On locally bounded above solutions of an equation of the Goła̧b–Schinzel type

We characterize solutions ${f, g : \mathbb{R} \to \mathbb{R}}$ of the functional equation f(x + g(x)y) = f(x)f(y) under the assumption that f is locally bounded above at each point ${x \in \mathbb{R}}$ . Our result refers to Go?a?b and Schinzel (Publ Math Debr 6:113–125, 1959) and Wo?od?ko (Aequationes Math 2:12–29, 1968).     

Christensen measurability and some functional equation

Let X be a real separable F-space. We characterize solutions ${f:X\to\mathbb{R}}$ and ${M:\mathbb{R}\to\mathbb{R}}$ of the equation f(x?+?M(f(x))y)?=?f(x)f(y) such that f is bounded on a nonzero Christensen measurable set. Our result generalizes [Jab?o??ska in Acta Math Hung 125(1?C2):113?C119 2009, Theorem 1].     

On the asymptotic behavior of the solutions of semilinear nonautonomous equations

We consider nonautonomous semilinear evolution equations of the form $$\frac{dx}{dt}= A(t)x+f(t,x) . $$ Here A(t) is a (possibly unbounded) linear operator acting on a real or complex Banach space $\mathbb{X}$ and $f: \mathbb{R}\times\mathbb {X}\to\mathbb{X}$ is a (possibly nonlinear) continuous function. We assume that the linear equation (1) is well-posed (i.e. there exists a continuous linear evolution family {U(t,s)}_(t,s)∈Δ such that for every s∈?₊ and x∈D(A(s)), the function x(t)=U(t,s)x is the uniquely determined solution of Eq. (1) satisfying x(s)=x). Then we can consider the mild solution of the semilinear equation (2) (defined on some interval [s,s+δ),δ>0) as being the solution of the integral equation $$x(t) = U(t, s)x + \int_s^t U(t, \tau)f\bigl(\tau, x(\tau)\bigr) d\tau,\quad t\geq s . $$ Furthermore, if we assume also that the nonlinear function f(t,x) is jointly continuous with respect to t and x and Lipschitz continuous with respect to x (uniformly in t∈?₊, and f(t,0)=0 for all t∈?₊) we can generate a (nonlinear) evolution family {X(t,s)}_(t,s)∈Δ , in the sense that the map $t\mapsto X(t,s)x:[s,\infty)\to\mathbb{X}$ is the unique solution of Eq. (4), for every $x\in\mathbb{X}$ and s∈?₊. Considering the Green’s operator $(\mathbb{G}{f})(t)=\int_{0}^{t} X(t,s)f(s)ds$ we prove that if the following conditions hold

• the map $\mathbb{G}{f}$ lies in $L^{q}(\mathbb{R}_{+},\mathbb{X})$ for all $f\in L^{p}(\mathbb{R}_{+},\mathbb{X})$ , and
• $\mathbb{G}:L^{p}(\mathbb{R}_{+},\mathbb{X})\to L^{q}(\mathbb {R}_{+},\mathbb{X})$ is Lipschitz continuous, i.e. there exists K>0 such that $$\|\mathbb{G} {f}-\mathbb{G} {g}\|_{q} \leq K\|f-g\|_{p} , \quad\mbox{for all}\ f,g\in L^p(\mathbb{R}_+,\mathbb{X}) , $$

then the above mild solution will have an exponential decay.     

Long-time asymptotics for fully nonlinear homogeneous parabolic equations

We study the long-time asymptotics of solutions of the uniformly parabolic equation $$ u_t + F(D^2u) = 0 \quad{\rm in}\, {\mathbb{R}^{n}}\times \mathbb{R}_{+},$$ for a positively homogeneous operator F, subject to the initial condition u(x, 0) =  g(x), under the assumption that g does not change sign and possesses sufficient decay at infinity. We prove the existence of a unique positive solution Φ⁺ and negative solution Φ^?, which satisfy the self-similarity relations $$\Phi^\pm (x,t) = \lambda^{\alpha^\pm}\Phi^\pm ( \lambda^{1/2} x,\lambda t ).$$ We prove that the rescaled limit of the solution of the Cauchy problem with nonnegative (nonpositive) initial data converges to ${\Phi^+}$ ( ${\Phi^-}$ ) locally uniformly in ${\mathbb{R}^{n} \times \mathbb{R}_{+}}$ . The anomalous exponents α⁺ and α^? are identified as the principal half-eigenvalues of a certain elliptic operator associated to F in ${\mathbb{R}^{n}}$ .     

