PERIODIC SOLUTIONS OF NON-AUTONOMOUS SECOND ORDER SYSTEMS. 1. Introduction

Transcription

1 ao DOI:.2478/s Math. Slovaca 64 (24), No. 4, PERIODIC SOLUIONS OF NON-AUONOMOUS SECOND ORDER SYSEMS WIH (,)-LAPLACIAN Daniel Paşca* Chun-Lei ang** (Communicated by Christian Pötzsche ) ABSRAC. Using the least action principle in critical point theory we obtain some existence results of periodic solutions for (,)-Laplacian systems which generalize some existence results. c 24 Mathematical Institute Slovak Academy of Sciences. Introduction In the last two decades many authors starting with Mawhin and Willem (see 3]) proved the existence of solutions for problem: { ü(t) = F (t, u(t)) a.e. t,], () u() u( )= u() u( )=, under suitable conditions on the potential F (see 4] 28]). In a series of papers (see 4], 5], 6]) we have generalized some of those results for the case when the potential F is just locally Lipschitz in the second variable x, not continuously differentiable: { ü(t) F(t, u(t)) a.e. t,], (2) u() u( )= u() u( )=, where >, F :,] R N R, denotes the Clarke subdifferential (see ]) and F (t, x) ismeasurableint for each x R N, and locally Lipschitz and regular (see ]) in x for each t,]. 2 M a t h e m a t i c s Subject Classification: Primary 34C25. K e y w o r d s: periodic solutions, (,)-Laplacian systems, generalized Lebesgue and Sobolev spaces. Download Date /9/8 2:54 AM

2 DANIEL PAŞCA CHUN-LEI ANG In 7], 8] and 9] we have considered the second order Hamiltonian inclusions systems with p-laplacian: { d ( dt u(t) p 2 u(t) ) F(t, u(t)) a.e. t,], (3) u() u( )= u() u( )=, where p>, >, F :,] R N R. Also we have considered the extensions to second-order differential systems with (q, p)-laplacian: d dt( u (t) q 2 u (t) ) = u F (t, u (t),u 2 (t)), a.e. t,], d dt( u2 (t) p 2 u 2 (t) ) = u2 F (t, u (t),u 2 (t)), a.e. t,], (4) u () u ( )= u () u ( )=, u 2 () u 2 ( )= u 2 () u 2 ( )=, where <p,q<, >, and F :,] R N R N R and they satisfy some assumptions (see ], ], 2]). his kind of systems has been considered also in 2] and 35]. Recently (see 3]) we have proved some existence results of periodic solutions for d dt( u (t) q 2 u (t) ) u F (t, u (t),u 2 (t)), a.e. t,], d dt( u2 (t) p 2 u 2 (t) ) u2 F (t, u (t),u 2 (t)), a.e. t,], (5) u () u ( )= u () u ( )=, u 2 () u 2 ( )= u 2 () u 2 ( )=, where <p,q<, >, and F :,] R N R N R and they satisfy some assumptions. In the last years some authors (see 32] 34]) have considered the existence of periodic solutions for { d ( dt u(t) 2 u(t) ) = F (t, u(t)) a.e. t,], (6) u() u( )= u() u( )=, where >, p C(,]; R), p(t )= for all t R, F :,] R N R. he aim of this paper is to consider a more general problem. More exactly our results represent the extensions to non-autonomous second-order differential systems with (,)-Laplacian: d dt( u (t) 2 u (t) ) = u F (t, u (t),u 2 (t)), a.e. t,] d dt( u2 (t) 2 u 2 (t) ) = u2 F (t, u (t),u 2 (t)), a.e. t,] (7) u () u ( )= u () u ( )=, u 2 () u 2 ( )= u 2 () u 2 ( )=, 94 Download Date /9/8 2:54 AM

