Approximately Linear Mappings in Banach Modules over a C^＊-algebra

Let X and Y be vector spaces. The authors show that a mapping f ： X →Y satisfies the functional equation 2d f（∑^2d j=1（-1）^j＋1xj/2d）=∑^2dj=1（-1）^j＋1f（xj） with f（0） = 0 if and only if the mapping f ： X→ Y is Cauchy additive, and prove the stability of the functional equation （≠） in Banach modules over a unital C^＊-algebra, and in Poisson Banach modules over a unital Poisson C＊-algebra. Let A and B be unital C^＊-algebras, Poisson C^＊-algebras or Poisson JC^＊- algebras. As an application, the authors show that every almost homomorphism h ： A →B of A into is a homomorphism when h（（2d-1）^nuy） =- h（（2d-1）^nu）h（y） or h（（2d-1）^nuoy） = h（（2d-1）^nu）oh（y） for all unitaries u ∈A, all y ∈ A, n = 0, 1, 2,.... Moreover, the authors prove the stability of homomorphisms in C^＊-algebras, Poisson C^＊-algebras or Poisson JC^＊-algebras.     

On Homomorphisms between <Emphasis Type="Italic">C</Emphasis>*-Algebras and Linear Derivations on <Emphasis Type="Italic">C</Emphasis>*-Algebras

It is shown that every almost linear Pexider mappings f, g, h from a unital C*-algebra into a unital C*-algebra ℬ are homomorphisms when f(2 ⁿ _uy) = f(2 ⁿ u)f(y), g(2 ⁿ uy) = g(2 ⁿ u)g(y) and h(2 ⁿ uy) = h(2 ⁿ u)h(y) hold for all unitaries u ∈ , all y ∈ , and all n ∈ ℤ, and that every almost linear continuous Pexider mappings f, g, h from a unital C*-algebra of real rank zero into a unital C*-algebra ℬ are homomorphisms when f(2 ⁿ uy) = f(2 ⁿ u)f(y), g(2 ⁿ uy) = g(2 ⁿ u)g(y) and h(2 ⁿ uy) = h(2 ⁿ u)h(y) hold for all u ∈ {v ∈ : v = v* and v is invertible}, all y ∈ and all n ∈ ℤ. Furthermore, we prove the Cauchy-Rassias stability of *-homomorphisms between unital C*-algebras, and ℂ-linear *-derivations on unital C*-algebras. This work was supported by Korea Research Foundation Grant KRF-2003-042-C00008. The second author was supported by the Brain Korea 21 Project in 2005.     

Almost homomorphisms between unital <Emphasis Type="Italic">C</Emphasis>*-algebras: A fixed point approach

Let A , B be two unital C*-algebras. By using fixed pint methods, we prove that every almost unital almost linear mapping h : A → B which satisfies h(2 n uy) = h(2 n u)h(y) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, … , is a homomorphism. Also, we establish the generalized Hyers-Ulam-Rassias stability of *-homomorphisms on unital C*-algebras.     

A generalized Jensen’s mapping and linear mappings between Banach modules

Let X and Y be vector spaces. It is shown that a mapping f : X → Y satisfies the functional equation

Isomorphisms between unital C-algebras

It is shown that every almost linear bijection of a unital C^∗-algebra A onto a unital C^∗-algebra B is a C^∗-algebra isomorphism when h(n²uy)=h(n²u)h(y) for all unitaries u∈A, all y∈A, and n=0,1,2,…, and that almost linear continuous bijection of a unital C^∗-algebra A of real rank zero onto a unital C^∗-algebra B is a C^∗-algebra isomorphism when h(n²uy)=h(n²u)h(y) for all , all y∈A, and n=0,1,2,…. Assume that X and Y are left normed modules over a unital C^∗-algebra A. It is shown that every surjective isometry , satisfying T(0)=0 and T(ux)=uT(x) for all x∈X and all unitaries u∈A, is an A-linear isomorphism. This is applied to investigate C^∗-algebra isomorphisms between unital C^∗-algebras.     

Bilinear isometries on subspaces of continuous functions

Let A and B be strongly separating linear subspaces of C₀(X) and C₀(Y), respectively, and assume that ?A ≠ ?? (?A stands for the set of generalized peak points for A) and ?B ≠ ??. Let T: A × B → C₀(Z) be a bilinear isometry. Then there exist a nonempty subset Z₀ of Z, a surjective continuous mapping h: Z₀ → ?A × ?B and a norm‐one continuous function a: Z₀ → K such that T (f, g)(z) = a (z)f (π_x (h (z))g (π_y (h (z)) for all z ∈ Z₀ and every pair (f, g) ∈ A × B. These results can be applied, for example, to non‐unital function algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The existence of positive solutions to a non‐local singular boundary value problem

We consider the non‐local singular boundary value problem (1) where q ∈ C⁰([0,1]) and f, h ∈ C⁰((0,∞)), limf(x)=?∞, limh(x)=∞. We present conditions guaranteeing the existence of a solution x ∈ C¹([0,1]) ∩ C²((0,1]) which is positive on (0,1]. The proof of the existence result is based on regularization and sequential techniques and on a non‐linear alternative of Leray–Schauder type. Copyright © 2005 John Wiley & Sons, Ltd.     