Homoclinic orbits for an infinite dimensional Hamiltonian system with periodic potential

In this paper, we study the following diffusion system where $z:(u,v):\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{R}^{2}$ , $V(x)\in C(\mathbb{R}^{N},\mathbb{R})$ is a general periodic function, g(t,x,v), f(t,x,u) are periodic in t,x and superquadratic in v,u at infinity. By using much more direct methods to prove all Cerami sequences for the energy functional are bounded and establish the existence of homoclinic orbits, which are ground state solutions for above system.     

Completeness of the Grid Translates of Functions

If $f\in L^{p}(\mathbb{R}^{d})$ is a bounded real valued continuous function which has a unique maximum or a unique minimum at a point $x_{0}\in \mathbb{R}^{d}$ and if the inverse image of the neighborhoods of f(x ₀) shrinks regularly to x ₀, then $\mathrm{ span }\{f^{m}(x-2^{-m}\varSigma_{i=1}^{d} j_{i} e_{i})\mid m\in\mathbb{N}, j_{i}\in\mathbb{Z}\}$ is a dense subset of $L^{p}(\mathbb{R}^{d}), 1\le p<\infty$ where f ^m (x)=f(x) ^m and {e _i } is the natural basis of $\mathbb{R}^{d}$ . The result extends to all homogeneous groups, Riemannian symmetric spaces of noncompact type, Damek-Ricci spaces etc.     

Classification of rotational special Weingarten surfaces of minimal type in {\mathbb{S}^2 \times \mathbb{R}} and {\mathbb{H}^2 \times \mathbb{R}}

In this paper we classify the complete rotational special Weingarten surfaces in ${\mathbb{S}^2 \times \mathbb{R}}$ and ${\mathbb{H}^2 \times \mathbb{R}}$ ; i.e. rotational surfaces in ${\mathbb{S}^2 \times \mathbb{R}}$ and ${\mathbb{H}^2 \times \mathbb{R}}$ whose mean curvature H and extrinsic curvature K _e satisfy H = f(H ² ? K _e ), for some function ${f \in \mathcal{C}^1([0,+\infty))}$ such that f(0) = 0 and 4x(f′(x))² < 1 for any x ≥ 0. Furthermore we show the existence of non-complete examples of such surfaces.     

Regularity for non-autonomous functionals with almost linear growth

We consider non-autonomous functionals ${\mathcal{F}(u; \Omega)=\int_{\Omega}f(x, Du)\ dx}$ , where the density ${f:\Omega\times\mathbb{R}^{nN}\rightarrow\mathbb{R}}$ has almost linear growth, i.e., $$f(x,\xi)\approx |\xi|\log(1+|\xi|).$$ We prove partial C ^1,?? -regularity for minimizers ${u:\mathbb{R}^n\supset\Omega\rightarrow \mathbb{R}^N}$ under the assumption that D _?? f (x, ??) is H?lder continuous with respect to the x-variable. If the x-dependence is C ¹ we can improve this to full regularity provided additional structure conditions are satisfied.     

On restricting Cauchy–Pexider functional equations to submanifolds

Sufficient geometric conditions are given which determine when the Cauchy–Pexider functional equation f(x)g(y) = h(x + y) restricted to x, y lying on a hypersurface in ${\mathbb{R}^d}$ has only solutions which extend uniquely to exponential affine functions ${\mathbb{R}^d \to \mathbb{C}}$ (when f, g, h are assumed to be measurable and non-trivial). The Cauchy–Pexider-type functional equations ${\prod_{j=0}^df_j(x_j)=F(\sum_{j=0}^dx_j)}$ for ${x_0, \ldots,x_d}$ lying on a curve and ${f_1(x_1)f_2(x_2)f_3(x_3)=F(x_1+x_2+x_3)}$ for x ₁, x ₂, x ₃ lying on a hypersurface are also considered.     

