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

 Access to full text

 Full (PDF)

1. [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 Zbl1196.26015
2. [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 Zbl1253.47038
3. [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 Zbl1204.26014
4. [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 Zbl1229.47087
5. [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 Zbl1202.47027
6. [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 Zbl1216.47090
7. [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 Zbl0997.47051
8. [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. [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 Zbl1027.47046
10. [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. [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 Zbl1033.26019
12. [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. [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 Zbl1225.47078
14. [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 Zbl1229.47088
15. [15] Knop J., On globally Lipschitzian Nemytskiĭ operator in a special Banach space of functions, Fasc. Math., 1990, 21, 79–85 Zbl0728.39005
16. [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. [17] Kuczma M., Functional Equations in a Single Variable, Monogr. Mat., 46, PWN, Warsaw, 1968 
18. [18] Kuczma M., An Introduction to the Theory of Functional Equations and Inequalities, Prace Nauk. Uniw. Slask. Katowic., 489, PWN, Warsaw, 1985 
19. [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 Zbl0762.26015
20. [20] Lupa M., Form of Lipschitzian operator of substitution in some class of functions, Zeszyty Nauk. Politech. Łódz. Mat., 1989, 21, 87–96 
21. [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 Zbl0599.46032
22. [22] Matkowska A., Matkowski J., Merentes N., Remark on globally Lipschitzian composition operators, Demonstratio Math., 1995, 28(1), 171–175 Zbl0842.47042
23. [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 Zbl1173.47043
24. [24] Matkowski J., Uniformly bounded composition operators between general Lipschitz function normed spaces, Topol. Methods Nonlinear Anal., 2011, 38(2), 395–406 Zbl1272.47070
25. [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 Zbl0785.47033
26. [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. [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 Zbl0566.47033
28. [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 Zbl1153.47048
29. [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 Zbl1173.47044
30. [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. [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 Zbl0808.47050
32. [32] Merentes N., On the composition operator in RVϕ[a, b], Collect. Math., 1995, 46(3), 231–238 
33. [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 Zbl0856.47042
34. [34] Merentes N., Rivas S., Sanchez J.L., On functions of bounded (p, k)-variation, J. Funct. Spaces Appl., 2012, #202987 Zbl1241.26007
35. [35] Popoviciu T., Sur les Fonctions Convexes d’une Fonctions d’une Variable Réelle, DSc thesis, Paris, 1933 
36. [36] Riesz F., Untersuchungen über Systeme integrierbarer Funktionen, Math. Ann., 1910, 69, 449–497 http://dx.doi.org/10.1007/BF01457637 Zbl41.0383.01
37. [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 Zbl0254.26017
38. [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 Zbl0333.46025
39. [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 Zbl0679.47038
40. [40] Sharkovsky A.N., Maĭstrenko Yu.L., Romanenko E.Yu., Difference Equations and their Applications, Math. Appl., 250, Kluwer, Dordrecht, 1993 
41. [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 Zbl39.0331.01
42. [42] Wróbel M., On functions of bounded n-th variation, Ann. Math. Sil., 2001, 15, 79–86 Zbl1085.26004
43. [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 Zbl1182.47045
44. [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 Zbl1225.47079
45. [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 Zbl1208.47052
46. [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 Zbl1266.47083
47. [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 Zbl1321.47124