Search

We study shortest curves in proximally smooth subset of a finite or infinite dimensional Hilbert space. We consider an R -proximally smooth set A in a Hilbert space with points a and b satisfying | a − b | < 2 R . We provide a simple geometric algorithm of constructing a curve inside A connecting a and b whose length is at most 2 R arcsin | a − b | 2 R , which corresponds to the shortest curve inside the model set – a Euclidean sphere of radius R passing through a and b . Using this construction, we show that there exists a unique shortest curve inside A connecting a and b . This result is tight since two points of A at distance 2 R are not necessarily connected in A ; the bound on the length cannot be improved since the equality is attained on the Euclidean sphere of radius R . [ABSTRACT FROM AUTHOR]

We present an analysis of generalized Nash equilibrium problems in infinite-dimensional spaces with possibly non-convex objective functions of the players. Such settings arise, for instance, in games that involve nonlinear partial differential equation constraints. Due to non-convexity, we work with equilibrium concepts that build on first order optimality conditions, especially Quasi-Nash Equilibria (QNE), i.e. first-order optimality conditions for (Generalized) Nash Equilibria, and Variational Equilibria (VE), i.e. first-order optimality conditions for Normalized Nash Equilibria. We prove existence of these types of equilibria and study characterizations of them via regularized (and localized) Nikaido-Isoda merit functions. We also develop continuity and (continuous) differentiability results for these merit functions under quite weak assumptions, using a generalization of Danskin's theorem. They provide a theoretical foundation for, e.g. using globalized descent methods for computing QNE or VE. [ABSTRACT FROM AUTHOR]

Being informed that one of our articles is cited in the paper mentioned in the title, we downloaded it, and we were surprised to see that, practically, all the results from our paper were reproduced in Section 3 of Meghea and Stamin’s article. Having in view the title of the article, one is tempted to think that the remarks mentioned in the paper are original and there are examples given as to where and how (at least) some of the reviewed results are effectively applied. Unfortunately, a closer look shows that most of those remarks in Section 3 are, in fact, extracted from our article, and it is not shown how a specific result is used in a certain application. So, our aim in the present note is to discuss the content of Section 3 of Meghea and Stamin’s paper, emphasizing their Remark 8, in which it is asserted that the proof of Lemma 7 in our article is “full of errors.”

In this paper we investigate a class of time-dependent quasi-variational-hemivariational inequalities (TDQVHVIs) of elliptic type in a reflexive separable Banach space, which is characterized by a constraint set depending on a solution. The solvability of the TDQVHVIs is obtained by employing a measurable selection theorem for measurable set-valued mappings, while the uniqueness of solution to the TDQVHVIs is guaranteed by enhancing the assumptions on the data. Then, under additional hypotheses, we deliver a continuous dependence result when all the data are subjected to perturbations. Finally, the applicability of the abstract results is illustrated by a frictional elastic contact problem with locking materials for which the existence and stability of the weak solutions is proved. [ABSTRACT FROM AUTHOR]

We extend the result of Lisini (Calc Var Partial Differ Equ 28:85–120, 2007) on the superposition principle for absolutely continuous curves in p-Wasserstein spaces to the special case of p = 1 . In contrast to the case of p > 1 , it is not always possible to have lifts on absolutely continuous curves. Therefore, one needs to relax the notion of a lift by considering curves of bounded variation, or shortly BV-curves, and replace the metric speed by the total variation measure. We prove that any BV-curve in a 1-Wasserstein space can be represented by a probability measure on the space of BV-curves which encodes the total variation measure of the Wasserstein curve. In particular, when the curve is absolutely continuous, the result gives a lift concentrated on BV-curves which also characterizes the metric speed. The main theorem is then applied for the characterization of geodesics and the study of the continuity equation in a discrete setting. [ABSTRACT FROM AUTHOR]

