PERIODIC SOLUTIONS FOR NON-AUTONOMOUS SECOND-ORDER DIFFERENTIAL SYSTEMS WITH (q, p)-laplacian

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1 Electronic Journal of Differential Euations, Vol. 24 (24), No. 64, pp. 3. ISSN: URL: or ftp ejde.math.txstate.edu PERIODIC SOLUIONS FOR NON-AUONOMOUS SECOND-ORDER DIFFERENIAL SYSEMS WIH (, p)-laplacian CHUN LI, ZENG-QI OU, CHUN-LEI ANG Abstract. Some existence theorems are obtained for periodic solutions of nonautonomous second-order differential systems with (, p)-laplacian by using the least action principle and the saddle point theorem.. Introduction Consider the second-order system d ( u (t) 2 u (t) ) = u F (t, u (t), u 2 (t)), dt d dt ( u 2(t) p 2 u 2 (t)) = u2 F (t, u (t), u 2 (t)), a.e. t [, ], u () u ( ) = u () u ( ) =, u 2 () u 2 ( ) = u 2 () u 2 ( ) =, (.) where < p, <, > and denotes the Euclidean norm in R N. F : [, ] R N R N R satisfies the following assumption (A) 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 ) and b 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 [, ]. We denote by W,p the Sobolev space of functions u L p (, ; R N ) having a weak derivative u L p (, ; R N ). he norm in W,p is defined by u W,p ( ( = u(t) p u(t) p) dt he corresponding functional ϕ : W R given is ϕ(u, u 2 ) = u (t) dt p u 2 (t) p dt ) /p. F (t, u (t), u 2 (t))dt, 2 Mathematics Subject Classification. 34C25, 35B38, 47J3. Key words and phrases. Periodic solution; differential systems; (, p)-laplacian; least action principle; saddle point theorem. c 24 exas State University - San Marcos. Submitted March 8, 22. Published March 5, 24.

2 2 C. LI, Z.-Q. OU, C.-L. ANG EJDE-24/64 where W = W, W,p is a reflexive Banach space and endowed with the norm (u, u 2 ) W = u W, u 2 W,p. It follows from assumption (A) that the functional ϕ is continuously differentiable and weakly lower semicontinuous on W. Moreover, ϕ (u, u 2 ), (v, v 2 ) = [( u (t) 2 u (t), v (t)) ( u F (t, u (t), u 2 (t)), v (t))]dt [( u 2 (t) p 2 u 2 (t), v 2 (t)) ( u2 F (t, u (t), u 2 (t)), v 2 (t))]dt for all (u, u 2 ), (v, v 2 ) W. For each u W,p can be written as u(t) = ū ũ(t) with ū = u(t)dt, ũ(t)dt =. We have the Sobolev s ineuality (for a proof and details see [4]) ũ C u p, ṽ C v for each u W,p, v W,, and Wirtinger s ineuality (see [4]) where ũ p C 2 u p, ṽ C 2 v for each u W,p, v W,, ( u p = u(t) p dt) /p, u = max t [, ] u(t). 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 > and all x, y R N (see [9]). he existence of periodic solutions for the second-order Hamiltonian system ü(t) = F (t, u), a.e. t [, ], u() u( ) = u() u( ) =, (.2) has been extensively investigated in papers, such as [, 2, 3, 4, 2, 3, 4, 5, 6, 8, 9] and the reference therein. Many solvability conditions are given, such as the coercive condition (see []), the periodicity condition (see [8]), the convexity condition (see [3]), the boundedness condition (see [4]), the subadditive condition (see [2]), and the sublinear condition (see [4]). When the gradient F (t, x) is bounded; that is, there exists g L (, ; R ) such that F (t, x) g(t) for all x R N and a.e. t [, ]. Mawhin and Willem [4] obtained the existence of solutions for problem (.2) under the condition F (t, x)dt (or ), as x. ang [4] proved the existence of solutions for problem (.2) when F (t, x) f(t) x α g(t) (.3)