3 PERIODIC SOLUIONS OF NON-AUONOMOUS SYSEMS WIH (,)-LAPLACIAN where >, q, p C(,]; R), and F :,] R N R N R satisfied the following assumptions (A): q(t )=, p(t )=, for all t R; q = min >, p t, ] = min > ; t, ] F is measurable in t for each (x,x 2 ) R N R N ; F is continuously differentiable in (x,x 2 ) for a.e. t,]; there exist a,a 2 C(R, R )andb L (,]; R ) such that F (t, x,x 2 ), x F (t, x,x 2 ), x2 F (t, x,x 2 ) a ( x )a 2 ( x 2 ) ] b(t) for all (x,x 2 ) R N R N and a.e. t,]. he corresponding functional ϕ: W, ϕ(u,u 2 )= u (t) dt W, u 2(t) dt 2. Main results R is given by F (t, u (t),u 2 (t)) dt. We suppose that there exist f i,g i L (,]; R ), i =, 2andα,q ), α 2,p ) such that x F (t, x,x 2 ) f (t) x α g (t) x2 F (t, x,x 2 ) f 2 (t) x 2 α 2 g 2 (t) for all (x,x 2 ) R N R N and a.e. t,]. heorem. Suppose that (8) and assumption (A) hold. Let q and p be such that q q =and p p =. Assume that x qα x2 pα 2 (8) F (t, x,x 2 )dt (9) as x = x 2 x 2 2. hen problem (7) has at least one solution which minimizes the functional ϕ on W, W,. Remark. heorem generalizes other results as follows: If q (, ) and p (, ) then we get the corresponding : heorem ]; If = 2andF (t, x,x 2 )= F (t, x ), we get the corresponding 6: heorem ]; If F (t, x,x 2 )= F (t, x 2 ), α = α 2 =andg (t), we get 34: heorem 3.]. 95 Download Date /9/8 2:54 AM

4 DANIEL PAŞCA CHUN-LEI ANG Definition. (see 9]) A function G: R N R is called to be (λ, µ)-subconvex if G(λ(x y)) µ(g(x)g(y)) for some λ, µ > andallx, y R N. Remark 2. (see 9]) When λ = µ = 2, a function ( 2, 2 )-subconvex is called convex. When λ = µ =, a function (, )-subconvex is called subadditive. When λ =,µ >, a function (,µ)-subconvex is called µ-subadditive. heorem 2. Assume that F = F F 2,whereF, F 2 satisfy assumption (A) and the following conditions: (i) F (t,, ) is (λ, µ)-subconvex with λ>/2 and <µ<2 r λ r for a.e. t,] where r =min(q,p ); (ii) there exist f i,g i L (,]; R ), i =, 2 and α,q ), α 2,p ) such that x F 2 (t, x,x 2 ) f (t) x α g (t) x2 F 2 (t, x,x 2 ) f 2 (t) x 2 α 2 g 2 (t) for all (x,x 2 ) R N R N and a.e. t,]; (iii) ] x qα x2 pα F 2 (t, λx,λx 2 )dt F 2 (t, x,x 2 )dt µ as x = x 2 x 2 2,where q q =and p p =. hen problem (7) has at least one solution which minimizes the functional ϕ on W, W,. heorem 3. Assume that F = F F 2,whereF, F 2 satisfy assumption (A) and the following conditions: (iv) F (t,, ) is (λ, µ)-subconvex with λ>/2 and <µ<2 r λ r for a.e. t,] where r =min(q,p ),andthereexistγ L (,]; R), h,h 2 L (,]; R N ) with h i (t)dt =, i =, 2 such that F (t, x,x 2 ) ((h (t),h 2 (t)), (x,x 2 )) γ(t) for all (x,x 2 ) R N R N and a.e. t,]; (v) there exist g,g 2 L (,]; R ), c R such that x F 2 (t, x,x 2 ) g (t) x2 F 2 (t, x,x 2 ) g 2 (t) 96 for all (x,x 2 ) R N R N and a.e. t,], Download Date /9/8 2:54 AM

