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

PERIODIC SOLUTIONS OF SECOND-ORDER DIFFERENTIAL INCLUSIONS SYSTEMS WITH p–LAPLACIAN DANIEL PAS¸CA Abstract. Some existence results are obtained for periodic solutions of nonautonomous second-order differential inclusions systems with p–Laplacian. 1. Introduction Consider the second order system ¨u(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→Rsatisfies the following assumption: (A) F(t, x) is measurable in tfor each x∈Rnand continuously differentiable in xfor a.e. t∈[0, T], and there exist a∈C(R+,R+), b∈ L1(0, T;R+) such that |F(t, x)| ≤ a(kxk)b(t),k∇F(t, x)k ≤ a(kxk)b(t) for all x∈Rnand 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 conditions on the potential F(see [6]-[17]). Also in a series of papers (see [3]-[5]) we have generalized some of these results for the case when the potential Fis just locally Lipschitz in the second variable xnot continuously differentiable. The aim of this paper is to consider the problem (1) in a more general sense. More exactly our results represent the extensions to systems with p– Laplacian and also with discontinuity (we consider the generalized gradients unlike 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 Educacio´n y Ciencia, grant number SB2003-0337. 1 2 DANIEL PAS¸CA Consider the second order differential inclusions system d dt¡k˙u(t)kp−2˙u(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→Rand ∂denotes the Clarke subdifferential. We suppose that F=F1+F2and F1,F2satisfy the following assumption: (A’) F1,F2are measurable in tfor each x∈Rn, at least F1or F2are strictly differentiable in xand 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∈Rnand all t∈[0, T]. The corresponding functional ϕ:W1,p T→Ris given by ϕ(u) = 1 pZT 0 k˙u(t)kpdt +ZT 0 F(t, u(t))dt. Definition 1. A function G:Rn→Ris called to be (λ, µ)–subconvex if G(λ(x+y)) ≤µ(G(x) + G(y)) for some λ, µ > 0and all x, y ∈Rn. Remark 1. When λ=µ=1 2, a function (1 2,1 2)–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 results Theorem 1. Assume that F=F1+F2, where F1,F2satisfy assumption (A’) and the following conditions: (i) F1(t, ·)is (λ, µ)–subconvex with λ > 1/2and 0< µ < 2p−1λpfor a.e. t∈[0, T]; (ii) there exist c1, c2>0and α∈[0, p −1) such that ζ∈∂F2(t, x)⇒ kζk ≤ c1kxkα+c2 for all x∈Rnand a.e. t∈[0, T]; (iii) for q=p p−1, 1 kxkqα h1 µZT 0 F1(t, λx)dt +ZT 0 F2(t, x)dti→ ∞,as kxk → ∞. PERIODIC SOLUTIONS OF INCLUSIONS SYSTEMS WITH p–LAPLACIAN 3 Then the problem (2) has at least one solution which minimizes ϕon W1,p T. Theorem 2. Assume that F=F1+F2, where F1,F2satisfy 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 RT 0h(t)dt = 0 such that F1(t, x)≥ hh(t), xi+γ(t) for all x∈Rnand a.e. t∈[0, T]; (v) there exist c1>0,c0∈Rsuch that ζ∈∂F2(t, x)⇒ kζk ≤ c1 for all x∈Rnand all t∈[0, T ], and ZT 0 F2(t, x)dt ≥c0 for all x∈Rn; (vi) 1 µZT 0 F1(t, λx)dt +ZT 0 F2(t, x)dt → ∞,as kxk → ∞. Then the problem (2) has at least