This work concerns Ekeland variational principles for scalar and vector cyclically antimonotone bifunctions on complete metric spaces. The scalar results work for extended bifunctions and they are obtained by a generalized version of the Dancs–Hegedüs–Medvegyev's fixed point theorem. As a result, weaker lower-semicontinuity assumptions have been considered, that generalize the concept of strictly decreasingly lower-semicontinuous real-valued function. The vector results are derived from the previous ones by a scalarization approach and are based on new notions of cyclical antimonotonicity, lower boundedness and strictly decreasingly lower-semicontinuity for vector bifunctions. Several results in the literature are improved since they are stated by weaker assumptions. [ABSTRACT FROM AUTHOR]

Learning processes play an important role in enhancing understanding and analyzing real phenomena. Most of these methodologies revolve around solving penalized optimization problems. A significant challenge arises in the choice of the penalty hyperparameter, which is typically user-specified or determined through Grid search approaches. There is a lack of automated tuning procedures for the estimation of these hyperparameters, particularly in unsupervised learning scenarios. In this paper, we focus on the unsupervised context and propose a bi-level strategy to address the issue of tuning the penalty hyperparameter. We establish suitable conditions for the existence of a minimizer in an infinite-dimensional Hilbert space, along with presenting some theoretical considerations. These results can be applied in situations where obtaining an exact minimizer is unfeasible. Working on the estimation of the hyperparameter with the gradient-based method, we also introduce a modified version of Ekeland's principle as a stopping criterion for these methods. Our approach distinguishes from conventional techniques by reducing reliance on random or black-box strategies, resulting in stronger mathematical generalization. [ABSTRACT FROM AUTHOR]

This paper gives verifiable conditions for the existence of solutions of some set quasi-optimization problems involving Minkowski difference, where the objective maps are defined in complete metric spaces. Our proof method is not based on any scalarizing approach, and our existence results are written in terms of the given data of the problems. Several examples are provided. As applications of the main results of this paper, new results on the existence of robust solutions for uncertain multi-objective quasi-optimization problems with set-valued maps are formulated. [ABSTRACT FROM AUTHOR]

In this paper, we deal with vector equilibrium problems. We prove the nonemptiness of the solution set for this type of problems in the sequential compactness case and in the absence of convexity and lower semicontinuity assumptions. Some examples are presented and an existence result for countable systems of vector equilibrium problems is stated. [ABSTRACT FROM AUTHOR]

We deal with a class of elliptic quasivariational inequalities with constraints in a reflexive Banach space. We use arguments of monotonicity, convexity and compactness in order to prove a convergence criterion for such inequalities. This criterion allows us to consider a new well-posedness concept in the study of the corresponding inequalities, which extends the classical Tykhonov and Levitin–Polyak well-posedness concepts used in the literature. Then, we introduce a new penalty method, for which we provide a convergence result. Finally, we consider a variational inequality which describes the equilibrium of a spring–rods system, under the action of external forces. We apply our abstract results in the study of this inequality and provide the corresponding mechanical interpretations. We also present numerical simulations which validate our convergence results. [ABSTRACT FROM AUTHOR]

In this paper, we extend the definition of the qx-asymptotic functions, for an extended real-valued function defined on the infinite-dimensional topological normed spaces without lower semicontinuity or quasi-convexity condition. As the main result, by using some asymptotic conditions, we obtain sufficient optimality conditions for the existence of solutions to equilibrium problems, under weaker assumptions of continuity and convexity, when the feasible set is an unbounded subset of infinite-dimensional space. Also, as a corollary, we obtain necessary and sufficient optimality conditions for the existence of solutions to equilibrium problems with an unbounded feasible set. Finally, as an application, we establish a result for the existence of solutions to minimization problems. [ABSTRACT FROM AUTHOR]

