Nonlocal problem with integral conditions for a system of hyperbolic equations in characteristic rectangle

We consider a nonlocal problem with integral conditions for a system of hyperbolic equations in rectangular domain. We investigate the questions of existence of unique classical solution to the problemunder consideration and approaches of its construction. Sufficient conditions of unique solvability to the investigated problem are established in the terms of initial data. The nonlocal problem with integral conditions is reduced to an equivalent problem consisting of the Goursat problem for the system of hyperbolic equations with functional parameters and functional relations. We propose algorithms for finding a solution to the equivalent problem with functional parameters on the characteristics and prove their convergence. We also obtain the conditions of unique solvability to the auxiliary boundary-value problem with an integral condition for the system of ordinary differential equations. As an example, we consider the nonlocal boundary-value problem with integral conditions for a two-dimensional system of hyperbolic equations.     

A problem with a nonlocal,with respect to time,condition for multidimensional hyperbolic equations

We study a boundary-value problem for a hyperbolic equation with a nonlocal with respect to time-variable integral condition. We obtain sufficient conditions for unique solvability of the nonlocal problem. The proof is based on reduction of the nonlocal first-type condition to the second-type one. This allows to reduce the nonlocal problem to an operator equation. We show that unique solvability of the operator equation implies the existence of a unique solution to the problem.     

On the existence and uniqueness of a generalized solution of the mixed problem for the wave equation with the second and third boundary conditions

We prove the uniqueness of a generalized solution of an initial-boundary value problem for the wave equation with boundary conditions of the third and second kind. In addition, we find a closed-form expression for the analytic solution of that problem with zero initial data. The result plays an important role in the investigation of the boundary control problem. We show how to use the obtained solution for the investigation of the boundary control problem in the case of subcritical time intervals for which the solution of the boundary control problem, if it exists at all, is unique. We obtain necessary and sufficient conditions for the existence of a unique solution in a class admitting the existence of finite energy.     

A nonlocal problem for a hyperbolic equation with integral conditions of the 1st kind with time-dependent kernels

We consider a nonlocal problem with integral conditions of the 1st kind. The main goal is to prove the unique solvability of this problem under the assumption that kernels of nonlocal conditions depend both on spatial and time variables. To this end we propose a technique based on the proved equivalence between the nonlocal problem with integral conditions of the 1st kind and a nonlocal problem with integral conditions of the 2nd kind in a special form. We formulate requirements to the initial data guaranteeing the unique existence of a generalized solution to the stated problem.     

A certain boundary-value problem with shifts in a four-dimensional space

We consider a problem with shifts in boundary conditions for the Bianchi equation in a four-dimensional space. We establish sufficient conditions which allow one to evaluate its unique solution in terms of the Riemann function.     

On the solvability of the Tricomi problem with a generalized Frankl transmission condition

We study the solvability of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions in the elliptic part of the domain. On the type change line of the equation, the solution gradient is subjected to a condition that is usually referred to as the generalized Frankl transmission condition. We show that the inhomogeneous Tricomi problem either has a unique solution or is conditionally solvable and the homogeneous problem has only the trivial solution. We write out an integral representation of the solution of this problem.     

Criteria of unique solvability of nonlocal boundary-value problem for systems of hyperbolic equations with mixed derivatives

We consider nonlocal boundary-value problem for a system of hyperbolic equations with two independent variables. We investigate questions of existence of unique classical solution to problem under consideration. In terms of initial data we propose criteria of unique solvability and suggest algorithms of finding of solutions to nonlocal boundary-value problem. As an application we give conditions of solvability of periodic boundary-value problem for a system of hyperbolic equations.     

Nonzero solutions to a two-point boundary-value periodic problem for differential equations with maxima

We prove a theorem on the unique existence of a solution to a nonlinear equation with maxima and demonstrate its continuous dependence on the initial function and the parameter of the problem. We also establish conditions for the existence of a nonzero solution to a two-point boundary-value periodic problem in dependence of both linear and nonlinear terms of the equation.     

On analytic solutions to the generalized Cauchy problem with data on three surfaces for a quasilinear system

The generalized Cauchy problem with data on three surfaces is under consideration for a quasilinear analytic system of the third order. Under some simplifying assumption, we find necessary and sufficient conditions for existence of a solution in the form of triple series in the powers of the independent variables. We obtain convenient sufficient conditions under which the data of the generalized Cauchy problem has a unique locally analytic solution. We give counterexamples demonstrating that in the case we study it is impossible to state necessary and sufficient conditions for analytic solvability of the generalized Cauchy problem. We also show that the analytic solution can fail to exist even if the generalized Cauchy problem with data on three surfaces has a formal solution since the series converge only at a sole point, the origin.     

