Weak and strong convergence theorems for mixed equilibrium problems in Banach spaces

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problems and the set of fixed points for a $\phi $ -nonexpansive mapping in Banach spaces by using sunny generalized nonexpansive retraction in Banach spaces. Moreover, we also apply our result for finding a zero point of maximal monotone operators. Finally, we give a numerical example to illustrate our main theorem.     

Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces

The paper proposes two new iterative methods for solving pseudomonotone equilibrium problems involving fixed point problems for quasi-\(\phi \)-nonexpansive mappings in Banach spaces. The proposed algorithms combine the extended extragradient method or the linesearch method with the Halpern iteration. The strong convergence theorems are established under standard assumptions imposed on equilibrium bifunctions and mappings. The results in this paper have generalized and enriched existing algorithms for equilibrium problems in Banach spaces.     

Strong convergence theorems for variational inequality problems and quasi-{\phi}-asymptotically nonexpansive mappings

In this paper, we introduce an iterative process which converges strongly to a common solution of finite family of variational inequality problems for ??-inverse strongly monotone mappings and fixed point of two continuous quasi- ${\phi}$ -asymptotically nonexpansive mappings in Banach spaces. Our theorems extend and unify most of the results that have been proved for the class of monotone mappings.     

Viscosity iterative scheme for generalized mixed equilibrium problems and nonexpansive semigroups

In this paper, we propose a new general iterative scheme based on the viscosity approximation method for finding a common element of the set of solutions of the generalized mixed equilibrium problem and the set of all common fixed points of a finite family of nonexpansive semigroups. Then, we prove the strong convergence of the iterative scheme to find a unique solution of the variational inequality that is the optimality condition for the minimization problem. Our results extend and improve some recent results of Cianciaruso et al. (J. Optim. Theory Appl. 146:491–509, 2010), Kamraksa and Wangkeeree (J. Glob. Optim. 51:689–714, 2011), and many others.     

An iterative algorithm for common fixed points for nonexpansive semigroups and strictly pseudo-contractive mappings with optimization problems

In this paper, we introduce an iterative algorithm for finding a common element of the set of solutions of a system of mixed equilibrium problems, the set of solutions of a variational inclusion problems for inverse strongly monotone mappings, the set of common fixed points for nonexpansive semigroups and the set of common fixed points for an infinite family of strictly pseudo-contractive mappings in Hilbert spaces. Furthermore, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions which solves some optimization problems. Our results extend and improve the recent results of Chang et al. (Appl Math Comput 216:51–60, 2010), Hao (Appl Math Comput 217(7):3000–3010, 2010), Jaiboon and Kumam (Nonlinear Anal 73:1180–1202, 2010) and many others.     

The generalization of Minkowski problems for polytopes

The generalization of Minkowski problems, such as the $L_p$ and Orlicz Minkowski problems, have caused wide concern recently. In this paper, we will establish the existence of the Orlicz Minkowski problem for polytopes. In particular, a solution to the $L_p$ Minkowski problem for polytopes with $p>1$ is given. By the uniqueness of this solution, we present a new proof of the $L_p$ Minkowski inequality that demonstrates the relationship between these two fundamental theorems of the $L_p$ Brunn–Minkowski theory.     

Continuity of the solution mapping to parametric generalized vector equilibrium problems

In this paper, two kinds of parametric generalized vector equilibrium problems in normed spaces are studied. The sufficient conditions for the continuity of the solution mappings to the two kinds of parametric generalized vector equilibrium problems are established under suitable conditions. The results presented in this paper extend and improve some main results in Chen and Gong (Pac J Optim 3:511–520, 2010), Chen and Li (Pac J Optim 6:141–152, 2010), Chen et al. (J Glob Optim 45:309–318, 2009), Cheng and Zhu (J Glob Optim 32:543–550, 2005), Gong (J Optim Theory Appl 139:35–46, 2008), Li and Fang (J Optim Theory Appl 147:507–515, 2010), Li et al. (Bull Aust Math Soc 81:85–95, 2010) and Peng et al. (J Optim Theory Appl 152(1):256–264, 2011).     

Common coupled fixed point theorems for w^*-compatible mappings without mixed monotone property

In this paper, we show that the mixed $g$ -monotone property in coupled coincidence point theorems can be replaced by generalized property. Hence, these results can be applied in a much wider class of problems. We also study the condition for the uniqueness of a common coupled fixed point and give some example of nonlinear contraction mappings where the existence of the common coupled fixed point cannot be obtained by the mixed monotone property, but it follows by our results. At the end of this paper, we give an open problems for further investigation.     

