On everywhere divergence of the strong Φ-means of Walsh–Fourier series

Almost everywhere strong exponential summability of Fourier series in Walsh and trigonometric systems was established by Rodin in 1990. We prove, that if the growth of a function Φ(t):[0,∞)→[0,∞)$\Phi (t):[0,\infty )\to [0,\infty )$ is bigger than the exponent, then the strong Φ-summability of a Walsh–Fourier series can fail everywhere. The analogous theorem for trigonometric system was proved before by one of the authors of this paper.     

Some remarks on the asymptotic profiles of solutions for strongly damped wave equations on the 1-D half space

We consider a mixed problem with the Dirichlet null boundary conditions on the 1-dimensional half space (0,+∞)$(0,+\infty )$. We will derive an asymptotic profiles of the corresponding solutions in the case when the initial data belong to a weighted L^1,2(0,∞)$L^{1,2}(0,\infty )$ space by employing a method recently introduced in [7] or [8].     

Families of completely positive maps associated with monotone metrics

An operator convex function on (0,∞)$(0,\infty )$ which satisfies the symmetry condition k(x⁻¹)=xk(x)$k(x^{-1})=xk(x)$ can be used to define a type of non-commutative multiplication by a positive definite matrix (or its inverse) using the primitive concepts of left and right multiplication and the functional calculus. The operators for the inverse can be used to define quadratic forms associated with Riemannian metrics which contract under the action of completely positive trace-preserving maps.     

Global existence and nonexistence of solutions for second-order nonlinear differential equations

This paper deals with the global existence and nonexistence of solutions of the second-order nonlinear differential equation (φ(x^′))^′+λφ(x)=0${(\phi (x^{\prime }))}^{\prime }+\lambda \phi (x)=0$ satisfying x(0)=_x0$x(0)=x_0$ and x^′(0)=_x1$x^{\prime }(0)=x_1$, where λ   is a positive parameter and φ:(−ρ,ρ)→(−σ,σ)$\phi :(-\rho ,\rho )\to (-\sigma ,\sigma )$ with 0<ρ?∞$0<\rho ?\infty$ and 0<σ?∞$0<\sigma ?\infty$ is strictly increasing odd bijective and continuous on (−ρ,ρ)$(-\rho ,\rho )$. Necessary and sufficient conditions are obtained for the initial value problem to have a unique global solution which is oscillatory and periodic. Examples are given to illustrate our main result. Finally, a nonexistence result for the equation with a damping term is discussed as an application to our result.     

Minimal time of controllability of two parabolic equations with disjoint control and coupling domains

We consider two parabolic equations coupled by a matrix A(x)=q(x)A₀$A(x)=q(x)A_0$, where A₀$A_0$ is a Jordan block of order 1, and controlled by a single localized function, or by a single boundary control. The support of the coupling coefficient, q  , and the control domain may be disjoint. We exhibit an explicit minimal time of null-controllability, T₀(q)∈[0,+∞]$T_0(q)\in [0,+\infty ]$.     

Asymptotic solutions for large time of Hamilton–Jacobi equations in Euclidean n space

We study the large time behavior of solutions of the Cauchy problem for the Hamilton–Jacobi equation ut+H(x,Du)=0$u_t+H(x,Du)=0$ in Rn×(0,∞)${\mathbf{R}}^n\times (0,\infty )$, where H(x,p)$H(x,p)$ is continuous on Rn×Rn${\mathbf{R}}^n\times {\mathbf{R}}^n$ and convex in p  . We establish a general convergence result for viscosity solutions u(x,t)$u(x,t)$ of the Cauchy problem as t→∞$t\to \infty$.     

Real-variables characterization of generalized Stieltjes functions

We obtain a characterization of generalized Stieltjes functions of any order λ>0$\lambda >0$ in terms of inequalities for their derivatives on (0,∞)$(0,\infty )$. When λ=1$\lambda =1$, this provides a new and simple proof of a characterization of Stieltjes functions first obtained by Widder in 1938.     

Infinitely many solutions for a class of semilinear elliptic equations

In this paper we find some new conditions to ensure the existence of infinitely many nontrivial solutions for the Dirichlet boundary value problems of the form −Δu+a(x)u=g(x,u)$-\mathrm{\Delta }u+a(x)u=g(x,u)$ in a bounded smooth domain. Conditions (_S1)$(S_1)$–(_S3)$(S_3)$ in the present paper are somewhat weaker than the famous Ambrosetti–Rabinowitz-type superquadratic condition. Here, we assume that the primitive of the nonlinearity g   is either asymptotically quadratic or superquadratic as |u|→∞$|u|\to \infty$.