Characterizations Of Well-posedness Research Articles

Well-posedness for multi-time variational inequality problems via generalized monotonicity and for variational problems with multi-time variational inequality constraints

The present paper investigates the well-posedness associated with multi-time variational inequality problems and the corresponding variational problems involving aforesaid inequality as a constraint. Firstly, we introduce the multi-time variational inequality problems determined by curvilinear integral functionals. Thereafter, we present the metric characterization of well-posedness in terms of approximate solution by defining the generalized monotonicity for the considered multi-time functional. Also, we establish that the well-posedness is equivalent to the existence and uniqueness of solution for the problems under consideration. Moreover, the mathematical development is accompanied by various illustrative examples.

On characterization of well-posedness for split hemivariational inequalities

The aim of the present work is to investigate a generalization of well-posedness for a class of split hemivariational inequalities. Also, some equivalent formulation of split hemivariational inequalities are obtained under different monotonicity conditions. Moreover, some metric characterization are given for them.

Well-posedness for generalized (eta ,g,varphi )-mixed vector variational-type inequality and optimization problems

The purpose of this paper is to focus on the well-posedness for a generalized (eta ,g,varphi )-mixed vector variational-type inequality and optimization problems with a constraint. We establish a metric characterization of well-posedness in terms of an approximate solution set. Also we prove that well-posedness of optimization problem is closely related to that of generalized (eta ,g,varphi )-mixed vector variational-type inequality problems.

Metric characterizations for well-posedness of split hemivariational inequalities

In this paper, we generalize the concept of well-posedness to a class of split hemivariational inequalities. By imposing very mild assumptions on involved operators, we establish some metric characterizations of the well-posedness for the split hemivariational inequality. The obtained results generalize some related theorems on well-posedness for hemivariational inequalities and variational inequalities in the literature.

Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian

We study the equations\begin{document}$\begin{align}\partial_t u(t, n) = L u(t, n) + f(u(t, n), n); \partial_t u(t, n) = iL u(t, n) + f(u(t, n), n)\end{align}$\end{document}and\begin{document}$\begin{align}\partial_{tt} u(t, n) =Lu(t, n) + f(u(t, n), n), \end{align}$\end{document}where $n∈ \mathbb{Z}$, $t∈ (0, ∞)$, and $L$ is taken to be either the discrete Laplacian operator $Δ_\mathrm{d} f(n)=f(n+1)-2f(n)+f(n-1)$, or its fractional powers $-(-Δ_{\mathrm{d}})^{σ}$, $0<σ<1$. We combine operator theory techniques with the properties of the Bessel functions to develop a theory of analytic semigroups and cosine operators generated by $Δ_\mathrm{d}$ and $-(-Δ_\mathrm{d})^{σ}$. Such theory is then applied to prove existence and uniqueness of almost periodic solutions to the above-mentioned equations. Moreover, we show a new characterization of well-posedness on periodic Hölder spaces for linear heat equations involving discrete and fractional discrete Laplacians. The results obtained are applied to Nagumo and Fisher-KPP models with a discrete Laplacian. Further extensions to the multidimensional setting $\mathbb{Z}^N$ are also accomplished.

Pointwise and global well-posedness in set optimization: a direct approach

The aim of this paper is to characterize some of the pointwise and global well-posedness notions available in literature for a set optimization problem completely by compactness or upper continuity of an appropriate minimal solution set maps. The characterizations of compactness of set-valued maps, lead directly to many characterizations for well-posedness. Sufficient conditions are also given for global well-posedness.

Some metric characterizations of well-posedness for hemivariational-like inequalities

Some metric characterizations of well-posedness for hemivariational-like inequalities

Well-posedness for general parametric quasi-variational inclusion problems

In this paper, we aim to suggest the new concept of well-posedness for the general parametric quasi-variational inclusion problems (QVIP). The corresponding concepts of well-posedness in the generalized sense are also introduced and investigated for QVIP. Some metric characterizations of well-posedness for QVIP are given. We prove that under suitable conditions, the well-posedness is equivalent to the existence of uniqueness of solutions. As applications, we obtain immediately some results of well-posedness for the parametric quasi-variational inclusion problems, parametric vector quasi-equilibrium problems and parametric quasi-equilibrium problems.