Generalized weak peripheral multiplicativity in algebras of Lipschitz functions

Let (X, d _X ) and (Y,d _Y ) be pointed compact metric spaces with distinguished base points e _X and e _Y . The Banach algebra of all $\mathbb{K}$ -valued Lipschitz functions on X — where $\mathbb{K}$ is either?or ? — that map the base point e _X to 0 is denoted by Lip₀(X). The peripheral range of a function f ∈ Lip₀(X) is the set Ran_µ(f) = {f(x): |f(x)| = ‖f‖_∞} of range values of maximum modulus. We prove that if T ₁, T ₂: Lip₀(X) → Lip₀(Y) and S ₁, S ₂: Lip₀(X) → Lip₀(X) are surjective mappings such that $Ran_\pi (T_1 (f)T_2 (g)) \cap Ran_\pi (S_1 (f)S_2 (g)) \ne \emptyset $ for all f, g ∈ Lip₀(X), then there are mappings φ₁φ₂: Y → $\mathbb{K}$ with φ₁(y)φ₂(y) = 1 for all y ∈ Y and a base point-preserving Lipschitz homeomorphism ψ: Y → X such that T _j (f)(y) = φ _j (y)S _j (f)(ψ(y)) for all f ∈ Lip₀(X), y ∈ Y, and j = 1, 2. In particular, if S ₁ and S ₂ are identity functions, then T ₁ and T ₂ are weighted composition operators.     

Generalized Hyers–Ulam–Rassias stability of n-sesquilinear-quadratic mappings on Banach modules over -algebras

Assume that X is a left Banach module over a unital C^*-algebra A. It is shown that almost every n-sesquilinear-quadratic mapping h:X×X×Xⁿ→A is an n-sesquilinear-quadratic mapping when holds for all x,y,z₁,…,z_nX.Moreover, we prove the generalized Hyers–Ulam–Rassias stability of an n-sesquilinear-quadratic mapping on a left Banach module over a unital C^*-algebra.     

Compact homomorphisms of regular banach algebras

Let A be a complex commutative Banach algebra and let M_A be the maximal ideal space of A. We say that A has the bounded separating property if there exists a constant C > 0 such that for every two distinct points ?₁, ?₂ ∈ M_A, there is an element a ∈ A for which , and ‖a‖ ? C, where is the Gelfand transform of a ∈ A. We show that if A is a strongly regular Banach algebra with the bounded separating property, then every compact homomorphism from A into another Banach algebra is of finite dimensional range. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Inverse closedness ofC *-algebras in Banach algebras

LetA denote a unital Banach algebra, and letB denote aC ^*-algebra which is contained as a unital subalgebra inA. We prove thatB is inverse closed inA if the norms ofA andB coincide. This generalizes well known result about inverse closedness ofC ^*-subalgebras inC ^*-algebras.     

Maximal L p regularity for parabolic difference equations

In this paper, we consider a family of finite difference operators {Ah }_{h >0} on discrete L _q ‐spaces L _q (?^N _h ). We show that the solution u _h to u ′_h (t) – A _h u _h(t) = f _h (t), t > 0, u _h (0) = 0 satisfies the estimate ‖A _h u _h ‖equation/tex2gif-inf-15.gif ≤ C ‖f _h ‖equation/tex2gif-inf-21.gif, where C is independent of h and f _h . In this case, the family {A _h }_{h >0} is said to have discrete maximal L _p regularity on the discrete L _q ‐space. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Banach–Stone Theorems for maps preserving common zeros

Let X and Y be completely regular spaces and E and F be Hausdorff topological vector spaces. We call a linear map T from a subspace of C(X, E) into C(Y, F) a Banach–Stone map if it has the form T f (y) =  S _y (f (h(y))) for a family of linear operators S _y : E → F, \({y \in Y}\) , and a function h: Y → X. In this paper, we consider maps having the property:

$\bigcap^{k}_{i=1}Z(f_{i}) \neq\emptyset \iff \bigcap^{k}_{i=1}Z(Tf_{i})\neq\emptyset , \quad({\rm Z}) $

where Z(f) =  {f =  0}. We characterize linear bijections with property (Z) between spaces of continuous functions, respectively, spaces of differentiable functions (including C ^∞), as Banach–Stone maps. In particular, we confirm a conjecture of Ercan and Önal: Suppose that X and Y are realcompact spaces and E and F are Hausdorff topological vector lattices (respectively, C ^*-algebras). Let T: C(X, E) → C(Y, F) be a vector lattice isomorphism (respectively, *-algebra isomorphism) such that

$ Z(f) \neq\emptyset\iff Z(Tf) \neq\emptyset. $

Then X is homeomorphic to Y and E is lattice isomorphic (respectively, C ^*-isomorphic) to F. Some results concerning the continuity of T are also obtained.

