 = { Z(1, mt ) u (i) < R 1}. f(, u (i) )u(i). M p u(i) 2 F (, u (i) ) 1 f(, u (i) p I 1 = F (, u (i) ) α 2 ( 1 α p I 2 ) mt F (, u (i) I 1 F (, u (i) ) [ α pf (, u(i) 2 )u(i) ) f(, u(i) 1 f(, u (i) p I 2 ) 1 f(, u (i) p I 2 )u(i) ]. )u(i) The continuity of (α/2)pf (, v) f(, v)v with respect to the second variable implies that there exists a constant M 4 > 0 such that α 2 pf (, v) f(, v)v M 6, for Z(1, mt ) and v R 1. Therefore, By (F 5 ), we get M p u(i) 2 ( 1 α ) F (, u (i) 2 ) 1 p mt M 4. M ( p u(i) 2 1 α ( )a 1 u (i) 2 (γ/2)p 1 α ) a 2 mt 1 2 p mt M 4 )u(i)

14 1692 HAIPING SHI, XIA LIU AND YUANBIAO ZHANG ( 1 α )a 1 u (i) 2 (γ/2)p M 5, where M 5 = (1 (α/2))a 2 mt + (1/p)mT M 4. Combining with (2.4), we have M ( p u(i) 2 1 α ) a 1 c (γ/2)p 1 u (i) (γ/2)p 2 M 5. 2 Thus, ( 1 α ) a 1 c (γ/2)p 1 u (i) (γ/2)p p u(i) 2 M 3 + M 5. This implies that { u (i) 2 } is bounded on the finite-dimensional space E mt. As a consequence, it has a convergent subsequence. 3. Proof of the main results. In this section, we shall prove our main results by using the critical point method. Proof of Theorem 1.1. By Lemma 2.2, we now that J satisfies the PS condition. In order to prove Theorem 1.1 by using the Saddle theorem, we shall prove conditions (J 1 ) and (J 2 ). From (2.8) and (F 2), for any v V, J(v)= H(v) + F (, v ) p cp 1 λ(np)/2 min + q p cp 1 λp/2 min p cp 1 λ(np)/2 min + q p cp 1 λp/2 min ) v p2 +mt M 1+M 0 v ) v p 2 +mt M 1+M 0 mt v 2 as v 2 +. Therefore, it is easy to see that condition (J 1 ) is satisfied. In the following, we shall verify condition (J 2 ). For any w W, w = (w 1, w 2,..., w mt ), there exists a z R such that w = z, for all Z(1, mt ). By (F 3 ), we now that there exists a constant R 0 > 0 such that F (, z) > 0 for Z and z > R 0. Let

15 EXISTENCE OF PERIODIC SOLUTIONS 1693 M 6 = min Z, z R0 F (, z), M 7 = min{0, M 6 }. Then So, we have F (, z) M 7, for all (, z) Z R. J(w) = F (, w ) = F (, z) mt M 7, for all w W. The conditions of (J 1 ) and (J 2 ) are satisfied. Proof of Theorem 1.2. By Lemma 2.3, J satisfies the PS condition. To apply the Saddle Point theorem, it suffices to prove that J satisfies conditions (J 1 ) and (J 2 ). For any w W, since H(w) = 0, we have By (F 5 ), J(w) = F (, w ). J(w) a 1 w (γ/2)p a 2 mt a 2 mt. Combining with (F 4), (2,4) and (2.8), for any v V, we get, as before, J(v) p cp 1 λ(np)/2 min p cp 1 λ(np)/2 min + q p cp 1 λp/2 min + q ) p cp 1 λp/2 min v p 2 + a 3 v (α/2)p + a 4 mt ) v p 2 + a 3c (α/2)p 2 v (α/2)p 2 + a 4 mt. Let µ = a 2 mt. Since 1 < α < 2, there exists a constant ρ > 0 large enough such that J(v) µ 1 < µ, for all v V, v 2 = ρ. Thus, by Lemma 2.1, equation (1.1) has at least one mt -periodic solution. Proof of Theorem 1.3. Similarly to the proof of Lemma 2.3, we can prove that J satisfies the PS condition. We shall prove this theorem