Tykhonov well-posedness for lexicographic equilibrium problems

In this paper, we consider the vector equilibrium problems involving lexicographic cone in Banach spaces. We introduce the new concepts of the Tykhonov well-posedness for such problems. The corresponding concepts of the Tykhonov well-posedness in the generalized sense are also proposed and studied. Some metric characterizations of well-posedness for such problems are given. As an application of the main results, several results on well-posedness for the class of lexicographic variational inequalities are derived.

Well-Posedness for Parametric Generalized Strong Vector Quasi-Equilibrium Problem

In this paper, well-posedness for parametric generalized strong vector quasi-equilibrium problems is studied. The corresponding concept of well-posedness in the generalized sense is also investigated for the parametric generalized strong vector quasi-equilibrium problem. Under some suitable conditions, we establish some characterizations of well-posedness for the parametric generalized strong vector quasi-equilibrium problem.

Well-posedness and stability in vector optimization problems using Henig proper efficiency

In this article, we study well-posedness and stability aspects for vector optimization in terms of minimizing sequences defined using the notion of Henig proper efficiency. We justify the importance of set convergence in the study of well-posedness of vector problems by establishing characterization of well-posedness in terms of upper Hausdorff convergence of a minimizing sequence of sets to the set of Henig proper efficient solutions. Under certain compactness assumptions, a convex vector optimization problem is shown to be well-posed. Finally, the stability of vector optimization is discussed by considering a perturbed problem with the objective function being continuous. By assuming the upper semicontinuity of certain set-valued maps associated with the perturbed problem, we establish the upper semicontinuity of the solution map.

Perturbation theory, stability, boundedness and asymptotic behaviour for second order evolution equation in discrete time

This work deals with the existence, uniqueness and stability of solutions for semilinear discrete second order evolution equations in Banach spaces by using recent characterization of well-posedness for second order evolution equations in terms of R-boundedness and l p -multipliers. We also give results on boundedness and asymptotic behaviour for semilinear discrete evolution equations.

Well-posedness of Systems of Equilibrium Problems

In this paper we introduce the concepts of well-posedness and generalized well-posedness for a system of equilibrium problems. We derive a metric characterization of well-posedness by considering the diameter of approximating solution set and a Furi-Vignoli type characterization of generalized well-posedness by considering the Kuratowski noncompactness measure of approximating solution set. Under suitable conditions, we prove that the well-posedness of a system of equilibrium problems is equivalent to the existence and uniqueness of its solution.

Well-posedness for parametric quasivariational inequality problems and for optimization problems with quasivariational inequality constraints

In this article, the concepts of well-posedness and well-posedness in the generalized sense are introduced for parametric quasivariational inequality problems with set-valued maps. Metric characterizations of well-posedness and well-posedness in the generalized sense, in terms of the approximate solutions sets, are presented. Characterization of well-posedness under certain compactness assumptions and sufficient conditions for generalized well-posedness in terms of boundedness of approximate solutions sets are derived. The study is further extended to discuss well-posedness for an optimization problem with quasivariational inequality constraints.

Well-posedness for set optimization problems

In this paper, three kinds of well-posedness for set optimization are first introduced. By virtue of a generalized Gerstewitz’s function, the equivalent relations between the three kinds of well-posedness and the well-posedness of three kinds of scalar optimization problems are established, respectively. Then, sufficient and necessary conditions of well-posedness for set optimization problems are obtained by using a generalized forcing function, respectively. Finally, various criteria and characterizations of well-posedness are given for set optimization problems.

線形フィードバックの well-posedness の特徴づけ

線形フィードバックの well-posedness の特徴づけ