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In this paper, we obtain a version of Ekeland's variational principle for interval-valued functions by means of the Dancs-Hegedus-Medvegyev theorem (Dancs et al. (1983) [14] ). We also derive two versions of Ekeland's variational principle involving the generalized Hukuhara Gâteaux differentiability of interval-valued functions as well as a version of Ekeland's variational principle for interval-valued bifunctions. Finally, we apply these new versions of Ekeland's variational principle to fixed point theorems, to interval-valued optimization problems, to the interval-valued Mountain Pass Theorem, to noncooperative interval-valued games, and to interval-valued optimal control problems described by interval-valued differential equations.

Fuzzy variational inequalities (FVIs) are modeling tools used to characterize a variety of fuzzy decision making problems arising in mathematical optimization, control theory, operations research, and game theory. Fractional order delay differential equations are also finding applications in all disciplines including chemistry, physics, and finance. The aim of this paper is to introduce and study a new dynamical fuzzy system, named the fractional differential fuzzy variational inequality (FDFVI) with delay, which bridges these two areas of research and broadens the class of problems amenable to be studied under the fuzzy environments. By using the KKM theorem and monotonicity arguments, we show that the solution set of the FVI is nonempty, convex and compact. We also establish the upper semicontinuity of the solution mapping U of the FVI involved in the FDFVI with delay. Moreover, we obtain an existence of the mild solution for the FDFVI with delay by employing Bohnenblust-Karlin's fixed point theorem. In addition, we provide an approximating algorithm to find a solution of the FDFVI with delay. Finally, we give two numerical examples to illustrate our main results.

The goal of this paper is to introduce and study a new class of fractional differential hemivariational inequality formulated by an evolutionary hemivariational inequality and a fractional differential equation in Banach spaces. By employing the Rothe method and the surjectivity result, we derive the existence of unique solution for such a problem under some mild conditions. Moreover, we use the fully discrete scheme to approximate the fractional differential hemivariational inequality and provide an error estimate for the approximation. Finally, the main results are applied to obtain the unique solvability as well as the numerical analysis for a viscoelastic frictional contact problem with adhesion.

The goal of this paper is to consider a new fuzzy differential system consisting of a fuzzy fractional differential inclusion combined with a variational inequality, which is called fuzzy fractional differential variational inequality (FFDVI). Such a model captures the desired features of both fuzzy fractional differential inclusions and fractional differential variational inequalities within the same framework. By employing the set-valued version of the Krasnoselskii fixed point theorem, an existence of solutions is obtained for the FFDVI under some mild conditions. Moreover, an approximating algorithm is provided to find a solution of the FFDVI. Finally, two numerical examples are given to illustrate our main results.

In this paper, a pricing formula for volatility swaps is delivered when the underlying asset follows the stochastic volatility model with jumps and stochastic intensity. By using Feynman-Kac theorem, a partial integral differential equation is obtained to derive the joint moment generating function of the previous model. Moreover, discrete and continuous sampled volatility swap pricing formulas are given by employing transform techniques and the relationship between two pricing formulas is discussed. Finally, some numerical simulations are reported to support the results presented in this paper., 15PAGES

The bird swarm algorithm (BSA) is a very important bionic intelligence algorithm which can be used to solve many optimization problems. The main idea of this paper is to increase the effectiveness of BSA by improving the flight behaviour. This paper provides an adaptive bird swarm algorithm with the irregular random flight (AI-BSA) for solving the portfolio optimization problems with cardinality constraints. We prove the local convergence of AI-BSA under mild conditions and verify the effectiveness of AI-BSA via some numerical tests. Moreover, we give a detailed process for solving the cardinality constrained portfolio optimization problem by using AI-BSA and provide a numerical example to compare with both the bird swarm algorithm and particle swarm optimization.

In this paper, we establish the arcwise connectedness of the sets of l -minimal solutions and weak l -minimal solutions for set optimization problems under the assumption of the strictly natural quasi cone-convexity. Finally, our main result is applied to the arcwise connectedness of the solution sets for vector optimization problems.