one solution which minimizes ϕon W1,p T. Theorem 3. Assume that F=F1+F2, where F1,F2satisfy 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 RT 0h(t)dt = 0 such that F1(t, x)≥ hh(t), xi+γ(t) for all x∈Rnand a.e. t∈[0, T]; (viii) there exist c1, c2>0and α∈[0, p −1) such that ζ∈∂F2(t, x)⇒ kζk ≤ c1kxkα+c2 for all x∈Rnand a.e. t∈[0, T]; (ix) for q=p p−1, 1 kxkqα ZT 0 F2(t, x)dt → ∞,as kxk → ∞. Then the problem (2) has at least one solution which minimizes ϕon W1,p T. Remark 2. Theorems 1, 2 and 3, generalizes the corresponding Theorems 2.1, 2.3 and 2.4 of [5]. In fact, it follows from these theorems letting p= 2. 4 DANIEL PAS¸CA 3. The preliminary results We introduce some functional spaces. Let [0, T] be a fixed real interval (0 < T < ∞) and 1 < p < ∞. We denote by W1,p Tthe Sobolev space of functions u∈Lp(0, T;Rn) having a weak derivative ˙u∈Lp(0, T;Rn). The norm over W1,p Tis defined by kukW1,p T=³ZT 0 ku(t)kpdt +ZT 0 k˙u(t)kpdt´ 1 p. We recall that kukLp=³ZT 0 ku(t)kpdt´ 1 pand kuk∞= max t∈[0,T ]ku(t)k. For our aims it is necessary to recall some very well know results (for proof and details see [2]). Proposition 4. If u∈W1,p Tthen kuk∞≤ckukW1,p T. If u∈W1,p Tand RT 0u(t)dt = 0 then kuk∞≤ck˙ukLp. Proposition 5. If the sequence (uk)kconverges weakly to uin W1,p T, then (uk)kconverges uniformly to uon [0, T]. Let Xbe a Banach space. Now follows [1], for each x, v ∈X, we define the generalized directional derivative at xin the direction vof a given f∈ Liploc(X, R) as f0(x;v) = lim sup y→x,λ&0 f(y+λv)−f(y) λ and we denote by ∂f(x) = {x∗∈X∗:f0(x;v)≥ hx∗, vi,for all v∈X} the generalized gradient of fat 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 on open set containing the line segment [x,y]. Then there exists a point u in (x,y) such that f(y)−f(x)∈ h∂f(u), y −xi. 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 Ybe a separable Banach space; •let Zbe a closed subspace of Lp(T;Y) (for some pin [1,∞)), where Lp(T;Y) is the space of pintegrable functions from Tto Y; •define a functional fon Zvia f(x) = ZT ft(x(t))µ(dt), where ft:Y→R, (t∈T) is a given family of functions; •suppose that for each yin Ythe function t→ft(y) is measurable, and that xis a point at which f(x) is defined (finitely). Hypothesis 1: There is a function kin Lq(T, R), ³1 p+1 q= 1´such that, for all t∈T, |ft(y1)−ft(y2)| ≤ k(t)ky1−y2kYfor all y1, y2∈Y Hypothesis 2: Each function ftis Lipschitz (of some rank) near each point of Y, and for some constant c, for all t∈T,y∈Y, one has ζ∈∂ft(y)⇒ kζkY∗≤c{1 + kykp−1 Y}. Under this conditions described above Clarke prove (see [1], Theorem 2.7.5): Theorem 7. Under the conditions described above, under either of Hypothesis 1 or 2, f is uniformly Lipschitz on bounded subsets of Z, and one has ∂f(x)⊂ZT ∂ft(x(t))µ(dt). Further, if each ftis regular at x(t)then fis regular at xand equality holds. Remark 3. fis globally Lipschitz on Zwhen Hypothesis 1 hold. Now we can prove the following