5 PERIODIC SOLUIONS OF NON-AUONOMOUS SYSEMS WIH (,)-LAPLACIAN (vi) and F 2 (t, x,x 2 )dt c for all x R N ; F (t, λx,λx 2 )dt F 2 (t, x,x 2 )dt µ as x = x 2 x 2 2. hen problem (7) has at least one solution which minimizes ϕ on W, W,. heorem 4. Assume that F = F F 2,whereF, F 2 satisfy assumption (A) and the following conditions: (vii) there exists γ L (,]; R), h,h 2 L (,]; R N ) with h i (t)dt =, i =, 2 such that F (t, x,x 2 ) ((h (t),h 2 (t)), (x,x 2 )) γ(t) for all (x,x 2 ) R N R N and a.e. t,]; (viii) there exist f i,g i L (,]; R ), i =, 2 and α,q ), α 2,p ) such that (ix) x F 2 (t, x,x 2 ) f (t) x α g (t) x2 F 2 (t, x,x 2 ) f 2 (t) x 2 α 2 g 2 (t) for all (x,x 2 ) R N R N and a.e. t,]; x qα x2 pα 2 as x = x 2 x 2 2. F 2 (t, x,x 2 )dt hen problem (7) has at least one solution which minimizes ϕ on W, W,. Remark 3. heorems 2, 3 and 4 generalized other results as follows: If q (, ) and p (, ) then we get the corresponding : heorems, 2, 3]; If = 2andF (t, x,x 2 )= F (t, x ) we get the corresponding 9: heorems, 2, 3]; Also heorem 2 and 3 generalized 34: heorems 3.5, 3.6]. 97 Download Date /9/8 2:54 AM

6 DANIEL PAŞCA CHUN-LEI ANG 3. he functional framework: generalized Lebesgue and Sobolev spaces In order to study the second order systems with (,)-Laplacian using variational methods, we will need the generalized Lebesgue and Sobolev spaces (also known as Lebesgue and Sobolev spaces with variable exponent). For more details on the basic properties of these spaces, we refer the reader to 29] 32]. Let >andp C(,]; R) with Define with the norm p = min, p t, ] L (,]; R N )= { u L (,]; R N ): { u =inf λ>: = max t, ]. } u(t) dt< u(t) } dt. λ For u L loc (,]; RN ), let u denote the weak derivative of u, i.e., u L loc (,]; RN ) such that u (t)ϕ(t)dt = u(t)ϕ (t)dt, for all ϕ C (,]; RN ). Define W, (,]; R N )= {u L (,]; R N) : u L (,]; R N)} with the norm u = u u. Lemma 5. (3]) (i) he space (L, ) is a separable, uniform convex Banach space, and its conjugate space is L p (t),where p (t) =. For any u L and v L p (t), we have u(t)v(t)dt 2 u v p (t). (ii) If p (t),p 2 (t) C(,]; R) and p (t) p 2 (t) for any t,], then L p 2(t) L p (t), and the embedding is continuous. 98 Download Date /9/8 2:54 AM

7 PERIODIC SOLUIONS OF NON-AUONOMOUS SYSEMS WIH (,)-LAPLACIAN Lemma 6. (3]) If we denote then ρ(u) = u(t) dt, for all u L, (i) u < (=, > ) if ρ(u) < (=, > ); (ii) u > = u p ρ(u) u p ; u < = u p ρ(u) u p ; (iii) u ρ(u) ; u ρ(u). Lemma 7. (3]) If u, u n L (n =, 2,...), then the following statements are equivalent to each other () lim n u n u L =; (2) lim n ρ(u n u) =; (3) u n u in measure in,] and lim n ρ(u n)=ρ(u). Lemma 8. (29], 3]) L and W, are both Banach spaces with the norms defined above. When p > they are reflexive. Now, we recall the definition of space W, (,]; R N ) and some properties on it. For this purpose, let C = C (R, RN ) = { u C (R, R N ) : u is periodic }. For a constant p, ), using another concept of weak derivative which was called -weak derivative in 32], Mawhin and Willem (see 3]) gave the definition of the space W,p by the following way: Definition 2. (3], 32]) Let u, v L (,]; R N ). If v(t)ϕ(t)dt = u(t)ϕ (t)dt, for all ϕ C, then v is called a -weak derivative of u and is denoted by u. Definition 3. (3], 32]) Define W,p with the norm u W,p (,]; RN )= { u L p (,]; R N ): u L p (,]; R N ) } Definition 4. (32]) Define = ( u p p u p p ) p. W, (,]; R N )= { u L (,]; R N ): u L (,]; R N ) } and H, (,]; R N ) to be the closure of C in W, (,]; R N ) 99 Download Date /9/8 2:54 AM

