Some Geometric Characterizations of a Fractional Banach Set

Abstract

This paper is devoted to investigate the modular structure of a fractional Banach set of sequences and prove that this set is reflexive and convex and it possesses uniform Opial, $( \beta )$, $ (L) $ and $ (H) $ properties. The convexity of the set is investigated by the notion of extreme points. These properties play an important role both in the study of fixed point theory and in the geometric characterizations of the Banach sets of sequences. This study extends the scope of the fractional calculus and it is related with fixed point and approximation theories.

Keywords

• reflexivity,convexity,fixed point property,approximative compactness

References

1. Altay, B. and Başar, F., Some paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math. 30(5) (2006), 591-608.
2. Altay, B. and Başar, F., The fine spectrum and the matrix domain of the difference operator Δ on the sequence space ℓ_{p}, (0
3. Aydın, C. and Başar, F., Some generalizations of the sequence space a_{p}^{r}, Iran. J. Sci. Technol. Trans. A Sci. 30(A2) (2006), 175-190.
4. Baliarsingh, P., Kadak, U. and Mursaleen, M., On statistical convergence of difference sequences of fractional order and related Korovkin type approximation theorems, Quaest. Math. (2018), doi:10.2989/16073606.2017.1420705.
5. Baliarsingh, P. and Dutta, S., On the classes of fractional order of difference sequence spaces and matrix transformations, Appl. Math. Comput. 250 (2015), 665-674.
6. Banaś, J., On modulus of noncompact convexity and its properties, Canad. Math. Bull. 30(2) (1987), 186-192.
7. Başar, F., Summability theory and its applications, Bentham Science Publishers. e-books, Monographs, Istanbul, 2012 ISBN: 978-1-60805-420-6.
8. Başar, F. and Altay, B., On the space of sequences of p-bounded variation and related matrix mappings, (English, Ukrainian summary) Ukrain. Mat. Zh. 55(1) (2003), 108--118; reprinted in Ukrainian Math. J. 55(1) (2003), 136-147.

9. Başar, F. and Kirişçi, M., Almost convergence and generalized difference matrix, Comput. Math. Appl. 61 (2011), 602-611.
10. Başar, F. and Braha, N.L., Euler-Cesàro difference spaces of bounded, convergent and null sequences, Tamkang J. Math. 47(4) (2016), 405-420.
11. Başarır, M. and Kayıkçı, M., On the generalized B^{m}-Riesz difference sequence space and β-property, J. Inequal. Appl. (2009), doi:10.1155/2009/385029.
12. Braha, N.L., Geometric properties of the second-order Cesàro spaces, Banach J. Math. Anal. 10(1) (2016), 1-14.
13. Candan, M., Almost convergence and double sequential band matrix, Acta Math. Sci. Ser. B Engl. Ed. 34(2) (2014), 354-366.
14. Cui, Y., Hudzik, H., Kaczmarek, R., Ma, H., Wang, Y. and Zhang, M., On some applications of geometry of Banach spaces and somen new results related to the fixed point theory in Orlicz sequence spaces, J. Math. Study 49(4) (2016), 325-378.
15. Cui, Y. and Hudzik, H., On the uniform Opial property in some modular sequence spaces, Functiones et Approximatio Commentarii Mathematici 26 (1998), 93-102.
16. Ercan, S. and Bektaş, Ç., On new convergent difference BK-spaces, J. Comput. Anal. Appl. 23(5) (2017), 793-801.
17. Et, M., Karataş, M. and Karakaya, V., Some geometric properties of a new difference sequence space defined by de la Vallée-Poussin mean, Appl. Math. Comput. 234 (2014), 237-244.
18. Gurarii, V.I., Differential properties of the convexity moduli of Banach spaces, Matematicheskie Issledovaniya 2(1) (1967), 141-148.
19. Hudzik, H., Kowalewski, W. and Lewicki, G., Approximative compactness and full rotundity in Musielak-Orlicz spaces and Lorentz-Orlicz spaces, Z. Anal. Anwend. 25(2) (2006), 163-192.
20. Huff, R., Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), 473-749.
21. Karaisa, A. and Özger, F., Almost difference sequence space derived by using a generalized weighted mean, J. Comput. Anal. Appl. 19(1) (2015), 27-38.
22. Karaisa, A. and Özger, F., On almost convergence and difference sequence spaces of order m with core theorems, Gen. Math. Notes 26(1) (2015), 102-125.
23. Malkowsky, E. and Özger, F., A note on some sequence spaces of weighted means, Filomat 26(3) (2012), 511-518.
24. Malkowsky, E., Özger, F. and Alotatibi, A., Some notes on matrix mappings and their Hausdorff measure of noncompactness, Filomat 28(5) (2014), 1059-1072.
25. Malkowsky, E. and Özger, F., Compact operators on spaces of sequences of weighted means, AIP Conf. Proc. 1470 (2012), 179-182.
26. Malkowsky, E., Özger, F. and Veličković, V., Some spaces related to Cesàro sequence spaces and an application to crystallography, MATCH Commun. Math. Comput. Chem. 70(3) (2013), 867-884.
27. Malkowsky, E., Özger, F. and Veličković, V., Some mixed paranorm spaces, Filomat 31(4) (2017), 1079-1098.
28. Malkowsky, E., Özger, F. and Veličković, V., Matrix transformations on mixed paranorm spaces, Filomat 31(10) (2017), 2957-2966.
29. Mongkolkeha, C. and Kumam, P., Some geometric properties of Lacunary sequence spaces related to fixed point property, Abstr. Appl. Anal. (2011), doi:10.1155/2011/903736
30. Montesinos, V., Drop property equals reflexivity, Studia Math. 87(1) (1987), 93-100.
31. Mursaleen, M., Başar, F. and Altay, B., On the Euler sequence spaces which include the spaces ℓ_{p} and ℓ₁ II, Nonlinear Anal. 65(3) (2006), 707-717.
32. Özger, F., Compact operators on the sets of fractional difference sequences, Bull. Math. Anal. Appl. (2018), to be appear.
33. Özger, F. and Başar, F., Domain of the double sequential band matrix B(r,s) on some Maddox's spaces, Acta Math. Sci. Ser. B Engl. Ed. 34(2) (2014), 394-408.
34. Özger, F. and Başar, F., Domain of the double sequential band matrix B(r,s) on some Maddox's spaces, AIP Conf. Proc. 1470 (2012), 152-155.
35. Radon, J., Theorie und Anwendungen der absolut additiven Mengenfunktionen, Sitz. Akad. Wiss. Wien 122 (1913), 1295-1438.
36. Rolewicz, S., On drop property, Studia Math., 85 (1987) 25-35. rolew : Rolewicz, S., On D-uniform convexity and drop property. Studia Math. 87 (1987), 181-191.
37. Sanhan, W. and Suantai, S., Some geometric properties of Cesàro sequence space, Kyungpook Math. J. 43(2) (2003), 191--197.
38. Veličković, V., Malkowsky, E. and Özger, F., Visualization of the spaces W(u,v;ℓ_{p}) and their duals, AIP Conf. Proc. 1759 (2016), doi: 10.1063/1.4959634.
39. Yeşilkayagil, M. and Başar, F., Some topological properties of almost null and almost convergent sequences. Turk J. Math. 40 (2016), 624-630.