By invoking the

Implicit Function Theorem, there exists a function f such that the identity [y'.sub.1] - E'Qy = f(t,Qy) holds.

Assume that the

implicit function [PSI] : [R.sup.l] x [R.sup.r] [right arrow] [R.sup.l] is continuously differentiable at each point ([??], [sigma]) on the open set Y [subset] [R.sup.l] x [R.sup.r].

In the above formula, [g.sub.i] (i = 1, 2, ..., m) is a real continuous function while g is an

implicit function of a geometric object.

By using the

implicit function theorem we will prove that the solution of (10) given in Lemma 4 can be continued to a solution with [M.sub.l+2] > 0.

The above follows from the

implicit function theorem (IFT) [3, Theorem 13.7, page 374] where k [member of] [K.sub.0] is the independent variable for the existence of the P(k) function.

A local

implicit function is blended with neighbor functions in [C.sup.2] continuity by partition of unity.

We also show that numerous contrastive conditions of the existing literature enjoy the format of our newly introduced

implicit function besides admitting several new and natural contrastive conditions.

If the improvement of the solution for an one-criterion sub-problem in the complex optimisation, using special mathematical tools for acceleration of the computing process GMRG, is called complex optimal correction of ESS of EPS, the improvement of the solution of multi-objective sub-problems of the complex optimisation using special mathematical tools for acceleration of the computing process [application instead of the gradient of the one

implicit function the array of the multi-objective descent or rise (dependent on the minimisation or maximisation of the partners objective functions)], is called complex multiobjective or Pareto-optimal correction of ESS of IPS.

Since J is holomorphic and [[partial derivative].sub.d]J(c) [not equal to] 0, there exists a holomorphic function g guaranteed by the

implicit function theorem such that in some open ball around c, J(X, g(X)) = 0.

(14) This

implicit function can conceptually be rearranged to an explicit form of Firm i's reaction function.

After a chapter on general preliminaries, chapters cover differential calculus of boundary perturbations, the

implicit function theorem, bifurcation problems, the transversality theorem, generic perturbation of the boundary, boundary operators for second-order elliptic equations, and the method of rapidly oscillating solutions.