Uniformly bounded composition operators in the banach space of bounded (p, k)-variation in the sense of Riesz-Popoviciu

• Autorzy: Francy Armao , Dorota Głazowska , Sergio Rivas , Jessica Rojas
• Języki publikacji: EN

Abstrakty

EN

We prove that if the composition operator F generated by a function f: [a, b] × ℝ → ℝ maps the space of bounded (p, k)-variation in the sense of Riesz-Popoviciu, p ≥ 1, k an integer, denoted by RV(p,k)[a, b], into itself and is uniformly bounded then RV(p,k)[a, b] satisfies the Matkowski condition.

Słowa kluczowe

EN

Nemytskij (composition; superposition) operator; Uniformly bounded mapping; Uniformly continuous mapping; de la Vallée Poussin second-variation; Popoviciu k-th variation

Kategorie tematyczne

• 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytski{\u\i}, Uryson, etc.)
• 47B38: Operators on function spaces (general)
• 26A45: Functions of bounded variation, generalizations

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 2
• Strony: 357-367

Daty

wydano

2013-02-01

online

2012-11-21

Twórcy

autor

Francy Armao

• Universidad Central de Venezuela

autor

Dorota Głazowska

• University of Zielona Góra

autor

Sergio Rivas

• Universidad Nacional Abierta

autor

Jessica Rojas

• Universidad Central de Venezuela

Bibliografia

• [1] Acosta A., Aziz W., Matkowski J., Merentes N., Uniformly continuous composition operator in the space of φ-variation functions in the sense of Riesz, Fasc. Math., 2010, 43, 5–11
• [2] Appell J., Guanda N., Väth M., Function spaces with the Matkowski property and degeneracy phenomena for composition operators, Fixed Point Theory, 2011, 12(2), 265–284
• [3] Aziz W., Azocar A., Guerrero J., Merentes N., Uniformly continuous composition operators in the space of functions of φ-variation with weight in the sense of Riesz, Nonlinear Anal., 2011, 74(2), 573–576 http://dx.doi.org/10.1016/j.na.2010.09.010
• [4] Aziz W., Giménez J., Merentes N., Sánchez J.L., Uniformly continuous set-valued composition operators in the space of total φ-bidimensional variation in the sense of Riesz, Opuscula Math., 2010, 30(3), 241–248
• [5] Aziz W., Guerrero J.A., Merentes N., Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Riesz, Bull. Pol. Acad. Sci. Math., 2010, 58(1), 39–45 http://dx.doi.org/10.4064/ba58-1-5
• [6] Azócar A., Guerrero J.A., Matkowski J., Merentes N., Uniformly continuous set-valued composition operators in the space of functions of bounded variation in the sense of Wiener, Opuscula Math., 2010, 30(1), 53–60
• [7] Chistyakov V.V., Lipschitzian superposition operators between spaces of functions of bounded generalized variation with weight, J. Appl. Anal., 2000, 6(2), 173–186 http://dx.doi.org/10.1515/JAA.2000.173
• [8] Chistyakov V.V., On mapping of finite generalized variation and nonlinear operators, In: Report on the Summer Symposium in Real Analysis XXIV, Denton, May 23–27, 2000, Real Anal. Exchange, Summer Symposium 2000, suppl., Michigan State University Press, East Lansing, 2000, 39–43
• [9] Chistyakov V.V., Generalized variation of mappings with applications to composition operators and multifunctions, Positivity, 2001, 5(4), 323–358 http://dx.doi.org/10.1023/A:1011879221347
• [10] Chistyakov V.V., The algebra of functions of two variables with finite variation and Lipschitzian superposition, In: Proceedings of the 12th Baikal International Conference “Optimization Methods and their Applications”, Irkutsk, June 24–July 1, 2001, 53–58 (in Russian)
• [11] Chistyakov V.V., Superposition operators in the algebra of functions of two variables with finite total variation, Monatsh. Math., 2002, 137(2), 99–114 http://dx.doi.org/10.1007/s006050200048
• [12] Chistyakov V.V., Solycheva O. M., Lipschitzian operators of substitution in the Algebra ΛBV, J. Difference Equ. Appl., 2003, 9(3–4), 407–416 http://dx.doi.org/10.1080/1023619021000047824
• [13] Guerrero J.A., Leiva H., Matkowski J., Merentes N., Uniformly continuous composition operators in the space of bounded φ-variation functions, Nonlinear Anal., 2010, 72(6), 3119–3123 http://dx.doi.org/10.1016/j.na.2009.11.051
• [14] Guerrero J.A., Matkowski J., Merentes N., Uniformly continuous composition operators in the space of functions of two variables of bounded φ-variation in the sense of Wiener, Comment. Math., 2010, 50(1), 41–48
• [15] Knop J., On globally Lipschitzian Nemytskiĭ operator in a special Banach space of functions, Fasc. Math., 1990, 21, 79–85
• [16] Krasnosel’skiĭ M.A., Pokrovskiĭ A.V., Systems with Hysteresis, Springer, Berlin-New York, 1989 http://dx.doi.org/10.1007/978-3-642-61302-9
• [17] Kuczma M., Functional Equations in a Single Variable, Monogr. Mat., 46, PWN, Warsaw, 1968
• [18] Kuczma M., An Introduction to the Theory of Functional Equations and Inequalities, Prace Nauk. Uniw. Slask. Katowic., 489, PWN, Warsaw, 1985
