Advanced Search
Article Contents
Article Contents

Global Kneser solutions to nonlinear equations with indefinite weight

  • * Corresponding author: Serena Matucci

    * Corresponding author: Serena Matucci

The first author is supported by the grant GA 17-03224S of the Czech Grant Agency. The third author is partially supported by Gnampa, National Institute for Advanced Mathematics (INdAM)

Abstract Full Text(HTML) Related Papers Cited by
  • The paper deals with the nonlinear differential equation

    $\bigl(a(t)\Phi(x^{\prime})\bigr)^{\prime}+b(t)F(x)=0,\ \ \ t\in\lbrack1,\infty),$

    in the case when the weight $b$ has indefinite sign. In particular, the problem of the existence of the so-called globally positive Kneser solutions, that is solutions $x$ such that $x(t)>0, {{x}'}(t)<0$ on the whole closed interval $[1,\infty )$, is considered. Moreover, conditions assuring that these solutions tend to zero as $t\rightarrow\infty$ are investigated by a Schauder's half-linearization device jointly with some properties of the principal solution of an associated half-linear differential equation. The results cover also the case in which the weight $b$ is a periodic function or it is unbounded from below.

    Mathematics Subject Classification: Primary: 34B40; Secondary: 34B18.


    \begin{equation} \\ \end{equation}
  • 加载中
  •   J. Andres , G. Gabor  and  L. Górniewicz , Boundary value problems on infinite intervals, Trans. Amer. Math. Soc., 351 (1999) , 4861-4903.  doi: 10.1090/S0002-9947-99-02297-7.
      A. Boscaggin  and  F. Zanolin , Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem, Ann. Mat. Pura Appl., 194 (2015) , 451-478.  doi: 10.1007/s10231-013-0384-0.
      M. Cecchi , Z. Došlá  and  M. Marini , Principal solutions and minimal sets of quasilinear differential equations, Dynam. Systems Appl., 13 (2004) , 221-232. 
      M. Cecchi , Z. Došlá  and  M. Marini , Half-linear differential equations with oscillating coefficient, Differential Integral Equations, 18 (2005) , 1243-1256. 
      M. Cecchi , Z. Došlá , I. Kiguradze  and  M. Marini , On nonnegative solutions of singular boundary-value problems for Emden-Fowler-type differential systems, Differential Integral Equations, 20 (2007) , 1081-1106. 
      M. Cecchi , M. Furi  and  M. Marini , On continuity and compactness of some nonlinear operators associated with differential equations in noncompact intervals, Nonlinear Anal., 9 (1985) , 171-180.  doi: 10.1016/0362-546X(85)90070-7.
      J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries. Vol. I. Elliptic equations, Research Notes in Mathematics, 106, Pitman (Advanced Publishing Program), Boston, MA, 1985.
      Z. Došlá , M. Marini  and  S. Matucci , On some boundary value problems for second order nonlinear differential equations, Math. Bohem., 137 (2012) , 113-122. 
      Z. Došlá , M. Marini  and  S. Matucci , A boundary value problem on a half-line for differential equations with indefinite weight, Commun. Appl. Anal., 15 (2011) , 341-352. 
      Z. Došlá , M. Marini  and  S. Matucci , Positive solutions of nonlocal continuous second order BVP's, Dynam. Systems Appl., 23 (2014) , 431-446. 
      Z. Došlá , M. Marini  and  S. Matucci , A Dirichlet problem on the half-line for nonlinear equations with indefinite weight, Ann. Mat. Pura Appl., 196 (2017) , 51-64.  doi: 10.1007/s10231-016-0562-y.
      O. Došlý  and  A. Elbert , Integral characterization of principal solution of half-linear differential equations, Studia Sci. Math. Hungar., 36 (2000) , 455-469.  doi: 10.1556/SScMath.36.2000.3-4.16.
      O. Došlý and P. Řehák, Half-linear Differential Equations, North-Holland Mathematics Studies 202, Elsevier Sci. B. V., Amsterdam, 2005.
      P. Drábek  and  A. Kufner , Discreteness and symplicity of the spectrum of a quasilinear Sturm-Liouville-type problem on an infinite interval, Proc. Amer. Math. Soc., 134 (2006) , 235-242.  doi: 10.1090/S0002-9939-05-07958-X.
      P. Drábek , A. Kufner  and  K. Kuliev , Half-linear Sturm Liouville problem with weights: Asymptotic behavior of eigenfunctions, Proc. Steklov Inst. Math., 284 (2014) , 148-154.  doi: 10.1134/S008154381401009X.
      A. Elbert  and  T. Kusano , Principal solutions of non-oscillatory half-linear differential equations, Adv. Math. Sci. Appl., 8 (1998) , 745-759. 
      P. Hartman, Ordinary Differential Equations, Reprint of the second edition, Birkäuser, Boston, Mass., 1982.
      J. Jaroš  and  T. Kusano , Decreasing regularly varying solutions of sublinearly perturbed superlinear Thomas-Fermi equation, Results Math., 66 (2014) , 273-289.  doi: 10.1007/s00025-014-0376-4.
      K. Kamo , Asymptotic equivalence for positive decaying solutions of the generalized Emden-Fowler equations and its application to elliptic problems, Arch. Math. (Brno), 40 (2004) , 209-217. 
      I. T. Kiguradze and T. A. Chanturia, Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Acad. Publ. G., Dordrecht, 1993. doi: 10.1007/978-94-011-1808-8.
      T. Kusano , V. Marić  and  T. Tanigawa , Regularly varying solutions of generalized Thomas-Fermi equations, Bull. Cl. Sci. Math. Nat. Sci. Math., 34 (2009) , 43-73. 
      M. Marini and S. Matucci, A boundary value problem on the half-line for superlinear differential equations with changing sign weight, Rend. Istit. Mat. Univ. Trieste, 44 (2012), 117–132.
      S. Matucci , A new approach for solving nonlinear BVP's on the half-line for second order equations and applications, Mathematica Bohemica, 140 (2015) , 153-169. 
      V. Marić, Regular Variation and Differential Equations, Lecture Notes in Mathematics, 1726, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103952.
      D. D. Mirzov , Principal and nonprincipal solutions of a nonlinear system, Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy, 31 (1988) , 100-117. 
      J. R. L. Webb  and  G. Infante , Positive solutions of nonlocal boundary value problems: a unified approach, J. London Math. Soc., 74 (2006) , 673-693.  doi: 10.1112/S0024610706023179.
  • 加载中

Article Metrics

HTML views(1256) PDF downloads(304) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint