Topological Methods in Nonlinear Analysis

Almost-periodicity problem as a fixed-point problem for evolution inclusions

Jan Andres and Alberto M. Bersani

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Existence of almost-periodic solutions to quasi-linear evolution inclusions under a Stepanov almost-periodic forcing is nontraditionally examined by means of the Banach-like and the Schauder-Tikhonov-like fixed-point theorems. These multivalued fixed-point principles concern condensing operators in almost-periodic function spaces or their suitable closed subsets. The Bohr-Neugebauer-type theorem jointly with the Bochner transform are employed, besides another, for this purpose. Obstructions related to possible generalizations are discussed.

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Topol. Methods Nonlinear Anal., Volume 18, Number 2 (2001), 337-349.

First available in Project Euclid: 22 August 2016

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Andres, Jan; Bersani, Alberto M. Almost-periodicity problem as a fixed-point problem for evolution inclusions. Topol. Methods Nonlinear Anal. 18 (2001), no. 2, 337--349.

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  • J. Andres, Bounded, almost-periodic and periodic solutions of quasi-linear differential inclusions , Differential Inclusions and Optimal Control, Lecture Notes in Nonlin. Anal. (J. Andres, L. Górniewicz and P. Nistri, eds.), 2 (1998), 35–50 \ref\key 2 ––––, Almost-periodic and bounded solutions of Carathéodory differential inclusions , Differential Inttegral Equations, 12 (1999), 887–912 \ref\key 3
  • J. Andres and R. Bader, Asymptotic boundary value problems in Banach spaces , Preprint (2001) \ref\key 4
  • J. Andres, A. M. Bersani and K. Leśniak, On some almost-periodicity problems in various metrics , Acta Appl. Math., 65(2001), 35–57 \ref\key 5
  • J. Andres, G. Gabor and L. Górniewicz, Boundary value problems on infinite intervals , Trans. Amer. Math. Soc., 351(1999), 4861–4903 \ref\key 6
  • J. Andres and L. Górniewicz, On the Banach contraction principle for multivalued mappings , Approximation, Optimization and Mathematical Economics (M. Lassonde, ed.), Physica–Verlag, Springer, Berlin (2001), 1–23 \ref\key 7
  • J. Andres and B. Krajc, Unified approach to bounded, periodic and almost-periodic solutions of differential systems , Ann. Math. Sil., 11(1997), 39–53 \ref\key 8
  • L. Amerio and G. Prouse, Almost-Periodic Functions and Functional Equations, Van Nostrand Reinhold Co., New York (1971) \ref\key 9
  • W. M. Bogdanowitz, On the existence of almost periodic solutions for systems of ordinary differential equations in Banach spaces , Arch. Rational Mech. Anal., 13 (1963), 364–370 \ref\key 10
  • B. F. Bylov, R. E. Vinograd, V. Ya. Lin and O. O. Lokutsievskiĭ, On the topological reasons for the anomalous behaviour of certain almost periodic systems , Problems in the Asymptotic Theory of Nonlinear Oscillations, Naukova Dumka, Kiev (1977), 54–61, (Russian) \ref\key 11
  • W. A Coppel, Almost periodic properties of ordinary differential equations , Ann. Mat. Pura Appl. (4), 76(1967), 27–50 \ref\key 12
  • C. Corduneanu, Almost periodic solutions to differential equations in abstract spaces , Rev. Roumaine Math. Pures Appl., 42(1997), 9–10 \ref\key 13
  • G. L. Cain, Jr. and M. Z. Nashed, Fixed points and stability for a sum of two operators in locally convex spaces , Pacific J. Math., 39(1971), 581–592 \ref\key 14
  • H. Covitz and S. B. Nadler, Multivalued contraction mappings in generalized metric spaces, Israel J. Math., 8 (1970), 5–11 \ref\key 15
  • L. I. Danilov, Measure-valued almost periodic functions and almost periodic selections of multivalued maps , Mat. Sb., 188 (1997), 3–24, (Russian);, Sbornik: Mathematics, 188 (1997), 1417–1438 \ref\key 16
  • Ju. L. Daleckiĭ and M. G. Krein, Stability of Solutions of Differential Equations and Banach Space , Trans. Math. Monogr., 43 , Amer. Math. Soc., Providence, R. I. (1974) \ref\key 17
  • A. M. Dolbilov and I. Ya. Shneiberg, Almost periodic multifunctions and their selections , Sibirsk. Math. Zh., 32 (1991), 172–175, (Russian) \ref\key 18
  • A. M. Fink, Compact families of almost periodic functions and an application of the Schauder fixed point theorem , SIAM J. Appl. Math., 17(1969), 1258–1262 \ref\key 19 ––––, Almost Periodic Differential Equations, LNM 377, Springer, Berlin (1974) \ref\key 20
  • A. M. Fink and G. Seifert, Non-resonance conditions for the existence of almost periodic solutions of almost periodic systems , SIAM J. Appl. Math., 21(1971), 362–366 \ref\key 21
  • G. Gabor, On the acyclicity of fixed point sets of multivalued maps, Topol. Methods Nonlinear Anal., 14 (1999), 327–343 \ref\key 22
  • A. Haraux, Asymptotic behavior for two-dimensional, quasi-autonomous, almost-periodic evolution equations , J. Differential Equations, 66 (1987), 62–70 \ref\key 23
  • S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis Theory, 1 , Kluwer, Dordrecht (1997) \ref\key 24
  • M. Kamenskiĭ, V. Obukhovskiĭ and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter, Berlin, to appear \ref\key 25
  • J. L. Massera and J. J. Schäfer, Linear Differential Equations and Function Spaces, Academic Press, New York (1966) \ref\key 26
  • G. Mehta, K.-K. Tan and X.-Z. Yuan, Fixed points, maximal elements and equilibria of generalized games , Nonlinear Anal., 28(1997), 689–699 \ref\key 27
  • A. A. Pankov, Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations, Kluwer, Dordrecht (1990) \ref\key 28
  • S. Park, Generalized Leray-Schauder principles for condensing admissible multifunctions , Ann. Mat. Pura Appl. (4), 172 (1997), 65–85 \ref\key 29 ––––, A unified fixed point theory of multimaps on topological vector spaces , J. Korean Math. Soc., 35 (1998), 803–829 \ref\key 30
  • B. N. Sadovskiĭ, Limit compact and condensing operators, Russian Math. Surveys, 27(1972), 85–155