3 EJDE-24/64 PERIODIC SOLUIONS 3 for all x R N and a.e. t [, ], where f, g L (, ; R ) and α [, ). And, F satisfies the condition x 2α F (t, x)dt (or ), as x. ang and Meng [6] studied the existence of solutions for problem (.2) under the conditions (.3) or F (t, x) f(t) x g(t) for all x R N and a.e. t [, ], where f, g L (, ; R ). he results in [6] complement those in [4, heorem and 2]. Recently, Paşca and ang [9] established the existence results for problem (.) which extend [4, heorems and 2]. By applying the least action principle, Paşca [7] proved some existence theorems for problem (.) which generalize the corresponding heorems of [9]. Using the Saddle Point heorem, Paşca and ang [] obtained some existence results for problem (.). Paşca [8] studied the existence of periodic solutions for nonautonomous second-order differential inclusions systems with (, p)-laplacian which extend the results of [5, 6, 9, 4]. In this paper, motivated by references [7, 9, 4, 6], we consider the existence of periodic solutions for problem (.) by using the least action principle and the Saddle Point heorem. Our main results are the following theorems. heorem.. Suppose that F = F F 2, where F and F 2 satisfy assumption (A) and the following conditions: (H) F (t,, ) is (λ, µ)-subconvex with λ > /2 and /2 < µ < 2 r λ r for a.e.t [, ], where r = min{p, }; (H) there exist f i, g i, h i L (, ; R ), i =, 2, α [, ), α 2 [, p ), β [, p/ ), β 2 [, / ), = /( ) and = p/(p ) (H2) lim x x F 2 (t, x, x 2 ) f (t) x α g (t) x 2 β h (t) x2 F 2 (t, x, x 2 ) f 2 (t) x 2 α2 g 2 (t) x β2 h 2 (t) for all (x, x 2 ) R N R N and a.e. t [, ]; ( ) F (t, λx, λx 2 )dt F 2 (t, x, x 2 )dt > 2K, x γ x 2 γ2 µ where x = x 2 x 2 2, γ = max{ α, β 2 }, γ 2 = max{ α 2, β } and { 4 / (2 f K = max L C ), 4p /p (2 p f 2 L C, 4 / (2 β g L C ), 4p /p (2 β2 g 2 L C ) } p. hen problem (.) has at least one solution in W. Corollary.2. Suppose that F = F F 2, satisfies (H), (H) and (H2 ) ( ) F (t, λx, λx 2 )dt F 2 (t, x, x 2 )dt x γ x 2 γ2 µ

4 4 C. LI, Z.-Q. OU, C.-L. ANG EJDE-24/64 as x. hen problem (.) has at least one solution in W. Remark.3. Corollary.2 generalizes heorem of [7]. In fact, it follows from Corollary.2 by letting β = β 2 =. here are functions satisfying the assumptions of our Corollary.2 and not satisfying the assumptions in [7, 9]. For example, Let α = α 2 = 5/4, β = β 2 = /4, p = = 5, = = 5/4, and F (t, x ) = 5 sin( x 6 x 2 6 ), F 2 (t, x, x 2 ) = ( 2 3 t) ( x 9/4 x 2 9/4 x 5/4 x 2 5/4 ). heorem.4. Suppose that F (t, x, x 2 ) satisfies (H) and (H3) lim x x γ x 2 γ2 hen problem (.) has at least one solution in W. F (t, x, x 2 )dt < (2 2 )2K. Corollary.5. Suppose that F (t, x, x 2 ) satisfies (H) and (H3 ) F (t, x, x 2 )dt x γ x 2 γ2 as x. hen problem (.) has at least one solution in W. Remark.6. Corollary.5 extends [9, heorem 2]. In fact, it follows from Corollary.5 by letting β = β 2 =. here are functions satisfying the assumptions of our Corollary.5 and not satisfying the assumptions in [9]. For example, Let α = α 2 = 5/4, β = β 2 = /4, p = = 5, = = 5/4, and F (t, x, x 2 ) = ( 3 t) ( x 9/4 x 2 9/4 x 5/4 x 2 5/4 ). 2. Proofs of main results ian and Ge [7] proved the following result which generalizes a very well known result proved by Jean Mawhin and Michel Willem [4, heorem.4]. Lemma 2. ([7]). Let L : [, ] 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 [, ]. If there exist a i C(R, R ), i =, 2, b L (, ; R ), and c L p (, ; R ), c 2 L (, ; R ), < p, <, 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 p ), D x L(t, x, x 2, y, y 2 ) (a ( x ) a 2 ( x 2 ))(b(t) y 2 p ), 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 p ), then the function ϕ : W, W,p R defined by ϕ(u, u 2 ) = L(t, u (t), u 2 (t), u (t), u 2 (t))dt

