Nemytskii Operator in the Space of Set-Valued Functions of Bounded *φ*-Variation

ABSTRACT

In this paper we consider the Nemytskii
operator, *i.e*., the composition
operator defined by (*Nf*)(*t*)=*H*(*t*,*f*(*t*)), where *H* is a given set-valued function. It is
shown that if the operator *N* maps the space of functions bounded *φ*_{}_{1}-variation in the sense of
Riesz with respect to the weight function *α*into the space of
set-valued functions of bounded *φ*_{}_{2}-variation in the sense of
Riesz with respect to the weight, if it is globally Lipschitzian, then it has
to be of the form (*Nf*)(*t*)=*A*(*t*)*f*(*t*)+*B*(*t*), where *A*(*t*) is a linear continuous set-valued function and *B* is a set-valued
function of bounded *φ*_{}_{2}-variation in the sense of Riesz with
respect to the weight.

Cite this paper

W. Aziz, "Nemytskii Operator in the Space of Set-Valued Functions of Bounded*φ*-Variation," *Advances in Pure Mathematics*, Vol. 3 No. 6, 2013, pp. 563-575. doi: 10.4236/apm.2013.36072.

W. Aziz, "Nemytskii Operator in the Space of Set-Valued Functions of Bounded

References

[1] A. Smajdor and W. Smajdor, “Jensen Equation and Nemytskii Operator for Set-Valued Functions,” Radovi Matematicki, Vol. 5, 1989, pp. 311-319.

[2] J. Matkwoski, “Functional Equation and Nemytskii Operators,” Funkcialaj Ekvacioj, Vol. 25, No. 2, 1982, pp. 127-132.

[3] G. Zawadzka, “On Lipschitzian Operators of Substitution in the Space of Set-Valued Functions of Bounded Variation,” Radovi Matematicki, Vol. 6, 1990, pp. 179-193.

[4] J. Matkwoski and J. Mis, “On a Characterization of Lipschitzian Operators of Substitution in the Space BV(a,b),” Mathematische Nachrichten. Vol. 117, No. 1, 1984, pp. 155-159. doi:10.1002/mana.3211170111

[5] N. Merentes and K. Nikodem, “On Nemytskii Operator and Set-Valued Functions of Bounded P-Variation,” Radovi Matematicki, Vol. 8, 1992, pp. 139-145.

[6] N. Merentes and S. Rivas, “On Nemytskii Operator in the Space of Set-Valued Functions of Bounded P-Variation in the Sense of Riesz,” Publicationes Mathematicae Debrecen, Vol. 47, No. 1-2, 1995, pp. 15-27.

[7] N. Merentes and J. L. Sánchez, “Characterization of Globally Lipschitz Nemytskii Operator between Spaces of Set-Valued Functions of Bounded φ-Variation in Sense of Riesz,” Bulletin of the Polish Academy of Sciences Mathematics, Vol. 52, No. 4, 2004, pp. 417-430. doi:10.4064/ba52-4-8

[8] W. Aziz, J. A. Guerrero, N. Merentes and J. L. Sánchez, “Nemytskii Operator in the Space of Set-Valued Functions of Bounded P-Variation,” JMCSA, Vol. 4, No. 1, 2011, pp. 85-94.

[9] V. V. Chistyakov, “Lipschitzian Superposition Operators between Spaces of Functions of Bounded Generalized Variation with Weight,” Journal of Applied Analysis, Vol. 6, No. 2, 2000, pp. 173-186. doi:10.1515/JAA.2000.173

[10] F. Riesz, “Untersuchugen über Systeme Integrierbarer Funktionen,” Mathematische Annalen, Vol. 69, No. 4, 1910, pp. 449-497. doi:10.1007/BF01457637

[11] Yu. T. Medvedev, “A Generalization of a Theorem of F. Riesz,” Uspekhi Matematicheskikh Nauk, Vol. 6, No. 8, 1953, pp. 115-118.

[12] Z. Cibertowicz and W. Matuszewska, “Functions of Bounded Generalized Variations,” Commentationes Mathematicae (Prace Matematyczne), Vol. 20, No. 1, 1977, pp. 29-52.

[13] W. A. J. Luxemburg, “Banach Function Spaces,” Ph.D. Dissertation, Technische Hogeschool te Delft, 1955.

[14] H. Nakano, “Modulared Semi-Ordered Spaces,” Tokyo, 1950.

[15] L. Maligranda and W. Orlicz, “On Some Properties of Functions of Generalized Variation,” Monatshefte für Mathematik, Vol. 104, No. 1, 1987, pp. 53-65.

[16] F. Riesz and B. Sz. Nagy, “Functional Analysis,” Ungar, New York, 1955.

[17] M. C. Chakrabarty, “Some Results on AC-ω Functions,” Fundamenta Mathematicae, Vol. 64, No. 2, 1969, pp. 219-230.

[18] R. L. Jeffery, “Generalized Integrals with Respect to Bounded Variation,” Canadian Journal of Mathematics, Vol. 10, 1958, pp. 617-628.

