Çift Küme Dizilerinin f-Asimptotik ?? ? -Denkliği
Bu çalışmada, ilk olarak, çift küme dizilerinin kuvvetli asimptotik ℐ2 ? -denkliği, ?-asimptotik ℐ2 ? -denkliği, kuvvetli ?-asimptotik ℐ2 ? -denkliği kavramları tanımlandı. Daha sonra bu kavramlar arasındaki ilişkiler ve bazı özellikler incelendi. İkinci olarak, yine çift küme dizileri için asimptotik ℐ2 ? -istatistiksel denklik kavramı tanımlandı. Ayrıca, asimptotik ℐ2 ? -istatistiksel denklik kavramı ve kuvvetli ?-asimptotik ℐ2 ? - denkliği kavramı arasındaki ilişkiler incelendi.
f-Asymptotically ?? ? -Equivalence of Double Sequences of Sets
In this study, first, we present the concepts of strongly asymptotically ℐ2 ? -equivalence, ?-asymptotically ℐ2 ? -equivalence, strongly ?-asymptotically ℐ2 ? -equivalence for double sequences of sets. Then, we investigated some properties and relationships among this new concepts. After, we present asymptotically ℐ2 ? -statistical equivalence for double sequences of sets. Also we investigate relationships between asymptotically ℐ2 ? -statistical equivalence and strongly ?-asymptotically ℐ2 ? -equivalence.
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- Baronti M., and Papini P., 1986. Convergence of
sequences of sets. In: Methods of functional
analysis in approximation theory (pp. 133-155),
ISNM 76, Birkhäuser, Basel.
- Beer G., 1985. On convergence of closed sets in a
metric space and distance functions. Bulletin of
the Australian Mathematical Society, 31, 421-
432.
- Beer G., 1994. Wijsman convergence: A survey. SetValued
Analysis, 2 , 77-94.
- Das, P., Kostyrko, P., Wilczyński, W. and Malik, P.,
2008. ℐ and ℐ
∗-convergence of double
sequences. Mathematica Slovaca, 58(5),
605-620.
- Dündar, E., Ulusu, U. and Pancaroğlu, N., 2016.
Strongly ℐ2-convergence and ℐ2-lacunary Cauchy
double sequences of sets. The Aligarh Bulletin of
Mathematics, 35(1-2), 1-15.
- Dündar, E., Ulusu, U. and Aydın, B., 2017. ℐ2-
lacunary statistical convergence of double
sequences of sets. Konuralp Journal of
Mathematics, 5(1), 1-10.
- Dündar, E., Ulusu, U. and Nuray, F., On
asymptotically ideal invariant equivalence of
double sequences, (In review).
Fast, H., 1951. Sur la convergence statistique.
Colloquium Mathematicum, 2, 241-244.
- Khan, V. A. and Khan, N., 2013. On Some
ℐ-Convergent Double Sequence Spaces Defined
by a Modulus Function. Engineering, 5, 35-40.
- Kılınç, G. and Solak, İ., 2014. Some Double Sequence
Spaces Defined by a Modulus Function. General
Mathematics Notes, 25(2), 19-30.
- Kişi Ö., Gümüş H. and Nuray F., 2015. ℐAsymptotically
lacunary equivalent set
sequences defined by modulus function. Acta
Universitatis Apulensis, 41, 141-151.
- Kişi, Ö. and Nuray, F., 2013. A new convergence for
sequences of sets. Abstract and Applied
Analysis, 2013, 6 pages.
- Kostyrko P., Šalát T. and Wilczyński W., 2000.
ℐ-Convergence. Real Analysis Exchange, 26(2),
669-686.
- Kumar V. and Sharma A., 2012. Asymptotically
lacunary equivalent sequences defined by ideals
and modulus function. Mathematical Sciences,
6(23), 5 pages.
- Lorentz G., 1948. A contribution to the theory of
divergent sequences. Acta Mathematica, 80,
167-190.
- Maddox J., 1986. Sequence spaces defined by a
modulus. Mathematical Proceedings of the
Cambridge Philosophical Society, 100, 161-166.
- Marouf, M., 1993. Asymptotic equivalence and
summability. Int. J. Math. Math. Sci., 16(4), 755-
762.
- Mursaleen, M., 1983. Matrix transformation
between some new sequence spaces. Houston
Journal of Mathematics, 9, 505-509.
- Mursaleen, M. and Edely, O. H. H., 2009. On the
invariant mean and statistical convergence.
Applied Mathematics Letters, 22(11), 1700-
1704.
- Nakano H., 1953. Concave modulars. Journal of the
Mathematical Society Japan, 5 ,29-49.