5 EJDE-24/64 PERIODIC SOLUIONS 5 is continuously differentiable on W, W,p and ϕ (u, u 2 ), (v, v 2 ) = ((D x 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 2.2. Let L : [, ] R N R N R N R N R be defined by L(t, x, x 2, y, y 2 ) = y p y 2 p F (t, x, x 2 ) where F : [, ] R N R N R satisfies condition (A). If (u, u 2 ) W, W,p is a solution of the corresponding Euler euation ϕ (u, u 2 ) =, then (u, u 2 ) is a solution of problem (.). Remark 2.3. he function ϕ is weakly lower semi-continuous (w.l.s.c.) on W as the sum of two convex continuous functions and of a weakly continuous one. We will prove heorem. by using the least action principle [4, heorem.], and heorem.4 by using the saddle point theorem [, heorem 4.6]. Proof of heorem.. Let β = log 2λ (2µ). hen < β < r. For x >, there exists a positive integer n such that n < log 2λ x n. So, we have x β > (2λ) (n )β = (2µ) n and x (2λ) n. hen, by (A) and (H), one has F (t, x, x 2 ) 2µF (t, x /(2λ), x 2 /(2λ))... (2µ) n F (t, x /(2λ), x 2 /(2λ)) 2µ x β (a a 2 )b(t) for a.e. t [, ] and all x >, where a i = max s a i (s), i =, 2. herefore, F (t, x, x 2 ) (2 β/2 µ( x β x 2 β ) )(a a 2 )b(t) (2.) for a.e. t [, ] and all (x, x 2 ) R N R N. It follows from (H), Sobolev s ineuality and Young s ineuality that (F 2 (t, u (t), ū 2 ) F 2 (t, ū, ū 2 ))dt = ( x F 2 (t, ū sũ (t), ū 2 ), ũ (t))dsdt f (t) ū sũ (t) α ũ (t) dsdt h (t) ũ (t) dsdt g (t) ū 2 β ũ (t) dsdt 2 ( ū α ũ α ) ũ f L ū 2 β ũ g L ũ h L 2 C α f L u α 2 f L C ū α u

6 6 C. LI, Z.-Q. OU, C.-L. ANG EJDE-24/64 and C g L ū 2 β u C h L u 2 C α f L u α 4 u 4 / (2 f L C ) ū α 4 u 4 / ( g L C ) ū 2 β C h L u = 2 u 2 C α f L u α 4 / (2 f L C ) ū α / 4 ( g L C ) ū 2 β C h L u (F 2 (t, u (t), u 2 (t)) F 2 (t, u (t), ū 2 ))dt = ( x2 F 2 (t, u (t), ū 2 sũ 2 (t)), ũ 2 (t))dsdt f 2 (t) ū 2 sũ 2 (t) α2 ũ 2 (t) dsdt h 2 (t) ũ 2 (t) dsdt g 2 (t) u β2 ũ 2 (t) dsdt 2 p ( ū 2 α2 ũ 2 α2 ) ũ 2 f 2 L 2 β2 ( ū β2 ũ β2 ) ũ 2 g 2 L ũ 2 h 2 L 2 p C α2 f 2 L u 2 α2 p 2 p C f 2 L ū 2 α2 u 2 p C h 2 L u 2 p 2 β2 g 2 L C β2 u 2 p u β2 2 β2 g 2 L C u 2 p ū β2 2 p C α2 f 2 L u 2 p α2 4p u 2 p p 4p /p (2 p f 2 L C ū 2 α 2 4p u 2 p p 4p /p (2 β2 g 2 L C β2 u β2 4p u 2 p p /p (2 4p β2 g 2 L C ū β2 C h 2 L u 2 p = 3 4p u 2 p p 2 p C α2 f 2 L u 2 α2 p 4p /p (2 p f 2 L C ū 2 α 2 4p /p (2 β2 g 2 L C β2 C h 2 L u 2 p for all (u, u 2 ) W. So, one has (F 2 (t, u (t), u 2 (t)) F 2 (t, ū, ū 2 ))dt (F 2 (t, u (t), ū 2 ) F 2 (t, ū, ū 2 ))dt u β2 4p /p (2 β2 g 2 L C ū β2

