in this paper, we solve the quadratic $alpha$-functional equations $2f(x) + 2f(y) = f(x + y) + alpha^{-2}f(alpha(x-y)); (0.1)$ where $alpha$ is a fixed non-archimedean number with $alpha^{-2}neq 3$. using the fixed point method and the direct method, we prove the hyers-ulam stability of the quadratic $alpha$-functional equation (0.1) in non-archimedean banach spaces.

and Applied Analysis 3 The condition of right K-completeness for X, p leaves outside the scope of this theorem an important class of asymmetric normed spaces, the asymmetric normed spaces associated to normed lattices because these spaces are right K-complete only for the trivial case 13 . In this paper, we give a uniform boundedness type theorem in the setting of asymmetric normed spaces which...

For simplicity, we follow the rules: X, X1 denote sets, r, s denote real numbers, z denotes a complex number, R1 denotes a real normed space, and C1, C2, C3 denote complex normed spaces. Let X be a set, let C2, C3 be complex normed spaces, and let f be a partial function from C2 to C3. We say that f is uniformly continuous on X if and only if the conditions (Def. 1) are satisfied. (Def. 1)(i) X...

We show that if the Banach-Mazur distance between an n-dimensional normed space X and l∞ is at most 3/2, then there exist n+ 1 equidistant points in X. By a well-known result of Alon and Milman, this implies that an arbitrary n-dimensional normed space admits at least e √ log n equidistant points, where c > 0 is an absolute constant. We also show that there exist n equidistant points in spaces ...

A new constantWD(X) is introduced into any real 2n-dimensional symmetric normed space X . By virtue of this constant, an upper bound of the geometric constant D(X), which is used to measure the difference between Birkhoff orthogonality and isosceles orthogonality, is obtained and further extended to an arbitrarym-dimensional symmetric normed linear space (m≥ 2). As an application, the result is...

The purpose of this paper is to establish the Hyers-UlamRassias stability of quartic functional equation (3 ) ( 3 ) 64 ( ) 64 ( ) 24 ( ) 6 ( ) f x y f x y f x f y f x y f x y          in the setting of random normed space and intuitionistic random normed space. The stability of the equation is proved by using the fixed point method and direct method.

‎it is well known that every (real or complex) normed linear space $l$ is isometrically embeddable into $c(x)$ for some compact hausdorff space $x$‎. ‎here $x$ is the closed unit ball of $l^*$ (the set of all continuous scalar-valued linear mappings on $l$) endowed with the weak$^*$ topology‎, ‎which is compact by the banach--alaoglu theorem‎. ‎we prove that the compact hausdorff space $x$ can ...

1.1. Introducton to Banach Spaces Definition 1.1. Let X be a K–vector space. A functional p ∶ X → [0,+∞) is called a seminorm, if (a) p(λx) = ∣λ∣p(x), ∀λ ∈ K, x ∈X, (b) p(x + y) ≤ p(x) + p(y), ∀x, y ∈X. Definition 1.2. Let p be a seminorm such that p(x) = 0 ⇒ x = 0. Then, p is a norm (denoted by ∥ ⋅ ∥). Definition 1.3. A pair (X, ∥ ⋅ ∥) is called a normed linear space. Lemma 1.4. Each normed sp...