8 DANIEL PAŞCA CHUN-LEI ANG From Definition 2 we see that, for u L (,]; R N ), the weak derivative u and the -weak derivative u are two different concepts (for details see 32]). Although the two derivatives are distinct, we have: Lemma 9. (32]) (i) C (,]; RN ) is dense in W, (,]; R N ); (ii) W, (,]; R N )=H, (,]; R N ) = { u W, (,]; R N ): u() = u( ) } ; (iii) If u H,, then the weak derivative u is also the -weak derivative u, i.e. u = u. Lemma. (3]) Assume that u W,,then () u(t)dt =; (2) u has its continuous representation, which is still denoted by u u(t) = t u(s)ds u(), u() = u( ); (3) u is the classical derivative of u if u C(,]; R N ). Since every closed linear subspace of a reflexive Banach space is also reflexive, we have: Lemma. (32]) H, (,]; R N ) is a reflexive Banach space if p >. Obviously, there are continuous embeddings L L p, W, W,p and H, H p. By the classical Sobolev embedding theorem we obtain: Lemma 2. (32]) here is a continuous embedding W, C(,]; R N )(or C(,]; R N )). Whenp > the embedding is compact. H, In what follows, we set ū = u(t)dt and ũ(t) =u(t) ū for all u W,. Lemma 3 (Sobolev inequality). here is a constant c independent of u such that ũ c u for all u W, Lemma 4. (32]) Each of the following two norms is equivalent to the norm in W,. () u u q, for all q, (2) u ū. 92 Download Date /9/8 2:54 AM

9 PERIODIC SOLUIONS OF NON-AUONOMOUS SYSEMS WIH (,)-LAPLACIAN Lemma 5. (33]) he space W, = W, { W, = u W, : W, R N,where } u(t)dt =. Moreover, in space, u and u u q are two equivalent norms, for any constant q, ). Proposition 6. (3]) Let f :,] R N R N be a Carathéodory function, i.e., f(t, x) is measurable in t for each x R N, continuously differentiable in x for a.e. t,], and satisfies the following growth condition: f(t, x) a(t)b x p (t) p 2 (t) for a.e. t,] and for all x R N, some nonnegative function a L p 2(t) (,]; R) and some constant b. hen the Nemytsky operator defined by f, i.e., L p (t) (,]; R) u f(t, u(t)), froml p (t) (,]; R) into L p 2(t) (,]; R), is continuous and bounded. he following results generalize 27: Lemma 3.] and 3: heorem.4], and theproofcanbedoneinthesameway. Lemma 7. Let p, q C(,]; R), p(t )=, q(t )=, for all t R and p >, q >. LetL:,] R N R N R N R N R, (t, x,x 2,y,y 2 ) L(t, x,x 2,y,y 2 ) be measurable in t for each (x,x 2,y,y 2 ), and continuously differentiable in (x,x 2,y,y 2 ) for a.e. t,]. Ifthereexista i C(R, R ), b L (,]; R ),andc L q (t) (,]; R ), c 2 L p (t) (,]; R ), q (t) =, p (t) =such that for a.e. t,] and every (x,x 2,y,y 2 ) R N R N R N R N, one has L(t, x,x 2,y,y 2 ) a ( x )a 2 ( x 2 ) ] b(t) y y 2 ], D x L(t, x,x 2,y,y 2 ) a ( x )a 2 ( x 2 ) ] b(t) y 2 ], D x2 L(t, x,x 2,y,y 2 ) a ( x )a 2 ( x 2 ) ] b(t) y ], D y L(t, x,x 2,y,y 2 ) a ( x )a 2 ( x 2 ) ] c (t) y ], D y2 L(t, x,x 2,y,y 2 ) a ( x )a 2 ( x 2 ) ] c 2 (t) y 2 ], then the function ϕ: W, ϕ(u,u 2 )= W, R defined by L(t, u (t),u 2 (t), u (t), u 2 (t)) dt 92 Download Date /9/8 2:54 AM