- Nuray, F. and Savaş, E., 1994. Invariant statistical
convergence and ?-invariant statistical
convergence. Indian Journal of Pure and Applied
Mathematics, 25(3), 267-274.
- Nuray, F., Gök, H. and Ulusu, U., 2011.
ℐ?-convergence. Mathematical Communications,
16, 531-538.
- Nuray F. and Rhoades B. E., 2012. Statistical
convergence of sequences of sets. Fasiciculi
Mathematici, 49 , 87-99.
- Nuray, F., Ulusu, U. and Dündar, E., 2016. Lacunary
statistical convergence of double sequences of
sets. Soft Computing, 20, 2883-2888.
- Pancaroğlu Akın, N. and Dündar, E., 2018.
Asymptotica ℐ-Invariant Statistical Equivalence
of Sequences of Set Defined by a Modulus
Function. AKU Journal of Science Engineering,
18(2), 477-485.
- Pancaroğlu Akın, N., Dündar, E., and Ulusu, U., 2018.
Asymptotically ℐσθ-statistical Equivalence of
Sequences of Set Defined By A Modulus
Function. Sakarya University Journal of
Science, 22(6), 1857-1862.
- Pancaroğlu Akın, N., Wijsman lacunary ℐ2-invariant
convergence of double sequences of sets, (In
review).
- Pancaroğlu, N. and Nuray, F., 2013a. Statistical
lacunary invariant summability. Theoretical
Mathematics and Applications, 3(2), 71-78.
- Pancaroğlu N. and Nuray F., 2013b. On Invariant
Statistically Convergence and Lacunary Invariant
Statistically Convergence of Sequences of Sets.
Progress in Applied Mathematics, 5(2), 23-29.
- Pancaroğlu N. and Nuray F. and Savaş E., 2013. On
asymptotically lacunary invariant statistical
equivalent set sequences. AIP Conf. Proc.
1558(780) http://dx.doi.org/10.1063/1.4825609
- Pancaroğlu N. and Nuray F., 2014. Invariant Statistical
Convergence of Sequences of Sets with
respect to a Modulus Function. Abstract and
Applied Analysis, 2014, 5 pages.
- Patterson, R. F., 2003. On asymptotically
statistically equivalent sequences. Demostratio
Mathematica, 36(1), 149-153.
- Pehlivan S., and Fisher B., 1995. Some sequences
spaces defined by a modulus. Mathematica
Slovaca, 45, 275-280.
- Raimi, R. A., 1963. Invariant means and invariant
matrix methods of summability. Duke
Mathematical Journal, 30(1), 81-94.
- Savaş, E., 2013. On ℐ-asymptotically lacunary
statistical equivalent sequences. Advances in
Difference Equations, 111(2013), 7 pages
doi:10.1186/1687-1847-2013-111
- Savaş, E., 1989a. Some sequence spaces involving
invariant means. Indian Journal of Mathematics,
31, 1-8.
- Savaş, E., 1989b. Strongly ?-convergent sequences.
Bulletin of Calcutta Mathematical Society, 81,
295-300.
- Savaş, E. and Nuray, F., 1993. On ?-statistically
convergence and lacunary ?-statistically
convergence. Mathematica Slovaca, 43(3),
309-315.
- Schaefer, P., 1972. Infinite matrices and invariant
means. Proceedings of the American Mathematical
Society, 36, 104-110.
- Schoenberg I. J., 1959. The integrability of certain
functions and related summability methods.
American Mathematical Monthly, 66, 361-375.
- Tortop, Ş. and Dündar, E., 2018. Wijsman I2-
invariant convergence of double sequences of
sets. Journal of Inequalities and Special
Functions, 9(4), 90-100.
- Ulusu U. and Nuray F., 2013. On asymptotically
lacunary statistical equivalent set sequences.
Journal of Mathematics, Article ID 310438, 5
pages.
- Ulusu, U. and Dündar, E., 2014. I-lacunary statistical
convergence of sequences of sets. Filomat,
28(8), 1567-1574.
- Ulusu, U., Dündar, E. and Nuray, F., 2018. Lacunary
I_2-invariant convergence and some properties.
International Journal of Analysis and
Applications, 16(3), 317-327.
- Ulusu U. and Gülle E., 2019. Asymptotically ℐσequivalence
of sequences of sets, (In review).
- Wijsman R. A., 1964. Convergence of sequences of
convex sets, cones and functions. Bulletin
American Mathematical Society, 70, 186-188.
- Wijsman R. A., 1966. Convergence of Sequences of
Convex sets, Cones and Functions II.
Transactions of the American Mathematical
Society, 123(1) , 32-45.