In this paper, we introduce two new matrix stochastic processes: fractional Wishart processes and e -fractional Wishart processes with integer indices which are based on the fractional Brownian motions and then extend e -fractional Wishart processes to the case with non-integer indices. Both processes include classic Wishart processes (if the Hurst index H equals 1 2 ) and present serial correlation of stochastic processes. Applying e -fractional Wishart processes to financial volatility theory, the financial models account for the stochastic volatilities of the assets and for the stochastic correlations not only between the underlying assets’ returns but also between their volatilities and for stochastic serial correlation of the relevant assets.

The purpose of this paper is to introduce and investigate a new system of global fractional-order interval implicit projection neural networks. An existence and uniqueness theorem of the equilibrium point for the system of global fractional-order interval implicit projection neural networks is obtained under some suitable assumptions. Moreover, Mittag–Leffler stability for the system of global fractional-order interval implicit projection neural networks is also proved. Finally, two numerical examples are given to illustrate the validity of our results.

This paper is devoted to study a new nonlinear system involving a parabolic variational inequality, a history-dependent hemi-variational inequality and a differential equation in Banach spaces. By employing the surjectivity argument and Banach’s fixed point theorem, we derive a unique solvability theorem for such a problem under some mild conditions. Moreover, the main results are applied to obtain the unique solvability of a long memory elastic frictional contact problem with wear and damage. Finally, we use the fully discrete scheme to approximate the contact problem and provide the error estimates for numerical solutions.

This paper considers a mean–variance–utility portfolio problem in which the investor manages her/his wealth by consuming and investing in a financial market consisting of a risk-free bond and a risky asset whose price process follows Heston’s stochastic volatility (SV) model. The objective of the investor is to obtain an equilibrium investment and consumption strategy by maximizing the terminal wealth and the accumulated binary utility function incorporating consumption and income. The semi-closed form expressions for the equilibrium investment and consumption strategy as well as the corresponding value function are derived by solving an extended Hamilton–Jacobi–Bellman system within the game-theoretic framework. Moreover, some numerical experiments are provided to show the effects of model parameters on the equilibrium strategy by adopting the Cobb–Douglas utility function.

In this paper, we establish the lower semicontinuity of the solution mapping and the approximate solution mapping for parametric fixed point problems under some suitable conditions. As applications, the lower semicontinuity result applies to the parametric vector quasi-equilibrium problem, and allows to prove the existence of solutions for generalized Stackelberg games.

In this note, we investigate the global stability for a tuberculosis model with vaccination and treatment. The model exhibits the traditional threshold behavior. We prove that the basic reproduction number characterizes the global dynamics of the model, that is, the disease endemic equilibrium of the model is globally asymptotically stable whenever it exists, which accords with the corresponding conjecture presented in Liu and Zhang (2011).

This paper focuses on the variance and volatility swaps pricing in the framework of a liquidity-adjusted underlying assets model. The pricing formulas of discretely-sampled variance and volatility swaps are obtained under the stochastic liquidity risk by deriving the characteristic function and solving the governing PDEs. The limiting properties for the pricing formulas of discretely-sampled variance and volatility swaps, namely the pricing formulas of continuous sampling, are also given. Finally, to illustrate the performance of the pricing formulas, some detailed numerical experiments are provided, including the comparisons with Monte Carlo (MC) simulations results, the effects of stochastic liquidity and parameters variation on derivative prices as well as the empirical studies from real-market data.

In this paper, a new class of impulsive differential variational inequalities (IDVIs) are introduced and studied in finite dimensional Euclidean spaces. Some existence results on the solutions for the IDVIs are presented under suitable conditions and a convergence theorem of the discrete Euler time-dependent procedure for solving the IDVI is proved by using constructed discrete approximation methods for the impulsive differential inclusions (IDIs). The stability results concerned with the solutions of the IDVIs are also considered when the variation of initial data, impulsive perturbation and right-hand sides happens.

This paper studies a class of global fractional-order projective dynamical systems involving set-valued perturbations in real separable Hilbert spaces. We prove that the set of solutions for this type of systems is nonempty and closed under some suitable conditions. Furthermore, we show that the set of solutions is continuous with respect to initial value in the sense of Hausdorff metric. Finally, an interesting numerical example is given to illustrate the validity of the main theorem presented in this paper.

