Non-Newtonian Sequence Spaces with Applications

Feyzi Başar is a Professor Emeritus since July 2016, at Inönü University, Türkiye. He received his PhD 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 α-, β-, and γ-duals, and some topological properties of the domains of some two- and four-dimensional triangles in certain spaces of single and double sequences, sets of the sequences of fuzzy numbers, and multiplicative calculus. He has guided 17 master and 10 PhD students and served as a referee for 148 international scientific journals. He is the member of editorial board of 21 scientific journals. Feyzi Başar is also a member of scientific committee of 17 mathematics conference. He gave talks at 14 different universities as invited speaker and participated more than 70 mathematics symposium with a paper.Bipan Hazarika is Professor in the Department of Mathematics at Gauhati University, Guwahati, Assam. He worked at Rajiv Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh, India from 2005 to 2017. He was Professor at Rajiv Gandhi University up to August 10, 2017. He received his PhD from Gauhati University, Guwahati, India. His main research interests include the field of sequences spaces, summability theory, applications fixed point theory, fuzzy analysis, and 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 published from Springer, Elsevier, Taylor and Francis, Wiley, IOS Press, World Scientific, American Mathematical Society, and De Gruyter. He has published books on Differential Equations, Differential Calculus, and Integral Calculus. He has edited books in CRC press on Sequence Spaces, Advances in Mathematical Analysis and its Applications, Dynamic Equations on Time Scales and Applications, and in Springer Nature 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 five International journals and a Guest Editor of the special issue Sequence Spaces, Function Spaces and Approximation Theory, Journal of Function Spaces.

• Preface viiAcknowledgements ixList of Abbreviations and Symbols x1 Sequence and Function Spaces over the Non-newtonian ... 11.1 Some Basic Results on the Spaces of Sequences ... . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Preliminaries, background and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Geometric complex field and related properties . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 Geometric metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.4 Convergence and completeness in (GC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.5 Sequence spaces over C(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2 Some Results on Sequence Spaces with ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.1 Preliminaries, backround and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.2 Non-newtonian real field and related properties . . . . . . . . . . . . . . . . . . . . . . . 131.2.3 Non-newtonian metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.4 Convergence and completeness in (NC) . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3 Sequence Spaces Over the Non-newtonian ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4 Certain Non-newtonian Complex Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 231.4.1 Preliminaries, background and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.6 Some Sequence Spaces and Matrix Transformations in ... . . . . . . . . . . . . . . . . . . . . . . 291.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.6.2 Preliminaries, background and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.6.3 Characterizations of some matrix classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.6.4 Multiplicative dual summability methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 Application of Geometric Calculus in Numerical Analysis and Difference Sequence Spaces 392.1 Introduction and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.2 α-generator and Geometric Real Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.2.1 Some useful relations between geometric operations and ordinary arithmetic operations . 402.3 Geometric Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.4 Dual Spaces of ℓG∞(□G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.4.1 Geometric form of Abel’s partial summation formula . . . . . . . . . . . . . . . . . . . . 462.5 α-, β- and γ-duals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.6 Some Applications of Geometric Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.6.1 Geometric Newton-Gregory backward interpolation formula . . . . . . . . . . . . . . . . 532.6.2 Advantages of geometric interpolation formulae over ordinary interpolation formulae . . 552.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 Bigeometric Integral Calculus 563.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2 Geometric Arithmetic and Geometric Real Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.3 Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.3.1 G-derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.3.2 Some standard G-derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57iv3.4 G-Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4.1 Some standard G-integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.4.2 Integration by transforming the function to the form ex f′(x)f(x) . . . . . . . . . . . . . . . . 583.4.3 Integration by the relation between G-integral and ordinary integral . . . . . . . . . . . 583.4.4 Properties of G-integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.5 Definite Bigeometric Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.5.1 Properties of definite G-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.5.2 Definite bigeometric integral as a limit of geometric sum . . . . . . . . . . . . . . . . . . 633.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664 Bigeometric Calculus and Its Applications 674.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.1.1 Some useful relations between geometric operations and ordinary arithmetic operations . 674.2 Definitions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2.1 Geometric binomial formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2.2 Geometric real number line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2.3 Geometric coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2.4 Geometric factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2.5 Generalized geometric forward difference operator □nG . . . . . . . . . . . . . . . . . . . . 694.2.6 Generalized Geometric Backward Difference Operator ∇nG . . . . . . . . . . . . . . . . . 694.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3.1 Geometric Pythagorean triplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3.2 Geometric trigonometric ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3.3 Relation between geometric trigonometry and ordinary trigonometry . . . . . . . . . . . 714.3.4 Geometric trigonometric identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.3.5 G-limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.3.6 G-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.4 Basic Properties of G-Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.4.1 G-derivative and its interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.4.2 Relation between G-derivative and ordinary derivative . . . . . . . . . . . . . . . . . . . 774.4.3 G-derivatives of some standard functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.4.4 Geometric Taylor’s series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.5 Some Applications of G-Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.5.1 Expansion of some useful functions in Taylor’s product . . . . . . . . . . . . . . . . . . . 834.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865 Solution of Bigeometric-Differential Equations by Numerical Methods 875.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.2 Basic Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.2.1 Geometric factorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.2.2 Geometric Newton-Gregory formula for forward interpolation . . . . . . . . . . . . . . . 885.2.3 Geometric Newton-Gregory formula for backward interpolation . . . . . . . . . . . . . . 885.2.4 G-derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.2.5 Some standard G-derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2.6 Geometric Taylor’s series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.3 Numerical Methods and Solution of G-Differential Equations . . . . . . . . . . . . . . . . . . . . 895.3.1 G-Euler’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.3.2 Taylor’s G-series method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.3.3 G-Runge-Kutta method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.3.4 G-Runge-Kutta method of order four . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996 Certain Spaces of Functions over the Set of Non-Newtonian Complex Numbers 1006.1 Preliminaries, Backround and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.2 The Set of ∗-Complex Numbers and ∗-Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.3 Continuous Function Space over the Field C∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 Multiplicative Type Complex Calculus 1107.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.2 Definitions, Methods, and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.2.1 A multiplicative group, an additive group, and an isomorphism . . . . . . . . . . . . . . 1117.2.2 Remoteness of two values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.2.3 Change rate of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147.2.4 Derivative and integral operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147.2.5 Euler’s simple method in differential equation solving . . . . . . . . . . . . . . . . . . . . 1177.2.6 Some fundamental theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1238 Function Sequences and Series ... 1248.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248.2 ∗-Function Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1268.2.1 ∗-function sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1268.2.2 ∗-function series and consequences of ∗-uniform convergence . . . . . . . . . . . . . . . . 1298.2.3 ∗-uniform convergence and ∗-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1328.2.4 ∗-uniform convergence and ∗-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1338.2.5 ∗-Uniform Convergence and ∗-Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 1369 On Non-newtonian Power Series and its Applications 1399.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1399.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409.2.1 ∗-Dirichlet’s and ∗-Abel’s tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409.2.2 ∗-power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Bibliography 150Index 153