Strong singularities in mixed boundary value problems

Irena Rachůnková

Mathematica Bohemica (2006)

  • Volume: 131, Issue: 4, page 393-409
  • ISSN: 0862-7959

Abstract

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We study singular boundary value problems with mixed boundary conditions of the form ( p ( t ) u ' ) ' + p ( t ) f ( t , u , p ( t ) u ' ) = 0 , lim t → 0 + p ( t ) u ' ( t ) = 0 , u ( T ) = 0 , where [ 0 , T ] ⊂ ℝ . We assume that ℝ 2 , f satisfies the Carathéodory conditions on ( 0 , T ) × p ∈ C [ 0 , T ] and 1 / p need not be integrable on [ 0 , T ] . Here f can have time singularities at t = 0 and/or t = T and a space singularity at x = 0 . Moreover, f can change its sign. Provided f is nonnegative it can have even a space singularity at y = 0 . We present conditions for the existence of solutions positive on [ 0 , T ) .

How to cite

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Rachůnková, Irena. "Strong singularities in mixed boundary value problems." Mathematica Bohemica 131.4 (2006): 393-409. <http://eudml.org/doc/249898>.

@article{Rachůnková2006,
abstract = {We study singular boundary value problems with mixed boundary conditions of the form \[ (p(t)u^\{\prime \})^\{\prime \}+ p(t)f(t,u,p(t)u^\{\prime \})=0, \quad \lim \_\{t\rightarrow 0+\}p(t)u^\{\prime \}(t)=0, \quad u(T)=0, \] where $[0,T]\subset \{\mathbb \{R\}\}.$ We assume that $\{\mathbb \{R\}\}^2,$$f$ satisfies the Carathéodory conditions on $(0,T)\times $$p\in C[0,T]$ and $\{1/p\}$ need not be integrable on $[0,T].$ Here $f$ can have time singularities at $t=0$ and/or $t=T$ and a space singularity at $x=0$. Moreover, $f$ can change its sign. Provided $f$ is nonnegative it can have even a space singularity at $y=0.$ We present conditions for the existence of solutions positive on $[0,T).$},
author = {Rachůnková, Irena},
journal = {Mathematica Bohemica},
keywords = {singular mixed boundary value problem; positive solution; lower function; upper function; convergence of approximate regular problems; positive solution; lower function; upper function},
language = {eng},
number = {4},
pages = {393-409},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Strong singularities in mixed boundary value problems},
url = {http://eudml.org/doc/249898},
volume = {131},
year = {2006},
}

TY - JOUR
AU - Rachůnková, Irena
TI - Strong singularities in mixed boundary value problems
JO - Mathematica Bohemica
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 131
IS - 4
SP - 393
EP - 409
AB - We study singular boundary value problems with mixed boundary conditions of the form \[ (p(t)u^{\prime })^{\prime }+ p(t)f(t,u,p(t)u^{\prime })=0, \quad \lim _{t\rightarrow 0+}p(t)u^{\prime }(t)=0, \quad u(T)=0, \] where $[0,T]\subset {\mathbb {R}}.$ We assume that ${\mathbb {R}}^2,$$f$ satisfies the Carathéodory conditions on $(0,T)\times $$p\in C[0,T]$ and ${1/p}$ need not be integrable on $[0,T].$ Here $f$ can have time singularities at $t=0$ and/or $t=T$ and a space singularity at $x=0$. Moreover, $f$ can change its sign. Provided $f$ is nonnegative it can have even a space singularity at $y=0.$ We present conditions for the existence of solutions positive on $[0,T).$
LA - eng
KW - singular mixed boundary value problem; positive solution; lower function; upper function; convergence of approximate regular problems; positive solution; lower function; upper function
UR - http://eudml.org/doc/249898
ER -

References

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  1. 10.1006/jdeq.1996.0147, J. Differ. Equations 130 (1996), 333–355. (1996) MR1410892DOI10.1006/jdeq.1996.0147
  2. 10.1006/jdeq.1997.3353, J. Differ. Equations 143 (1998), 60–95. (1998) MR1604959DOI10.1006/jdeq.1997.3353
  3. 10.1023/A:1025045626822, Acta Math. Hungar. 100 (2003), 245–256. (2003) MR1990185DOI10.1023/A:1025045626822
  4. 10.1017/S0013091503000105, Proc. Edinb. Math. Soc. 47 (2004), 1–13. (2004) MR2064733DOI10.1017/S0013091503000105
  5. 10.1016/S0898-1221(03)90239-2, Comp. Math. Appl. 46 (2003), 1827–1837. (2003) MR2018768DOI10.1016/S0898-1221(03)90239-2
  6. 10.1137/0522030, SIAM J. Math. Anal. 22 (1991), 463–479. (1991) Zbl0719.34038MR1084968DOI10.1137/0522030
  7. 10.1006/jdeq.1995.1022, J. Differ. Equations 115 (1995), 441–457. (1995) MR1310940DOI10.1006/jdeq.1995.1022
  8. 10.1016/S0895-7177(00)00160-6, Math. Comp. Modelling 32 (2000), 631–641. (2000) MR1791171DOI10.1016/S0895-7177(00)00160-6
  9. 10.1512/iumj.1981.30.30012, Indiana Univ. Math. J. 30 (1981), 141–157. (1981) MR0600039DOI10.1512/iumj.1981.30.30012
  10. 10.1090/qam/1012280, Quart. Appl. Math. 47 (1989), 571–581. (1989) Zbl0683.73022MR1012280DOI10.1090/qam/1012280
  11. 10.1016/0022-0396(89)90113-7, J. Differ. Equations 79 (1989), 62–78. (1989) MR0997609DOI10.1016/0022-0396(89)90113-7
  12. Symmetry of positive solutions of nonlinear elliptic equations in ℝ N , Adv. Math. Suppl. Studies 7A (1981), 369–402. (1981) MR0634248
  13. 10.1090/qam/1466147, Quart. Appl. Math. 55 (1997), 537–550. (1997) Zbl0885.73027MR1466147DOI10.1090/qam/1466147
  14. Singular and nonsingular boundary value problems with sign changing nonlinearities, J. Inequal. Appl. 5 (2000), 621–637. (2000) MR1812574
  15. 10.1016/S0362-546X(98)00025-X, Nonlin. Anal. 36 (1999), 481–494. (1999) MR1675264DOI10.1016/S0362-546X(98)00025-X
  16. On some singular boundary value problems for ordinary differential equations, Tbilis. Univ. Press, Tbilisi, 1975. (Russian) (1975) MR0499402
  17. Singular boundary value problems for second order ordinary differential equations, Itogi Nauki Tekh., Ser. Sovrm. Probl. Mat., Viniti 30 (1987), 105–201. (Russian) (1987) MR0925830
  18. Theory of Singular Boundary Value Problems, World Scientific, Singapore, 1994. (1994) MR1286741
  19. Singular mixed boundary value problem, (to appear). (to appear) MR2225980
  20. Superlinear mixed BVP with time and space singularities, Submitted. 
  21. 10.1006/jmaa.1997.5516, J. Math. Anal. Appl. 212 (1997), 443–451. (1997) Zbl0883.34021MR1464889DOI10.1006/jmaa.1997.5516
  22. 10.1006/jdeq.1994.1112, J. Differ. Equations 113 (1994), 1–16. (1994) Zbl0810.34019MR1296159DOI10.1006/jdeq.1994.1112

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