The results of the superposition operator on sequence space $bv_{p}$

Document Type : Research Paper


Department of Mathematics, University of Hormozgan, Bandar Abbas, Iran


In this paper, the conditions for the superposition operators were provided to map the space $bv_{p}$ into $b v_{q}$, where $1 \leq p$, $q<\infty$. Additionally, we presented the necessary and sufficient conditions under which superposition operators become bounded, continuous and uniformly continuous on the sequence space $b v_{p}$.


[1] R. Adams, Sobolev spaces, Academic Press, New York, 1976.
[2] J. Appell and P.P. Zabrejko, Nonlinear superposition operators, Cambridge University Press, 1990.
[3] J. Appell and P. P. Zabrejko, Remarks on the superposition operator problem in various function spaces, Complex Var. Elliptic Equ. 55 (2010), 727-–737.
[4] F. Basar and B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, Ukrain. Math. J. 55 (2003), 136-–147.
[5] V.A. Bondarenko and P.P. Zabrejko, The superposition operator in Holder functions spaces (Russian), Dokl. Akad. Nauk SSSR 222 (1975), 1265–1268.
[6] G. Bourdaud, M. Lanza de Cristoforis and W. Sickel, Superposition operators and functions of bounded p-variation, Rev. Mat. Iberoam. 22 (2006) 455–487.
[7] D. Bugajewska, On the superposition operator in the space of functions of bounded variation, Math. Comput. Model. 52 (2010), 791–796.
[8] F. Dedagic and N. Okicic, On the compactness and condensing of nonlinear superposition operators, Kragujevac J. Math. 28 (2005), 173–183.
[9] F. Dedagic and P.P. Zabrejko, On the superposition operator in ℓp spaces, (in Russian), Sibir. Mat. Zhurn. (1987), 86–98.
[10] A.I. Gusejnov and H.Sh. Muhtarov, Introduction to the theory of nonlinear singular integral equations (Russian), Nauka, Moskva, 1980.
[11] R. Lashkaripour and J. Fathi, Norms of matrix operators on bvp, J. Math. Inequal. 4 (2012), 589—592.
[12] M. Marcus and V.J. Mizel, Nemytskij operators on Sobolev spaces, Arch. Rat. Mech. Anal. 51 (1973), 347–370.
[13] R. Pluciennik, Continuity of superposition operators on ω0 and W0, Commentat. Math. Univ. Carol. 31 (1990), 529-–542.
[14] J. Robert, Continuite d’un operateur non lineaire sur certains espaces de suites, C. R. Acad. Sci. 259 (1964) 1287–1290.
[15] A. Sama-ae, Boundedness and continuity of superposition operator on Er(p) and Fr(p), Songklanakarin J. Sci. Technol. 24 (2002), 451—466.
[16] P.P. Zabrejko and A.I. Povolotskij, The Hammerstein operator and Orlicz spaces (Russian), Jaroslav. Gos. Univ. Kach. Priblizh. Metody Issled. Oper. Uravn. 2 (1977), 39–51.
Volume 14, Issue 7
July 2023
Pages 309-320
  • Receive Date: 29 July 2022
  • Revise Date: 02 September 2022
  • Accept Date: 27 November 2022