On a criterion for the existence of at least four solutions of functional boundary value problems

Staněk, Svatoslav

Archivum Mathematicum (1997)

  • Volume: 033, Issue: 4, page 335-348
  • ISSN: 0044-8753

Abstract

top
A class of functional boundary conditions for the second order functional differential equation x ' ' ( t ) = ( F x ) ( t ) is introduced. Here F : C 1 ( J ) → L 1 ( J ) is a nonlinear continuous unbounded operator. Sufficient conditions for the existence of at least four solutions are given. The proofs are based on the Bihari lemma, the topological method of homotopy, the Leray-Schauder degree and the Borsuk theorem.

How to cite

top

Staněk, Svatoslav. "On a criterion for the existence of at least four solutions of functional boundary value problems." Archivum Mathematicum 033.4 (1997): 335-348. <http://eudml.org/doc/248037>.

@article{Staněk1997,
abstract = {A class of functional boundary conditions for the second order functional differential equation $x^\{\prime \prime \}(t)=(Fx)(t)$ is introduced. Here $F:C^1(J) \rightarrow L_1(J)$ is a nonlinear continuous unbounded operator. Sufficient conditions for the existence of at least four solutions are given. The proofs are based on the Bihari lemma, the topological method of homotopy, the Leray-Schauder degree and the Borsuk theorem.},
author = {Staněk, Svatoslav},
journal = {Archivum Mathematicum},
keywords = {functional boundary conditions; functional differential equation; existence; multiplicity; Bihari lemma; homotopy; Leray Schauder degree; Borsuk theorem; functional boundary conditions; functional-differential equations; existence; multiplicity; Bihari lemma; homotopy; Leray Schauder degree; Borsuk theorem},
language = {eng},
number = {4},
pages = {335-348},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On a criterion for the existence of at least four solutions of functional boundary value problems},
url = {http://eudml.org/doc/248037},
volume = {033},
year = {1997},
}

TY - JOUR
AU - Staněk, Svatoslav
TI - On a criterion for the existence of at least four solutions of functional boundary value problems
JO - Archivum Mathematicum
PY - 1997
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 033
IS - 4
SP - 335
EP - 348
AB - A class of functional boundary conditions for the second order functional differential equation $x^{\prime \prime }(t)=(Fx)(t)$ is introduced. Here $F:C^1(J) \rightarrow L_1(J)$ is a nonlinear continuous unbounded operator. Sufficient conditions for the existence of at least four solutions are given. The proofs are based on the Bihari lemma, the topological method of homotopy, the Leray-Schauder degree and the Borsuk theorem.
LA - eng
KW - functional boundary conditions; functional differential equation; existence; multiplicity; Bihari lemma; homotopy; Leray Schauder degree; Borsuk theorem; functional boundary conditions; functional-differential equations; existence; multiplicity; Bihari lemma; homotopy; Leray Schauder degree; Borsuk theorem
UR - http://eudml.org/doc/248037
ER -

References

top
  1. Ambrosetti A., Prodi G., On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. 93, 1972, 231–247. (1972) Zbl0288.35020MR0320844
  2. Azbelev N. V., Maksimov V. P., Rakhmatullina L. F., Introduction to the Theory of Functional Differential Equations, Moscow, Nauka, 1991 (in Russian). (1991) Zbl0725.34071MR1144998
  3. Bihari I., A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Sci. Hungar. 7, 1956, 71–94. (1956) Zbl0070.08201MR0079154
  4. Brüll L., Mawhin J., Finiteness of the set of solutions of some boundary- value problems for ordinary differential equations, Arch. Math. (Brno) 24, 1988, 163–172. (1988) Zbl0678.34023MR0983234
  5. Brykalov S. A., Solvability of a nonlinear boundary value problem in a fixed set of functions, Diff. Urav. 27, 1991, 2027–2033 (in Russian). (1991) Zbl0788.34070MR1155041
  6. Brykalov S. A., Solutions with given maximum and minimum, Diff. Urav. 29, 1993, 938–942 (in Russian). (1993) MR1254551
  7. Brykalov S. A., A second-order nonlinear problem with two-point and integral boundary conditions, Proceedings of the Georgian Academy of Science, Math. 1, 1993, 273–279. (1993) Zbl0798.34021MR1262564
  8. Deimling K., Nonlinear Functional Analysis, Springer, Berlin Heidelberg 1985. (1985) Zbl0559.47040MR0787404
  9. Ermens B., Mawhin J., Higher order nonlinear boundary value problems with finitely many solutions, Séminaire Mathématique, Université de Louvain, No. 139, 1988, 1–14, (preprint). (1988) MR1065638
  10. Filatov A. N., Sharova L. V., Integral Inequalities and the Theory of Nonlinear Oscillations, Nauka, Moscow 1976 (in Russian). (1976) Zbl0463.34001MR0492576
  11. Mawhin J., Topological Degree Method in Nonlinear Boundary Value Problems, CMBS Reg. Conf. in Math., No. 40, AMS, Providence, 1979. (1979) MR0525202
  12. Mawhin J., Willem M., Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations 52, 1984, 264–287. (1984) Zbl0557.34036MR0741271
  13. Nkashama M. N., Santanilla J., Existence of multiple solutions for some nonlinear boundary value problems, J. Differential Equations 84, 1990, 148–164. (1990) Zbl0693.34011MR1042663
  14. Rachůnková I., Staněk S., Topological degree method in functional boundary value problems, Nonlinear Analysis 27, 1996, 153–166. (1996) 
  15. Rachůnková I., On the existence of two solutions of the periodic problem for the ordinary second-order differential equation, Nonlinear Analysis 22, 1994, 1315–1322. (1994) Zbl0808.34023
  16. Staněk S., Existence of multiple solutions for some functional boundary value problems, Arch. Math. (Brno) 28, 1992, 57–65. (1992) Zbl0782.34074MR1201866
  17. Staněk S., Multiple solutions for some functional boundary value problems, Nonlinear Analysis, to appear. Zbl0945.34049MR1610598
  18. Staněk S., Multiplicity results for second order nonlinear problems with maximum and minimum, Math. Nachr., to appear. Zbl0920.34058MR1626344
  19. Šeda V., Fredholm mappings and the generalized boundary value problem, Differential and Integral Equations 8, 1995, 19–40. (1995) MR1296108

NotesEmbed ?

top

You must be logged in to post comments.