Optimal transport maps for Monge-Kantorovich problem on loop groups

Let G be a compact Lie group, and consider the loop group LeG:={?∈C([0,1],G); ?(0)=?(1)=e}. Let ν be the heat kernel measure at the time 1. For any density function F on LeG such that Entν(F)<∞, we shall prove that there exists a unique optimal transportation map which pushes ν forward to Fν.     

Existence and multiplicity of solutions to 2mth-order ordinary differential equations

In this paper, the existence and multiplicity of solutions are obtained for the 2mth-order ordinary differential equation two-point boundary value problems u(2(m−i))(t)=f(t,u(t)) for all t∈[0,1] subject to Dirichlet, Neumann, mixed and periodic boundary value conditions, respectively, where f is continuous, ai∈R for all i=1,2,…,m. Since these four boundary value problems have some common properties and they can be transformed into the integral equation of form , we firstly deal with this nonlinear integral equation. By using the strongly monotone operator principle and the critical point theory, we establish some conditions on f which are able to guarantee that the integral equation has a unique solution, at least one nonzero solution, and infinitely many solutions. Furthermore, we apply the abstract results on the integral equation to the above four 2mth-order two-point boundary problems and successfully resolve the existence and multiplicity of their solutions.     

On Generic Uniqueness of Optimal Solutions for the General Monge-Kantorovich Problem

We show that a multifunction a, which assignes to a cost function c the set a(c) of optimal solutions for the corresponding Monge-Kantorovich problem, is generically single-valued.     

A-Monotonicity and Its Role in Nonlinear Variational Inclusions

The notion of A-monotonicity in the context of solving a new class of nonlinear variational inclusion problems is presented. Since A-monotonicity generalizes not only the well-explored maximal monotone mapping, but also a recently introduced and studied notion of H-monotone mapping, the results thus obtained are general in nature.Communicated by G. Leitmann     

New approach to the η-proximal point algorithm and nonlinear variational inclusion problems

A new approach using the over-relaxed proximal point algorithm in the context of solving a class of inclusion problems based on the notion of maximal (η)-monotonicity is developed and examined. Convergence analysis seems to be reasonable, and finally, some specializations are included. Furthermore, the model developed in this communication seems to be appropriate to the Yosida approximation in the sense that it can be applied to first-order evolution equations/inclusions as well.     

Approximating Solution Of ∈ T(x) for an H-Monotone Operator in Hilbert Spaces

The purpose of this paper is to study the solution of 0 ∈ T(x) for an H-monotone operator introduced in [Fang and Huang, Appl. Math. Comput. 145(2003)795-803] in Hilbert spaces, which is the first proposal of it's kind. Some strong and weak convergence results are presented and the relations between maximal monotone operators and H-monotone operators are analyzed. Simultaneously, we apply these results to the minimization problem for T = ∂f and provide some numerical examples to support the theoretical findings.     

Hyperinvariant subspaces for quasinilpotent operators on Hilbert spaces

In this paper, we employ the model theory due to C. Foias and C. Pearcy and the notion of Enflo's extremal vectors of quasinilpotent operators to study the hyperinvariant subspace problem for quasinilpotent operators. Our main result is that if a quasinilpotent quasiaffinity T has a sequence of “c-eigenvectors” xn of TnT^∗n such that the set is compact, then T has a nontrivial hyperinvariant subspace.     

Globally strongly convex cost functional for a coefficient inverse problem

A Carleman Weight Function (CWF) is used to construct a new cost functional for a Coefficient Inverse Problems for a hyperbolic PDE. Given a bounded set of an arbitrary size in a certain Sobolev space, one can choose the parameter of the CWF in such a way that the constructed cost functional will be strongly convex on that set. Next, convergence of the gradient method, which starts from an arbitrary point of that set, is established. Since restrictions on the size of that set are not imposed, then this is the global convergence.     

An η-approximation approach to duality in mathematical programming problems involving r-invex functions

An η-approximation approach introduced by Antczak [T. Antczak, A new method of solving nonlinear mathematical programming problems involving r-invex functions, J. Math. Anal. Appl. 311 (2005) 313-323] is used to obtain a solution Mond-Weir dual problems involving r-invex functions. η-Approximated Mond-Weir dual problems are introduced for the η-approximated optimization problem constructed in this method associated with the original nonlinear mathematical programming problem. By the help of η-approximated dual problems various duality results are established for the original mathematical programming problem and its original Mond-Weir duals.     

Existence and iterative approximation of solutions of generalized mixed quasi-variational-like inequality problem in Banach spaces

In this paper, some existence theorems for the mixed quasi-variational-like inequalities problem in a reflexive Banach space are established. The auxiliary principle technique is used to suggest a novel and innovative iterative algorithm for computing the approximate solution for the mixed quasi-variational-like inequalities problem. Consequently, not only the existence of theorems of the mixed quasi-variational-like inequalities is shown, but also the convergence of iterative sequences generated by the algorithm is also proven. The results proved in this paper represent an improvement of previously known results.     

Lipschitz compact operators

We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.     

