Some existence results are obtained for periodic solutions of nonautonomous second-order differential inclusions systems with p-Laplacia

Paşca, Daniel

Universitat Autònoma de Barcelona. Centre de Recerca Matemàtica

Centre de Recerca Matemàtica

Diposit Digital de Documents de la UAB

PERIODIC SOLUTIONS OF SECOND-ORDERDIFFERENTIAL INCLUSIONS SYSTEMS WITHp–LAPLACIANDANIEL PAS¸CAAbstract. Some existence results are obtained for periodic solutionsof nonautonomous second-order differential inclusions systems withp–Laplacian.1. IntroductionConsider the second order systemu¨(t) = ∇F (t, u(t)) a.e. t ∈ [0, T ]u(0)− u(T ) = u˙(0)− u˙(T ) = 0(1)where T > 0 and F : [0, T ]× Rn → R satisfies the following assumption:(A) F (t, x) is measurable in t for each x ∈ Rn and continuously dif-ferentiable in x for a.e. t ∈ [0, T ], and there exist a ∈ C(R+,R+), b ∈L1(0, T ;R+) such that|F (t, x)| ≤ a(‖x‖)b(t), ‖∇F (t, x)‖ ≤ a(‖x‖)b(t)for all x ∈ Rn and a.e. t ∈ [0, T ].In the last years many authors starting with Mawhin and Willem (see [2])proved the existence of solutions for problem (1) under suitable conditionson the potential F (see [6]-[17]). Also in a series of papers (see [3]-[5]) wehave generalized some of these results for the case when the potential F isjust locally Lipschitz in the second variable x not continuously differentiable.The aim of this paper is to consider the problem (1) in a more generalsense. More exactly our results represent the extensions to systems with p–Laplacian and also with discontinuity (we consider the generalized gradientsunlike continuously gradient in classical results).1991 Mathematics Subject Classification. 34C25, 49J24.Key words and phrases. p–Laplacian, inclusions systems, periodic solutions.† The author was supported by Ministerio de Educacion´ y Ciencia, grant numberSB2003-0337.12 DANIEL PAS¸CAConsider the second order differential inclusions systemddt(‖u˙(t)‖p−2u˙(t)) ∈ ∂F (t, u(t)) a.e. t ∈ [0, T ],u(0) = u(T ), u˙(0) = u˙(T ),(2)where p > 1, T > 0, F : [0, T ] × Rn → R and ∂ denotes the Clarkesubdifferential.We suppose that F = F1+F2 and F1, F2 satisfy the following assumption:(A’) F1, F2 are measurable in t for each x ∈ Rn, at least F1 or F2 arestrictly differentiable in x and there exist k1, k2 ∈ Lq(0, T ;R) such that|F1(t, x1)− F1(t, x2)| ≤ k1(t)||x1 − x2||,|F2(t, x1)− F2(t, x2)| ≤ k2(t)||x1 − x2||,for all x1, x2 ∈ Rn and all t ∈ [0, T ].The corresponding functional ϕ :W 1,pT → R is given byϕ(u) =1p∫ T0‖u˙(t)‖pdt+∫ T0F (t, u(t))dt.Definition 1. A function G : Rn → R is called to be (λ, µ)–subconvex ifG(λ(x+ y)) ≤ µ(G(x) +G(y))for some λ, µ > 0 and all x, y ∈ Rn.Remark 1. When λ = µ = 12 , a function (12 ,12 )–subconvex is called convex.When λ = µ = 1, a function (1, 1)–subconvex is called subadditive.When λ = 1, µ > 0, a function (1, µ)–subconvex is called µ–subadditive.2. Main