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].     

Ekeland’s variational principle for vectorial multivalued mappings in a uniform space

In this paper, we establish Ekeland’s variational principle and an equilibrium version of Ekeland’s variational principle for vectorial multivalued mappings in the setting of separated, sequentially complete uniform spaces. Our approaches and results are different from those in Chen et al. (2008), Hamel (2005), and Lin and Chuang (2010) [13], [14] and [15]. As applications of our results, we study vectorial Caristi’s fixed point theorems and Takahashi’s nonconvex minimization theorems for multivalued mappings and their equivalent forms in a separated, sequentially complete uniform space. We also apply our results to study maximal element theorems, which are unified methods of several variational inclusion problems. Our results contain many known results in the literature Fang (1996) [21], and will have many applications in nonlinear analysis.     

Coderivatives of normal cone mappings and Lipschitzian stability of parametric variational inequalities

In this paper, we provide a comprehensive study of coderivative formulas for normal cone mappings. This allows us to derive necessary and sufficient conditions for the Lipschitzian stability of parametric variational inequalities in reflexive Banach spaces. Our development not only gives an answer to the open questions raised in Yao and Yen (2009) [11], but also establishes generalizations and complements of the results given in Henrion et al. (2010) [4] and Yao and Yen (2009) [11] and [12].     

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.     

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].     

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.     

Approximation of signals of class Lip(α, p) by linear operators

Mittal, Rhoades [5], [6], [7] and [8] and Mittal et al. [9] and [10] have initiated a study of error estimates E_n(f) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper we continue the work. Here we extend two theorems of Leindler [4], where he has weakened the conditions on {p_n} given by Chandra [2], to more general classes of triangular matrix methods. Our Theorem also partially generalizes Theorem 4 of Mittal et al. [11] by dropping the monotonicity on the elements of matrix rows, which in turn generalize the results of Quade [15].     

On the Secant method

We present a new semilocal convergence analysis for the Secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis is based on the weaker center-Lipschitz concept instead of the stronger Lipschitz condition which has been ubiquitously employed in other studies such as Amat et al. (2004)  [2], Bosarge and Falb (1969)  [9], Dennis (1971)  [10], Ezquerro et al. (2010)  [11], Hernández et al. (2005, 2000)   and , Kantorovich and Akilov (1982)  [14], Laasonen (1969)  [15], Ortega and Rheinboldt (1970)  [16], Parida and Gupta (2007)  [17], Potra (1982, 1984–1985, 1985)  ,  and , Proinov (2009, 2010)   and , Schmidt (1978) [23], Wolfe (1978)  [24] and Yamamoto (1987)  [25] for computing the inverses of the linear operators. We also provide lower and upper bounds on the limit point of the majorizing sequences for the Secant method. Under the same computational cost, our error analysis is tighter than that proposed in earlier studies. Numerical examples illustrating the theoretical results are also given in this study.     

Inexact Newton-type methods

We introduce the new idea of recurrent functions to provide a semilocal convergence analysis for an inexact Newton-type method, using outer inverses. It turns out that our sufficient convergence conditions are weaker than in earlier studies in many interesting cases (Argyros, 2004 [5] and [6], Argyros, 2007 [7], Dennis, 1971 [14], Deuflhard and Heindl, 1979 [15], Gutiérrez, 1997 [16], Gutiérrez et al., 1995 [17], Häubler, 1986 [18], Huang, 1993 [19], Kantorovich and Akilov, 1982 [20], Nashed and Chen, 1993 [21], Potra, 1982 [22], Potra, 1985 [23]).     

Ergodic theorems for hybrid sequences in a Hilbert space with applications

In this paper, we introduce the notion of generalized hybrid sequences, extending the notion of nonexpansive sequences introduced and studied in our previous work Djafari Rouhani (1981, 1990, 1990, 1997, 2002, 2004, 2002)  , , , , ,  and , and prove ergodic and convergence theorems for such sequences in a Hilbert space H$H$. Subsequently, we apply our results to prove new fixed point theorems for generalized hybrid mappings, first introduced in Kocourek et al. (2010)  [14], Takahashi and Takeuchi (2011)  [20], defined on arbitrary nonempty subsets of H$H$.     

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.     

