Localization of nonsmooth lower and upper functions for periodic boundary value problems

Irena Rachůnková; Milan Tvrdý

Mathematica Bohemica (2002)

  • Volume: 127, Issue: 4, page 531-545
  • ISSN: 0862-7959

Abstract

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In this paper we present conditions ensuring the existence and localization of lower and upper functions of the periodic boundary value problem u ' ' + k u = f ( t , u ) , u ( 0 ) = u ( 2 π ) , u ' ( 0 ) = u ' ( 2 π ) , k ∈ ℝ , k ≠ 0 . These functions are constructed as solutions of some related generalized linear problems and can be nonsmooth in general.

How to cite

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Rachůnková, Irena, and Tvrdý, Milan. "Localization of nonsmooth lower and upper functions for periodic boundary value problems." Mathematica Bohemica 127.4 (2002): 531-545. <http://eudml.org/doc/249021>.

@article{Rachůnková2002,
abstract = {In this paper we present conditions ensuring the existence and localization of lower and upper functions of the periodic boundary value problem $u^\{\prime \prime \}+k\,u=f(t,u)$, $ u(0)=u(2\,\pi )$, $u^\{\prime \}(0)=u^\{\prime \}(2\pi )$, $k\in \mathbb \{R\}\hspace\{0.56905pt\}$, $k\ne 0.$ These functions are constructed as solutions of some related generalized linear problems and can be nonsmooth in general.},
author = {Rachůnková, Irena, Tvrdý, Milan},
journal = {Mathematica Bohemica},
keywords = {second order nonlinear ordinary differential equation; periodic problem; lower and upper functions; generalized linear differential equation; second-order nonlinear ordinary differential equation; periodic problem; lower and upper functions; generalized linear differential equation},
language = {eng},
number = {4},
pages = {531-545},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Localization of nonsmooth lower and upper functions for periodic boundary value problems},
url = {http://eudml.org/doc/249021},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Rachůnková, Irena
AU - Tvrdý, Milan
TI - Localization of nonsmooth lower and upper functions for periodic boundary value problems
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 4
SP - 531
EP - 545
AB - In this paper we present conditions ensuring the existence and localization of lower and upper functions of the periodic boundary value problem $u^{\prime \prime }+k\,u=f(t,u)$, $ u(0)=u(2\,\pi )$, $u^{\prime }(0)=u^{\prime }(2\pi )$, $k\in \mathbb {R}\hspace{0.56905pt}$, $k\ne 0.$ These functions are constructed as solutions of some related generalized linear problems and can be nonsmooth in general.
LA - eng
KW - second order nonlinear ordinary differential equation; periodic problem; lower and upper functions; generalized linear differential equation; second-order nonlinear ordinary differential equation; periodic problem; lower and upper functions; generalized linear differential equation
UR - http://eudml.org/doc/249021
ER -

References

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  1. Lower and upper solutions in the theory of ODE boundary value problems: Classical and recent results, F. Zanolin (ed.) Non Linear Analysis and Boundary Value Problems for Ordinary Differential Equations. CISM Courses Lect. 371, Springer, Wien, 1996, pp. 1–78. 
  2. Coincidence Degree and Nonlinear Differential Equations, Lect. Notes Math. 568, Springer, Berlin, 1977. (1977) MR0637067
  3. Periodic solutions of some Liénard equations with singularities, Proc. Amer. Math. Soc. 109 (1990), 1035–1044. (1990) MR1009991
  4. Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 7 (1957), 418–449. (1957) Zbl0090.30002MR0111875
  5. Topological degree methods in nonlinear boundary value problems, Regional Conf. Ser. Math. 40, AMS, Rhode Island, 1979. (1979) Zbl0414.34025MR0525202
  6. Topological degree and boundary value problems for nonlinear differential equations, M. Furi (ed.) Topological Methods for Ordinary Differential Equations. Lect. Notes Math. 1537, Springer, Berlin, 1993, pp. 73–142. Zbl0798.34025MR1226930
  7. Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences 74, Springer, Berlin, 1989. (1989) MR0982267
  8. Nonordered lower and upper solutions and solvability of the periodic problem for the Liénard and the Reyleigh equations, Rend. Ist. Mat. Univ. Trieste 20 (1988), (Fasc. supplementare), 54–64. (1988) MR1113247
  9. Necessary and sufficient conditions for the existence of periodic solutions of second order ordinary differential equations with singular nonlinearities, Differential Integral Equations 8 (1995), 1843–1858. (1995) Zbl0831.34048MR1347985
  10. 10.1006/jmaa.1999.6375, J. Math. Anal. Appl. 234 (1999), 311–327. (1999) DOI10.1006/jmaa.1999.6375
  11. 10.1016/S0377-0427(99)00241-1, J. Comput. Appl. Math. 113 (2000), 27–34. (2000) DOI10.1016/S0377-0427(99)00241-1
  12. Nonlinear systems of differential inequalities and solvability of certain nonlinear second order boundary value problems, J. Inequal. Appl. 6 (2001), 199–226. (2001) MR1835526
  13. Method of lower and upper functions and the existence of solutions to singular periodic problems for second order nonlinear differential equations, Mathematical Notes, Miskolc 1 (2000), 135–143. (2000) 
  14. Construction of lower and upper functions and their application to regular and singular boundary value problems, Nonlinear Analysis, Theory, Methods Appl. 47 (2001), 3937–3948. (2001) MR1972337
  15. 10.1006/jdeq.2000.3995, J. Differential Equations 176 (2001), 445–469. (2001) MR1866282DOI10.1006/jdeq.2000.3995
  16. Resonance and multiplicity in singular periodic boundary value problems, (to appear). (to appear) 
  17. Differential and Integral Equations: Boundary Value Problems and Adjoints, Academia and D. Reidel, Praha and Dordrecht, 1979. (1979) MR0542283
  18. Generalized Ordinary Differential Equations, World Scientific, Singapore, 1992. (1992) Zbl0781.34003MR1200241
  19. Generalized differential equations in the space of regulated functions (Boundary value problems and controllability), Math. Bohem. 116 (1991), 225–244. (1991) Zbl0744.34021MR1126445
  20. A relationship between the periodic and the Dirichlet BVP’s of singular differential equations, Proc. Royal Soc. Edinburgh 128A (1998), 1099–1114. (1998) MR1642144

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