resultsTheorem 1. Assume that F = F1 + F2, where F1, F2 satisfy assumption(A’) and the following conditions:(i) F1(t, ·) is (λ, µ)–subconvex with λ > 1/2 and 0 < µ < 2p−1λp fora.e. t ∈ [0, T ];(ii) there exist c1, c2 > 0 and α ∈ [0, p− 1) such thatζ ∈ ∂F2(t, x)⇒ ‖ζ‖ ≤ c1‖x‖α + c2for all x ∈ Rn and a.e. t ∈ [0, T ];(iii) for q = pp−1 ,1‖x‖qα[ 1µ∫ T0F1(t, λx)dt+∫ T0F2(t, x)dt]→∞, as ‖x‖ → ∞.PERIODIC SOLUTIONS OF INCLUSIONS SYSTEMS WITH p–LAPLACIAN 3Then the problem (2) has at least one solution which minimizes ϕ on W 1,pT .Theorem 2. Assume that F = F1 + F2, where F1, F2 satisfy assumption(A’) and the following conditions:(iv) F1(t, ·) is (λ, µ)-subconvex for a.e. t ∈ [0, T ], and there exists γ ∈L1(0, T ;R), h ∈ L1(0, T ;Rn) with ∫ T0h(t)dt = 0 such thatF1(t, x) ≥ 〈h(t), x〉+ γ(t)for all x ∈ Rn and a.e. t ∈ [0, T ];(v) there exist c1 > 0, c0 ∈ R such thatζ ∈ ∂F2(t, x)⇒ ‖ζ‖ ≤ c1for all x ∈ Rn and all t ∈ [0, T ], and∫ T0F2(t, x)dt ≥ c0for all x ∈ Rn;(vi)1µ∫ T0F1(t, λx)dt+∫ T0F2(t, x)dt→∞, as ‖x‖ → ∞.Then the problem (2) has at least one solution which minimizes ϕ on W 1,pT .Theorem 3. Assume that F = F1 + F2, where F1, F2 satisfy assumption(A’) and the following conditions:(vii) F1(t, ·) is (λ, µ)-subconvex for a.e. t ∈ [0, T ], and there exists γ ∈L1(0, T ;R), h ∈ L1(0, T ;Rn) with ∫ T0h(t)dt = 0 such thatF1(t, x) ≥ 〈h(t), x〉+ γ(t)for all x ∈ Rn and a.e. t ∈ [0, T ];(viii) there exist c1, c2 > 0 and α ∈ [0, p− 1) such thatζ ∈ ∂F2(t, x)⇒ ‖ζ‖ ≤ c1‖x‖α + c2for all x ∈ Rn and a.e. t ∈ [0, T ];(ix) for q = pp−1 ,1‖x‖qα∫ T0F2(t, x)dt→∞, as ‖x‖ → ∞.Then the problem (2) has at least one solution which minimizes ϕ on W 1,pT .Remark 2. Theorems 1, 2 and 3, generalizes the corresponding Theorems2.1, 2.3 and 2.4 of [5]. In fact, it follows from these theorems letting p = 2.4 DANIEL PAS¸CA3. The preliminary resultsWe introduce some functional spaces. Let [0, T ] be a fixed real interval(0 < T < ∞) and 1 < p < ∞. We denote by W 1,pT the Sobolev space offunctions u ∈ Lp(0, T ;Rn) having a weak derivative u˙ ∈ Lp(0, T ;Rn). Thenorm over W 1,pT is defined by‖u‖W 1,pT =(∫ T0‖u(t)‖pdt+∫ T0‖u˙(t)‖pdt) 1p.We recall that‖u‖Lp =(∫ T0‖u(t)‖pdt) 1pand ‖u‖∞ = maxt∈[0,T ]‖u(t)‖.For our aims it is necessary to recall some very well know results (for proofand details see [2]).Proposition 4. If u ∈W 1,pT then‖u‖∞ ≤ c‖u‖W 1,pT .If u ∈W 1,pT and∫ T0u(t)dt = 0 then‖u‖∞ ≤ c‖u˙‖Lp .Proposition 5. If the sequence (uk)k converges weakly to u in W1,pT , then(uk)k converges uniformly to u on [0, T ].Let X be a Banach space. Now follows [1], for each x, v ∈ X, we definethe generalized directional derivative at x in the direction v of a given f ∈Liploc(X,R) asf0(x; v) = lim supy→x,λ↘0f(y + λv)− f(y)λand we denote by∂f(x) = {x∗ ∈ X∗ : f0(x; v) ≥ 〈x∗, v〉, for all v ∈ X}the generalized gradient of f at x (the Clarke subdifferential).We