We suggest and analyze a predictor-corrector method for solving nonsmooth convex equilibrium problems based on the auxiliary problem principle. In the main algorithm each stage of computation requires two proximal steps. One step serves to predict the next point; the other helps to correct the new prediction. At the same time, we present convergence analysis under perfect foresight and imperfect one. In particular, we introduce a stopping criterion which gives rise to $\mathrm{\Delta }$-stationary points. Moreover, we apply this algorithm for solving the particular case: variational inequalities.

This paper presents some results concerning the properties of distances and existence and uniqueness of best proximity points of p-cyclic proximal, weak proximal contractions, and some of their generalizations for the non-self-mapping $T:{\bigcup }_{i\in \stackrel{-}{p}}{A}_i\to {\bigcup }_{i\in \stackrel{-}{p}}{B}_i(p\ge 2)$, where $A_i$ and $B_i$, $\forall i\in \stackrel{-}{p}=\{\mathrm{1,2},\dots ,p\}$, are nonempty subsets of $X$ which satisfy $T\left(A_i\right)\subseteq B_i,\forall i\in \stackrel{-}{p}$, such that $(X,d)$ is a metric space. The boundedness and the convergence of the sequences of distances in the domains and in their respective image sets of the cyclic proximal and weak cyclic proximal non-self-mapping, and of some of their generalizations are investigated. The existence and uniqueness of the best proximity points and the properties of convergence of the iterates to such points are also addressed.

This paper is concerned with the nonlinear Klein-Gordon equation with damping term and nonnegative potentials. We introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions. Using the potential well argument, we obtain a new existence theorem of global solutions and a blow-up result for solutions in finite time.

Hussain et al. (2013) established new fixed point results in complete metric space. In this paper, we prove fixed point results of α-admissible mappings with respect to η, for modified contractive condition in complete metric space. An example is given to show the validity of our work. Our results generalize/improve several recent and classical results existing in the literature.

An off-line optimization approach of high precision minimum time feedrate for CNC machining is proposed. Besides the ordinary considered velocity, acceleration, and jerk constraints, dynamic performance constraint of each servo drive is also considered in this optimization problem to improve the tracking precision along the optimized feedrate trajectory. Tracking error is applied to indicate the servo dynamic performance of each axis. By using variable substitution, the tracking error constrained minimum time trajectory planning problem is formulated as a nonlinear path constrained optimal control problem. Bang-bang constraints structure of the optimal trajectory is proved in this paper; then a novel constraint handling method is proposed to realize a convex optimization based solution of the nonlinear constrained optimal control problem. A simple ellipse feedrate planning test is presented to demonstrate the effectiveness of the approach. Then the practicability and robustness of the trajectory generated by the proposed approach are demonstrated by a butterfly contour machining example.

We consider the fourth-order difference equation: $\mathrm{\Delta }(z(k+1){\mathrm{\Delta }}^{3}u(k-1))=w(k)f(k,u(k))$, $k\in \{\mathrm{1,2},\dots ,n-1\}$ subject to the boundary conditions: $u(0)=u(n+2)={\sum }_{i=1}^{n+1}g(i)u(i)$, $a{\mathrm{\Delta }}^{2}u(0)-bz(2){\mathrm{\Delta }}^{3}u(0)={\sum }_{i=3}^{n+1}h(i){\mathrm{\Delta }}^{2}u(i-2)$, $a{\mathrm{\Delta }}^{2}u(n)-bz(n+1){\mathrm{\Delta }}^{3}u(n-1)={\sum }_{i=3}^{n+1}h(i){\mathrm{\Delta }}^{2}u(i-2)$, where $a,b>0$ and $\mathrm{\Delta }u(k)=u(k+1)-u(k)$ for $k\in \{\mathrm{0,1},\dots ,n-1\}$, $f:\{\mathrm{0,1},\dots ,n\}\times [0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ is continuous. $h(i)$ is nonnegative $i\in \{\mathrm{2,3},\dots ,n+2\}$; $g(i)$ is nonnegative for $i\in \{\mathrm{0,1},\dots ,n\}$. Using fixed point theorem of cone expansion and compression of norm type and Hölder’s inequality, various existence, multiplicity, and nonexistence results of positive solutions for above problem are derived, which extends and improves some known recent results.

