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.
Citation: |
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.![]() ![]() ![]() |