A symmetry result for a general class of divergence form PDEs in fibered media

In R^m×R^n−m, endowed with coordinates X=(x,y), we consider the PDE

     

Strong convergence theorems for nonexpansive semigroup in Banach spaces

Let K be a nonempty closed convex subset of a reflexive and strictly convex Banach space E with a uniformly Gâteaux differentiable norm, and a nonexpansive self-mappings semigroup of K, and a fixed contractive mapping. The strongly convergent theorems of the following implicit and explicit viscosity iterative schemes {xn} are proved.

xn=αnf(xn)+(1−αn)T(tn)xn,     

Iterative approximation of fixed points of nonexpansive mappings

Let K be a nonempty closed convex subset of a real Banach space E which has a uniformly Gâteaux differentiable norm and be a nonexpansive mapping with F(T):={x∈K:Tx=x}≠∅. For a fixed δ∈(0,1), define by Sx:=(1−δ)x+δTx, ∀x∈K. Assume that {zt} converges strongly to a fixed point z of T as t→0, where zt is the unique element of K which satisfies zt=tu+(1−t)Tzt for arbitrary u∈K. Let {αn} be a real sequence in (0,1) which satisfies the following conditions: ; . For arbitrary x₀∈K, let the sequence {xn} be defined iteratively by

xn+1=αnu+(1−αn)Sxn.     

An inequality for sums of binary digits, with application to Takagi functions

Let ?(x)=2inf{|x−n|:n∈Z}, and define for α>0 the function

     

A mean value theorem for cubic fields

Let r(n) denote the number of integral ideals of norm n in a cubic extension K of the rationals, and define and Δ(x)=S(x)−αx where α is the residue of the Dedekind zeta function ζ(s,K) at 1. It is shown that the abscissa of convergence of

     

Self-similar singular solutions of a p-Laplacian evolution equation with absorption

We consider, for p∈(1,2) and q>1, self-similar singular solutions of the equation v_t=div(|∇v|^p−2∇v)−v^q in Rn×(0,∞); here by self-similar we mean that v takes the form v(x,t)=t^−αw(|x|t^−αβ) for α=1/(q−1) and β=(q+1−p)/p, whereas singular means that v is non-negative, non-trivial, and for all x≠0. That is, we consider the ODE problem

(0.1)     

Solution of Hyers-Ulam stability problem for generalized Pappus' equation

The purpose of this paper is to solve the stability problem of Ulam for an approximate mapping of the following generalized Pappus' equation:

n²Q(x+my)+mnQ(x−ny)=(m+n)[nQ(x)+mQ(ny)]     

Invariance of quasi-arithmetic means with function weights

Let I⊂R be a non-trivial interval and let . We present some results concerning the following functional equation, generalizing the Matkowski-Sutô equation,

λ(x,y)φ⁻¹(μ(x,y)φ(x)+(1−μ(x,y))φ(y))+(1−λ(x,y))ψ⁻¹(ν(x,y)ψ(x)+(1−ν(x,y))ψ(y))=λ(x,y)x+(1−λ(x,y))y,     

Strong convergence theorems for fixed points of asymptotically pseudocontractive semi-groups

Let K be a nonempty closed convex and bounded subset of a real Banach space E. Let be a strongly continuous uniformly asymptotically regular and uniformly L-Lipschitzian semi-group of asymptotically pseudocontractive mappings from K into K. Then for a given u∈K there exists a sequence {y_n}∈K satisfying the equation y_n=(1−α_n)(T(t_n))ⁿy_n+α_nu for each , where α_n∈(0,1) and t_n>0 satisfy appropriate conditions. Suppose further that E is uniformly convex and has uniformly Gâteaux differentiable norm, under suitable conditions on the mappings T, the sequence {y_n} converges strongly to a fixed point of . Furthermore, an explicit sequence {x_n} generated from x₁∈K by x_n+1:=(1−λ_n)x_n+λ_n(T(t_n))ⁿx_n−λ_nθ_n(x_n−x₁) for all integers n?1, where {λ_n}, {θ_n} are positive real sequences satisfying appropriate conditions, converges strongly to a fixed point of .     

Deduction of L. Hörmander's extension of Ásgeirsson's mean value theorem

L. Hörmander's extension of Ásgeirsson's mean value theorem states that if u is a solution of the inhomogeneous ultrahyperbolic equation (Δ_x−Δ_y)u=f, , , then

     

Convergence theorems for fixed points of uniformly continuous generalized Φ-hemi-contractive mappings

Let E be a real normed linear space, K be a nonempty subset of E and be a uniformly continuous generalized Φ-hemi-contractive mapping, i.e., , and there exist x^∗∈F(T) and a strictly increasing function , Φ(0)=0 such that for all x∈K, there exists j(x−x^∗)∈J(x−x^∗) such that

〈Tx−x^∗,j(x−x^∗)〉?‖x−x^∗‖2−Φ(‖x−x^∗‖).     

Viscosity approximation methods for nonexpansive nonself-mappings

Let E a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^∗, and K be a closed convex subset of E which is also a sunny nonexpansive retract of E, and be nonexpansive mappings satisfying the weakly inward condition and F(T)≠∅, and be a fixed contractive mapping. The implicit iterative sequence {xt} is defined by for t∈(0,1)

xt=P(tf(xt)+(1−t)Txt).     

Interior estimates for solutions of a fourth order nonlinear partial differential equation

Let be a locally strongly convex hypersurface, given by the graph of a convex function xn+1=f(x₁,…,xn) defined in a convex domain Ω⊂Rn. M is called a α-extremal hypersurface, if f is a solution of