The Dirichlet boundary value problems for strongly singular higher-order nonlinear functional-differential equations

Sulkhan Mukhigulashvili

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 1, page 235-263
  • ISSN: 0011-4642

Abstract

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The a priori boundedness principle is proved for the Dirichlet boundary value problems for strongly singular higher-order nonlinear functional-differential equations. Several sufficient conditions of solvability of the Dirichlet problem under consideration are derived from the a priori boundedness principle. The proof of the a priori boundedness principle is based on the Agarwal-Kiguradze type theorems, which guarantee the existence of the Fredholm property for strongly singular higher-order linear differential equations with argument deviations under the two-point conjugate and right-focal boundary conditions.

How to cite

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Mukhigulashvili, Sulkhan. "The Dirichlet boundary value problems for strongly singular higher-order nonlinear functional-differential equations." Czechoslovak Mathematical Journal 63.1 (2013): 235-263. <http://eudml.org/doc/252464>.

@article{Mukhigulashvili2013,
abstract = {The a priori boundedness principle is proved for the Dirichlet boundary value problems for strongly singular higher-order nonlinear functional-differential equations. Several sufficient conditions of solvability of the Dirichlet problem under consideration are derived from the a priori boundedness principle. The proof of the a priori boundedness principle is based on the Agarwal-Kiguradze type theorems, which guarantee the existence of the Fredholm property for strongly singular higher-order linear differential equations with argument deviations under the two-point conjugate and right-focal boundary conditions.},
author = {Mukhigulashvili, Sulkhan},
journal = {Czechoslovak Mathematical Journal},
keywords = {higher order functional-differential equation; Dirichlet boundary value problem; strong singularity; Fredholm property; a priori boundedness principle; higher order functional-differential equation; two-point boundary value problem; strong singularity; solvability},
language = {eng},
number = {1},
pages = {235-263},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The Dirichlet boundary value problems for strongly singular higher-order nonlinear functional-differential equations},
url = {http://eudml.org/doc/252464},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Mukhigulashvili, Sulkhan
TI - The Dirichlet boundary value problems for strongly singular higher-order nonlinear functional-differential equations
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 1
SP - 235
EP - 263
AB - The a priori boundedness principle is proved for the Dirichlet boundary value problems for strongly singular higher-order nonlinear functional-differential equations. Several sufficient conditions of solvability of the Dirichlet problem under consideration are derived from the a priori boundedness principle. The proof of the a priori boundedness principle is based on the Agarwal-Kiguradze type theorems, which guarantee the existence of the Fredholm property for strongly singular higher-order linear differential equations with argument deviations under the two-point conjugate and right-focal boundary conditions.
LA - eng
KW - higher order functional-differential equation; Dirichlet boundary value problem; strong singularity; Fredholm property; a priori boundedness principle; higher order functional-differential equation; two-point boundary value problem; strong singularity; solvability
UR - http://eudml.org/doc/252464
ER -

