Let I be an interval. I need not be compact or bounded. Let f : I → I be a continuous map, and (x0, x1, · · · , xn) be a trajectory of f with xn ≤ x0 < x1 or x1 < x0 ≤ xn. Then there is a point v ∈ I such that min{x0, · · · , xn} < v = f(v) < max{x0, · · · , xn}. A point y ∈ I is called a centripetal point of f relative to v if y < f(y) < v or v < f(y) < y, and y is centrifugal if f(y) < y < v ...

Let q ≥ 1 be an integer, Sq be the unit sphere embedded in Rq+1, and μq be the volume element of Sq. For x0 ∈ Sq, and α ∈ (0, π), let Sα(x0) denote the cap {ξ ∈ Sq : x0 · ξ ≥ cosα}. We prove that for any integer m ≥ 1, there exists a positive constant c = c(q,m), independent of α, with the following property. Given an arbitrary set C of points in Sα(x0), satisftying the mesh norm condition max ...

The articles [8], [1], [4], [2], [3], [5], [7], [12], [13], [6], [9], and [10] provide the notation and terminology for this paper. We follow the rules: h, r, r1, r2, x0, x1, x2, x3, x4, x5, x, a, b, c, k denote real numbers and f , f1, f2 denote functions from R into R. Next we state a number of propositions: (1)1 ∆[f ](x, x+ h) = ( ~ ∆h[f ])(1)(x) h . (2) If h 6= 0, then ∆[f ](x, x+ h, x+ 2 ·...

A class of multiobjective fractional programmings (MFP) are first formulated, where the involved functions are local Lipschitz and Clarke subdifferentiable. In order to deduce our main results, we give the definitions of the generalized (F,ρ) convex class about the Clarke subgradient. Under the above generalized convexity assumption, the alternative theorem is obtained, and some sufficient and ...

We show that various analogs of Hindman’s Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring c : R → Q, such that for every X ⊆ R with |X| = |R|, and every colour γ ∈ Q, there are two distinct elements x0, x1 of X for which c(x0+x1) = γ. This forms a simultaneous generalization of a theorem of Hindman, Leader and Straus...

A ubiquitous problem that has found many applications, from signal processing to machine learning, is to estimate a highdimensional vector x0 ∈ R from a set of incomplete linear observations y = Mx0 ∈ R. This is an ill-posed problem which admits infinitely many solutions, hence solving it is hopeless unless we can use additional prior knowledge on x0. The assumption that x0 admits a sparse repr...

Given K real d-by-d nonsingular matrices S 1, . . . , S K , by extending the well-known Li-Yorke chaotic description of a deterministic nonlinear dynamical system to a discrete-time linear inclusion dynamical system: xn ∈ {S k xn−1}1≤k≤K with x0 ∈ Rd and n ≥ 1, we study the chaotic characteristic of the state trajectory (xn(x0, σ))n≥1, governed by a switching law σ : N → {1, . . . , K}, for any...

where LAu = div(A(x)∇u) and La,pu = div(a(x)|∇u| ∇u) are, respectively, linear and quasilinear uniformly elliptic operators in divergence form in a non-smooth bounded open subset Ω of R, 1 < p < n, p∗ = np/(n − p) is the critical Sobolev exponent and λ is a real parameter. Both problems have been quite studied when the ellipticity of LA and La,p concentrate in the interior of Ω. We here focus o...

and Applied Analysis 3 x0, y0 ∈ L StrD F, S resp., x0, y0 ∈ LW MinD F, S , if there exists a neighborhood U of x0 in X such that y0 ∈ StrD F U ∩ S ( resp., y0 ∈ W minD F U ∩ S ) , that is, ∀x ∈ S∩U, F x − y0 ∩ −D\{0} ∅ resp. ∀x ∈ S∩U, F x − y0 ∩ − intD ∅. 2.4 We will say that x0, y0 is a global strict global weak minimizers when U X. The set of all global strict minimizers resp., weak minimizer...