On solvability recognition for interval linear systems of equations

The paper considers the problem of recognizing solvability (nonemptiness of the solution set) for interval systems of linear algebraic equations. We introduce a quantitative measure of the membership of a point in the solution set, the so-called “recognizing functional” of the solution set. As the result, the decision on solvability of the interval linear systems reduces to global maximization of the recognizing functional. Additionally, the specific value of this maximum and its argument provide us with important quantitative information of the solvability supply or its deficiency, which can used for the correction of the interval system in a desired sense.     

Nonstandard decision methods for the solvability of real polynomial equations

For a multivariate polynomial equation with coefficients in a computable ordered field, two criteria of this equation having real solutions are given. Based on the criteria, decision methods for the existence of real zeros and the semidefiniteness of binaly polynomials are provided. With the aid of computers, these methods are used to solve several examples. The technique is to extend the original field involved in the question to a computable non-Archimedean ordered field containing infinitesimal elements. Project supported by the National Natural Science Foundation of China (Grant No. 19661002) and the Climbing Project.     

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.     

On polynomial equations

The set of all triples of positive integers (α, β, γ) for which there exist polynomials f, g, h (with or without common factors) which satisfy the equationf ^α + ^β =h ^γ , is determined.     

On polynomial solvability of two multiprocessor scheduling problems

We discuss a new approach for solving multiprocessor scheduling problems by using and improving results for guillotine pallet loading problem. We introduce a new class of schedules by analogy with the guillotine restriction for cutting stock problem and show that many well-known algorithms from classical scheduling theory construct schedules only from this class. We also consider two multiprocessor scheduling problems and prove that they can be solved in polynomial time.     

On coercive solvability of parabolic equations

In this paper we study the coercive solvability of abstract differential equations of parabolic type in the spaces of L. N. Slobodetskii W _p . It is established that the solution of equations with a constant operator A which generates an analytic semigroup belongs to the trace space E(, p, A). The results obtained are applied to the study of equations with a variable operator.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 409–419, April, 1972.     

On the computation of solutions of systems of interval polynomial equations

Systems of algebraic equations with interval coefficients are very common in several areas of engineering sciences. The computation of the solution of such systems is a central problem when the characterization of the variables related by such systems is desired.In this paper we characterize the solution of systems of algebraic equations with real interval coefficients. The characterization is obtained considering the approach introduced in J. Comput. Appl. Math. 136 (2001) 271.     

On the solvability of the systems of equations arising in implicit Runge-Kutta methods

In this paper we present a new condition under which the systems of equations arising in the application of an implicit Runge-Kutta method to a stiff initial value problem, has unique solutions. We show that our condition is weaker than related conditions presented previously. It is proved that the Lobatto IIIC methods fulfil the new condition.     

On nonnegative solvability of linear operator equations

LetE be a Banach lattice having order continuous norm. Suppose, moreover,T is a nonnegative reducible operator having a compact iterate and which mapsE into itself. The purpose of this work is to extend the previous results of the authors, concerning nonnegative solvability of (kernel) operator equations on generalL ^p-spaces. In particular, we provide necessary and sufficient conditions for the operator equation x=T x+y to possess a nonnegative solutionxE wherey is a given nonnegative and nontrivial element ofE and is any given positive parameter.     

On the local solvability of semilinear equations

We prove that complex semilinear equations whose linear principal part satisfies condition (p) are locally solvable. This extends the local solvability theorem of Beals and Fefferman to semilinear equations.     

On coercive solvability of operator-differential equations inB-spaces

We study the coercive solvability of operator-differential equations in anisotropicB-spaces of vector-functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 8, pp. 1084–1087, August, 1994.The author expresses his deep gratitude to Prof. Shakhmurov for helpful discussions.     

Positive solvability of systems of nonlinear Hammerstein integral equations

The paper deals with the existence of positive continuous solutions to systems of nonlinear Hammerstein integral equations. The main tool used in the proofs is fixed point index theory in a cone. The results obtained here are essentially different from existing ones in the literature.     

Necessary solvability conditions of systems of linear extremal equations

Systems of linear equations of the form A?X = B?X and of the form A?X = A?Y over the structure based on linearly ordered commutative group (G, ?, ≤) where the role of ⊕ plays the maximum are treated. Necessary solvability conditions are derived using known results concerning eigenvectors of matrices in such structures. In the special case of idempotent, increasing matrices A and B a condition is given which is necessary and sufficient for the existence of a non-trivial solution.     

On solvability of some special systems of one-dimensional loaded parabolic equations and composite-type systems with Cauchy data

We study a one-dimensional system of two parabolic loaded equations of a special kind and a one-dimensional composite-type system of a special kind. Both systems are considered with Cauchy data. We obtain sufficient conditions for the existence of a solution in the class of bounded smooth functions. The method of splitting on the differential level (the weak approximation method) is used in the proof.     

On the polynomial limit cycles of polynomial differential equations

In this paper we deal with ordinary differential equations of the form dy/dx = P(x, y) where P(x, y) is a real polynomial in the variables x and y, of degree n in the variable y. If y = φ(x) is a solution of this equation defined for x ∈ [0, 1] and which satisfies φ(0) = φ(1), we say that it is a periodic orbit. A limit cycle is an isolated periodic orbit in the set of all periodic orbits. If φ(x) is a polynomial, then φ(x) is called a polynomial solution.     

On the unique solvability of fuzzy relational equations

The solution set of a consistent system of fuzzy relational equations with max-min composition can be characterized by one maximum solution and a finite number of minimal solutions. A polynomial-time method of O(mn) complexity is proposed to determine whether such a system has a unique minimal solution and/or a unique solution, where m, n are the dimensions of the input data. The proposed method can be extended to examining a system of fuzzy relational equations with max-T composition where T is a continuous triangular norm.     

Some computational methods for systems of nonlinear equations and systems of polynomial equations

This paper gives a brief survey and assessment of computational methods for finding solutions to systems of nonlinear equations and systems of polynomial equations. Starting from methods which converge locally and which find one solution, we progress to methods which are globally convergent and find an a priori determinable number of solutions. We will concentrate on simplicial algorithms and homotopy methods. Enhancements of published methods are included and further developments are discussed.     

On solvability of singular integral-differential equations with convolution

In this paper, we study a class of singular integral-different equations of convolution type with Cauchy kernel. By means of the classical boundary value theory, of the theory of Fourier analysis, and of the principle of analytic continuation, we transform the equations into the Riemann-Hilbert problems with discontinuous coefficients and obtain the general solutions and conditions of solvability in class $\{0\}$. Thus, the result in this paper generalizes the classical theory of integral equations and boundary value problems.