Open Access

Equivalent Solutions of Nonlinear Equations in a Topological Vector Space with a Wedge

Journal of Inequalities and Applications20072007:046041

DOI: 10.1155/2007/46041

Received: 31 December 2006

Accepted: 28 May 2007

Published: 5 July 2007


We obtain efficient conditions under which some or all solutions of a nonlinear equation in a topological vector space preordered by a closed wedge are comparable with respect to the corresponding preordering. Conditions sufficient for the equivalence of comparable solutions are also given. The wedge under consideration is not assumed to be a cone, nor any continuity conditions are imposed on the mappings considered.


Authors’ Affiliations

Institute of Mathematics, Academy of Sciences of the Czech Republic


  1. Stecenko VY, Filin VA: Test for the solvability of a nonlinear eigenvalue problem and a nonlinear resolvent equation. Doklady Akademii Nauk Tadzhikskoĭ SSR 1974,17(8):12–16.MathSciNetGoogle Scholar
  2. Stecenko VY, Esajan AR: Theorems on positive solutions of equations of the second kind with nonlinear operators. Matematicheskij Sbornik 1965,68 (110)(4):473–486.MathSciNetGoogle Scholar
  3. Rontó A: Bounds for the spectrum of compact linear operators in a preordered Banach space. Miskolc Mathematical Notes 2006,7(1):53–89.MathSciNetMATHGoogle Scholar
  4. Rontó A: On the unique solvability of linear equations determined by monotone decomposable operators. Miskolc Mathematical Notes 2004,5(1):71–82.MathSciNetMATHGoogle Scholar
  5. Rontó A: On equivalent solutions of nonlinear equations in a topological vector space with a wedge. Dopovīdī Natsīonal'noï Akademīï Nauk Ukraïni 2006, (1):30–35.Google Scholar
  6. Krasnosel'skiĭ MA, Zabreĭko PP: Geometrical Methods of Nonlinear Analysis, Fundamental Principles of Mathematical Sciences. Volume 263. Springer, Berlin, Germany; 1984:xix+409.View ArticleGoogle Scholar
  7. Krasnosel'skiĭ MA, Lifshits JeA, Sobolev AV: Positive Linear Systems. The Method of Positive Operators, Sigma Series in Applied Mathematics. Volume 5. Heldermann, Berlin, Germany; 1989:viii+354.Google Scholar
  8. Kreĭn MG, Rutman MA: Linear operators leaving invariant a cone in a Banach space. American Mathematical Society Translations 1950,1950(26):128.MathSciNetGoogle Scholar
  9. Krasnosel'skiĭ MA: Positive Solutions of Operator Equations. P. Noordhoff, Groningen, The Netherlands; 1964:381.Google Scholar
  10. Krasnosel'skiĭ MA, Vaĭnikko GM, Zabreĭko PP, Rutitskii YB, Stetsenko VY: Approximate Solution of Operator Equations. Wolters-Noordhoff, Groningen, The Netherlands; 1972:xii+484.View ArticleGoogle Scholar
  11. Reich S, Zaslavski AJ: Generic existence and uniqueness of positive eigenvalues and eigenvectors. Integral Equations and Operator Theory 2001,41(4):455–471. 10.1007/BF01202104MathSciNetView ArticleMATHGoogle Scholar
  12. Reich S, Zaslavski AJ: Generic convergence of infinite products of positive linear operators. Integral Equations and Operator Theory 1999,35(2):232–252. 10.1007/BF01196385MathSciNetView ArticleMATHGoogle Scholar
  13. Akilov GP, Kutateladze SS: Ordered Vector Spaces. "Nauka" Sibirsk. Otdel., Novosibirsk, Russia; 1978:368.MATHGoogle Scholar
  14. Fel'dman MM: Sublinear Operators Defined on a Cone. Sibirskiĭ Matematičeskiĭ Žurnal 1975,16(6):1308–1321, 1372.Google Scholar
  15. Kusraev AG, Kutateladze SS: Subdifferentials: Theory and Applications, Mathematics and Its Applications. Volume 323. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1995:x+398.View ArticleGoogle Scholar
  16. Rubinov AM: Sublinear operators and their applications. Russian Mathematical Surveys 1977,32(4):115–175. 10.1070/RM1977v032n04ABEH001640MathSciNetView ArticleMATHGoogle Scholar
  17. Kiguradze IT: Some Singular Boundary Value Problems for Ordinary Differential Equations. Izdat. Tbilis. Univ., Tbilisi, Russia; 1975:352.Google Scholar
  18. Hakl R: On periodic-type boundary value problems for functional differential equations with a positively homogeneous operator. Miskolc Mathematical Notes 2004,5(1):33–55.MathSciNetMATHGoogle Scholar
  19. Hakl R: On a periodic type boundary value problem for functional differential equations with a positively homogeneous operator. Memoirs on Differential Equations and Mathematical Physics 2007, 40: 17–54.MathSciNetMATHGoogle Scholar
  20. Hakl R: On a boundary value problem for nonlinear functional differential equations. Boundary Value Problems 2005,2005(3):263–288.MathSciNetView ArticleMATHGoogle Scholar


© A. Rontó and J. ŠSremr. 2007

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