The goal of this paper is to develop new fixed points for quasi upper semicontinuous set-valued mappings and compact continuous (single-valued) mappings, and related applications for useful tools in nonlinear analysis by applying the best approximation approach for classes of semiclosed 1-set contractive set-valued mappings in locally p-convex and p-vector spaces for p ∈ (0 , 1 ] . In particular, we first develop general fixed point theorems for quasi upper semicontinuous set-valued and single-valued condensing mappings, which provide answers to the Schauder conjecture in the affirmative way under the setting of locally p-convex spaces and topological vector spaces for p ∈ (0 , 1 ] ; then the best approximation results for quasi upper semicontinuous and 1-set contractive set-valued mappings are established, which are used as tools to establish some new fixed points for nonself quasi upper semicontinuous set-valued mappings with either inward or outward set conditions under various boundary situations. The results established in this paper unify or improve corresponding results in the existing literature for nonlinear analysis, and they would be regarded as the continuation of the related work by Yuan (Fixed Point Theory Algorithms Sci. Eng. 2022:20, 2022)–(Fixed Point Theory Algorithms Sci. Eng. 2022:26, 2022) recently. [ABSTRACT FROM AUTHOR]

We consider the forward Kolmogorov equation corresponding to measure-valued processes stemming from a class of interacting particle systems in population dynamics, including variations of the Bolker–Pacala–Dieckmann-Law model. Under the assumption of detailed balance, we provide a rigorous generalized gradient structure, incorporating the fluxes arising from the birth and death of the particles. Moreover, in the large population limit, we show convergence of the forward Kolmogorov equation to a Liouville equation, which is a transport equation associated with the mean-field limit of the underlying process. In addition, we show convergence of the corresponding gradient structures in the sense of Energy-Dissipation Principles, from which we establish a propagation of chaos result for the particle system and derive a generalized gradient-flow formulation for the mean-field limit. [ABSTRACT FROM AUTHOR]

This paper concerns with vector saddle point problems where the image space of the objective bifunction is not endowed with any topology and the orders in the image space are defined from general sets. Some new existence results of vector saddle points are established based on using notions of vector-cyclic quasimonotonicity together with notions of "algebraic" semicontinuity, without assuming convexity assumptions. [ABSTRACT FROM AUTHOR]

We study a variational approach introduced by S.D. Fisher and the author in the 1970s in the context of norm minimization for differentiable mappings occurring in nonlinear elliptic boundary value problems. It may be viewed as an abstract version of the calculus of variations. A strong hypothesis, initially limiting the scope of this approach, is the assumption of a bounded minimizing sequence in the least squares formulation. In this article, we employ regularization and invariant regions to overcome this obstacle. A consequence of the framework is the convergence of approximations for regularized problems to a desired solution. The variational method is closely associated with the implicit function theorem, and it can be jointly studied, so that continuous parameter stability is naturally deduced. A significant aspect of the theory is that the reaction term in a reaction-diffusion equation can be selected to act globally as in the steady Schrödinger-Hartree equation. Local action, as in the non-equilibrium Poisson-Boltzmann equation, is also included. Both cases are studied at length prior to the development of a general theory. [ABSTRACT FROM AUTHOR]

This paper presents some variants of minimal point theorem together with corresponding variants of Ekeland variational principle. In the second part of this article, there is a discussion on Ekeland variational principle and minimal point theorem relative to it in uniform spaces. The aim of these series of important results is to highlight relations between them, some improvements and specific applications.

The goal of this paper is to develop some new and useful tools for nonlinear analysis by applying the best approximation approach for classes of semiclosed 1-set contractive set-valued mappings in locally p-convex or p-vector spaces for p ∈ (0 , 1 ]. In particular, we first develop general fixed point theorems for both set-valued and single-valued condensing mappings which provide answers to the Schauder conjecture in the affirmative way under the setting of (locally p-convex) p-vector spaces, then the best approximation results for upper semi-continuous and 1-set contractive set-valued are established, which are used as tools to establish some new fixed points for non-self set-valued mappings with either inward or outward set conditions under various situations. These results unify or improve corresponding results in the existing literature for nonlinear analysis. [ABSTRACT FROM AUTHOR]

In this paper, we deal with vector optimization problems where the images of the objective maps are given in linear spaces not endowed with any topological structure. Firstly, we propose new concepts of semicontinuity of vector-valued maps acting from metric spaces into linear ones and discuss their properties and characterizations. Next, using Zorn's Lemma and these properties, we formulate existence conditions for such problems. Then, we introduce two notions of Levitin-Polyak well-posedness for the reference problems, and study sufficient conditions for their fulfillment, as well as relationships of these notions. Finally, using the Kuratowski measure of noncompactness, several metric characterizations of such well-posedness properties for vector optimization problems are investigated. Many examples are given to analyze our results. [ABSTRACT FROM AUTHOR]

