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Periodic solutions of second order differential equations in Banach spaces. (English) Zbl 1104.34041

The authors consider the maximal regularity for the second order periodic boundary value problem

$\begin{array}{c}{u}^{\text{'}\text{'}}\left(t\right)-aAu\left(t\right)-\alpha A{u}^{\text{'}}\left(t\right)=f\left(t\right),\phantom{\rule{4pt}{0ex}}0\le t\le 2\pi ,\\ u\left(0\right)=u\left(2\pi \right),\phantom{\rule{4pt}{0ex}}{u}^{\text{'}}\left(0\right)={u}^{\text{'}}\left(2\pi \right),\end{array}$

on a Banach space $X$. Necessary and sufficient conditions for the existence and uniqueness of periodic solutions in the spaces ${L}_{2\pi }^{p}\left(ℝ,X\right)$ ($1) and ${C}_{2\pi }^{s}\left(ℝ,X\right)$ $\left(0 are given. Moreover, results on mild solutions are also presented. Two types of mild periodic solutions are considered. When the operator $A$ is the generator of a strongly continuous cosine function, characterizations are given in terms of Fourier multipliers and spectral properties of the cosine function. Throughout the paper, examples involving elliptic partial differential operators with Dirichlet boundary conditions are discussed.

##### MSC:
 34G10 Linear ODE in abstract spaces 35L90 Abstract hyperbolic equations 47D06 One-parameter semigroups and linear evolution equations 47D09 Operator sine and cosine functions and higher-order Cauchy problems 47N20 Applications of operator theory to differential and integral equations 47F05 Partial differential operators 35L70 Nonlinear second-order hyperbolic equations 35G20 General theory of nonlinear higher-order PDE
##### References:
 [1] Aizicovici, S., Pavel, N.: Anti-periodic solutions to a class of nonlinear differential equations in Hilbert space. J. Functional Analysis. 99, 387–408 (1991) · Zbl 0743.34067 · doi:10.1016/0022-1236(91)90046-8 [2] Amann, H.: Linear and Quasilinear Parabolic Problems. Monographs in Mathematics. 89, Basel: Birkhäuser Verlag, 1995 [3] Amann, H.: Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Math. Nach. 186, 5–56 (1997) · Zbl 0880.42007 · doi:10.1002/mana.3211860102 [4] Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics. 96, Basel: Birkhäuser Verlag, 2001 [5] Arendt, W., Batty, C., Bu, S.: Fourier multipliers for Hölder continuous functions and maximal regularity. Studia Math. 160(1), 23–51 (2004) · Zbl 1073.42005 · doi:10.4064/sm160-1-2 [6] Arendt, W., Bu, S.: The operator-valued Marcinkiewicz multiplier theorem and maximal regularity. Math. Z. 240, 311–343 (2002) · Zbl 1018.47008 · doi:10.1007/s002090100384 [7] Arendt, W., Bu, S.: Operator-valued Fourier multipliers on periodic Besov spaces and applications. Proc. Edinburgh Math. Soc. 47(1), 15–33 (2004) · Zbl 1083.42009 · doi:10.1017/S0013091502000378 [8] Bourgain, J.: Vector-valued singular integrals and the H1 - BMO duality. Probability Theory and Harmonic Analysis, 1–19 Marcel Dekker, New York, 1986 [9] Burkholder, D.L.: Martingales and Fourier analysis in Banach spaces. Probability and analysis (Varenna, 1985), 61–108, Lecture Notes in Math., 1206, Springer, Berlin, 1986 [10] Chill, R., Srivastava, S.: Lp maximal regularity for second order Cauchy problems. Math. Z., to appear [11] Cioranescu, I., Lizama, C.: Spectral properties of cosine operator functions. Aequationes Mathematicae 36, 80–98 (1988) · Zbl 0675.47029 · doi:10.1007/BF01837973 [12] Clément, Ph., Prüss, J.: An operator valued transference principle and maximal regularity on vector valued Lp spaces. In: Lumer, Weis (eds.), Evolution Equations and their Applications in Physics and Life Sciences, pp. 67–87. Marcel Dekker, 2000 [13] Clément, Ph., de Pagter, B., Sukochev, F.A., Witvliet, M.: Schauder decomposition and multiplier theorems. Studia Math. 138, 135–163 (2000) [14] Clements, J.: On the existence and uniqueness of the equation . Canad. Math. Bull. 18, 181–187 (1975) · Zbl 0312.35017 · doi:10.4153/CMB-1975-036-1 [15] Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge, 1989 [16] Denk, R., Hieber, M., Prüss, J. R-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type. Memoirs Amer. Math. Soc., 166, Amer. Math. Soc., Providence, R.I., 2003 [17] De Pagter, B., Witvliet, H.: Unconditional decompositions and UMD spaces. Publ. Math. Besançon, Fasc. 16, 79–111 (1998) [18] Dostanic, M.R.: Marcinkiewicz’s theorem on operator multipliers of Fourier series. Proc. Amer. Math. Soc. 132, 391–396 (2004) · Zbl 1034.42011 · doi:10.1090/S0002-9939-03-07017-5 [19] Ebihara, Y.: On some nonlinear evolution equations with the strong dissipation. J. Differential Equations 30, 149–164 (1978) · Zbl 0387.35019 · doi:10.1016/0022-0396(78)90011-6 [20] Ebihara, Y.: On some nonlinear evolution equations with the strong dissipation, II. J. Differential Equations 34, 339–352 (1979) · Zbl 0414.35053 · doi:10.1016/0022-0396(79)90024-X [21] Fattorini, H.O.: The Cauchy Problem . Addison-Wesley. Reading (Mass.) 1983 [22] Fattorini, H.O.: Second Order Linear Differential Equations in Banach Spaces. North Holland, Amsterdam, 1985 [23] Girardi, M., Weis, L.: Operator-valued Fourier multiplier theorems on Besov spaces. Math. Nachr. 251, 34–51 (2003) · Zbl 1077.46024 · doi:10.1002/mana.200310029 [24] Girardi, M., Weis, L.: Criteria for R-boundedness of operator families. Evolution equations, 203–221, Lecture Notes in Pure and Appl. Math., 234 Dekker, New York, 2003 [25] Keyantuo, V., Lizama, C.: Fourier multipliers and integro-differential equations in Banach spaces. J. London Math. Soc. 69(3), 737–750 (2004) · Zbl 1053.45008 · doi:10.1112/S0024610704005198 [26] Lions, J.L.: Une remarque sur les applications du théorème de Hille Yosida. J. Math. Soc. Japan. 9, 62–70 (1957) · Zbl 0078.08304 · doi:10.2969/jmsj/00910062 [27] Martinez, C., Sanz, M.: The Theory of Fractional Powers of Operators. Math. Studies 187, North-Holland, 2002 [28] Nakao, M., Okochi, H.: Anti-periodic solution for utt - ($\sigma$(ux))x -uxxt = f(x,t). J. Math. Anal. Appl. 197, 796–809 (1996) · Zbl 0863.35066 · doi:10.1006/jmaa.1996.0054 [29] Pazy, A.: Semigroups of Operators and Applications to Partial Differential Equations. Springer Verlag, New York, 1983 [30] Prüss, J.: Evolutionary Integral Equations and Applications. Monographs Math. 87, Birkhäuser Verlag, 1993 [31] Prüss, J.: On the spectrum of C0-semigroups. Trans. Amer. Math. Soc. 284, 847–857 (1984) [32] Schüler, E.: On the spectrum of cosine functions. J. Math Anal. Appl. 229, 376–398 (1999) · Zbl 0921.34073 · doi:10.1006/jmaa.1998.6140 [33] Schweiker, S.: Mild solutions of second-order differential equations on the line. Math. Proc. Cambridge Philos. Soc. 129, 129–151 (2000) · Zbl 0958.34043 · doi:10.1017/S0305004199004351 [34] Sobolevskii, P.E.: On second order differential equations in a Banach space (Russian). Doklady Akad. Nauk. SSSR 146, 774–777 (1962) [35] Travis, C.C., Webb, G.F.: Cosine families and abstract nonlinear second order differential equations. Acta Math. Acad. Sci. Hungar. 32, 75–96 (1978) · Zbl 0388.34039 · doi:10.1007/BF01902205 [36] Tsutsumi, M.: Some evolution equations of second order. Proc. Japan Acad. 47, 950–955 (1970) · Zbl 0258.35017 · doi:10.3792/pja/1195526303 [37] Weis, L.: Stability theorems for semi-groups via multiplier theorems. Differential equations, asymptotic analysis and mathematical physics (Postdam, 1996), Akademie Verlag, Berlin, 1997, pp. 407–411 [38] Weis, L.: Operator-valued Fourier multiplier theorems and maximal Lp-regularity. Math. Ann. 319, 735–758 (2001) · Zbl 0989.47025 · doi:10.1007/PL00004457 [39] Weis, L.: A new approach to maximal Lp-regularity. Lect. Notes Pure Appl. Math. Marcel Dekker, New York 215, 195–214 (2001) [40] Xiao, T.J., Liang, J.: Differential operators and C-wellposedness of complete second order abstract Cauchy problems. Pacific J. Math. 186(1), 167–200 (1998)