result. Theorem 8. Let F: [0, T ]×Rn→Rsuch that F=F1+F2where F1, F2are measurable in tfor each x∈Rn, and there exist k1∈Lq(0, T;R), a∈C(R+,R+),b∈L1(0, T;R+),c1,c2>0and α∈[0, p −1) such that |F1(t, x1)−F1(t, x2)| ≤ k1(t)kx1−x2k |F2(t, x)| ≤ a(kxk)b(t) ζ∈∂F2(t, x)⇒ kζk ≤ c1kxkα+c2 (3) 6 DANIEL PAS¸CA for 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) = 1 pkykp+F(t, x). Then, the functional f:Z∈R, where Z=n(u, v)∈Lp(0, T;Y) : u(t) = Zt 0 v(s)ds +c, c ∈Rno given by f(u, v) = RT 0L(t, u(t), v(t))dt, is uniformly Lipschitz on bounded subsets of Zand one has ∂f(u, v)⊂ZT 0 {∂F1(t, u(t)) + ∂F2(t, u(t))} × {kv(t)kp−2v(t)}dt. Proof. Let L1(t, x, y) = F1(t, x), L2(t, x, y) = 1 pkykp+F2(t, x) and f1,f2:Z→Rgiven by f1(u, v) = RT 0L1(t, u(t), v(t))dt,f2(u, v) = RT 0L2(t, u(t), v(t))dt. For f1we can apply Theorem 7 under Hypothesis 1, with the following cast of characters: •(T,T,µ)=[0,T] with Lebesgue measure, Y=Rn×Rnbe the Hilbert product space (hence is separable); •p > 1 and Z=n(u, v)∈Lp(0, T;Y) : u(t) = Zt 0 v(s)ds +c, c ∈Rno be a closed subspace of Lp(0, T;Y); •ft(x, y) = L1(t, x, y) = F1(t, x); in our assumptions it results that the integrand L1(t, x, y) is measurable in tfor a given element (x, y) of Yand 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)kx1−x2k ≤ k1(t)¡kx1−x2k+ky1−y2k¢= =k1(t)k(x1, y1)−(x2, y2)kY (4) for all t∈[0, T] and all (x1, y1), (x2, y2)∈Y. Hence f1is uniformly Lipschitz on bounded subsets of Zand one has ∂f1(u, v)⊂ZT 0 ∂L1(t, u(t), v(t))dt. For f2we can apply Theorem 7 under Hypothesis 2 with the same cast of characters, but now ft(x, y) = L2(t, x, y) = 1 pkykp+F2(t, x). In our assumptions it results that the integrand L2(t, x, y) is measurable in tfor a given element (x, y) of Yand 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)× {kykp−2y}. PERIODIC SOLUTIONS OF INCLUSIONS SYSTEMS WITH p–LAPLACIAN 7 Using (3) and (4), if ζ= (ζ1, ζ2)∈∂L2(t, x, y) it results ζ1∈∂F2(t, x) and ζ2=kykp−2y, and hence kζk=kζ1k+kζ2k ≤ c1kxkα+c2+kykp−1≤˜c{1 + k(x, y)kp−1} for each t∈[0, T], since α < p−1 and p > 1. Hence f2is uniformly Lipschitz on bounded subsets of Zand one has ∂f2(u, v)⊂ZT 0 ∂L2(t, u(t), v(t))dt. It follows that f=f1+f2is uniformly Lipschitz on the bounded subsets of Z. Proposition 2.3.3 and Proposition 2.3.15 from [1] implies ∂f(u, v)⊂∂f1(u, v) + ∂f2(u, v)⊂ ⊂ZT 0h∂L1(t, u(t), v(t)) + ∂L2(t, u(t), v(t))idt ⊂ ⊂ZT 0h³∂xL1(t, u(t), v(t)) ×∂yL1(t, u(t), v(t))´+ +³∂xL2(t, u(t), v(t)) ×∂yL2(t, u(t), v(t))´idt ⊂ ⊂ZT 0h³∂xL1(t, u(t), v(t)) + ∂xL2(t, u(t), v(t))´× ׳∂yL1(t, u(t), v(t)) + ∂yL2(t, u(t), v(t))´idt = =ZT 0³∂F1(t, u(t)) + ∂F2(t, u(t))´× {kv(t)kp−2v(t)}dt. Moreover, Corollary 1 of Proposition 2.3.3 from from [1] imply that, if at least of the functions F1,F2is strictly differentiable in xfor all t∈[0, T] then ∂f(u, v)⊂ZT 0 ∂F(t, u(t)) × {kv(t)kp−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= ˙u0) and if ζ∈∂f(u0, v0), we deduce the existence of a measurable function (q(t), p(t)) such that q(t)∈∂F(t, u0(t)) and p(t) = kv0(t)kp−2v0(t)a.e. on [0, T](6) 8 DANIEL PAS¸CA and for any (u, v)in Z, one has hζ, (u, v)i=ZT 0 {hq(t), u(t)i+hp(t), v(t)i}dt. In particular, if ζ= 0 (so that u0is critical point for ϕ(u) = RT 0h1 pk˙u(t)kp+ F(t, u(t))idt), it then follows easily that q(t) = ˙p(t)a.e., or taking into account (6) d dt¡k˙u0(t)kp−2˙u0(t)¢∈∂F(t, u0(t)) a.e. on [0, T], so that u0satisfies the inclusions system (2). Remark 5. Of course if p= 2 and Fis continuously differentiable in x, then the system (2) becomes system (1). 4. Proofs of the Theorems Proof of Theorem 1. From (A’) it follows immediately there exist a∈C(R+,R+), b∈L1(0, T;R+) such that |F1(t, x)| ≤ a(kxk)b(t), for all x∈Rnand all t∈[0, T]. Like in [11] we obtain F1(t, x)≤(2µkxkβ+ 1)a0b(t) for all x∈Rnand all t∈[0, T], where β < p and a0= max0≤s≤1a(s). For u∈W1,p T, let ¯u=1 TRT 0u(t)dt and ˜u=u−¯u. From Lebourg’s mean value 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) = hζ, ˜u(t)i.It follows from (ii) and H¨older’s inequality that   ZT 0 [F2(t, u(t)) −F2(t, ¯u)]dt  ≤ZT 0 |F2(t, u(t)) −F2(t, ¯u)|dt ≤ ≤ZT 0 kζkk˜u(t)kdt ≤ZT 0h2c1(k¯ukα+k˜u(t)kα) + c2ik˜u(t)kdt ≤ ≤C1k˜ukα+1 ∞+C2k˜uk∞k¯ukα+C3k˜uk∞≤ ≤C4k˙ukα+1 Lp+1 2pk˙ukp Lp+C5k˙ukLp+C6k¯ukqα for all u∈W1,p Tand some positive constants C4,C5and C6. Hence we have ϕ(u)≥1 pZT 0 k˙u(t)kpdt +1 µZT 0 F1(t, λ¯u)dt −ZT 0 F1(t, −˜u(t))dt+ PERIODIC SOLUTIONS OF INCLUSIONS SYSTEMS WITH p–LAPLACIAN 9 +ZT 0 F2(t, ¯u)dt +ZT 0 [F2(t, u(t)) −F2(t, ¯u)]dt ≥ ≥1 2pk˙ukp Lp−C4k˙ukα+1 Lp−C5k˙ukLp−C6k¯ukqα −(2µk˜ukβ ∞+ 1) ZT 0 a0b(t)dt+ +1 µZT 0 F1(t, λ¯u)dt +ZT 0 F2(t, ¯u)dt ≥1 2pk˙ukp Lp−C4k˙ukα+1 Lp−C5k˙ukLp− −C7k˙ukβ Lp−C8+k¯ukqαn1 k¯ukqα h1 µZT 0 F1(t, λ¯u)dt +ZT 0 F2(t, ¯u)dti−C6o for all u∈W1,p T, which implies that ϕ(u)→ ∞ as kuk → ∞ by (iii) because α < p −1, β < p, and the norm kuk= (k¯ukp+k˙ukp Lp)1 pis an equivalent norm on W1,p T. Now we write ϕ(u) = ϕ1(u) + ϕ2(u) where ϕ1(u) = 1 pZT 0 k˙u(t)kpdt and ϕ2(u) = ZT 0 F(t, u(t))dt. The function ϕ1is weakly lower semi-continuous (w.l.s.c.) on W1,p T. From (i), (ii) and Theorem 7, taking to account Remark 3 and Proposition 5, it follows that ϕ2is w.l.s.c. on W1,p T. By Theorem 1.1 in [2] it follows that ϕhas a minimum u0on W1,p T. Evidently Z≃W1,p Tand ϕ(u) = f(u, v) for all (u, v)∈Z. From Theorem 8, it results that fis uniformly Lipschitz on bounded subsets of Z, and therefore ϕpossesses the same properties relative to W1,p T. Proposition 2.3.2 in [1] implies that 0 ∈∂ϕ(u0) (so that u0is critical point for ϕ). Now from Theorem 8 and Remark 4 it follows that the problem (2) has at least one solution u∈W1,p T. Proof of Theorem 2. Let (uk) be a minimizing sequence of ϕ. It follows from (iv), (v), Lebourg’s mean value theorem and Proposition 4 that ϕ(uk)≥1 pk˙ukkp Lp+ZT 0 hh(t), uk(t)idt +ZT 0 γ(t)dt+ +ZT 0 F2(t, ¯uk)dt −ZT 0 kζkk˜uk(t)kdt ≥1 pk˙ukkp Lp− −k˜ukk∞ZT 0 kh(t)kdt +ZT 0 γ(t)dt −c1k˜ukk∞+c0≥ ≥1 pk˙ukkp Lp−c2k˙ukkLp−c3