Existence of periodic solutions for nonlinear evolution equations with pseudo monotone operators

Shioji, Naoki

doi:10.1090/S0002-9939-97-03984-1

Existence of periodic solutions for nonlinear evolution equations with pseudo monotone operators
HTML articles powered by AMS MathViewer

by Naoki Shioji PDF

Proc. Amer. Math. Soc. 125 (1997), 2921-2929 Request permission

Abstract:

In this paper, we study the existence of $T$-periodic solutions for the problem \[ u’ (t) +A(t)u(t) =0, \quad t\in \mathbb R, \] where $A(t)$ is a $T$-periodic, pseudo monotone mapping from a reflexive Banach space into its dual.

References

Herbert Amann, Periodic solutions of semilinear parabolic equations, Nonlinear analysis (collection of papers in honor of Erich H. Rothe), Academic Press, New York, 1978, pp. 1–29. MR 499089
Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR 0390843
Ronald I. Becker, Periodic solutions of semilinear equations of evolution of compact type, J. Math. Anal. Appl. 82 (1981), no. 1, 33–48. MR 626739, DOI 10.1016/0022-247X(81)90223-7
Felix E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1976, pp. 1–308. MR 0405188
Radu Caşcaval and Ioan I. Vrabie, Existence of periodic solutions for a class of nonlinear evolution equations, Rev. Mat. Univ. Complut. Madrid 7 (1994), no. 2, 325–338. MR 1297518
R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR 0221256
Jean-Pierre Gossez, Existence of periodic solutions for some first order nonlinear ordinary differential equations at resonance, Ordinary differential equations and functional equations (Florence, 1978), CNR, Rome, 1978, pp. 361–379. MR 690624
Norimichi Hirano, Abstract nonlinear Volterra equations with positive kernels, SIAM J. Math. Anal. 17 (1986), no. 2, 403–414. MR 826701, DOI 10.1137/0517030
Norimichi Hirano, Nonlinear Volterra equations with positive kernels, Nonlinear and convex analysis (Santa Barbara, Calif., 1985) Lecture Notes in Pure and Appl. Math., vol. 107, Dekker, New York, 1987, pp. 83–98. MR 892784
Norimichi Hirano, Nonlinear evolution equations with nonmonotonic perturbations, Nonlinear Anal. 13 (1989), no. 6, 599–609. MR 998507, DOI 10.1016/0362-546X(89)90081-3
Norimichi Hirano, Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces, Proc. Amer. Math. Soc. 120 (1994), no. 1, 185–192. MR 1174494, DOI 10.1090/S0002-9939-1994-1174494-8
J. Mawhin, Topological degree methods in nonlinear boundary value problems, CBMS Regional Conference Series in Mathematics, vol. 40, American Mathematical Society, Providence, R.I., 1979. Expository lectures from the CBMS Regional Conference held at Harvey Mudd College, Claremont, Calif., June 9–15, 1977. MR 525202
Jan Prüss, Periodic solutions of semilinear evolution equations, Nonlinear Anal. 3 (1979), no. 5, 601–612. MR 541871, DOI 10.1016/0362-546X(79)90089-0
Michael Renardy and Robert C. Rogers, An introduction to partial differential equations, Texts in Applied Mathematics, vol. 13, Springer-Verlag, New York, 1993. MR 1211418
Ioan I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc. 109 (1990), no. 3, 653–661. MR 1015686, DOI 10.1090/S0002-9939-1990-1015686-4
Eberhard Zeidler, Nonlinear functional analysis and its applications. II/A, Springer-Verlag, New York, 1990. Linear monotone operators; Translated from the German by the author and Leo F. Boron. MR 1033497, DOI 10.1007/978-1-4612-0985-0
Eberhard Zeidler, Nonlinear functional analysis and its applications. II/B, Springer-Verlag, New York, 1990. Nonlinear monotone operators; Translated from the German by the author and Leo F. Boron. MR 1033498, DOI 10.1007/978-1-4612-0985-0