Weak Compactness and Variational Characterization of the Convexity

We prove that if X is a Banach space and ${f : X \rightarrow \mathbb{R} \cup \{+\infty\}}$ is a proper function such that f ? ? attains its minimum for every ? ε X ^*, then the sublevels of f are all relatively weakly compact in X. As a consequence we show that a Banach space X where there exists a function ${f : X \rightarrow \mathbb{R}}$ such that f ? ? attains its minimum for every ? ε X ^* is reflexive. We also prove that if ${f : X \rightarrow \mathbb{R} \cup \{+\infty\}}$ is a weakly lower semicontinuous function on the Banach space X and if for every continuous linear functional ? on X the set where the function f ? ? attains its minimum is convex and non-empty then f is convex.     

Dimension theory and superpositions of continuous functions

The main result of this paper is the following: IfX is a compact two dimensional metric space, and {φ _i} _i ^{= 1/4} are four functions inC(X), then there exists a functionf inC(X) which cannot be represented in the form: $$f(x) = \sum\limits_{i = 1}^4 {g_\iota (\varphi _i (x))} $$ , with $$g_\iota \in C(R)$$ .     

On certain generalizations of the Gagliardo–Nirenberg inequality and their applications to capacitary estimates and isoperimetric inequalities

We derive the inequality $$\int_\mathbb{R}M(|f'(x)|h(f(x))) dx\leq C(M,h)\int_\mathbb{R}M\left({\sqrt{|f''(x)\tau_h(f(x))|}\cdot h(f(x))}\right)dx$$ with a constant C(M, h) independent of f, where f belongs locally to the Sobolev space ${W^{2,1}(\mathbb{R})}$ and f′ has compact support. Here M is an arbitrary N-function satisfying certain assumptions, h is a given function and ${\tau_h(\cdot)}$ is its given transform independent of M. When M(λ) =  λ ^p and ${h \equiv 1}$ we retrieve the well-known inequality ${\int_\mathbb{R}|f'(x)|^{p}dx \leq (\sqrt{p - 1})^{p}\int_\mathbb{R}(\sqrt{|f''(x) f(x)|})^{p}dx}$ . We apply our inequality to obtain some generalizations of capacitary estimates and isoperimetric inequalities due to Maz’ya (1985).     

Planar nearrings on the Euclidean plane

Planar near-rings are generalized rings which can serve as coordinate domains for geometric structures in which each pair of nonparallel lines has a unique point of intersection. It is known that all planar nearrings can be constructed from regular groups of automorphisms of groups which can be viewed as the “action groups” of the planar nearring. In this article, we study planar nearrings whose additive group is \({(\mathbb{R}^n,+)}\) , in particular, n = 1 and 2. It is natural to study topological planar nearrings in this context, following ideas of the late Kenneth D. Magill, Jr. In the case of n = 1, we characterize all topological planar nearrings by their action groups \({(\mathbb{R}^*, \cdot)}\) or \({(\mathbb{R}^+, \cdot)}\) . For n = 2, these action groups and the circle group \({(\mathbb{U}, \cdot)}\) seem to be the most interesting cases, but the last case can be excluded completely. As a consequence, we obtain characterizations of the semi-homogeneous continuous mappings from \({\mathbb{R}^n}\) to \({\mathbb{R}}\) for n = 1 and 2. Such a mapping f enjoys the property that f(f(u)v) = f(u)f(v) for all \({u,v \in \mathbb{R}^n}\) . When \({f(\mathbb{R}^n) = \mathbb{R}^+}\) , f is a positive homogeneous mapping of degree 1.     

