Uniform Continuity Condition Research Articles

Existence, uniqueness and comparison theorem on unbounded solutions of scalar super-linear BSDEs

This paper is devoted to the existence, uniqueness and comparison theorem on unbounded solutions of a scalar backward stochastic differential equation (BSDE) whose generator grows (with respect to both unknown variables y and z) in a super-linear way like |y||ln|y||δ+|z||ln|z||λ for some δ∈[0,1] and λ≥0. Let k be the maximum of δ, λ+1/2 and 2λ. For the following four different ranges of the growth power parameter k: k=1/2, k∈(1/2,1), k=1 and k>1, we give reasonably weakest possible different integrability conditions on the terminal value for ensuring existence and uniqueness of the unbounded solution to the BSDE. In the first two cases, they are stronger than the LlnL-integrability and weaker than any Lp-integrability with p>1; in the third case, the integrability condition is just some Lp-integrability for p>1; and in the last case, the integrability condition is stronger than any Lp-integrability with p>1 and weaker than any exp(Lɛ)-integrability with ɛ∈(0,1). We also establish three comparison theorems, which yield immediately the uniqueness, when either one of generators of both BSDEs is convex (or concave) in both unknown variables (y,z), or satisfies a one-sided Osgood condition in the first unknown variable y and a uniform continuity condition in the second unknown variable z. Finally, we give an application of our results in mathematical finance.

Lp solutions of general time interval BSDEs with generators satisfying a p-order weak stochastic-monotonicity condition

This article is interested in solutions of general time interval backward stochastic differential equations (BSDEs) with generator g satisfying a p-order weak stochastic-monotonicity condition and a general growth condition in y. An existence and uniqueness result of the Lp solution is proved for the multidimensional BSDEs where the generator g is stochastic-Lipschitz in z. And, a general comparison theorem and an existence and uniqueness theorem of the Lp solution is established for the one-dimensional BSDEs under a stochastic uniform-continuity condition of the generator g in z. These results strengthen some known works. The stochastic Gronwall-type inequality and the stochastic Bihari-type inequality play an important role in the whole article.

$L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under general assumptions

We establish several existence, uniqueness and comparison results for $L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under the assumptions that the generator $g$ satisfies a one-sided Osgood condition together with a very general growth condition in $y$, a uniform continuity condition and/or a sub-linear growth condition in $z$, and a generalized Mokobodzki condition for reflected BSDEs which relates the growth of $g$ and that of the barriers. This generalized Mokobodzki condition is proved to be necessary for existence of $L^{1}$ solutions of the reflected BSDEs. We also prove that the $L^{1}$ solutions of reflected BSDEs can be approximated by a penalization method and by some sequences of $L^{1}$ solutions of reflected BSDEs. The approach is based on a combination between existing methods, their refinement and perfection, but also on some novel ideas and techniques. These results strengthen some existing work on the $L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs.

Eigenvectors of homogeneous order-bounded order-preserving maps

The existence of eigenvectors associated with the cone spectral radius is shown for homogenous, order-preserving, continuous maps that have compact and order-bounded powers (iterates). The order-boundedness makes it possible to show the existence of eigenvectors for perturbations of the maps using Hilbert's projective metric, while the power compactness or similar compactness properties together with a uniform continuity condition let the eigenvectors of the perturbations converge to an eigenvector of the original map.

Existence, uniqueness and approximation for L p solutions of reflected BSDEs with generators of one-sided Osgood type

We prove several existence and uniqueness results for Lp (p > 1) solutions of reflected BSDEs with continuous barriers and generators satisfying a one-sided Osgood condition together with a general growth condition in y and a uniform continuity condition or a linear growth condition in z. A necessary and sufficient condition with respect to the growth of barrier is also explored to ensure the existence of a solution. And, we show that the solutions may be approximated by the penalization method and by some sequences of solutions of reflected BSDEs. These results are obtained due to the development of those existing ideas and methods together with the application of new ideas and techniques, and they unify and improve some known works.

Some functional inequalities in variable exponent spaces with a more generalization of uniform continuity condition

Some functional inequalities in variable exponent Lebesgue spaces are presented. The bi-weighted modular inequality with variable exponent p(:) for the Hardy operator restricted to non-increasing function which is Z 1 0 f(x) p(x) u(x)dx; is studied. We show that the exponent p(:) for which these modular inequalities hold must have constant oscillation. Also we study the boundedness of integral operator Tf(x) = R K(x;y)f(x)dy on L p(:) when the variable exponent p(:) satises some uniform continuity condition that is named -controller condition and so multiple interesting results which can be seen as a generalization of the same classical results in the constant exponent case, derived.