16 1694 HAIPING SHI, XIA LIU AND YUANBIAO ZHANG by means of the Saddle Point theorem. Firstly, we verify the condition (J 1 ). In fact, (F 4 ) clearly implies (F 4). For any v V, by (F 4) and equation (2.4), we again have J(v) as v 2 +. Next, we show that J satisfies the condition (J 2 ) for any given v 0 V and w W. Let u = v 0 + w. So J(u) = H(u) + F (, u ) = H(v 0 ) + F (, (v 0 ) + w ) p cp 2 λ(np)/2 max + q ) p cp 2 λp/2 max v 0 p 2 + a 5 p cp 2 λ(np)/2 max + q ) p cp 2 λp/2 max v 0 p 2 + a 5 = ) p cp 2 λ(np)/2 max + q p cp 2 λp/2 max p cp 2 λ(np)/2 max p cp 2 λ(np)/2 max + q p cp 2 λp/2 max + q p cp 2 λp/2 max + a 5 c (γ/2)p 1 w (γ/2)p 2. (v 0 ) + w ) (γ/2)p v 0 p 2 +a 5c (γ/2)p 1 ) ) (v 0 ) + w (γ/2)p [ mt v 0 p 2 +a 5c (γ/2)p 1 ] (γ/4)p (v 0 ) +w 2 [ v w 2 2] (γ/4)p v 0 p 2 + a 5c (γ/2)p 1 v 0 (γ/2)p 2 Since 1 < γ < 2, there exists a constant δ > 0 small enough such that [ ( J(v 0 +w) δ (γ/2)p a 5 c (γ/2)p r 1 p cp 2 λ(np)/2 max + q ] p cp 2 λp/2 max )δ p (γ/2)p > 0, for v 0 V, v 0 2 = δ and for any w W. Tae ν = δ (γ/2)p [ a 5 c (γ/2)p 1 p cp 2 λ(np)/2 max + q ] p cp 2 λp/2 max )δ p (γ/2)p.

17 EXISTENCE OF PERIODIC SOLUTIONS 1695 Then, for v 0 V and for any w W, we get v 0 2 = δ and J(v 0 + w) ν > 0. By the Saddle Point theorem, a critical point u E mt exists which corresponds to an mt -periodic solution of (1.1). In the following, we shall prove that u is nontrivial, i.e., u / W. Otherwise, u W. Since J (u) = 0, then n (r n φ p ( n u 1 )) + q φ p ( u 1 ) = ( 1) n f(, u ). On the other hand, u W implies that there is a point z R such that u = z, for all Z(1, mt ), that is, u 1 = u 2 = = u = = z. Thus, f(, u ) = f(, z) = 0, for all Z(1, mt ). now that z = 0. Therefore, by (F 6 ), we have From (F 6 ), we J(u) = F (, u ) = F (, 0) = 0. This contradicts J(u) ν > 0. The proof of Theorem 1.6 is finished. Remar 3.1. The techniques of the proof of Theorem 1.7 are exactly the same as those carried out in the proof of Theorem 1.6. We do not repeat them here. Remar 3.2. Due to Theorems 1.6 and 1.7, the conclusion of Theorems 1.8 and 1.9 is obviously true. 4. Examples. As an application of the main theorems, we give two examples to illustrate our results. Example 4.1. For all n Z(1) and Z, assume that (4.1) n (r n φ p ( n u 1 )) + ( 1) n q φ p ( u 1 ) = ( 1) n αpψ()u αp 1, where {r } and {q } are real sequences, ψ is continuously differentiable and ψ() > 0, T is a given positive integer, r +T = r > 0, q +T =

18 1696 HAIPING SHI, XIA LIU AND YUANBIAO ZHANG q 0, ψ( + T ) = ψ(), 1 < p <, 1 < α < 2. We have f(, v) = αpψ()v αp 1 and F (, v) = ψ()v αp. Then F (, v) = αpψ()v αp 1. v It is easy to verify that all assumptions of Theorem 1.6 are satisfied. Consequently, for any given positive integer m > 0, equation (4.1) has at least one nontrivial mt -periodic solution. Example 4.2. For all n Z(1) and Z, assume that (4.2) n (r n φ p ( n u 1 )) + ( 1) n q φ p ( u 1 ) ( = ( 1) n θpu 3 + cos 2 π ) u θp 1, T where {r } and {q } are real sequences, r +T = r > 0, q +T = q 0, 1 < p <, 0 < θ < 2. We have ( f(, v) = θp 3 + cos 2 π ) v θp 1 T and F (, v) = ( 3 + cos 2 π ) v θp. T Then F (, v) v ( = θp 3 + cos 2 π ) v θp 1. T It is easy to verify that all of the assumptions of Theorem 1.7 are satisfied. Consequently, for any given positive integer m > 0, equation (4.2) has at least one nontrivial mt -periodic solution. REFERENCES 1. R.P. Agarwal, Difference equations and inequalities: Theory, methods and applications, Marcel Deer, New Yor, R.P. Agarwal and P.J.Y. Wong, Advanced topics in difference equations, Kluwer Academic Publishers, Dordrecht, 1997.

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