The nonlinear boundary layer to the Boltzmann equation for cutoff soft potential with physical boundary condition

We consider the nonlinear boundary layer to the Boltzmann equation for cutoff soft potential with physical boundary condition, i.e., the Dirichlet boundary condition with weak diffuse effect. Under the assumption that the distribution function of gas particles tends to a global Maxwellian in the far field, we will show the boundary layer exist if the boundary data satisfy the solvability condition. Moreover, the codimensions of the boundary data which satisfies the solvability condition change with the Mach number of the far field Maxwellian like Chen et al. (2004) [5], Ukai et al. (2003) [6] and Wang et al. (2007) [7].     

Fractional weak discrepancy and split semiorders

The fractional weak discrepancywd_F(P) of a poset P=(V,?) was introduced in Shuchat et al. (2007) [6] as the minimum nonnegative k for which there exists a function f:V→R satisfying (i) if a?b then f(a)+1≤f(b) and (ii) if a∥b then |f(a)−f(b)|≤k. In this paper we generalize results in Shuchat et al. (2006, 2009) [5] and [7] on the range of wd_F for semiorders to the larger class of split semiorders. In particular, we prove that for such posets the range is the set of rationals that can be represented as r/s for which 0≤s−1≤r<2s.     

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].     

Rotundity structure of local nature for generalized Orlicz-Lorentz function spaces

Criteria for extreme points and rotund points in generalized Orlicz-Lorentz function spaces, which were introduced in Foralewski (2011) [27] are given. Some examples show that in these spaces the notion of rotund point is essentially stronger than the notion of extreme point. Finally, the criteria obtained in this paper are interpreted in the case of classical Orlicz-Lorentz spaces. Results of this paper are related to the results from Carothers et al. (1992) [9], Kamińska (1990) [4], Foralewski et al. (2008) [26].     

Criteria for univalence of the Dziok-Srivastava and the Srivastava-Wright operators in the class A

In the geometric function theory (GFT) much attention is paid to various linear integral operators mapping the class S of the univalent functions and its subclasses into themselves. In [12] and [13] Hohlov obtained sufficient conditions that guarantee such mappings for the operator defined by means of Hadamard product with the Gauss hypergeometric function. In our earlier papers as [20], [19], [17] and [18], etc., we extended his method to the operators of the generalized fractional calculus (GFC, [16]). These operators have product functions of the forms _m+1F_m and _m+1Ψ_m and integral representations by means of the Meijer G- and Fox H-functions. Here we propose sufficient conditions that guarantee mapping of the univalent, respectively of the convex functions, into univalent functions in the case of the celebrated Dziok-Srivastava operator ([8] : J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput.103, No 1 (1999), pp. 1-13) defined as a Hadamard product with an arbitrary generalized hypergeometric function _pF_q. Similar conditions are suggested also for its extension involving the Wright _pΨ_q-function and called the Srivastava-Wright operator (Srivastava, [36]). Since the discussed operators include the above-mentioned GFC operators and many their particular cases (operators of the classical FC), from the results proposed here one can derive univalence criteria for many named operators in the GFT, as the operators of Hohlov, Carlson and Shaffer, Saigo, Libera, Bernardi, Erdélyi-Kober, etc., by giving particular values to the orders p ? q + 1 of the generalized hypergeometric functions and to their parameters.     

A note on strong matrix summability via ideals

Following the line of recent works of Das et al. (2011) [15] and Savas and Das (2011) [19] we apply the notion of ideals to strong matrix summability, and in this note, we introduce the notion of strong A^I-summability with respect to an Orlicz function. We make certain investigations on the classes of sequences arising out of this new summability method.     

A simpler analysis of a hybrid numerical method for time-dependent convection-diffusion problems

A finite difference method for a time-dependent convection-diffusion problem in one space dimension is constructed using a Shishkin mesh. In two recent papers (Clavero et al. (2005) [2] and Mukherjee and Natesan (2009) [3]), this method has been shown to be convergent, uniformly in the small diffusion parameter, using somewhat elaborate analytical techniques and under a certain mesh restriction. In the present paper, a much simpler argument is used to prove a higher order of convergence (uniformly in the diffusion parameter) than in [2] and [3] and under a slightly less restrictive condition on the mesh.     

Monotonic positive solution of a nonlinear quadratic functional integral equation

The existence of positive monotonic solutions, in the class of continuous functions, for some nonlinear quadratic integral equation have been studied in [4], [5], [6], [7] and [8]. Here we are concerning with a nonlinear quadratic integral equation of Volterra type and we shall prove the existence of at least one L₁-positive monotonic solution for that equation under Carathèodory condition.