Strong convergence of Mann’s type iteration method for an infinite family of generalized asymptotically nonexpansive nonself mappings in Hilbert spaces

Let $C$ be a nonempty closed convex subset of a real Hilbert space $H$ . Let $\{T_i\}^{\infty }_{i=1}:C\rightarrow H$ be an infinite family of generalized asymptotically nonexpansive nonself mappings. By using a specific way of choosing the indexes of the involved mappings, we prove strong convergence of Mann’s type iteration to a common fixed point of $\{T_i\}^{\infty }_{i=1}$ without the compactness assumption imposed either on $T$ or on $C$ provided that the interior of common fixed points is nonempty. The results extend previous results restricted to the situation of at most finite families of such mappings.     

Application of the multipole method to direct and inverse problems for the Grad-Shafranov equation with a nonlocal condition

Two homogeneous Dirichlet problems for the Grad-Shafranov equation with an affine right-hand side are considered in plane simply connected domains with a piecewise smooth boundary. The problems are denoted by and $(\mathfrak{D})$ , and $(\mathfrak{A})$ the second involves a nonlocal condition. The corresponding inverse problems $(\mathfrak{D}^{ - 1} )$ and $(\mathfrak{A}^{ - 1} )$ of finding unknown parameters on the right-hand side of the equation from a given normal derivative of the solution to the respective direct problem are also considered. These problems arise in the computation of plasma flow characteristics in a tokamak. It is shown that the sought parameters can be found from two given quantities: (i) the normal derivative in the corresponding direct problem, which is physically interpreted as the magnetic field at an arbitrary single point $\tilde x$ of a special subset $\tilde \Gamma $ of the boundary Γ, and (ii) the integral of the normal derivative over Γ, which is physically interpreted as the total current through the tokamak cross section. Both problems are shown to be uniquely solvable, and necessary and sufficient conditions for unique solvability are given. A method for finding the desired parameters is proposed, which includes a technique for finding the subset $\tilde \Gamma $ . The results are based, first, on the multipole method, which ensures the high-order accurate computation of the normal derivatives of the solution to direct problems $(\mathfrak{D})$ and $(\mathfrak{A})$ , and, second, on the asymptotics of these derivatives as one of the parameters on the right-hand side of the equation tends to infinity.     

\epsilon -Henig proper efficiency of set-valued optimization problems in real ordered linear spaces

The aim of this paper is to investigate \(\epsilon \) -Henig proper efficiency of set-valued optimization problems in linear spaces. Firstly, a new notion of \(\epsilon \) -Henig properly efficient point is introduced in linear spaces. Secondly, scalarization theorems of set-valued optimization problems are established in the sense of \(\epsilon \) -Henig proper efficiency. Finally, under the assumption of generalized cone subconvexlikeness, Lagrange multiplier theorems are obtained. Our results generalize some known results in the literature from topological spaces to linear spaces.     

On efficiency and mixed duality for a new class of nonconvex multiobjective variational control problems

In this paper, we extend the notions of \((\Phi ,\rho )\) -invexity and generalized \((\Phi ,\rho )\) -invexity to the continuous case and we use these concepts to establish sufficient optimality conditions for the considered class of nonconvex multiobjective variational control problems. Further, multiobjective variational control mixed dual problem is given for the considered multiobjective variational control problem and several mixed duality results are established under \((\Phi ,\rho )\) -invexity.     

Positive solutions for a system of generalized Lidstone problems

In this paper, we study the existence and multiplicity of positive solutions for the system of the generalized Lidstone problems We use fixed point index theory to establish our main results based on a priori estimates achieved by utilizing some properties of concave functions, so that the nonlinearities f and g are allowed to grow in distinct manners, with one of them growing superlinearly and the other growing sublinearly.     

Existence theorem for a class of generalized quasi-variational inequalities

In this paper we consider a class of generalized quasi-variational inequalities. The variational problem is studied in the convex set \(X\times Y\) , with \(Y\) bounded and \(X\) unbounded. In the latter settings, we investigate about the solvability of the problem. In particular, by using the perturbation theory, we give an existence result of the solution without requesting any coercivity hypothesis on the operator. Finally, we give an application to the obtained theoretical results in terms of an economic equilibrium problem.     