recall the Lebourg’s mean value theorem (see [1], Theorem 2.3.7).Theorem 6. Let x and y be points in X, and suppose that f is Lipschitz onopen set containing the line segment [x,y]. Then there exists a point u in(x,y) such thatf(y)− f(x) ∈ 〈∂f(u), y − x〉.Clarke consider in [1] the following abstract framework:PERIODIC SOLUTIONS OF INCLUSIONS SYSTEMS WITH p–LAPLACIAN 5• let (T ,T , µ) be a positive complete measure space with µ(T ) <∞,and let Y be a separable Banach space;• let Z be a closed subspace of Lp(T ;Y ) (for some p in [1,∞)), whereLp(T ;Y ) is the space of p- integrable functions from T to Y ;• define a functional f on Z viaf(x) =∫Tft(x(t))µ(dt),where ft : Y → R, (t ∈ T ) is a given family of functions;• suppose that for each y in Y the function t→ ft(y) is measurable,and that x is a point at which f(x) is defined (finitely).Hypothesis 1: There is a function k in Lq(T,R),(1p +1q = 1)such that, forall t ∈ T ,|ft(y1)− ft(y2)| ≤ k(t)‖y1 − y2‖Y for all y1, y2 ∈ YHypothesis 2: Each function ft is Lipschitz (of some rank) near each pointof Y , and for some constant c, for all t ∈ T , y ∈ Y , one hasζ ∈ ∂ft(y)⇒ ‖ζ‖Y ∗ ≤ c{1 + ‖y‖p−1Y }.Under this conditions described above Clarke prove (see [1], Theorem 2.7.5):Theorem 7. Under the conditions described above, under either of Hy-pothesis 1 or 2, f is uniformly Lipschitz on bounded subsets of Z, and onehas∂f(x) ⊂∫T∂ft(x(t))µ(dt).Further, if each ft is regular at x(t) then f is regular at x and equality holds.Remark 3. f is globally Lipschitz on Z when Hypothesis 1 hold.Now we can prove the following result.Theorem 8. Let F : [0, T ] × Rn → R such that F = F1 + F2 where F1,F2 are measurable in t for each x ∈ Rn, and there exist k1 ∈ Lq(0, T ;R),a ∈ C(R+,R+), b ∈ L1(0, T ;R+), c1, c2 > 0 and α ∈ [0, p− 1) such that|F1(t, x1)− F1(t, x2)| ≤ k1(t)‖x1 − x2‖|F2(t, x)| ≤ a(‖x‖)b(t)ζ ∈ ∂F2(t, x)⇒ ‖ζ‖ ≤ c1‖x‖α + c2(3)6 DANIEL PAS¸CAfor all t ∈ [0, T ] and all x, x1, x2 ∈ Rn. We suppose that L : [0, T ]× Rn ×Rn → R, is given by L(t, x, y) = 1p‖y‖p + F (t, x).Then, the functional f : Z ∈ R, whereZ ={(u, v) ∈ Lp(0, T ;Y ) : u(t) =∫ t0v(s)ds+ c, c ∈ Rn}given by f(u, v) =∫ T0L(t, u(t), v(t))dt, is uniformly Lipschitz on boundedsubsets of Z and one has∂f(u, v) ⊂∫ T0{∂F1(t, u(t)) + ∂F2(t, u(t))} × {‖v(t)‖p−2v(t)}dt.Proof. Let L1(t, x, y) = F1(t, x), L2(t, x, y) = 1p‖y‖p + F2(t, x) andf1,f2 : Z → R given by f1(u, v) =∫ T0L1(t, u(t), v(t))dt, f2(u, v) =∫ T0L2(t, u(t), v(t))dt. For f1 we can apply Theorem 7 under Hypothesis1, with the following cast of characters:• (T ,T , µ)=[0,T] with Lebesgue measure, Y = Rn × Rn be theHilbert product space (hence is separable);• p > 1 andZ ={(u, v) ∈ Lp(0, T ;Y ) : u(t) =∫ t0v(s)ds+ c, c ∈ Rn}be a closed subspace of Lp(0, T ;Y );• ft(x, y) = L1(t, x, y) = F1(t, x); in our assumptions it results thatthe integrand L1(t, x, y) is measurable