• [19] Lichawski K., Matkowski J., Mis J., Locally defined operators in the space of differentiable functions, Bull. Polish Acad. Sci. Math., 1989, 37(1–6), 315–325
• [20] Lupa M., Form of Lipschitzian operator of substitution in some class of functions, Zeszyty Nauk. Politech. Łódz. Mat., 1989, 21, 87–96
• [21] Matkowska A., On characterization of Lipschitzian operators of substitution in the class of Hölder’s functions, Zeszyty Nauk. Politech. Łódz. Mat., 1984, 17, 81–85
• [22] Matkowska A., Matkowski J., Merentes N., Remark on globally Lipschitzian composition operators, Demonstratio Math., 1995, 28(1), 171–175
• [23] Matkowski J., Uniformly continuous superposition operators in the Banach spaces Hölder functions, J. Math. Anal. Appl., 2009, 359(1), 56–61 http://dx.doi.org/10.1016/j.jmaa.2009.05.020
• [24] Matkowski J., Uniformly bounded composition operators between general Lipschitz function normed spaces, Topol. Methods Nonlinear Anal., 2011, 38(2), 395–406
• [25] Matkowski J., Merentes N., Characterization of globally Lipschitzian composition operators in the Banach space BV p2[a, b], Arch. Math. (Brno), 1992, 28(3–4), 181–186
• [26] Matkowski J., Merentes N., Characterization of globally Lipschitzian composition operators in the Sobolev space W p n [a, b], Zeszyty Nauk. Politech. Łódz. Mat., 1993, 24, 91–99
• [27] Matkowski J., Miś J., On a characterization of Lipschitzian operators of substitution in the space BV〈a; b〉, Math. Nachr., 1984, 117, 155–159 http://dx.doi.org/10.1002/mana.3211170111
• [28] Matkowski J., Wróbel M., Locally defined operators in the space of Whitney differentiable functions, Nonlinear Anal., 2008, 68, 2933–2942 http://dx.doi.org/10.1016/j.na.2007.02.037
• [29] Matkowski J., Wróbel M., Representation theorem for locally defined operators in the space of Whitney differentiable functions, Manuscripta Math., 2009, 129(4), 437–448 http://dx.doi.org/10.1007/s00229-009-0283-2
• [30] Matkowski J., Wróbel M., The bounded local operators in the Banach space of Hölder functions, Pr. Nauk. Akad. Jana Długosza Czest. Mat., 2010, 15, 91–98
• [31] Merentes N., On a characterization of Lipschitzian operators of substitution in the space of bounded Riesz φ-variation, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 1991, 34, 139–144
• [32] Merentes N., On the composition operator in RVϕ[a, b], Collect. Math., 1995, 46(3), 231–238
• [33] Merentes N., Rivas S., On characterization of the Lipschitzian composition operator between spaces of functions of bounded p-variation, Czechoslovak Math. J., 1995, 45(120)(4), 627–637
• [34] Merentes N., Rivas S., Sanchez J.L., On functions of bounded (p, k)-variation, J. Funct. Spaces Appl., 2012, #202987
• [35] Popoviciu T., Sur les Fonctions Convexes d’une Fonctions d’une Variable Réelle, DSc thesis, Paris, 1933
• [36] Riesz F., Untersuchungen über Systeme integrierbarer Funktionen, Math. Ann., 1910, 69, 449–497 http://dx.doi.org/10.1007/BF01457637
• [37] Russell A.M., Functions of bounded kth variation, Proc. London Math. Soc., 1973, 26, 547–563 http://dx.doi.org/10.1112/plms/s3-26.3.547
• [38] Russell A.M., A Banach space of functions of generalized variation, Bull. Aust. Math. Soc., 1976, 15(3), 431–438 http://dx.doi.org/10.1017/S0004972700022863
• [39] Sieczko A., Characterization of globally Lipschitzian Nemytskiĭ operator in the Banach space ACr−1, Math. Nachr., 1989, 141, 7–11 http://dx.doi.org/10.1002/mana.19891410102
• [40] Sharkovsky A.N., Maĭstrenko Yu.L., Romanenko E.Yu., Difference Equations and their Applications, Math. Appl., 250, Kluwer, Dordrecht, 1993
• [41] de la Vallée Poussin Ch.J., Sur la convergence des formules d’interpolation entre ordennées équidistantes, Bull. Acad. R. Belg. Cl. Sci., 1908, 314–410
• [42] Wróbel M., On functions of bounded n-th variation, Ann. Math. Sil., 2001, 15, 79–86
• [43] Wróbel M., Lichawski-Matkowski-Mis theorem of locally defined operators for functions of several variables, Ann. Acad. Pedagog. Crac. Stud. Math., 2008, 7, 15–22
• [44] Wróbel M., Locally defined operators and a partial solution of a conjecture, Nonlinear Anal., 2010, 72(1), 495–506 http://dx.doi.org/10.1016/j.na.2009.06.093
• [45] Wróbel M., Representation theorem for local operators in the space of continuous and monotone functions, J. Math. Anal. Appl., 2010, 372(1), 45–54 http://dx.doi.org/10.1016/j.jmaa.2010.06.013
• [46] Wróbel M., Locally defined operators in Hölder’s spaces, Nonlinear Anal., 2011, 74(1), 317–323 http://dx.doi.org/10.1016/j.na.2010.08.046
• [47] Wróbel M., Uniformly bounded Nemytskij operators between the Banach spaces of functions of bounded n-th variation, J. Math. Anal. Appl., 2012, 391(2), 451–456 http://dx.doi.org/10.1016/j.jmaa.2012.02.053

Identyfikatory

DOI

10.2478/s11533-012-0051-5