Uniformly Continuous Research Articles

Fuzzy Continuous Mappings in Fuzzy Normed Linear Spaces

In this paper we continue the study of fuzzy continuous mappings in fuzzy normed linear spaces initiated by T. Bag and S.K. Samanta, as well as by I. Sadeqi and F.S. Kia, in a more general settings. Firstly, we introduce the notion of uniformly fuzzy continuous mapping and we establish the uniform continuity theorem in fuzzy settings. Furthermore, the concept of fuzzy Lipschitzian mapping is introduced and a fuzzy version for Banach’s contraction principle is obtained. Finally, a special attention is given to various characterizations of fuzzy continuous linear operators. Based on our results, classical principles of functional analysis (such as the uniform boundedness principle, the open mapping theorem and the closed graph theorem) can be extended in a more general fuzzy context.

Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation

We consider a statement of the Dirichlet problem which generalizes the notions of classical and weak solutions, in which a solution belongs to the space of -dimensionally continuous functions with values in the space . The property of -dimensional continuity is similar to the classical definition of uniform continuity; however, instead of the value of a function at a point, it looks at the trace of the function on measures in a special class, that is, elements of the space with respect to these measures. Up to now, the problem in the statement under consideration has not been studied in sufficient detail. This relates first to the question of conditions on the right-hand side of the equation which ensure the solvability of the problem. The main results of the paper are devoted to just this question. We discuss the terms in which these conditions can be expressed. In addition, the way the behaviour of a solution near the boundary depends on the right-hand side is investigated. Bibliography: 47 titles.

ϕ -statistically quasi Cauchy sequences

Abstract Let P denote the space whose elements are finite sets of distinct positive integers. Given any element σ of P, we denote by p(σ) the sequence {pn(σ)} such that p n ( σ ) = 1 for n ∈ σ and p n ( σ ) = 0 otherwise. Further P s = { σ ∈ P : ∑ n = 1 ∞ p n ( σ ) ≤ s } , i.e. Ps is the set of those σ whose support has cardinality at most s. Let (ϕn) be a non-decreasing sequence of positive integers such that n ϕ n + 1 ≤ ( n + 1 ) ϕ n for all n ∈ N and the class of all sequences (ϕn) is denoted by Φ. Let E ⊆ N . The number δ ϕ ( E ) = lim s → ∞ 1 ϕ s | { k ∈ σ , σ ∈ P s : k ∈ E } | is said to be the ϕ-density of E. A sequence (xn) of points in R is ϕ-statistically convergent (or Sϕ-convergent) to a real number l for every e > 0 if the set { n ∈ N : | x n − l | ≥ ɛ } has ϕ-density zero. We introduce ϕ-statistically ward continuity of a real function. A real function is ϕ-statistically ward continuous if it preserves ϕ-statistically quasi Cauchy sequences where a sequence (xn) is called to be ϕ-statistically quasi Cauchy (or Sϕ-quasi Cauchy) when ( Δ x n ) = ( x n + 1 − x n ) is ϕ-statistically convergent to 0. i.e. a sequence (xn) of points in R is called ϕ-statistically quasi Cauchy (or Sϕ-quasi Cauchy) for every e > 0 if { n ∈ N : | x n + 1 − x n | ≥ ɛ } has ϕ-density zero. Also we introduce the concept of ϕ-statistically ward compactness and obtain results related to ϕ-statistically ward continuity, ϕ-statistically ward compactness, statistically ward continuity, ward continuity, ward compactness, ordinary compactness, uniform continuity, ordinary continuity, δ-ward continuity, and slowly oscillating continuity.

