We study nonsmooth multiobjective fractional programming problem containing local Lipschitz exponential $B\text{-}\left(p,r\right)$-invex functions with respect to $\eta$ and $b$. We introduce a new concept of nonconvex functions, called exponential $B\text{-}\left(p,r\right)$-invex functions. Base on the generalized invex functions, we establish sufficient optimality conditions for a feasible point to be an efficient solution. Furthermore, employing optimality conditions to perform Mond-Weir type duality model and prove the duality theorems including weak duality, strong duality, and strict converse duality theorem under exponential $B\text{-}\left(p,r\right)$-invexity assumptions. Consequently, the optimal values of the primal problem and the Mond-Weir type duality problem have no duality gap under the framework of exponential $B\text{-}\left(p,r\right)$-invexity.

A generalized Caristi type coincidence point theorem and its equivalences in the setting of topological spaces by using a kind of nonmetric type function are obtained. These results are used to establish variational principle and its equivalences in $d$-complete spaces, bornological vector space, seven kinds of completed quasi-semimetric spaces equipped with $Q$-functions, uniform spaces with $q$-distance, generating spaces of quasimetric family, and fuzzy metric spaces.

A common fixed point theorem for a pair of maps satisfying condition (C) is proved under certain conditions. We extend the well-knownDeMarr's fixed point theorem to the case of noncommuting family of maps satisfying condition (C). As for an application, an invariant approximation theorem is also derived.

We prove a common fixed point theorem for two pairs of compatible and subsequentially continuous (alternately subcompatible and reciprocally continuous) mappings satisfying a general contractive condition in a metric space. Some illustrative examples are furnished to highlight the realized improvements. Our result improves the main result of Sedghi and Shobe (2007).

This paper is concerned with $H_{\infty }$ control for linear positive discrete-time systems. Positive systems are characterized by nonnegative restriction on systems’ variables. This restriction results in some remarkable results which are available only for linear positive systems. One of them is the celebrated diagonal positive definite matrix solutions to some existed well-known results for linear systems without nonnegative restriction. We provide an alternative proof for criterion of $H_{\infty }$ norm by using separating hyperplane theorem and Perron-Frobenius theorem for nonnegative matrices. We also consider $H_{\infty }$ control problem for linear positive discrete-time systems via state feedback. Necessary and sufficient conditions for such problem are presented under controller gain with and without nonnegative restriction, and then the desired controller gains can be obtained from the feasible solutions.

We will combine linear successive overrelaxation method with nonlinear monotone iterative scheme to obtain a new iterative method for solving nonlinear equations. The basic idea of this method joining traditional monotone iterative method (known as the method of lower and upper solutions) which depends essentially on the monotone parameter is that by introducing an acceleration parameter one can construct a sequence to accelerate the convergence. The resulting increase in the speed of convergence is very dramatic. Moreover, the sequence can accomplish monotonic convergence behavior in the iterative process when some suitable acceleration parameters are chosen. Under some suitable assumptions in aspect of the nonlinear function and the matrix norm generated from this method, we can prove the boundedness and convergence of the resulting sequences. Application of the iterative scheme is given to a logistic model problem in ecology, and numerical results for a test problem with known analytical solution are given to demonstrate the accuracy and efficiency of the present method.

Let $F:ℝ\times ℝ\to ℝ$ be a real-valued polynomial function of the form $F(x,y)=a_s(x)y^s+a_{s-\mathrm{1}}(x)y^{s-\mathrm{1}}+\cdots +a_{\mathrm{0}}(x)$, where the degree $s$ of $y$ in $F(x,y)$ is greater than or equal to $\mathrm{1}$. For arbitrary polynomial function $f(x)\in ℝ[x]$, $x\in ℝ$, we will find a polynomial solution $y(x)\in ℝ[x]$ to satisfy the following equation: ($\mathrm{*}$): $F(x,y(x))=af(x)$, where $a\in ℝ$ is a constant depending on the solution $y(x)$, namely, a quasi-coincidence (point) solution of ($\mathrm{*}$), and $a$ is called a quasi-coincidence value. In this paper, we prove that (i) the leading coefficient $a_s(x)$ must be a factor of $f(x)$, and (ii) each solution of ($\mathrm{*}$) is of the form $y(x)=-a_{s-\mathrm{1}}(x)/s{a}_s(x)+\lambda p(x)$, where $\lambda$ is arbitrary and $p(x)=c(f(x)/a_s(x){)}^{\mathrm{1}/s}$ is also a factor of $f(x)$, for some constant $c\in ℝ$, provided the equation $(\mathrm{*})$ has infinitely many quasi-coincidence (point) solutions.

We prove new coincidence point theorems for the $(\phi ,\psi ,\varphi ,\xi )$-contractions and generalized Meir-Keeler-type $\alpha$-$\psi$-contractions in partially ordered metric spaces. Our results generalize many recent coincidence point theorems in the literature.

A dual quaternion is associated with two quaternions that have basis elements $e_0$, $e_1$, $e_2$, $e_3$, and $\epsilon$. Dual numbers are often written in the form $z=\zeta +\epsilon {\zeta }^{*}$, where $\epsilon$ is the dual identity and has the properties ${\epsilon }^2=0(\epsilon \ne 0)$. We research the properties of some regular functions with values in dual quaternion and give applications of the extension problem for dual quaternion functions.