The idea of $[\lambda , \mu ]$-almost convergence (briefly, ${\mathcal{F}}_{[\lambda , \mu ]}$-convergence) has been recently introduced and studied by Mohiuddine and Alotaibi (2014). In this paper first we define a norm on ${\mathcal{F}}_{[\lambda , \mu ]}$ such that it is a Banach space and then we define and characterize those four-dimensional matrices which transform ${\mathcal{F}}_{[\lambda , \mu ]}$-convergence of double sequences $x=(x_{jk})$ into ${\mathcal{F}}_{[\lambda , \mu ]}$-convergence. We also define a ${\mathcal{F}}_{[\lambda , \mu ]}$-core of $x=(x_{jk})$ and determine a Tauberian condition for core inclusions and core equivalence.

We introduce two proximal iterative algorithms with errors which converge strongly to the common solution of certain variational inequality problems for a finite family of pseudocontractive mappings and a finite family of monotone mappings. The strong convergence theorems are obtained under some mild conditions. Our theorems extend and unify some of the results that have been proposed for this class of nonlinear mappings.

In the setting of Hilbert spaces, inspired by Iemoto and Takahashi (2009), we study a Mann’s method with viscosity to approximate strongly (common) fixed points of a nonexpansive mapping and a nonspreading mapping. A crucial tool in our results is the nonspreading-average type mapping.

Let $K$ be a nonempty closed convex subset of a real Banach space $E$, let $S:K\to K$ be nonexpansive, and let $T:K\to K$ be Lipschitz strongly pseudocontractive mappings such that $p\in F\left(S\right)\cap F\left(T\right)=\left\{x\in K:Sx=Tx=x\right\}$ and $∥x-Sy∥\le ∥Sx-Sy∥ \text{and} ∥x-Ty∥\le ∥Tx-Ty∥$ for all $x, y\in K$. Let $\left\{{\beta }_n\right\}$ be a sequence in $\left[0, 1\right]$ satisfying (i) ${\sum }_{n=1}^{\infty }{\beta }_n=\infty$; (ii) ${\text{lim}}_{n\to \infty }{\beta }_n=0.$ For arbitrary $x_0\in K$, let $\left\{x_n\right\}$ be a sequence iteratively defined by $x_n=S{y}_n, y_n=\left(1-{\beta }_n\right)x_{n-1}+{\beta }_{n}T{x}_n, n\ge 1.$ Then the sequence $\left\{x_n\right\}$ converges strongly to a common fixed point $p$ of $S$ and $T$.

By using the weaker Meir-Keeler function $\varphi$ and the triangular $\alpha$-admissible mapping $\alpha$, we introduce the notion of $(\alpha -\varphi )$-weaker Meir-Keeler contractive mappings and prove a theorem which assures the existence of a periodic point for these mappings on generalized quasimetric spaces.

We introduce and analyze a hybrid iterative algorithm by combining Korpelevich's extragradient method, the hybrid steepest-descent method, and the averaged mapping approach to the gradient-projection algorithm. It is proven that, under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of finitely many nonexpansive mappings, the solution set of a generalized mixed equilibrium problem (GMEP), the solution set of finitely many variational inclusions, and the solution set of a convex minimization problem (CMP), which is also a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solving a hierarchical variational inequality problem with constraints of the GMEP, the CMP, and finitely many variational inclusions.

A class of α-admissible contractions defined via altering distance functions is introduced. The existence and uniqueness conditions for fixed points of such maps on complete metric spaces are investigated and related fixed point theorems are presented. The results are reconsidered in the context of partially ordered metric spaces and applied to boundary value problems for differential equations with periodic boundary conditions.

