Mathematische Zeitschrift

, Volume 266, Issue 1, pp 43–63 | Cite as

Positive solutions for nonlinear operator equations and several classes of applications

Article

Abstract

In this paper, we study a class of nonlinear operator equations x = Ax + x 0 on ordered Banach spaces, where A is a monotone generalized concave operator. Using the properties of cones and monotone iterative technique, we establish the existence and uniqueness of solutions for such equations. In particular, we do not demand the existence of upper-lower solutions and compactness and continuity conditions. As applications, we study first-order initial value problems and two-point boundary value problems with the nonlinear term is required to be monotone in its second argument. In the end, applications to nonlinear systems of equations and to nonlinear matrix equations are also considered.

Keywords

Positive solution Nonlinear operator equation Normal cone Initial value problem Boundary value problem Nonlinear algebra systems 

Mathematics Subject Classification (2000)

47H10 47H07 34B18 35F25 15A30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agarwal R.P., Wong P.J.Y., Regan D.O.: Positive Solutions of Differential, Difference, Integral Equations. Kluwer Academic, Boston (1999)Google Scholar
  2. 2.
    Agarwal R.P., Regan D.O.: A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem. Appl. Math. Comput. 161, 433–439 (2005)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Amann H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18(4), 620–709 (1976)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Avery R.I., Henderson J.: Three symmetric positive solutions for a second order boundary value problem. Appl. Math. Lett. 13(3), 1–7 (2000)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Avery R.I., Peterson A.C.: Three positive fixed points of nonlinear operators on ordered Banach spaces. Comput. Math. Appl. 42, 313–322 (2001)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Bushell P.J.: On a class of Volterra and Fredholm nonlinear integral equations. Math. Proc. Camb. Phil. Soc. 79, 329–335 (1976)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Chen Y.Z.: Continuation method for α-sublinear mappings. Proc. Am. Math. Soc. 129, 203–210 (2001)CrossRefMATHGoogle Scholar
  8. 8.
    Deimling K.: Nonlinear Functional Analysis. Springer-Varlag, Berlin (1985)MATHGoogle Scholar
  9. 9.
    Du S.W., Lakshmikantham V.: Monotone iterative technique for differential equations in Banach space. J. Math. Anal. Appl. 87, 454–459 (1982)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Guo D.J.: Fixed points and eigenelements of a class of concavex and convex operator (in Chinese). Chinese Sci. Bull. 15, 1132–1135 (1985)Google Scholar
  11. 11.
    Guo D.J.: Existence and uniqueness of positive fixed points for mixed monotone operators and applications. Appl. Anal. 46, 91–100 (1992)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Guo D.J.: Existence and nuiqueness of positive fixed points for noncompact decreasing operators. Indian J. Pure. Appl. Math. 31(5), 551–562 (2000)MathSciNetMATHGoogle Scholar
  13. 13.
    Guo D.J., Lakshmikantham V.: Nonlinear Problems in Abstract Cones. Academic Press Inc (2), Boston (1988)MATHGoogle Scholar
  14. 14.
    Guo D.J., Lakshmikantham V., Liu X.Z.: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic Publishers, Dordrecht (1996)MATHGoogle Scholar
  15. 15.
    Krasnosel’skii M.A.: Positive Solutions of Operators Equations. Noordoff, Groningen (1964)Google Scholar
  16. 16.
    Ladde, G.S., Lakshmikantham, V., Vatsala, A.S.: Monotone Iterative Techniques for Differential Equations. Pitman (1985)Google Scholar
  17. 17.
    Lakshmikantham V., Leela S., Oguztoreli M.N.: Quasi-solutions, veter Lyapunov functions and monotone methods. IEEE. Trans. Autom. Control. 26, 1149–1153 (1981)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Li F.Y.: Existence and uniqueness of positive solutions of some nonlinear equations. Acta. Math. Appl. Sinica. 20(4), 609–615 (1997)MathSciNetMATHGoogle Scholar
  19. 19.
    Li K., Liang J., Xiao T.J.: Positive fixed points for nonlinear operators. Comput. Math. Appl. 50, 1569–1578 (2005)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Li K., Liang J., Xiao T.J.: A fixed point theorem for convex and decreasing operators. Nonlinear Anal. 63, e209–e206 (2005)CrossRefMATHGoogle Scholar
  21. 21.
    Liang Z.D., Wang W.X., Li S.J.: On concave operators. Acta. Math. Sinica (Engl. Ser.) 22(2), 577–582 (2006)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Loewner C.: Uber monotone matrixfunktionen. Math. Z. 38, 177–216 (1934)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Ly I., Seck D.: Isoperimetric inequality for an interior free boundary problem with p-Laplacian operator. Electron. J. Differ. Equ. 109, 1–12 (2004)MathSciNetGoogle Scholar
  24. 24.
    Ma D., Du Z., Ge W.: Existence and iteration of monotone positive solutions for multipoint boundary value problem with p-Laplacian operator. Comput. Math. Appl. 50, 729–739 (2005)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Regan D.O.: Some general existence principles results for (ψ p(y′))′ = q(t) f (t, y, y′), 0 < t < 1. SIAM J. Math. Anal. 24(3), 648–668 (1993)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Potter A.J.B.: Applications of Hilbert’s projective metric to certain class of nonhomogeneous operators. Quart. J. Math. Oxford. 28(2), 93–99 (1977)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Wan W.X.: Conditions for contraction of mappings and Banach type fixed point theorem (in Chinese). Acta. Math. Sinica. 27, 35–52 (1984)MathSciNetMATHGoogle Scholar
  28. 28.
    Wang W.X., Liang Z.D.: Fixed point theorems of a class of nonlinear operators and applications (in Chinese). Acta. Math. Sinica. 48(4), 789–800 (2005)MathSciNetMATHGoogle Scholar
  29. 29.
    Wang Y., Ge W.: Positive solutions for multipoint boundary value problems with a one-dimensional p-Laplacian. Nonlinear Anal. 66, 1246–1256 (2007)CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    Yang C., Zhai C.B., Yan J.R.: Positive solutions of three-point boundary value problem for second order differential equations with an advanced argument. Nonlinear Anal. 65, 2013–2023 (2006)CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Zhai C.B., Guo C.M.: On α-convex operators. J. Math. Anal. Appl. 316, 556–565 (2006)CrossRefMathSciNetMATHGoogle Scholar
  32. 32.
    Zhai C.B., Yang C., Guo C.M.: Positive solutions of operator equation on ordered Banach spaces and applications. Comput. Math. Appl. 56, 3150–3156 (2008)CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    Zhai C.B., Wang W.X., Zhang L.L.: Generalization for a class of concave and convex operators (in Chinese). Acta. Math. Sinica. 51(3), 529–540 (2008)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.School of Mathematical SciencesShanxi UniversityTaiyuanPeople’s Republic of China
  2. 2.Department of Mathematics and Electronic ScienceBusiness College of Shanxi UniversityTaiyuanPeople’s Republic of China

Personalised recommendations