Existence and uniqueness theorem

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. 

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,... 

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... 

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. 

Why is this the general solution First of all, it is a linear combination of solutions to X = Ax, and hence is itself a solution. Second, it satisfies the initial condition x(0) = Xq, and so by the existence and uniqueness theorem, it is the only solution. (See Section 6.2 for a general statement of the existence and uniqueness theorem.)... 

We have been a bit optimistic so far—at this stage, we have no guarantee that the general nonlinear system x = f(x) even has solutions Fortunately the existence and uniqueness theorem given in Section 2.5 can be generalized to two-dimen-... 

Existence and Uniqueness Theorem Consider the initial value problem X = f(x), x(0) = Xq. Suppose that f is continuous and that all its partial derivatives /dXj, J, J = 1,..., n, are continuous for x in some open connected set D c R". Then for x, e D, the initial value problem has a solution x(t) on some time interval... 

The existence and uniqueness theorem has an important corollary different trajectories never intersect. If two trajectories did intersect, then there would be two solutions starting from the same point (the crossing point), and this would violate the uniqueness part of the theorem. In more intuitive language, a trajectory can t move in two directions at once. 

Arimoto, S., Spivakovsky, M., Ohno, H., Zizler, P., Taylor, K.F., Yamabe, T. and Mezey, P.G. (2001) Structural analysis of certain linear operators representing chemical network systems via the existence and uniqueness theorems of spectral resolution. VI. Int. J. Quant. Chem., 84, 389-400. 

The modeling of stationarily operating enzyme electrodes is a special case of nonstationary modeling. In this context, stationary means the consideration of the behaviour after infinite time (steady state), i.e., solutions have to be found for which the time derivative has to be zero. For all models considered, the existence of a unique and stable mathematical solution is assumed. Since the respective partial differential equations are linear initial boundary problems, known existence and uniqueness theorems may be used (Kamke, 1956 Fife, 1979 Ozisik, 1980). 

State and prove existence and uniqueness theorems on the continuous component model. 

For those first-order equations that cannot be expressed in polynomial form, there is no single analytical method to produce a solution as seen earlier in Section 2.1. This difficulty increases the importance of the issues of existence and uniqueness of a solution. For a very lucid discussion on the existence and uniqueness theorem for nonlinear first-order differential equations, many excellent texts are available [1,2]. 

Renardy, M., A local existence and uniqueness theorem for a K-BKZ fluid, Univ. of Wise. MRC technical summary 2530. 

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. 

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. 

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. 

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

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. 

This equation is weakly singular if 0 < p < 1 and regular for p > 1. In the former case, we must give explicit proofs for existence and uniqueness of the solution. Hence, results that are very similar to the corresponding classical theorems of existence and uniqueness, known in the scalar case of first-order equations, are discussed subsequently [49]. 

The existence and uniqueness of the solution of mass-action-type kinetic differential equations (or, more precisely, initial value or Cauchy problems for this type of differential equations) are ensured by general theorems, such as the Picard-Lindelof theorem (see the textbooks cited above). 

In this section a very important fundamental theorem will be discussed. This theorem is important because it resolves two of the issues raised at the end of Section 1.1. Specifically, the theorem addresses existence and uniqueness of a... 

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. 

Theorem 3.11 Existence and uniqueness) [25] Consider the initial value problem given by... 

We are interested in the case where n is sufficiently large so that x can be replaced by /u. The following theorem demonstrates the existence and uniqueness of the RE equilibrium in the secondary market ... 

This property obviously implies coercivity and strict monotonicity of A. The right-hand side of (1.105) belongs to V since H c V. Then, by Theorem 1.14, there exists a unique solution V,n = 0,1,2,..., to... 

Theorem 1.18. Under the above assumptions, there exists a unique solution u gV of the problem (1.104) and... 

Theorem 1.19. There exists a unique solution u G K of the inequality (1.102), and the convergence... 

Theorem 1.23. If A V V is a linear, self-conjugate, strongly monotonous and Lipschitz continuous operator in a Hilbert space V, then there exists a unique solution u G K of the variational inequality (1.126) given by the formula... 

The second axiom, which is reminiscent of Mach s principle, also contains the seeds of Leibniz s Monads [reschQl]. All is process. That is to say, there is no thing in the universe. Things, objects, entities, are abstractions of what is relatively constant from a process of movement and transformation. They are like the shapes that children like to see in the clouds. The Einstein-Podolsky-Rosen correlations (see section 12.7.1) remind us that what we empirically accept as fundamental particles - electrons, atoms, molecules, etc. - actually never exist in total isolation. Moreover, recalling von Neumann s uniqueness theorem for canonical commutation relations (which asserts that for locally compact phase spaces all Hilbert-space representations of the canonical commutation relations are physically equivalent), we note that for systems with non-locally-compact phase spaces, the uniqueness theorem fails, and therefore there must be infinitely many physically inequivalent and... 

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