Compactness Assumption Research Articles

Saddle Point Problems, Bilevel Problems, and Mathematical Program with Equilibrium Constraint on Complete Metric Spaces

We apply an existence theorem of variational inclusion problem on metric spaces to study optimization problems, set-valued vector saddle point problems, bilevel problems, and mathematical programs with equilibrium constraint on metric spaces. We study these problems without any convexity and compactness assumptions. Our results are different from any existence results of these types of problems in topological vector spaces.

Some critical point theorems and their applications to periodic solution for second order Hamiltonian systems

Some critical point theorems without the compactness assumptions are obtained by the reduction method, the perturbation argument and the least action principle. As applications, some existence results of periodic solutions are obtained for nonautonomous second order Hamiltonian systems, which unify and generalize some recent corresponding results.

TENSOR PRODUCT REPRESENTATION OF THE (PRE)DUAL OF THE Lp-SPACE OF A VECTOR MEASURE

AbstractThe duality properties of the integration map associated with a vector measure m are used to obtain a representation of the (pre)dual space of the space Lp(m) of p-integrable functions (where 1<p<∞) with respect to the measure m. For this, we provide suitable topologies for the tensor product of the space of q-integrable functions with respect to m (where p and q are conjugate real numbers) and the dual of the Banach space where m takes its values. Our main result asserts that under the assumption of compactness of the unit ball with respect to a particular topology, the space Lp(m) can be written as the dual of a suitable normed space.

On compact holomorphically pseudosymmetric Kählerian manifolds

Abstract For compact Kählerian manifolds, the holomorphic pseudosymmetry reduces to the local symmetry if additionally the scalar curvature is constant and the structure function is non-negative. Similarly, the holomorphic Ricci-pseudosymmetry reduces to the Ricci-symmetry under these additional assumptions. We construct examples of non-compact essentially holomorphically pseudosymmetric Kählerian manifolds. These examples show that the compactness assumption cannot be omitted in the above stated theorem. Recently, the first examples of compact, simply connected essentially holomorphically pseudosymmetric Kählerian manifolds are discovered in [4]. In these examples, the structure functions change their signs on the manifold.

Peano's theorem for rough differential equations in infinite-dimensional Banach spaces

We present a proof for Peano's theorem for rough differential equations, which is valid in infinite dimensions under an appropriate compactness assumption on the vector fields. Our approach makes full use of Lyons’ Universal Limit Theorem and is based on the construction of a family of rough polynomial approximations, each of which is a concatenation of rough path solutions of different equations.

Existence of equilibria of set-valued maps on bounded epi-Lipschitz domains in Hilbert spaces without invariance conditions

In this paper, we provide a new result of the existence of equilibria for set-valued maps on bounded closed subsets K of Hilbert spaces. We do not impose either convexity or compactness assumptions on K but we assume that K has epi-Lipschitz sections, i.e. its intersection with suitable finite dimensional spaces is locally the epigraph of Lipschitz functions. In finite dimensional spaces, the famous Brouwer theorem asserts the existence of a fixed point for a continuous function from a compact convex set K to itself. Our result could be viewed as a kind of generalization of this classical result in the context of Hilbert spaces and when the function (or the set-valued map) does not necessarily map K into itself ( K is not invariant under the map). Our approach is based firstly on degree theory for compact and for condensing set-valued maps and secondly on flows generated by trajectories of differential inclusions.

Generalized [formula omitted] property in a hyperconvex metric space and its applications

In this paper, we first define the family 2 - g K K M ( X , Y ) in a hyperconvex metric space, and then we get a 2 - g K K M theorem and a fixed point theorem without compactness assumption. Next, by using the 2 - g K K M theorem, we get the matching theorems, coincidence theorems, variational inequality theorems and minimax inequality theorems.

New differential properties of sup-type functions

In this paper, we study the directional derivative, subderivative, and subdifferential of sup-type functions without any compactness assumption on the index set. As applications, we provide an estimate of the Lipschitz modulus for sup-type functions.

The 2-KKM property and its applications in hyperconvex metric spaces

In this work, we first define the generalized 2- KKM mapping and the family 2- KKM ( X , Y ) on a metric space, and then we apply these properties to get a 2- KKM theorem and a fixed point theorem without a compactness assumption. Next, by using these 2- KKM theorems, we get some variational inequality theorems and minimax inequality theorems.

Existence of solutions for semilinear differential equations with nonlocal conditions in Banach spaces

This paper is concerned with semilinear differential equations with nonlocal conditions in Banach spaces. Using the tools involving the measure of noncompactness and fixed point theory, existence of mild solutions is obtained without the assumption of compactness or equicontinuity on the associated linear semigroup.

Semi-Markov control processes with non-compact action spaces and discontinuous costs

We establish the average cost optimality equation and show the existence of an ($\varepsilon $-)optimal stationary policy for semi-Markov control processes without compactness and continuity assumptions. The only condition we impose on the model is the $V

Characterizations of the Nonemptiness and Boundedness of Weakly Efficient Solution Sets of Convex Vector Optimization Problems in Real Reflexive Banach Spaces

Under a weak compactness assumption on the functions involved, which always holds in finite-dimensional normed linear spaces, this paper extends various characterizations of the nonemptiness and boundedness of weakly efficient solution sets of convex vector optimization problems, obtained previously by the author (Deng in J. Optim. Theory Appl. 96:123–131, 1998) in the real finite-dimensional normed linear space setting, to those in the real reflexive Banach space setting.