10 is continuously differentiable on W, ϕ (u,u 2 ), (v,v 2 ) = DANIEL PAŞCA CHUN-LEI ANG W, and (Dx L(t, u (t),u 2 (t), u (t), u 2 (t)),v (t)) (D y L(t, u (t),u 2 (t), u (t), u 2 (t)), v (t)) (D x2 L(t, u (t),u 2 (t), u (t), u 2 (t)),v 2 (t)) (D y2 L(t, u (t),u 2 (t), u (t), u 2 (t)), v 2 (t)) ] dt. Corollary 8. Let L:,] R N R N R N R N R be defined by L(t, x,x 2,y,y 2 )= y y 2 F (t, x,x 2 ) where q, p C(,]; R) and F :,] R N R N R satisfy condition (A). If (u,u 2 ) W, W, is a solution of the corresponding Euler equation ϕ (u,u 2 )=,then(u,u 2 ) is a solution of (7). 4. Proofs of the theorems In what follows we will denote W =W, u, W u 2, W. W, with the norm (u,u 2 ) W = P r o o f o f h e o r e m. It follows from (8) and Sobolev inequality that 922 = F (t, u (t),u 2 (t)) F (t, ū, ū 2 ) ] dt F (t, u (t),u 2 (t)) F (t, u (t), ū 2 ) ] dt F (t, u (t), ū 2 ) F (t, ū, ū 2 ) ] dt ( x2 F (t, u (t), ū 2 sũ 2 (t)), ũ 2 (t)) ds dt ( x F (t, ū sũ (t), ū 2 ), ũ (t)) ds dt Download Date /9/8 2:54 AM

11 PERIODIC SOLUIONS OF NON-AUONOMOUS SYSEMS WIH (,)-LAPLACIAN f 2 (t) ū 2 sũ 2 (t) α 2 ũ 2 (t) ds dt f (t) ū sũ (t) α ũ (t) ds dt 2 ( ) p ū 2 α 2 ũ 2 α 2 ũ2 f 2 (t)dt ũ 2 2 ( ) q ū α ũ α ũ f (t)dt ũ g 2 (t) ũ 2 (t) ds dt = c ũ α c 2 ū α ũ c 2 ũ 2 α 2 c 3 ũ c 22 ū 2 α 2 ũ 2 c 23 ũ 2 g (t) ũ (t) ds dt g 2 (t)dt g (t)dt c u α c 2 ū α u c 3 u c 2 u 2 α 2 c 22 ū 2 α 2 u 2 c 23 u 2 c u α (2q ) u q c 3 u c 2 ū qα c 2 u 2 α 2 (2p ) u 2 p c 23 u 2 c 22 ū 2 pα 2 for all (u,u 2 ) W and some positive constants c,...,c 23, c,..., c 23. Hence we have ϕ(u,u 2 )= u (t) dt u 2(t) dt q F (t, u (t),u 2 (t)) F (t, ū, ū 2 ) ] dt F (t, ū, ū 2 )dt u (t) dt u q 2q c u α c 3 u c 2 ū qα p c 23 u 2 c 22 ū 2 pα 2 u 2 (t) dt u 2 p 2p c 2 u 2 α 2 F (t, ū, ū 2 )dt 923 Download Date /9/8 2:54 AM

12 DANIEL PAŞCA CHUN-LEI ANG 2q 2p u (t) dt c u α u 2 (t) dt c 2 u 2 α 2 ( ū qα ū 2 pα 2 )( ū qα ū2 pα 2 c 3 u c 23 u 2 ) F (t, ū, ū 2 )dt max( c 2, c 22 ) for all (u,u 2 ) W, which implies that ϕ(u,u 2 ) as (u,u 2 ) W due to (9) and Lemma 4. By 3: heorem.] and Corollary 8 we complete our proof. Proof of heorem 2. Like in 9] weobtain ( F (t, x,x 2 ) 2 β 2 µ ( x β x 2 β) ) (a a 2 )b(t) for all (x,x 2 ) R N R N and a.e. t,], where β<r,anda i = max s a i(s), i =, 2. Like in the proof of heorem we get F2 (t, u (t),u 2 (t)) F 2 (t, ū, ū 2 ) ] dt c u α (2q ) u q c 3 u c 2 u 2 α 2 c 2 ū qα (2p ) u 2 p c 23 u 2 c 22 ū 2 pα 2 for all (u,u 2 ) W and some positive constants c,..., c 23. Hence we have 924 = ϕ(u,u 2 ) u (t) dt 2q 2p u 2(t) dt F2 (t, u (t),u 2 (t)) F 2 (t, ū, ū 2 ) ] dt F (t, u (t),u 2 (t)) dt F 2 (t, ū, ū 2 )dt u (t) dt c u α c 3 u c 2 ū qα u 2 (t) dt c 2 u 2 α 2 c 23 u 2 Download Date /9/8 2:54 AM