On bounded continuous solutions of the archetypal equation with rescaling

The ‘archetypal’ equation with rescaling is given by y ( x ) = ∬ R 2 y ( a ( x − b ) ) μ ( d a , d b ) ( x ∈ R ), where μ is a probability measure; equivalently, y ( x ) = E { y ( α ( x − β ) ) } , with random α , β and E denoting expectation. Examples include (i) functional equation y ( x ) = ∑ i p i y ( a i ( x − b i ) ) ; (ii) functional–differential (‘pantograph’) equation y ′ ( x ) + y ( x ) = ∑ i p i y ( a i ( x − c i ) ) ( p i >0, ∑ i p i = 1 ). Interpreting solutions y ( x ) as harmonic functions of the associated Markov chain ( X n ), we obtain Liouville-type results asserting that any bounded continuous solution is constant. In particular, in the ‘critical’ case E { ln ⁡ | α | } = 0 such a theorem holds subject to uniform continuity of y ( x ); the latter is guaranteed under mild regularity assumptions on β , satisfied e.g. for the pantograph equation (ii). For equation (i) with a i = q m i ( m i ∈ Z , ∑ i p i m i = 0 ), the result can be proved without the uniform continuity assumption. The proofs exploit the iterated equation y ( x ) = E { y ( X τ ) | X 0 = x } (with a suitable stopping time τ ) due to Doob's optional stopping theorem applied to the martingale y ( X n ).

Strong continuity for the 2D Euler equations

We prove two results of strong continuity with respect to the initial datum for bounded solutions to the Euler equations in vorticity form. The first result provides sequential continuity and holds for a general bounded solution. The second result provides uniform continuity and is restricted to Hölder continuous solutions.

A Phenomenological Model of Brittle Rocks under Uniaxial Compression

A quantitative model was recently proposed by Peng et al (2015) to characterize the crack closure behavior of rocks. The model can simulate the initial nonlinearity of stress–strain curves in uniaxial and triaxial compressions. However, the crack propagation behavior near peak and in the post-peak deformation stage cannot be captured by the model. This paper extends the model to simulate the complete stress–strain curves of rocks under uniaxial compression. A phenomenological damage model, which uses a logistic function to describe the damage evolution, is adopted to model the pre-peak and post-peak deformation stages beyond the crack closure stage. Combining the crack closure model and the damage model yields a new phenomenological model. A uniform continuity condition is used to ensure that the stress–strain curve is smooth and continuous at the junction point of the crack closure model and the damage model. The proposed model has four model parameters, which can be calibrated using laboratory test data. The uniaxial compression tests of the Carrara marble under different heating cycles are simulated to verify the proposed model. The simulated stress–strain curves agree well with the test data, from initial crack closure to near peak and post peak, suggesting that the model can be used for simulating the entire deformation stage of brittle rocks with different degrees of initial microcrack damage.

Uniform Continuity of Continuous Functions on Compact Metric Spaces

A basic theorem asserts that a continuous function on a compact metric space with values in another metric space is uniformly continuous. The usual proofs based on a contradiction argument involving sequences or on the covering property of compact sets are quite sophisticated for students taking a first course on real analysis. We present a direct proof only using results that are established anyway in such an introductory course. Let f ∶ X → Y be a continuous function from the compact metric space (X, dX) into the metric space (Y , dY ). The function F ∶ X ×X → R given by F (x, y) ∶= dY (

METHODICAL ASPECTS OF FORMATION OF COMPLEX SYSTEM OF AN ASSESSMENT OF A SUSTAINABLE DEVELOPMENT OF TITANIC BRANCH

The titanic branch is one of most successfully developing directions of industrial production of the Russian Federation, its development is connected with creation of a titanic cluster. On the one hand, competitive advantages are a driving force of development of clusters. On the other hand, the market competition can lead to infringement in a cluster of social and ecological interests. The parity of economic, ecological and social interests is reached on the basis of a sustainable development of society. Successful realization in practice of a paradigm of a sustainable development considerably depends on possibilityof a quantitative assessment of stability. For the solution of this task in article the analysis of methodical approaches, now in use for an assessment of a sustainable development is made. It is established that the system of the indicators reflecting this approach, is insufficiently developed. The author offered complex system of indicators of a sustainable development (branch, the enterprise). It qualitatively differs from approaches now in use: – for productions and the branches using renewable resources (metallurgy and metal working concerns to them), it is offered to allocate the fourth direction – responsibility in a zone of rational use of resources; – the system of indicators characterizing zones of responsibility for the account of introduction is expanded: a) the indicators based on application of DS. It provided uniform approach, comparability and continuity of indicators both on micro and mesolevels, and on responsibility zones (economic, social, ecological); b) indicators of formation and metalfund use (the general, extensive, intensive) in the sphere of rational use of resources. Such approach allowed for the branches using renewable resources (for example, metal) to reflect level of their rational use.