We mainly present several equivalent characterizations of the strong metric subregularity of the Mordukhovich subdifferential for an extended-real-valued lower semicontinuous, prox-regular, and subdifferentially continuous function acting on an Asplund space.

We introduce two new types of fixed point theorems in the collection of multivalued and single-valued mappings in complete metric spaces.

The nonlinear minimax problems without constraints are discussed. Due to the expensive computation for solving QP subproblems with inequality constraints of SQP algorithms, in this paper, a QP-free algorithm which is also called sequential systems of linear equations algorithm is presented. At each iteration, only two systems of linear equations with the same coefficient matrix need to be solved, and the dimension of each subproblem is not of full dimension. The proposed algorithm does not need any penalty parameters and barrier parameters, and it has small computation cost. In addition, the parameters in the proposed algorithm are few, and the stability of the algorithm is well. Convergence property is described and some numerical results are provided.

The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation, which includes second-order cone as a special case when the rotation angle is 45 degrees. Let ${\mathcal{L}}_{\theta }$ denote the circular cone in ${\mathbb{R}}^n$. For a function $f$ from $\mathbb{R}$ to $\mathbb{R}$, one can define a corresponding vector-valued function $f^{{\mathcal{L}}_{\theta }}$ on ${\mathbb{R}}^n$ by applying $f$ to the spectral values of the spectral decomposition of ${x\in \mathbb{R}}^n$ with respect to ${\mathcal{L}}_{\theta }$. In this paper, we study properties that this vector-valued function inherits from $f$, including Hölder continuity, $B$-subdifferentiability, $\rho$-order semismoothness, and positive homogeneity. These results will play crucial role in designing solution methods for optimization problem involved in circular cone constraints.

We propose a strictly convex functional in which the regular term consists of the total variation term and an adaptive logarithm based convex modification term. We prove the existence and uniqueness of the minimizer for the proposed variational problem. The existence, uniqueness, and long-time behavior of the solution of the associated evolution system is also established. Finally, we present experimental results to illustrate the effectiveness of the model in noise reduction, and a comparison is made in relation to the more classical methods of the traditional total variation (TV), the Perona-Malik (PM), and the more recent D-α-PM method. Additional distinction from the other methods is that the parameters, for manual manipulation, in the proposed algorithm are reduced to basically only one.

The aim of this work is to explore the optimal exploitation way for a biological resources model incorporating individual’s size difference and spatial effects. The existence of a unique nonnegative solution to the state system is shown by means of Banach’s fixed point theorem, and the continuous dependence of the population density with the harvesting effort is given. The optimal harvesting strategy is established via normal cone and adjoint system technique. Some conditions are found to assure that there is only one optimal policy.

In the framework of a real Banach space with uniformly Gateaux differentiable norm, some new viscosity iterative sequences $\{x_n\}$ are introduced for an infinite family of asymptotically nonexpansive mappings ${\left\{T_i\right\}}_{i=1}^{\infty }$ in this paper. Under some appropriate conditions, we prove that the iterative sequences $\{x_n\}$ converge strongly to a common fixed point of the mappings ${\left\{T_i\right\}}_{i=1}^{\infty }$, which is also a solution of a variational inequality. Our results extend and improve some recent results of other authors.

In this paper a HBV infection model with impulsive vaccination is considered. By using fixed point theorem and stroboscopic map we prove the existence of disease-free T-periodic solution. Also by comparative theorem of impulsive differential equation we get the global asymptotic stability of the disease-free periodic solution and permanence of the disease. Numerical simulations show the influence of parameters on the dynamics of HBV, which provided references for seeking optimal measures to control the transmission of HBV.

We study a class of resonant cooperative elliptic systems and replace the Ambrosetti-Rabinowitz superlinear condition with general superlinear conditions. We obtain ground state solutions and infinitely many nontrivial solutions of this system by a generalized Nehari manifold method developed recently by Szulkin and Weth.

