Existence-uniqueness theorem

Theorem 2.6. Under the above hypotheses there exist unique functions w, M = Mij such that ... 

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. 

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... 

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,... 

Kyf — 0. This is the fixed-point equation for every positive normalized r there exists unique positive normalized steady state c (r) KrC (r) — 0, c >0 and c (r) = 1. We have to solve the equation r = c (r). The solution exists because the Brauer fixed point theorem. 

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... 

ORDINARY DIFFERENTIAL EQUATIONS, I.G. Petrovski. Covers basic concepts, some differential equations and such aspects of the general theory as Euler lines, Ariel s theorem, Beano s existence theorem, Osgood s uniqueness theorem, more. 45 figures. Problems. Bibliography. Index, xi + 232pp. 5X 8H. 

Note that the solution of equation (10.62) and, correspondingly, equation (10.65), can be nonunique in a general case, since there may exist m (r) distributions that produce a zero external field, similar to the existence of the current distributions producing zero external field demonstrated in the beginning of this chapter. However, according to the uniqueness theorem by Gusarov (1981, see Chapter 1), the excess conductivity Aa (r) can be found uniquely in the class of piecewise-analytic functions. Therefore,... 

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). 

Theorem 2,2.11 (Division Algorithm). For any two integers x and y, where y > 0, there exist unique integers q and r such that... 

Orlov Rozonoer (1984a, b) present a general phenomenological approach to the macroscopic description of the dynamics of open systems. They prove theorems on existence, uniqueness and stability of the stationary slates. Singularly perturbed equations are also considered. The variational principle for the studied equations is formulated as well. As an application of the general results, nonisothermal kinetic differential equations are considered, detailed balanced and balanced systems are studied and the method of quasistationary concentrations is discussed. For open isothermal systems a theorem on the existence of a positive stationary state is proved providing a solution to an old basic problem. 

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. 

Theorem 1.13. Let the above assumptions he fulfilled. Then there exists a unique solution to the problem (1.81). 

The existence of a unique duality mapping could easily be proved. Indeed, let 1 = u G y I u = 1 be a unit sphere in V. According to the Hahn-Banach theorem, for every fixed u G E there exists a unique element u G V such that u = 1, u, u) = 1 due to the strict convexity of V. Let us define... 

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... 

To verify this theorem, it suffices to note that a unique solution Uq GV of (1.134) always exists due to the mentioned properties of the operator A and Theorem 1.14. 

Thus, all conditions of Theorem 1.11 are fulfilled, hence there exists a unique solution u G K oi the problem (1.149). One can calculate the derivative... 

Theorem 3.1. There exists a unique function x satisfying the variational inequality... 

---