A mean-value theorem and its applications

For a function f defined in an interval I, satisfying the conditions ensuring the existence and uniqueness of the Lagrange mean L[f], we prove that there exists a unique two variable mean M[f] such that

     

A NOVEL FIXED POINT THEOREM AND ITS APPLICATIONS

In this article, a novel fixed point theorem in C[0,1] space is established by using the properties of fixed point index. This theorem is then applied to prove the existence of positive solutions for three-point boundary value problems and generalizes some previous results.     

A theorem on the transcendency and its applications

A theorem on the transcendency, for a limit of a sequence of algebraic numbers, is given and used to test the transcendencies of some classes of numbers. Project supported by the National Natural Science Foundation of China and the Natural Science Foundation of Zhejiang Province.     

A new fixed point theorem and its applications

In this paper, we first give a new fixed theorem of lower semicontinuous multivalued mappings, and then, as its applications we obtain some new equilibrium theorems for abstract economies and qualitative games.

     

A substitution theorem for graceful trees and its applications

A graceful labeling of a graph G=(V,E) assigns |V| distinct integers from the set {0,…,|E|} to the vertices of G so that the absolute values of their differences on the |E| edges of G constitute the set {1,…,|E|}. A graph is graceful if it admits a graceful labeling. The forty-year old Graceful Tree Conjecture, due to Ringel and Kotzig, states that every tree is graceful.We prove a Substitution Theorem for graceful trees, which enables the construction of a larger graceful tree through combining smaller and not necessarily identical graceful trees. We present applications of the Substitution Theorem, which generalize earlier constructions combining smaller trees.     

A normal comparison inequality and its applications

 Let $\xi=(\xi_i, 1 \leq i \leq n)$ and $\eta= (\eta_i, 1 \leq i \leq n)$ be standard normal random variables with covariance matrices $R^1=(r_{ij}^1)$ and $R^0=(r_{ij}^0)$, respectively. Slepian's lemma says that if $r_{ij}^1 \geq r_{ij}^0$ for $1 \leq i, j \leq n$, the lower bound $\P(\xi_i \leq u \mb{for} 1 \leq i \leq n ) /\P(\eta_i \leq u \mb{for} 1 \leq i \leq n ) $ is at least $1$. In this paper an upper bound is given. The usefulness of the upper bound is justified with three concrete applications: (i) the new law of the iterated logarithm of Erdős and Révész, (ii) the probability that a random polynomial does not have a real zero and (iii) the random pursuit problem for fractional Brownian particles. In particular, a conjecture of Kesten (1992) on the random pursuit problem for Brownian particles is confirmed, which leads to estimates of principal eigenvalues. Received: 17 October 2000 / Revised version: 30 May 2001 / Published online: 22 February 2002     

A local Hahn–Banach theorem and its applications

An important consequence of the Hahn–Banach theorem says that on any locally convex Hausdorff topological space X, there are sufficiently many continuous linear functionals to separate points of X. In the paper, we establish a “local” version of this theorem. The result is applied to study the uo-dual of a Banach lattice that was recently introduced in Gao et al. (Positivity 22(3):711–725, 2018). We also provide a simplified approach to the measure-free characterization of uniform integrability established in Kardaras (J Funct Anal 266:1913–1927, 2014).     

A Hahn-Banach theorem for separation of semigroups and its applications

Summary In the first part of this paper a separation theorem is proved for disjoint subsemigroups of a given abelian semigroup. Applying this result, separation theorems and characterization theorems are obtained for semiinternal functions.     

A transference theorem for the Dunkl transform and its applications

For a family of weight functions invariant under a finite reflection group, we show how weighted Lp multiplier theorems for Dunkl transform on the Euclidean space Rd can be transferred from the corresponding results for h-harmonic expansions on the unit sphere Sd of Rd+1. The result is then applied to establish a Hörmander type multiplier theorem for the Dunkl transform and to show the convergence of the Bochner-Riesz means of the Dunkl transform of order above the critical index in weighted Lp spaces.     

A monotonicity theorem and its applications to weighted elliptic equations

We study the equation w_(tt) + ?_(S~(N-1))w-μw_t-δw + h(t, ω)w~p= 0,(t, ω) ∈ R × S~(N-1), and under some conditions we prove a monotonicity theorem for its positive solutions. Applying this monotonicity theorem,we obtain a Liouville-type theorem for some nonlinear elliptic weighted singular equations. Moreover, we obtain the necessary and sufficient condition for-div(|x|~θ▽u) = |x|~lu~p, x ∈ R~N\{0} having positive solutions which are bounded near 0, which is also a positive answer to Souplet's conjecture(see Phan and Souplet(2012)) on the weighted Lane-Emden equation-?u = |x|~au~p, x ∈ R~N.     

A renewal theorem and its applications to some sequential procedures

Probability Theory and Related Fields - A renewal theorem of the elementary type for some stopping times which arise from some statistical estimation problems has been established. It is applied to...     

An automatic adjoint theorem and its applications

In this paper, we prove the following automatic adjoint theorem: For any sequence spaces and , if has the signed-weak gliding hump property and is an infinite matrix which transforms into , then the transpose matrix of transforms into , and for any and , . That is, the adjoint operator of automatically exists and is just the transpose matrix of . From the theorem we obtain a class of infinite matrix topological algebras , and prove also a -multiplier convergence theorem of Orlicz-Pettis type. The theorem improves substantially the famous Stiles' Orlicz-Pettis theorem.

     

A comparison theorem

The existence of an explosive singularity in the unsteady boundary layer on a rotating disc in a counter-rotating fluid has now been shown incontrovertibly following the work of K. Stewartson, C. J. Simpson, and R. J. Bodonyi (J. Fluid. Mech.121 (1982), 507–515). Here, we develop some asymptotic results for the governing differential equations to help gain an understanding of the mechanism behind this phenomenon. No definite conclusions are possible, but the presence of inertial oscillations at the edge of the boundary layer could well play a definite role.     

A theorem on asymptotic behaviour of functional equations and its applications

A method for the determination of rate of decay of a solution of a stable functional equation is discussed. A method for establishing a bound on an unstable, but bounded, solution is also discussed. Some applications are considered as examples.     

Generalized Obata theorem and its applications on foliations

We prove the generalized Obata theorem on foliations. Let M be a complete Riemannian manifold with a foliation F of codimension q?2 and a bundle-like metric gM. Then (M,F) is transversally isometric to (Sq(1/c),G), where Sq(1/c) is the q-sphere of radius 1/c in (q+1)-dimensional Euclidean space and G is a discrete subgroup of the orthogonal group O(q), if and only if there exists a non-constant basic function f such that for all basic normal vector fields X, where c is a positive constant and ∇ is the connection on the normal bundle. By the generalized Obata theorem, we classify such manifolds which admit transversal non-isometric conformal fields.