非紧域上集值映象的重合点与不动点

在赋范线性空间的非空非紧凸集上建立了集值映象对的一个重合点定理,然后用这一定理改进了文献［１］中的集值映象内向集定理与外向集定理,并得到几个集值映象不动点定理．     

Concepts of lower semicontinuity and continuous selections for convex valued multifunctions

We discuss several concepts of continuity, weaker than lower semicontinuity, but still implying the existence of a continuous selection for a closed convex valued multifunction from a paracompact Hausdorff topological space into a Banach space. In this way, an extension of Michael's celebrated selection theorem is given. The behavior of -envelop approximations, as well as the localization of continuous selections, is also discussed.     

Zur minimierung konvexer subdifferenzierbarer funktionen Über dem R n

A minimization method is described for convex subdifferentiable functions and the convergence proof is given. The method is based on an extension of the subdifferential in such a way that this point to set map is continuous. Different methods follow for a special choice of the used norm and the used extension of the subdifferential. For certain classes of functions the method is explained in detail and corresponding realisations of the extensions of the subdiffential are given.     

Banach空间上集值测度与模糊数测度的扩张

在Banach空间上,给出集值测度的扩张定理并借助集测度的扩张给出了模糊数测度的扩张定理。     

无套利预算对应的连续性

当消费者的偏好是凸的 (不一定严格凸 )时 ,不完全市场的预算约束是价格单纯形与Grassman流形的乘积空间到商品空间的非线性集合值映射 ,简称为预算对应 .为了研究该模型的一般经济均衡的存在性 ,作者研究了无套利预算对应的性质 ,得到如下主要结果 :无套利预算对应是连续的 .     

Fuzzy convex sets in median algebras

The fuzzification of convex sets in median algebras is considered, and some of their properties are investigated. A characterization of finite valued fuzzy convex set is given.     

Stability and convergence in discrete convex monotone dynamical systems

We study the stable behaviour of discrete dynamical systems where the map is convex and monotone with respect to the standard positive cone. The notion of tangential stability for fixed points and periodic points is introduced, which is weaker than Lyapunov stability. Among others we show that the set of tangentially stable fixed points is isomorphic to a convex inf-semilattice, and a criterion is given for the existence of a unique tangentially stable fixed point. We also show that periods of tangentially stable periodic points are orders of permutations on n letters, where n is the dimension of the underlying space, and a sufficient condition for global convergence to periodic orbits is presented.     

Efficient solution of interval optimization problem

In this paper the interval valued function is defined in the parametric form and its properties are studied. A methodology is developed to study the existence of the solution of a general interval optimization problem, which is expressed in terms of the interval valued functions. The methodology is applied to the interval valued convex quadratic programming problem.     

THE EXISTENCE THEOREM OF THE CONE-WEAK SUBDIFFERENTIAL OF SET-VALUED MAPPING

In this paper, the existence theorem of the cone-weak subdiflerential of set-valued mapping in locally convex topological vector space is proved.     

一种反向集值鞅型序列的收敛性

本文研究了在非空有界闭凸集的上鞅收敛性和有界闭凸集的适应可积集值序列的收敛性.引入了反向集值mil的基础上,得到了收敛性定理及定理证明.     

A Survey on Existence Results for Boundary Value Problems of Nonlinear Fractional Differential Equations and Inclusions

In this survey paper, we shall establish sufficient conditions for the existence and uniqueness of solutions for various classes of initial and boundary value problem for fractional differential equations and inclusions involving the Caputo fractional derivative. The both cases of convex and nonconvex valued right hand side are considered. The topological structure of the set of solutions is also considered.     

Convex cubic HERMITE-spline interpolation

We present a method to construct convex cubic C¹-splines which interpolate a given convex data set. The problem is reduced to the solution of a system of linear inequalities. The existence of such convex interpolation splines is assured if the data fulfill slight additional conditions. For stronger conditions some easier methods are developed. Finally, error estimations are given.     

A convergence for infinite dimensional vector valued functions

By using the definition of Γ-convergence for vector valued functions given in Oppezzi and Rossi (Optimization, to appear), we obtain a property of infimum values of the Γ-limit. By generalizing Mosco convergence to vector valued functions, we also obtain, in the convex case, the extension of some stability results analogous to the ones in Oppezzi and Rossi (optimization, to appear), when domain and value space are infinite dimensional.      

