Research ArticleOpen AccessGoogle Scholar indexed
Locally Defined Operators and Locally Lipschitz Composition Operators in the Space <i>WBV</i><i>p</i>(·)([a, b])
Departamento de Matemática y Fsica, Universidad Nacional Experimental del Táchira, San Cristóbal, Venezuela
Departamento de Matemática, Universidad Central de Venezuela, Caracas, Venezuela
Departamento de Matemática, Universidad Central de Venezuela, Caracas, Venezuela
- 1 Departamento de Matemática y Fsica, Universidad Nacional Experimental del Táchira, San Cristóbal, Venezuela
- 2 Departamento de Matemática, Universidad Central de Venezuela, Caracas, Venezuela
- 3 Departamento de Matemática, Universidad Central de Venezuela, Caracas, Venezuela
Advances in Pure Mathematics·Volume 06 (2016)·Pages 727–744·Published 12 September 2016·DOI10.4236/apm.2016.610059
Copy link · social · email
Abstract
We give a neccesary and sufficient condition on a function such that the composition operator (Nemytskij Operator) H defined by acts in the space and satisfies a local Lipschitz condition. And, we prove that every locally defined operator mapping the space of continuous and bounded Wiener p ( · )-variation with variable exponent functions into itself is a Nemytskij com-position operator.
KeywordsGeneralized Variation<i>p</i>(·)-Variation in Wiener’s SenseVariable ExponentConvergenceHelly’s TheoremLocal Operator
- Castillo, R., Merentes, N. and Rafeiro, H. (2014) Bounded Variation Spaces with p-Variable. Mediterranean Journal of Mathematics, 11, 1069-1079. http://dx.doi.org/10.1007/s00009-013-0342-5
- Mejía, O., Merentes, N. and Sánchez, J.L. (2015) The Space of Bounded -Variation in Wiener’s Sense with Variable Exponent. Journal Advances in Pure Mathematics, 5, 703-716. http://dx.doi.org/10.4236/apm.2015.511064
- Orlicz, W. (1931) über konjugierte exponentenfolgen. Studia Mathematica, 3, 200-211.
- Nakano, H. (1950) Modulared Semi-Ordered Linear Spaces. Maruzen Co., Ltd., Tokyo.
- Kovácik, O. and Rákosník, J. (1991) On Spaces and . Czechoslovak Mathematical Journal, 41, 592-618.
- Fan, X., Zhao, Y. and Zhao, D. (2001) Compact Imbedding Theorems with Symmetry of Strauss-Lions Type for the Space . Journal of Mathematical Analysis and Applications, 255, 333-348. http://dx.doi.org/10.1006/jmaa.2000.7266
- Appell, J. and Zabreiko, P.P. (1990) Nonlinear Superposition Operators. Cambridge University Press, Cambridge.
- Appell, J., Banas, J. and Merentes, N. (2014) Bounded Variation and Around. De Gruyter, Berlin, Boston.
- Sovolevskij, E.P. (1984) The Superposition Operator in Hölder Spaces. (Russian). VINITI No. 3765-84, Voronezh.
- Appell, J., Merentes, N. and Sánchez, J.L. (2011) Locally Lipschitz Composition Operator in Spaces of Functions Of bounded Variation. Annali di Matematica, 190, 33-43. http://dx.doi.org/10.1007/s10231-010-0135-4
- Merentes, N., Rivas, S. and Sánchez, J.L. (2012) Locally Lipschitz Composition Operator in Spaces of . Nonlinear Analysis Series A: Theory, Methods & Applications, 75, 1751-1757.
- Mejía, O., Merentes, N. and Rzepka, B. (2014) Locally Lipschitz Composition Operator in Space of the Functions of Bounded -Variation. Journal of Function Spaces, 2014, 1-8.
- Lichawski, K., Matkowski, J. and Mis, J. (1989) Locally Defined Operators in the Space of Differentiable Functions. Bulletin of the Polish Academy of Sciences—Mathematics, 37, 315-325.
- Matkowski, J. and Wróbel, M. (2008) Locally Defined Operators in the Space of Whitney Differentiable Functions. Nonlinear Analysis: Theory, Methods & Applications, 68, 2933-2942.
- Matkowski, J. and Wróbel, M. (2009) Representation Theorem for Locally Defined Operators in the Space of Whitney Differentiable Functions. Manuscripta Mathematica, 129, 437-448. http://dx.doi.org/10.1007/s00229-009-0283-2