in t for a given element (x, y)of Y and there exists k ∈ Lq(0, T ;R) such that|L1(t, x1, y1)− L1(t, x2, y2)| = |F1(t, x1)− F1(t, x2)| ≤≤ k1(t)‖x1 − x2‖ ≤ k1(t)(‖x1 − x2‖+ ‖y1 − y2‖) == k1(t)‖(x1, y1)− (x2, y2)‖Y(4)for all t ∈ [0, T ] and all (x1, y1), (x2, y2) ∈ Y . Hence f1 is uniformlyLipschitz on bounded subsets of Z and one has∂f1(u, v) ⊂∫ T0∂L1(t, u(t), v(t))dt.For f2 we can apply Theorem 7 under Hypothesis 2 with the same castof characters, but now ft(x, y) = L2(t, x, y) = 1p‖y‖p + F2(t, x). In ourassumptions it results that the integrand L2(t, x, y) is measurable in t for agiven element (x, y) of Y and locally Lipschitz in (x, y) for each t ∈ [0, T ].Proposition 2.3.15 from [1] implies∂L2(t, x, y) ⊂ ∂xL2(t, x, y)× ∂yL2(t, x, y) = ∂F2(t, x)× {‖y‖p−2y}.PERIODIC SOLUTIONS OF INCLUSIONS SYSTEMS WITH p–LAPLACIAN 7Using (3) and (4), if ζ = (ζ1, ζ2) ∈ ∂L2(t, x, y) it results ζ1 ∈ ∂F2(t, x) andζ2 = ‖y‖p−2y, and hence‖ζ‖ = ‖ζ1‖+ ‖ζ2‖ ≤ c1‖x‖α + c2 + ‖y‖p−1 ≤ c˜{1 + ‖(x, y)‖p−1}for each t ∈ [0, T ], since α < p−1 and p > 1. Hence f2 is uniformly Lipschitzon bounded subsets of Z and one has∂f2(u, v) ⊂∫ T0∂L2(t, u(t), v(t))dt.It follows that f = f1+ f2 is uniformly Lipschitz on the bounded subsets ofZ.Proposition 2.3.3 and Proposition 2.3.15 from [1] implies∂f(u, v) ⊂ ∂f1(u, v) + ∂f2(u, v) ⊂⊂∫ T0[∂L1(t, u(t), v(t)) + ∂L2(t, u(t), v(t))]dt ⊂⊂∫ T0[(∂xL1(t, u(t), v(t))× ∂yL1(t, u(t), v(t)))++(∂xL2(t, u(t), v(t))× ∂yL2(t, u(t), v(t)))]dt ⊂⊂∫ T0[(∂xL1(t, u(t), v(t)) + ∂xL2(t, u(t), v(t)))××(∂yL1(t, u(t), v(t)) + ∂yL2(t, u(t), v(t)))]dt ==∫ T0(∂F1(t, u(t)) + ∂F2(t, u(t)))× {‖v(t)‖p−2v(t)}dt.Moreover, Corollary 1 of Proposition 2.3.3 from from [1] imply that, if atleast of the functions F1, F2 is strictly differentiable in x for all t ∈ [0, T ]then∂f(u, v) ⊂∫ T0∂F (t, u(t))× {‖v(t)‖p−2v(t)}dt.(5)¤Remark 4. The interpretation of expression (5) is as follows: if (u0, v0)is an element of Z (so that v0 = u˙0) and if ζ ∈ ∂f(u0, v0), we deduce theexistence of a measurable function (q(t), p(t)) such thatq(t) ∈ ∂F (t, u0(t)) and p(t) = ‖v0(t)‖p−2v0(t) a.e. on [0, T ](6)8 DANIEL PAS¸CAand for any (u, v) in Z, one has〈ζ, (u, v)〉 =∫ T0{〈q(t), u(t)〉+ 〈p(t), v(t)〉}dt.In particular, if ζ = 0 (so that u0 is critical point for ϕ(u) =∫ T0[1p‖u˙(t)‖p+F (t, u(t))]dt), it then follows easily that q(t) = p˙(t) a.e., or taking intoaccount (6)ddt(‖u˙0(t)‖p−2u˙0(t)) ∈ ∂F (t, u0(t)) a.e. on [0, T ],so that u0 satisfies the inclusions system (2).Remark 5. Of course if p = 2 and F is continuously differentiable in x,then the system (2) becomes system (1).4. Proofs of the TheoremsProof of Theorem 1. From (A’) it follows immediately there exista ∈ C(R+,R+), b ∈ L1(0, T ;R+) such that|F1(t, x)| ≤ a(‖x‖)b(t),for all x ∈ Rn and all t ∈ [0, T ]. Like in [11] we obtainF1(t, x) ≤ (2µ‖x‖β + 1)a0b(t)for all x ∈ Rn and all t ∈ [0, T ], where β < p