Heat flow, weighted Bergman spaces, and real analytic Lipschitz approximation

Abstract We show that, for f any uniformly continuous (UC) complex-valued function on real Euclidean n-space ℝ n , the heat flow f˜(t) is Lipschitz for all t > 0 and f˜(t) converges uniformly to f as t → 0. Analogously, let Ω be any irreducible bounded symmetric (Cartan) domain in complex n-space ℂ n and consider the Bergman metric β(·, ·) on Ω. For f any β-uniformly continuous function on Ω, we show that there is a Berezin–Harish-Chandra flow of real analytic functions B λ f which is β-Lipschitz for each λ ≥ p (p, the genus of Ω) and B λ f converges uniformly to f as λ → ∞. For a certain subspace of UC we obtain stronger approximation results and we study the asymptotic behaviour of the Lipschitz constants.

On Filter $(\alpha)$-convergence and Exhaustiveness of Function Nets in Lattice Groups and Applications

We consider (strong uniform)continuity of thelimit of a pointwise convergent net of latticegroup-valued functions, (strong weak)ex\-hau\-sti\-ve\-ness and (strong)$(\alpha)$-con\-ver\-gen\-ce with respect to a pairof filters, which in the setting of nets aremore natural than the corresponding notionsformulated with respect to a single filter. Somecomparison results are givenbetween such concepts, inconnection with suitable properties of filters.Moreover, some modes of filter(strong uniform) continuity for lattice group-valuedfunctions are investigated, givingsome characterization.As an application, we getsome Ascoli-type theorem in an abstract setting,extending earlier results to the context of filter$(\alpha)$-con\-ver\-gen\-ce.Furthermore, we pose some open problems.

Uniform equicontinuity, multiplier topology and continuity of convolution

We characterise bounded uniformly equicontinuous sets of functions on locally compact groups in terms of uniform factorisation. We apply this result to study the continuity of the convolution product on the dual LUC(G)* of the space of bounded left uniformly continuous functions with the topology of uniform convergence on bounded uniformly equicontinuous sets. When restricted to the space of finite Radon measures on a locally compact group, this is the right multiplier topology. For any topological group, the convolution is jointly continuous on bounded sets in the measure algebra. It is jointly continuous on all of LUC(G)* when G is a locally compact SIN group.

Uniform positivity and continuity of Lyapunov exponents for a class of [formula omitted] quasiperiodic Schrödinger cocycles

We show that for a class of C2 quasiperiodic potentials and for any fixed Diophantine frequency, the Lyapunov exponent of the corresponding Schrödinger cocycles, as a function of energies, are uniformly positive and weakly Hölder continuous. As a corollary, we obtain that the corresponding integrated density of states is weakly Hölder continuous as well. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to more general SL(2,R) cocycles, which in turn can be applied to get uniform positivity and continuity of Lyapunov exponents around unique nondegenerate extremal points of any smooth potential, and to a certain class of C2 Szegő cocycles.

Lp Solutions of -Backward Stochastic Differential Equations with Continuous Coefficients

Abstract. In this paper, we study -backward stochastic differential equations with continuous coefficients. We give existence and uniqueness results for G-backward stochastic differential equations, when the generator is uniformly continuous in , and the terminal value with . We consider the G-backward stochastic differential equations driven by a G-Brownian motion in the following form: (1) where and are unknown and the random function , called the generator, and the random variable , called terminal value, are given. Our main result of this paper is the existence and uniqueness of a solution for (1) in the G-framework.

