| Title:
|
Singular Dirichlet boundary value problems. II: Resonance case
(English)
|
| Author:
|
O'Regan, Donal |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 |
| Volume:
|
48 |
| Issue:
|
2 |
| Year:
|
1998 |
| Pages:
|
269-289 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Existence results are established for the resonant problem $y^{\prime \prime }+\lambda _m \,a\,y=f(t,y)$ a.e. on $[0,1]$ with $y$ satisfying Dirichlet boundary conditions. The problem is singular since $f$ is a Carathéodory function, $a\in L_{{\mathrm loc}}^1(0,1)$ with $a>0$ a.e. on $[0,1]$ and $\int ^1_0 x(1-x)a(x)\,\mathrm{d}x <\infty $. |
| MSC:
|
34B15 |
| MSC:
|
34L30 |
| idZBL:
|
Zbl 0957.34016 |
| idMR:
|
MR1624319 |
| . |
| Date available:
|
2009-09-24T10:13:21Z |
| Last updated:
|
2012-05-31 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127416 |
| . |
| Reference:
|
[1] Atkinson, F.V.: Discrete and continuous boundary problems.(1964), Academic Press, New York. Zbl 0117.05806, MR 0176141 |
| Reference:
|
[2] Bobisud, L.E., and O’Regan, D.: Positive solutions for a class of nonlinear singular boundary value problems at resonance.Jour. Math. Anal. Appl. 184 (1994), 263–284. MR 1278388 |
| Reference:
|
[3] Bobisud, L.E., O’Regan, D., and Royalty, W.D.: Singular boundary value problems.Appl. Anal. 23 (1986), 233–243. MR 0870490 |
| Reference:
|
[4] Everitt, W.N., Kwong, M.K., and Zettl, A.: Oscillations of eigenfunctions of weighted regular Sturm Liouville problems.J. London Math. Soc. 27 (1983), 106–120. MR 0686509 |
| Reference:
|
[5] Habets, P., and Zanolin, F.: Upper and lower solutions for a generalized Emden-Fowler equation.J. Math. Anal. Appl. 181 (1994), 684–700. MR 1264540 |
| Reference:
|
[6] Iannacci, R., and Nkashama, M.N.: Unbounded perturbations of forced second order ordinary differential equations at resonance.Jour. Diff. Eq. 69 (1987), 289–309. MR 0903389 |
| Reference:
|
[7] Mawhin, J.: Topological degree methods in nonlinear boundary value problems.AMS Regional Conf. Series in Math. 40, Providence, 1978. MR 0525202 |
| Reference:
|
[8] Mawhin, J., and Ward, J.R.: Nonuniform nonresonance conditions at the first two eigenvalues for periodic solutions of forced Liénard and Duffing equations.Rocky M.J. Math. 112 (1982), 643–654. |
| Reference:
|
[9] Naimark, M.A.: Linear differential operators, Part II.Ungar Publ. Co., London, 1968. Zbl 0227.34020, MR 0262880 |
| Reference:
|
[10] O’Regan, D.: Theory of singular boundary value problems.World Scientific Press, Singapore, 1994. |
| Reference:
|
[11] O’Regan, D.: Existence principles and theory for singular Dirichlet boundary value problems.Diff. Eqms. and Dynamical Systems 3 (1995), 289–304. MR 1386750 |
| Reference:
|
[12] O’Regan, D.: Singular Dirichlet boundary value problems I: Superlinear and nonresonance case.Nonlinear Analysis 29 (1997), 221–245. MR 1446226 |
| . |
