Solvability of a periodic boundary-value problem for systems of functional differential equations with cyclic matrices

We consider first-order systems of linear functional differential equations with regular operators. For families of systems of two equations we obtain the general necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem. For families of systems of n linear functional differential equations with cyclic matrices we obtain effective necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem.     

Solvability of the Cauchy problem for higher-order linear functional differential equations

We obtain necessary and sufficient conditions for the unique solvability of the Cauchy problem for higher-order linear non-Volterra functional differential equations.     

Unique solvability of a non-linear non-local boundary-value problem for systems of non-linear functional differential equations

General conditions for the unique solvability of a non-linear nonlocal boundary-value problem for systems of non-linear functional differential equations are obtained.     

Solution of Kartsatos' Problem on Nonlinear Boundary Value Problems for Ordinary Differential Equations on Infinitive Intervals

We solve in the affirmative a problem raised by Kartsatos about the solvability of functional boundary value problems for ordinary differential equations on infinite intervals.     

Solvability of Inhomogeneous Autonomous Differential Equation With Aftereffect on the Negative Semi-Axis

Russian Mathematics - We obtain a necessary and sufficient solvability condition for a linear autonomous inhomogeneous functional differential equation with aftereffect and find the representation...     

Inverse problem of the variational calculus for differential equations of second order with deviating argument

We obtain conditions for the solvability of the inverse problem of the variational calculus for differential equations of second order with deviating argument of special form as well as the formula for the functional of the inverse problem defined by the integral that differs from the standard one by that the required function has a retarded argument.     

One class of singular linear functional differential equations

We consider one class of first-order functional differential equations with a singularity in the independent variable. We obtain conditions for the Fredholm property and the solvability of the mentioned equations.     

On the Construction and Growth of Solutions of Degenerate Functional Differential Equations of Neutral Type

We consider degenerate linear functional differential equations in Banach spaces and construct solutions of exponential and hyperexponential growth. We establish conditions for the unique solvability of an initial-value problem and describe the set of initial functions. The results are applied to partial differential equations with time delay     

共振情形下四点泛函边值问题解的存在性

运用Mawhin重合度理论,讨论了一类二阶四点泛函边值问题解的存在性和多解性.分别在非线性项f有界和无界的条件下,获得了此类泛函边值问题解的存在性结果.     

Positivity of solutions to boundary value problems for infinite functional differential systems

In this paper the problems of unique solvability of boundary value problems for infinite functional differential systems and positivity of their solutions are reduced to analysis of scalar functional equations in the space of measurable essentially bounded functions y:[0,+∞)→(−∞,+∞). Efficient tests of the solvability and the positivity are obtained.     

Linear Forward—Backward Stochastic Differential Equations

The problem of finding adapted solutions to systems of coupled linear forward—backward stochastic differential equations (FBSDEs, for short) is investigated. A necessary condition of solvability leads to a reduction of general linear FBSDEs to a special one. By some ideas from controllability in control theory, using some functional analysis, we obtain a necessary and sufficient condition for the solvability of a class of linear FBSDEs. Then a Riccati-type equation for matrix-valued (not necessarily square) functions is derived using the idea of the Four-Step Scheme (introduced in [11] for general FBSDEs). The solvability of such a Riccati-type equation is studied which leads to a representation of adapted solutions to linear FBSDEs. Accepted 29 April 1997     

On the solvability of perturbations of linear boundary value problems at resonance for functional differential equations

Necessary and sufficient conditions of the unique solvability of perturbations of linear boundary value problems at resonance for functional differential equations are obtained. These conditions are effective and expressed in terms of norms of the positive and negative parts of regular operators.     

Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations

This paper is concerned with the solvability of periodic boundary value problems for nonlinear impulsive functional differential equations

(∗)     

On solutions of differential inclusions in homogeneous spaces of functions

In this paper we study the solvability of some classes of differential inclusions with multivalued linear operators in homogeneous spaces of functions. These spaces include a large number of functional spaces like periodic functions and Bohr and Stepanov almost periodic functions. As an application, we consider some existence results for feedback control systems governed by degenerate differential equations of Sobolev type in a Banach space.     

Tropical differential equations

Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in the absolute values of its coefficients. Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability). This extends a similar complexity bound established for tropical linear systems. In case of tropical linear differential systems in one variable a polynomial complexity algorithm for testing its solvability is designed.We prove also that the problem of solvability of a system of tropical non-linear differential equations in one variable is NP-hard, and this problem for arbitrary number of variables belongs to NP. Similar to tropical algebraic equations, a tropical differential equation expresses the (necessary) condition on the dominant term in the issue of solvability of a differential equation in power series.     

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.     

Programmed Iteration Method in Differential Games with a Functional Target Set

A variant of the programmed iteration method is developed for solving game problems of realizing trajectories of a nonlinear conflict controlled system in a given set of functions. Solving these open-loop control problems yields an iterative process in a space of sets whose elements are functional positions, and the corresponding limit determines the set of successful solvability of the original problem in the class of set-valued quasi-strategies (nonanticipating responses to disturbances). For the conflict controlled system, the conditions of generalized uniqueness and uniform boundedness of programmed motions are assumed to hold. The use of an infinite-dimensional space of functional positions is essential, since the arising differential game generally does not satisfy the alternative solvability conditions in classes of feedback strategies.     

A Multipoint Problem with Complex Coefficients for Partial Differential Equations That Are Unsolvable for the Highest Derivative

Conditions of correct solvability in various functional spaces of a problem with multipoint conditions in the time variable and periodicity conditions in the space coordinate are established for partial differential equations that are unsolvable for the highest derivative with time. Metric theorems on lower estimates of small denominators that arise in the construction of solutions are proved.     

General conditions for the unique solvability of initial-value problem for nonlinear functional differential equations

We establish general conditions for the unique solvability of the Cauchy problem for systems of nonlinear functional differential equations. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 167–172, February, 2008.     

Variational method for proving the existence of a nonlocal solution of a boundary value problem for a quasilinear partial differential equation

We study the solvability of a boundary value problem for a quasilinear partial differential equation of the second kind. To this end, we use a variational method; namely, we prove the existence of a point of absolute minimum of a functional, and this point is a solution of the original problem.