Topological Methods in Nonlinear Analysis

On second-order boundary value problems in Banach spaces: a bound sets approach

Jan Andres, Luisa Malaguti, and Martina Pavlačková

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Abstract

The existence and localization of strong (Carathéodory) solutions is obtained for a second-order Floquet problem in a Banach space. The combination of applied degree arguments and bounding (Liapunov-like) functions allows some solutions to escape from a given set. The problems concern both semilinear differential equations and inclusions. The main theorem for upper-Carathéodory inclusions is separately improved for Marchaud inclusions (i.e. for globally upper semicontinuous right-hand sides) in the form of corollary. Three illustrative examples are supplied.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 37, Number 2 (2011), 303-341.

Dates
First available in Project Euclid: 20 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461184788

Mathematical Reviews number (MathSciNet)
MR2849825

Zentralblatt MATH identifier
1242.34116

Citation

Andres, Jan; Malaguti, Luisa; Pavlačková, Martina. On second-order boundary value problems in Banach spaces: a bound sets approach. Topol. Methods Nonlinear Anal. 37 (2011), no. 2, 303--341. https://projecteuclid.org/euclid.tmna/1461184788


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References

  • S. Aizicovici and M. Fečkan, Forced symmetric oscillations of evolution equations , Nonlin. Anal., 64(2006), 1621–1640 \ref\key 2
  • J. Andres and R. Bader, Asymptotic boundary value problems in Banach spaces , J. Math. Anal. Appl., 247 (2002), 437–457 \ref\key 3
  • J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Topological Fixed Point Theory and Its Applications, 1 , Kluwer, Dordrecht (2003) \ref\key 4
  • J. Andres, M. Kožušníková and L. Malaguti, Bound sets approach to boundary value problems for vector second-order differential inclusions , Nonlin. Anal., 71 (2009), 28–44 \ref\key 5
  • J. Andres, L. Malaguti and V. Taddei, On boundary value problems in Banach spaces , Dynam. Syst. Appl., 18(2009), 275–302 \ref\key 6
  • J.-P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin (1984) \ref\key 7
  • J. Chandra, V. Lakshmikantham and A. R. Mitchell, Existence of solutions of boundary value problems for nonlinear second order systems in Banach space , Nonlinear Anal., 2(1978), 157–168 \ref\key 8
  • Ju. L. Daleckiĭ and M.G. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, Translation of Mathematical Monographs, Amer. Math. Soc., Providence, R. I. (1974) \ref\key 9
  • K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin (1992) \ref\key 10
  • R. Deville, G. Godefroy and V. Zizler, Smoothness and Renorming in Banach Spaces, Longman Scientific and Technical, Harlow (1993) \ref\key 11
  • J. Diestel, Geometry of Banach Spaces – Selected Topics , Springer, Berlin (1975) \ref\key 12
  • R. Dragoni, J. W. Macki, P. Nistri and P. Zecca, Solution Sets of Differential Equations in Abstract Spaces, Pitman Research Notes in Mathematics Series, 342 , Longman, Harlow (1996) \ref\key 13
  • S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I: Theory, Kluwer, Dordrecht (1997) \ref\key 14 ––––, Handbook of Multivalued Analysis, Vol. II, Kluwer, Dordrecht (2000) \ref\key 15
  • M. I. Kamenskiĭ, V. V. Obukhovskiĭ and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, W. de Gruyter, Berlin (2001) \ref\key 16
  • J. L. Massera and J. J. Schäffer, Linear Differential Equations and Functional Spaces, Academic Press, New York (1966) \ref\key 17
  • J. Mawhin, Two point boundary value problems for nonlinear second order differential equations in Hilbert spaces , Tôhoku Math. J., 32 (1980), 225–233 \ref\key 18
  • H. Mönch, Boundary value problems for nonlinear differential equations of second order in Banach spaces , Nonlinear Anal. TMA, 4(1980), 225–233 \ref\key 19
  • N. S. Papageorgiou and S. Th. Kyritsi-Yiallourou, Handbook of Applied Analysis, Springer, Berlin (2009) \ref\key 20
  • K. Schmitt and R. Thompson, Boundary value problems for infinite systems of second-order differential equations , J. Differential Equations, 18(1975), 277–295 \ref\key 21
  • K. Schmitt and P. Volkmann, Boundary value problems for second order differential equations in convex subsets in a Banach space , Trans. Amer. Math. Soc., 218(1976), 397–405 \ref\key 22
  • G. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, Berlin (2002) \ref\key 23
  • P. Volkmann, Über die positive Invarianz einer abgeschlossenen Teilmenge eines Banachschen Raumes bezüglich der Differentialgleichung $u' = f(t, u)$ , J. Reine Angew. Math., 285(1976), 59–65 \ref\key 24
  • P. Volkmann and L. Yiping, The positive invariant set of the differential equation on Banach space , Ann. Differential Equations, 14(1998), 267–270 \ref\key 25
  • I. I. Vrabie, Compactness Methods for Nonlinear Evolutions, 2nd ed., Longman House, Burn Mill, Harlow (1990) \ref\key 26
  • Z. Wang and F. Zhang, Two points boundary value problems in Banach spaces , Appl. Math. Mech., 17 (1996), 275–280