On recent developments in the theory of abstract differential equations with fractional derivatives

This note is motivated from some recent papers treating the problem of the existence of a solution for abstract differential equations with fractional derivatives. We show that the existence results in [Agarwal et al. (2009) [1], Belmekki and Benchohra (2010) [2], Darwish et al. (2009) [3], Hu et al. (2009) [4], Mophou and N’Guérékata (2009) [6] and [7], Mophou (2010) [8] and [9], Muslim (2009) [10], Pandey et al. (2009) [11], Rashid and El-Qaderi (2009) [12] and Tai and Wang (2009) [13]] are incorrect since the considered variation of constant formulas is not appropriate. In this note, we also consider a different approach to treat a general class of abstract fractional differential equations.     

Fixed point theorems for some generalized multivalued nonexpansive mappings

In this paper, we introduce a condition on multivalued mappings which is a multivalued version of condition (C_λ) defined by Garcia-Falset et al. (2011) [3]. It is shown here that some of the classical fixed point theorems for multivalued nonexpansive mappings can be extended to mappings satisfying this condition. Our results generalize the results in Lim (1974), Lami Dozo (1973), Kirk and Massa (1990), Garcia-Falset et al. (2011), Dhompongsa et al. (2009) and Abkar and Eslamian (2010) [4], [5], [6], [3], [7] and [8] and many others.     

Solitary waves for the Klein-Gordon-Maxwell system with critical exponent

In this paper we study the existence of solutions for nonlinear Klein-Gordon-Maxwell equations coupled with Maxwell’s equations when the nonlinearity exhibits critical growth. We improve some previous existence results in Azzollini et al. (2009) [5], Carrião et al. (2009) [4] and Cassani (2004) [3].     

Linearly perturbed polyhedral normal cone mappings and applications

Under a mild regularity assumption, we derive an exact formula for the Fréchet coderivative and some estimates for the Mordukhovich coderivative of the normal cone mappings of perturbed polyhedra in reflexive Banach spaces. Our focus point is a positive linear independence condition, which is a relaxed form of the linear independence condition employed recently by Henrion et al. (2010) [1], and Nam (2010) [3]. The formulae obtained allow us to get new results on solution stability of affine variational inequalities under linear perturbations. Thus, our paper develops some aspects of the work of Henrion et al. (2010) [1] Nam (2010) [3] Qui (in press) [12] and Yao and Yen (2009) [6] and [7].     

The number of excellent discrete Morse functions on graphs

In Nicolaescu (2008) [7] the number of non-homologically equivalent excellent Morse functions defined on S² was obtained in the differentiable setting. We carried out an analogous study in the discrete setting for some kinds of graphs, including S¹, in Ayala et al. (2009) [1]. This paper completes this study, counting excellent discrete Morse functions defined on any infinite locally finite graph.     

Fixed point theorems for random pseudo-contractive mappings

Very recently in Fierro et al. (2009) [6], we obtained a general principle to prove the existence of Random Fixed Point Theorems. As a consequence of this, we have been able to obtain various generalizations for pseudo-contractive mappings with rather simple proofs. In addition, while we were deriving these extensions for random operators, some deterministic results arose, which also appear to be new.     

Common fixed point results in CAT(0) spaces

Let X be a complete CAT(0) space, T be a generalized multivalued nonexpansive mapping, and t be a single valued quasi-nonexpansive mapping. Under the assumption that T and t commute weakly, we shall prove the existence of a common fixed point for them. In this way, we extend and improve a number of recent results obtained by Shahzad (2009) [7] and [12], Shahzad and Markin (2008) [6], and Dhompongsa et al. (2005) [5].     

Ramsey numbers for a disjoint union of good graphs

We give the Ramsey number for a disjoint union of some G-good graphs versus a graph G generalizing the results of Stahl (1975) [5] and Baskoro et al. (2006) [1] and the previous result of the author Bielak (2009) [2]. Moreover, a family of G-good graphs with s(G)>1 is presented.     

Approximations for nonlinear mappings by the hybrid method in Hilbert spaces

The hybrid method in mathematical programming was introduced by Haugazeau (1968) [1] and he proved a strong convergence theorem for finding a common element of finite nonempty closed convex subsets of a real Hilbert space. Later, Bauschke and Combettes (2001) [2] proposed some condition for a family of mappings (the so-called coherent condition) and established interesting results by the hybrid method. The authors (Nakajo et al., 2009) [10] extended Bauschke and Combettes’s results. In this paper, we introduce a condition weaker than the coherent condition and prove strong convergence theorems which generalize the results of Nakajo et al. (2009) [10]. And we get strong convergence theorems for a family of asymptotically κ-strict pseudo-contractions, a family of Lipschitz and pseudo-contractive mappings and a one-parameter uniformly Lipschitz semigroup of pseudo-contractive mappings.     

Archimedean copulas in finite and infinite dimensions—with application to ruin problems

In this paper we discuss the link between Archimedean copulas and L₁ Dirichlet distributions for both finite and infinite dimensions. With motivation from the recent papers Weng et al. (2009) and Albrecher et al. (2011) we apply our results to certain ruin problems.     

On strong convergence in real Banach spaces obtained by dropping the Lipshitz condition

We consider an implicit iterative process for two finite families of mappings in a real Banach space and prove strong convergence results without using the Lipschitz condition on mappings. Our results mainly improve and extend the recent results of Chang et al. (2001, 2009, 2007) [1], [2], [3], Cho et al. (2005) [4], Gu (2008) [14], Ofoedu (2006) [9], Schu (1991) [13], Zeng (2003, 2005) [20], [21], and Qin et al. (2008) [11], [12].     