In this paper, an estimate of convergence rate concerned with an inexact proximal point algorithm for the singularity of maximal monotone vector fields on Hadamard manifolds is discussed. We introduce a weaker growth condition, which is an extension of that of Luque from Euclidean spaces to Hadamard manifolds. Under the growth condition, we prove that the inexact proximal point algorithm has linear/superlinear convergence rate. The main results presented in this paper generalize and improve some corresponding known results.

In this paper, for dealing with the portfolio model from stocks market, a new particle swarm optimization algorithm (NPSO) is presented, in which the optimal and sub-optimal positions of each particle are considered in the iteration process, and the crossover operation is used to avoid premature. It is demonstrated from optimization tests that NPSO outperforms existed PSO. Then NPSO is used to solve a discontinuous programming model, and four different optimal portfolio selections are displayed which are denoted by S 1 , S 2 , S 3 and S 4 , respectively. Finally, actual return rates of these portfolios are obtained, and it is analyzed from related graphs that S 2 and S 3 gain better results.

In this paper, the pricing of foreign exchange (FX) options is studied under the Moretto–Pasquali–Trivellato (MPT) stochastic volatility model by introducing an approximative fractional stochastic volatility and jumps, in which the FX rate has log-normal jump amplitudes, the volatility has asymmetric double exponential jump amplitudes, and the domestic and foreign interest rates are governed by Cox–Ingersoll–Ross (CIR) dynamics. By employing a suitable version of the Fourier inversion technique for corresponding conditional characteristic functions, a semi-analytical formula for the price of FX European call options is obtained under mild conditions. The behavior of the newly derived pricing formula is further demonstrated through some numerical experiments.

In this paper, by applying the comparison theorem on the differential equation and constructing a suitable Lyapunov function, a Lotka-Volterra model with mutual interference and Holling III type functional response is introduced and studied. Some sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the model are obtained.

In this paper, an inexact proximal point algorithm concerned with the singularity of maximal monotone vector fields is introduced and studied on Hadamard manifolds, in which a relative error tolerance with squared summable error factors is considered. It is proved that the sequence generated by the proposed method is convergent to a solution of the problem. Moreover, an application to the optimization problem on Hadamard manifolds is given. The main results presented in this paper generalize and improve some corresponding known results given in the literature.

In this paper, we consider stochastic affine variational inequality problems with nonlinear perturbation (for short, SVIPP). Firstly, we formulate SVIPP as an ERM problem that minimizes the expected residual of the so-called gap function. Furthermore, we study some properties of the ERM problem under some suitable conditions. By means of quasi-Monte Carlo method, we obtain a discrete approximation of ERM problem. Finally, we consider the convergence of optimal solutions and stationary points of the approximation problem as sample size increases.

In this paper, characterizations of the set of solutions for VVI are presented by using scalarization approaches. The set of solutions of VVI is shown to be the set of weak sharpness for gap functions of some scalarization of VVI and for gap functions of VVI under semi-strong monotonicity. Some examples are given to illustrate these results.

In this paper, firstly a modified particle swarm optimization algorithm (MPSO) is developed, in which the mean value of past optimal positions for each particle and the mutation operation are considered for avoiding premature. In the optimization test, MPSO performs better than particle swarm optimization algorithm (PSO). Then MPSO is applied to solve four portfolio optimization models with the real data from the Hong Kong Stock Market, and optimal values are obtained when the number of swarm n = 80 , 160 , respectively. Finally, actual return rates of these models are calculated in numerical experiments, and it is illustrated from these graphs of actual return rates that when considering higher return, Cai’s model performs better in short-term investment.

In this paper, combining the new splitting method proposed by Eckstein and Svaiter and the forcing strong convergence method of Solodov and Svaiter, we propose a splitting proximal projection method for the sum of two maximal monotone operators. We prove that the proposed method is well-defined whether the solution set of the problem is nonempty or not and the sequence generated by the method converges strongly to an extended solution of the problem.

We study the strong convergence of two viscosity iteration processes for pseudo-contractive semigroup and for ϕ -strongly pseudo-contractive mapping in uniformly convex Banach spaces with uniformly Gâteaux differentiable norm. As special cases, we get strong convergence of two viscosity iteration processes for approximating common fixed points of nonexpansive semigroups in certain Banach spaces. The results presented in this paper extend and generalize previous results.