Our purpose in this paper is to present a better method of parametric estimation for a bivariate nonlinear regression model, which takes the performance indicator of rubber aging as the dependent variable and time and temperature as the independent variables. We point out that the commonly used two-step method (TSM), which splits the model and estimate parameters separately, has limitation. Instead, we apply the Marquardt’s method (MM) to implement parametric estimation directly for the model and compare these two methods of parametric estimation by random simulation. Our results show that MM has better effect of data fitting, more reasonable parametric estimates, and smaller prediction error compared with TSM.

By using the formula of the ε-subdifferential for the sum of a convex function with a composition of convex functions, some necessary and sufficient optimality conditions for a DC programming problem involving a composite function are obtained. As applications, a composed convex optimization problem, a DC optimization problem, and a convex optimization problem with a linear operator are examined at the end of this paper.

Multistage voting is a common voting form through which the winners are selected. By virtue of weighted multistage voting rules, in this paper, we establish a weighted voting model by analyzing the correlation between individual preference and group preference. The weights of voters in each voting stage are adjusted through preference deviation degrees between individual preferences and group preference, and the ranking among candidates in each stage is determined according to weighted Borda function value. Examples are given to verify our model, which shows that weighted information aggregation model can mine more useful information from different individual preferences of voters to quicken the aggregation of group preference.

Now, it is known that the split common fixed point problem is a generalization of the split feasibility problem and of the convex feasibility problem. In this paper, the split common fixed point problem associated with the pseudocontractions is studied. An iterative algorithm has been presented for solving the split common fixed point problem. Strong convergence result is obtained.

We consider a strong mixed vector equilibrium problem in topological vector spaces. Using generalized Fan-Browder fixed point theorem (Takahashi 1976) and generalized pseudomonotonicity for multivalued mappings, we provide some existence results for strong mixed vector equilibrium problem without using KKM-Fan theorem. The results in this paper generalize, improve, extend, and unify some existence results in the literature. Some special cases are discussed and an example is constructed.

Two new iterations with Cesàro's means for nonexpansive mappings are proposed and the strong convergence is obtained as $n\to \infty$. Our main results extend and improve the corresponding results of Xu (2004), Song and Chen (2007), and Yao et al. (2009).

We show that some definitions of multidimensional coincidence points are not compatible with the mixed monotone property. Thus, some theorems reported in the recent publications (Dalal et al., 2014 and Imdad et al., 2013) have gaps. We clarify these gaps and we present a new theorem to correct the mentioned results. Furthermore, we show how multidimensional results can be seen as simple consequences of our unidimensional coincidence point theorem.

Based on the ideas of the classical Benson proper efficiency, a new kind of unified proper efficiency named S-Benson proper efficiency is introduced by using Assumption (B) proposed by Flores-Bazán and Hernández, which unifies some known exact and approximate proper efficiency including $(C,\epsilon )$-proper efficiency and E-Benson proper efficiency in vector optimization. Furthermore, a characterization of S-Benson proper efficiency is established via a kind of nonlinear scalarization functions introduced by Göpfert et al.

We study extended mixed vector equilibrium problems, namely, extended weak mixed vector equilibrium problem and extended strong mixed vector equilibrium problem in Hausdorff topological vector spaces. Using generalized KKM-Fan theorem (Ben-El-Mechaiekh et al.; 2005), some existence results for both problems are proved in noncompact domain.

The equilibrium and fixed point problems are considered. An iterative algorithm is presented. Convergence analysis of the algorithm is provided.

The purpose of this paper is introduce several types of Levitin-Polyak well-posedness for bilevel vector equilibrium and optimization problems with equilibrium constraints. Base on criterion and characterizations for these types of Levitin-Polyak well-posedness we argue on diameters and Kuratowski’s, Hausdorff’s, or Istrǎtescus measures of noncompactness of approximate solution sets under suitable conditions, and we prove the Levitin-Polyak well-posedness for bilevel vector equilibrium and optimization problems with equilibrium constraints. Obtain a gap function for bilevel vector equilibrium problems with equilibrium constraints using the nonlinear scalarization function and consider relations between these types of LP well-posedness for bilevel vector optimization problems with equilibrium constraints and these types of Levitin-Polyak well-posedness for bilevel vector equilibrium problems with equilibrium constraints under suitable conditions; we prove the Levitin-Polyak well-posedness for bilevel equilibrium and optimization problems with equilibrium constraints.

