Unique solution

This problem has uot a unique solution [5], i.e. there may exist temperature stress distributions for which rr z = 0. 

The consistency condition for this set of equations to possess a (unique) solution is that the field intensity tensor defined in Eq. (99) is zero [72], which is also known as the curl condition and is written in an abbreviated form as... 

The degree of the least polynomial of a square matr ix A, and henee its rank, is the number of linearly independent rows in A. A linearly independent row of A is a row that eannot be obtained from any other row in A by multiplieation by a number. If matrix A has, as its elements, the eoeffieients of a set of simultaneous nonhomo-geneous equations, the rank k is the number of independent equations. If A = , there are the same number of independent equations as unknowns A has an inverse and a unique solution set exists. If k < n, the number of independent equations is less than the number of unknowns A does not have an inverse and no unique solution set exists. The matrix A is square, henee k > n is not possible. 

In what immediately follows, we will obtain eigenvalues i and 2 for //v / = Ei ) from the simultaneous equation set (6-38). Each eigenvalue gives a n-election energy for the model we used to generate the secular equation set. In the next chapter, we shall apply an additional equation of constr aint on the minimization parameters ai, 2 so as to obtain their unique solution set. 

As a single equation with three variables, equation 6.26 does not have a unique solution for the concentrations of CHaCOOH, CHaCOQ-, and HaO+. At constant temperature, different solutions of acetic acid may have different values for [HaO+], [CHaCOQ-] and [CHaCOOH], but will always have the same value ofiQ. 

Equation 7.8 does not have a unique solution because different combinations of and give the same overall error. The choice of how many samples to collect and how many times each sample should be analyzed is determined by other concerns, such as the cost of collecting and analyzing samples, and the amount of available sample. 

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

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... 

Repeating the proof of Theorem 1.19 for this case, one deduces that equation (1.119) has a unique solution gV which satisfies... 

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... 

We proceed with an investigation of the contact problem for a plate under creep conditions. We know that for every fixed / G L Q) there exists a unique solution w,M satisfying (2.35)-(2.37). Let G L Q) be a given element and F c (Q) be a closed convex and bounded set. We introduce the cost functional... 

This result enables us to investigate the extreme crack shape problem. The formulation of the last one is as follows. Let C Hq 0, 1) be a convex, closed and bounded set. Assume that for every -0 G the graph y = %j) x) describes the crack shape. Consequently, for a given -0 G there exists a unique solution of the problem... 

The angular brackets ( , ) denote the integration over flc- In virtue of the linearity, boundedness, and coercivity of the form a(-, ), there exists a unique solution to (2.165). 

By the above properties for the operator of (2.167), there exists a unique solution G H Qc) to the problem (2.167). Then, for sufficiently... 

We first note that the coercivity and weak lower semicontinuity of the functional n imply that the problem (2.248) has a (unique) solution The coercivity is provided by the following two inequalities,... 

There exists a unique solution IK G if of the variational inequality (2.265). 

It is obvious that there exists a unique solution G The... 

There exists a unique solution of the optimal control problem,... 

Observe that variational inequality (3.106) is valid for every function X G 82- It means that a solution % to problem (3.106) with 9 G Si coincides with the unique solution to problem (3.100) with the same 9] i.e. problems (3.100) and (3.106) are equivalent. For small 5, we write down an extra variational inequality for which a solution exists, and demonstrate that the solution coincides with the solution of variational inequality (3.98). 

Because g is nonnegative, one can see that ff is a coercive, strongly convex and lower semicontinuous functional. Therefore, there exists a unique solution w G H Qc) of the problem (3.204) or (3.205) (see Section 1.2). 

---