An equilibrium chance-constrained multiobjective programming model with birandom parameters is proposed. A type of linear model is converted into its crisp equivalent model. Then a birandom simulation technique is developed to tackle the general birandom objective functions and birandom constraints. By embedding the birandom simulation technique, a modified genetic algorithm is designed to solve the equilibrium chance-constrained multiobjective programming model. We apply the proposed model and algorithm to a real-world inventory problem and show the effectiveness of the model and the solution method.

We propose an explicit iterative scheme for finding a common element of the set of fixed points of infinitely many strict pseudo-contractive mappings and the set of solutions of an equilibrium problem by the general iterative method, which solves the variational inequality. In the setting of real Hilbert spaces, strong convergence theorems are proved. The results presented in this paper improve and extend the corresponding results reported by some authors recently. Furthermore, two numerical examples are given to demonstrate the effectiveness of our iterative scheme.

We consider the regularity for nonadditive measures. We prove that the non-additive measures which satisfy Egoroff's theorem and have pseudometric generating property possess Radon property (strong regularity) on a complete or a locally compact, separable metric space.

In a topological sup-semilattice, we established a new existence result for vector quasiequilibrium problems. By the analysis of essential stabilities of maximal elements in a topological sup-semilattice, we prove that for solutions of each vector quasi-equilibrium problem, there exists a connected minimal essential set which can resist the perturbation of the vector quasi-equilibrium problem.

We consider multivalued nonself-weak contractions on convex metric spaces and establish the existence of a fixed point of such mappings. Presented theorem generalizes results of M. Berinde and V. Berinde (2007), Assad and Kirk (1972), and many others existing in the literature.

Using a fixed point theorem in a partially ordered set, we give a new proof of the Hahn-Banach theorem in the case where the range space is a partially ordered vector space.

We introduce composite implicit and explicit iterative algorithms for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a real smooth and uniformly convex Banach space. These composite iterative algorithms are based on Korpelevich's extragradient method and viscosity approximation method. We first consider and analyze a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another composite explicit iterative algorithm in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures.

By using a fixed point theorem in a cone and the nonlocal third-order BVP's Green function, the existence of at least one positive solution for the third-order boundary-value problem with the integral boundary conditions $x^{\mathrm{\prime \prime \prime }}$ $(t)+f(t,x(t),x^{\prime }(t))=0$, $t\in J$, $x(0)=0$, $x^{\mathrm{\prime \prime }}(0)=0$, and $x(1)={\displaystyle {\int }_0^{1}g(t)x(t)dt}$ is considered, where $f$ is a nonnegative continuous function, $J=[0,1]$, and $g\in L[0,1].$ The emphasis here is that $f$ depends on the first-order derivatives.

By using a new hybrid method, a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of Bregman strongly nonexpansive mappings in a reflexive Banach space is proved.

We introduce and study the notion of common coupled fixed points for a pair of mappings in complex valued metric space and demonstrate the existence and uniqueness of the common coupled fixed points in a complete complex-valued metric space in view of diverse contractive conditions. In addition, our investigations are well supported by nontrivial examples.

We consider a vector variational inequality in a finite-dimensional space. A new gap function is proposed, and an equivalent optimization problem for the vector variational inequality is also provided. Under some suitable conditions, we prove that the gap function is directionally differentiable and that any point satisfying the first-order necessary optimality condition for the equivalent optimization problem solves the vector variational inequality. As an application, we use the new gap function to reformulate a stochastic vector variational inequality as a deterministic optimization problem. We solve this optimization problem by employing the sample average approximation method. The convergence of optimal solutions of the approximation problems is also investigated.

We give new results of a cyclic generalized weakly $C$-contraction in partial metric space. The results of this paper extend, generalize, and improve some fixed point theorems in the literature.

The aim of this paper is to prove some best proximity point theorems for new classes of cyclic mappings, called pointwise cyclic orbital contractions and asymptotic pointwise cyclic orbital contractions. We also prove a convergence theorem of best proximity point for relatively nonexpansive mappings in uniformly convex Banach spaces.

Some new limit contractive conditions in fuzzy metric spaces are introduced, by using property (E.A), some common fixed point theorems for four maps are proved in GV-fuzzy metric spaces. As an application of our results, some new contractive conditions are presented, and some common fixed point theorems are proved under these contractive conditions. The contractive conditions presented in this paper contain or generalize many contractive conditions that appeared in the literatures. Some examples are given to illustrate that our results are real generalizations for the results in the references and to show that our limit contractive conditions are important for the existence of fixed point.

We prove two new fixed point theorems in the framework of partially ordered metric spaces. Our results generalize and improve many recent fixed point theorems in the literature.

We consider an auxiliary operator, defined in a real Hilbert space in terms of $K$ and $F$, that is, monotone and Lipschitz mappings (resp., monotone and bounded mappings). We use an explicit iterative process that converges strongly to a solution of equation of Hammerstein type. Furthermore, our results improve related results in the literature.

We introduce a new class of non-self-contractive mappings. For such mappings, we study the existence and uniqueness of best proximity points. Several applications and interesting consequences of our obtained results are derived.

We introduce a new iterative algorithm for finding a common element of a fixed point problem of amenable semigroups of nonexpansive mappings, the set solutions of a system of a general system of generalized equilibria in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative algorithm to a common element of the above three sets under some suitable conditions. As applications, at the end of the paper, we apply our results to find the minimum-norm solutions which solve some quadratic minimization problems. The results obtained in this paper extend and improve many recent ones announced by many others.