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Convex solutions to polynomial-like iterative equations on open intervals

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Abstract

The paper deals with the polynomial-like iterative functional equation

$$\lambda_1 f(x)+\lambda_2 f^2(x)+\cdots+\lambda_n f^n(x)=F(x).$$

By using Schauder’s fixed point theorem and a version of the uniform boundedness principle for families of convex (respectively higher order convex) functions as basic tools, the existence of nondecreasing convex (respectively higher order convex) solutions to this equation on open (possibly unbounded) intervals is investigated. The results of the paper complement similar ones established by other authors, concerning the existence of monotonic or convex solutions to the above equation on compact intervals. Some examples illustrating their applicability are provided.

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References

  1. Agarwal R.P., Meehan M., O’Regan D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  2. Baron K., Jarczyk W.: Recent results on functional equations in a single variable, perspectives and open problems. Aequationes Math. 61, 1–48 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Breckner W.W.: Equicontinuous families of generalized convex mappings. Math. Rev. Anal. Numér. Théor. Approx., Sér. Math. 26(49), 9–20 (1984)

    MathSciNet  Google Scholar 

  4. Breckner W.W., Trif T.: On the singularities of certain families of nonlinear mappings. Pure Math. Appl. 6, 121–137 (1995)

    MATH  MathSciNet  Google Scholar 

  5. Breckner W.W., Trif T.: Convex Functions and Related Functional Equations, Selected Topics. Cluj University Press, Cluj-Napoca (2008)

    Google Scholar 

  6. Chen J., Zhang W.: Leading coefficient problem for polynomial-like iterative equations. J. Math. Anal. Appl. 349, 413–419 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gradshteyn I.S., Ryzhik I.M.: Tables of Integrals, Series and Products. Academic Press, New York (1980)

    Google Scholar 

  8. Jarczyk W.: On an equation of linear iteration. Aequationes Math. 51, 303–310 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kuczma, M.: An introduction to the theory of functional equations and inequalities. Cauchy’s equation and Jensen’s inequality. Second edn. Edited and with a preface by Attila Gilányi. Birkhäuser, Basel (2009)

  10. Kuczma, M., Choczewski, B., Ger, R.: Iterative Functional Equations. Encyclopedia Math. Appl. 32, Cambridge University Press, Cambridge (1990)

  11. Neumann M.M.: Uniform boundedness and closed graph theorems for convex operators. Math. Nachr. 120, 113–125 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  12. Rassias Th.M., Trif T.: Log-convex solutions of the second order to the functional equation f(x + 1) = g(x)f(x). J. Math. Anal. Appl. 331, 1440–1451 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Roberts A.W., Varberg D.E.: Convex Functions. Academic Press, New York (1973)

    MATH  Google Scholar 

  14. Tabor J., Tabor J.: On a linear iterative equation. Results Math. 27, 412–421 (1995)

    MATH  MathSciNet  Google Scholar 

  15. Xu B., Zhang W.: Construction of continuous solutions and stability for the polynomial-like iterative equation. J. Math. Anal. Appl. 325, 1160–1170 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Xu B., Zhang W.: Decreasing solutions and convex solutions of the polynomial-like iterative equation. J. Math. Anal. Appl. 329, 483–497 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Zhang W.: Discussion on the iterated equation \({\sum_{i=1}^n\lambda_i f^i(x)=F(x)}\). Chinese Sci. Bull. 32, 1444–1451 (1987)

    MATH  Google Scholar 

  18. Zhang W.: Discussion on the differentiable solutions of the iterated equation \({\sum_{i=1}^n\lambda_i f^i(x)=F(x)}\). Nonlinear Anal. 15, 387–398 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  19. Zhang W., Nikodem K., Xu B.: Convex solutions of polynomial-like iterative equations. J. Math. Anal. Appl. 315, 29–40 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Tiberiu Trif.

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Trif, T. Convex solutions to polynomial-like iterative equations on open intervals. Aequat. Math. 79, 315–325 (2010). https://doi.org/10.1007/s00010-010-0020-7

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  • DOI: https://doi.org/10.1007/s00010-010-0020-7

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