Covering mappings and well-posedness of nonlinear Volterra equations

The statements on solvability, solution estimates, and well-posedness of equations with conditionally covering mappings are proved. The results obtained are applied to the investigation of Volterra equations (including integral equations) unsolved for the unknown function.     

Covering mappings in a product of metric spaces and boundary value problems for differential equations unsolved for the derivative

We prove some assertions about solvability, solution estimates, and well-posed solvability of equations with covering mappings in a product of metric spaces. The results are applied to the analysis of boundary value problems for differential equations unsolved for the derivative.     

Covering mappings and their applications to differential equations unsolved for the derivative

We continue to study the properties of covering mappings of metric spaces and present their applications to differential equations. To extend the applications of covering mappings, we introduce the notion of conditionally covering mapping. We prove that the solvability and the estimates for solutions of equations with conditionally covering mappings are preserved under small Lipschitz perturbations. These assertions are used in the solvability analysis of differential equations unsolved for the derivative.     

On coincidence points for vector mappings

For mappings acting in the product of metric spaces we propose a concept of vector covering. This concept is a natural extension of the notion of covering formappings inmetric spaces. The statements on the solvability of systems of operator equations are proved for the case when the left-hand side of an equation is a value of a vector covering mapping and the right-hand side is Lipschitzian vector mapping. In the scalar case the obtained statements are equivalent to the coincidence point theorems by A. V. Arutyunov. As an application, we prove a statement on the existence of n-fold coincidence points and obtain estimates of the points. The sufficient conditions for n-fold fixed points existence, including the well-known theorems on double fixed point, follow from the obtained results.     

On the well-posedness of differential equations unsolved for the derivative

We prove some results concerning solvability, estimates for solutions, and well-posed solvability of equations with conditionally covering mappings. These results are applied to the analysis of differential equations unsolved for the derivative.     

Chain rules for linear openness in metric spaces and applications

In this work we present a general theorem concerning chain rules for linear openness of set-valued mappings acting between metric spaces. As particular cases, we obtain classical and also some new results in this field of research, including the celebrated Lyusternik–Graves Theorem. The applications deal with the study of the well-posedness of the solution mappings associated to parametric systems. Sharp estimates for the involved regularity moduli are given.     

Perturbations of vectorial coverings and systems of equations in metric spaces

E. R. Avakov, A. V. Arutyunov, S. E. Zhukovski?, and E. S. Zhukovski? studied the problem of Lipschitz perturbations of conditional coverings of metric spaces. Here we propose some extension of the concept of conditional covering to vector-valued mappings; i.e., the mappings acting in products of metric spaces. The idea is that, to describe a mapping, we replace the covering constant by the matrix of covering coefficients of the components of the vector-valued mapping with respect to the corresponding arguments. We obtain a statement on the preservation of the property of conditional and unconditional vectorial coverings under Lipschitz perturbations; the main assumption is that the spectral radius of the product of the covering matrix and the Lipschitz matrix is less than one. In the scalar case this assumption is equivalent to the traditional requirement that the covering constant be greater than the Lipschitz constant. The statement can be used to study various simultaneous equations. As applications we consider: some statements on the solvability of simultaneous operator equations of a particular form arising in the problems on n-fold coincidence points and n-fold fixed points; as well as some conditions for the existence of periodic solutions to a concrete implicit difference equation.     

Perturbation of the mapping and solvability theorems in Banach space

Here we consider perturbations of continuous mappings on Banach spaces, and investigate their images under various conditions. Consequently we study the solvability of some classes of equations and inclusions. For these we start by the investigation of local properties of the considered mapping and local comparisons of this mapping with certain smooth mappings. Moreover, we study different mixed problems.     

一类k-次增生型变分包含解的存在性与Noor迭代逼近

在实自反Banach空间中,引入并研究一类k-次增生型变分包含问题,证明了这类变分包含解的存在性、唯一性以及带混合误差的Noor三步迭代序列的收敛性,给出了收敛率的估计式,从而本质改进,统一和发展了谷峰教授的新近的结果.     

集优化问题的广义适定性与解的稳定性

本文研究了集优化问题的适定性与解的稳定性. 首次利用嵌入技术引入了集优化问题的广义适定性概念, 得到了此类适定性的一些判定准则和特征, 并给出其充分条件. 此外, 借助一类广义Gerstewitz 函数, 建立了此类适定性与一类标量优化问题广义适定性之间的等价关系. 最后, 在适当条件下研究了含参集优化问题弱有效解映射的上半连续性和下半连续性.     

