Criteria for the well-posedness of a linear two-point boundary value problem for systems of integro-differential equations

We consider a linear two-point boundary value problem for systems of integro-differential equations. By using the parametrization method and an approximation of the integro-differential equation by a loaded differential equation, we establish coefficient tests for the well-posedness of the considered problem and suggest an algorithm for finding the solution.     

Application of the averaging method for a two-point boundary problem for systems of integro-differential equations of fredholm type

The authors justify a version of the averaging method for solving boundary problems for systems of integro-differential equations of Fredholm type with fast and slow variables.     

Fundamental solution for linear two-point boundary value problem

The quadratic functional minimization with differential restrictions represented by the command linear systems is considered. The optimal solution determination implies the solving of a linear problem with two points boundary values. The proposed method implies the construction of a fundamental solution S(t)—a n×n matrix—and of a vector h(t) defining an adjoint variable λ(t) depending of the state variable x(t). From the extremum necessary conditions it is obtained the Riccati matrix differential equation having the S(t) as unknown fundamental solution is obtained. The paper analyzes the existence of the Riccati equation solution S(t) and establishes as the optimal solution of the proposed optimum problem. Also a superior limit of the minimum for the considered quadratic functionals class are evaluated.     

On the unique solvability of a nonlocal boundary value problem with data on intersecting lines for systems of hyperbolic equations

We consider a nonlocal boundary value problem for a system of hyperbolic equations with two independent variables with data on intersecting lines one of which is a characteristic. In terms of the data of the nonlocal boundary value problem, we obtain sufficient coefficient conditions for its unique solvability.     

On a boundary value problem for integro-differential equations on the halfline

The following boundary value problem

     

A priori estimates and solvability of the third two-point boundary value problem

We study a priori estimates and solvability of a nonlinear two-point boundary value problem for systems of second-order ordinary differential equations with leading positively homogeneous nonlinearity of order > 1 vanishing on a single surface. Assuming that an a priori estimate holds, we prove the invariance of the solvability of the problem under a continuous change of the leading nonlinear homogeneous terms and under arbitrary perturbations that do not affect the behavior of the leading nonlinear homogeneous terms at infinity.     

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.     

Impulsive anti-periodic boundary value problem of first-order integro-differential equations

This paper is concerned with the anti-periodic boundary value problem of first-order nonlinear impulsive integro-differential equations. We first establish a new comparison principle, and then obtain the existence of extremal solutions by upper-lower solution and monotone iterative techniques. Some examples are presented to illustrate the main results.     

Maximum principles for the periodic boundary value problem for impulsive integro-differential equations

This paper studies a first-order periodic boundary value problem for impulsive integro-differential equations and establishes some new maximum principles. Some points of a recent paper are also corrected and extended.     

On the unique solvability of a family of two-point boundary-value problems for systems of ordinary differential equations

We consider a family of two-point boundary-value problems for systems of ordinary differential equations with functional parameters. This family is the result of the reduction of a boundary-value problem with nonlocal condition for a system of second-order, quasilinear hyperbolic equations by the introduction of additional functions. Using the parametrization method, we establish necessary and sufficient conditions of the unique solvability of the family of two-point boundary-value problems for a linear system in terms of the initial data. We also prove sufficient conditions of the unique solvability of the problem considered and propose an algorithm for its solution. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 21–39, 2006.     

New results for the periodic boundary value problem for impulsive integro-differential equations

This paper discusses the periodic boundary value problem for first-order nonlinear impulsive integro-differential equations. We prove some new comparison principles, and then establish new existence results for extremal solutions by using these principles and the monotone iterative technique. These results improve and extend all known relevant conclusions from the literature. Some examples are also given to illustrate the advantage of our results.     

Block boundary value methods for linear weakly singular Volterra integro-differential equations

A class of block boundary value methods (BBVMs) is constructed for linear weakly singular Volterra integro-differential equations (VIDEs). The convergence and stability of these methods is analysed. It is shown that optimal convergence rates can be obtained by using special graded meshes. Numerical examples are given to illustrate the sharpness of our theoretical results and the computational effectiveness of the methods. Moreover, a numerical comparison with piecewise polynomial collocation methods for VIDEs is given, which shows that the BBVMs are comparable in numerical precision.

     

On boundary value problem solvability theory for a class of high-order operator-differential equations

In this note, we establish sufficient conditions for the correct and unique solvability of various boundary value problems for a class of even-order operator-differential equations on the half-axis. These conditions are unimprovable in terms of operator coefficients of the equation. We note that the principal part of the equation under study suffers a discontinuity.     

Integral boundary value problem for impulsive integro-differential equations in Banach spaces

In this paper, by using the cone theory and monotone iterative technique, we investigate the existence of extremal solutions and unique solution of the integral boundary value problem for a class of first-order impulsive integro-differential equations in a real Banach space. An explicit iterative scheme for the unique solution and an error estimate of the approximation sequence are also derived.     

On solvability of boundary value problems for systems of differential equations

We consider systems of differential equations (1)x=g(t, x) together with boundary conditions of the form (2)x(0)=x ₀,x(T)=x ₁; (3)x(0)=Qx(T), x(0)=Qx(T); and (4)B ₁ x(0)–B ₂ x(0)=0=C ₁ x(T)+C ₂ x(T). Herex=(x ₁,...,x _n )^T andg=g(t, x) are realn-vectors andQ, B _i ,C _i ,i=1, 2, denoten×n matrices withQ nonsingular andB ₂,C ₂ positive definite. We examine the existence of solutions of (1) satisfying (3) or (4) and which also stay in a certain regionin(t, x) space. Conditions in terms of the Jacobian matrixG (t, x)=g _x (t, x) and an auxiliary positive definite symmetric matrixP=P(t) C ² [0,T] are given which yield the existence of the desired solution of (1), (3) or (1), (4).Dedicated to Professor Hans W. Knobloch on the occassion of his sixtieth birthdayResearch supported by NSERC Canada Grant A7673 and NSF Grant DMS-8501311.     

Multiple solutions for a two-point boundary value problem

In this paper, using a recent result of Ricceri, we prove two multiplicity theorems for the problem −u^″=λf(u)+μg(x,u), u(0)=u(1)=0, extending a previous result that G. Bonanno obtained for μ=0.     

A method for solving the linear boundary value problem for an integro-differential equation

A method for solving the linear boundary value problem for an integro-differential equation is proposed that is based on interval partition and the introduction of additional parameters. Necessary and sufficient conditions for the solvability of the problem are obtained.     

An algorithm for solving a linear two-point boundary value problem for an integrodifferential equation

family of algorithms for solving a linear two-point boundary value problem is constructed in terms of the data of the integrodifferential equation and the boundary condition involved. The convergence conditions for the algorithms are established, and necessary and sufficient conditions for the well-posedness of the problem are found.