Advanced differential equations with nonlinear boundary conditions

We investigate the existence of solutions for advanced differential equations with nonlinear boundary conditions. Sufficient conditions when the problem has extremal solutions or a unique solution are formulated. Linear advanced differential inequalities are also discussed.     

On the conditional solvability of the Tricomi problem with mixed boundary conditions

In the present paper, we consider the Tricomi problem with mixed boundary conditions. One of these conditions specifies a directional derivative with constant inclination angle. We show that the problem is either conditionally solvable or has a unique solution depending on the inclination angle.     

A Problem with Nonlocal Conditions for Partial Differential Equations with Variable Coefficients

We establish conditions for the unique solvability of a problem for partial differential equations with coefficients dependent on variables t and x in a rectangular domain with nonlocal two-point conditions with respect to t and local boundary conditions with respect to x. We prove metric statements related to lower bounds of small denominators appearing in the course of solution of the problem.     

Mild solutions of evolution quantum stochastic differential equations with nonlocal conditions

We study existence of a unique mild solution of evolution quantum stochastic differential equations with nonlocal conditions under the strong topology. Using the method of successive approximations, we do not need to transform the nonlocal problem to a fixed point form. The evolution operator A generates a family of semigroup that are continuous. Nonlocal conditions allow additional measurements of certain phenomena that cannot be captured by the traditional initial conditions. We show that under some given conditions, the mild solution is unique and also stable. The method applied here is much easier when compared with previous methods used in literature.     

On the nonuniqueness of the solution of the Tricomi problem with the generalized Frankl matching condition

We obtain an integral representation of the solution of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions in the elliptic part of the domain and with zero posed on one characteristic of the equation. The gradient of the solution is not continuous but satisfies some condition referred to as the “generalized Frankl matching condition.” We state theorems implying that the inhomogeneous Tricomi problem either has a unique solution or is determined modulo a solution of the homogeneous Tricomi problem.     

Riemann problem for Chaplygin gas with combustion

We study the Riemann problem of the Chapman–Jouguet model for an ideal combustible Chaplygin gas. By analyzing the wave curves in the phase plane, we obtain constructively the unique solution of the Riemann problem under the global entropy conditions.     

New existence theorems for quasi-equilibrium problems and a minimax theorem on complete metric spaces

Motivated by the Suzuki’s type fixed point theorems, we give several new existence theorems for scalar quasi-equilibrium problems, and vector quasi-equilibrium problem on complete metric spaces. We give important examples for our results. Note that the solution of quasi-equilibrium problem (resp. vector quasi-equilibrium problem) is unique under suitable conditions, and we can find the unique solution by the Picard iteration. Besides, we also give a new coincidence theorem on complete metric spaces. Finally, we give a new minimax theorem on complete metric spaces. Note that the solution of minimax theorem is unique under suitable conditions, and we can find the unique solution by the Picard iteration.     

Necessary optimality conditions for a special class of bilevel programming problems with unique lower level solution

We consider a bilevel programming problem in Banach spaces whose lower level solution is unique for any choice of the upper level variable. A condition is presented which ensures that the lower level solution mapping is directionally differentiable, and a formula is constructed which can be used to compute this directional derivative. Afterwards, we apply these results in order to obtain first-order necessary optimality conditions for the bilevel programming problem. It is shown that these optimality conditions imply that a certain mathematical program with complementarity constraints in Banach spaces has the optimal solution zero. We state the weak and strong stationarity conditions of this problem as well as corresponding constraint qualifications in order to derive applicable necessary optimality conditions for the original bilevel programming problem. Finally, we use the theory to state new necessary optimality conditions for certain classes of semidefinite bilevel programming problems and present an example in terms of bilevel optimal control.     

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.     

A Mixed Problem for the Heat Equation with Advanced Time in Boundary Conditions

We study a one-dimensional mixed problem for the heat equation, with time advance in nonlocal and non-self-adjoint boundary conditions, describing a real physical process. Under minimal conditions on the initial data, we prove its unique solvability and obtain an explicit representation for the solution.     

Boundary value problem for the Laplace equation outside cuts on the plane with different conditions of the third kind on opposite sides of the cuts

We consider a boundary value problem for the Laplace equation outside cuts on a plane. Boundary conditions of the third kind, which are in general different on different sides of each cut, are posed on the cuts. We show that the classical solution of the problem exists and is unique. We obtain an integral representation for the solution of the problem in the form of potentials whose densities are found from a uniquely solvable system of Fredholm integral equations of the second kind.