We study conjugate and Lagrange dualities for composite optimization problems within the framework of abstract convexity. We provide conditions for zero duality gap in conjugate duality. For Lagrange duality, intersection property is applied to obtain zero duality gap. Connection between Lagrange dual and conjugate dual is also established. Examples related to convex and weakly convex functions are given. [ABSTRACT FROM AUTHOR]

We give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space (X , d , μ) satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum u 0 on the parabolic boundary of a space-time-cylinder Ω × (0 , T) with Ω ⊂ X an open set and T > 0 , we prove existence in the weak parabolic function space L w 1 (0 , T ; BV (Ω)) . In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for BV -valued parabolic function spaces. We argue completely on a variational level. [ABSTRACT FROM AUTHOR]

The goal of this paper is to develop some fundamental and important nonlinear analysis for single-valued mappings under the framework of p-vector spaces, in particular, for locally p-convex spaces for 0 < p ≤ 1 . More precisely, based on the fixed point theorem of single-valued continuous condensing mappings in locally p-convex spaces as the starting point, we first establish best approximation results for (single-valued) continuous condensing mappings, which are then used to develop new results for three classes of nonlinear mappings consisting of 1) condensing; 2) 1-set contractive; and 3) semiclosed 1-set contractive mappings in locally p-convex spaces. Next they are used to establish the general principle for nonlinear alternative, Leray–Schauder alternative, fixed points for nonself mappings with different boundary conditions for nonlinear mappings from locally p-convex spaces, to nonexpansive mappings in uniformly convex Banach spaces, or locally convex spaces with the Opial condition. The results given by this paper not only include the corresponding ones in the existing literature as special cases, but are also expected to be useful tools for the development of new theory in nonlinear functional analysis and applications to the study of related nonlinear problems arising from practice under the general framework of p-vector spaces for 0 < p ≤ 1 . Finally, the work presented by this paper focuses on the development of nonlinear analysis for single-valued (instead of set-valued) mappings for locally p-convex spaces. Essentially, it is indeed the continuation of the associated work given recently by Yuan (Fixed Point Theory Algorithms Sci. Eng. 2022:20, 2022); therein, the attention is given to the study of nonlinear analysis for set-valued mappings in locally p-convex spaces for 0 < p ≤ 1 . [ABSTRACT FROM AUTHOR]

We first provide new sufficient conditions for the existence of minima of functions defined on 0-complete partial metric spaces. We then apply the obtained results to derive some fixed point results for single-valued and set-valued mappings that improve and generalize several well-known results in the literature. Examples are given to illustrate our findings. [ABSTRACT FROM AUTHOR]

It is known that the class of p-vector spaces (0 < p ≤ 1) is an important generalization of the usual norm spaces with rich topological and geometrical structure, but most tools and general principles with nature in nonlinearity have not been developed yet. The goal of this paper is to develop some useful tools in nonlinear analysis by applying the best approximation approach for the classes of 1-set contractive set-valued mappings in p-vector spaces. In particular, we first develop general fixed point theorems of compact (single-valued) continuous mappings for closed p-convex subsets, which also provide an answer to Schauder’s conjecture of 1930s in the affirmative way under the setting of topological vector spaces for 0 < p ≤ 1 . Then one best approximation result for upper semicontinuous and 1-set contractive set-valued mappings is established, which is used as a useful tool to establish fixed points of nonself set-valued mappings with either inward or outward set conditions and related various boundary conditions under the framework of locally p-convex spaces for 0 < p ≤ 1 . In addition, based on the framework for the study of nonlinear analysis obtained for set-valued mappings with closed p-convex values in this paper, we conclude that development of nonlinear analysis and related tools for singe-valued mappings in locally p-convex spaces for 0 < p ≤ 1 seems even more important, and can be done by the approach established in this paper. [ABSTRACT FROM AUTHOR]