Approximation of periodic solutions for a dissipative hyperbolic equation

This paper studies the numerical approximation of periodic solutions for an exponentially stable linear hyperbolic equation in the presence of a periodic external force $f$ . These approximations are obtained by combining a fixed point algorithm with the Galerkin method. It is known that the energy of the usual discrete models does not decay uniformly with respect to the mesh size. Our aim is to analyze this phenomenon’s consequences on the convergence of the approximation method and its error estimates. We prove that, under appropriate regularity assumptions on $f$ , the approximation method is always convergent. However, our error estimates show that the convergence’s properties are improved if a numerically vanishing viscosity is added to the system. The same is true if the nonhomogeneous term $f$ is monochromatic. To illustrate our theoretical results we present several numerical simulations with finite element approximations of the wave equation in one or two dimensional domains and with different forcing terms.     

Bilevel invex equilibrium problems with applications

In this paper, bilevel invex equilibrium problems of Hartman-Stampacchia type and Minty type [resp., in short, (HSBEP) and (MBEP)] are firstly introduced in finite Euclidean spaces. The relationships between (HSBEP) and (MBEP) are presented under some suitable conditions. By using fixed point technique, the nonemptiness and compactness of solution sets to (HSBEP) and (MBEP) are established under the invexity, respectively. As applications, we investigate the existence of solution and the behavior of solution set to the bilevel pseudomonotone variational inequalities of [Anh et al. J Glob Optim 2012, doi:10.1007/s10898-012-9870-y] and the solvability of minimization problem with variational inequality constraint.     

A hybrid viscosity iterative method with averaged mappings for split equilibrium problems and fixed point problems

In this paper, with the help of averaged mappings, we introduce and study a hybrid iterative method to approximate a common solution of a split equilibrium problem and a fixed point problem of a finite collection of nonexpansive mappings. We prove that the sequences generated by the iterative scheme strongly converges to a common solution of the above-said problems. We give some numerical examples to ensure that our iterative scheme is more efficient than the methods of Plubtieng and Punpaeng (J. Math Anal. Appl. 336(1), 455–469, 15), Liu (Nonlinear Anal. 71(10), 4852–4861, 10) and Wen and Chen (Fixed Point Theory Appl. 2012(1), 1–15, 18). The results presented in this paper are the extension and improvement of the recent results in the literature.     

Iterative reweighted minimization methods for l_p regularized unconstrained nonlinear programming

In this paper we study general \(l_p\) regularized unconstrained minimization problems. In particular, we derive lower bounds for nonzero entries of the first- and second-order stationary points and hence also of local minimizers of the \(l_p\) minimization problems. We extend some existing iterative reweighted \(l_1\) ( \(\mathrm{IRL}_1\) ) and \(l_2\) ( \(\mathrm{IRL}_2\) ) minimization methods to solve these problems and propose new variants for them in which each subproblem has a closed-form solution. Also, we provide a unified convergence analysis for these methods. In addition, we propose a novel Lipschitz continuous \({\epsilon }\) -approximation to \(\Vert x\Vert ^p_p\) . Using this result, we develop new \(\mathrm{IRL}_1\) methods for the \(l_p\) minimization problems and show that any accumulation point of the sequence generated by these methods is a first-order stationary point, provided that the approximation parameter \({\epsilon }\) is below a computable threshold value. This is a remarkable result since all existing iterative reweighted minimization methods require that \({\epsilon }\) be dynamically updated and approach zero. Our computational results demonstrate that the new \(\mathrm{IRL}_1\) method and the new variants generally outperform the existing \(\mathrm{IRL}_1\) methods (Chen and Zhou in 2012; Foucart and Lai in Appl Comput Harmon Anal 26:395–407, 2009).     

Morrey-type regularity of solutions to parabolic problems with discontinuous data

We consider regular oblique derivative problem in cylinder Q _T ?=????× (0, T), ${\Omega\subset {\mathbb R}^n}$ for uniformly parabolic operator ${{{\mathfrak P}}=D_t- \sum_{i,j=1}^n a^{ij}(x)D_{ij}}$ with VMO principal coefficients. Its unique strong solvability is proved in Manuscr. Math. 203?C220 (2000), when ${{{\mathfrak P}}u\in L^p(Q_T)}$ , ${p\in(1,\infty)}$ . Our aim is to show that the solution belongs to the generalized Sobolev?CMorrey space ${W^{2,1}_{p,\omega}(Q_T)}$ , when ${{{\mathfrak P}}u\in L^{p,\omega} (Q_T)}$ , ${p\in (1, \infty)}$ , ${\omega(x,r):\,{\mathbb R}^{n+1}_+\to {\mathbb R}_+}$ . For this goal an a priori estimate is obtained relying on explicit representation formula for the solution. Analogous result holds also for the Cauchy?CDirichlet problem.