On the stability theorem of $L^{p}$ solutions for multidimensional BSDEs with uniform continuity generators in $z$

In this paper, we first establish an existence and uniqueness result of $L^{p}$ ($p>1$) solutions for multidimensional backward stochastic differential equations (BSDEs) whose generator $g$ satisfies a certain one-sided Osgood condition with a general growth in $y$ as well as a uniform continuity condition in $z$, and the $i$th component ${}^{i}g$ of $g$ depends only on the $i$th row ${}^{i}z$ of matrix $z$ besides $(\omega,t,y)$. Then we put forward and prove a stability theorem for $L^{p}$ solutions of this kind of multidimensional BSDEs. This generalizes some known results.

Results on weighted stepanov-like pseudo periodic functions and applications

In this article, we first established some new results on composition of weighted Stepanov-like pseudo periodic function of class $r$ under a uniform continuity condition with respect to $L^p$ norm. And then, we proved the existence and uniqueness of weighted pseudo periodic solutions to a semi-linear functional differential equation with finite delay under Stepanov-like nonlinear term.

Stability Preserving Simulations and Bisimulations for Hybrid Systems

Pre-orders and equivalence relations between processes, like simulation and bisimulation, have played a central role in the minimization and abstraction based verification and analysis of discrete-state systems for modal and temporal properties. In this paper, we investigate the pre-orders and equivalence relations on hybrid systems which preserve stability. We first show that stability with respect to reference trajectories is not preserved by either the traditional notion of bisimulation or the more recently proposed stronger notions with additional continuity constraints. We introduce the concept of uniformly continuous simulation and bisimulation—namely, simulation and bisimulation with some additional uniform continuity conditions on the relation—that can be used to reason about stability of trajectories. Finally, we show that uniformly continuous simulations and bisimulations are widely prevalent, by recasting many classical results on proving stability of dynamical and hybrid systems as establishing the existence of a simple, obviously stable system that (bi)-simulates the given system through uniformly continuous (bi)-simulations. We also discuss briefly a new abstraction method for stability analysis which is based on the foundations developed in the paper.

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.

WEIGHTED PSEUDO-ALMOST PERIODIC SOLUTIONS FOR SOME ABSTRACT DIFFERENTIAL EQUATIONS WITH UNIFORM CONTINUITY

AbstractIn this work, we give some theorems on (mild) weighted pseudo-almost periodic solutions for some abstract semilinear differential equations with uniform continuity. To facilitate this we give a new composition theorem of weighted pseudo-almost periodic functions. Our composition theorem improves the known one by making use of a uniform continuity condition instead of the Lipschitz condition.

Homogenization of a class of quasilinear elliptic equations with non-standard growth in high-contrast media

We study the asymptotic behaviour of solutions to a quasilinear equation with high-contrast coefficients. The energy formulation of the problem leads to work with variable exponent Lebesgue spaces Lpε (·) in a domain Ω with a complex microstructure depending on a small parameter ε. Assuming only that the functions pε converge uniformly to a limit function p0 and that p0 satisfy certain logarithmic uniform continuity conditions, we rigorously derive the corresponding homogenized problem which is completely described in terms of local energy characteristics of the original domain. In the framework of our method we do not have to specify the geometrical structure Ω. We illustrate our result with periodical examples, extending, in particular, the classical extension results to variable exponent Sobolev spaces.

Formula omitted] continuity conditions of adjacent NURBS surfaces

For the special NURBS surfaces such as bicubic or biquartic B-spline surfaces with single interior knots, some results of the G 1 continuity conditions for two adjacent surfaces appeared recently. But for the NURBS surfaces with arbitrary degrees and generally structured knots, the G 1 continuity conditions are still unsolved. Therefore, this paper pays attention to studying this problem. In this paper, we obtain the necessary and sufficient conditions of G 1 continuity, and present two kinds of sufficient conditions of G 1 continuity for two adjacent NURBS surfaces with arbitrary degrees and generally structured knots. In particular, we obtain the intrinsic conditions for the control points and weights of the common boundary curve (so-called the uniform continuity conditions of G 1 continuity). It should be pointed out that this case is unique in the discussion of G 1 continuity of adjacent NURBS surfaces. At the end, we give the algorithm for constructing adjacent G 1 NURBS surfaces.

