Strong and weak convergences in 2-probabilistic normed spaces

Yıl 2021, , 454 - 466, 30.12.2021

Harikrishnan Panackal , Bernardo Guillen Ravi Agarwal , Hamid Moradi

https://doi.org/10.31197/atnaa.921900

Öz

In this paper, we have introduced the notions of strong and weak convergences in 2-probabilistic normed spaces (2-PN spaces) and established some of its properties. Later, we have defined the strong and weak boundedness of a linear map between two 2-PN spaces and proved a necessary and sufficient condition for the linear map between two 2-PN spaces to be strongly and weakly bounded.

Anahtar Kelimeler

Menger, linear 2-normed spaces, triangle functions, t-norms

Kaynakça

• [1] C. Alsina, B. Schweizer and A. Sklar, On the definition of a probabilistic normed space, Aequationes Math., 46, (1993) 91-98.
• [2] C. Alsina, B. Schweizer and A. Sklar, Continuity properties of probabilistic norms, J. Math. Anal. Appl., 208, (1997) 446-452.
• [3] N. Eghbali, Frechet differentiation between Menger probabilistic normed spaces, Proyecciones Journal of Mathematics, 33(4), (2014) 415-435.
• [4] L. Fatemeh and N. Kourosh, Compact operators defined on 2-normed and 2- probabilistic normed spaces, Mathematical Problems in Engineering, (2009).
• [5] I. Golet, On probabilistic 2-normed spaces, Novi Sad J. Math, vol. 35, no. 1,(2005) 95-102.
• [6] R. Haloi and M. Sen, µ-statistically convergent multiple sequences in probabilistic normed spaces, in Advances in Algebra and Analysis, Springer, (2018) 353-360.
• [7] P.K. Harikrishnan, B. Lafuerza-Guillen, K.T. Ravindran.: Compactness and D− boundedness in Menger's 2-Probabilistic Normed Spaces, FILOMAT, 30(5), 1263-1272 (2016).
• [8] P.K. Harikrishnan K. T. Ravindran.: Some Results Of Accretive Operators and Convex Sets in 2-Probabilistic Normed Space, Journal of Prime Research in Mathematics, 8, 76-84 (2012).
• [9] P.K. Harikrishnan, B. Lafuerza-Guillén,Yeol Je Cho, K. T. Ravindran, New classes of generalized PN Spaces and their Normability, Acta Mathematica Vietnamica, 42 (3) (2017), 727-746.
• [10] P.K. Harikrishnan Bernardo Lafuerza - Guillen, K. T. Ravindran Accretive operators and Banach Alogolu Theorem in Linear 2-normed spaces, Proyeccions Journal of Mathematics, Vol 30, No.3, (2011) 319-327.
• [11] B. Lafuerza-Guillén, A. Rodríguez Lallena and C. Sempi,A study of boundedness in probabilistic normed spaces, J. Math. Anal. Appl., 232, 183-196 (1999).
• [12] B. Lafuerza-Guillén.: D-bounded sets in probabilistic normed spaces and their products, Rend. Mat., Serie VII, 21, 17-28 (2001).
• [13] B. Lafuerza-Guillen, Carlo Sempi, Gaoxun Zhang.: A Study of Boundedness in Probabilistic Normed Spaces, Nonlinear Analysis, 73 , 1127-1135 (2010).
• [14] B. Lafuerza-Guillén, Panackal Harikrishnan.: Probabilistic Normed Spaces, Imperial College Press, World Scientific, UK, London (2014).
• 15] A. Pourmoslemi, M. Ferrara, B. A. Pansera, and M. Salimi, Probabilistic norms on the homeomorphisms of a group, Soft Computing, (2020) 1-8.
• [16] Raymond W. Freese,Yeol Je Cho, Geometry of linear 2-normed spaces, Nova Science publishers, Inc, Newyork,(2001). [17] M. Sen, S. Nath, and B.C. Tripathy, Best approximation in quotient probabilistic normed space, Journal of Applied Analysis, vol. 23, no. 1, (2017) 5-57.
• [18] B.C. Tripathy, M. Sen, and S. Nath, I-convergence in probabilistic n-normed space, Soft computing, 16 (6) (2012) 1021-1027.
• [19] B. Tripathy and R. Goswami, Statistically convergent multiple sequences in probabilistic normed spaces, Scientific Bulletin- Politehnica University of Bucharest Series A, Applied mathematics and physics, 78 (4), (2016) 83-94.