13 PERIODIC SOLUIONS OF NON-AUONOMOUS SYSEMS WIH (,)-LAPLACIAN c 22 ū pα 2 2 2q 2p F 2 (t, ū, ū 2 )dt µ F (t, ũ (t), ũ 2 (t)) dt F (t, λū,λū 2 )dt u (t) dt c u α c 3 u c 2 ū qα u 2 (t) dt c 2 u 2 α 2 c 23 u 2 c 22 ū 2 pα 2 F 2 (t, ū, ū 2 )dt µ F (t, λū,λū 2 )dt 2 β 2 µ ( ũ β ũ 2 β ) ] (a a 2 ) b(t)dt u (t) dt c u α 2q c 3 u c 3 u β 2p u 2 (t) dt c 2 u 2 α 2 c 23 u 2 c 3 u 2 β c 32 F 2 (t, ū, ū 2 )dt F (t, λū,λū 2 )dt µ max( c 2, c 22 ) ( ) ū qα ū pα 2 2 = u (t) dt c u α 2q c 3 u c 3 u β 2p u 2 (t) dt c 2 u 2 α 2 c 23 u 2 c 3 u 2 β c Download Date /9/8 2:54 AM

14 DANIEL PAŞCA CHUN-LEI ANG ( ) { ū qα ū pα 2 2 F ū qα ū2 pα (t, λū,λū 2 )dt 2 µ ] } F 2 (t, ū, ū 2 )dt max( c 2, c 22 ) for all (u,u 2 ) W, which implies that ϕ(u,u 2 ) as (u,u 2 ) W due to (iii). By 3: heorem.] and Corollary 8 we complete our proof. Proof of heorem 3. Let (u k,u 2k ) be a minimizing sequence of ϕ. follows from (iv), (v) and Sobolev s inequality that It = 926 ϕ(u k,u 2k ) q q u k(t) dt u 2k(t) dt ũ 2k F (t, u k (t),u 2k (t)) dt u k (t) dt u 2k (t) dt p ( (h (t),h 2 (t)), (u k (t),u 2k (t)) ) dt F 2 (t, u k (t),u 2k (t)) dt γ(t)dt ( x2 F 2 (t, u k (t), ū 2k sũ 2k (t)), ũ 2k (t)) ds dt ( x F 2 (t, ū k sũ k (t), ū 2k ), ũ k (t)) ds dt u k (t) dt p h 2 (t) dt u 2k (t) dt ũ k γ(t)dtc ũ k F 2 (t, ū k, ū 2k )dt h (t) dt g (t)dt ũ 2k g 2 (t)dt Download Date /9/8 2:54 AM

15 PERIODIC SOLUIONS OF NON-AUONOMOUS SYSEMS WIH (,)-LAPLACIAN u k (t) dt u 2k (t) dt c u k c 2 u 2k c 3 q p for all k and some constants c, c 2, c 3, which implies that (ũ k, ũ 2k ) is bounded. On the other hand, in a way similar to the proof of heorem, one has F2 (t, u (t),u 2 (t)) F 2 (t, ū, ū 2 ) ] dt c 3 u c 23 u 2 for all (u,u 2 ) W and some positive constants c 3, c 23, which implies that ϕ(u k,u 2k ) u k (t) dt u 2k (t) dt q p F (t, λū k,λū 2k )dt F (t, ũ k (t), ũ 2k (t)) dt µ q F2 (t, u k (t),u 2k (t) F 2 (t, ū k, ū 2k ) ] dt u k (t) dt c 3 u k u 2k (t) dt p c 23 u 2k a ( ũ k )a 2 ( ũ 2k ) ] F 2 (t, ū k, ū 2k )dt µ F (t, λū k,λū 2k )dt F 2 (t, ū k, ū 2k )dt b(t)dt for all k. It follows from (vi) and the boundedness of (ũ k, ũ 2k )that(ū k, ū 2k ) is bounded. Hence ϕ has a bounded minimizing sequence (u k,u 2k ). Now heorem 2 follows from 3: heorem.] and Corollary 8. P r o o f o f h e o r e m 4. From (vii) and Sobolev s inequality it follows like in the proof of heorem that 927 Download Date /9/8 2:54 AM