References

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  1. Agarwal, R. P., Focal Boundary Value Problems for Differential and Difference Equations, Mathematics and Its Applications Dordrecht: Kluwer Academic Publishers (1998). (1998) Zbl0914.34001MR1619877
  2. Agarwal, R. P., Kiguradze, I., Two-point boundary value problems for higher-order linear differential equations with strong singularities, Bound. Value Probl., Article ID 83910 (2006), pp. 32. (2006) Zbl1137.34006MR2211396
  3. Agarwal, R. P., O'Regan, D., Singular Differential and Integral Equations with Applications, Dordrecht: Kluwer Academic Publishers (2003). (2003) Zbl1055.34001MR2011127
  4. Bravyi, E., A note on the Fredholm property of boundary value problems for linear functional-differential equations, Mem. Differ. Equ. Math. Phys. 20 (2000), 133-135. (2000) Zbl0968.34049MR1789344
  5. Kiguradze, T. I., 10.1134/S0012266110020047, Differ. Equ. 46 187-194 (2010), translation from Differ. Uravn. 46 (2010), 183-190. (2010) Zbl1203.34037MR2677145DOI10.1134/S0012266110020047
  6. Kiguradze, I., On two-point boundary value problems for higher order singular ordinary differential equations, Mem. Differ. Equ. Math. Phys. 31 (2004), 101-107. (2004) Zbl1074.34020MR2061760
  7. Kiguradze, I. T., 10.1007/BF02415727, Ann. Mat. Pura Appl., IV. Ser. 86 (1970), 367-399. (1970) Zbl0251.34012MR0271449DOI10.1007/BF02415727
  8. Kiguradze, I., Some optimal conditions for the solvability of two-point singular boundary value problems, Funct. Differ. Equ. 10 (2003), 259-281. (2003) Zbl1062.34017MR2017411
  9. Kiguradze, I. T., Some Singular Boundary Value Problems for Ordinary Differential Equations, Tbilisi University Press (1975), Russian. (1975) MR0499402
  10. Kiguradze, I., The Dirichlet and focal boundary value problems for higher order quasi-halflinear singular differential equations, Mem. Differential Equ. Math. Phys. 54 (2011), 126-133. (2011) MR2919691
  11. Kiguradze, I. T., Chanturiya, T. A., Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Translation from the Russian, Mathematics and Its Applications. Soviet Series. 89 Dordrecht, Kluwer Academic Publishers (1993). (1993) MR1220223
  12. Kiguradze, I., Půža, B., Stavroulakis, I. P., On singular boundary value problems for functional-differential equations of higher order, Georgian Math. J. 8 (2001), 791-814. (2001) Zbl1011.34056MR1884501
  13. Kiguradze, I., Tskhovrebadze, G., 10.1007/BF02315301, Georgian Math. J. 1 (1994), 31-45. (1994) Zbl0804.34023DOI10.1007/BF02315301
  14. Lomtatidze, A. G., A boundary value problem for nonlinear second order ordinary differential equations with singularities, Differ. Equations 22 (1986), 301-310 translation from Differ. Uravn. 22 (1986), 416-426. (1986) Zbl0625.34012MR0835774
  15. Mukhigulashvili, S., On one estimate for solutions of two-point boundary value problems for strongly singular higher-order linear differential equations, Mem. Differential Equ. Math. Phys. 58 (2013). (2013) MR3288192
  16. Mukhigulashvili, S., Two-point boundary value problems for second order functional differential equations, Mem. Differ. Equ. Math. Phys. 20 (2000), 1-112. (2000) Zbl0994.34052MR1789341
  17. Mukhigulashvili, S., Partsvania, N., On two-point boundary value problems for higher order functional differential equations with strong singularities, Mem. Differ. Equ. Math. Phys. 54 (2011), 134-138. (2011) MR3026940
  18. Mukhigulashvili, S., Partsvania, N., 10.14232/ejqtde.2012.1.38, Electron. J. Qual. Theory Differ. Equ. 38 (2012), 1-34. (2012) MR2920961DOI10.14232/ejqtde.2012.1.38
  19. Parstvania, N., On extremal solutions of two-point boundary value problems for second order nonlinear singular differential equations, Bull. Georgian Natl. Acad. Sci. (N.S.) 5 (2011), 31-36. (2011) MR2883772
  20. Partsvania, N., On solvability and well-posedness of two-point weighted singular boundary value problems, Mem. Differ. Equ. Math. Phys. 54 (2011), 139-146. (2011) MR3026941
  21. Půža, B., 10.1023/A:1022107625905, Georgian Math. J. 4 (1997), 557-566. (1997) MR1478554DOI10.1023/A:1022107625905
  22. Půža, B., Rabbimov, A., On a weighted boundary value problem for a system of singular functional-differential equations, Mem. Differ. Equ. Math. Phys. 21 (2000), 125-130. (2000) Zbl0984.34057
  23. Rachůnková, I., Staněk, S., Tvrdý, M., Singularities and Laplacians in Boundary Value Problems for Nonlinear Ordinary Differential Equations, Handbook of differential equations: ordinary differential equations. Vol. III Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam (2006). (2006) MR2457638
  24. Rachůnková, I., Staněk, S., Tvrdý, M., Solvability of Nonlinear Singular Problems for Ordinary Differential Equations, Contemporary Mathematics and Its Applications, 5 Hindawi Publishing Corporation, New York (2008). (2008) MR2572243
  25. Schwabik, Š., Tvrdý, M., Vejvoda, O., Differential and Integral Equations. Boundary Value Problems and Adjoints, D. Reidel Publishing Co. Dordrecht, Boston, London (1979). (1979) Zbl0417.45001MR0542283

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