The Levitin-Polyak well-posedness by perturbations for systems of general variational inclusion and disclusion problems

In this paper, the notions of the Levitin-Polyak well-posedness by perturbations for system of general variational inclusion and disclusion problems (shortly, (SGVI) and (SGVDI)) are introduced in Hausdorff topological vector spaces. Some sufficient and necessary conditions of the Levitin-Polyak well-posedness by perturbations for (SGVI) (resp., (SGVDI)) are derived under some suitable conditions. We also explore some relations among the Levitin-Polyak well-posedness by perturbations, the existence and uniqueness of solution of (SGVI) and (SGVDI), respectively. Finally, the lower (upper) semicontinuity of the approximate solution mappings of (SGVI) and (SGVDI) are established via the Levitin-Polyak well-posedness by perturbations.     

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.     

The Discontinuous Riemann–Hilbert Problem for Elliptic Complex Equations of First Order

In this article, we first transform the general uniformly elliptic systems of first order equations with certain conditions into the complex equations, and propose the discontinuous Riemann- Hilbert problem and its modified well-posedness for the complex equations. Then we give a priori estimates of solutions of the modified discontinuous Riemann-Hilbert problem for the complex equations and verify its solvability. Finally the solvability results of the original discontinuous Riemann-Hilbert boundary value problem can be derived. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.     

Levitin–Polyak well-posedness for strong bilevel vector equilibrium problems and applications to traffic network problems with equilibrium constraints

In this paper we consider strong bilevel vector equilibrium problems and introduce the concepts of Levitin–Polyak well-posedness and Levitin–Polyak well-posedness in the generalized sense for such problems. The notions of upper/lower semicontinuity involving variable cones for vector-valued mappings and their properties are proposed and studied. Using these generalized semicontinuity notions, we investigate sufficient and/or necessary conditions of the Levitin–Polyak well-posedness for the reference problems. Some metric characterizations of these Levitin–Polyak well-posedness concepts in the behavior of approximate solution sets are also discussed. As an application, we consider the special case of traffic network problems with equilibrium constraints.     

An iterative method for finding coincidence points of two mappings

The problem of finding coincidence points of two mappings of which one is a covering, while the other satisfies the Lipschitz condition, is examined. An iterative method for finding an approximate solution to this problem is discussed. It is based on the a priori estimates derived in the paper.     

On the solvability of one class of parameterized operator inclusions

We consider a class of parameterized operator inclusions with set-valued mappings of type. Sufficient conditions for the solvability of these inclusions are obtained and the dependence of the sets of their solutions on functional parameters is investigated. Examples that illustrate the results obtained are given. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1619–1630, December, 2008.     

On coincidence points of multivalued vector mappings of metric spaces

The notion of metric regularity can be extended to multivalued mappings acting in the products of metric spaces. A vector analog of Arutyunov’s coincidence-point theorem for two multivalued mappings is proved. Statements on the existence and estimates of solutions of systems of inclusions of special form occurring in the multiple fixed-point problem are obtained. In particular, these results imply some well-known double-point theorems.     

Boundary value problems on infinite intervals

We present two methods, both based on topological ideas, to the solvability of boundary value problems for differential equations and inclusions on infinite intervals. In the first one, related to the rich family of asymptotic problems, we generalize and extend some statements due to the Florence group of mathematicians Anichini, Cecchi, Conti, Furi, Marini, Pera, and Zecca. Thus, their conclusions for differential systems are as well true for inclusions; all under weaker assumptions (for example, the convexity restrictions in the Schauder linearization device can be avoided). In the second, dealing with the existence of bounded solutions on the positive ray, we follow and develop the ideas of Andres, Górniewicz, and Lewicka, who considered periodic problems. A special case of these results was previously announced by Andres. Besides that, the structure of solution sets is investigated. The case of l.s.c. right hand sides of differential inclusions and the implicit differential equations are also considered. The large list of references also includes some where different techniques (like the Conley index approach) have been applied for the same goal, allowing us to envision the full range of recent attacks on the problem stated in the title.

     

Initial value problems for creeping flow of Maxwell fluids

We consider the flow of nonlinear Maxwell fluids in the unsteady quasistatic case, where the effect of inertia is neglected. We study the well-posedness of the resulting PDE initial-boundary value problem locally in time. This well-posedness depends on the unique solvability of an elliptic boundary value problem. We first present results for the 3D case with sufficiently small initial data and for a simple shear flow problem with arbitrary initial data; after that we extend our results to some 3D flow problems with large initial data.We solve our problem using an iteration between linear subproblems. The limit of the iteration provides the solution of our original problem.     

模糊映射的完全广义混合型强变分包含

研究一类新的关于模糊映射的完全广义混合型强变分包含问题，给出解的逼近算法，证明这类问题解的一个存在定理和序列收敛定理。