APPROXIMATION OF A CAUCHY-JENSEN ADDITIVE FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN NORMED SPACES

Using the fixed point and direct methods, we prove the Hyers-Ulam stability of the following Cauchy-Jensen additive functional equation 2f（p∑i=1xi＋q∑j=1yj＋2d∑k=1zk/2）=p∑i=1f（xi）＋q∑j=1f（yj）＋2d∑k=1f（zk）,where p, q, d are integers greater than 1, in non-Archimedean normed spaces.

Stability of a cubic functional equation in intuitionistic random normed spaces

In this paper, the stability of a cubic functional equation in the setting of intuitionistic random normed spaces is proved. We first introduce the notation of intuitionistic random normed spaces. Then, by virtue of this notation, we study the stability of a cubic functional equation in the setting of these spaces under arbitrary triangle norms. Furthermore, we present the interdisciplinary relation among the theory of random spaces, the theory of intuitionistic spaces, and the theory of functional equations.

ISOMETRY AND PHASE-ISOMETRY OF NON-ARCHIMEDEAN NORMED SPACES

In this paper,we study isometries and phase-isometries of non-Archimedean normed spaces.We show that every isometry f:Sr(X)→Sr(X),where X is a finite-dimensional non-Archimedean normed space and Sr(X)is a sphere with radius r∈||X||,is surjective if and only if is spherically complete and k is finite.Moreover,we prove that if X and Y are non-Archimedean normed spaces over non-trivially non-Archimedean valued fields with|2|=1,any phase-isometry f:X→Y is phase equivalent to an isometric operator.

Nonlinear Approximation in Linear 2 Normed Spaces

The best approximation, best Σ approximation and best sup approximation in linear 2 normed spaces are discussed in this paper. The characterization and uniqueness theorems for these nonlinear approximations are given.

ON L^p-MATRICIALLY NORMED SPACES

It is proved that there is only one L^P-matricially normed space of dimension 1 and that quotient spaces of L^P-matricially normed spaces are also L^P-matricially normed spaces. Some properties of L^P-matricially normed spaces are given.

NONLINEAR SEMIGROUPS AND DIFFERENTIAL INCLUSIONSIN PROBABILISTIC NORMED SPACES

The purpose of this paper is to introduce and study the semi-groups of nonlinear contractions in probabilistic normed spaces and to establish the Crandall-Liggett's exponential formula for some kind of accretive mappings in probabilistic normed spaces. As applications, these results are utilized to study the Cauchy problem for a kind of differential inclusions with accretive mappings in probabilistic normed spaces.

Solution and stability of mixed type functional equation in non-Archimedean random normed spaces

The generalized stability of the Euler-Lagrange quadratic mappings in the framework of non-Archimedean random normed spaces is proved. The interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean spaces, and the theory of functional equations is presented.

VARIOUS NOTIONS OF ORTHOGONALITY IN NORMED SPACES

In this paper, we present various notions and aspects of orthogonality in normed spaces. Characterizations and generalizations of orthogonality are also consid- ered. Results on orthogonality of the range and the kernel of elementary operators and the operators implementing them also are given.

ON THE EXISTENCE OF SOLUTIONS FOR EQUATIONS WITH ACCRETIVE MAPPINGS IN PROBABILISTIC NORMED SPACES

The purpose of this paper is to introduce the concept of accretive mapping in probabilistic normed space (PN space, in short) and to study the existence problem of solutions for equations with accretive mappings in PN space and some existence theorems are shown.

Fixed Point Theorem of{a, b, c}Contraction and Nonexpansive Type Mappings in Weakly Cauchy Normed Spaces

Let C be a closed convex weakly Cauchy subset of a normed space X. Then we define a new {a,b,c} type nonexpansive and {a,b,c} type contraction mapping T from C into C. These types of mappings will be denoted respectively by {a,b,c}-ntype and {a,b,c}-ctype. We proved the following： 1. If T is {a,b,c}-ntype mapping, then inf{ ｜｜ T（x）-x｜｜ ：x C C} =0, accordingly T has a unique fixed point. Moreover, any sequence {Xn}n∈NN in C with limn→∞｜｜T（xn） - Xn｜｜ = 0 has a subsequence strongly convergent to the unique fixed point of T. 2. If T is {a,b,c}-ctype mapping, then T has a unique fixed point. Moreover, for any x∈C the sequence of iterates {Tn （x）}n∈N has subsequence strongly convergent to the unique fixed point of T. This paper extends and generalizes some of the results given in [2,4, 7] and [13].

