Singular Dirichlet problem for ordinary differential equations with φ -Laplacian

Vladimír Polášek; Irena Rachůnková

Mathematica Bohemica (2005)

  • Volume: 130, Issue: 4, page 409-425
  • ISSN: 0862-7959

Abstract

top
We provide sufficient conditions for solvability of a singular Dirichlet boundary value problem with - L a p l a c i a n . ((u)) = f(t, u, u), u(0) = A, u(T) = B, . w h e r e is an increasing homeomorphism, ( ℝ ) = ℝ , ( 0 ) = 0 , f satisfies the Carathéodory conditions on each set [ a , b ] × ℝ 2 with [ a , b ] ⊂ ( 0 , T ) and f is not integrable on [ 0 , T ] for some fixed values of its phase variables. We prove the existence of a solution which has continuous first derivative on [ 0 , T ] .

How to cite

top

Polášek, Vladimír, and Rachůnková, Irena. "Singular Dirichlet problem for ordinary differential equations with $\phi $-Laplacian." Mathematica Bohemica 130.4 (2005): 409-425. <http://eudml.org/doc/249593>.

@article{Polášek2005,
abstract = {We provide sufficient conditions for solvability of a singular Dirichlet boundary value problem with \[-Laplacian \] . ((u)) = f(t, u, u), u(0) = A, u(T) = B, . \[ where \] is an increasing homeomorphism, $(\mathbb \{R\})=\mathbb \{R\}$, $(0)=0$, $f$ satisfies the Carathéodory conditions on each set $[a, b]\times \mathbb \{R\}^\{2\}$ with $[a, b]\subset (0, T)$ and $f$ is not integrable on $[0, T]$ for some fixed values of its phase variables. We prove the existence of a solution which has continuous first derivative on $[0, T]$.},
author = {Polášek, Vladimír, Rachůnková, Irena},
journal = {Mathematica Bohemica},
keywords = {existence of smooth solution; lower function; upper function},
language = {eng},
number = {4},
pages = {409-425},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Singular Dirichlet problem for ordinary differential equations with $\phi $-Laplacian},
url = {http://eudml.org/doc/249593},
volume = {130},
year = {2005},
}

TY - JOUR
AU - Polášek, Vladimír
AU - Rachůnková, Irena
TI - Singular Dirichlet problem for ordinary differential equations with $\phi $-Laplacian
JO - Mathematica Bohemica
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 130
IS - 4
SP - 409
EP - 425
AB - We provide sufficient conditions for solvability of a singular Dirichlet boundary value problem with \[-Laplacian \] . ((u)) = f(t, u, u), u(0) = A, u(T) = B, . \[ where \] is an increasing homeomorphism, $(\mathbb {R})=\mathbb {R}$, $(0)=0$, $f$ satisfies the Carathéodory conditions on each set $[a, b]\times \mathbb {R}^{2}$ with $[a, b]\subset (0, T)$ and $f$ is not integrable on $[0, T]$ for some fixed values of its phase variables. We prove the existence of a solution which has continuous first derivative on $[0, T]$.
LA - eng
KW - existence of smooth solution; lower function; upper function
UR - http://eudml.org/doc/249593
ER -

References

top
  1. 10.1016/0022-247X(78)90022-7, J. Math. Anal. Appl. 64 (1978), 96–105. (1978) MR0478973DOI10.1016/0022-247X(78)90022-7
  2. 10.1137/0138024, SIAM J. Appl. Math. 38 (1980), 275–281. (1980) MR0564014DOI10.1137/0138024
  3. 10.1017/S030500410005636X, Proc. Camb. Phil. Soc. 86 (1979), 495–510. (1979) MR0542697DOI10.1017/S030500410005636X
  4. 10.1016/0362-546X(86)90068-4, Nonlinear Anal. 10 (1986), 1303–1325. (1986) MR0866262DOI10.1016/0362-546X(86)90068-4
  5. 10.1016/S0362-546X(99)00308-9, Nonlinear Anal. 44 (2001), 791–809. (2001) Zbl1042.34526MR1825782DOI10.1016/S0362-546X(99)00308-9
  6. Some Singular Boundary Value Problems for Ordinary Differential Equations, Izdat. Tbilis. Univ., Tbilisi, 1975. (Russian) (1975) MR0499402
  7. Some optimal conditions for solvability of two-point singular boundary value problem, Functional Differential Equations 10 (2003), 259–281. (2003) MR2017411
  8. Singular boundary value problems for second order ordinary differential equations, Itogi Nauki Tekh., Ser. Sovrm. Probl. Mat., Viniti 30 (1987), 105–201. (1987) MR0925830
  9. Theory of Singular Boundary Value Problems, World Scientific, Singapore, 1995. (1995) MR1286741
  10. Existence principles and theory for singular Dirichlet boundary value problems, Differential Equations and Dynamical Systems 3 (1995), 289–304. (1995) MR1386750
  11. 10.1016/S0362-546X(96)00026-0, Nonlinear Anal., Theory Methods Appl. 29 (1997), 221–245. (1997) MR1446226DOI10.1016/S0362-546X(96)00026-0
  12. 10.1023/A:1022837420342, Czechoslovak Math. J. 48 (1998), 269–289. (1998) MR1624319DOI10.1023/A:1022837420342
  13. 10.1006/jmaa.1994.1199, J. Math. Anal. Appl. 184 (1994), 263–284. (1994) MR1278388DOI10.1006/jmaa.1994.1199
  14. 10.1016/S0362-546X(99)00139-X, Nonlinear Anal. 42 (2000), 949–960. (2000) MR1780446DOI10.1016/S0362-546X(99)00139-X
  15. 10.1016/S0362-546X(01)00148-1, Nonlinear Anal. 52 (2003), 1553–1567. (2003) MR1951507DOI10.1016/S0362-546X(01)00148-1
  16. Numerical solution of singular nonlinear boundary value problems, Proceedings of the Third International Colloquium in Numerical Analysis. Plovdiv, Bulgaria, August 13–17, 1994, 1995, pp. 15–24. (1995) Zbl0843.65055MR1455945
  17. Boundary behavior and computation of solutions of singular nonlinear boundary value problems, Communications on Appl. Analysis 4 (2000), 207–226. (2000) MR1752847
  18. 10.1016/0362-546X(94)90245-3, Nonlinear Anal., Theory Methods Appl. 23 (1994), 669–681. (1994) Zbl0813.34021MR1297285DOI10.1016/0362-546X(94)90245-3
  19. Existence results for the problem ( φ ( u ' ) ) ' = f ( t , u , u ' ) with nonlinear boundary conditions, Nonlinear Anal. 35 (1999), 221–231. (1999) MR1643240
  20. 10.2748/tmj/1178225520, Tôhoku Math. J. 47 (1995), 327–344. (1995) MR1344906DOI10.2748/tmj/1178225520

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.