Extending the applicability of the Gauss–Newton method under average Lipschitz–type conditions

We extend the applicability of the Gauss–Newton method for solving singular systems of equations under the notions of average Lipschitz–type conditions introduced recently in Li et al. (J Complex 26(3):268–295, 2010). Using our idea of recurrent functions, we provide a tighter local as well as semilocal convergence analysis for the Gauss–Newton method than in Li et al. (J Complex 26(3):268–295, 2010) who recently extended and improved earlier results (Hu et al. J Comput Appl Math 219:110–122, 2008; Li et al. Comput Math Appl 47:1057–1067, 2004; Wang Math Comput 68(255):169–186, 1999). We also note that our results are obtained under weaker or the same hypotheses as in Li et al. (J Complex 26(3):268–295, 2010). Applications to some special cases of Kantorovich–type conditions are also provided in this study.     

Comments on two Notes by L. Ma and X. Xu

In this Note I discuss some assertions made by L. Ma and X. Xu (2009) [6] and L. Ma (2010) [5], which need to be corrected and supplemented with additional references.     

On Fractional Evolution Inclusion Coupled with a Time and State Dependent Maximal Monotone Operator

Set-Valued and Variational Analysis - The paper deals with second-order evolution problems driven by time and state dependent maximal monotone operators with non-Lipschitz perturbations. Systems...     

Peripheral spectrum of positive quasi-compact operators onC 0(X)

LetX be a locally compact space, andT, a quasi-compact positive operator onC ₀(X), with positive spectral radius,r. Then the peripheral spectrum ofT is a finite set of poles containingr, and the residue of the resolvent ofT at each peripheral pole is of finite rank. Using the concept of closed absorbing set, we develop an iterative process that gives the order,p, ofr, some special bases of the algebraic eigenspaces ker(T-r) ^p and ker(T ^*-r) ^p , and finally the dimension of the algebraic eigenspace associated to each peripheral pole.     

Diverse Imprecise Goal Programming Model Formulations

The goal programming (GP) model is probably the best known in mathematical programming with multiple objectives. Available in various versions, GP is one of the most powerful multiple objective methods which has been applied in much varied fields. It has also been the target of many criticisms among which are those related to the difficulty of determining precisely the goal values as well as those concerning the decision-maker's near absence in this modelling process. In this paper, we will use the concept of indifference thresholds for modelling the imprecision related to the goal values. Many classical imprecise and fuzzy GP model formulations can be considered as a particular case of the proposed formulation.     

Analytical model for thermal resistance due to multiple moving circular contacts on a coated bodyModèle analytique pour la résistance thermique due à des contacts circulaires mobiles multiples à la surface d'un solide revêtu

The heat transfer at the interface of two solids in sliding/rolling contact depends on the constriction phenomenon which occurs at the vicinity of asperities. In order to study this problem, the micro-contacts are represented by multiple moving circular heat sources on the surface of a body. The studied body is constituted of a substrate and a surface coating. The thermal constriction resistance due to those contacts is determined analytically in this paper. The solution is developed by using the integral Fourier transforms, and it is valid regardless of the velocity and the relative contact size values. To cite this article: A. Baïri, C. R. Mecanique 331 (2003).     

New estimates for the Laplacian,the div–curl,and related Hodge systems

We establish new estimates for the Laplacian, the div–curl system, and more general Hodge systems in arbitrary dimension, with an application to minimizers of the Ginzburg–Landau energy. To cite this article: J. Bourgain, H. Brezis, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Existence Result for a Superlinear Fractional Navier Boundary Value Problems

In this paper, we study the following fractional Navier boundary value problem

$$\begin{aligned} \left\{ \begin{array}{lllc} D^{\beta }(D^{\alpha }u)(x)=u(x)g(u(x)),\quad x\in (0,1), \\ \displaystyle \lim _{x\longrightarrow 0}x^{1-\beta }D^{\alpha }u(x)=-a,\quad \,\,u(1)=b, \end{array} \right. \end{aligned}$$

where \(\alpha ,\beta \in (0,1]\) such that \(\alpha +\beta >1\), \(D^{\beta }\) and \(D^{\alpha }\) stand for the standard Riemann–Liouville fractional derivatives and a, b are nonnegative constants such that \(a+b>0\). The function g is a nonnegative continuous function in \([0,\infty )\) that is required to satisfy some suitable integrability condition. Using estimates on the Green’s function and a perturbation argument, we prove the existence of a unique positive continuous solution, which behaves like the unique solution of the homogeneous problem.