Existence and Uniqueness

Our treatment of vectorf ields has been very informal. In particular, we have taken a cavalier attitude toward questions of existence and uniqueness of solutions to 

Show that the solution to x = starting from Xg =0 is nor unique. 

Solution The point x = 0 is a fixed point, so one obvious solution is x t) = 0 

When uniqueness fails, our geometric approach collapses because the phase point doesn t know how to move if a phase point were started at the origin, would it stay there or would it move according to x(t) = (Or as my friends in el- 

Actually, the situation in Example 2.5.1 is even worse than we ve let on—there are infinitely many solutions starting from the same initial condition (Exercise 2.5.4). 

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Ciarlet P.G., Sanchez-Palencia F. (1996) An existence and uniqueness theorem for the two-dimensional linear membrane shell equations. J. Math. Pures Appl. 75 (1), 51-67. 

It may be noted that an elastic material for which potentials of this sort exist is called a hyperelastic material. Hyperelasticity ensures the existence and uniqueness of solutions to intial/boundary value problems for an elastic material undergoing small deformations, and also implies that all acoustic wave speeds in the material are real and positive. 

It should be noted that the normality conditions, arising from the work assumption applied to inelastic loading, ensure the existence and uniqueness of solutions to initial/boundary value problems for inelastic materials undergoing small deformations. Uniqueness of solutions is not always desirable, however. Inelastic deformations often lead to instabilities such as localized deformations. It is quite possible that the work assumption, which is essentially a stability postulate, is too strong in these cases. Normality is a necessary condition for the work assumption. Instabilities, while they may occur in real deformations, are therefore likely to be associated with loss of normality and violation of the work assumption. 

Interest in developing and refining the mathematical methods of operations research has become intensified and sophisticated. Attention is generally given to a priori upper bounds on the number of solutions of a problem, the existence and uniqueness of solutions,... 

Classical methods of mathematical physics are employed at the first stage. Numerous physical problems lead to mathematical models having no advanced methods for solving them. Quite often in practice, the user is forced to. solve such nonlinear problems of mathematical physics for which even the theorems of existence and uniqueness have not yet been proven and some relevant issues are still open. 

In the treatment of steady-state pipeline network problems so far we have tacitly assumed that there is a unique solution for each problem. For certain types of networks the existence of a unique solution can indeed be rigorously established. The existence and uniqueness theorems for formulation C were proved by Duffin (DIO) and later extended by Warga (Wl). In Warga s derivation the governing relation for each network element assumes the form,... 

Remark 1 The selection of weight functions u and ui is basically arbitrary. However, these functions have to satisfy certain conditions in order to ensure the existence and uniqueness of a solution. The weight functions u i and U2 utilized in this approach have been verified as meeting all these conditions (Maronna, 1976). Since u and U2 solve the minimax problem (10.36), they are also known as Huber type weights. 

M. Feinberg, The existence and uniqueness of steady states for a class of chemical reaction networks. Arch, Rational Mech. Anal. 132(4), 311 370 (1995). 

Detonation, Strong and Weak. This subject is discussed by Evans 8t Ablow (Ref 2, pp 141-42), but prior to this it. is necessary to discuss the existence and uniqueness of classes of reaction waves for specific boundary conditions , as given in the book of Courant 8c Friedrichs (Ref 1, pp 215-22) and in Ref 2... 

Existence and uniqueness of the particular solution of (5.1) for an initial value y° can be shown under very mild assumptions. For example, it is sufficient to assume that the function f is differentiable and its derivatives are bounded. Except for a few simple equations, however, the general solution cannot be obtained by analytical methods and we must seek numerical alternatives. Starting with the known point (tD,y°), all numerical methods generate a sequence (tj y1), (t2,y2),. .., (t. y1), approximating the points of the particular solution through (tQ,y°). The choice of the method is large and we shall be content to outline a few popular types. One of them will deal with stiff differential equations that are very difficult to solve by classical methods. Related topics we discuss are sensitivity analysis and quasi steady state approximation. 