We consider error bound issue for conic inequalities in Hilbert spaces. In terms of proximal subdifferentials of vector-valued functions, we provide sufficient conditions for the existence of a local error bound for a conic inequality. In the Hilbert space case, our result improves and extends some existing results on local error bounds.

This paper is concerned with the existence of positive solutions of semilinear biharmonic problem whose associated functionals do not satisfy the Palais-Smale condition.

We use a three-step iterative process to prove some strong and $\mathrm{\Delta }$-convergence results for nonexpansive mappings in a uniformly convex hyperbolic space, a nonlinear domain. Three-step iterative processes have numerous applications and hyperbolic spaces contain Banach spaces (linear domains) as well as CAT(0) spaces. Thus our results can be viewed as extension and generalization of several known results in uniformly convex Banach spaces as well as CAT(0) spaces.

We first introduce the notion of $\eta$-upper sign property which is an extension of the upper sign property introduced in Castellani and Giuli, 2013, by relaxing convexity on the set. Afterwards, we establish a link between the solution sets of local dual equilibrium problem (Minty local equilibrium problem) and equilibrium problem for mappings whose domains are not necessarily convex by relaxing the upper sign continuity on the map, as it is assumed in the literature (Bianchi and Pini, 2005; Castellani and Giuli, 2013; Farajzadeh and Zafarani, 2010). Accordingly, it allows us to extend and obtain some existence results for equilibrium-like problems.

We first introduce and analyze one iterative algorithm by using the composite shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. We prove a strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions. Our results improve and extend the corresponding results in the earlier and recent literature.

This paper introduces and analyzes a viscosity iterative algorithm for an infinite family of nonexpansive mappings ${\{T_i\}}_{i=1}^{\infty }$ in the framework of a strictly convex and uniformly smooth Banach space. It is shown that the proposed iterative method converges strongly to a common fixed point of ${\{T_i\}}_{i=1}^{\infty }$, which solves specific variational inequalities. Necessary and sufficient convergence conditions of the iterative algorithm for an infinite family of nonexpansive mappings are given. Results shown in this paper represent an extension and refinement of the previously known results in this area.

We introduce and analyze a relaxed extragradient-like viscosity iterative algorithm for finding a solution of a generalized mixed equilibrium problem with constraints of several problems: a finite family of variational inequalities for inverse strongly monotone mappings, a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings, and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we derive the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems which also solves a variational inequality problem.

We introduce and analyze a hybrid extragradient-like viscosity iterative algorithm for finding a common solution of a systems of generalized equilibrium problems and a generalized mixed equilibrium problem with the constraints of two problems: a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we prove the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems.

$G$-metric spaces proved to be a rich source for fixed point theory; however, the best proximity point problem has not been considered in such spaces. The aim of this paper is to introduce certain new classes of proximal contraction mappings and establish the best proximity point theorems for such kind of mappings in $G$-metric spaces. As a consequence of these results, we deduce certain new best proximity and fixed point results. Moreover, we present an example to illustrate the usability of the obtained results.

We establish certain fixed point results for $\alpha \text{-}\eta$-generalized convex contractions, $\alpha \text{-}\eta$-weakly Zamfirescu mappings, and $\alpha \text{-}\eta$-Ćirić strong almost contractions. As an application, we derive some Suzuki type fixed point theorems and certain new fixed point theorems in metric spaces endowed with a graph and a partial order. Moreover, we discuss some illustrative examples to highlight the realized improvements.

An explicit projection algorithm with viscosity technique is constructed for finding the fixed points of the pseudocontractive mapping in Hilbert spaces. Strong convergence theorem is demonstrated. Consequently, as an application, we can approximate to the minimum-norm fixed point of the pseudocontractive mapping.