Two-step Newton methods

We present sufficient convergence conditions for two-step Newton methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. The advantages of our approach over other studies such as Argyros et al. (2010) [5], Chen et al. (2010) [11], Ezquerro et al. (2000) [16], Ezquerro et al. (2009) [15], Hernández and Romero (2005) [18], Kantorovich and Akilov (1982) [19], Parida and Gupta (2007) [21], Potra (1982) [23], Proinov (2010) [25], Traub (1964) [26] for the semilocal convergence case are: weaker sufficient convergence conditions, more precise error bounds on the distances involved and at least as precise information on the location of the solution. In the local convergence case more precise error estimates are presented. These advantages are obtained under the same computational cost as in the earlier stated studies. Numerical examples involving Hammerstein nonlinear integral equations where the older convergence conditions are not satisfied but the new conditions are satisfied are also presented in this study for the semilocal convergence case. In the local case, numerical examples and a larger convergence ball are obtained.     

Global existence and nonexistence for diffusive polytropic filtration equations with nonlinear boundary conditions

In this paper, we deal with the global existence and nonexistence of solutions to a diffusive polytropic filtration system with nonlinear boundary conditions. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions, which extend the recent results of Li et al. (Z Angew Math Phys 60:284–298, 2009) and Wang et al. (Nonlinear Anal 71:2134–2140, 2009) to more general equations and simplify their proofs slightly.     

Visibilité dans le modèle Booléen

The aim of this Note is to give an estimate on the tail probability of the visibility function in a Boolean model: this function is defined as the length of the longest ray starting at the origin that does not intersect an obstacle. Convergence results are added, when the size of obstacles goes to zero and when the distance between the origin and the closest obstacle goes to infinity. To cite this article: P. Calka et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Acyclic 4-choosability of planar graphs with neither 4-cycles nor triangular 6-cycles

Every planar graph is known to be acyclically 7-choosable and is conjectured to be acyclically 5-choosable (Borodin et al. 2002) [7]. This conjecture if proved would imply both Borodin’s acyclic 5-color theorem (1979) and Thomassen’s 5-choosability theorem (1994). However, as yet it has been verified only for several restricted classes of graphs.Some sufficient conditions are also obtained for a planar graph to be acyclically 4-choosable and 3-choosable. In particular, acyclic 4-choosability was proved for the following planar graphs: without 3-cycles and 4-cycles (Montassier, 2006 [23]), without 4-cycles, 5-cycles and 6-cycles (Montassier et al. 2006 [24]), and either without 4-cycles, 6-cycles and 7-cycles, or without 4-cycles, 6-cycles and 8-cycles (Chen et al. 2009 [14]).In this paper it is proved that each planar graph with neither 4-cycles nor 6-cycles adjacent to a triangle is acyclically 4-choosable, which covers these four results.     

No end of the world in 2012 for 4OR

The purpose of this note is to give the readers of 4OR information on the state of the journal and its future, abiding by a triennial tradition which started with Bouyssou et al. (4OR 1(1):1–6, 2003) and continued with Bouyssou et al. (4OR 4(1):1–9, 2006) and Bouyssou et al. (4OR Q J Oper Res 7(1):15, 2009). In the 3 years that have passed since the last editorial note, three volumes of the journal have been published, each containing four issues: vol. 7 (2009), vol. 8 (2010) and vol. 9 (2011).     

Iterative algorithms for variational inequality and equilibrium problems with applications

In this paper, we introduce an iterative method for finding a common element of the set of solutions of equilibrium problems, of the set of variational inequalities and of the set of common fixed points of an infinite family of nonexpansive mappings in the framework of real Hilbert spaces. Strong convergence of the proposed iterative algorithm is obtained. As an application, we utilize the main results which improve the corresponding results announced in Chang et al. (Nonlinear Anal, 70:3307–3319, 2009), Colao et al. (J Math Anal Appl, 344:340–352, 2008), Plubtieng and Punpaeng (Appl Math Comput, 197:548–558, 2008) to study the optimization problem.     

Abstract Hopf bifurcation theorem and further extensions via second variation

Sufficient conditions of bifurcation stated in Arutyunov et al. (2009) [5] are investigated in order to reconsider celebrated Hopf bifurcation as the simplest bifurcation of Fredholm operators of zero index. In several examples abstract result is applied to both finite and infinite dynamical systems exhibiting classical Hopf bifurcation as well as double Hopf bifurcation.     

Nonexistence of solutions for singular elliptic equations with a quadratic gradient term

This paper concerns the nonexistence of solutions for singular elliptic equations with a quadratic gradient term. The main results complement and partly extend some works by Arcoya et al. (2009) [1]. As a by-product of the main results, we fill in a gap in one of their works.     

An upper bound for the competition numbers of graphs

A hole of a graph is an induced cycle of length at least 4. Kim (2005) [2] conjectured that the competition number k(G) is bounded by h(G)+1 for any graph G, where h(G) is the number of holes of G. In Lee et al. [3], it is proved that the conjecture is true for a graph whose holes are mutually edge-disjoint. In Li et al. (2009) [4], it is proved that the conjecture is true for a graph, all of whose holes are independent. In this paper, we prove that Kim’s conjecture is true for a graph G satisfying the following condition: for each hole C of G, there exists an edge which is contained only in C among all induced cycles of G.