Some local Forms of Known Convergences of Sequence of Real Valued Functions

  • Doris Doda Albanian
  • Agron Tato

Abstract

Using the notions of local uniform and strong local uniform con-vergence for the sequence of real valued functions or with value in metric space, the class of locally equally and strong locally equally convergences are studied. We are concern to dependence of type of some convergences from the neighborhood of the limit point. The known locally uniformly convergence is a key of some applications of this idea. We can reformulate one type of Arzela Theorem and nd relations of this convergence with quasi-uniformly by segments of Alexandro off convergence. Beside this type of convergence, we focus to another convergence which is nearer the well known a-convergence.

Author Biography

Agron Tato

Wisdom University, Albania zTirana Polytechnic University, Department of Mathematics, Albania

References

C. Arzel`a, Intorno alla continuit´a della somma di infinit´a di funzioni continue, Rend. R. Accad. Sci. Istit. Bologna (1883/1884), 79-84.

V. Gregoriades, N. Papanastassiou, “The notion of exhaustiveness and Ascoli-type theorems”, Topology and its Applications 155 (2008), 1111- 1128.

R. Das, N. Papanastassiou, Some types of convergence of sequences of real valued functions, Real Anal. Exchange 28 (2) (2002/2003) 1-16.

P.S. Alexandroff, Einf¨uhrung in die Mengenlehre und die Theorie der reellen Funktionem, Deutsch Verlag Wissenschaft, 1956, translated from the 1948, Russian edition.

R.G. Bartle, On compactness in functional analysis, Trans. Amer. Math. Soc. 79 (1955), 35-57.

R.F. Arens, A topology for spaces of transformations, Ann. of Math. (2) 47 (3) (1946), 480-495.

Kelley, J., General topology, Springer-Verlag, 1975.

L. Hol`a, T. Sal´at, Graph convergence, uniform, quasi-uniform and continuous convergence and some characterizarions of compactness, Acta Math. Unic. Comenian, 54-55 (1998), 121-132.

Agata Caserta, Giuseppe Di Maio , Lubica Hol´a, Arzel`a’s Theorem and strong uniform convergence on bornologies, J. Math. Anal. Appl. 371 (2010), 384-392.

N. Papanastassiou, On a new type of convergence of sequences of functions, submitted.

A. Cs´asz´ar and M. Laczkovich, Discrete and equal convergence, Studia Sci. Math. Hungar., 10 (1975), 463-472.

S. Stoilov, Continuous convergence, Rev. Math. Pures Appl. 4 (1959), 341-344.

D. Doda, A. Tato, Some local uniform convergences and its applications on Integral theory, 2018. (submitted)

Published
2018-12-31
How to Cite
Doda, D., & Tato, A. (2018). Some local Forms of Known Convergences of Sequence of Real Valued Functions. JOURNAL OF ADVANCES IN MATHEMATICS, 15, 8185-8198. https://doi.org/10.24297/jam.v15i0.7995
Section
Articles