[19] H. Radstrom, “An Embedding Theorem for Space of Convex Sets,” Proceedings of the American Mathematical Society, Vol. 3, No. 1, 1952, pp. 165-169.

[20] K. Nikodem, “K-Convex and K-Concave Set-Valued Functions,” Zeszyty Naukowe Politechniki Lódzkiej, Vol. 559, 1989, pp. 210-225.

[21] W. Aziz, J. A. Guerrero and N. Merentes, “On Nemytskii Operator in the Space of Set-Valued Functions of Bounded p-Variation in the Sense of Riesz with Respect to the Weight Function,” Fasciculi Mathematici, Vol. 50, 2013, in press.

[1] A. Smajdor and W. Smajdor, “Jensen Equation and Nemytskii Operator for Set-Valued Functions,” Radovi Matematicki, Vol. 5, 1989, pp. 311-319.

[2] J. Matkwoski, “Functional Equation and Nemytskii Operators,” Funkcialaj Ekvacioj, Vol. 25, No. 2, 1982, pp. 127-132.

[3] G. Zawadzka, “On Lipschitzian Operators of Substitution in the Space of Set-Valued Functions of Bounded Variation,” Radovi Matematicki, Vol. 6, 1990, pp. 179-193.

[4] J. Matkwoski and J. Mis, “On a Characterization of Lipschitzian Operators of Substitution in the Space BV(a,b),” Mathematische Nachrichten. Vol. 117, No. 1, 1984, pp. 155-159. doi:10.1002/mana.3211170111

[5] N. Merentes and K. Nikodem, “On Nemytskii Operator and Set-Valued Functions of Bounded P-Variation,” Radovi Matematicki, Vol. 8, 1992, pp. 139-145.

[6] N. Merentes and S. Rivas, “On Nemytskii Operator in the Space of Set-Valued Functions of Bounded P-Variation in the Sense of Riesz,” Publicationes Mathematicae Debrecen, Vol. 47, No. 1-2, 1995, pp. 15-27.

[7] N. Merentes and J. L. Sánchez, “Characterization of Globally Lipschitz Nemytskii Operator between Spaces of Set-Valued Functions of Bounded φ-Variation in Sense of Riesz,” Bulletin of the Polish Academy of Sciences Mathematics, Vol. 52, No. 4, 2004, pp. 417-430. doi:10.4064/ba52-4-8

[8] W. Aziz, J. A. Guerrero, N. Merentes and J. L. Sánchez, “Nemytskii Operator in the Space of Set-Valued Functions of Bounded P-Variation,” JMCSA, Vol. 4, No. 1, 2011, pp. 85-94.

[9] V. V. Chistyakov, “Lipschitzian Superposition Operators between Spaces of Functions of Bounded Generalized Variation with Weight,” Journal of Applied Analysis, Vol. 6, No. 2, 2000, pp. 173-186. doi:10.1515/JAA.2000.173

[10] F. Riesz, “Untersuchugen über Systeme Integrierbarer Funktionen,” Mathematische Annalen, Vol. 69, No. 4, 1910, pp. 449-497. doi:10.1007/BF01457637

[11] Yu. T. Medvedev, “A Generalization of a Theorem of F. Riesz,” Uspekhi Matematicheskikh Nauk, Vol. 6, No. 8, 1953, pp. 115-118.

[12] Z. Cibertowicz and W. Matuszewska, “Functions of Bounded Generalized Variations,” Commentationes Mathematicae (Prace Matematyczne), Vol. 20, No. 1, 1977, pp. 29-52.

[13] W. A. J. Luxemburg, “Banach Function Spaces,” Ph.D. Dissertation, Technische Hogeschool te Delft, 1955.

[14] H. Nakano, “Modulared Semi-Ordered Spaces,” Tokyo, 1950.

[15] L. Maligranda and W. Orlicz, “On Some Properties of Functions of Generalized Variation,” Monatshefte für Mathematik, Vol. 104, No. 1, 1987, pp. 53-65.

[16] F. Riesz and B. Sz. Nagy, “Functional Analysis,” Ungar, New York, 1955.

[17] M. C. Chakrabarty, “Some Results on AC-ω Functions,” Fundamenta Mathematicae, Vol. 64, No. 2, 1969, pp. 219-230.

[18] R. L. Jeffery, “Generalized Integrals with Respect to Bounded Variation,” Canadian Journal of Mathematics, Vol. 10, 1958, pp. 617-628.

[19] H. Radstrom, “An Embedding Theorem for Space of Convex Sets,” Proceedings of the American Mathematical Society, Vol. 3, No. 1, 1952, pp. 165-169.

[20] K. Nikodem, “K-Convex and K-Concave Set-Valued Functions,” Zeszyty Naukowe Politechniki Lódzkiej, Vol. 559, 1989, pp. 210-225.

[21] W. Aziz, J. A. Guerrero and N. Merentes, “On Nemytskii Operator in the Space of Set-Valued Functions of Bounded p-Variation in the Sense of Riesz with Respect to the Weight Function,” Fasciculi Mathematici, Vol. 50, 2013, in press.