Gronwall's inequality for a nabla fractional difference system with a retarded argument and an application

ABSTRACT

We study the nabla fractional difference system with retarded argument. There are two major ingredients. A Gronwall's inequality for the nabla case is given. This allows us to evaluate the solution of nabla fractional difference system. We shall illustrate the validity of our results by means of examples.     

A new integral inequality with power nonlinearity and its discrete analogue

1. IntroductionIt is well known that integral inequalities are very crucial in the study of differelltialequations, integral equations, functional-differential equations, iniegro-differential equationsand evolution equations. Besides the famous Gronwall-Bellman inequality and its first nonlinear generalization due to i. Bihari (of. [1]), there is another very useful integral inequalityestablished by L. On-fang[2]:Lemma 1. Let u(t) and h(t) be real-valued, nonnegative and continuous functions…     

A projected Newton method for minimization problems with nonlinear inequality constraints

Summary Recently developed projected Newton methods for minimization problems in polyhedrons and Cartesian products of Euclidean balls are extended here to general convex feasible sets defined by finitely many smooth nonlinear inequalities. Iterate sequences generated by this scheme are shown to be locally superlinearly convergent to nonsingular extremals , and more specifically, to local minimizers satisfying the standard second order Kuhn-Tucker sufficient conditions; moreover, all such convergent iterate sequences eventually enter and remain within the smooth manifold defined by the active constraints at . Implementation issues are considered for large scale specially structured nonlinear programs, and in particular, for multistage discrete-time optimal control problems; in the latter case, overall per iteration computational costs will typically increase only linearly with the number of stages. Sample calculations are presented for nonlinear programs in a right circular cylinder in ³.Investigation supported by NSF Research Grant #DMS-85-03746     

A discrete convolution involving Fourier sine and cosine series and its applications

ABSTRACT

In this paper, we introduce a discrete convolution involving both the Fourier sine and cosine series. We study Young's type inequality and a discrete transform related to this convolution and solve in closed form a class of discrete Toeplitz plus Hankel equations.     

Remez-type inequality on sets with cusps

A Remez-type inequality is proved for a large family of sets with cusps in R^N${\mathbb{R}}^N$, including compact, fat and semialgebraic (or subanalytic) sets.     

退化(K_1,K_2)拟正则映射的充分条件

研究了退化弱(k1,k2)拟正则映射的正则性.利用H lder不等式、Sobolev空间的空间分析方法,以及内插定理等工具,给出了退化弱(k1,k2)拟正则映射事实上为退化(k1,k2)拟正则映射的一个充分条件,其结果对非退化情形也成立.     

A generalized h-fractional Gronwall's inequality and its applications for nonlinear h-fractional difference systems with ‘maxima’

ABSTRACT

This paper gives a generalized h-fractional Gronwall's inequality. Applying this result, we prove the uniqueness and give bounds on solutions for a nonlinear h-fractional difference system with ‘maxima’. Finally, we give an example to illustrate one of our main results.     

A modified Korpelevich's method convergent to the minimum-norm solution of a variational inequality

In this article, we propose a modified Korpelevich's method for solving variational inequalities. Under some mild assumptions, we show that the suggested method converges strongly to the minimum-norm solution of some variational inequality in an infinite-dimensional Hilbert space.     

Inverse inequality estimates with symbolic computation

In the convergence analysis of numerical methods for solving partial differential equations (such as finite element methods) one arrives at certain generalized eigenvalue problems, whose maximal eigenvalues need to be estimated as accurately as possible. We apply symbolic computation methods to the situation of square elements and are able to improve the previously known upper bound, given in “p- and hp-finite element methods” (Schwab, 1998), by a factor of 8. More precisely, we try to evaluate the corresponding determinant using the holonomic ansatz, which is a powerful tool for dealing with determinants, proposed by Zeilberger in 2007. However, it turns out that this method does not succeed on the problem at hand. As a solution we present a variation of the original holonomic ansatz that is applicable to a larger class of determinants, including the one we are dealing with here. We obtain an explicit closed form for the determinant, whose special form enables us to derive new and tight upper resp. lower bounds on the maximal eigenvalue, as well as its asymptotic behaviour.     

A pointwise inequality for fractional laplacians

The fractional laplacian is an operator appearing in several evolution models where diffusion coming from a Lévy process is present but also in the analysis of fluid interphases. We provide an extension of a pointwise inequality that plays a rôle in their study. We begin recalling two scenarios where it has been used. After stating the results, for fractional Laplace–Beltrami and Dirichlet–Neumann operators, we provide a sketch of their proofs, unravelling the underlying principle to such inequalities.     

Erratum to \"More Results on Hermite-Hadamard type inequality through (α,m)-preinvexity\"

In this paper, we present some corrections to definitions of $m$-preinvex, $left( alpha,mright) $-preinvex functions and statements of the theorems of the results proved in [7].     

Regularity property of solution to two-parameter stochastic volterra equation with non-lipschitz coefficients

This article proves the existence and uniqueness of solution to two-parameter stochastic Volterra equation with non-Lipschitz coefficients and driven by Brownian sheet, where the main tool is Bihari's inequality in the plane. Moreover, we also discuss the time regularity property of the solution by Kolmogorov's continuity criterion.     

Regularity for quasi-linear degenerate elliptic equations with VMO coefficients

In this paper we establish an interior regularity of weak solution for quasi-linear degenerate elliptic equations under the subcritical growth if its coefficient matrix A（x, u） satisfies a VMO condition in the variable x uniformly with respect to all u, and the lower order item B（x, u, △↓u） satisfies the subcritical growth （1.2）. In particular, when F（x） ∈ L^q（Ω） and f（x） ∈ L^γ（Ω） with q,γ 〉 for any 1 〈 p 〈 ＋∞, we obtain interior HSlder continuity of any weak solution of （1.1） u with an index κ = min{1 - n/q, 1 - n/γ}.     

A Faber-Krahn inequality for the Laplacian with Generalised Wentzell boundary conditions

We give an extension of the Faber-Krahn inequality to the Laplacian Δ on bounded Lipschitz domains , with generalised Wentzell boundary conditions on ∂Ω, where β, γ are nonzero real constants. We prove that when β, γ > 0, the ball B minimises the first eigenvalue with respect to all Lipschitz domains Ω of the same volume as B, and that B is the unique minimiser amongst C ²-domains. We also consider β, γ not both positive, and slightly extend what is known about the associated Wentzell operator and its resolvent in addition to considering an analogue of the Faber-Krahn inequality. This is based on the recent extension of the Faber-Krahn inequality to the Robin Laplacian. We also give a version of Cheeger’s inequality for the Wentzell Laplacian when β, γ > 0.      

A New Approach to the Creation and Propagation of Exponential Moments in the Boltzmann Equation

We study the creation and propagation of exponential moments of solutions to the spatially homogeneous d-dimensional Boltzmann equation. In particular, when the collision kernel is of the form |v ? v _*|^β b(cos (θ)) for β ∈ (0, 2] with cos (θ) = |v ? v _*|^?1(v ? v _*)·σ and σ ∈ 𝕊 ^d?1, and assuming the classical cut-off condition b(cos (θ)) integrable in 𝕊 ^d?1, we prove that there exists a > 0 such that moments with weight exp (amin {t, 1}|v|^β) are finite for t > 0, where a only depends on the collision kernel and the initial mass and energy. We propose a novel method of proof based on a single differential inequality for the exponential moment with time-dependent coefficients.