# SUBSTITUTION OPERATORS IN THE SPACES OF FUNCTIONS OF BOUNDED VARIATION BV2α(I)

• Aziz, Wadie (Universidad de Los Andes Departamento de Fisica y Matematica) ;
• Guerrero, Jose Atilio (Universidad Nacional Experimental del Tachira Departamento de Matematica y Fisica) ;
• Merentes, Nelson (Universidad Central de Venezuela Escuela de Matematicas)
• Received : 2014.05.01
• Published : 2015.03.31

#### Abstract

The space $BV^2_{\alpha}(I)$ of all the real functions defined on interval $I=[a,b]{\subset}\mathbb{R}$, which are of bounded second ${\alpha}$-variation (in the sense De la Vall$\acute{e}$ Poussin) on I forms a Banach space. In this space we define an operator of substitution H generated by a function $h:I{\times}\mathbb{R}{\rightarrow}\mathbb{R}$, and prove, in particular, that if H maps $BV^2_{\alpha}(I)$ into itself and is globally Lipschitz or uniformly continuous, then h is an affine function with respect to the second variable.

#### References

1. A. Acosta, W. Aziz, J. Matkowski, and N. Merentes, Uniformly continuous composition operator in the space of $\varphi$-variation functions in the sense of Riesz, Fasc. Math. 43 (2010), 5-11.
2. W. Aziz, A. Azocar, J. Guerrero, and N. Merentes, Uniformly continuous composition operator in the space of functions of $\varphi$-variation with weight in the sense of Riesz, Nonlinear Anal. 74 (2011), no. 2, 573-576. https://doi.org/10.1016/j.na.2010.09.010
3. W. Aziz, J. Guerrero, and N. Merentes, Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Riesz, Bull. Pol. Acad. Sci. Math. 58 (2010), no. 1, 39-45. https://doi.org/10.4064/ba58-1-5
4. R. Castillo and E. Trousselot, An application of the generalized Maligranda-Orlicz's Lemma, J. Inequal. Pure Appl. Math. 9 (2008), no. 3, Article 84, 6 pp.
5. J. A. Guerrero, H. Leiva, J. Matkowski, and N. Merentes, Uniformly continuous composition operators in the space of bounded $\varphi$-variation functions, Nonlinear Anal. 72 (2010), no. 6, 3119-3123. https://doi.org/10.1016/j.na.2009.11.051
6. T. Kostrzewski, Globally Lipschitz operators of substitution in Banach space BCI, Sci. Bull. Lodzka Technical Univ. 602, (1993), 17-25.
7. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Polish Scientific Editors and Silesian University, Warszawa-Krakow-Katowice, 1985.
8. A. Matkowska, On characterization of Lipschitizian operators of substitution in the class of Holder functions, Sci. Bull. Lodzka Technical Univ. 17 (1984), 81-85.
9. A. Matkowska, J. Matkowski, and N. Merentes, Remark on globally Lipschitizian composition operators, Demonstratio Math. 28 (1995), no. 1, 171-175.
10. J. Matkowski, Functional equations and Nemytskii operators, Funkcial. Ekvac. 25 (1982), no. 2, 127-132.
11. J. Matkowski, Form of Lipschitz operators of substitution in Banach spaces of differentiable functions, Sci. Bull. Lodzka Tech. Univ. 17, (1984), 5-10.
12. J. Matkowski, Uniformly continuous superposition operators in space of differentiable function and absolutely continuous functions, Internat. Ser. Numer. Math. 157 (2008), 155-166.
13. J. Matkowski, Uniformly continuous superposition operators in Banach space of Holder functions, J. Math. Anal. Appl. 359 (2009), no. 1, 56-61. https://doi.org/10.1016/j.jmaa.2009.05.020
14. J. Matkowski, Uniformly continuous superposition operators in the space of bounded variation functions, Math. Nachr. 283 (2010), no. 7, 1060-1064. https://doi.org/10.1002/mana.200710126
15. J. Matkowski and J. Mis, On a characterization of Lipschitzian operators of substitution in the space BV[a, b], Math. Nachr. 117 (1984), 155-159.
16. N. Merentes, On a characterization of Lipschitzian operators of substitution in the space of bounded Riesz $\varphi$-variation, Ann. Univ. Sci. Budapest. Eotvos Sect. Math. 34 (1991), 139-144.
17. N. Merentes and S. Rivas, On a characterization of the Lipschitzian composition operators between spaces of functions of bounded p-variation, Czechoslov. Math. J. 45 (1995), no. 4, 43-48.
18. A. W. Roberts and D. E. Varberg, Convex Functions, New York and London, 1973.
19. A. M. Russell, Functions of bounded second variation and Stieltjes-type integrals, J. Lond. Math. Soc. (2) 2 (1970), 193-208.
20. Ch. J. de la Vallee Poussin, Sur la convergence des formules d'interpolation entre ordon ees equidistantes, Acad. Roy. Belge. Bull. Cl. Sci. (6) 4 (1908), 319-410.