Essential spectra of self-adjoint relations under relatively compact perturbations

This paper is concerned with the stability of essential spectra of self-adjoint relations under relatively compact perturbation in Hilbert spaces. Relationships between relative boundedness and relative compactness of linear relations are established, and some necessary and sufficient conditions of relative compactness and relative boundedness of linear relations are given. It is shown that the essential spectra of self-adjoint relations are invariant under either relatively compact perturbation or a more general perturbation. The results obtained in the present paper generalize the corresponding results for operators to relations, and some of which weaken certain assumptions of the related existing results.     

Global Attractor of Hindmarsh-Rose Equations in Neurodynamics

Global dynamics for a new mathematical model in neurodynamics of the diffusive Hindmarsh-Rose equations on a bounded domain is investigated in this paper. The existence of a global attractor and its regularity are proved through uniform estimates showing the dissipative properties and the asymptotically compact and smoothing characteristics.     

Positive solutions for some non-autonomous Schrödinger-Poisson systems

In this paper we study the Schrödinger-Poisson system

(SP)     

无穷区间上两类抽象无界函数族相对紧的判定及其应用

给出了无穷区间上抽象无界连续函数族和抽象无界可微函数族相对紧性判定的充要条件,并应用它获得了无穷区间上二阶微分方程两点边值问题无界解的存在性.     

Existence of global solutions to isentropic gas dynamics equations with a source term

In this paper we prove existence of isentropic gas dynamic equations with a source term (1.2). To this end we construct a sequence of regular hyperbolic systems (1.1) to approximate the inhomogeneous system of isentropic gas dynamics (1.2). First,for each fixed approximation parameter δ and very general condition on P (ρ),we establish the existence of entropy solutions for the Cauchy problem (1.1) with bounded initial date (1.4). Second,letting=o(δ),we obtain a complete proof of the H-1loc compactness of weak entropy pairs of system (1.2) in the form η(ρ,u) =ρH(ρ,u) given in Chen-LeFloch (2003). Finally,for the conditions of P(ρ) given in Chen-LeFloch (2003),applied to the results in Theorems 1 and 2,we obtain the global existence of entropy solutions for the Cauchy problem (1.2) with bounded initial date (1.4).     

Shared value in a finite-dimensional Banach manifold

Sufficient conditions are given to assert that two C¹-mappings share only one value in a connected compact Banach manifold modelled over ?ⁿ. The proof of the result, which is based upon continuation methods, is constructive.     

Existence and uniqueness of an optimum in the infinite-horizon portfolio-cum-saving model with semimartingale investments

The model considered here is essentially that formulated in the author's previous paper Conditions for Optimality in the Infinite-Horizon Portfolio-cum-Saving Problem with Semimartingale Investments, Stochastics and Stochastics Reports 29 (1990), 133-171. In this model, the vector process representing returns to investments is a general semimartingale. Processes defining portfolio plans arc here required only to be predictable and non-negative. Existence of an optimal portfolio-cum-saving plan is proved under slight conditions of integrability imposed on the welfare functional; the proofs rely on properties of weak precompactness of portfolio and utility sequences in suitable L _p spaces together with dominated and monotone convergence arguments. Conditions are also obtained for the uniqueness of the portfolio plan generating a given returns process (i.e. for the uniqueness of the integrands generating a given sum of semimartingale integrals) and for the uniqueness of an optimal plan; here use is made of random measures associated with the jumps of a semimartingale     

一类带积分边界条件的三阶微分方程边值问题的解

运用Banach压缩映射原理以及Leray-Schauder连续性原理,在非线性项为L1-Caratheodory函数的条件下,研究了一类带积分边界条件的三阶微分方程边值问题解的唯一性、存在性以及解集的紧性.     

Multiple solutions of a quasilinear elliptic system involving the nonlinear boundary conditions on exterior domain

In this paper, we consider the multiplicity results of nontrivial nonnegative solutions of the quasilinear p‐Laplacian system with the nonlinear boundary conditions: (0.1) where Ω is a smooth exterior domain in is the outward normal derivative on the boundary Γ = ?Ω, and . By the Nehari manifold and variational methods, we prove that the problem (0.1) has at least two nontrivial nonnegative solutions when the pair of the parameters (λ,μ) belongs to a certain subset of . Copyright © 2012 John Wiley & Sons, Ltd.     

Solutions of nonperiodic super quadratic Hamiltonian systems

This paper concerns solutions for the Hamiltonian system: , where H(t,u) = 1 ∕ 2Lu ? u + W(t,u), L is a 2N × 2N symmetric matrix, and . We consider the case that 0 ? σ_c( ? (Jd ∕ dt + L)) and W satisfies some new generalized super quadratic condition different from the type of Ambrosetti–Rabinowitz. The method is variational: By virtue of some auxiliary system related to the ‘limit equation’ of the Hamiltonian system, we first establish that (C)_c‐condition holds true for all c less than the least energy of the limit equation. Then, using some weak linking theorem recently developed, we obtain one least energy solution of the Hamiltonian system. Copyright © 2013 John Wiley & Sons, Ltd.