7 EJDE-24/64 PERIODIC SOLUIONS 7 (F 2 (t, u (t), u 2 (t)) F 2 (t, u (t), ū 2 ))dt 2 u 3 4p u 2 p p C h L u C h 2 L u 2 p 2 C α f L u α 2 p C α2 f 2 L u 2 α2 p / 4 (2 f L C ) ū α / 4 ( g L C ) ū 2 β /p 4p (2 p f 2 L C ū 2 α /p 2 4p (2 β2 g 2 L C ū β 2 4p /p (2 β2 g 2 L C β2 u β2 for all (u, u 2 ) W. Hence, we obtain from (H), (2.) and the above expression that ϕ(u, u 2 ) = u (t) dt p u 2 (t) p dt F (t, u (t), u 2 (t))dt (F 2 (t, u (t), u 2 (t)) F 2 (t, ū, ū 2 ))dt F 2 (t, ū, ū 2 )dt 2 u 4p u 2 p p C h L u C h 2 L u 2 p 2 C α f L u α 2 p C α2 f 2 L u 2 p α2 4p /p (2 β2 g 2 L C β2 u β 2 4 / (2 f L C ) ū α / ( g 4 L C ) ū 2 β /p (2 p f 4p 2 L C 4p /p (2 β2 g 2 L C µ F (t, λū, λū 2 )dt ū β2 F 2 (t, ū, ū 2 )dt F (t, ũ, ũ 2 )dt 2 u 4p u 2 p p 4p /p (2 β2 g 2 L C β2 u β 2 ū 2 α 2 2 C α f L u α 2 p C α2 f 2 L u 2 p α2 C h L u C h 2 L u 2 p (2 β/2 C β µ( u β u 2 β p ) )(a a 2 ) ( ū γ ū 2 γ2 )( ū γ ū 2 ( γ2 µ F 2 (t, ū, ū 2 )dt) 2K) K b(t)dt F (t, λū, λū 2 )dt

