An approximation method for strictly pseudocontractive mappings

Let K$K$ be a closed convex subset of a q$q$-uniformly smooth separable Banach space, T:K→K$T:K\to K$ a strictly pseudocontractive mapping, and f:K→K$f:K\to K$ an L$L$-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1)$t\in (0,1)$, let _xt$x_t$ be the unique fixed point of tf+(1-t)T$\mathit{tf}+(1-t)T$. We prove that if T$T$ has a fixed point, then {_xt}$\{x_t\}$ converges to a fixed point of T$T$ as t$t$ approaches to 0.     

p-Laplacian boundary value problems on exterior regions

This paper treats some variational principles for solutions of inhomogeneous p  -Laplacian boundary value problems on exterior regions U?R^N$U?{\mathbb{R}}^N$ with dimension N?3$N?3$. Existence-uniqueness results when p∈(1,N)$p\in (1,N)$ are provided in a space E^1,p(U)$E^{1,p}(U)$ of functions that contains W^1,p(U)$W^{1,p}(U)$. Functions in E^1,p(U)$E^{1,p}(U)$ are required to decay at infinity in a measure theoretic sense. Various properties of this space are derived, including results about equivalent norms, traces and an L^p$L^p$-imbedding theorem. Also an existence result for a general variational problem of this type is obtained.     

Concentration and multiplicity of solutions for a fourth-order equation with critical nonlinearity

In this paper, we consider the problem (_Pε)$(P_{\epsilon })$ : Δ²u=u^n+4/n-4+εu,u>0${\mathrm{\Delta }}^{2}u=u^{n+4/n-4}+\epsilon u,u>0$ in Ω,u=Δu=0$\Omega ,u=\mathrm{\Delta }u=0$ on ∂Ω$\mathrm{\partial }\Omega$, where Ω$\Omega$ is a bounded and smooth domain in Rⁿ,n>8${\mathbb{R}}^n,n>8$ and ε>0$\epsilon >0$. We analyze the asymptotic behavior of solutions of (_Pε)$(P_{\epsilon })$ which are minimizing for the Sobolev inequality as ε→0$\epsilon \to 0$ and we prove existence of solutions to (_Pε)$(P_{\epsilon })$ which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε$\epsilon$ small, (_Pε)$(P_{\epsilon })$ has at least as many solutions as the Ljusternik–Schnirelman category of Ω$\Omega$.     

Existence and separation of positive radial solutions for semilinear elliptic equations

We consider the semilinear elliptic equation Δu+K(|x|)u^p=0$\mathrm{\Delta }u+K(|x|)u^p=0$ in R^N${\mathbf{R}}^N$ for N>2$N>2$ and p>1$p>1$, and study separation phenomena of positive radial solutions. With respect to intersection and separation, we establish a classification of the solution structures, and investigate the structures of intersection, partial separation and separation. As a consequence, we obtain the existence of positive solutions with slow decay when the oscillation of the function r^−?K(r)$r^{-?}K(r)$ with ?>−2$?>-2$ around a positive constant is small near r=∞$r=\infty$ and p   is sufficiently large. Moreover, if the assumptions hold in the whole space, the equation has the structure of separation and possesses a singular solution as the upper limit of regular solutions. We also reveal that the equation changes its nature drastically across a critical exponent _pc$p_c$ which is determined by N   and the order of the behavior of K(r)$K(r)$ as r=|x|→0$r=|x|\to 0$ and ∞. In order to understand how subtle the structure is on K   at p=_pc$p=p_c$, we explain the criticality in a similar way as done by Ding and Ni (1985) [6] for the critical Sobolev exponent p=(N+2)/(N−2)$p=(N+2)/(N-2)$.     

Derivations and local derivations on Freudenthal almost f-algebras

Let A be an Archimedean f  -algebra and let N(A)$N(A)$ be the set of all nilpotent elements of A. Colville et al. [4] proved that a positive linear map d:A→A$d:A\to A$ is a derivation if and only if d(A)⊂N(A)$d(A)\subset N(A)$ and d(A²)={0}$d(A^2)=\{0\}$, where A²$A^2$ is the set of all products ab in A.     

Sur la répartition des puissances modulo 1

For almost all x>1$x>1$, (xⁿ)$(x^n)$(n=1,2,…)$(n=1,2,\dots )$ is equidistributed modulo 1, a classical result. What can be said on the exceptional set? It has Hausdorff dimension one. Much more: given an (b_n)$(b_n)$ in [0,1[$[0,1[$ and ε>0$\epsilon >0$, the x  -set such that |xⁿ−b_n|<ε$|x^n-b_n|<\epsilon$ modulo 1 for n   large enough has dimension 1. However, its intersection with an interval [1,X]$[1,X]$ has a dimension <1, depending on ε and X. Some results are given and a question is proposed.     

Maximum principles and isoperimetric inequalities for some Monge–Ampère-type problems

In this note we derive a maximum principle for an appropriate functional combination of u(x)$u(\mathbf{x})$ and |∇u|²${|\mathrm{\nabla }u|}^2$, where u(x)$u(\mathbf{x})$ is a strictly convex classical solution to a general class of Monge–Ampère equations. This maximum principle is then employed to establish some isoperimetric inequalities of interest in the theory of surfaces of constant Gauss curvature in R^N+1${\mathbb{R}}^{N+1}$.     

Multiplicity of positive and nodal solutions for scalar field equations

In this paper the question of finding infinitely many solutions to the problem −Δu+a(x)u=|u|^p−2u$-\mathrm{\Delta }u+a(x)u={|u|}^{p-2}u$, in R^N${\mathbb{R}}^N$, u∈H¹(R^N)$u\in H^1({\mathbb{R}}^N)$, is considered when N≥2$N\ge 2$, p∈(2,2N/(N−2))$p\in (2,2N/(N-2))$, and the potential a(x)$a(x)$ is a positive function which is not required to enjoy symmetry properties. Assuming that a(x)$a(x)$ satisfies a suitable “slow decay at infinity” condition and, moreover, that its graph has some “dips”, we prove that the problem admits either infinitely many nodal solutions or infinitely many constant sign solutions. The proof method is purely variational and allows to describe the shape of the solutions.     

On Waringʼs problem for systems of skew-symmetric forms

This paper is devoted to a problem of finding the smallest positive integer s(m,n,k)$s(m,n,k)$ such that (m+1)$(m+1)$ generic skew-symmetric (k+1)$(k+1)$-forms in (n+1)$(n+1)$ variables as linear combinations of the same s(m,n,k)$s(m,n,k)$ decomposable skew-symmetric (k+1)$(k+1)$-forms.     

The spherical ergodic theorem revisited

In this paper, we give a new proof of a result of R. Jones showing almost everywhere convergence of spherical means of actions of R^d${\mathbb{R}}^d$ on L^p(X)$L^p(X)$-spaces are convergent for d?3$d?3$ and p>d/(d-1)$p>d/(d-1)$.