New types of convergence and asymptotically equivalent functions on time scales

In this study, some concepts related to statistical convergence and summability theory given for number sequences in the literature are studied on arbitrary time scales. This thesis consists of six chapters. The first chapter is devoted to the introduction, and brief information is given about the historical development of summability theory and time scale theory. In the second chapter, some basic definitions, theorems and examples regarding the theory of summability and time scales are given. The other chapters contain the original results of the thesis. In the third chapter, new convergence types are presented for the real valued delta measurable functions defined on time scales. In the first part of this chapter, the concepts of αβ-statistical convergence and strong αβ-Cesàro summability are introduced and the relationships between these concepts are examined. In the second part of this chapter, the concepts of f-lacunary statistical convergence and strong f-lacunary Cesàro summability are defined by using modulus functions and some results related to these concepts are obtained. In the fourth chapter, by using two positive real valued delta measurable functions defined on time scales, the concepts of asymptotically statistical equivalence, asymptotically lacunary statistical equivalence, asymptotically f-statistical equivalence, asymptotically f-lacunary statistical equivalence, strong asymptotically lacunary equivalence and strong asymptotically f-lacunary equivalence are introduced. In addition, some results related to these concepts are obtained. In the fifth chapter, the concept of lacunary statistical boundedness on time scales is introduced. In addition, the relationships of this concept with the concept of lacunary statistical convergence and the concept of statistical boundedness has been examined. The last section is devoted to the conclusion and recommendations.