Continuous Dependence Theorems Research Articles

Continuous dependence theorems in the theory of linear elastic materials with microstructure

It is shown that the solution to the equations of a linear micropolar elastic solid (see Eringen [1]), in an exterior domain in R 3, depends continuously on initial and boundary data, body forces and material coefficients. The linear dipolar elastic theory of Green and Rivlin [2] is also briefly examined.

A priori bounds for solutions to nonlinear two-point boundary value problems

The boundary value problem is considered. The function f is assumed to be continuous on [a, b]× R2, and g 1 and g 2 are assumed to be continuous and nondecreasing on R. The condition (A)There exists M 1 >0 and a positive, continuous function Ψ(p) defined on [0, ∞) such that and for , is used to obtain an a priori bound on solutions, As applications of the a priori bound, an existence theorem is obtained using the Leray-Schauder Theorem, and a “weak continuous dependence theorem” is proved, In both cases, the well-known Nagumo condition is also imposed in order to obtain a priori bounds on derivatives of solutions.

A class of Neutral Functional Differential Equations

Formulation and study of the initial value problem for neutral functional differential equations. The existence, uniqueness, and continuation of solutions to this problem are investigated, and an analysis is made of the dependence of the solutions on the initial conditions and parameters, resulting in the derivation of a continuous dependence theorem in which the fundamental mathematical principles underlying the continuous dependence problem for a very general system of nonlinear neutral functional differential equations are separated out.

Gronwall’s inequality for systems of partial differential equations in two independent variables

This paper presents a generalization for systems of partial differential equations of Gronwall’s classical integral inequality for ordinary differential equations. The proof is by reducing the vector integral inequality to a vector partial differential inequality and then using a vector generalization of Riemann’s method to obtain the final inequality. The final inequality involves a matrix function in the integrand which is a generalization of the scalar Riemann function. The proof includes a successive approximations argument to guarantee the existence and positivity property of this matrix function. The inequality is applied to prove a uniqueness theorem for a nonlinear vector hyperbolic partial differential equation, a comparison theorem for a linear hyperbolic vector partial differential equation, and a continuous dependence theorem for a nonlinear vector boundary value problem. The inequality also appears to have many applications in stability problems and in numerical solutions of partial differential equations. All of these results hold for the corresponding Volterra integral equations and the method of proof of the main result shows that the function on the right-hand side of the final inequality is the solution of the integral equation and hence is the maximal solution of the original inequality.

Free boundary problems for the heat equation in which the moving interface coincides initially with the fixed face

A semi-infinite solid at the melting temperature 0 occupies the region 0 ⩽ x < ∞. At the fixed face x = 0 there is prescribed either ( a) the temperature f( t) ⩾ 0 or ( b) the heat flux f( t) directed into the solid, where f( t) is not identically zero in the vicinity of t = 0. There is a free boundary s( t) separating the liquid phase from the solid phase, and a temperature distribution u( x, t) in the liquid phase. We prove existence, uniqueness, and continuous dependence theorems for problems ( a) and ( b). Let F( t) be the integral, from 0 to t, of f(τ). Then we prove that for problem ( a) s( t) behaves, in the vicinity of t = 0, essentially like the square root of F( t) while for problem ( b) s( t) behaves essentially like F( t). We obtain also upper bounds for s′( t) for both problems. The existence theorems are proved as follows: the region 0 ⩽ x ⩽ a is taken to be liquid at the melting temperature and the region a < x < ∞ solid at the melting temperature. Problems ( a) and ( b) are known to have solutions u a and s a in this case which are bounded, equicontinuous families. By Arzela's theorem uniformly convergent subsequences can be extracted. These subsequences converge to the solutions of ( a) and ( b) for the case a = 0.

Existence, uniqueness and continuity of solutions of integral equations. An addendum

This note corrects au error in the continuous dependence theorem of an earlier paper on the subject stated in the title. Here it is shows that the topology of one of the spaces of functions can be modified in order to obtain the desired continuity results.

Some Continuous Dependence Theorems for Viscous Fluid Motions

Some Continuous Dependence Theorems for Viscous Fluid Motions