On a functional equation of Bruce Ebanks

In the paper Brillouët-Belluot and Ebanks (Aequationes Math 60:233–242, 2000), the authors found all continuous functions f: [0, 1] → [0, + ∞) which verify f(0) = f(1) = 0 and the functional equation $$f(xy +c f(x) f(y)) = x f(y) + y f(x) +d \, f(x) f(y)$$ where c and d are given real numbers with c ≠ 0. In the present paper we obtain all continuous solutions ${f: \mathbb{R} \rightarrow \mathbb{R}}$ of the functional equation (1).     

A proof of a theorem by Fried and MacRae and applications to the composition of polynomial functions

Fried and MacRae (Math. Ann. 180, 220?C226 (1969)) proved that for univariate polynomials ${p,q, f, g \in \mathbb{K}[t]}$ ( ${\mathbb{K}}$ a field) with p, q nonconstant, p(x) ? q(y) divides f(x) ? g(y) in ${\mathbb{K}[x,y]}$ if and only if there is ${h \in \mathbb{K}[t]}$ such that f?=?h(p(t)) and g?=?h(q(t)). Schicho (Arch. Math. 65, 239?C243 (1995)) proved this theorem from the viewpoint of category theory, thereby providing several generalizations to multivariate polynomials. In the present note, we give a new proof of one of these generalizations. The theorem by Fried and MacRae yields a way to prove the following fact for nonconstant functions f, g from ${\mathbb{C}}$ to ${\mathbb{C}}$ : if both the composition ${f \circ g}$ and g are polynomial functions, then f has to be a polynomial function as well. We give an algebraic proof of this fact and present a generalization to multivariate polynomials over algebraically closed fields. This provides a way to prove a generalization of a result by Carlitz (Acta Sci. Math. (Szeged) 24, 196?C203 (1963)) that describes those univariate polynomials over finite fields that induce bijective functions on all of their finite extensions.     

Difference Inequalities for the Groups and Semigroups of Operators on Banach Spaces

Let ${\mathbf{T}=\{T(t)\} _{t\in\mathbb{R}}}$ be a ??(X, F)-continuous group of isometries on a Banach space X with generator A, where ??(X, F) is an appropriate local convex topology on X induced by functionals from ${ F\subset X^{\ast}}$ . Let ?? _A (x) be the local spectrum of A at ${x\in X}$ and ${r_{A}(x):=\sup\{\vert\lambda\vert :\lambda \in \sigma_{A}(x)\},}$ the local spectral radius of A at x. It is shown that for every ${x\in X}$ and ${\tau\in\mathbb{R},}$ $$\left\Vert T(\tau) x-x\right\Vert \leq \left\vert \tau \right\vert r_{A}(x)\left\Vert x\right\Vert.$$ Moreover if ${0\leq \tau r_{A}(x)\leq \frac{\pi}{2},}$ then it holds that $$\left\Vert T(\tau) x-T(-\tau)x\right\Vert \leq 2\sin \left(\tau r_{A}(x)\right)\left\Vert x\right\Vert.$$ Asymptotic versions of these results for C ₀-semigroup of contractions are also obtained. If ${\mathbf{T}=\{T(t)\}_{t\geq 0}}$ is a C ₀-semigroup of contractions, then for every ${x\in X}$ and ????? 0, $$\underset{t\rightarrow \infty }{\lim } \left\Vert T( t+\tau) x-T(t) x\right\Vert\leq\tau\sup\left\{ \left\vert \lambda \right\vert :\lambda \in\sigma_{A}(x)\cap i \mathbb{R} \right\} \left\Vert x\right\Vert. $$ Several applications are given.     

Everywhere divergent Φ-means of Fourier series

For a function ? ∈, L ¹( $\mathbb{T}$ ), we investigate the sequence (C, 1) of mean values Φ(¦S _k (x, ?) ? ?(x)¦), where Φ(t): [0, +∞) → [0,+∞), Φ(0) = 0, is a continuous increasing function. We prove that if Φ increases faster than exponentially, then these means can diverge everywhere. Divergence almost everywhere of such means was established earlier.     

