Topological Methods in Nonlinear Analysis

A note on bounded solutions of second order differential equations at resonance

Wioletta Karpińska

Full-text: Open access

Abstract

In the paper we study the existence of bounded solutions for differential equations of the form: $x''-Ax= f(t,x)$, where $A\in L(H)$, $f\colon {\mathbb R}\times H \to H$ ($H$ - a Hilbert space) is a continuous mapping. Using a perturbation of the equation, the Leray-Schauder topological degree and fixed point theory, we overcome the difficulty that the linear problem is non-Fredholm in any resonable Banach space.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 14, Number 2 (1999), 371-384.

Dates
First available in Project Euclid: 29 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1475179851

Mathematical Reviews number (MathSciNet)
MR1766179

Zentralblatt MATH identifier
0960.34042

Citation

Karpińska, Wioletta. A note on bounded solutions of second order differential equations at resonance. Topol. Methods Nonlinear Anal. 14 (1999), no. 2, 371--384. https://projecteuclid.org/euclid.tmna/1475179851


Export citation

References

  • J. Andres, G. Gabor and L. Górniewicz, Boundary value problems on infinite intervals , Trans. Amer. Math. Soc., 351 (1999), 4861–4903 \ref \key2
  • J. W. Bebernes and L. K. Jackson, Infinite interval boundary value problems for $y''=f(x,y)$ , Duke Math. J., 34 (1967), 39–47 \ref \key3
  • J. Blot, P. Cieutat and J. Mawhin, Almost periodic oscillations of monotone second order systems , Adv. Differential Equations, 2 (1997), 693–714 \ref \key4
  • Y. L. Daletskiĭ, M. G. Kreĭn, Stability of Solutions of Differential Equations in a Banach Space, Nauka, Moscow (1970 (in Russian)) \ref \key5 ––––, On differential equations in a Hilbert space , Ukrainian Math. J., 2, 4 (1950), (Russian) \ref \key6
  • J. Dugundji and A. Granas, Fixed Point Theory, I , PWN, Warszawa (1981) \ref \key7
  • M. Frigon, Application de la théorie de la transversalité topologique á des problémes non linéaires pour des équations différentielles ordinares , Dissertationes Math., 296 (1990), 1–79 \ref \key8
  • M. Furi and P. Pera, A continuation method on locally convex spaces and aplications to ordinary differential equations on noncompact intervals , Ann. Polon. Math., XLVII (1987), 331–346 \ref \key9
  • R. E. Gaines and J. L. Mawhin, Coincidence degree and nonlinear differencial equations , Lecture Notes in Math., 568 , Springer-Verlag, New York–Berlin (1977) \ref \key10
  • O. A. Gross, The boundary value problems on an infinite interval: existence, uniqueness and asyptotic behavior of bounded solutions to a class of nonlinear second order differential equations , J. Math. Anal. Appl., 7 (1963), 100–109 \ref \key11
  • W. Karpińska, On bounded solutions of nonlinear differential equations at resonance , to appear, J. Nonlinear Anal., Theory, Meth. Appl. \ref \key12
  • M.G. Kreĭn, Linear differential equations in Banach space, Nauka, Moscow (1968 (in Russian)) \ref \key13
  • N. G. Lloyd, Degree theory, Cambridge University Press, Cambridge, London–New York–Melbourne (1978) \ref \key14
  • R. H. Martin, jr, Nonlinear operators and differential equations in Banach spaces, A Wiley-Interscience Publication, John Wiley & Sons, New York–London–Sydney–Toronto (1976) \ref \key15
  • J. Mawhin, Topological degree methods in nonlinear boundary value problems, Regional Conf. Series in Math., Amer. Math. Soc., 40 , Providence, R. I. (1979) \ref \key16 ––––, Bounded solutions of nonlinear ordinary differential equations Nonlineary Analysis and Boundary Value Problems for Ordinary Differential Equations, Udine 1995 , CISM Courses and Lectures, 371 , Springer, Vienne (1996), 121–147 \ref \key17 ––––, Bounded and almost periodic solutions of nonlinear differential equations: variational vs. nonvariational approach , Nonlineary Analysis and Boundary Value Problems for Ordinary Differential Equations, Udine 1995, Calculus of Variational and Differenrial Equations (Technion, 1998) (A. Ioffe, S. Reich and I. Shafrir, eds.), Research Notes in Math. No. 410, Chapman and Hall/CRC, Boca Raton (1999), 167–184 \ref \key18
  • J. Mawhin and J. R. Ward Jr, Bounded solutions of some second order nonlinear differentials equations , J. London Math. Soc., 58 (1998), 733–747 \ref \key19
  • R. Ortega and A. Tineo, Resonance and non-resonance in a problem of boundedness , Proc. Amer. Math. Soc., 124 (1996), 2089–2096 \ref \key20
  • B. Przeradzki, The existence of bounded solutions for differential eguations in Hilbert spaces , Annal. Polon. Math., LVI.2 (1992), 106–108 \ref \key21 ––––, On a two-point boundary value problem for differential equations on the half-line , Annal. Polon. Math., L (1989), 53–61 \ref \key22
  • D. O'Regan, Solvability of some singular boundary value problems on the semi-infinite interval , Canad. J. Math., 48 (1) (199), 143–158 \ref \key23 ––––, Theory of Singular Boundary Value Problems, World Scientific Publishing Co. Pte. Ltd., Singapore–New Jersey–London–Hong Kong (1994) \ref \key24 ––––, Positive solutions for a class of boundary value problems on infinite intervals , Nonlinear Differential Equations Appl., 3 (1994), 203–228 \ref \key25
  • K. Schmitt, Bounded solutions of nonlinear second order differential equations , Duke Math. J., 36 (1969), 237–243 \ref \key26
  • R. Stańczy, Hammerstein equations with integral over a noncompact domain , Ann. Polon. Math., LXIX.1 (1998), 49–60 \ref \key27
  • K. Yosida, Functional Analysis, Springer-Verlag, Berlin–Heidelberg–New York (1974)