We obtain several characterizations of the existence of greatest elements of a total preorder. The characterizations pertain to the existence of unconstrained greatest elements of a total preorder and to the existence of constrained greatest elements of a total preorder on every nonempty compact subset of its ground set. The necessary and sufficient conditions are purely topological and, in the case of constrained greatest elements, are formulated by making use of a preorder relation on the set of all topologies that can be defined on the ground set of the objective relation. Observing that every function into a totally ordered set can be naturally conceived as a total preorder, we then reformulate the mentioned characterizations in the more restrictive case of an objective function with a totally ordered codomain. The reformulations are expressed in terms of upper semi- and pseudo-continuity by showing a topological connection between the two notions of generalized continuity. [ABSTRACT FROM AUTHOR]

We give the correct statements of Theorem 4.4 and its consequences in Yuan, "Fixed point theorem and related nonlinear analysis by the best approximation method in p-vector spaces" (Numer. Funct. Anal. Optimiz. 44(4): 221 - 295. DOI: 10.1080/01630563.2023.2167088) [1]. [ABSTRACT FROM AUTHOR]

We consider direct and inverse abstract Cauchy problems with time-dependent coefficients. We give an explicit representation of the unique solution operator of the direct abstract Cauchy problem. We use the obtained solution and a new method to recover time-dependent coefficients for the inverse abstract Cauchy problem. [ABSTRACT FROM AUTHOR]

An invariant-point theorem and its equivalent formulation are established in which distance functions are replaced by minimal time functions. It is worth emphasizing here that the class of minimal time functions can be interpreted as a general type of directional distance functions recently used to develop new applications in optimization theory. The obtained results are applied in two directions. First, we derive sufficient conditions for the existence of solutions to optimization-related problems without convexity. As an easy corollary, we get a directional Ekeland variational principle. Second, we propose a new type of global error bounds for inequalities which allows us to simultaneously study nonconvex and convex functions. Several examples and comparison remarks are included as well to explain advantages of our results with existing ones in the literature. [ABSTRACT FROM AUTHOR]

Let X be a completely regular topological space and f a real-valued bounded from above lower semicontinuous function in it. Let C(X) be the space of all bounded continuous real-valued functions in X endowed with the usual sup-norm. We show that the following two properties are equivalent: X is α-favourable (in the sense of the Banach-Mazur game); The set of functions h in C(X) for which f + h attains its supremum in X contains a dense and Gδ-subset of the space C(X). In particular, property (b) has place if X is a compact space or, more generally, if X is homeomorphic to a dense Gδ subset of a compact space. We show also the equivalence of the following stronger properties: X contains some dense completely metrizable subset; the set of functions h in C(X) for which f + h has strong maximum in X contains a dense and Gδ-subset of the space C(X). If X is a complete metric space and f is bounded, then the set of functions h from C(X) for which f + h has both strong maximum and strong minimum in X contains a dense Gδ-subset of C(X). [ABSTRACT FROM AUTHOR]

In 2013, Masson and Siljander determined a method to prove that the minimal p-weak upper gradient g f ε for the time mollification f ε , ε > 0 , of a parabolic Newton–Sobolev function f ∈ L loc p (0 , τ ; N loc 1 , p (Ω)) , with τ > 0 and Ω open domain in a doubling metric measure space (X , d , μ) supporting a weak (1, p)-Poincaré inequality, p ∈ (1 , ∞) , is such that g f - f ε → 0 as ε → 0 in L loc p (Ω τ) , Ω τ being the parabolic cylinder Ω τ : = Ω × (0 , τ) . Their approach involved the use of Cheeger's differential structure, and therefore exhibited some limitations; here, we shall see that the definition and the formal properties of the parabolic Sobolev spaces themselves allow to find a more direct method to show such convergence, which relies on p-weak upper gradients only and which is valid regardless of structural assumptions on the ambient space, also in the limiting case when p = 1 . [ABSTRACT FROM AUTHOR]