On Ideal Convergence of Double Sequences in Probabilistic Normed Spaces

The notion of ideal convergence is a generalization of statistical convecgence which has been intensively investigated in last few years. For an admissible ideal ∮ C N × N, the aim of the present paper is to introduce the concepts of ∮-convergence and ：∮＊-convergence for double sequences on probabilistic normed spaces （PN spaces for short）. We give some relations related to these notions and find condition on the ideal ∮ for which both the notions coincide. We also define ∮-Cauchy and ：∮＊- Cauchy double sequences on PN spaces and show that ∮-convergent double sequences are ∮-Cauchy on these spaces. We establish example which shows that our method of convergence for double sequences on PN spaces is more general.

The structure of weak Pareto solution sets in piecewise linear multiobjective optimization in normed spaces

In general normed spaces, we consider a multiobjective piecewise linear optimization problem with the ordering cone being convex and having a nonempty interior. We establish that the weak Pareto optimal solution set of such a problem is the union of finitely many polyhedra and that this set is also arcwise connected under the cone convexity assumption of the objective function. Moreover, we provide necessary and sufficient conditions about the existence of weak (sharp) Pareto solutions.

Functional Inequalities in Non-Archimedean Normed Spaces

In this paper, we introduce an additive functional inequality and a quadratic functional inequality in normed spaces, and prove the Hyers-Ulam stability of the functional inequalities in Banach spaces. Furthermore, we introduce an additive functional inequality and a quadratic functional inequality in non-Archimedean normed spaces, and prove the Hyers-Ulam stability of the functional inequalities in non-Archimedean Banach spaces.

The Related Properties of Generalized Orthogonal Group in Specific Normed Linear Spaces

Firstly, in the general normed linear space, the concepts of generalized isosceles orthogonal group, generalized Birkhoff orthogonal group, generalized Roberts orthogonal group, strong Birkhoff orthogonal group and generalized orthogonal basis are introduced. Secondly, the conclusion that any two nonzero generalized orthogonal groups must be linearly independent group is proven. And the existence of nonzero generalized orthogonal group and its linear correlation are discussed preliminarily, as well as some related properties of nonempty generalized orthogonal group in specific normed linear space namely the lp space.

CHEBYSHEV CENTERS PROXIMINALITY AND FARTHEST POINTS IN STRONG NORMED ALMOST LINEAR SPACES

Some results from the theory of best (or best simultaneous) approximation in a narmed linear space have been extended to a normed almost linear space [strong normed almost linear space].

Volume of unit ball in an n-dimensional normed space and its asymptotic properties

Let Bp^n={x∈R^b｜‖x‖p≤1} be the unit ball of p norm in the n-dimensional normed space εp＆n.The formula for the volume of Bp^n was obtained and its asymptotic properties were found out as n→∞and p→∞.

Generalized contraction mapping principle in intuitionistic Menger spaces and application to differential equations

Using the idea of Atanassov, we define the notion of intuitionistic Menger spaces as a netural generalizations of Menger spaces due to Menger. We also obtain a new generalized contraction mapping and utilize this contraction mapping to prove the existance theorems of solutions to differential equations in intuitionistic Menger spaces.

BANACH SPACES AND INEQUALITIES ASSOCIATED WITH NEW GENERALIZATION OF CESàRO MATRIX

Let the triangle matrix A^(ru)be a generalization of the Cesàro matrix and U∈{c_(0),c,l_(∞)}.In this study,we essentially deal with the space U(A^(ru))defined by the domain of A^(ru)in the space U and give the bases,and determine the Kothe-Toeplitz,generalized K?theToeplitz and bounded-duals of the space U(A^(ru)).We characterize the classes(l_(∞)(A^(ru)):l_(∞)),(l_(∞)(A^(ru)):c),(c(A^(ru)):c),and(U:V(A^(ru)))of infinite matrices,where V denotes any given sequence space.Additionally,we also present a Steinhaus type theorem.As an another result of this study,we investigate the l_(p)-norm of the matrix A^(ru)and as a result obtaining a generalized version of Hardy's inequality,and some inclusion relations.Moreover,we compute the norm of well-known operators on the matrix domain l_(p)(A^(ru)).

DERIVATIVES OF HARMONIC MIXED NORM AND BLOCH-TYPE SPACES IN THE UNIT BALL OF R^n

Let H（B） be the set of all harmonic functions f on the unit ball B of Rn. For 0 p, q ≤ ∞ and a normal weight φ, the mixed norm space Hp,q, φ（B） consists of all functions f in H（B） for which the mixed norm p,q, φ ∞. In this article, we obtain some characterizations in terms of radial, tangential, and partial derivative norms in Hp,q, φ（B）. The parallel results for the Bloch-type space are also obtained. As an application, the analogous problems for polyharmonic functions are discussed.

Angles of Norm ed Linear Spaces and Characterizations of Inner Product Spaces

In [1],a family of angles are defined in normed linear spaces. In this paper,it is shown that if anyone of the angles satisfies certain euclidean triangle congruence properties,the space must be an inner product space.