The notation and terminology used here are introduced in the following papers: [21], [22], [23], [3], [4], [2], [12], [18], [6], [1], [20], [7], [5], [8], [16], [14], [13], [15], [11], [19], [17], [9], and [10]. The following propositions are true: (1) Let R be a non empty subset of R and r0 be a real number. If for every real number r such that r ∈ R holds r ¬ r0, then supR ¬ r0. (2) Let X be ...

The definitions of new type separated subsets are given in ideal topological spaces. By using these definitions, we introduce types connectedness. It is shown that connectedness more general than some previously defined concepts compared with well-known point-set topology. Then, the intermediate value theorem for spaces given. Also, special cases, it and coincide.

In this note we present new proofs of three basic theorems in calculus. Although these theorems are well-known, in each proof we obtain something which seems to be unknown. We start with the Heine-Cantor theorem about uniform continuity and obtain explicitly the optimal delta for the given epsilon. We then proceed with the Weierstrass extreme value theorem and present two proofs of it: the “env...

The proofs of the extreme value theorem, the mean value theorem and the inverse function theorem for analytic functions on the Levi-Civita field will be presented. After reviewing convergence criteria for power series [15], we review their analytical properties [18, 20]. Then we derive necessary and sufficient conditions for the existence of relative extrema for analytic functions and use that ...

An introduction to recent work on analysis over the non-Archimedean Levi-Civita field related to applications for common numerical tasks is provided. After studying the algebraic, order, and topological properties, a calculus is developed under which central concepts like the intermediate value theorem, mean value theorem, and Taylor’s theorem with remainder hold under slightly stronger conditi...

in this study, we introduce the classes of $phi$-strongly pseudocontractive mappings in the intermediate sense and generalized $phi$-pseudocontractive mappings in the intermediate sense and prove the existence of fixed points for those maps. the results generalise the results of several authors in literature including xiang [chang he xiang, fixed point theorem for generalized $phi$-pseudocontra...

In this study, we introduce the classes of $phi$-strongly pseudocontractive mappings in the intermediate sense and generalized $Phi$-pseudocontractive mappings in the intermediate sense and prove the existence of fixed points for those maps. The results generalise the results of several authors in literature including Xiang [Chang He Xiang, Fixed point theorem for generalized $Phi$-pseudocontra...