     

Young‐measure solutions to a generalized Benjamin–Bona–Mahony equation

We consider the evolution of microstructure under the dynamics of the generalized Benjamin–Bona–Mahony equation (1) with u: ?² → ?. If we model the initial microstructure by a sequence of spatially faster and faster oscillating classical initial data vⁿ, we obtain a sequence of spatially highly oscillatory classical solutions uⁿ. By considering the Young measures (YMs) ν and µ generated by the sequences vⁿ and uⁿ, respectively, as n → ∞, we derive a macroscopic evolution equation for the YM solution µ, and show exemplarily how such a measure‐valued equation can be exploited in order to obtain classical evolution equations for effective (macroscopic) quantities of the microstructure for suitable initial data vⁿ and non‐linearities f. Copyright © 2005 John Wiley & Sons, Ltd.     

Subsolutions for abstract evolution equations

We study different notions of subsolutions for an abstract evolution equation du/dt+Auf where A is an m-accretive nonlinear operation in an ordered Banach space X with order-preserving resolvents. A first notion is related to the operator d/dt+A in the ordered Banach space L ¹(0, T; X); a second one uses the evolution equation du/dt+A uf where A :x{y;zy for some zAx}; other notions are also considered.     

Homomorphisms between JC＊-algebras and Lie C＊-algebras

It is shown that every almost ＊-homomorphism h ： A→B of a unital JC＊-algebra A to a unital JC＊-algebra B is a ＊-homomorphism when h（rx） = rh（x） （r 〉 1） for all x∈A, and that every almost linear mapping h ： A→B is a ＊-homomorphism when h（2^nu o y） - h（2^nu） o h（y）, h（3^nu o y） - h（3^nu） o h（y） or h（q^nu o y） = h（q^nu） o h（y） for all unitaries u ∈A, all y ∈A, and n = 0, 1,.... Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings. We prove that every almost ＊-homomorphism h ： A→B of a unital Lie C＊-algebra A to a unital Lie C＊-algebra B is a ＊-homomorphism when h（rx） = rh（x） （r 〉 1） for all x ∈A.     

Infinitely many solutions for p‐Kirchhoff equation with concave–convex nonlinearities in

In this paper, we study the existence of infinitely many solutions to p‐Kirchhoff‐type equation (0.1) where f(x,u) = λh₁(x)|u|^{m ? 2}u + h₂(x)|u|^{q ? 2}u,a≥0,μ > 0,τ > 0,λ≥0 and . The potential function verifies , and h₁(x),h₂(x) satisfy suitable conditions. Using variational methods and some special techniques, we prove that there exists λ₀>0 such that problem 0.1 admits infinitely many nonnegative high‐energy solutions provided that λ∈[0,λ₀) and . Also, we prove that problem 0.1 has at least a nontrivial solution under the assumption f(x,u) = h₂|u|^{q ? 2}u,p < q< min{p*,p(τ + 1)} and has infinitely many nonnegative solutions for f(x,u) = h₁|u|^{m ? 2}u,1 < m < p. Copyright © 2015 John Wiley & Sons, Ltd.     

The Helmholtz–Weyl decomposition in weighted Sobolev spaces

Let f∈L_{2, ? µ}(?³), where where x = (x₁, x₂, x₃) is the Cartesian system in ?³, x′ = (x₁, x₂), , µ∈?₊\?. We prove the decomposition f = ? ?u + g, with g divergence free and u is a solution to the problem in ?³ Given f∈L_{2, ? µ}(?³) we show the existence of u∈H(?³) such that where Since f, u, g are defined in ?³ we need a sufficiently fast decay of these functions as |x|→∞. Copyright © 2010 John Wiley & Sons, Ltd.     

Quantitative form of the Luzin <Emphasis Type="Italic">C</Emphasis>-property

We prove the following statement, which is a quantitative form of the Luzin theorem on C-property: Let (X, d, μ) be a bounded metric space with metric d and regular Borel measure μ that are related to one another by the doubling condition. Then, for any function f measurable on X, there exist a positive increasing function η ∈ Ω (η(+0) = 0 and η(t)t ^−a decreases for a certain a > 0), a nonnegative function g measurable on X, and a set E ⊂ X, μE = 0 , for which

Lower-semicontinuity result for the two-scale convergence

Let (m, n) ∈ ℕ², Ω an open bounded domain in ℝ^m , Y = [0, 1]^m ; u^ε in (L²(Ω))ⁿ which is two-scale converges to some u in (L²(Ω × Y))ⁿ . Let φ: Ω × ℝ^m × ℝⁿ → ℝ such that: φ(x, ·, ·) is continuous a.e. x ∈ Ω φ(·, y, z) is measurable for all (y, z) in ℝ^m × ℝⁿ , φ(x, ·, z) is 1-periodic in y, φ(x, y, ·) is convex in z. Assume that there exist a constant C₁ > 0 and a function C₂ ∈ L²(Ω) such that