Given two functions f, g : {0, 1}n → {0, 1} a mapping ψ : {0, 1}n → {0, 1}n is said to be a mapping from f to g if it is a bijection and f(z) = g(ψ(z)) for every z ∈ {0, 1}n. In this paper we study Lipschitz mappings between boolean functions. Our first result gives a construction of a C-Lipschitz mapping from the Majority function to the Dictator function for some universal constant C. On the ...

We use Baire categorical arguments to construct dramatically pathological locally Lipschitz functions. The origins of this approach can be traced back to Banach and Mazurkiewicz (1931) who independently used similar categorical arguments to show that \almost every continuous real-valued function deened on 0,1] is nowhere diierentiable". As with the results of Banach and Mazurkiewicz, it appears...

We prove that a Lipschitz (or uniformly continuous) mapping f : X → Y can be approximated by smooth Lipschitz (resp. uniformly continuous) mapping, if X is a separable Banach space admitting a smooth Lipschitz bump and either X or Y is a C(K) space (resp. super-reflexive space). As a corollary we obtain also smooth approximation of C1-smooth mappings together with their first derivatives.

The aim of paper is to prove a weak convergenceresult for finding a common of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. Using an example in C++, validity of the result will be proved. Also, we shall find a common element of the set of fixed points of a nonexpansive mapping and the ...

In [11] the author proved that every quasiconformal harmonic mapping between two Jordan domains with C, 0 < α ≤ 1, boundary is biLipschitz, providing that the domain is convex. In this paper we avoid the restriction of convexity. More precisely we prove: any quasiconformal harmonic mapping between two Jordan domains Ωj , j = 1, 2, with C, j = 1, 2 boundary is bi-Lipschitz.

and Applied Analysis 3 2. Main Results In this paper, we are interested in the following classes of mappings. Definition 8. Let T : H → H be a mapping.Then T is called (a) β-strongly monotone if there exists β > 0 such that ⟨T (x) − T (y) , x − y⟩ ≥ β 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 2 , ∀x, y ∈ H, (12) (b) ξLipschitz if there exists ξ > 0 such that 󵄩󵄩󵄩󵄩T (x) − T (y) 󵄩󵄩󵄩󵄩 ≤ ξ 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 , ∀x, y ∈ H. (13) Tha...

In a paper of 1950 L. M. Graves proved that, for a function f acting between Banach spaces and an interior point x̄ in its domain, if there exists a continuous linear mapping A which is surjective and the Lipschitz modulus of the difference f − A at x̄ is sufficiently small, then f is (linearly) open at x̄. This is an extension of the Banach open mapping principle from continuous linear mappings t...

We show that S.Vavasis’ sufficient condition for global invertibility of a polynomial mapping can be easily generalized to the case of a general Lipschitz mapping.

In a paper of 1950 Graves proved that for a function f acting between Banach spaces and an interior point x̄ in its domain, if there exists a continuous linear mapping A which is surjective and the Lipschitz modulus of the difference f −A at x̄ is sufficiently small, then f is (linearly) open at x̄. This is an extension of the Banach open mapping principle from continuous linear mappings to Lipsch...