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.     

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.     

跨越共振的非线性常微分方程的第二边值问题

使用最优控制理论方法和Schauder不动点定理证明了非线性常微分方程第二边值问题解的存在性和唯一性,并且给出了其可解性的最优估计,特别地,我们考虑的问题可以是跨越共振的.     

On the initial boundary value problem for a nonlinear nonhomogeneous equation of Sobolev type

The initial boundary value problem for a nonlinear nonhomogeneous equation of Sobolev type used for modeling nonstationary processes in semiconductors is examined. It is proved that this problem is uniquely solvable at least locally in time. Sufficient conditions for the problem to be solvable globally in time are found, as well as sufficient conditions for the local (but not global) solvability. In the case of only local solvability, upper and lower estimates for the time when a solution exists are determined in the form of either explicit or quadrature formulas.     

The problem of Dirichlet for anisotropic quasilinear degenerate elliptic equations

We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations. New a priori estimates for solutions and for the gradient of solutions are established. Based on these estimates sufficient conditions guaranteeing the solvability of the problem are formulated. The results are new even in the semilinear case when the principal part is the Laplace operator.     

A majorant-minorant criterion for the total preservation of global solvability of a functional operator equation

We study a nonlinear controlled functional operator equation in an ideal Banach space. We establish sufficient conditions for the global solvability for all controls from a given set, and obtain a pointwise estimate for solutions. Using upper and lower estimates of the functional component in the right-hand side of the initial equation (with a fixed operator component), we obtain majorant and minorant equations. We prove the stated theorem, assuming the monotonicity of the operator component in the right-hand side and the global solvability of both majorant andminorant equations. We give examples of the reduction of controlled initial boundary value problems to the equation under consideration.     

On a problem with nonlinear boundary conditions for systems of functional-differential equations

We use the method of a priori estimates to obtain effective sufficient solvability conditions for systems of nonlinear functional-differential equations with nonlinear boundary conditions of periodic type. The results are in a sense optimal.     

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.     

Gradient Estimates for Solutions of Ultraparabolic Equations

We establish a priori estimates for solutions to ultraparabolic equations which play a crucial role in the solvability of the initial value problem. A class of these equations came from population dynamics, namely from a fish larvae model.      

Inverse problems for elliptic equations in the space: I

We consider inverse problems of finding the right-hand side of a linear secondorder elliptic equation of general form. We study the first boundary value problem. Two ways of representation of the additional information (over-determination) are considered: specification of the trace of the solution of the boundary value problem inside the domain on some manifold of smaller dimension and specification of values of the normal derivative on a part of the boundary. The Fredholm alternative is proved for the considered problems. Stability estimates are given for the case of unique solvability. The analysis is performed in classes of continuous functions whose derivatives satisfy the Hölder condition.     

Estimates for the Cauchy function of linear singular differential equations and some applications

For higher-order linear singular equations, we find sharp estimates for the Cauchy function and its partial derivatives. We use these estimates to study the properties of solutions of singular differential inequalities and obtain optimal sufficient conditions for the unique solvability of the linear singular Cauchy problem.     

On the solvability of initial boundary-value problems for a class of operator-differential equations of third order

We obtain sufficient conditions for the regular solvability of initial boundary-value problems for a class of operator-differential equations of third order with variable coefficients on the semiaxis. These conditions are expressed only in terms of the operator coefficients of the equations under study. We obtain estimates of the norms of intermediate derivative operators via the discontinuous principal parts of the equations and also find relations between these estimates and the conditions for regular solvability.     

A uniqueness theorem for solution of one Dirichlet problem

We consider the Dirichlet problem in a four-dimensional domain formed by characteristic surfaces of an equation of the 8th order with the double major partial derivative. We state sufficient conditions for the unique solvability of this problem in terms of control coefficients, based on the method of a priori estimates.     

The uniqueness of a solution to the inverse Cauchy problem for a fractional differential equation in a Banach space

We consider linear fractional differential operator equations involving the Caputo derivative. The goal of this paper is to establish conditions for the unique solvability of the inverse Cauchy problem for these equations. We use properties of the Mittag-Leffler function and the calculus of sectorial operators in a Banach space. For equations with operators in a general form we obtain sufficient conditions for the unique solvability, and for equations with densely defined sectorial operators we obtain necessary and sufficient unique solvability conditions.     

Averaging of a multipoint problem with parameters for an impulsive oscillation system

By using the averaging method, we prove the solvability of a multipoint problem with parameters for a nonlinear oscillation system with pulse influence at fixed times. We establish estimates for the deviation of solutions of the original and averaged problems.     

Boundary-value problems with parameters for a multifrequency oscillation system

By using the averaging method, we prove the solvability of boundary-value problems with parameters for nonlinear oscillation systems. We obtain estimates for the deviation of solutions of averaged problems from solutions of original problems.     

On the initial boundary-value problem for a nonlinear Sobolev-type equation with variable coefficient

We study the initial boundary-value problem for a nonlinear Sobolev-type equation with variable coefficient. We obtain sufficient conditions for both global and local (in time) solvability. In the case of local (but not global) solvability, we obtain upper and lower bounds for the existence time of the solution in the form of explicit and quadrature formulas.     

On a nonclassical fractional boundary-value problem for the Laplace operator

In this paper we consider a boundary-value problem for the Poisson equation with a boundary condition comprising the fractional derivative in time and the right-hand sides dependent on time. We prove the one-valued solvability of this problem, and provide the coercive estimates of the solution.     

Averaging of Boundary-Value Problems with Parameters for Multifrequency Impulsive Systems

By using the averaging method, we prove the solvability of boundary-value problems with parameters for nonlinear oscillating systems with pulse influence at fixed times. We also obtain estimates for the deviation of solutions of the averaged problem from solutions of the original problem.     

Continuous dependence of the solution of a parabolic equation in a Hilbert space on the parameters and initial data

We present classes of equations for which weak a priori estimates hold and prove the global strong solvability of one system of equations.