For a bounded closed convex set K, in this note, we study the FPP for α-Hölder nonexpansive maps, i.e. mappings T:K→K which ‖Tx−Ty‖≤‖x−y‖α all x,y∈K, α∈(0,1). First, note that only finite-dimensional spaces have Hölder-FPP. Moreover, unit ball BX of any infinite-dimensional space fails Hölder maps with d(T,BX)>0, where d(T,K) denotes minimal displacement T. We further show reflexivity and weak ...

We obtain a Harnack type inequality for solutions to elliptic equations in divergence form with non-standard p(x)-type growth. A model equation is the inhomogeneous p(x)-Laplacian. Namely, ∆p(x)u := div ( |∇u|p(x)−2∇u ) = f(x) in Ω, for which we prove Harnack’s inequality when f ∈ Lq0 (Ω) if max{1, N p1 } < q0 ≤ ∞. The constant in Harnack’s inequality depends on u only through ‖|u|p(x)‖p2−p1 L1...

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In this paper, we establish pointwise estimates for supersolutions of quasilinear elliptic equations with structural conditions involving a generalized Orlicz growth in terms Wolff type potential. As consequence, under an extra assumption, obtain that the satisfy Harnack inequality and local Hölder continuity.

The integrated density of states (IDS) for random operators is an important function describing many physical characteristics of a random system. Properties of the IDS are derived from the Wegner estimate that describes the influence of finite-volume perturbations on a background system. In this paper, we present a simple proof of the Wegner estimate applicable to a wide variety of random pertu...

In this paper we consider various regularity results for discrete quasiperiodic Schrödinger equations −ψn+1 − ψn−1 + V (θ + nω)ψn = Eψn with analytic potential V . We prove that on intervals of positivity for the Lyapunov exponent the integrated density of states is Hölder continuous in the energy provided ω has a typical continued fraction expansion. The proof is based on certain sharp large d...

The Hölder continuity of the truncated moment map a shade function in Euclidean space is established vicinity principal semi-algebraic set. proof combines volume bounds sets and convex optimization methods. main estimate applied to potential type transform specific two real variables, for perturbations quadrature domains.

Abstract We will show the Cheeger–Colding segment inequality for manifolds with integral Ricci curvature bound. By using this inequality, almost rigidity structure results be derived by a similar method as in [1]. And sharp Hölder continuity result of [7] holds limit space