The existence of a generalized fourier transform for a solution as a radiation condition for a class of problems generalizing oscillation excitation problems in regular waveguides

For a second-order inhomogeneous differential equation defined on the real axis and such that its right-hand side and solutions are functions in a Hilbert space, it is shown that the existence of a generalized Fourier transform of the solution is a correct radiation condition if the right-hand side is sufficiently smooth and compactly supported.     

On the algebraic solution of fuzzy linear systems based on interval theory

In this paper, fuzzy linear systems involving a crisp square matrix and a fuzzy right-hand side vector are considered. A new approach to solve such systems based on interval theory and the new concept “interval inclusion linear system” is proposed. Also, new necessary and sufficient conditions are derived for obtaining the unique algebraic solution. Numerical examples are given to illustrate the efficiency of the proposed method.     

Semilinear evolution inclusions with nonlocal conditions

We discuss conditions for the existence of at least one solution of semilinear evolution inclusions with a nonlocal condition and a nonconvex right-hand side. Our technique is based on fixed point theorems for multivalued maps.     

Inverse problem of simultaneously determining the right-hand side and the coefficient of a lower order derivative for a parabolic equation on the plane

We obtain existence and uniqueness theorems for the solution of the inverse problem of simultaneously determining the right-hand side and the coefficient of a lower-order derivative in a parabolic equation under an integral observation condition. We give explicit estimates for the maximum absolute value of the unknown right-hand side and the unknown coefficient of the equation with constants expressed via the input data of the problem. We present a nontrivial example of an inverse problem to which our theorems apply.     

Morrey Regularity of Solutions of Nonlinear Elliptic Systems of Arbitrary Order with Restrictions on the Modulus of Ellipticity

We investigate the dependence of the regularity of generalized solutions of nonlinear elliptic systems on the modulus of ellipticity and regularity of the right-hand side. We establish Morrey regularity with limit exponent determined by the modulus of ellipticity in the case where the right-hand side belongs to a space with a norm stronger than the Dini function. These conditions are exact for second-order systems, namely, for any violation of the Dini condition, we construct a solution that does not belong to the Morrey space with limit exponent.     

A Delay Second-Order Set-Valued Differential Equation with Hukuhara Derivatives

In this article, we study a second-order differential equation with three-point boundary conditions with the notion of Hukuhara derivatives. The existence and uniqueness of a solution is given under a Lipschitz condition on the right-hand side in the second and third variables.     

The solution sets of infinite fuzzy relational equations with sup-conjunctor composition on complete distributive lattices

This paper deals with sup-conjunctor composition fuzzy relational equations in infinite domains and on complete distributive lattices. When its right-hand side is a continuous join-irreducible element or has an irredundant continuous join-decomposition, a necessary and sufficient condition describing an attainable solution (resp. an unattainable solution) is formulated and some properties of the attainable solution (resp. the unattainable solution) are shown. Further, the structure of solution sets is investigated.     

On first and second order stationarity of random coefficient models

We give conditions for first and second order stationarity of mixture autoregressive processes. We obtain a simple condition for positive definiteness of the solution of a generalisation of the Stein’s equation with semidefinite right-hand side and apply it to second order stationarity. The said condition may be of independent interest.     

On the asymptotic stability of discontinuous systems analysed via the averaging method

The averaging method is one of the most powerful methods used to analyse differential equations appearing in the study of nonlinear problems. The idea behind the averaging method is to replace the original equation by an averaged equation with simple structure and close solutions. A large number of practical problems lead to differential equations with discontinuous right-hand sides. In a rigorous theory of such systems, developed by Filippov, solutions of a differential equation with discontinuous right-hand side are regarded as being solutions to a special differential inclusion with upper semi-continuous right-hand side. The averaging method was studied for such inclusions by many authors using different and rather restrictive conditions on the regularity of the averaged inclusion. In this paper we prove natural extensions of Bogolyubov’s first theorem and the Samoilenko-Stanzhitskii theorem to differential inclusions with an upper semi-continuous right-hand side. We prove that the solution set of the original differential inclusion is contained in a neighbourhood of the solution set of the averaged one. The extension of Bogolyubov’s theorem concerns finite time intervals, while the extension of the Samoilenko-Stanzhitskii theorem deals with solutions defined on the infinite interval. The averaged inclusion is defined as a special upper limit and no additional condition on its regularity is required.     

Solution of the Bitsadze–Samarskii problem for a parabolic system in a semibounded domain

We consider a nonlocal initial–boundary value Bitsadze–Samarskii problem for a spatially one-dimensional parabolic second-order system in a semibounded domain with nonsmooth lateral boundary. The boundary integral equation method is used to construct a classical solution of this problem under the condition that the vector function on the right-hand side in the nonlocal boundary condition only has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.     

