Lipschitz continuous

We will also use the theorem on contraction mappings. A mapping S y —> y is called a contraction mapping if it is Lipschitz continuous,... 

Take the norm of both parts of this equality and use the Lipschitz continuity of P (see Lemma 1.2). By the linearity of I, it provides... 

Theorem 1.22. For strongly monotonous and Lipschitz continuous operator A satisfying (1.129), (1.130), there exists a unique solution u G K of the variational inequality (1.126) and... 

Here is the space of functions having k Lipschitz continuous deriva-... 

A priori estimates in the case of a variable operator A. So far we have established stability in Ha under the agreement that operator A is constant, that is, independent of t. In the case when A(t) = A t) > 0 depends on t, this obstacle necessitates imposing the Lipschitz continuity of the operator A(t) in the variable t... 

Schemes with variable operators. If operators A and R depend on the variable t, the extra property of the Lipschitz continuity of A and R with respect to the variable t is needed in this connection ... 

Armijo, L. Minimization of Functions Having Lipschitz Continuous First Partial Derivatives. Pac J Math 16 1-3 (1966). 

Remark 2 The property of stability can be interpreted as a Lipschitz continuity condition on the perturbation function v(y). 

The objective function /( ) and the inequality constraint g(x) are convex since f(x) is separable quadratic (sum of quadratic terms, each of which is a linear function of xi, x2,X3, respectively) and g(x) is linear. The equality constraint h(x) is linear. The primal problem is also stable since v(0) is finite and the additional stability condition (Lipschitz continuity-like) is satisfied since f(x) is well behaved and the constraints are linear. Hence, the conditions of the strong duality theorem are satisfied. This is why... 

We hope the reader will appreciate the elegance and simplicity of the arguments supporting Theorem 3.2, which are based on the LaSalle corollary. In particular, a linearized stability analysis about each of the rest points of (3.3), required in Chapter 1, was completely avoided. A careful reading of the proof of Theorem 3.2 reveals that assumption (iii) on f is not crucial to the proof we will have more to say about this later. Finally, it should be noted that the assumption (iv) on f can be relaxed somewhat. It can be weakened to requiring only that f be locally Lipschitz continuous... 

Abadie, J., and Carpentier, J. (1969), Generalization of the Wolfe Reduced Gradient Method to the Case of Nonlinear Constraints, in Optimization, R. Fletcher, Ed., Academic Press, New York. Armijo, L. (1966), Minimization of Functions having Lipschitz Continuous First-Partial Derivatives, Pacific J. Mathematics, Vol. 16, No. 1, pp. 1-3. 

Armijo, L. (1966) Minimization of functions having Lipschitz continuous first-partial derivatives. Padjic Journal of Mathematics, 16 (1), 1-3. 

Note that this assumption also implies that the potential production rate of one reservoir does not depend on e volumes produced from the other reservoirs. We will also assume for i = ,... that fi is nonnegative and nonincreasing as a function of Qi t) for all t and that the recoverable volume of each reservoir is finite. Finally, to ensure uniqueness of potential production pro files we will also assume that is Lipschitz continuous in Qi, i =... 

Initial value problems with ordinary differential equations (ODE) have well-defined conditions (based on Lipschitz continuity of the time derivatives) that guarantee unique solutions. Conditions for unique solutions of DAEs (Equations 14.2 and 14.3) are less well defined. One way to guarantee existence and uniqueness of DAE solutions is to confirm that the DAE can be converted (at least implicitly) to an initial value ODE. A general analysis of these DAE properties can be found in [5] and is beyond the scope of this chapter. On the other hand, for a workable analysis, one needs to ensure a regularity condition on the DAE characterized by its index. 

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