State Dependent Sweeping Process in p-Uniformly Smooth and q-Uniformly Convex Banach Spaces

We prove the existence of solutions for the state depending convex sweeping processes in p-uniformly smooth and q-uniformly convex Banach spaces via new recent results on proximal analysis in reflexive smooth Banach spaces.     

On Hanner type inequalities with a weight for Banach spaces

We shall present several Hanner type inequalities with a weight constant and characterize 2-uniformly smooth and 2-uniformly convex Banach spaces with these inequalities. p-Uniformly smooth and q-uniformly convex Banach spaces will be also characterized with another Hanner type inequalities with a weight in the other side term. The best value of the weight in these inequalities will be determined for Lp spaces. Also we shall present a duality theorem between these inequalities in a generalized form.     

The p-step iterative algorithm for a system of generalized mixed quasi-variational inclusions with -accretive operators in q-uniformly smooth banach spaces

In this paper, we introduce and study a new system of generalized mixed quasi-variational inclusions with (A,η)-accretive operators in q-uniformly smooth Banach spaces. By using the resolvent operator technique associated with (A,η)-accretive operators, we construct a new p-step iterative algorithm for solving this system of generalized mixed quasi-variational inclusions in real q-uniformly smooth Banach spaces. We also prove the existence of solutions for the generalized mixed quasi-variational inclusions and the convergence of iterative sequences generated by algorithm. Our results improve and generalize many known corresponding results.     

Convergence analysis of an iterative method for solving multiple-set split feasibility problems in certain Banach spaces

Our purpose in this paper is to introduce an iterative scheme for solving multiple-set split feasiblity problems in p-uniformly convex Banach spaces which are also uniformly smooth using Bregman distance techniques. We further obtain a strong convergence result for approximating solutions of multiple-set split feasiblity problems in the framework of p-uniformly convex Banach spaces which are also uniformly smooth.     

Strong Convergence Theorem for Multiple Sets Split Feasibility Problems in Banach Spaces

The purpose of this article is to study the iterative approximation of solution to multiple sets split feasibility problems in p-uniformly convex real Banach spaces that are also uniformly smooth. We propose an iterative algorithm for solving multiple sets split feasibility problems and prove a strong convergence theorem of the sequence generated by our algorithm under some appropriate conditions in p-uniformly convex real Banach spaces that are also uniformly smooth.     

Second-order subdifferentials of C 1,1 functions and optimality conditions

We present second-order subdifferentials of Clarke's type of C ^1,1 functions, defined in Banach spaces with separable duals. One of them is an extension of the generalized Hessian matrix of such functions in ? ⁿ , considered by J. B. H.-Urruty, J. J. Strodiot and V. H. Nguyen. Various properties of these subdifferentials are proved. Second-order optimality conditions (necessary, sufficient) for constrained minimization problems with C ^1,1 data are obtained.     

A system of generalized variational inclusions involving generalized H(·, ·)-accretive mapping in real q-uniformly smooth Banach spaces

In this paper, we consider a class of accretive mappings called generalized H(·, ·)-accretive mappings in Banach spaces. We prove that the proximal-point mapping of the generalized H(·, ·)-accretive mapping is single-valued and Lipschitz continuous. Further, we consider a system of generalized variational inclusions involving generalized H(·, ·)-accretive mappings in real q-uniformly smooth Banach spaces. Using proximal-point mapping method, we prove the existence and uniqueness of solution and suggest an iterative algorithm for the system of generalized variational inclusions. Furthermore, we discuss the convergence criteria of the iterative algorithm under some suitable conditions. Our results can be viewed as a refinement and improvement of some known results in the literature.     

An iterative algorithm for solving split feasibility problems and fixed point problems in Banach spaces

The purpose of this paper is to study split feasibility problems and fixed point problems concerning left Bregman strongly relatively nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. We suggest an iterative scheme for the problem and prove strong convergence theorem of the sequences generated by our scheme under some appropriate conditions in real p-uniformly convex and uniformly smooth Banach spaces. Finally, we give numerical examples of our result to study its efficiency and implementation. Our result complements many recent and important results in this direction.     

Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type

In this paper we study hybrid iterative scheme for finding a common element of set of solutions of generalized mixed equilibrium problem, set of common fixed points of finite family of weak relatively nonexpansive mapping and null spaces of finite family of γ-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. Our results extend, improve and generalize the results of several authors which are announced recently. Application of our theorem to solution of equations of Hammerstein-type is of independent interest.     

Mean-Value Theorem with Small Subdifferentials

We prove a mean-value theorem for lower semicontinuous functions on a large class of Banach spaces which contains the class of Asplund spaces, in particular reflexive Banach spaces and Banach spaces with a separable dual. It involves the lower subdifferential (or contingent subdifferential) and the Fréchet subdifferentials, which are among the smallest subdifferentials known to date. It follows that the estimates which it provides require weak assumptions and are accurate. When the function is locally Lipschitzian, we get a simple statement which refines the Lebourg mean-value theorem.     

Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mappings in q-uniformly smooth Banach spaces

In this paper, we introduce the new iterative sequences with errors approximating the common fixed point for a couple of quasi-contractive mappings and show the stability of the iterative procedures with errors in q-uniformly smooth Banach spaces. Our results extend, improve and unify the corresponding results of Chidume, Osilike, Liu, and others.     

Iterative approximation of zeroes of monotone operators and system of generalized mixed equilibrium problems

In this paper, we construct a new iterative scheme by hybrid method for approximation of common element of set of zeroes of a finite family of ??-inverse-strongly monotone operators and set of common solutions to a system of generalized mixed equilibrium problems in a 2-uniformly convex real Banach space which is also uniformly smooth. Then, we prove strong convergence of the scheme to a common element of the two sets.     

On a proximal point method for convex optimization in banach spaces

We analyze the behavior of a parallel proximal point method for solving convex optimization problems in reflexive Banach spaces. Similar algorithms were known to converge under the implicit assumption that the norm of the space is Hilbertian. We extend the area of applicability of the proximal point method to solving convex optimization problems in Banach spaces on which totally convex functions can be found. This includes the class of all smooth uniformly convex Banach spaces. Also, our convergence results leave more flexibility for the choice of the penalty function involved in the algorithm and, in this way, allow simplification of the computational procedure.     

Approximate Rolle's theorems for the proximal subgradient and the generalized gradient

We establish approximate Rolle's theorems for the proximal subgradient and for the generalized gradient. We also show that an exact Rolle's theorem for the generalized gradient is completely false in all infinite-dimensional Banach spaces (even when they do not possess smooth bump functions).     

Lower Semicontinuity of the Solution Map to a Parametric Generalized Variational Inequality in Reflexive Banach Spaces

This paper is concerned with the study of solution stability of a parametric generalized variational inequality in reflexive Banach spaces. Under the requirements that the operator of a unperturbed problem is of class (S)_?+? and operators under consideration are pseudo-monotone and demicontinuous, we show that the solution map of a parametric generalized variational inequality is lower semicontinuous. The obtained results are proved without conditions related to the degree theory and the metric projection.     

Convergence theorems for λ-strict pseudo-contractions in q-uniformly smooth Banach spaces

In this paper, we continue to discuss the properties of iterates generated by a strict pseudo- contraction or a finite family of strict pseudo-contractions in a real q-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [Marino, G., Xu, H. K.： Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl., 324, 336-349 （2007）]. In order to get a strong convergence theorem, we modify the normal Mann＇s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in C. This result extends a recent result of Kim and Xu [Kim, T. H., Xu, H. K.： Strong convergence of modified Mann iterations. Nonl. Anal., 61, 51- 60 （2005）] both from nonexpansive mappings to λ-strict pseudo-contractions and from Hilbert spaces to q-uniformly smooth Banach spaces.     

Existence theorems for generalized set-valued mixed (quasi-)variational inequalities in Banach spaces

In this paper, utilizing the properties of the generalized f -projection operator and the well-known KKM and Kakutani–Fan–Glicksberg theorems, under quite mide assumptions, we derive some new existence theorems for the generalized set-valued mixed variational inequality and the generalized set-valued mixed quasi-variational inequality in reflexive and smooth Banach spaces, respectively. The results presented in this paper can be viewed as the supplement, improvement and extension of recent results in Wu and Huang (Nonlinear Anal 71:2481–2490, 2009).     

Necessary conditions for constrained optimization problems with semicontinuous and continuous data

We consider nonsmooth constrained optimization problems with semicontinuous and continuous data in Banach space and derive necessary conditions without constraint qualification in terms of smooth subderivatives and normal cones. These results, in different versions, are set in reflexive and smooth Banach spaces.

     

Strong convergence theorems of modified Ishikawa iterative process with errors for an infinite family of strict pseudo-contractions

In this paper, we study modified Ishikawa iterative process with errors to have strong convergence for an infinite family of strict pseudo-contractions in the framework of q-uniformly smooth Banach spaces. Our results improve and extend the recent ones announced by many others.     

自反Banach空间内一类新的广义混合平衡问题组的辅助原理和逼近可解性

在自反Banach空间内引入和研究了一类新的涉及广义混合似变分不等式问题的广义混合平衡问题组(SGMEP)．首先,为了求解 SGMEP,引入了一类辅助广义混合平衡问题组(SAGMEP)．在没有任何强制条件的相当温和假设下, 对SAGMEP证明了解的存在性和唯一性．其次, 利用辅助原理技巧,对求解SGMEP建议和分析了一类新的迭代算法．最后,在没有任何强制条件的相当温和假设下,证明了由算法生成的迭代序列的强收敛性．这些结果改进、统一和推广了这一领域内某些最近结果．