Об аппроксимации свермками и баэисах иэ сдвигов функции

For 2π-periodic functions f ∈ L ^p ( $ \mathbb{T} $ ), 1 ≤ p < ∞, σ ∈ V ( $ \mathbb{T} $ ) and g ∈ L( $ \mathbb{T} $ ), we consider the convolutions $$ (f*d\sigma )_T (x) = \int_0^{2\pi } {f(x - t)d\sigma (t), } (f*g)_T (x) = \int_0^{2\pi } {f(x - t)g(t)dt.} $$ For fixed functions σ ∈ V ( $ \mathbb{T} $ ) and g ∈ L( $ \mathbb{T} $ ), necessary and sufficient conditions are obtained for the density of the ranges of these operators in L ^p . Similar result is proved for the dyadic convolution $$ (f*g)_2 (x) = \int_0^1 {f(x \oplus t)g(t)dt,} $$ where ⊕ is the operation of dyadic addition on [0, 1). Moreover, it is proved that in the spaces L ^p ( $ \mathbb{T} $ ), 1 ≤ p ∞, and C( $ \mathbb{T} $ ) there exist no bases of shifts of a function. Similar results are obtained for the spaces L ^p [0, 1]*, 1 ≤ p < ∞, and C[0, 1]* relative to dyadic shifts, where [0, 1]* is the modified segment [0, 1]. It is also proved that in the space L(?₊) there exists no basis of dyadic shifts of a function.     

Algebraic equations on the adèlic closure of a Drinfeld module

Let k be a field of positive characteristic and K = k(V) a function field of a variety V over k and let A _K be the ring of adèles of K with respect to the places on K corresponding to the divisors on V. Given a Drinfeld module $\Phi :\mathbb{F}[t] \to End_K (\mathbb{G}_a )$ over K and a positive integer g we regard both K ^g and A _K ^g as $\Phi \left( {\mathbb{F}_p [t]} \right)$ -modules under the diagonal action induced by Φ. For Γ ? K ^g a finitely generated $\Phi \left( {\mathbb{F}_p [t]} \right)$ -submodule and an affine subvariety $X \subseteq \mathbb{G}_a^g$ defined over K, we study the intersection of X(A _K ), the adèlic points of X, with $\bar \Gamma$ , the closure of Γ with respect to the adèlic topology, showing under various hypotheses that this intersection is no more than X(K) ∩ Γ.     

On Uniform Approximation in R2 of Continuous Functions of Two Variables with Bounded Variation in the Sense of Hardy

Introduce the notation: $\mathbb{Z}$ is the set of integers, $\bar {\mathbb{Z}}={\mathbb{Z}} \cup \{-\infty, +\infty\},{\mathbb{R}}_+^2 =\{x=(x_1,x_2) \in {\mathbb{R}}^2; x_1>0,x_2>0\}$ , $g_{k,m} (x,\alpha,h)= \int\limits_0^1 {g_1 (\frac{(k+u)h_1 - x_1}{\alpha_1})g_2(\frac{(m+u)h_2 - x_2}{\alpha_2})}du$ , where $g_i :\mathbb{R} \to \mathbb{R},x \in \mathbb{R}^2 ,\alpha ,h \in \mathbb{R}_ + ^2 $ . Under certain conditions on the functions g ₁, g ₂, we prove that the system of functions $g_{k,m} (x,\alpha^(n), h^(n)) (k,m \in \bar {\mathbb{Z}})$ , where $\alpha ^{\left( n \right)} ,h^{\left( n \right)} \in \mathbb{R}_ + ^2 $ are arbitrary infinitesimal sequences, is complete in the space C $\mathbb{R}^2 $ of uniformly continuous bounded functions f equipped with the norm $||f|| = \mathop {\sup }\limits_{x \in \mathbb{R}^2 } |f(x)|$ . Starting with the functions g _k,m , it is possible to construct a method for uniform approximating in $\mathbb{R}^2 $ any continuous function of bounded variation in the sense of Hardy. An error estimate is derived in terms of the second order moduli of continuity. Based on the obtained results, we discuss in detail the accuracy of uniform approximation of functions of several variables by linear functions. The error estimates are derived by using second order moduli of continuity. We pay a particular attention to sharpness of constants. Bibliography: 8 titles.