Using tools provided by the theory of abstract convexity, we extend conditions for zero duality gap to the context of non-convex and nonsmooth optimization. Mimicking the classical setting, an abstract convex function is the upper envelope of a family of abstract affine functions (being conventional vertical translations of the abstract linear functions). We establish new conditions for zero duality gap under no topological assumptions on the space of abstract linear functions. In particular, we prove that the zero duality gap property can be fully characterized in terms of an inclusion involving (abstract) ϵ -subdifferentials. This result is new even for the classical convex setting. Endowing the space of abstract linear functions with the topology of pointwise convergence, we extend several fundamental facts of functional/convex analysis. This includes (i) the classical Banach–Alaoglu–Bourbaki theorem (ii) the subdifferential sum rule, and (iii) a constraint qualification for zero duality gap which extends a fact established by Borwein, Burachik and Yao (2014) for the conventional convex case. As an application, we show with a specific example how our results can be exploited to show zero duality for a family of non-convex, non-differentiable problems. [ABSTRACT FROM AUTHOR]

Sufficient conditions are given for the existence of solutions of bifunction-set optimization problems, whose underlying sets are complete metric spaces. The main result of this paper is formulated in Theorem 3.1, with the help of some new data. The advantage of introducing the new data is that, by choosing suitable new data, we can obtain existence results with assumptions, that are imposed directly on the objective maps, or that are formulated via scalarization. The main result is applied successfully to Kuroiwa set optimization problems and Ky Fan vector inequality problems. We also discuss equivalences between the existence of solutions of the bifunction-set optimization problem, an Ekeland-type variational principle and a Caristi-type fixed point theorem. Examples are provided. [ABSTRACT FROM AUTHOR]

This paper addresses an evolution inclusion of subdifferential type with a multivalued perturbation. The values of the latter are closed, not necessarily convex sets. Our inclusion is implicit in the sense that the velocity enters it implicitly: the subdifferential is evaluated not at the state, but at a function depending both on the state and the velocity. We prove the existence of a solution to the inclusion by using a fixed point theorem for an auxiliary multivalued mapping with closed, nonconvex, decomposable values. This multivalued mapping is related to an ordinary differential equation containing the resolvent of the subdifferential operator. In the case when the perturbation is single-valued the solution is unique. We also introduce an explicit ordinary differential equation with the solution set coinciding with that for our implicit evolution inclusion. The application of our general result to implicit sweeping processes with nonconvex perturbations yields the existence of solutions to these processes generalizing a number of recent existence results for implicit sweeping processes. This existence result is further illustrated for a quasistatic evolution variational inequality arising in contact mechanics. Our results on implicit subdifferential inclusions are completely new and have no analogs in the existence literature. [ABSTRACT FROM AUTHOR]

In this work, we consider the initial-value problem associated with a coupled system of generalized Korteweg–de Vries equations. We present a relationship between the best constant for a Gagliardo–Nirenberg type inequality and a criterion for the existence of global solutions in the energy space. We prove that such a constant is directly related to the existence problem of solitary wave solutions with minimal mass, the so-called ground state solutions. A characterization of the ground states and the orbital instability of the solitary waves are also established. [ABSTRACT FROM AUTHOR]

We prove the applicability of the Weighted Energy-Dissipation (WED) variational principle to nonlinear parabolic stochastic partial differential equations in abstract form. The WED principle consists in the minimization of a parameter-dependent convex functional on entire trajectories. Its unique minimizers correspond to elliptic-in-time regularizations of the stochastic differential problem. As the regularization parameter tends to zero, solutions of the limiting problem are recovered. This in particular provides a direct approach via convex optimization to the approximation of nonlinear stochastic partial differential equations. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of solutions to equilibrium and quasi-equilibrium problems without any convexity assumption. Coverage includes some equivalences to the Ekeland variational principle for bifunctions and basic facts about transfer lower continuity. An application is given to systems of quasi-equilibrium problems. [ABSTRACT FROM AUTHOR]

