Positive periodic solutions of functional differential equations

We consider the existence, multiplicity and nonexistence of positive ω-periodic solutions for the periodic equation x′(t)=a(t)g(x)x(t)−λb(t)f(x(t−τ(t))), where are ω-periodic, , , f,g∈C([0,∞),[0,∞)), and f(u)>0 for u>0, g(x) is bounded, τ(t) is a continuous ω-periodic function. Define , , i₀=number of zeros in the set and i_∞=number of infinities in the set . We show that the equation has i₀ or i_∞ positive ω-periodic solution(s) for sufficiently large or small λ>0, respectively.     

Existence and multiplicity of positive solutions for the p-Laplacian with nonlocal coefficient

We consider the Dirichlet problem with nonlocal coefficient given by in a bounded, smooth domain Ω⊂Rn (n?2), where Δp is the p-Laplacian, w is a weight function and the nonlinearity f(u) satisfies certain local bounds. In contrast with the hypotheses usually made, no asymptotic behavior is assumed on f. We assume that the nonlocal coefficient (q?1) is defined by a continuous and nondecreasing function satisfying a(t)>0 for t>0 and a(0)?0. A positive solution is obtained by applying the Schauder Fixed Point Theorem. The case a(t)=tγ/q (0<γ<p−1) will be considered as an example where asymptotic conditions on the nonlinearity provide the existence of a sequence of positive solutions for the problem with arbitrarily large sup norm.     

On a parabolic problem with nonlinear term in a half space and global behavior of solutions

We take up the existence and global behavior of positive continuous solutions of the following nonlinear parabolic equation in (n?2) with boundary conditions u=0 on and u(x,0)=u₀(x). The nonlinear term is required to satisfy some conditions related to a functional class , which we introduce in this paper and will be called parabolic Kato class in the half space. Our approach is based on potential theory.     

Global existence and asymptotic behavior of solutions for nonlinear parabolic equations on unbounded domains

We study the existence and the asymptotic behavior of positive solutions for the parabolic equation on D×(0,∞), where is a some unbounded domain in and V belongs to a new parabolic class J^∞ of singular potentials generalizing the well-known parabolic Kato class at infinity P^∞ introduced recently by Zhang. We also show that the choice of this class is essentially optimal.     

Positive solutions and eigenvalue intervals of nonlocal boundary value problems with delays

The paper is concerned with the delay differential equation u^″+λb(t)f(u(t−τ))=0 satisfying u(t)=0 for −τ?t?0 and , where denotes the Riemann-Stieltjes integral. By applying the fixed point theorem in cones, we show the relationship between the asymptotic behaviors of the quotient (at zero and infinity) and the open intervals (eigenvalue intervals) of the parameter λ such that the problem has zero, one and two positive solution(s). If g(t)=t, by using a property of the Riemann-Stieltjes integral, the above nonlocal boundary value problem educes a three-point boundary value problem with delay, for which some similar results are established.     

Exponential elliptic boundary value problems on a solid torus in the critical of supercritical case

In this paper we investigate the behavior and the existence of positive and non-radially symmetric solutions to nonlinear exponential elliptic model problems defined on a solid torus of R³, when data are invariant under the group G=O(2)×I⊂O(3). The model problems of interest are stated below:

     

Identities for sums of characteristic values of entire matrix pencils

The paper deals with an entire matrix-valued function of a complex argument (an entire matrix pencil) f of order ρ(f)<∞. Identities for the following sums of the characteristic values of f are established:

     

A remark on Ricceri's conjecture for a class of nonlinear eigenvalue problems

Consider the eigenvalue problem : −Δu=λf(x,u) in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN. Denote by the set of all Carathéodory functions f:Ω×R→R such that for a.e. x∈Ω, f(x,⋅) is Lipschitzian with Lipschitz constant L, f(x,0)=0 and , and denote by (resp. ) the set of λ>0 such that has at least one nonzero classical (resp. weak) solution. Let λ₁ be the first eigenvalue for the Laplacian-Dirichlet problem. We prove that and . Our result is a positive answer to Ricceri's conjecture if use f(x,u) instead of f(u) in the conjecture.     

Normal families of meromorphic functions with multiple zeros and poles

Let k be a positive integer with k?2 and let be a family of functions meromorphic on a domain D in , all of whose poles have multiplicity at least 3, and of whose zeros all have multiplicity at least k+1. Let a(z) be a function holomorphic on D, a(z)?0. Suppose that for each , f^(k)(z)≠a(z) for z∈D. Then is a normal family on D.     

Distance irredundance and connected domination numbers of a graph

Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number γ_k(G), the connected k-domination number ; the k-independent domination number and the k-irredundance number ir_k(G). The authors prove that if an ir_k-set X is a k-independent set of G, then , and that for k?2, if ir_k(G)=1, if ir_k(G) is odd, and if ir_k(G) is even, which generalize some known results.     

Spike solutions of a nonlinear Schrödinger equation with degenerate potential

We study the existence of positive solutions to the elliptic equation ε²Δu(x,y)−V(y)u(x,y)+f(u(x,y))=0 for (x,y) in an unbounded domain subject to the boundary condition u=0 whenever is nonempty. Our potential V depends only on the y variable and is a bounded or unbounded domain which may coincide with . The positive parameter ε is tending to zero and our solutions u_ε concentrate along minimum points of the unbounded manifold of critical points of V.     

Stability switches in a system of linear differential equations with diagonal delay

This paper deals with the stability problem of a delay differential system of the form x^′(t)=-ax(t-τ)-by(t), y^′(t)=-cx(t)-ay(t-τ), where a, b, and c are real numbers and τ is a positive number. We establish some necessary and sufficient conditions for the zero solution of the system to be asymptotically stable. In particular, as τ increases monotonously from 0, the zero solution of the system switches finite times from stability to instability to stability if ; and from instability to stability to instability if . As an application, we investigate the local asymptotic stability of a positive equilibrium of delayed Lotka-Volterra systems.     

On the asymptotic stability of equilibrium

The classical criterion of asymptotic stability of the zero solution of equations x^′=f(t,x) is that there exists a positive definite function V which has infinitesimal upper bound such that is negative definite. In this paper we prove that if is bounded then the condition that is negative definite can be weakened and replaced by that and is negative definite.     

Fixed points of dual quantum operations

Let ?A be a normal completely positive map on B(H) with Kraus operators . Denote M the subset of normal completely positive maps by . In this note, the relations between the fixed points of ?A and are investigated. We obtain that , where K(H) is the set of all compact operators on H and is the dual of ?A∈M. In addition, we show that the map is a bijection on M.     

A minimum problem with free boundary in Orlicz spaces

We consider the optimization problem of minimizing in the class of functions W1,G(Ω) with , for a given φ₀?0 and bounded. W1,G(Ω) is the class of weakly differentiable functions with . The conditions on the function G allow for a different behavior at 0 and at ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω∩∂{u>0}, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the C1,α regularity of their free boundaries near “flat” free boundary points.