On approximation properties of a family of linear operators at critical value of parameter

We introduce the family of linear operators

associated to a certain “admissible bunch” of operators S_t, t>0, acting on , and investigate the approximation properties of this family as α→0⁺. We give some applications to the Riesz and the Bessel potentials generated by the ordinary (Euclidean) and generalized translations.     

Infinitely many non-negative solutions for a Dirichlet problem involving -Laplacian

In this paper, we consider a Dirichlet problem involving the p(x)-Laplacian of the type

We prove the existence of infinitely many non-negative solutions of the problem by applying a general variational principle due to B. Ricceri and the theory of the variable exponent Sobolev spaces.     

Generalised twists, , and the p-energy over a space of measure preserving maps

Let be a bounded Lipschitz domain and consider the energy functional

with p]1,∞[ over the space of measure preserving maps

In this paper we introduce a class of maps referred to as generalised twists and examine them in connection with the Euler–Lagrange equations associated with over . The main result is a surprising discrepancy between even and odd dimensions. Here we show that in even dimensions the latter system of equations admit infinitely many smooth solutions, modulo isometries, amongst such maps. In odd dimensions this number reduces to one. The result relies on a careful analysis of the full versus the restricted Euler–Lagrange equations where a key ingredient is a necessary and sufficient condition for an associated vector field to be a gradient.     

Stationary twists and energy minimizers on a space of measure preserving maps

Let be a bounded Lipschitz domain, a suitably quasiconvex integrand and consider the energy functional

over the space of measure preserving maps

In this paper we discuss the question of existence of multiple strong local minimizers for over . Moreover, motivated by their significance in topology and the study of mapping class groups, we consider a class of maps, referred to as twists, and examine them in connection with the corresponding Euler–Lagrange equations and investigate various qualitative properties of the resulting solutions, the stationary twists. Particular attention is paid to the special case of the so-called p-Dirichlet energy, i.e., when .     

RELATIONS BETWEEN PACKING PREMEASURE AND MEASURE ON METRIC SPACE

Let X be a metric space andμa finite Borel measure on X. Let pμq,t and pμq,t be the packing premeasure and the packing measure on X, respectively, defined by the gauge (μB(x,r))q(2r)t, where q, t∈R. For any compact set E of finite packing premeasure the authors prove: (1) if q≤0 then pμq,t(E)=pμq,t(E);(2)if q>0 andμis doubling on E then pμq,t(E) and pμq,t(E) are both zero or neither.     

Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source

In this paper we investigate the global existence and finite time blow-up of solutions to the system of nonlinear viscoelastic wave equations

in Ω×(0,T) with initial and Dirichlet boundary conditions, where Ω is a bounded domain in . Under suitable assumptions on the functions g_i(), , the initial data and the parameters in the equations, we establish several results concerning local existence, global existence, uniqueness and finite time blow-up property.     

On solvability of boundary value problems for higher order nonlinear hyperbolic equations

In the rectangle Ω=[0,a]×[0,b] for the nonlinear hyperbolic equation

the boundary value problems of the type

are considered, where and are linear bounded functionals.Sufficient conditions of solvability and unique solvability of the general problem and its particular cases (Nicoletti type, Dirichlet, Lidstone and Periodic problems) are established.     

Classical and non-classical solutions of a prescribed curvature equation

We discuss existence and multiplicity of positive solutions of the one-dimensional prescribed curvature problem

depending on the behaviour at the origin and at infinity of the potential . Besides solutions in W^2,1(0,1), we also consider solutions in which are possibly discontinuous at the endpoints of [0,1]. Our approach is essentially variational and is based on a regularization of the action functional associated with the curvature problem.     

Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws

We consider a system of heat equations u_t=Δu and v_t=Δv in Ω×(0,T) completely coupled by nonlinear boundary conditions

We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on ∂Ω with

for p,q>0, 0≤α<1 and 0≤β<p.     

Three solutions for a differential inclusion problem involving the -Laplacian

In this paper we consider differential inclusion problem involving the p(x)-Laplacian of the type

Applying a version of the non-smooth three-critical-points theorem we obtain the existence of three solutions of the problem in .     

Characterisation of the weak lower semicontinuity for a type of nonlocal integral functional: The n-dimensional scalar case

In this work we are going to prove the functional J defined by

is weakly lower semicontinuous in W^1,p(Ω) if and only if W is separately convex. We assume that Ω is an open set in and W is a real-valued continuous function fulfilling standard growth and coerciveness conditions. The key to state this equivalence is a variational result established in terms of Young measures.     

The maximal range problem for a bounded domain

Let be a bounded domain such that 0Ω. Denote by , the set of all complex polynomials of degree at most n. Let

where . We relate the maximal polynomial range

to the geometry of Ω.     

New exact solutions for the classical Drinfel’d–Sokolov–Wilson equation

In this paper, we investigate the classical Drinfel’d–Sokolov–Wilson equation (DSWE)

where p, q, r, s are some nonzero parameters. Some explicit expressions of solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain solitary wave solutions, blow-up solutions, periodic solutions, periodic blow-up solutions and kink-shaped solutions. Some previous results are extended.     

Positivity of Szegö's rational function

We consider the problem of deciding whether a given rational function has a power series expansion with all its coefficients positive. Introducing an elementary transformation that preserves such positivity we are able to provide an elementary proof for the positivity of Szegö's function

which has been at the historical root of this subject starting with Szegö. We then demonstrate how to apply the transformation to prove a 4-dimensional generalization of the above function, and close with discussing the set of parameters (a,b) such that

has positive coefficients.     

Asymptotic behavior of solutions of nonlinear impulsive delay differential equations with positive and negative coefficients

This paper is concerned with the nonlinear impulsive delay differential equations with positive and negative coefficients

(*)

Sufficient conditions are obtained for every solution of equation (*) tending to a constant as t→∞.     

Marcinkiewicz–Zygmund strong laws for -statistics of weakly dependent observations

We prove the Marcinkiewicz–Zygmund Strong Law of Large Numbers for U-statistics of strictly stationary, absolutely regular observations (ξ_i)_i≥1. Under suitable moment conditions and conditions on the mixing rate, we show that

for some γ≥0, in the non-degenerate case, and

in the degenerate case.     

Stochastic functional Kolmogorov-type population dynamics

In this paper we stochastically perturb the functional Kolmogorov-type system

into the stochastic functional differential equation

This paper studies existence and uniqueness of the global positive solution, and its asymptotic bound properties and moment average in time. These properties are natural requirements from the biological point of view. As the special cases, we discuss the various stochastic Lotka–Volterra systems.     

Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay

We consider the existence and uniqueness of a Stepanov-like almost automorphic solution to the nonautonomous semilinear evolution equations with a constant delay:

where , generates an exponentially stable evolution family {U(t,s)} and satisfies a Lipschitz condition with respect to the second argument.     

A note on the distance between two consecutive zeros of m-orthogonal polynomials for a generalized Jacobi weight

Let , with

-1=x_0n<x_1n<<x_nn<x_n+1,n=1