16 DANIEL PAŞCA CHUN-LEI ANG ϕ(u,u 2 ) u (t) dt q p 2q 2p u 2 (t) dt ( (h (t),h 2 (t)), (u (t),u 2 (t)) ) dt F 2 (t, ū, ū 2 )dt γ(t)dt F2 (t, u (t),u 2 (t)) F 2 (t, ū, ū 2 ) ] dt u (t) dt c u α c 3 u c 2 ū qα ũ 2 2q u 2 (t) dt c 2 u 2 α 2 c 23 u 2 c 22 ū 2 pα 2 F 2 (t, ū, ū 2 )dt 2p h 2 (t) dt γ(t)dt ũ u (t) dt c u α u 2 (t) dt c 2 u 2 α 2 max( c 2, c 22 ) ( ū qα ū 2 pα 2 ) = 2q 2p u (t) dt c u α u 2 (t) dt c 2 u 2 α 2 c 4 u h (t) dt c 24 u 2 c ( ) ū qα ū pα 2 2 ū qα ū2 pα 2 F 2 (t, ū, ū 2 )dt c 4 u c 24 u 2 c ] F 2 (t, ū, ū 2 )dt max( c 2, c 22 ) for all (u,u 2 ) W and some positive constants c, c 4, c 2, c 24. Now follows like in the proof of heorem that ϕ is coercive by (ix), which completes the proof. 928 Download Date /9/8 2:54 AM

17 PERIODIC SOLUIONS OF NON-AUONOMOUS SYSEMS WIH (,)-LAPLACIAN REFERENCES ] CLARKE, F. H.: Optimization and Nonsmooth Analysis. Classics Appl. Math. 5, SIAM, Philadelphia, PA, 99. 2] JEBELEAN, P. PRECUP, R.: Solvability of p, q-laplacian systems with potential boundary conditions, Appl. Anal. 89 (2), ] MAWHIN, J. WILLEM, M.: Critical Point heory and Hamiltonian Systems, Springer- Verlag, Berlin-New York, ] PAŞCA, D.: Periodic solutions for second order differential inclusions, Comm. Appl. Nonlinear Anal. 6 (999), ] PAŞCA, D.: Periodic solutions for second order differential inclusions with sublinear nonlinearity, Panamer. Math. J. (2), ] PAŞCA, D.: Periodic solutions of a class of non-autonomous second order differential inclusions systems, Abstr. Appl. Anal. 6 (2), ] PAŞCA, D.: Periodic solutions of second-order differential inclusions systems with p-laplacian, J. Math. Anal. Appl. 325 (27), 9. 8] PAŞCA, D. ANG, CH.-L.: Subharmonic solutions for nonautonomous sublinear second order differential inclusions systems with p-laplacian, Nonlinear Anal. 69 (28), ] PAŞCA, D.: Periodic solutions for nonautonomous second order differential inclusions systems with p-laplacian, Comm. Appl. Nonlinear Anal. 6 (29), ] PAŞCA, D. ANG, CH.-L.: Some existence results on periodic solutions of nonautonomous second order differential systems with (q, p)-laplacian, Appl. Math. Lett. 23 (2), ] PAŞCA, D.: Periodic solutions of a class of nonautonomous second order differential systems with (q,p)-laplacian, Bull. Belg. Math. Soc. Simon Stevin 7 (2), ] PAŞCA, D. ANG, CH.-L.: Some existence results on periodic solutions of ordinary (q, p)-laplacian systems, J. Appl. Math. Inform. 29 (2), ] PAŞCA, D.: Periodic solutions of second-order differential inclusions systems with (q,p)-laplacian, Anal. Appl. 9 (2), ] ANG, CH.-L.: Periodic solutions of non-autonomous second-order systems with γ-quasisubadditive potential, J. Math. Anal. Appl. 89 (995), ] ANG, CH.-L.: Periodic solutions of non-autonomous second order systems, J.Math. Anal. Appl. 22 (996), ] ANG, CH.-L.: Periodic solutions for nonautonomous second order systems with sublinear nonlinearity, Proc. Amer. Math. Soc. 26 (998), ] ANG, CH.-L.: Existence and multiplicity of periodic solutions of nonautonomous second order systems, Nonlinear Anal. 32 (998), ] WU, X.-P.: Periodic solutions for nonautonomous second-order systems with bounded nonlinearity, J. Math. Anal. Appl. 23 (999), ] WU, X.-P. ANG, CH.-L.: Periodic solutions of a class of non-autonomous secondorder systems, J.Math.Anal.Appl.236 (999), ] ANG, CH.-L. WU, X.-P: Periodic solutions for second order systems with not uniformly coercive potential, J. Math. Anal. Appl. 259 (2), ] MA, J. ANG, CH.-L.: Periodic solutions for some nonautonomous second-order systems, J. Math. Anal. Appl. 275 (22), ] WU, X.-P. ANG, CH.-L.: Periodic solutions of nonautonomous second-order hamiltonian systems with even-typed potentials, Nonlinear Anal. 55 (23), ] ANG, CH.-L. WU, X.-P.: Notes on periodic solutions of subquadratic second order systems, J.Math.Anal.Appl.285 (23), Download Date /9/8 2:54 AM