In this article, the existence of heteroclinic solution of a class of generalized Hamiltonian system with potential $V : {\open R}^{n} \longmapsto {\open R}$ having a finite or infinite number of global minima is studied. Examples include systems involving the p-Laplacian operator, the curvature operator and the relativistic operator. Generalized conservation of energy is established, which leads to the property of equipartition of energy enjoyed by heteroclinic solutions. The existence problem of heteroclinic solution is studied using both variational method and the metric method. The variational approach is classical, while the metric method represents a more geometrical point of view where the existence problem of heteroclinic solution is reduced to that of geodesic in a proper length metric space. Regularities of the heteroclinic solutions are discussed. The results here not only provide alternative solution methods for Φ-Laplacian systems, but also improve existing results for the classical Hamiltonian system. In particular, the conditions imposed upon the potential are very mild and new proof for the compactness is given. Finally in ℝ2, heteroclinic solutions are explicitly written down in closed form by using complex function theory. [ABSTRACT FROM AUTHOR]

We consider an optimal control problem Q governed by an elliptic quasivariational inequality with unilateral constraints. We associate to Q a new optimal control problem Q ˜ , obtained by perturbing the state inequality (including the set of constraints and the nonlinear operator) and the cost functional, as well. Then, we provide sufficient conditions which guarantee the convergence of solutions of Problem Q ˜ to a solution of Problem Q. The proofs are based on convergence results for elliptic quasivariational inequalities, obtained by using arguments of compactness, lower semicontinuity, monotonicity, penalty and various estimates. Finally, we illustrate the use of the abstract convergence results in the study of optimal control associated with two boundary value problems. The first one describes the equilibrium of an elastic body in frictional contact with an obstacle, the so-called foundation. The process is static and the contact is modeled with normal compliance and unilateral constraint, associated to a version of Coulomb's law of dry friction. The second one describes a stationary heat transfer problem with unilateral constraints. For the two problems we prove existence, uniqueness and convergence results together with the corresponding physical interpretation. [ABSTRACT FROM AUTHOR]

We study vector optimization problems with solid non-polyhedral convex ordering cones, without assuming any convexity or quasiconvexity assumption. We state a Weierstrass-type theorem and existence results for weak efficient solutions for coercive and noncoercive problems. Our approach is based on a new coercivity notion for vector-valued functions, two realizations of the Gerstewitz scalarization function, asymptotic analysis and a regularization of the objective function. We define new boundedness and lower semicontinuity properties for vector-valued functions and study their properties. These new tools rely heavily on the solidness of the ordering cone through the notion of colevel and level sets. As a consequence of this approach, we improve various existence results from the literature, since weaker assumptions are required. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of subdifferential evolution inclusions involving history-dependent operators. First, we improve an existence and uniqueness theorem and prove the continuous dependence result in the weak topologies. Next, we establish the existence of optimal solution to an optimal control problem for the evolution inclusion. Finally, we illustrate the results by an example of an optimal control of a dynamic frictional contact problem in mechanics, whose weak formulation is the evolution variational inequality. [ABSTRACT FROM AUTHOR]

We consider an optimistic semivectorial bilevel programming problem in Banach spaces. The associated lower level multicriteria optimization problem is assumed to be convex w.r.t. its decision variable. This property implies that all its weakly efficient points can be computed applying the weighted-sum-scalarization technique. Consequently, it is possible to replace the overall semivectorial bilevel programming problem by means of a standard bilevel programming problem whose upper level variables comprise the set of suitable scalarization parameters for the lower level problem. In this note, we consider the relationship between this surrogate bilevel programming problem and the original semivectorial bilevel programming problem. As it will be shown, this is a delicate issue as long as locally optimal solutions are investigated. The obtained theory is applied in order to derive existence results for semivectorial bilevel programming problems with not necessarily finite-dimensional lower level decision variables. Some regarding examples from bilevel optimal control are presented. [ABSTRACT FROM AUTHOR]

We consider a continuous-time market with proportional transaction costs. Under appropriate assumptions, we prove the existence of optimal strategies for investors who maximise their worst-case utility over a class of possible models. We consider utility functions defined on either the positive axis or the whole real line. [ABSTRACT FROM AUTHOR]