Singular mixed boundary value problem

We study singular boundary value problems with mixed boundary conditions of the form

where , , f is a nonnegative function and satisfies the Carathéodory conditions on . Here, f can have a time singularity at t=0 and/or t=T and a space singularity at x=0 and/or y=0. We present conditions for the existence of solutions positive on [0,T) and having continuous first derivatives on [0,T].     

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.     

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.     

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.     

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.     

A new regularized method for two dimensional nonhomogeneous backward heat problem

We consider the problem of finding, from the final data u(x,y,T)=g(x,y), the initial data u(x,y,0) of the temperature function u(x,y,t),(x,y)I=(0,π)×(0,π),t[0,T] satisfying the following system

The problem is severely ill-posed. In this paper a simple and convenient new regularization method for solving this problem is considered. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively.     

Uniqueness of positive solutions for a boundary blow-up problem

In this paper we prove the uniqueness of the positive solution for the boundary blow-up problem

where Ω is a C² bounded domain in , under the hypotheses that f(t) is nondecreasing in t>0 and f(t)/t^p is increasing for large t and some p>1. We also consider the uniqueness of a related problem when the equation includes a nonnegative weight a(x).     

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→∞.     

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.     

Global stability of a difference equation with maximum

We prove that every positive solution to the difference equation

where , i=1,…,k, converges to the following quantity , confirming a quite recent conjecture of interest. We also prove another result on global convergence which concerns some cases when not all α_i,i=1,…,k belong to the interval (0, 1).     

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.     

Multi solitary waves for nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation in for any d1, with a nonlinearity such that solitary waves exist and are stable. Let R_k(t,x) be K arbitrarily given solitary waves of the equation with different speeds v₁,v₂,…,v_K. In this paper, we prove that there exists a solution u(t) of the equation such that

     

Periodic solutions of sublinear Liénard differential equations

In this paper, we study the existence of periodic solutions of the second order differential equations x^″+f(x)x^′+g(x)=e(t). Using continuation lemma, we obtain the existence of periodic solutions provided that F(x) () is sublinear when x tends to positive infinity and g(x) satisfies a new condition

where M, d are two positive constants.     

An improved upper bound for the Laplacian spectral radius of graphs

Let G be a simple graph with n vertices, m edges. Let Δ and δ be the maximum and minimum degree of G, respectively. If each edge of G belongs to t triangles (t≥1), then we present a new upper bound for the Laplacian spectral radius of G as follows:

Moreover, we give an example to illustrate that our result is, in some cases, the best.     

Christoffel-type functions for -orthogonal polynomials for Freud weights

This paper gives upper and lower bounds of the Christoffel-type functions , for the m-orthogonal polynomials for a Freud weight W=e^-Q, which are given as follows. Let a_n=a_n(Q) be the nth Mhaskar–Rahmanov–Saff number, φ_n(x)=max{n^-2/3,1-|x|/a_n}, and d>0. Assume that QC(R) is even, , and for some A,B>1

Then for xR

and for |x|a_n(1+dn^-2/3)

     

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.     

Positive periodic solutions of periodic neutral Lotka–Volterra system with state dependent delays

By using a fixed point theorem of strict-set-contraction, some new criteria are established for the existence of positive periodic solutions of the following periodic neutral Lotka–Volterra system with state dependent delays

where (i,j=1,2,…,n) are ω-periodic functions and (i=1,2,…,n) are ω-periodic functions with respect to their first arguments, respectively.     

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 .     

On the nonlinear wave equation with the mixed nonhomogeneous conditions: Linear approximation and asymptotic expansion of solutions

In this paper, we consider the following nonlinear wave equation

(1)

where , , μ, f, g are given functions. To problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method. In the case of , , μ(z)≥μ₀>0, μ₁(z)≥0, for all , and , , , a weak solution u_ε1,ε2(x,t) having an asymptotic expansion of order N+1 in two small parameters ε₁, ε₂ is established for the following equation associated to (1)_2,3:

(2)

     

On a filter for exponentially localized kernels based on Jacobi polynomials

Let , and for k=0,1,…, denote the orthonormalized Jacobi polynomial of degree k. We discuss the construction of a matrix H so that there exist positive constants c, c₁, depending only on H, α, and β such that

Specializing to the case of Chebyshev polynomials, , we apply this theory to obtain a construction of an exponentially localized polynomial basis for the corresponding L² space.