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Positive and maximal positive solutions of singular mixed boundary value problem

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Central European Journal of Mathematics

Abstract

The paper is concerned with existence results for positive solutions and maximal positive solutions of singular mixed boundary value problems. Nonlinearities h(t;x;y) in differential equations admit a time singularity at t=0 and/or at t=T and a strong singularity at x=0.

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References

  1. Agarwal R.P., O’Regan D., Singular differential and integral equations with applications, Kluwer, Dordrecht, 2003

    MATH  Google Scholar 

  2. Agarwal R.P., O’Regan D., Singular problems motivated from classical upper and lower solutions, Acta Math. Hungar., 2003, 100, 245–256

    Article  MATH  MathSciNet  Google Scholar 

  3. Agarwal R.P., O’Regan D., Staněek S., Existence of positive solutions for boundary-value problems with singularities in phase variables, Proc. Edinburgh Math. Soc., 2004, 47, 1–13

    Article  MATH  MathSciNet  Google Scholar 

  4. Agarwal R.P., Staněek S., Nonnegative solutions of singular boundary value problems with sign changing nonlinearities, Comput. Math. Appl., 2003, 46, 1827–1837

    Article  MATH  MathSciNet  Google Scholar 

  5. Bartle R.G., A modern theory of integrations, AMS Providence, Rhode Island, 2001

    Google Scholar 

  6. Berestycki H., Lions P.L., Peletier L.A., An ODE approach to the existence of positive solutions for semilinear problems in ℝN, Indiana Univ. Math. J., 1981, 30, 141–157

    Article  MATH  MathSciNet  Google Scholar 

  7. Bertsch M., Passo R.D., Ughi M., Discontinuous viscosity solutions of a degenerate parabolic equation, Trans. Amer. Math. Soc., 1990, 320, 779–798

    Article  MATH  MathSciNet  Google Scholar 

  8. Bertsch M., Ughi M., Positive properties of viscosity solutions of a degenerate parabolic equation, Nonlinear Anal., 1990, 14, 571–592

    Article  MATH  MathSciNet  Google Scholar 

  9. Cabada A., Cid J.A., Extremal solutions of ϕ-Laplacian-diffusion scalar problems with nonlinear functional boundary conditions in a unified way, Nonlinear. Anal., 2005, 63, 2515–2524

    Article  Google Scholar 

  10. Cabada A., Nieto J.J., Extremal solutions of second order nonlinear periodic boundary value problems, Appl. Math. Comput., 1990, 40, 135–145

    Article  MATH  MathSciNet  Google Scholar 

  11. Cabada A., Pouso R.P., Extremal solutions of strongly nonlinear discontinuous second-order equations with nonlinear functional boundary conditions, Nonlinear Anal., 2000, 42, 1377–1396

    Article  MATH  MathSciNet  Google Scholar 

  12. Carl S., Heikkilä S., Nonlinear differential equations in ordered spaces, Chapman & Hall/CRC, Monographs and Surveys in Pure and Applied Mathematics III, 2000

  13. Castro A., Sudhasree G., Uniqueness of stable and unstable positive solutions for semipositone problems, Nonlinear Anal., 1994, 22, 425–429

    Article  MATH  MathSciNet  Google Scholar 

  14. Cherpion M., De Coster C., Habets P., Monotone iterative method for boundary value problem, Differential integral equations, 1999, 12, 309–338

    MATH  MathSciNet  Google Scholar 

  15. Cid J.A., On extremal fixed point in Schauder’s theorem with application to differential equations, Bull. Belg. Math. Soc. Simon Stevin, 2004, 11, 15–20

    MATH  MathSciNet  Google Scholar 

  16. Gidas B., Ni W.M., Nirenberg L., Symmetry of positive solutions of nonlinear elliptic equations in ℝN, Adv. Math., Suppl. Stud. 7A, 1981, 369–402

  17. Kelevedjiev P., Nonnegative solutions to some singular second-order boundary value problems, Nonlinear Anal., 1999, 36, 481–494

    Article  MATH  MathSciNet  Google Scholar 

  18. Kiguradze I., Some optimal conditions for solvability of two-point singular boundary value problems, Funct. Differ. Equ., 2003, 10, 259–281

    MATH  MathSciNet  Google Scholar 

  19. Kiguradze I.T., Shekhter B.L., Singular boundary value problems for second order ordinary differential equations, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh., 1987, 30, 105–201 (in Russian), English translation: Journal of Soviet Mathematics, 1988, 43, 2340–2417

    MathSciNet  Google Scholar 

  20. Lang S., Real and functional analysis, Springer, New York, 1993

    MATH  Google Scholar 

  21. O’Regan D., Existence of positive solutions to some singular and nonsingular second order boundary value problems, J. Differential Equations, 1990, 84, 228–251

    Article  MATH  MathSciNet  Google Scholar 

  22. Rachůnková I., Singular mixed boundary value problem, J. Math. Anal. Appl., 2006, 320, 611–618

    Article  MATH  MathSciNet  Google Scholar 

  23. Rachůnková I., Staněk S., Connections between types of singularities in differential equations and smoothness of solutions for Dirichlet BVPs, Dyn. Contin. Discrete Impuls. Syst. Ser. A. Math. Anal., 2003, 10, 209–222

    MATH  MathSciNet  Google Scholar 

  24. Rachůnková I., Staněk S., Tvrdý M., Singularities and laplacians in boundary value problems for nonlinear differential equations, In: Cañada A., Drábek P., Fonda A. (Eds.), Handbook of differential equations, Ordinary differential equations, Vol. 3, 607–723, Elsevier, 2006

  25. Thompson H.B., Second order ordinary differential equations with fully nonlinear two point boundary conditions, Pacific J. Math., 1996, 172, 255–277

    MATH  MathSciNet  Google Scholar 

  26. Thompson H.B., Second order ordinary differential equations with fully nonlinear two point boundary conditions II, Pacific J. Math., 1996, 172, 259–297

    Google Scholar 

  27. Wang J., Gao W., A note on singular nonlinear two-point boundary value problems, Nonlinear Anal., 2000, 39, 281–287

    Article  MathSciNet  Google Scholar 

  28. Zheng L., Su X., Zhang X., Similarity solutions for boundary layer flow on a moving surface in an otherwise quiescent fluid medium, Int. J. Pure Appl. Math., 2005, 19, 541–552

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, Florida, USA

    Ravi P. Agarwal

  2. Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Ravi P. Agarwal

  3. Department of Mathematics, National University of Ireland, Galway, Ireland

    Donal O’Regan

  4. Department of Mathematical Analysis, Faculty of Science, Palacký University, Olomouc, Czech Republic

    Svatoslav Staněk

Authors
  1. Ravi P. Agarwal
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  2. Donal O’Regan
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  3. Svatoslav Staněk
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Correspondence to Ravi P. Agarwal.

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Agarwal, R.P., O’Regan, D. & Staněk, S. Positive and maximal positive solutions of singular mixed boundary value problem. centr.eur.j.math. 7, 694–716 (2009). https://doi.org/10.2478/s11533-009-0049-9

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  • DOI: https://doi.org/10.2478/s11533-009-0049-9

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