Asymptotic Stability of the Pexider–Cauchy Functional Equation in Non-Archimedean Spaces

In this paper, we investigated the asymptotic stability behaviour of the Pexider–Cauchy functional equation in non-Archimedean spaces. We also showed that, under some conditions, if ∥f(x+y)−g(x)−h(y)∥⩽ε, then f,g and h can be approximated by additive mapping in non-Archimedean normed spaces. Finally, we deal with a functional inequality and its asymptotic behaviour.

Interlingual competition and functional incompleteness of languages.

The state of mass multilingualism has been developed now in many countries of the world, and not only in post-colonial ones. Depending on the situation, most contemporaries use (actively or passively) two or three languages. The norm today is not monolingualism, but multilingualism. At the same time many societies are characterized by bilingualism not balanced, but vertical — diglossia. Since few people speak several languages equally well, the need to use them interchangeably requires additional effort and causes mental fatigue. The state of monolingualism is more usual and comfortable for a person. Language situations in society also favor the use of predominantly one language at the expense of others. Therefore, diglossia cannot last forever. Those sociolinguists are right who regard it as a temporary condition, an intermediate stage in the transition «primary monolingualism → bilingualism → secondary monolingualism». Such a transition is an inevitable consequence of the victory of a communicatively strong language over a communicatively weak one. If we evaluate the interaction of languages from a functional point of view, it should be recognized that their essence boils down to competition. It arises due to the fact that languages are not distributed once and for all in certain areas, are not ultimately tied to a certain circle of speakers. Languages always rise at the expense of the decline of other ones. If a language expands its area of use, it means that another language leaves this area, and therefore reduces the scope of its use. Of the two competing languages, the winner is the one with the greatest communicative power. The phenomenon when a language ceases to be used in a certain communicative sphere, it is appropriate to denote by the term loss of functionality. A language that does not fulfill all the functions that should be performed by a developed literary language should be recognized as incompletely functional. The current spread of English as a single world language, its dominance in the most prestigious spheres of communication (politics, economics, trade, science, education, culture, the Internet) leads to the fact that national languages are gradually displaced from these spheres, marginalized and eventually devalued. The process can become irreversible and lead to their complete decline as functionally weak. Even those languages that also claim to be global (French, Spanish) or interstate (German, Portuguese, Russian) are losing the competition. As a result of competition, languages begin to differ in the richness of their vocabulary, the elaboration of their syntactic structure, the development of the style system, their prevalence, and their social status. But the main result is functional inequality. A language that has a large communicative load is used in a larger number of areas (or in prestigious areas) and turns out to be functionally dominant. The one that exists in fewer spheres or is used with less intensity turns out to be functionally incomplete. As a result, it turns out to be unable to satisfy all the information needs of its speakers — they are forced to resort to languages with greater communicative power.

On new stability results for composite functional equations in quasi-β-normed spaces

In this article, we prove the generalized Hyers-Ulam-Rassias stability for the following composite functional equation: f ( f ( x ) − f ( y ) ) = f ( x + y ) + f ( x − y ) − f ( x ) − f ( y ) , f(f\left(x)-f(y))=f\left(x+y)+f\left(x-y)-f\left(x)-f(y), where f f maps from a ( β , p ) \left(\beta ,p) -Banach space into itself, by using the fixed point method and the direct method. Also, the generalized Hyers-Ulam-Rassias stability for the composite s s -functional inequality is discussed via our results.

The functional inequality for the mixed quermassintegral

In this paper, the functional Quermassintegrals of a log-concave function in $\mathbb{R}^{n}$ R n are discussed. The functional inequality for the ith mixed Quermassintegral is established. Moreover, as a special case, a weaker log-Quermassintegral inequality in $\mathbb{R}^{n}$ R n is obtained.

On the regularization of the Lagrange principle and on the construction of the generalized minimizing sequences in convex constrained optimization problems

We consider the regularization of the Lagrange principle (LP) in the convex constrained optimization problem with operator constraint-equality in a Hilbert space and with a finite number of functional inequality-constraints. The objective functional of the problem is not, generally speaking, strongly convex. The set of admissible elements of the problem is also embedded into a Hilbert space and is not assumed to be bounded. Obtaining a regularized LP is based on the dual regularization method and involves the use of two regularization parameters and two corresponding matching conditions at the same time. One of the regularization parameters is «responsible» for the regularization of the dual problem, while the other is contained in a strongly convex regularizing addition to the objective functional of the original problem. The main purpose of the regularized LP is the stable generation of generalized minimizing sequences that approximate the exact solution of the problem by function and by constraint, for the purpose of its practical stable solving.

Approximation Properties of Solutions of a Mean Value-Type Functional Inequality, II

Let X be a commutative normed algebra with a unit element e (or a normed field of characteristic different from 2), where the associated norm is sub-multiplicative. We prove the generalized Hyers-Ulam stability of a mean value-type functional equation, f(x)−g(y)=(x−y)h(sx+ty), where f,g,h:X→X are functions. The above mean value-type equation plays an important role in the mean value theorem and has an interesting property that characterizes the polynomials of degree at most one. We also prove the Hyers-Ulam stability of that functional equation under some additional conditions.