In this paper, we study the solution stability for a class of parametric generalized vector quasiequilibrium problems. By virtue of the parametric gap function, we obtain a sufficient and necessary condition for the Hausdorff lower semicontinuity of the solution mapping to the parametric generalized vector quasiequilibrium problem. The results presented in this paper generalize and improve some main results of Chen et al. (2010) [34], and Zhong and Huang (2011) [35].

In this paper, we study the economic model proposed by Anderlini and Canning, a parameterized class of generalized abstract fuzzy economies together with an associated abstract rationality function, and show that the structural stability of this model implies its robustness to @e-equilibria.

In this paper, a concept of graph convergence concerned with the H (·, ·)-accretive operator is introduced in Banach spaces and some equivalence theorems between of graph-convergence and resolvent operator convergence for the H (·, ·)-accretive operator sequence are proved. As an application, a perturbed algorithm for solving a class of variational inclusions involving the H (·, ·)-accretive operator is constructed. Under some suitable conditions, the existence of the solution for the variational inclusions and the convergence of iterative sequence generated by the perturbed algorithm are also given.

In this paper, some properties of the generalized f-projection operator are proved in Banach spaces. Using these results, the strong convergence theorems for relatively nonexpansive mappings are studied in Banach spaces. As applications, the strong convergence of general H-monotone mappings in Banach spaces is also given. The results presented in this paper generalize and improve the main results of Matsushita and Takahashi (2005) [9].

In this paper, the notion of weak sharp minima is employed to the investigation of set-valued vector variational inequalities. The gap function φ T for set-valued strong vector variational inequalities (for short, SVVI) is proved to be less than the gap function ϕ T for set-valued weak vector variational inequalities (for short, WVVI) under certain conditions, which implies that the solution set of SVVI is equivalent to the solution set of WVVI. Moreover, it is shown that weak sharp minima for the solution sets of SVVI and WVVI hold for min 1 ⩽ i ⩽ n p T i and for gap functions φ T and ϕ T under the assumption of strong pseudomonotonicity, where p T i is a gap function for i-th component of SVVI and WVVI. As an application, the weak Pareto solution set of vector optimization problems (for short, VOP) is proved to be weak sharp minimum for min 1 ⩽ i ⩽ n p ∇ g i when each component g i of objective function is strongly convex.

In this paper, we introduce and study a class of differential mixed variational inequalities in finite dimensional Euclidean spaces. Under various conditions, we obtain linear growth and bounded linear growth of the solution set for the mixed variational inequalities. Moreover, we present some conclusions which enrich the literature on the mixed variational inequalities and generalize the corresponding results of [4] . In particular we prove existence theorems for weak solutions of a differential mixed variational inequality in the weak sense of Caratheodory by using a result on differential inclusions involving an upper semicontinuous set-valued map with closed convex values. Also by employing the results from differential inclusions we establish a convergence result on Euler time-dependent procedure for solving initial-value differential mixed variational inequalities.

In this paper, a kind of Ishikawa type iterative scheme with errors for approximating a common fixed point of two sequences of uniformly quasi-Lipschitzian mappings is introduced and studied in convex metric spaces. Under some suitable conditions, the convergence theorems concerned with the Ishikawa type iterative scheme with errors to approximate a common fixed point of two sequences of uniformly quasi-Lipschitzian mappings were proved in convex metric spaces. The results presented in the paper generalize and improve some recent results of Wang and Liu (C. Wang, L.W. Liu, Convergence theorems for fixed points of uniformly quasi-Lipschitzian mappings in convex metric spaces, Nonlinear Anal., TMA 70 (2009), 2067-2071).

In this paper, we introduce and study a new class of (H,@f)[email protected] operators in Banach spaces. We define a proximal mapping associated with the (H,@f)[email protected] operator and show its Lipschitz continuity. As an application, we consider the solvability for a class of variational inclusions involving the (H,@f)[email protected] operators in Banach spaces. By using the technique of proximal mapping, an iterative algorithm for solving such class of variational inclusions is constructed in Banach spaces. Under some suitable conditions, we prove the convergence of iterative sequence generated by the algorithm.

