By Jean Mawhin

FACHGEB the decade has noticeable a massive improvement in severe element conception in limitless dimensional areas and its software to nonlinear boundary worth difficulties. specifically, extraordinary effects have been acquired within the classical challenge of periodic strategies of Hamiltonian structures. This ebook presents a scientific presentation of the main simple instruments of severe element conception: minimization, convex services and Fenchel remodel, twin least motion precept, Ekeland variational precept, minimax tools, Lusternik- Schirelmann conception for Z2 and S1 symmetries, Morse thought for very likely degenerate serious issues and non-degenerate serious manifolds. each one approach is illustrated by means of purposes to the dialogue of the life, multiplicity, and bifurcation of the periodic suggestions of Hamiltonian structures. one of the handled questions are the periodic strategies with fastened interval or fastened power of self reliant structures, the lifestyles of subharmonics within the non-autonomous case, the asymptotically linear Hamiltonian structures, loose and compelled superlinear difficulties. software of these effects to the equations of mechanical pendulum, to Josephson platforms of good kingdom physics and to questions from celestial mechanics are given. the purpose of the booklet is to introduce a reader universal to extra classical thoughts of standard differential equations to the strong procedure of recent severe aspect thought. the fashion of the exposition has been tailored to this aim. the recent topological instruments are brought in a innovative yet certain manner and instantly utilized to differential equation difficulties. The summary instruments can be utilized to partial differential equations and the reader also will locate the fundamental references during this course within the bibliography of greater than 500 goods which concludes the publication. ERSCHEIN

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**Example text**

E. 1 gives conditions for the existence of such a solution. If (19) is not satisfied with T replaced by T/k (k E R, k> 2), T is called the minimal period of u. 3. Subharmonics of Non-Autonomous Convex Hamiltonian Systems 51 for some k > 2 where minimal period is strictly greater than T. Such solutions are called subharmonic solutions or simply subharmonics. 3. 2. Assume that H(t, u)/Iu12 - 0 (20) H(t, u) --* +00 (21) and as Jul , oo uniformly in t E R. Then, for each k E N \ {0}, there exists a kT-periodic solution Uk of (18), such that (22) IIukiio - oo and such that the minimal period Tk of Uk tends to +oo when k -* oo.

Proof. Let cl = maxtER IH(t, )l. By condition (21) there exists R > 0 such that H(t, u) > 1 + ci for all t E R and u with IuI > R. By convexity we have, for all (t, u) E RxR 2N with Jul > R, 1 + ci < H (t,ju) < IRI H(t, u) + (1- RI) H(t, 0) IRIH(t,u)+cl and hence there is (3 > 0 and d > 0 such that H(t, u) > blul - 0 (23) for all (t, u) E R x R2N If k E N \ {0} is fixed, condition (20) implies that there exists a E ]0, 2ir/kT[ and -t > 0 such that H(t, u) < (a/2)lul2 +'Y (24) for all (t, u) E R x R2N.

The Direct Method of the Calculus of Variations 20 Proof. 1. If u is a solution of (5) then, integrating the differential equation over (0, T) and using the boundary conditions, we get T VF(t, u(t)) dt = 0. J0 (22) Let u = u + u where u = (11T) fo u(t) dt and define the strictly convex functions G and G on RN by G(x) = J T F(t, x) dt; 0 G(x) = f F(t, x + u(l)) dt. 5 implies that G(x) -+ +oo as jxj --} oo. ), we have G(x) < (1/2) J0 T F(t, 2x) dt + (1/2) J0 T F(t, 2u(t)) dt = (1/2)G(2x) + C. (24) It then follows from (23) and (24) that G(x) -+ +oo as jxj --r oo.