and a0 = max0≤s≤1 a(s).For u ∈ W 1,pT , let u¯ = 1T∫ T0u(t)dt and u˜ = u − u¯. From Lebourg’s meanvalue theorem it follows that for each t ∈ [0, T ] there exist z(t) in (u¯, u(t))and ζ ∈ ∂F2(t, z(t)) such that F2(t, u(t)) − F2(t, u¯) = 〈ζ, u˜(t)〉. It followsfrom (ii) and Ho¨lder’s inequality that∫ T0[F2(t, u(t))− F2(t, u¯)]dt ≤ ∫ T0|F2(t, u(t))− F2(t, u¯)|dt ≤≤∫ T0‖ζ‖‖u˜(t)‖dt ≤∫ T0[2c1(‖u¯‖α + ‖u˜(t)‖α) + c2]‖u˜(t)‖dt ≤≤ C1‖u˜‖α+1∞ + C2‖u˜‖∞‖u¯‖α + C3‖u˜‖∞ ≤≤ C4‖u˙‖α+1Lp +12p‖u˙‖pLp + C5‖u˙‖Lp + C6‖u¯‖qαfor all u ∈W 1,pT and some positive constants C4, C5 and C6. Hence we haveϕ(u) ≥ 1p∫ T0‖u˙(t)‖pdt+ 1µ∫ T0F1(t, λu¯)dt−∫ T0F1(t,−u˜(t))dt+PERIODIC SOLUTIONS OF INCLUSIONS SYSTEMS WITH p–LAPLACIAN 9+∫ T0F2(t, u¯)dt+∫ T0[F2(t, u(t))− F2(t, u¯)]dt ≥≥ 12p‖u˙‖pLp−C4‖u˙‖α+1Lp −C5‖u˙‖Lp−C6‖u¯‖qα−(2µ‖u˜‖β∞ + 1)∫ T0a0b(t)dt++1µ∫ T0F1(t, λu¯)dt+∫ T0F2(t, u¯)dt ≥ 12p‖u˙‖pLp − C4‖u˙‖α+1Lp − C5‖u˙‖Lp−−C7‖u˙‖βLp − C8 + ‖u¯‖qα{ 1‖u¯‖qα[ 1µ∫ T0F1(t, λu¯)dt+∫ T0F2(t, u¯)dt]− C6}for all u ∈W 1,pT , which implies that ϕ(u)→∞ as ‖u‖ → ∞ by (iii) becauseα < p − 1, β < p, and the norm ‖u‖ = (‖u¯‖p + ‖u˙‖pLp)1p is an equivalentnorm on W 1,pT . Now we write ϕ(u) = ϕ1(u) + ϕ2(u) whereϕ1(u) =1p∫ T0‖u˙(t)‖pdt and ϕ2(u) =∫ T0F (t, u(t))dt.The function ϕ1 is weakly lower semi-continuous (w.l.s.c.) on W1,pT . From(i), (ii) and Theorem 7, taking to account Remark 3 and Proposition 5, itfollows that ϕ2 is w.l.s.c. on W1,pT . By Theorem 1.1 in [2] it follows thatϕ has a minimum u0 on W1,pT . Evidently Z ' W 1,pT and ϕ(u) = f(u, v)for all (u, v) ∈ Z. From Theorem 8, it results that f is uniformly Lipschitzon bounded subsets of Z, and therefore ϕ possesses the same propertiesrelative to W 1,pT . Proposition 2.3.2 in [1] implies that 0 ∈ ∂ϕ(u0) (so thatu0 is critical point for ϕ). Now from Theorem 8 and Remark 4 it followsthat the problem (2) has at least one solution u ∈W 1,pT .Proof of Theorem 2. Let (uk) be a minimizing sequence of ϕ. Itfollows from (iv), (v), Lebourg’s mean value theorem and Proposition 4thatϕ(uk) ≥ 1p‖u˙k‖pLp +∫ T0〈h(t), uk(t)〉dt+∫ T0γ(t)dt++∫ T0F2(t, u¯k)dt−∫ T0‖ζ‖‖u˜k(t)‖dt ≥ 1p‖u˙k‖pLp−−‖u˜k‖∞∫ T0‖h(t)‖dt+∫ T0γ(t)dt− c1‖u˜k‖∞ + c0 ≥≥ 1p‖u˙k‖pLp − c2‖u˙k‖Lp − c310 DANIEL PAS¸CAfor all k and some constants c2, c3, which implies that (u˜k) is bounded. Onthe other hand, in a way similar to the proof of Theorem 1, one has∣∣∣ ∫ T0[F2(t, u(t))− F2(t, u¯)]dt∣∣∣ ≤ 12p‖u˙‖pLp + C1‖u˙‖Lpfor all k and some positive constant C1, which implies thatϕ(uk) ≥ 1p‖u˙k‖pLp +1µ∫ T0F1(t, λu¯k)dt−∫ T0F1(t,−u˜k(t))dt++∫ T0F2(t, u¯k)dt+∫ T0[F2(t, u(t))− F2(t, u¯k)]dt ≥≥ 12p‖u˙k‖pLp − a(‖u˜k‖∞)∫ T0b(t)dt− C1‖u˙k‖Lp++1µ∫ T0F1(t, λu¯k)dt+∫ T0F2(t, u¯k)dtfor all k and some positive constant C1. It follows from (vi) and the bound-edness of (u˜k) that (u¯k) is bounded. Hence ϕ has a bounded minimizingsequence (uk). Now Theorem 2 follows like