In this paper, we consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi for a minimization problem, to a mixed variational inequality problem in a Banach space. We establish some metric characterizations of the well-posedness by perturbations. We also show that under suitable conditions, the well-posedness by perturbations of a mixed variational inequality problem is equivalent to the well-posedness by perturbations of a corresponding inclusion problem and a corresponding fixed point problem. Also, we derive some conditions under which the well-posedness by perturbations of a mixed variational inequality is equivalent to the existence and uniqueness of its solution.

In this paper, we introduce and study a new class of variational inclusions in Banach spaces. For solving such class of variational inclusions, we introduce a new notion of B-monotone operator and prove the Lipschitz continuity of the proximal mapping associated with the B-monotone operator. By using the proximal mapping, an iterative algorithm for solving such class of variational inclusions is constructed in Banach spaces. Under some suitable conditions, we prove the convergence of iterative sequence generated by the algorithm.

In this paper, we further study some properties of the generalized f -projection operator introduced by Wu and Huang. By employing the properties of the generalized f -projection operator and the well-known KKM and Kakutani–Fan–Glicksberg theorems, we establish some new existence theorems of solutions for two classes of generalized set-valued variational and quasi-variational inequalities in reflexive and smooth Banach spaces under some suitable conditions.

In this paper, we introduce a total step method for solving a system of linear complementarity problems with perturbations and interval data. It is applied to two interval matrices [A] and [B] and two interval vectors [b] and [c]. We prove that the sequence generated by the total step method converges to ([x^*],[y^*]) which includes the solution set for the system of linear complementarity problems defined by any fixed [email protected]?[A],[email protected]?[B],[email protected]?[b] and [email protected]?[c]. We also consider a modification of the method and show that, if we start with two interval vectors containing the limits, then the iterates contain the limits. We close our paper with two examples which illustrate our theoretical results.

In this paper, we extend the extremality results for the variational inequality to a higher order evolution hemivariational inequality. More precisely, we give an existence theorem of solution for the higher order evolution hemivariational inequality by using the sub–super-solution method. We prove the compactness of the solution set within an order interval formed by the sub-solution and super-solution. We also show an existence theorem of the extremal solution for the higher order evolution hemivariational inequality under consideration.

In this paper we study the feasibility and solvability of vector variational inequalities with moving cones in Banach spaces. We show that the strict feasibility implies solvability of vector variational inequalities with moving cones under suitable conditions. Further we show that under suitable conditions, the homogeneous vector variational inequality with a moving cone is solvable whenever it is feasible. As consequences, we obtain the solvability of vector variational inequalities with feasibility assumptions in Banach spaces.

In this paper, some existence theorems of nonzero solutions for a system of bilinear variational inequalities are proved by using the coincidence degree theory in reflexive Banach spaces. The results presented in this paper improve and extend some known results in the literature.

In this work, we introduce a new notion of an exceptional family of elements for a J -completely continuous field, and utilize this notion to study the solvability for a class of generalized f -complementarity problems in Banach spaces. Employing the Leray–Schauder alternative theorem and the generalized f -projection operator introduced by Wu and Huang, we obtain some solvability results for the generalized f -complementarity problems in Banach spaces under suitable conditions.

In this work we introduce a system of inclusion problems, which can be regarded as a generalization of the system of equilibrium problems, the system of variational inequality problems, the system of optimization problems, and the inclusion problems. For suitable conditions, we prove an existence result for this system. As applications, we give some existence theorems for the system of equilibrium problems and the system of optimization problems.

In this paper, a new H ( · , · ) -accretive operator is introduced in Banach spaces and the Lipschitzian continuity of the resolvent operator for the H ( · , · ) -accretive operator is showed. By using the technique of resolvent operator, an iterative algorithm for solving a class of variational inclusions is constructed. Under some suitable conditions, the existence of the solution for the variational inclusions and the convergence of iterative sequence generated by the algorithm is proved. Some illustrative examples are also given.

In this paper, a new concept of η-proximal mapping for a proper subdifferentiable functional (which may not be convex) on a Banach space is introduced. An existence and Lipschitz continuity of the η-proximal mapping are proved. By using properties of the η-proximal mapping, a new class of general mixed variational inequalities is introduced and studied in Banach spaces. An existence theorem of solutions is established and a new iterative algorithm for solving the general mixed variational inequality is suggested. A convergence criteria of the iterative sequence generated by the new algorithm is also given.