8 8 C. LI, Z.-Q. OU, C.-L. ANG EJDE-24/64 for all (u, u 2 ) W and some positive constants K and K. It follows that ϕ(u, u 2 ) as (u, u 2 ) W due to (H2). By [4, heorem.] and Corollary 2.2, he proof is complete. Proof of heorem.4. Firstly, we prove that ϕ satisfies the (P S) condition. Suppose that {(u n, u 2n )} is a (P S) seuence for ϕ, that is, ϕ (u n, u 2n ) as n and {ϕ(u n, u 2n )} is bounded. In a way similar to the proof of heorem., we have and ( x F (t, u n (t), u 2n (t)), ũ n (t))dt 2 C α f L u n α 3 4 u n 4 ) 4 / (2 β g L C β C h L u n ( x2 F (t, u n (t), u 2n (t)), ũ 2n (t))dt 2 p C α2 f 2 L u 2n α2 p 3 4p u 2n p p 4p 4p /p (2 β2 g 2 L C β2 C h 2 L u 2n p for all n. Hence, one has / (2 f L C ) u 2n β p 4 / (2 β g L C ) ū 2n β /p (2 p f 2 L C u n β2 4p /p (2 β2 g 2 L C ū n β 2 ū n α ū 2n α 2 (ũ n, ũ 2n ) W ϕ (u n, u 2n ), (ũ n, ũ 2n ) = (( x F (t, u n (t), u 2n (t)), ũ n (t)) ( u n (t) 2 u n (t), u n (t)) ( x2 F (t, u n (t), u 2n (t)), ũ 2n (t)) ( u 2n (t) p 2 u 2n (t), u 2n (t)))dt 4 3 u n 4p 3 4 4p u 2n p p 2 C α f L u n α ) 4 / (2 β g L C β u 2n β p 4 / (2 f L C ) ū n α / 4 (2 β g L C ) ū 2n β 2 p C α2 f 2 L u 2n α2 p 4p /p (2 β2 g 2 L C β2 u n β2 4p /p (2 p f 2 L C ū 2n α 2 /p 4p (2 β2 g 2 L C ū n β 2 C h 2 L u 2n p C h L u n (2.2)

9 EJDE-24/64 PERIODIC SOLUIONS 9 for large n. It follows from Wirtinger s ineuality that (ũ n, ũ 2n ) W = ũ n W, ũ 2n W,p ( C 2 )/ u n ( C p 2 )/p u 2n p max { ( C 2 )/, ( C p 2 )/p}( u n u 2n p ) for all n. So, it follows from (2.2) and (2.3) that K( ū n p β 2 ū 2n p α 2 ū n α ū 2n β ) (2.3) / 4 (2 f L C ) ū n α / 4 (2 β g L C ) ū 2n β /p 4p (2 p f 2 L C ū 2n α /p 2 4p (2 β2 g 2 L C ū n β u n 4p 3 4p u 2n p p 2 C α f L u n α 4 / (2 β g L C β ) 2 p C α2 f 2 L u 2n p α2 4p /p (2 β2 g 2 L C β2 C h 2 L u 2n p C h L u n (ũ n, ũ 2n ) W u 2n β p u n β2 u n 4p 3 4p u 2n p p C h 2 L u 2n p C h L u n 2 C α f L u n α 4 / (2 β g L C β ) 2 p C α2 f 2 L u 2n p α2 4p /p (2 β2 g 2 L C β2 ( C p 2 )/p u 2n p ( C 2 )/ u n u n p p u 2n p p K = u n u 2n p p K u 2n β p u n β2 for large n and some positive constant K. Hence, by the above expression, we obtain 2K( ū n γ ū 2n γ2 ) min {, }( un u 2n p ) p K2 (2.4) for large n and some positive constant K 2. By the proof of heorem., we have (F (t, u n (t), u 2n (t)) F (t, ū n, ū 2n ))dt 2 u n 3 4p u 2n p p C h L u n C h 2 L u 2n p 2 C α f L u n α 2 p C α2 f 2 L u 2n α2 p / 4 (2 f L C ) ū n α / 4 ( g L C ) ū 2n β