18 DANIEL PAŞCA CHUN-LEI ANG 24] ZHAO, F. WU, X.-P.: Saddle point reduction method for some non-autonomous second order systems, J.Math.Anal.Appl.29 (24), ] ANG, CH.-L. WU, X.-P.: Subharmonic solutions for nonautonomous second order Hamiltonian systems, J. Math. Anal. Appl. 34 (25), ] XU, B. ANG, CH.-L.: Some existence results on periodic solutions of ordinary p-laplacian systems, J. Math. Anal. Appl. 333 (27), ] YU IAN, Y. GE, W.: Periodic solutions of non-autonomous second-order systems with a p-laplacian, Nonlinear Anal. 66 (27), ] ANG, CH.-L. WU, X.-P.: Some critical point theorems and their applications to periodic solution for second order Hamiltonian systems, J.DifferentialEquations248 (2), ] FAN, X. L. ZHAO, D.: On the space L p(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263 (2), ] FAN, X. L. ZHANG, Q. H.: Existence of solutions for p(x)-laplacian Dirichlet problem, Nonlinear Anal. 52 (23), ] FAN, X. L. ZHAO, Y. Z. ZHAO, D.: Compact embeddings theorems with symmetry of Strauss-Lions type for the space W m,p(x) (Ω), J. Math. Anal. Appl. 255 (2), ] FAN, X.-L. FAN, X.: A Knobloch-type result for -Laplacian systems, J.Math.Anal. Appl. 282 (23), ] WANG, X. J. YUAN, R.: Existence of periodic solutions for -Laplacian systems, Nonlinear Anal. 7 (29), ] ZHANG, L. YI CHEN, Y: Existence of periodic solutions of -Laplacian systems, Bull. Malays. Math. Sci. Soc. (2) 35 (22), ] YANG, X. CHEN, H.: Existence of periodic solutions for sublinear second order dynamical system with (q, p)-laplacian, Math. Slovaca 63 (23), Received.. 2 Accepted * Department of Mathematics and Informatics University of Oradea University Street 487 Oradea ROMANIA dpasca@uoradea.ro ** Department of Mathematics Southwest University Chongqing 475 PEOPLE S REPUBLIC OF CHINA tangcl@swu.edu.cn 93 Download Date /9/8 2:54 AM