non-Archimedean Normed Spaces Research Articles

Hyers-Ulam Stability of Cubic Mappings in Non-Archimedean Normed Spaces

We give a xed point approach to the generalized Hyers-Ulam stability of the cubic equation f(2x + y) + f(2x y) = 2f(x + y) + 2f(x y) + 12f(x) in non-Archimedean normed spaces. We will give an example to show that some known results in the stability of cubic functional equations in real normed spaces fail in non- Archimedean normed spaces. Finally, some applications of our results in non-Archimedean normed spaces over p-adic numbers will be exhibited.

Stability of Quartic Mappings in Non-Archimedean Normed Spaces

We establish a new method to prove Hyers-Ulam-Rassias stability of the quartic functional equation f(2x + y) + f(2x y) + 6f(y) = 4(f(x + y) + f(x y) + 6f(x)) in non-Archimedean normed linear spaces.

APPROXIMATELY ADDITIVE MAPPINGS IN NON-ARCHIMEDEAN NORMED SPACES

We establish a new strategy to study the Hyers-Ulam-Rassias stability of the Cauchy and Jensen equations in non-Archimedean normed spaces. We will also show that under some restrictions, every function which satisfies certain inequalities can be approximated by an additive mapping in non-Archimedean normed spaces. Some applications of our results will be exhibited. In particular, we will see that some results about stability and additive mappings in real normed spaces are not valid in non-Archimedean normed spaces.

Stability of Two Types of Cubic Functional Equations in Non-Archimedean Spaces

We prove the generalized stability of the cubic type functional equation $$f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x)$$ and another functional equation $$f(ax+y)+f(x+ay)=(a+1)(a-1)^{2}[f(x)+f(y)] +a(a+1)f(x+y),$$ where $a$ is an integer with $a \neq 0, \pm 1$ in the framework of non-Archimedean normed spaces.

A Mazur–Ulam theorem in non-Archimedean normed spaces

The classical Mazur–Ulam theorem which states that every surjective isometry between real normed spaces is affine is not valid for non-Archimedean normed spaces. In this paper, we establish a Mazur–Ulam theorem in the non-Archimedean strictly convex normed spaces.

On multi-orthogonal bases in finite-dimensional non-Archimedean normed spaces

The paper deals with the problem of the existence multi-orthogonal bases in finite-dimensional normed spaces over K, where K is a non-Archimedean complete valued field.

Stability of functional equations in non-Archimedean spaces

We prove the generalized Hyers-Ulam stability of the Cauchy functional equation f(x+y) = f(x)+f(y) and the quadratic functional equation f(x+ y) f(x - y) = 2f(x) + 2f(y) in non-Archimedean normed spaces.

A Fixed Point Theorem in Non-Archimedean Vector Spaces

A mapping $T$ defined on a normed linear space $X$ and taking values in $X$ is said to be contractive (nonexpansive) if whenever $x$ and $y$ are distinct points in $X, ||Tx - Ty|| < ||x - y||\;(||Tx - Ty|| \leqslant ||x - y||)$. In this paper we prove that every contractive mapping on a spherically complete non-Archimedean normed space has a unique fixed point.

A fixed point theorem in non-Archimedean vector spaces

A mapping T T defined on a normed linear space X X and taking values in X X is said to be contractive (nonexpansive) if whenever x x and y y are distinct points in X , | | T x − T y | | &gt; | | x − y | | ( | | T x − T y | | ⩽ | | x − y | | ) X,\,||Tx - Ty|| &gt; ||x - y||\;(||Tx - Ty|| \leqslant ||x - y||) . In this paper we prove that every contractive mapping on a spherically complete non-Archimedean normed space has a unique fixed point.

Nonarchimedean Normed Linear Spaces

Important theorems of valuation theory and classic functional analysis that have known analogues in the theory of nonarchimedean normed linear spaces over rank‐one valued fields are considered in the more general context of nonarchimedean normed linear spaces over fields of arbitrary rank. Maximal completeness, pseudocompleteness, and spherical completeness are shown to be equivalent. Versions of Riesz's lemma and the Hahn–Banach theorem are given. The utility of considering ordered (value) groups as being embedded in their Dedekind completions is demonstrated.

Probabilistic contractor couple and solutions for a system of nonlinear equations in non-archimedean menger probabilistic normed spaces

The Purpose of this paper is to introduce the concept of probabilistic contractor couple in non-Archimedean probabilistic normed spaces and to study the existence and uniqueness of solutions for a system of nonlinear operator equations with probabilistic contractor couples in non-Archimedean probabilistic normed spaces. The results presented in this paper improve and extend the corresponding results in [1–5].

Compact-like operators between non-archimedean normed spaces

The Fredholm theory for compact operators on a non-archimedean Banach space E, as recently developed by W. Schikhof, does not work if the hypothesis of the completeness of E is dropped. This observation led the authors to introduce two new ideals of operators between non-archimedean normed spaces which, in the case of Banach spaces coincide with the ideal of the compact operators. They also investigate in various ways the possible equality of the three operator ideals.

Non-archimedean Chebyshev centers

In a recent paper, M.Z.M.C. Soares [3] introduced the concepts of Chebyshev radii and centers in a non-archimedean normed space E and posed some of the main problems in the non-archimedean theory of best simultaneous approximation. In this paper we prove, within the more general context of ultrametric spaces, the following results about Soares's problems: E always admits Chebyshev centers. W is proximinal if and only if W verifies the relative Chebyshev center property in E. We show that the relative Chebyshev center property is closely related to spherical completeness. As a consequence of our results, we generalize some of the main theorems of [3].

On the topology of simple convergence in non-Archimedean function spaces

Let X be an ultraregular Hausdorff space, F a non-trivial complete nonArchimedean valued field and E a non-Archimedean F-locally convex space over F. Let C,(X, E) denote the space of all continuous E-valued functions on X equipped with the topology of simple convergence. In this paper, we study some of the properties of the space C,(X, E). We look at the problems: When is this space barrelled, quasi-barrelled or bornological? For the bornologicity of C,(X, E) we show that, when E is a non-Archimedean normed space, the space C,(X, E) is bornological iff X is N-replete. This is analogous to the result of Govaerts [4] for the space C,(X, F). We also characterize the strongly bounded subsets of the dual space of C,(X, E). In case E is a non-Archimedean normed space, it is shown that:

On the best approximation property in non-archimedean normed spaces

On the best approximation property in non-archimedean normed spaces