A new exact exponential penalty function method and nonconvex mathematical programming

A new exact penalty function method, called the l₁ exact exponential penalty function method, is introduced. In this approach, the so-called the exponential penalized optimization problem with the l₁ exact exponential penalty function is associated with the original optimization problem with both inequality and equality constraints. The l₁ exact exponential penalty function method is used to solve nonconvex mathematical programming problems with r-invex functions (with respect to the same function η). The equivalence between sets of optimal solutions of the original mathematical programming problem and of its associated exponential penalized optimization problem is established under suitable r-invexity assumption. Also lower bounds on the penalty parameter are given, for which above these values, this result is true.     

A new method of solving nonlinear mathematical programming problems involving r-invex functions

A new approach to a solution of a nonlinear constrained mathematical programming problem involving r-invex functions with respect to the same function η is introduced. An η-approximated problem associated with an original nonlinear mathematical programming problem is presented that involves η-approximated functions constituting the original problem. The equivalence between optima points for the original mathematical programming problem and its η-approximated optimization problem is established under r-invexity assumption.     

Strongly nonlinear second order differential inclusions with generalized boundary conditions

In this paper, we study second order differential inclusions in with a maximal monotone term and generalized boundary conditions. The nonlinear differential operator need not be necessary homogeneous and incorporates as a special case the one-dimensional p-Laplacian. The generalized boundary conditions incorporate as special cases well-known problems such as the Dirichlet (Picard), Neumann and periodic problems. As application to the proven results we obtain existence theorems for both “convex” and “nonconvex” problems when the maximal monotone term A is defined everywhere and when not (case of variational inequalities).     

Ordinary differential operator and some of its applications to certain meromorphically p-valent functions

By making use of the well-known assertions given in Miller and Mocanu (1978) [13] and Nunokawa (1993) [14], certain theorems concerning p-valently meromorphic (strongly) starlike and (strongly) convex functions obtained in this investigation are firstly proved and then their certain consequences which will be interesting or important for analytic and geometric function theory are pointed out.     

On α-times integrated C-cosine functions and abstract Cauchy problem I

In this paper we first investigate some basic properties concerning nondegenerate α-times integrated C-cosine functions on a Banach space X, and then characterize their generator A in terms of the unique existence of strong solutions of the following abstract Cauchy problem: for t>0, u(0)=x, u^′(0)=y.     

Porosity of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty closed subset of X. Let be a lower semicontinuous function bounded from below. This paper is concerned with the perturbed optimization problem infz∈Z{J(z)+‖x−z‖}, denoted by (x,J)-inf for x∈X. In the case when X is compactly fully 2-convex, it is proved in the present paper that the set of all points x in X for which there does not exist z₀∈Z such that J(z₀)+‖x−z₀‖=infz∈Z{J(z)+‖x−z‖} is a σ-porous set in X. Furthermore, if X is assumed additionally to be compactly locally uniformly convex, we verify that the set of all points x∈X?Z⁰ such that the problem (x,J)-inf fails to be approximately compact, is a σ-porous set in X?Z⁰, where Z⁰ denotes the set of all z∈Z such that z∈PZ(z). Moreover, a counterexample to which some results of Ni [R.X. Ni, Generic solutions for some perturbed optimization problem in nonreflexive Banach space, J. Math. Anal. Appl. 302 (2005) 417-424] fail is provided.     

A general iterative scheme for k-strictly pseudo-contractive mappings and optimization problems

In this paper, we introduce a new general iterative scheme for finding fixed points of a strictly pseudo-contractive mapping and then prove that the sequence generated by the proposed iterative scheme converges strongly to a fixed point of the mapping, which is a solution of a certain optimization problem related to a strongly positive bounded linear operator. Additional results of the main result are also obtained.     

On removable sets for convex functions

In the present article we provide a sufficient condition for a closed set F∈R^d$F\in {\mathbb{R}}^d$ to have the following property which we call c  -removability: Whenever a continuous function f:R^d→R$f:{\mathbb{R}}^d\to \mathbb{R}$ is locally convex on the complement of F  , it is convex on the whole R^d${\mathbb{R}}^d$. We also prove that no generalized rectangle of positive Lebesgue measure in R²${\mathbb{R}}^2$ is c-removable. Our results also answer the following question asked in an article by Jacek Tabor and Józef Tabor (2010) [5]: Assume the closed set F⊂R^d$F\subset {\mathbb{R}}^d$ is such that any locally convex function defined on R^d?F${\mathbb{R}}^d?F$ has a unique convex extension on R^d${\mathbb{R}}^d$. Is F   necessarily intervally thin (a notion of smallness of sets defined by their “essential transparency” in every direction)? We prove the answer is negative by finding a counterexample in R²${\mathbb{R}}^2$.     

General implicit variational inclusion problems based on A-maximal (m)-relaxed monotonicity (AMRM) frameworks

A general framework for an algorithmic procedure based on the variational convergence of operator sequences involving A-maximal (m)-relaxed monotone (AMRM) mappings in a Hilbert space setting is developed, and then it is applied to approximating the solution of a general class of nonlinear implicit inclusion problems involving A-maximal (m)-relaxed monotone mappings. Furthermore, some specializations of interest on existence theorems and corresponding approximation solvability theorems on H-maximal monotone mappings are included that may include several other results for general variational inclusion problems on general maximal monotonicity in the literature.