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Homoclinic solutions for a class of the second order Hamiltonian systems. (English) Zbl 1080.37067
Summary: We study the existence of homoclinic orbits for the second-order Hamiltonian system $\ddot q+ V_q(t,q)= f(t)$, where $q\in \Bbb R^n$ and $V\in C^1(\Bbb R\times\Bbb R^n,\Bbb R)$, and $V(t,q)=-K(t,q)+W(t,q)$ is $T$-periodic in $t$. A map $K$ satisfies the “pinching” condition $b_1|q|^2\le K(t,q)\le b_2|q|^2$, $W$ is superlinear at infinity and $f$ is sufficiently small in $L^2(\Bbb R,\Bbb R^n)$. A homoclinic orbit is obtained as a limit of $2kT$-periodic solutions of a certain sequence of the second-order differential equations.

##### MSC:
 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 58E05 Abstract critical point theory 34C37 Homoclinic and heteroclinic solutions of ODE 70H05 Hamilton’s equations
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##### References:
 [1] Ambrosetti, A.; Zelati, V. Coti: Multiple homoclinic orbits for a class of conservative systems. Rend. sem. Mat. univ. Padova 89, 177-194 (1993) · Zbl 0806.58018 [2] Ambrosetti, A.; Rabinowitz, P. H.: Dual variational methods in critical point theory and applications. J. funct. Anal. 14, 349-381 (1973) · Zbl 0273.49063 [3] Carrião, P. C.; Miyagaki, O. H.: Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems. J. math. Anal. appl. 230, 157-172 (1999) · Zbl 0919.34046 [4] V. Coti Zelati, I. Ekeland, E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann. 228 (1990) 133 -- 160. · Zbl 0731.34050 [5] Zelati, V. Coti; Rabinowitz, P. H.: Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. amer. Math. soc. 4, 693-727 (1991) · Zbl 0744.34045 [6] Ding, Y.: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear anal. 25, 1095-1113 (1995) · Zbl 0840.34044 [7] Ding, Y.; Girardi, M.: Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign. J. math. Anal. appl. 189, 585-601 (1995) [8] Ding, Y.; Li, S.: Homoclinic orbits for first order Hamiltonian systems. J. math. Anal. appl. 189, 585-601 (1995) · Zbl 0818.34023 [9] Ding, Y.; Willem, M.: Homoclinic orbits of a Hamiltonian system. Z. angew. Math. phys. 50, 759-778 (1999) · Zbl 0997.37041 [10] Hofer, H.; Wysocki, K.: First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Math. ann. 228, 483-503 (1990) · Zbl 0702.34039 [11] Omana, W.; Willem, M.: Homoclinic orbits for a class of Hamiltonian systems. Differential integral equations 5, 1115-1120 (1992) · Zbl 0759.58018 [12] Rabinowitz, P. H.: Homoclinic orbits for a class of Hamiltonian systems. Proc. roy. Soc. Edinburgh 114A, 33-38 (1990) · Zbl 0705.34054 [13] Rabinowitz, P. H.; Tanaka, K.: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206, 472-499 (1991) · Zbl 0707.58022 [14] Séré, E.: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 209, 561-590 (1993) [15] Szulkin, A.; Zou, W.: Homoclinic orbits for asymptotically linear Hamiltonian systems. J. funct. Anal. 187, 25-41 (2001) · Zbl 0984.37072 [16] Tanaka, K.: Homoclinic orbits for a singular second order Hamiltonian system. Ann. inst. H. Poincaré 7, No. 5, 427-438 (1990) · Zbl 0712.58026