Existence and uniqueness of solutions to the b.v.p. analogous to (2.2.1) has been proved in numerous contexts (see, e.g., [2]—[6]) and can be easily inferred for (2.2.1). We shall not do it here. Instead we shall assume the existence and uniqueness for (2.2.1) and similar formulations and, based on this assumption, we shall discuss some simple properties of the appropriate solutions. These properties will follow from those of the solution of the one-dimensional Poisson-Boltzmann equation, combined with two elementary comparison theorems for the nonlinear Poisson equation. These theorems follow from the Green s function representation for the solution of the nonlinear Poisson equation with a monotonic right-hand side (or from the maximum principle arguments [20]) and may be stated as follows. 

This together with Estimates 3.1 and 3.2 also provides the existence of uniform bounds for uxx in all such domains for 0 < t < oo. Analogous arguments demonstrate the uniform boundedness of uxxx which is sufficient for a reference to Gevrey s results [32]. All stated above proves that the length of any interval of the solution continuation depends only on the initial data, so that ATn is independent of N. Hence the series (3.2.41) diverges, which proves the existence and uniqueness of a global classical solution to the auxiliary problem under consideration and, moreover, confirms our assertion that this solution may be constructed by the method of continuation. Moreover, analyticity of u(x,t N) with respect to x for all N > 0 shows that... 

Existence and uniqueness of solutions to (4.2.12) and to the appropriate algebraic equations for ji, B have to be studied separately in any specific electro-diffusional set-up. 

Such purely mathematical problems as the existence and uniqueness of solutions of parabolic partial differential equations subject to free boundary conditions will not be discussed. These questions have been fully answered in recent years by the contributions of Evans (E2), Friedman (Fo, F6, F7), Kyner (K8, K9), Miranker (M8), Miranker and Keller (M9), Rubinstein (R7, R8, R9), Sestini (S5), and others, principally by application of fixed-point theorems and Green s function techniques. Readers concerned with these aspects should consult these authors for further references. 

A more complete model of equilibrium has been studied by J. H. Hancock T. S. Motzkin [25], who prove some valuable existence and uniqueness theorems. 

Blanchard and Briining [26] bring the history of the calculus of variations into the twentieth century, as the source of contemporary developments in pure mathematics. A search for existence and uniqueness theorems for variational problems engendered deep studies of the continuity and compactness of mathematical entities... 

The common underlying principle in the approaches for characterizing the solvability of a DAE system is to obtain, either explicitly, or implicitly, a local representation of an equivalent ODE system, for which available results on existence and uniqueness of solutions are applicable. The derivation of the underlying ODE system involves the repeated differentiation of the algebraic constraints of the DAE, and it is this differentiation process that leads to the concept of a DAE index that is widely used in the literature. For the semi-explicit DAE systems (A. 10) that are of interest to us here, the index has the following definition. 

To analyze a physical problem analytically, we must obtain the governing equations that model the phenomenon adequately. Additionally, if the auxiliary equations pertaining to initial and boundary conditions are prescribed those are also well-posed, then conceptually getting the solution of the problem is straightforward. Mathematicians are justifiably always concerned with the existence and uniqueness of the solution. Yet not every solution of the equation of motion, even if it is exact, is observable in nature. This is at the core of many physical phenomena where ohservahility of solution is of fundamental importance. If the solutions are not observable, then the corresponding basic flow is not stable. Here, the implication of stability is in the context of the solution with respect to infinitesimally small perturbations. 

Theorem 4.3 (Existence and uniqueness of one-dimensional global flows)... 

Equations (S)-(6) hold for any test function V in a subspace of including homogeneous boundary conditions on Fi and for any q in the L2(Q) space. A continuous inf-sup inequality ensures the existence and uniqueness of the solutitm of the weak problem (S)-(6). 

Chastened by this example, we state a theorem that provides sufficient conditions for existence and uniqueness of solutions to x = /(x). Existence and Uniqueness Theorem Consider the initial value problem X = fix), X(0) = Xg. ... 

For proofs of the existence and uniqueness theorem, see Borrelli and Coleman (1987), Lin and Segel(1988), or virtually any text on ordinary differential equations. 

There are various ways to extend the existence and uniqueness theorem. One can allow f to depend on time t, or on several variables x,..., x , One of the most useful generalizations will be discussed later in Section 6.2. 

From now on, we will not worry about issues of existence and uniqueness—our vector fields will typically be smooth enough to avoid trouble. If we happen to come across a more dangerous example, we ll deal with it then. 

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