Approximate controllability of nonlinear stochastic evolution systems with time-varying delays

In this paper, the approximate controllability of nonlinear stochastic evolution time-varying delay systems with preassigned responses is studied. The necessary conditions for controllability results of nonlinear systems are not associated with linear systems. No compactness assumptions are imposed in the main results.

Generalized nonlinear quasi-variational inclusions in Banach spaces

This paper introduces a new class of generalized nonlinear quasi-variational inclusions involving generalized m -accretive mappings in p -uniformly smooth real Banach spaces. By using the resolvent operator technique for generalized m -accretive mappings due to Huang et al. [N.J. Huang, Y.P. Fang, C.X. Deng, Nonlinear variational inclusions involving generalized m -accretive mappings, in: Proceedings of the Bellman Continuum: International Workshop on Uncertain Systems and Soft Computing, Beijing, China, July, 24–27, 2002, pp. 323–327] and Nadler Theorem [S.B. Nadler Jr., Multivalued contraction mappings, Pacific J. Math. 30 (1969) 475–488], we construct an iterative algorithm for solving generalized nonlinear quasi-variational inclusions with strongly accretive and relaxed accretive mappings in p -uniformly smooth real Banach spaces. Then we prove the existence of solutions for our inclusions without compactness assumption and convergence of the iterative sequences generated by the algorithm in p -uniformly smooth real Banach spaces. Some special cases are also discussed.

A note on negative dynamic programming for risk-sensitive control

Negative dynamic programming for risk-sensitive control is studied. Under some compactness and semicontinuity assumptions the following results are proved: the convergence of the value iteration algorithm to the optimal expected total reward, the Borel measurability or upper semicontinuity of the optimal value functions, and the existence of an optimal stationary policy.

The Guillemin–Sternberg conjecture for noncompact groups and spaces

AbstractThe Guillemin–Sternberg conjecture states that “quantisation commutes with reduction” in a specific technical setting. So far, this conjecture has almost exclusively been stated and proved forcompactLie groupsGacting oncompactsymplectic manifolds, and, largely due to the use of SpincDirac operator techniques, has reached a high degree of perfection under these compactness assumptions. In this paper we formulate an appropriate Guillemin–Sternberg conjecture in the general case, under the main assumptions that the Lie group action is proper and cocompact. This formulation is motivated by our interpretation of the “quantisation commuates with reduction” phenomenon as a special case of the functoriality of quantisation, and uses equivariantK-homology and theK-theory of the groupC*-algebraC*(G) in a crucial way. For example, the equivariant index – which in the compact case takes values in the representation ringR(G) – is replaced by the analytic assembly map – which takes values inK0(C*(G)) – familiar from the Baum–Connes conjecture in noncommutative geometry. Under the usual freeness assumption on the action, we prove our conjecture for all Lie groupsGhaving a discrete normal subgroup Γ with compact quotientG/Γ, but we believe it is valid for all unimodular Lie groups.

Existence of weak and strong solutions of nonlinear differential equations with delay in Banach space

Existence of weak and strong solutions of nonlinear differential equations with delay in Banach space is discussed. In the present work we give a generalization to recent results. We prove that, with certain conditions, every nonlinear differential equation with delay has at least one weak solution, furthermore, under suitable assumptions, these equations have solutions. Next under a generalization of the compactness assumptions, we show the same equations have solutions too.

Non--autonomous and random attractors for delay random semilinear equations without uniqueness

We first prove the existence and uniqueness of pullback and random attractors for abstract multi-valued non-autonomous and random dynamical systems. The standard assumption of compactness of these systems can be replaced by the assumption of asymptotic compactness. Then, we apply the abstract theory to handle a random reaction-diffusion equation with memory or delay terms which can be considered on the complete past defined by $\mathbb{R}^{-}$. In particular, we do not assume the uniqueness of solutions of these equations.

Generic Existence of Solutions in Vector Optimization on Metric Spaces

In this paper, we study a class of vector minimization problems on a complete metric space X which is identified with the corresponding complete metric space of objective functions \(\mathcal{A}\) . We do not impose any compactness assumption on X. We show that, for most (in the sense of Baire category) functions \(F\in \mathcal{A}\) , the corresponding vector optimization problem has a solution.

Viscosity Solutions of a Level-Set Method for Anisotropic Geometric Diffusion in Image Processing

We discuss the existence of viscosity solutions for a class of anisotropic level-set methods which can be seen as an extension of the mean-curvature motion with a nonlinear anisotropic diffusion tensor. In an earlier work (Mikula et al. in Comput. Vis. Sci. 6(4):197---209, [2004]; Preusser and Rumpf in SIAM J. Appl. Math. 62(5):1772---1793, [2002]) we have applied such methods for the denoising and enhancement of static images and image sequences. The models are characterized by the fact that--unlike the mean-curvature motion--they are capable of retaining important geometric structures like edges and corners of the level-sets. The article reviews the definition of the model and discusses its geometric behavior. The proof of the existence of viscosity solutions for these models is based on a fixed point argument which utilizes a compactness property of the diffusion tensor. For the application to image processing suitable regularizations of the diffusion tensor are presented for which the compactness assumptions of the existence proof hold. Finally, we consider the half relaxed limits of the solutions of auxiliary problems to show the compactness of the solution operator and thus the existence of a solution to the original problem.