Existence of Solutions for a Class of Nonlinear Evolution Inclusions with Nonlocal Conditions

This paper deals with the nonlocal problems for a class of nonlinear first-order evolution inclusions. Some existence results are established for the cases of a convex and of a nonconvex valued perturbation terms. Also, the existence of extremal solutions and a strong relaxation theorem are obtained. Subsequently a nonlinear hyperbolic optimal control problem is considered and the existence theorems based on the proven results are obtained. Then the nonlinear version of “bang–bang” principle for control systems is given as well by utilizing the relaxation theorem.     

The Caratheodory-Fejer extension theorem

A construction of Carathéodory and Fejér [1] produces a function which is bounded and analytic in the unit disk with specified initial coefficients. An operator generalization of the construction is now obtained for application to the invariant subspace problem. A formal proof [2] of the existence of invariant subspaces is given by the theory of square summable power series [3] in its vector formulation [4]. But the justification of the formal argument requires a determination of extreme points of a convex set [5]. A solution is now given to an extension problem for convex decompositions which arises in connection with the Carathéodory-Fejér theorem. A necessary condition for an extreme point is obtained as an application. The condition is conjectured to be sufficient.     

SOC-monotone and SOC-convex functions vs. matrix-monotone and matrix-convex functions

The SOC-monotone function (respectively, SOC-convex function) is a scalar valued function that induces a map to preserve the monotone order (respectively, the convex order), when imposed on the spectral factorization of vectors associated with second-order cones (SOCs) in general Hilbert spaces. In this paper, we provide the sufficient and necessary characterizations for the two classes of functions, and particularly establish that the set of continuous SOC-monotone (respectively, SOC-convex) functions coincides with that of continuous matrix monotone (respectively, matrix convex) functions of order 2.     

Carathéodory-Toeplitz and Nehari problems for matrix valued almost periodic functions

In this paper the positive and strictly contractive extension problems for almost periodic matrix functions are treated. We present necessary and sufficient conditions for the existence of extensions in terms of Toeplitz and Hankel operators on Besicovitch spaces and Lebesgue spaces. Furthermore, when a solution exists a special extension (the band extension) is constructed which enjoys a maximum entropy property. A linear fractional parameterization of the set of all extensions is also provided. The techniques used in the proofs include factorizations of matrix valued almost periodic functions and a general algebraic scheme called the band method.

     

On Generalizations of the Frank-Wolfe Theorem to Convex and Quasi-Convex Programmes

In this paper we are concerned with the problem of boundedness and the existence of optimal solutions to the constrained optimization problem. We present necessary and sufficient conditions for boundedness of either a faithfully convex or a quasi-convex polynomial function over the feasible set defined by a system of faithfully convex inequality constraints and/or quasi-convex polynomial inequalities, where the faithfully convex functions satisfy some mild assumption. The conditions are provided in the form of an algorithm, terminating after a finite number of iterations, the implementation of which requires the identification of implicit equality constraints in a homogeneous linear system. We prove that the optimal solution set of the considered problem is nonempty, this way extending the attainability result well known as the so-called Frank-Wolfe theorem. Finally we show that our extension of the Frank-Wolfe theorem immediately implies continuity of the solution set defined by the considered system of (quasi)convex inequalities.     

Euler-Lagrange and Hamiltonian formalisms in dynamic optimization

We consider dynamic optimization problems for systems governed by differential inclusions. The main focus is on the structure of and interrelations between necessary optimality conditions stated in terms of Euler-Lagrange and Hamiltonian formalisms. The principal new results are: an extension of the recently discovered form of the Euler-Weierstrass condition to nonconvex valued differential inclusions, and a new Hamiltonian condition for convex valued inclusions. In both cases additional attention was given to weakening Lipschitz type requirements on the set-valued mapping. The central role of the Euler type condition is emphasized by showing that both the new Hamiltonian condition and the most general form of the Pontriagin maximum principle for equality constrained control systems are consequences of the Euler-Weierstrass condition. An example is given demonstrating that the new Hamiltonian condition is strictly stronger than the previously known one.

     

On the strong convergence of derivatives in a time optimal problem

We consider a time optimal problem for a system described by a differential inclusion, whose right hand side is not necessarily convex valued. Under the assumption of strict convexity of the map obtained by convexifying the original, non-convex valued map, we obtain the strong convergence of the derivatives of any uniformly converging minimizing sequence. The assumptions required by this result are satisfied, for instance, by the classical brachystochrone problem and by Fermat’s principle.