Theorem 1.Proof of Theorem 3. From (vii), (viii) and Proposition 4 it followsthatϕ(u) ≥ 1p‖u˙‖pLp +∫ T0〈h(t), u(t)〉dt++∫ T0γ(t)dt+∫ T0F2(t, u¯)dt+∫ T0[F2(t, u(t))− F2(t, u¯)]dt ≥≥ 12p‖u˙‖pLp − ‖u˜‖∞∫ T0‖h(t)‖dt++∫ T0γ(t)dt− C1‖u˙‖α+1Lp − C2‖u˙‖Lp +∫ T0F2(t, u¯)dt− C3‖u¯‖qα ≥≥ 12p‖u˙‖pLp−C1‖u˙‖α+1Lp −C4(‖u˙‖Lp+1)+‖u¯‖qα[ 1‖u¯‖qα∫ T0F2(t, u¯)dt− C3]for all u ∈ W 1,pT and some positive constants C1, C3 and C4. Now followslike in the proof of Theorem 1 that ϕ is coercive by (ix), which completesthe proof.ACKNOWLEDGEMENTSPERIODIC SOLUTIONS OF INCLUSIONS SYSTEMS WITH p–LAPLACIAN 11The author thanks Centre de Recerca Matema`tica for the hospitality andfacilities for doing this work.References[1] F.H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Classics in Applied Math-ematics vol.5, Philadelphia, 1990.[2] J. Mawhin and M. 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AMS, vol. 126, nr. 11 (1998), 3263–3270.[9] Chun-Lei Tang - Existence and Multiplicity of Periodic Solutions of NonautonomousSecond Order Systems, Nonlinear Analysis, vol. 32, nr. 3 (1998), 299–304.[10] Xing-Ping Wu - Periodic Solutions for Nonautonomous Second-Order Systems withBounded Nonlinearity, J. Math. Anal. Appl. 230 (1999), 135–141.[11] Xing-Ping Wu and Chun-Lei Tang - Periodic Solutions of a Class of Non-autonomous Second-Order Systems, J. Math. Anal. Appl. 236 (1999), 227–235.[12] Chun-Lei Tang and Xing-Ping Wu - Periodic Solutions for Second Order Systemswith Not Uniformly Coercive Potential, J. Math. Anal. Appl. 259 (2001), 386–397.[13] Jian Ma and Chun-Lei Tang - Periodic Solutions for Some Nonautonomous Second-Order Systems, J. Math. Anal. Appl. 275 (2002), 482–494.[14] Chun-Lei Tang and Xing-Ping Wu - Notes on Periodic Solutions of SubquadraticSecond Order Systems, J. Math. Anal. Appl. 285 (2003), 8–16.[15] Xing-Ping Wu, Chun-Lei Tang - Periodic Solutions of Nonautonomous Second-Order Hamiltonian Systems with Even-Typed Potentials, Nonlinear Analysis 55(2003), 759–769.[16] Fukun Zhao and Xian Wu - Saddle Point Reduction Method for Some Non-autonomous Second Order Systems, J. Math. Anal. Appl. 291 (2004), 653–665.[17] Chun-Lei Tang, Xing-Ping Wu - Subharmonic Solutions for Nonautonomous SecondOrder Hamiltonian Systems, J. Math. Anal. Appl. 304 (2005), 383–393.Permanent address: Department of Mathematics and Informatics, Universityof Oradea, University Street 1, 410087 Oradea, RomaniaE-mail address: dpasca@uoradea.ro12 DANIEL PAS¸CACurrent address: Centre de Recerca Matema`tica, 08193 Bellaterra, Barcelona,SpainE-mail address: dpasca@crm.es

Periodic solutions of second-order differential inclusions systems with p-Laplacian

http://ddd.uab.cat/pub/prepub/2005/hdl_2072_1178/Pr652.pdf

Periodic solutions of second-order differential inclusions systems with p-Laplacian

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