Convergence of generalized proximal point algorithms
Giuseppe Marino Hong-Kun Xu
Communications on Pure & Applied Analysis 2004, 3(4): 791-808 doi: 10.3934/cpaa.2004.3.791
Weak and strong convergence for some generalized proximal point algorithms are proved. These algorithms include the Eckstein and Bertsekas generalized proximal point algorithm, a contraction-proximal point algorithm, and inexact proximal point algorithms. Convergence rate is also considered.
keywords: inexact) proximal point algorithm nonexpansive mapping Maximal monotone operator resolvent identity projection. generalized (contraction-
On the missing bound state data of inverse spectral-scattering problems on the half-line
Guangsheng Wei Hong-Kun Xu
Inverse Problems & Imaging 2015, 9(1): 239-255 doi: 10.3934/ipi.2015.9.239
The inverse spectral-scattering problems for the radial Schrödinger equation on the half-line are considered with a real-valued integrable potential with a finite moment. It is shown that if the potential is sufficiently smooth in a neighborhood of the origin and its derivatives are known, then it is uniquely determined on the half-line in terms of the amplitude or scattering phase of the Jost function without bound state data, that is, the bound state data is missing.
keywords: inverse problem Jost function Marchenko equation. Gel'fand-Levitan integral equation Bargmann-Jost-Kohn class scattering data
On qualitative analysis for a two competing fish species model with a combined non-selective harvesting effort in the presence of toxicity
Yunfeng Jia Jianhua Wu Hong-Kun Xu
Communications on Pure & Applied Analysis 2013, 12(5): 1927-1941 doi: 10.3934/cpaa.2013.12.1927
In this paper, a two competing fish species model with combined harvesting is concerned, both the species obey the law of logistic growth and release a toxic substance to the other. Use spectrum analysis and bifurcation theory, the stability of semi-trivial solution, positive constant solution and the bifurcation solutions of model are investigated. We discuss bifurcation solutions which emanate from positive constant solution and trivial solution by taking the growth rate as bifurcation parameter. By the monotonic method, the existence result of positive steady-state of the model is discussed. The possibility of existence of a bionomic equilibrium is also obtained by taking the economical factor into consideration. Finally, some numerical examples are given to illustrate the results.
keywords: Competing model stability bifurcation theory monotonic method bionomic equilibrium.

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