Duality theorems for joint source-channel coding

We consider joint source-channel coding for a memoryless Gaussian source and an additive white Gaussian noise (AWGN) channel. For a given code defined by an encoder-decoder pair (/spl alpha/, /spl beta/), its dual code is obtained by interchanging the encoder and decoder: (/spl beta/, /spl alpha/). It is shown that if a code (/spl alpha/, /spl beta/) is optimal at rate p channel uses per source sample and if it satisfies a certain uniform continuity condition, then its dual code (/spl beta/, /spl alpha/) is optimal for rate 1//spl rho/ channel uses per source sample. Further, it is demonstrated that there is a code which is optimal but its dual code is not optimal. Finally, using random coding, we show that there is an optimal code which has an optimal dual. The duality concept is also presented for the cases of (i) binary memoryless equiprobable source and binary-symmetric channel (BSC), and (ii) colored Gaussian source and additive colored Gaussian noise (ACGN) channel.

Harnack inequality for the linearized parabolic Monge-Ampère equation

In this paper we prove the Harnack inequality for nonnegative solutions of the linearized parabolic Monge-Ampère equation \[ u t − tr ( ( D 2 ϕ ( x ) ) − 1 D 2 u ) = 0 u_{t}-\text {tr}((D^{2}\phi (x))^{-1}D^{2}u)=0 \] on parabolic sections associated with ϕ ( x ) \phi (x) , under the assumption that the Monge-Ampère measure generated by ϕ \phi satisfies the doubling condition on sections and the uniform continuity condition with respect to Lebesgue measure. The theory established is invariant under the group A T ( n ) × A T ( 1 ) AT(n)\times AT(1) , where A T ( n ) AT(n) denotes the group of all invertible affine transformations on R n {\mathbf {R}}^{n} .

Properties of the solutions of the linearized Monge-Ampère equation

Let Φ: R n → R be a function strictly convex and smooth, and μ = det D 2 Φ is the Monge-Ampere generated by Φ. Given x ∈ R n and t > 0, let' S ( x, t ) = { y ∈ R n : Φ( y ) < Φ( x ) + ∇Φ( x ) · ( y - x ) + t }. The purpose of this paper is to study the properties of the solutions of the linearized Monge-Ampère equation given by a ij ( x ) D ij u = 0 where the coefficients a ij ( x ) are the cofactors of the matrix D 2 Φ( x ). It is assumed that μ satisfies a doubling condition on the sets S ( x, t ) and a uniform continuity condition at every scale with respect to Lebesgue measure. We establish that the distribution functions of nonnegative solutions u at altitude t decay like a negative power of t and prove an invariant Harnack's inequality on the sections S ( x, t ). All the estimates are independent of the regularity of Φ and depend only on the constants in the hypotheses made on the measure μ.

A uniform continuity condition which is equivalent to Coleman's conjecture

At the Fifth International Conference on Nonlinear Oscillations (Kiev, 1970), Coleman [l] raised the question of whether or not a flow with a topologically hyperbolic stationary point is necessarily locally conjugate with some flow near a differentiably hyperbolic (generic) stationary point. We shall prove an affirmative result in the case where either the stable manifold or the unstable manifold has dimension one and we shall prove the equivalence of a uniform continuity condition in the general case. These affirmative results are contrasted with the recent counterexample by Neumann [2], which has both the stable manifold and the unstable manifold with dimension 2. Using Hartman’s theorem [3], it is easy to show that every differentiably hyperbolic stationary point on [w N is locally conjugate to the flow F~;,,~-,~ of the Standard Example, where

Large-sample theory of sequential estimation

In a previous large-sample treatment of sequential estimation (1), it was shown that in certain circumstances, when there was only one unknown parameter in the distribution of the observations, an estimation formula valid for fixed sample sizes remained valid when the sample size was determined by a sequential stopping rule. The proof was heuristic, in that it depended on an application of the central limit theorem of which the justification was not obvious. Another proof has recently been given by Cox (2) (in the course of deriving a correction term to my result). Dr Cox pointed out to me that this work suggested that fixed-sample-size formulae might be valid generally for sequential sampling, provided the sample size was large. In establishing a proposition to that effect, I have now been able to by-pass some of the complexity of my previous approach by concentrating attention on a condition of ‘uniform continuity in probability’ to be satisfied by the statistic used. Theorem 1 is the basic result, which is applied in Theorem 2 to determine a sequential stopping rule giving required accuracy of estimation of an unknown parameter. Theorems 3–6 indicate some situations in which the uniform continuity condition postulated in Theorems 1 and 2 is satisfied. A few examples are discussed.