Reconstruction of a convolution operator from the right-hand side on the real half-axis

We study a Volterra convolution integral equation of the first kind on a semi-infinite interval. Under some rather natural constraints on the kernel and the right-hand side of the Volterra integral equation (the kernel has bounded support, while the support of the right-hand side may be unbounded), it is possible to reconstruct the integral operator of the equation (i.e., the solution and the kernel of the integral operator) from the right-hand side of the equation. The uniqueness theorem is proved, the necessary and sufficient conditions for solvability are found, and the explicit formulas for the solution and the kernel are obtained.     

Minimal trajectories of nonconvex differential inclusions

We consider an optimization problem with endpoint constraints associated with a nonconvex differential inclusion. We give a necessary condition of the maximum principle type for a solution of the problem. Following the approach from Ref. 1, the condition is stated in terms of single-valued selections of the convexified right-hand side of the inclusion.This work was supported in part by the National Science Foundation, Grant No. DMS-86-01774.     

Relaxation of a Bolza Problem Governed by a Time-Delay Sweeping Process

We study in an infinite dimensional Hilbert space a Bolza problem in which the dynamics are given by a time-delay perturbed sweeping process. This is a differential inclusion whose right-hand side involves a normal cone to a moving set, along with a time-delay perturbation. A relaxation result is established from which we deduce a sufficient condition ensuring the existence of an optimal solution.     

On the construction of solutions of terminal problems for multidimensional affine systems in quasicanonical form

We consider the construction of solutions of terminal problems for multidimensional affine systems. We show that the terminal problem for a regular system in quasicanonical form can be reduced to a boundary value problem for a system of ordinary differential equations of lower order with right-hand side depending on a vector parameter. We prove a sufficient condition for the existence of a solution of the above-mentioned boundary value problem. A method for constructing a numerical solution is developed.     

Solving Large-Scale Fuzzy and Possibilistic Optimization Problems

Fuzzy and possibilistic optimization methods are demonstrated to be effective tools in solving large-scale problems. In particular, an optimization problem in radiation therapy with various orders of complexity from 1000 to 62,250 constraints for fuzzy and possibilistic linear and nonlinear programming implementations possessing (1) fuzzy or soft inequalities, (2) fuzzy right-hand side values, and (3) possibilistic right-hand side is used to demonstrate that fuzzy and possibilistic optimization methods are tractable and useful. We focus on the uncertainty in the right side of constraints which arises, in the context of the radiation therapy problem, from the fact that minimal and maximal radiation tolerances are ranges of values, with preferences within the range whose values are based on research results, empirical findings, and expert knowledge, rather than fixed real numbers. The results indicate that fuzzy/possibilistic optimization is a natural and effective way to model various types of optimization under uncertainty problems and that large fuzzy and possibilistic optimization problems can be solved efficiently.     

On the classical and generalized solutions of boundary-value problems for difference-differential equations with variable coefficients

The first boundary-value problem for second-order difference-differential equations with variable coefficients on a finite interval (0, d) is considered. The following question is studied: Under what conditions will the boundary-value problem for a difference-differential equation have a classical solution for an arbitrary continuous right-hand side? It is proved that a necessary and sufficient condition for the existence of a classical solution is that certain coefficients of the difference operators on the orbits generated by the shifts be equal to zero.     

On determining the source function in heat and mass transfer problems under integral overdetermination conditions

We examine an inverse problem of determining the right-hand side (the source function) in a parabolic equation from integral overdetermination data. By a solution to a parabolic equation we mean a weak solution, and the right-hand side in this equation can be a distribution of a certain class. Under some conditions on the data of the problem, we demonstrate that this inverse problem is well posed and, in particular, some stability estimates hold.     

Sign-definiteness of solution to inhomogeneous higher-order equation of mixed parabolic-hyperbolic type

We study solutions of a polycaloric equation and an equation of mixed parabolichyperbolic type of the second order. We prove the sign-definiteness of the solution in dependence of the right-hand side of the equation. Based on these results we study the sign-definiteness of a solution to a higher-order inhomogeneous equation of mixed parabolic-hyperbolic type in dependence on the right-hand side of the equation.     

Finding an inner estimation of the solution set of a fuzzy linear system

In this paper, we consider the problem of finding an inner estimation of the solution set of a fuzzy linear system with a real-valued coefficient matrix and a fuzzy-valued right-hand side vector. The proposed idea is based on the utilization of interval Gaussian elimination procedure to produce an inner estimation of the solutions set. To this end, firstly we apply interval Gaussian elimination procedure to obtain the solution set of a fuzzy linear system and secondly, by limiting it via solving a crisp linear system, we find an inner estimation of the solutions set, such that it satisfies the related fuzzy linear system. Finally, several numerical examples are given to show the efficiency and ability of our method.     

On the entropy solution to an elliptic problem in anisotropic Sobolev–Orlicz spaces

For a certain class of anisotropic elliptic equations with the right-hand side from L ₁ in an arbitrary unbounded domains, the Dirichlet problem with an inhomogeneous boundary condition is considered. The existence and uniqueness of the entropy solution in anisotropic Sobolev–Orlicz spaces are proven.