10 C. LI, Z.-Q. OU, C.-L. ANG EJDE-24/64 /p 4p (2 p f 2 L C ū 2n α /p 2 4p (2 β2 g 2 L C ū n β 2 4p /p (2 β2 g 2 L C β2 u n β2 for all n. It follows from the boundedness of {ϕ(u n, u 2n )}, (2.4) and the above ineuality that K 3 ϕ(u n, u 2n ) = u n (t) dt u 2n (t) p dt p [ F (t, un (t), u 2n (t)) F (t, ū n, ū 2n ) ] dt F (t, ū n, ū 2n )dt 3 2 u n 7 4p u 2n p p C h L u n C h 2 L u 2n p 2 C α f L u n α 2 p C α2 f 2 L u 2n α2 p / (2 f 4 L C ) ū n α / ( g 4 L C ) ū 2n β /p (2 p f 4p 2 L C ū 2n α /p 2 (2 4p β2 g 2 L C 4p /p (2 β2 g 2 L C β2 u n β2 F (t, ū n, ū 2n )dt 2( u n u 2n p p) 2K( ū n γ ū 2n γ2 ) (2 max{, } )2K( ū n γ ū 2n γ2 ) (2 2 )2K( ū n γ ū 2n γ2 ) ( ū n γ ū 2n γ2 )( ū n γ ū 2n γ2 (2 2 )2K) K 5 ū n β 2 F (t, ū n, ū 2n )dt K 4 F (t, ū n, ū 2n )dt K 5 F (t, ū n, ū 2n )dt K 5 F (t, ū n, ū 2n )dt for large n and some real constants K 3, K 4 and K 5. he above ineuality and (H3) imply that ( ū n γ ū 2n γ2 ) is bounded. Hence, (u n, u 2n ) is bounded by (2.3) and (2.4). By the compactness of the embedding W,p, ( or W ) C(, ; RN ), the seuence {u n } (or {u 2n }) has a subseuence, still denoted by {u n } (or {u 2n }), such that u n (or u 2n ) u (or u 2 ) weakly in W,p (or in W, ), (2.5) u n (or u 2n ) u (or u 2 ) strongly in C(, ; R N ). (2.6)

11 EJDE-24/64 PERIODIC SOLUIONS Note that ϕ (u n, u 2n ), (u u n, ) = u n (t) p 2 ( u n (t), u u n (t))dt ( x F (t, u n (t), u 2n (t)), u (t) u n (t))dt as n. From (2.6), {u n } is bounded in C(, ; R N ). hen we have ( x F (t, u n (t), u 2n (t)), u (t) u n (t))dt x F (t, u n (t), u 2n (t)) u (t) u n (t) dt K 6 b(t) u (t) u n (t) dt K 6 b L u u n for some positive constant K 6, which combines with (2.6) implies that Hence, by (2.7), one has ( x F (t, u n (t), u 2n (t)), u (t) u n (t))dt as n. Moreover, from (2.6) we obtain Setting one obtains and ψ(u, u 2 ) = p u n (t) p 2 ( u n (t), u (t) u n (t))dt as n. u n (t) p 2 (u n (t), u (t) u n (t))dt as n. ψ (u n, u 2n ), (u u n, ) = By the Hölder s ineuality, we have ( u (t) p u (t) p )dt ( u 2 (t) u 2 (t) )dt, u n (t) p 2 (u n (t), u (t) u n (t))dt u n (t) p 2 ( u n (t), u (t) u n (t))dt (2.7) ψ (u n, u 2n ), (u u n, ) as n. (2.8) ( u n p u p )( u n u ) ψ (u n, u 2n ) ψ (u, u 2 ), (u u n, ), which together with (2.8) yields u n u. It follows that u n u strongly in W,p by the uniform convexity of W,p. Similarly, we have u 2n u 2 strongly in. Hence, the (P S) condition is satisfied. W,

