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Concavity of solutions of a 2n-th order problem with symmetry

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Języki publikacji
EN
Abstrakty
EN
In this article we apply an extension of a Leggett-Williams type fixed point theorem to a two-point boundary value problem for a 2n-th order ordinary differential equation. The fixed point theorem employs concave and convex functionals defined on a cone in a Banach space. Inequalities that extend the notion of concavity to 2n-th order differential inequalities are derived and employed to provide the necessary estimates. Symmetry is employed in the construction of the appropriate Banach space.
Rocznik
Strony
603--613
Opis fizyczny
Bibliogr. 23 poz.
Twórcy
autor
  • University of Benghazi Faculty of Arts & Sciences / Al Kufra Department of Mathematics Al Kufra, Libya
autor
  • University of Dayton Department of Mathematics Dayton, Ohio 45469-2316 USA
Bibliografia
  • [1] R.P. Agarwal, Boundary Value Problems for Higher Order Differential Equations,World Scientific, Singapore, 1986.
  • [2] R.P. Agarwal, On fourth-order boundary value problems arising in beam analysis, Differential Integral Equations 2 (1989), 91–110.
  • [3] A. Al Twaty, P. Eloe, The role of concavity in applications of Avery type fixed point theorems to higher order differential equations, J. Math. Inequal. 6 (2012), 79–90.
  • [4] D.R. Anderson, R.I. Avery, Fixed point theorem of cone expansion and compression of functional type, J. Difference Equations Appl. 8 (2002), 1073–1083.
  • [5] D.R. Anderson, R.I. Avery, J. Henderson, A topological proof and extension of the Leggett-Williams fixed point theorem, Comm. Appl. Nonlinear Anal. 16 (2009), 39–44.
  • [6] D.R. Anderson, R.I. Avery, J. Henderson, Existence of a positive solution to a right focal boundary value problem, Electron. J. Qual. Theory Differ. Equ. 16 (2010), 5, 6 pp.
  • [7] R.I. Avery, P. Eloe, J. Henderson, A Leggett-Williams type theorem applied to a fourth order problem, Proc. of Dynamic Systems and Applications 6 (2012), 579–588.
  • [8] R.I. Avery, J. Henderson, D. O’Regan, Dual of the compression-expansion fixed point theorems, Fixed Point Theory Appl. 2007 (2007), Article ID 90715, 11 pp.
  • [9] Z. Bai, The upper and lower solution method for some fourth-order boundary value problems, Nonlinear Anal. 67 (2007), 1704–1709.
  • [10] K.M. Das, A.S. Vatsala, On Green’s function of an n-point boundary value problem, Trans. Amer. Math. Soc. 182 (1973), 469–480.
  • [11] P. Eloe, J. Henderson, Inequalities based on a generalization of concavity, Proc. Amer. Math. Soc. 125 (1997), 2103–2107.
  • [12] D. Guo, A new fixed point theorem, Acta Math. Sinica 24 (1981), 444–450.
  • [13] D. Guo, Some fixed point theorems on cone maps, Kexeu Tongbao 29 (1984), 575–578.
  • [14] M.A. Krasnosel’skii, Positive Solutions of Operator Equations, P. Noordhoff, Groningen, The Netherlands, 1964.
  • [15] J.R. Graef, B. Yang, Positive solutions of a nonlinear fourth order boundary value problem, Comm. Appl. Nonlinear Anal. 1 (2007), 61–73.
  • [16] R.W. Leggett, L.R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979), 673–688.
  • [17] R. Ma, H. Wang, On the existence of positive solutions of fourth order-order ordinary differential equations, Appl. Anal. 59 (1995), 225–231.
  • [18] M. Pei, S.K. Chang, Monotone iterative technique and symmetric positive solutions for a fourth-order boundary value problem, Math. Comput. Modelling 51 (2010), 1260–1267.
  • [19] J. Sun, G. Zhang, A generalization of the cone expansion and compression fixed point theorem and applications, Nonlinear Anal. 67 (2007), 579-586.
  • [20] B. Yang, Positive solutions for the beam equation under certain boundary conditions, Electron. J. Differential Equations 2005 (78) (2005), 1–8.
  • [21] B. Yang, Upper estimate for positive solutions of the (p, n−p) conjugate boundary value problem, J. Math. Anal. Appl. 390 (2012), 535–548.
  • [22] C. Zhai, R. Song, Q. Han, The existence and uniqueness of symmetric positive solutions for a fourth-order boundary value problem, Comput. Math. Appl. 62 (2011), 2639–2647.
  • [23] E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems, Springer-Verlag, New York, 1986.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-3fc8aeaf-8ce7-423a-818c-e190e598f5e9
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