Existence and Uniqueness of Solutions

2 On the Existence and Uniqueness of Solutions of Multibody System Equations 

In this section we give necessary and sufficient conditions for the existence and uniqueness of solutions of the initial value problem 

that this definition requires that the initial values x to) = xq fulfill the constraint (5.1.3c). Some authors extend the definition of a solution and include functions with an initial impulse due to inconsistent initial values. 

A solution in the sense of Def. 5.1.1 only exists if the initial values satisfy these two equations. 

Note that [G p)M p) G p) ) is regular as we assumed G having full rank. 

In the present section we shall show that the initial value problem 

The basic definitions and results from fixed point theory used in this work have been taken from the excellent treatise of Krasnosel skii [23] and are summarized for convenience in an appendix. In order to use the fixed point method, Eqs.(1.3.1), (1.3.2) are written in the integral form,  

All possible solutions of Eq. (1.3.3), considered as elements of lie in the closed bounded region F(uq) = c defined by 

As with A oX l ( o) is bounded independently of the reaction rates. Any F( o) therefore satisfies 

The boundary 5K( o) of the region K(mq) consists of all points i such that (t) lies on the boundary of A o) for one or more values of t. We intend to prove the existence of a solution of Eq. (1.3.3) by showing that the vector field 1 =1—H has rotation +1 on 3V(uo). It may happen, however, that the solution lies on the boundary itself, in which case the rotation is not defined. To avoid this difficulty we enclose F(mo) in the interior of the ball 

We now have an upper triangular system to solve Ity backward substitution, 

X3 = -8 from the last equation. Then, from the second equation, 

With Gaussian elimination and partial pivoting, we have a method for solving linear systems that either finds a solution or fails under conditions in which no unique solution exists. In this section, we consider at more depth the question of when a linear system possesses a real solution (existence) and if so, whether there is exactly one (uniqueness). These questions are vitally important, for linear algebra is the basis upon which we build algorithms for solving nonlinear equations, ordinary and partial differential equations, and many other tasks. 

As a first step, we consider the equation Ax = b from a somewhat more abstract viewpoint. We note that is an A x A real matrix and x and b are both A-dimensional real vectors. 

We have introduced the notation for the set of all real iV-dimensional vectors. The term set merely refers to a collection of objects however, possesses many other properties, including 

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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,... 

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. 

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. 

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. 

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... 

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. ... 

In practice, such vector fields arise when we have a first-order system 0 = /(0), where /(0) is a real-valued, 27t-periodic function. That is, f d + 2it) = /(0) for all real 0. Moreover, we assume (as usual) that f(Q) is smooth enough to guarantee existence and uniqueness of solutions. Although this system could be regarded as a special case of a vector field on the line, it is usually clearer to think of it as a vector field on the circle (as in Example 4,1,1). This means that we don t distin-... 

In other words, existence and uniqueness of solutions are guaranteed if f is continuously differentiable. The proof of the theorem is similar to that for the case n = 1, and can be found in most texts on differential equations. Stronger versions of the theorem are available, but this one suffices for most applications. 

From now on, we ll assume that all our vector fields are smooth enough to ensure the existence and uniqueness of solutions, starting from any point in phase space. 

Although the problem defined by (3-95) and (3-96) is time dependent, it is linear in uJ and confined to the bounded spatial domain, 0 < r < 1. Thus it can be solved by the method of separation of variables. In this method we first find a set of eigensolutions that satisfy the DE (3-95) and the boundary condition at r = 1 then we determine the particular sum of those eigensolutions that also satisfies the initial condition at 7 = 0. The problem (3-95) and (3-96) comprises one example of the general class of so-called Sturm-Louiville problems for which an extensive theory is available that ensures the existence and uniqueness of solutions constructed by means of eigenfunction expansions by the method of separation of variables.14 It is assumed that the reader is familiar with the basic technique, and the solution of (3-95) and (3-96) is simply outlined without detailed proofs. We begin with the basic hypothesis that a solution of (3-95) exists in the separable form... 

Both DAEs and ODEs can be considered special cases of this structure. Brenan et al. [1] provide further reading on existence and uniqueness of solutions to these models, which are considerably more complex issues than in the case of simple ODEs, Initial conditions are required for dxjdt as well as x in this model,... 

One of the most important properties of a typical classical molecular Hamiltonian system is the existence and uniqueness of solutions started from a generic initial condition. The traditional study of existence and uniqueness for systems of nonlinear ordinary differential equations gives local results only, but for Hamiltonian systems considered in molecular dynamics the solutions are typically globally defined. The uniqueness of solutions is easily verified in the usual way (as for the local result for uniqueness of solutions). The key point is that, with the energy constraint, solutions typically remain bounded for all time. 

Gavalas (1968) was an early pioneer in the treatment of the deterministic models of chemical reaction kinetics. His book deals with homogeneous systems and systems with diffusion as well. Basing himself upon recent results in nonlinear functional analysis he treats such fundamental questions as stoichiometry, existence and uniqueness of solutions and the number and stability of equilibrium states. Up to that time this treatise might be considered the best (although brief and concise) summary of the topic. 

Similar to first-order equations, the issue of existence and uniqueness of a solution to second order equations must be dealt with. Below is a theorem [1,4] that addresses the existence and uniqueness of solutions of second-order differential equations. 

It is important to note that this theorem does not address the issue of existence and uniqueness of solutions for boundary value problems. Boundary value problems are discussed in Chapter 4. 

The solutions of Eqs.(2.1.34), (2.1.35) with appropriate boundary conditions such as Eqs. (2.4.3), (2.4.4) will be called the steady states of the system. Various properties of the steady states, such as the invariant manifolds and a priori bounds, the existence and uniqueness of solutions, the asymptotic behavior, and the stability will be treated in the sections that follow. There is a strong similarity in the properties of the uniform open systems investigated in Sections 1.8,1.9 and the distributed systems to be studied now. In both types of systems the interplay between reaction and transport rates (or flow rates) creates the possibility of multiple steady states for certain types of reaction kinetics. Furthermore, the conditions for uniqueness and stability of the steady state have a common mathematical and physical basis. 

It is interesting to note that, for the parameter values that existence and uniqueness of solution are violated, the system s apparent inertia, ImA, is negative. 

We must discuss one more topic before considering the existence and uniqueness of solutions to linear systems the use of basis sets. The set of vectors ft , in is said... 

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