12 2 C. LI, Z.-Q. OU, C.-L. ANG EJDE-24/64 hen Let W =,,p W W be the subspace of W given by W = {(u, u 2 ) W (ū, ū 2 ) = (, )}. ϕ(u, u 2 ) (2.9) as (u, u 2 ) W in W. In fact, by the proof of heorem., one has ϕ(u, u 2 ) = u (t) dt p u 2 (t) p dt (F (t, u (t), u 2 (t)) F (t, ū, ū 2 ))dt 2 u 4p u 2 p p 2 C α f L u α F (t, ū, ū 2 )dt 2 p C α2 f 2 L u 2 α2 p 4p /p (2 β2 g 2 L C β2 C h L u C h 2 L u 2 p for all (u, u 2 ) W. By Wirtinger s ineuality, the norm F (t, ū, ū 2 )dt (u, u 2 ) = ( u, u 2 ) L L p = u u 2 p u β2 is an euivalent norm on W. Hence, (2.9) follows from the above ineuality. On the other hand, one has ϕ(x, x 2 ) (2.) as (x, x 2 ) in R N R N, which follows from (H3). Now, heorem.4 is proved by (2.9), (2.) and the Saddle Point heorem (see [, heorem 4.6]). Acknowledgements. his research was supported by the National Natural Science Foundation of China (No. 798) and the Fundamental Research Funds for the Central Universities (No. XDJK2C55). References [] M. S. Berger, M. Schechter; On the solvability of semilinear gradient operator euations, Adv. Math. 25 (977), [2] Y.-M. Long; Nonlinear oscillations for classical Hamiltonian systems with bi-even subuadratic potentials, Nonlinear Anal. 24 (2) (995), [3] J. Mawhin; Semi-coercive monotone variational problems, Acad. Roy. Belg. Bull. Cl. Sci. 73 (987), 8-3. [4] J. Mawhin, M. Willem; Critical Point heory and Hamiltonian Systems, Springer-Verlag, Berlin/New York, 989. [5] D. Paşca; Periodic solutions for second order differential inclusions with sublinear nonlinearity, PanAmer. Math. J. (4) (2), [6] D. Paşca; Periodic solutions for nonautonomous second order differential inclusions systems with p-laplacian, Commun. Appl. Nonlinear Anal. 6(2) (29), [7] D. Paşca; Periodic solutions of a class of nonautonomous second order diffrential systems with (,p)-laplacian, Bull. Belg. Math. Soc. Simon Stevin 7 (2), no. 5, [8] D. Paşca; Periodic solutions of second-order diffrential inclusions systems with (, p)- Laplacian, Anal. Appl. (Singap.) 9 (2), no. 2, [9] D. Paşca, C.-L. ang; Some existence results on periodic solutions of nonautonomous secondorder differential systems with (, p)-laplacian, Appl. Math. Lett. 23 (2),

13 EJDE-24/64 PERIODIC SOLUIONS 3 [] D. Paşca, C.-L. ang; Some existence results on periodic solutions of ordinary (, p)- Laplacian systems, J. Appl. Math. Inform. 29 (2), no. -2, [] P. H. Rabinowitz; Minimax methods in critical point theory with applications to differential euations, CBMS Reg. Conf. Ser. in Math. No. 65, AMS, Providence, RI, 986. [2] C.-L. ang; Periodic solutions of nonautonomous second order systems with γ-uasisubadditive potential, J. Math. Anal. Appl. 89 (995), [3] C.-L. ang; Periodic solutions of nonautonomous second order systems, J. Math. Anal. Appl. 22 (996), [4] C.-L. ang; Periodic solutions for nonautonomous second order systems with sublinear nonlinearity, Proc. Amer. Math. Soc. 26() (998), [5] C.-L. ang, X. P. Wu; Periodic solutions for second order systems with not uniformly coercive potential, J. Math. Anal. Appl. 259 (2), [6] X. H. ang, Q. Meng; Solutions of a second-order Hamiltonian system with periodic boundary conditions, Nonlinear Anal. (2) [7] Y. ian, W. Ge; Periodic solutions of non-autonomous second-order systems with a p- Laplacian, Nonlinear Anal. 66 () (27), [8] M. Willem; Oscillations forcées de systèmes hamiltoniens, in: Public. Sémin. Analyse Non Linéaire, Univ. Besancon, 98. [9] X.-P. Wu, C.-L. ang; Periodic solutions of a class of nonautonomous second order systems, J. Math. Anal. Appl. 236 (999), Chun Li School of Mathematics and Statistics, Southwest University, Chonging 475, China address: Lch999@swu.edu.cn Zeng-Qi Ou School of Mathematics and Statistics, Southwest University, Chonging 475, China address: ouzeng77@sina.com Chun-Lei ang (corresponding author) School of Mathematics and Statistics, Southwest University, Chonging 475, China el , fax address: tangcl@swu.edu.cn