Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Nonresonance conditions on the potential for a second-order periodic boundary value problem

Authors: Pierpaolo Omari and Fabio Zanolin
Journal: Proc. Amer. Math. Soc. 117 (1993), 125-135
MSC: Primary 34B15; Secondary 47H15
MathSciNet review: 1143021
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the periodic problem

\begin{displaymath}\begin{array}{*{20}{c}} { - u'' = f(u) + h(t),} \\ {u(0) = u(2\pi ),\qquad u'(0) = u'(2\pi ),} \\ \end{array} \end{displaymath}

and prove its solvability for any given $ h$, under new assumptions on the asymptotic behaviour of the potential of the nonlinearity $ f$, with respect to two consecutive eigenvalues of the associated linear problem.

References [Enhancements On Off] (What's this?)

  • [ALL] S. Ahmad, A. C. Lazer, and J. L. Paul, Elementary critical point theory and perturbations of elliptic boundary value problems at resonance, Indiana Univ. Math. J. 25 (1976), 933-944. MR 0427825 (55:855)
  • [CO] D. Costa and A. Oliveira, Existence of solutions for a class of semilinear elliptic problems at double resonance, Boll. Soc. Brasil Mat. 19 (1988), 21-37. MR 1018926 (91c:35057)
  • [DFG] D. G. De Figueiredo and J. P. Gossez, Conditions de non-résonance pour certains problèmes elliptiques semi-linéaires, C. R. Acad. Sci. Paris 302 (1986), 543-545. MR 845644 (87j:35118)
  • [D] T. Ding, Nonlinear oscillations at a point of resonance, Sci. Sinica Ser. A 25 (1982), 918-931. MR 681856 (84c:34058)
  • [DIZ] T. Ding, R. Iannacci, and F. Zanolin, Existence and multiplicity results for periodic solutions of semilinear Duffing equations, preprint, 1990. MR 1240400 (94g:34060)
  • [DZ] T. Ding and F. Zanolin, Time-maps for the solvability of periodically perturbed nonlinear Duffing equations, Nonlinear Anal. T.M.A. (to appear). MR 1128965 (93j:34053)
  • [FF] C. Fabry and A. Fonda, Periodic solutions of nonlinear differential equations with double resonance, Ann. Mat. Pura Appl. 157 (1990), 99-116. MR 1108472 (92e:34029)
  • [GO1] J. P. Gossez and P. Omari, Nonresonance with respect to the Fucik spectrum for periodic solutions of second order ordinary differential equations, Nonlinear Anal. T.M.A. 14 (1990), 1079-1104. MR 1059615 (91f:34049)
  • [GO2] -, Periodic solutions of a second order ordinary differential equation: a necessary and sufficient condition for nonresonance, J. Differential Equations 94 (1991), 67-82. MR 1133541 (92m:34090)
  • [MW1] J. Mawhin and M. Willem, Critical points of convex perturbations of some indefinite quadratic forms and semilinear boundary value problems at resonance, Ann. Inst. Henri Poincaré 6 (1986), 431-453. MR 870864 (88a:35023)
  • [MW2] -, Critical Point Theory and Hamiltonian systems, Springer-Verlag, New York, 1989. MR 982267 (90e:58016)
  • [OZ] P. Omari and F. Zanolin, A note on nonlinear oscillations at resonance, Acta Math. Sinica (N.S.) 3 (1987), 351-361. MR 930765 (89c:34039)
  • [Q] D. Qian, An abundance of periodic solutions for Duffing equation with oscillatory time-map, preprint, 1990.
  • [R] M. Ramos, Remarks on resonance problems with unbounded perturbations, preprint, 1990. MR 1190173 (94a:35049)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34B15, 47H15

Retrieve articles in all journals with MSC: 34B15, 47H15

Additional Information

Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society