Multipoint Problem for Nonisotropic Partial Differential Equations with Constant Coefficients

We investigate the well-posedness of a problem with multipoint conditions with respect to a chosen variable t and periodic conditions with respect to coordinates x ₁,..., x _p for a nonisotropic (concerning differentiation with respect to t and x ₁,..., x _p) partial differential equation with constant complex coefficients. We establish conditions for the existence and uniqueness of a solution of this problem and prove metric theorems on lower bounds for small denominators appearing in the course of the construction of its solution.     

Multipoint problem for hyperbolic equations with variable coefficients

By using the metric approach, we study the problem of classical well-posedness of a problem with multipoint conditions with respect to time in a tube domain for linear hyperbolic equations of order 2n (n ≥ 1) with coefficients depending onx. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of the problem.     

A Multipoint Problem for Pseudodifferential Equations

We investigate the well-posedness of a problem with multipoint conditions with respect to a chosen variable t and periodic conditions with respect to coordinates x ₁,...,x _p for equations unsolved with respect to the leading derivative with respect to t and containing pseudodifferential operators. We establish conditions for the unique solvability of this problem and prove metric assertions related to lower bounds for small denominators appearing in the course of its solution.     

Cauchy-type problem for an abstract differential equation with fractional derivatives

The uniform well-posedness of a Cauchy-type problem with two fractional derivatives and bounded operator A is proved. For an unbounded operator A we present a test for the uniform well-posedness of the problem under consideration consistent with the test for the uniform well-posedness of the Cauchy problem for an equation of second order.     

The Cauchy problem for a model equation with triple characteristics

Summary We give necessary and sufficient conditions for the well-posedness of the Cauchy problem for hyperbolic operators with triple characteristics wlwse principal symbol is suitably factorized.     

Cauchy-type problem for an abstract differential equation with fractional derivatives

The uniform well-posedness of a Cauchy-type problem with two fractional derivatives and bounded operator A is proved. For an unbounded operator A we present a test for the uniform well-posedness of the problem under consideration consistent with the test for the uniform well-posedness of the Cauchy problem for an equation of second order.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 28–41.Original Russian Text Copyright © 2005 by A. V. Glushak.This revised version was published online in April 2005 with a corrected issue number.     

Boundary value problem for abstract first order differential equation with integral condition

The present paper is devoted to the study of a boundary value problem for abstract first order linear differential equation with integral boundary conditions. We obtain necessary and sufficient conditions for the unique solvability and well-posedness. We also study the Fredholm solvability. Finally, we obtain a result of the stability of solution with respect to small perturbation.     

On Schrödinger type evolution equations with non-Lipschitz coefficients

We prove results of well-posedness of the global Cauchy problem in Sobolev spaces for a class of evolution equations with real characteristics that contains an Euler– Bernoulli vibrating beam model. We consider non-Lipschitz coefficients with respect to the time variable t and study the sharp rate of their oscillations. This is coupled with some necessary decay conditions as the spatial variable x → ∞.     

Well-Posedness in Sobolev Spaces for Second-Order Strictly Hyperbolic Equations with Nondifferentiable Oscillating Coefficients

The goal of this paper is to study well-posedness to strictly hyperbolic Cauchyproblems with non-Lipschitz coefficients with low regularity with respect to timeand smooth dependence with respect to space variables. The non-Lipschitz conditionis described by the behaviour of the time-derivative of coefficients. This leads to a classification of oscillations, which has a strong influence on the loss of derivatives. To study well-posednesswe propose a refined regularizing technique. Two steps of diagonalizationprocedure basing on suitable zones of the phase spaceand corresponding nonstandard symbol classes allow to applya transformation corresponding to the effect of loss of derivatives.Finally, the application of sharp Gårding's inequality allows to derive a suitable energy estimate. From this estimatewe conclude a result about C well-posedness of the Cauchy problem.     

Nonlocal boundary-value problem for linear partial differential equations unsolved with respect to the higher time derivative

We study the well-posedness of the problem with general nonlocal boundary conditions in the time variable and conditions of periodicity in the space coordinates for partial differential equations unsolved with respect to the higher time derivative. We establish the conditions of existence and uniqueness of the solution of the considered problem. In the proof of existence of the solution, we use the method of divided differences. We also prove metric statements on the lower bounds of small denominators appearing in constructing the solution of the problem. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 3, pp. 370–381, March, 2007.     

Levitin–Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems

This paper is devoted to the Levitin–Polyak well-posedness by perturbations for a class of general systems of set-valued vector quasi-equilibrium problems (SSVQEP) in Hausdorff topological vector spaces. Existence of solution for the system of set-valued vector quasi-equilibrium problem with respect to a parameter (PSSVQEP) and its dual problem are established. Some sufficient and necessary conditions for the Levitin–Polyak well-posedness by perturbations are derived by the method of continuous selection. We also explore the relationships among these Levitin–Polyak well-posedness by perturbations, the existence and uniqueness of solution to (SSVQEP). By virtue of the nonlinear scalarization technique, a parametric gap function g for (PSSVQEP) is introduced, which is distinct from that of Peng (J Glob Optim 52:779–795, 2012). The continuity of the parametric gap function g is proved. Finally, the relations between these Levitin–Polyak well-posedness by perturbations of (SSVQEP) and that of a corresponding minimization problem with functional constraints are also established under quite mild assumptions.     

Stability in convex semi-infinite programming and rates of convergence of optimal solutions of discretized finite subproblems

We consider convex semiinfinite programming (SIP) problems with an arbitrary fixed index set T. The article analyzes the relationship between the upper and lower semicontinuity (lsc) of the optimal value function and the optimal set mapping, and the so-called Hadamard well-posedness property (allowing for more than one optimal solution). We consider the family of all functions involved in some fixed optimization problem as one element of a space of data equipped with some topology, and arbitrary perturbations are premitted as long as the perturbed problem continues to be convex semiinfinite. Since no structure is required for T, our results apply to the ordinary convex programming case. We also provide conditions, not involving any second order optimality one, guaranteeing that the distance between optimal solutions of the discretized subproblems and the optimal set of the original problem decreases by a rate which is linear with respect to the discretization mesh-size.     

Generalized Hadamard well-posedness for infinite vector optimization problems

In this paper, we study the generalized Hadamard well-posedness of infinite vector optimization problems (IVOP). Without the assumption of continuity with respect to the first variable, the upper semicontinuity and closedness of constraint set mappings are established. Under weaker assumptions, sufficient conditions of generalized Hadamard well-posedness for IVOP are obtained under perturbations of both the objective function and the constraint set. We apply our results to the semi-infinite vector optimization problem and the semi-infinite multi-objective optimization problem.     

On the Galerkin-Method in Intermediate Spaces for the Initial Boundary-Value Problem of the Navier-Stokes Equations in two Dimensions

We consider a GALERKIN scheme for the two-dimensional initial boundary-value problem (P) of the NAVIER-STOKES equations, derive a priori-estimates for the approximations in interpolation spaces between “standard spaces” as occuring in the theory of weak solutions and obtain well-posedness of (P) with respect to the interpolation spaces. As a by-product, we obtain various estimates for the solutions in a relatively simple way. The main tool is interpolation of non-linear operators.     

Scalarization of Levitin–Polyak well-posed set optimization problems

The aim of this paper is to study Levitin–Polyak (LP in short) well-posedness for set optimization problems. We define the global notions of metrically well-setness and metrically LP well-setness and the pointwise notions of LP well-posedness, strongly DH-well-posedness and strongly B-well-posedness for set optimization problems. Using a scalarization function defined by means of the point-to-set distance, we characterize the LP well-posedness and the metrically well-setness of a set optimization problem through the LP well-posedness and the metrically well-setness of a scalar optimization problem, respectively.     

Well-posedness for a Class of Variational–Hemivariational Inequalities with Perturbations

In this paper, we consider an extension of well-posedness for a minimization problem to a class of variational–hemivariational inequalities with perturbations. We establish some metric characterizations for the well-posed variational–hemivariational inequality and give some conditions under which the variational–hemivariational inequality is strongly well-posed in the generalized sense. Under some mild conditions, we also prove the equivalence between the well-posedness of variational–hemivariational inequality and the well-posedness of corresponding inclusion problem.     

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.     

On the Cauchy problem for nonlinear parabolic equations with variable density

We investigate the well-posedness of the Cauchy problem for a class of nonlinear parabolic equations with variable density. Sufficient conditions for uniqueness or nonuniqueness in L ^∞(IR ^N  × (0, T)) (N ≥ 3) are established in dependence of the behavior of the density at infinity. We deal with conditions at infinity of Dirichlet type, and possibly inhomogeneous.     

Optimal control problems for linear fractional-order systems defined by equations with Hadamard derivative

Two optimal control problems are studied for linear stationary systems of fractional order with lumped variables whose dynamics is described by equations with Hadamard derivative, a minimum-norm control problem and a time-optimal problem with a constraint on the norm of the control. The setting of the problem with nonlocal initial conditions is considered. Admissible controls are sought in the class of functions p-integrable on an interval for some p. The main approach to the study is based on the moment method. The well-posedness and solvability of the moment problem are substantiated. For several special cases, the optimal control problems under consideration are solved analytically. An analogy between the obtained results and known results for systems of integer and fractional order described by equations with Caputo and Riemann–Liouville derivatives is specified.

     

On the well-posedness of the cauchy problem and the mixed problem for some class of hyperbolic systems and equations with constant coefficients and characteristics of variable multiplicity

We study the well-posedness of the mixed problem for hyperbolic equations with constant coefficients and with characteristics of variable multiplicity. We single out a class of higher-order hyperbolic operators with constant coefficients and with characteristics of variable multiplicity, for which we obtain a generalization of the Sakamoto conditions for the well-posedness of the mixed problem in L ₂.