Non-Newtonian Sequence Spaces with Applications

Auteur

Feyzi Bäar is a Professor Emeritus since July 2016, at Inönü University, Türkiye. He received his PhD degree from Ankara University, Türkiye, in 1986. He has published four e-books for graduate students and researchers, and more than 160 scientific papers in the field of summability theory, sequence spaces, FK-spaces, Schauder bases, dual spaces, matrix transformations, spectrum of certain linear operators represented by a triangle matrix over some sequence space, the -, ß- and -duals, and some topological properties of the domains of some double and four-dimensional triangles in the certain spaces of single and double sequences, sets of the sequences of fuzzy numbers, multiplicative calculus. He has guided 17 master and 10 Ph.D. students., served as a referee 148 international scientific journals. He is the member of editorial board of 21 scientific journals. Feyzi Bäar is also a member of scientific committee of 17 mathematics conference, gave talks at 14 different universities as invited speaker and participated more than 70 mathematics symposium with a paper.

Bipan Hazarika is a Professor in the Department of Mathematics at Gauhati University, Guwahati, Assam. He has worked at Rajiv Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh, India from 2005 to 2017. He was Professor at Rajiv Gandhi University upto 10-08-2017. He received his PhD degree from Gauhati University, Guwahati, India. His main research interests are the _eld of sequences spaces, summability theory, applications fixed point theory, fuzzy analysis, function spaces of non absolute integrable functions. He has published over 200 research papers in several international journals. He is a regular reviewer of more than 50 different journals. He has published books on Differential Equations, Differential Calculus and Integral Calculus. He has edited books in CRC press on Sequence Space and Advances Mathematical Analysis, "Advances in Mathematical Analysis and its Applications", "Dynamic Equations on Time Scales and Applications" and a book on Fractional Differential Equations and Fixed point theory, Approximation Theory, Sequence Spaces and Applications (Industrial and Applied Mathematics), Advances in Functional Analysis and Fixed-Point Theory: An Interdisciplinary Approach (Industrial and Applied Mathematics). In 2022, 2023 and 2024 he was listed among the world's top 2% scientists by Stanford University. He is an editorial board member of more than 5 International journals and Guest Editor of the special issue Sequence spaces, Function spaces and Approximation Theory, Journal of Function Spaces.

Texte du rabat

Non-Newtonian Sequence Spaces with Applications presents an alternative to the usual calculus based on multiplication instead of addition.

Résumé
Non-Newtonian Sequence Spaces with Applications presents an alternative to the usual calculus based on multiplication instead of addition.

Contenu

Preface vii
Acknowledgements ix
List of Abbreviations and Symbols x
1 Sequence and Function Spaces over the Non-newtonian ... 1
1.1 Some Basic Results on the Spaces of Sequences ... . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Preliminaries, background and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Geometric complex field and related properties . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3 Geometric metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.4 Convergence and completeness in (GC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.5 Sequence spaces over C(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Some Results on Sequence Spaces with ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.1 Preliminaries, backround and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.2 Non-newtonian real field and related properties . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.3 Non-newtonian metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.4 Convergence and completeness in (NC) . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3 Sequence Spaces Over the Non-newtonian ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 Certain Non-newtonian Complex Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 Preliminaries, background and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.6 Some Sequence Spaces and Matrix Transformations in ... . . . . . . . . . . . . . . . . . . . . . . 29
1.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.6.2 Preliminaries, background and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.6.3 Characterizations of some matrix classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6.4 Multiplicative dual summability methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2 Application of Geometric Calculus in Numerical Analysis and Difference Sequence Spaces 39
2.1 Introduction and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 -generator and Geometric Real Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.1 Some useful relations between geometric operations and ordinary arithmetic operations . 40
2.3 Geometric Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Dual Spaces of G
( G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.1 Geometric form of Abel's partial summation formula . . . . . . . . . . . . . . . . . . . . 46
2.5 -, ß- and -duals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 Some Applications of Geometric Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6.1 Geometric Newton-Gregory backward interpolation formula . . . . . . . . . . . . . . . . 53
2.6.2 Advantages of geometric interpolation formulae over ordinary interpolation formulae . . 55
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3 Bigeometric Integral Calculus 56
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Geometric Arithmetic and Geometric Real Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.1 G-derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.2 Some standard G-derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
iv

3.4 G-Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.1 Some standard G-integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4.2 Integration by transforming the function to the form ex f (x)
f(x) . . . . . . . . . . . . . . . . 58
3.4.3 Integration by the relation between G-integral and ordinary integral . . . . . . . . . . . 58
3.4.4 Properties of G-integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5 Definite Bigeometric Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5.1 Propertie…