О задаче Дирихле для эллиптического уравнения

Хорошо известно, что естественно возникающее из вариационных принципов и удобное в применении понятие обобщeнного решения из соболевского пространства $W_2^1$ задачи Дирихле для эллиптического уравнения второго порядка не является в буквальном смысле обобщением понятия классического решения: не любая непрерывная на границе области функция является следом функции из $W_2^1$. Обобщение обоих этих понятий было предложено в 1976 году Валентином Петровичем Михайловым, памяти которого посвящена настоящая работа. В определении Михайлова граничное значение решения берется из $L_2$; естественно обобщается это понятие и на случай граничной функции из $L_p$, $p > 1$. Впоследствии автором настоящей работы было доказано, что при выполнении не слишком обременительных условий такие решения обладают свойством $(n-1)$-мерной непрерывности. Это свойство аналогично классическому определению равномерной непрерывности, но вместо значения функции в точке следует рассматривать еe следы на мерах из специального класса, немного более узкого, чем класс мер Карлесона. След функции на мере является элементом пространства $L_p$ по этой мере. $(n-1)$-мерная непрерывность означает, что следы на мерах близки, если близки эти меры. Определение близости мер учитывает близость (в специальном смысле) их носителей, а расстояние между следами (они элементы различных пространств) вводится с помощью погружения в пространство функций удвоенного числа переменных. Свойство $(n-1)$-мерной непрерывности позволило дать другое, по форме весьма близкое к классическому определение решения - $(n-1)$-мерно непрерывное решение. Как и понятия классического и обобщeнного решений оно не требует условий гладкости границы рассматриваемой области. В отличие от случаев классического и обобщeнного решений задача Дирихле в постановке Михайлова и тем более с $(n-1)$-мерно непрерывным решением исследована недостаточно полно. Прежде всего это относится к условиям на правую часть уравнения, при которых задача Дирихле разрешима. В работе приведeн ряд новых результатов в этом направлении. Кроме того, обсуждаются условия на коэффициенты уравнения, границу ограниченной области, в которой рассматривается задача, и заданные граничные значения решений. При этом результаты о разрешимости и о граничном поведении решений сравниваются с аналогичными теоремами, относящимися к случаю классического и обобщeнного решений, обсуждаются некоторые возникающие при таком сравнении нерешeнные задачи.

Uniform continuity of quasiconformal mappings onto generalized John domains

We study uniform continuity of quasiconformal mappings onto �-Gromov-hyperbolic '-John domains. The general '-John case is also investigated.

Constructive decidability of classical continuity

We show that the following instance of the principle of excluded middle holds: any function on the one-point compactification of the natural numbers with values on the natural numbers is either classically continuous or classically discontinuous. The proof does not require choice and can be understood in any of the usual varieties of constructive mathematics. Classical (dis)continuity is a weakening of the notion of (dis)continuity, where the existential quantifiers are replaced by negated universal quantifiers. We also show that the classical continuity of all functions is equivalent to the negation of the weak limited principle of omniscience. We use this to relate uniform continuity and searchability of the Cantor space.

Subseries convergence in abstract duality pairs

Let E, F be sets, G an Abelian topological group and b : ExF — G. Then (E, F, G) is called an abstract triple. Let w(F, E) be the weakest toplogy on F such that the maps {b(x, ·): x G E} from F into G are continuous. A subset B C F is w(F,E) sequentially conditionally compact if every sequence {yk} C B has a subsequence {ynk } such that limj; b(x, ynk) exists for every x G E. It is shown that if a formal series in E is subseries convergent in the sense that for every subsequence {xnj} there is an element x G E such that Xj=! b(xnj ,y) = b(x,y) for every y G F ,then the series Xj=! b(xnj ,y) converge uniformly for y belonging to w(F, E) sequentially conditionally compact subsets ofF. This result is used to establish Orlicz-Pettis Theorems in locall convex and function spaces. Applications are also given to Uniform Boundedness Principles and continuity results for bilinear mappings.

On a nonlocal problem for fractional differential equations via resolvent operators

Abstract Using the techniques of approximate solutions, the analytic resolvent method, and the uniform continuity of the resolvent, we discuss the existence of mild solutions for nonlocal fractional differential equations governed by a linear closed operator which generates a resolvent. An example is also given to illustrate the application of our theory. MSC:34K37, 47A10.

On the dynamics of a stochastic ratio-dependent predator–prey model with a specific functional response

In this paper, a new stochastic two-species predator–prey model which is ratio-dependent and a specific functional response is considered in, is proposed. The existence of a global positive solution to the model for any positive initial value is shown. Stochastically ultimate boundedness and uniform continuity are derived. Moreover, under some sufficient conditions, the stochastic permanence and extinction are established for the model. At last, numerical simulations are carried out to support our results.

О компактности и равномерной непрерывности разрешающего семейства для уравнения с дробными производными

О компактности и равномерной непрерывности разрешающего семейства для уравнения с дробными производными