Generalization of p-regularity notion and tangent cone description in the singular case

The theory of p-regularity has approximately twenty-five years’ history and many results have been obtained up to now. The main result of this theory is description of tangent cone to zero set in singular case. However there are numerous nonlinear objects for which the p-regularity condition fails, especially for p > 2. In this paper we generalize the p-regularity notion as a starting point for more detailed consideration based on different p-factor operators constructions.


Introduction.
In the setting of this article the local description of the solution set for curves and surfaces are essentially questions of p-regularity theory.In many classical cases these results can be viewed as the question about so-called singular points for the curve described by the equation The singular points (x 0 , y 0 ) are such points for which the first partial derivatives are zeros, i.e. ∂F ∂x | (x 0 ,y 0 ) = 0, ∂F ∂y | (x 0 ,y 0 ) = 0.
If additionally not all p-order partial derivatives equal 0 at points (x 0 , y 0 ), then we say that such points are p-times irregular.For description of irregular points it is useful to investigate the sign of the determinant: Depending on the test of the sign of ∆ we can classify and call irregular points.For ∆ = 0 the classification problems arise.
For this case we use the p-regularity theory where the basic apparatus is the so-called p-factor operator and p-factor method.The first result for description of zero set were obtained in [10], [11] for general case of p.It is necessary to point out that for p = 1, 2 the p-regularity notion is quite natural, but for p ≥ 3 there are numerous nonlinear mappings such that p-regularity condition fails.For example, F (x) = x 1 x 2 2 or F (x) = x 1 x p−1 2 and so on.
In this paper we generalize the p-regularity notion on much more nonlinear mappings and prove the theorem for description of the tangent cone to the zero set of the mappings in the singular case.
Finally we want to mention that p-factor method can be used to estimate the number of branch points with singularity of curves or surfaces, which we would like to realize in the present paper.
We begin with some notation.Throughout this paper we suppose that X and Y are Banach spaces.Let p be a natural number and let B : X × X × . . .× X (p-copies of X)→ Y be a continuous symmetric p-multilinear mapping.It means that B is defined on elements x 1 , x 2 , . . ., x p ∈ X as B(x 1 , x 2 , . . ., x p ).The p form associated to B is the map B[•] p : X → Y defined by B[x] p = B(x, x, . . ., x) for x ∈ X, where we assume that it is p + 1-times continuously differentiable on X and its pth order derivative at x ∈ X will be denoted as In a more detailed way, if L(X, Y ) is the space of all linear operators from X to Y , then The associated p-form, also called the pth order mapping, is Moreover, the following necessary general formula holds: For the mapping F : R n → R m , its derivatives F (p) (x) and their operations on elements h, we will use the notation as follows. Let T and generally Remark, that we can write , m, where for q = 1 we have We also assume that F is completely degenerate up to order p at the point x * , i.e. (1.7) Now we formulate the fundamental definitions and results for this case (see [5]).Definition 1.1.Linear operator Ψ p (h) ∈ L(X, Y ), for some fixed h ∈ X \ {0}, defined by is called a p-factor operator.
Definition 1.2.The mapping From these notions it is implied that for any fixed element h ∈ X there exists a linear operator In many of the most important applications are such situations when we consider the so-called p-kernel of p-derivatives.The p-kernel of F p (x * ) is the following set (1.12) It is shown by the two examples underneath.
Example 1.3.Consider the mapping x * = (0, 0).According to the following calculations Then we obtain the 2-kernel of the operator of the second derivative at the point x * as follows (1.15) and the 2-factor operators Ψ 2 (1.16) The image of the first 2-factor operator has the form Therefore the mapping F is 2-regular at the point x * = 0 along the elements of the 2-kernel of the second derivative.F : R 2 → R, It is well known that equation F (x, y) = 0 represents lemniscate of Bernoulli (treated as foot lines of equiaxial hyperbolas).We only have three points x * : (1.22) x * = (0, 0), (a, 0), (−a, 0) as solutions of the equation F (x) = 0 but only one of them x * = (0, 0) is interesting for our investigations.
The derivation of the map F gives us and for x * = (0, 0) we have in (1.7) p = 2 and For two another points (1.22) we get F (x * ) = 0.
Then the singularity exists only at the point x * = (0, 0).The second derivative of F (x) gives us (1.24) and at x * = 0 we get Next the 2-factor operator has the following form and (1.26) we obtain the linear 2-factor operators Ψ 2 (1.28) For example for (1.28) we have It means that the mapping F is 2-regular at the point x * = 0 along the elements of the 2-kernel of the second derivative.
In many nonlinear operator equations and geometric theories we can describe the solution set of mappings F with the help of the so-called tangent cone to the level set of the mapping F .Denote this solution set by Definition 1.5.An element h ∈ X is called tangent vector at a point x * ∈ X to some nonempty set M (x * ) if there exists a number ε > 0 and a mapping: where We can also say that the element h is a p-tangent vector to the set M (x * ) at the point x * if r(t) = o(t p ), p ≥ 1.The set of all p-tangent vectors at the point x * ∈ M (x * ) is called a p-tangent cone, and denoted as T p M (x * ).Now we are ready to generalize our considerations.
Theorem 1.6 (Tret'yakov -generalization of Lyusternik theorem).Let F be the mapping: If the mapping F is p-regular at the point x * along all elements h ∈ Ker p F (p) (x * ), then Applying this theorem to Example 1.4, we can show that in the 2-regular case the tangent cone T 1 M (x * ) = Ker 2 F (2) (x * ) is always two-sided.It means that there exist two tangent lines at point the x * .They are y = x and y = −x.
We conclude this section with a useful lemma which we need for our further consideration.Let ρ(x, y) = x − y be a distance between elements x and y in Banach space and ρ(x, M ) = inf{ x−z : z ∈ M } be the distance of element x from the subset M in this space.By dist H (A 1 , A 2 ) we mean the Hausdorff distance between sets A 1 and A 2 .
Lemma 1.7 (Contraction multimapping principle ( [3], [8])).Let Z be a Banach space.Assume that we are given a multimapping , where the sets Ψ(z) are nonempty and closed for any z ∈ U ε (z 0 ).Further, assume that there exists a number θ, Then, for every number ε 1 which satisfies the inequality

Generalization of p-regularity and description of tangent cones.
In this part we consider a generalization of the concept of p-regularity given in [1], [2].The new necessary optimality conditions for extremum problems with singularities (1.7) was derived by O. Brezhneva and A. A. Tret'yakov.They described the tangent cone T 1 M (x * ) to solution set M (x * ) in singular case (1.7) under an assumption that the mapping F does not satisfy the condition of p-regularity given in Definition 1.2.Under our generalization of p-regularity notion we describe the tangent cones of a new class of mappings.We start from theorem proved in [1] in a modified form.
Let us denote Y 1 (h) = ImF (p) (x * )[h] p−1 and Y 2 (h) is a linear subspace that complements Y 1 (h) with respect to Y and both are closed in Y and let P 2 (h) be the projection operator onto where F (k) (x * ) = 0, k = 1, . . ., p − 1 and there exists an element h ∈ X, for some 1 < α ≤ 3 2 , ε ∈ (0, 1), (2.4) where t ∈ (0, δ) and δ > 0 is sufficiently small.Then As a consequence of Theorem 2.1 we obtain the following corollary which is very important and useful for our next investigations for p = 4.

Corollary 2.2. Let the point x
0 and there exist an element h ∈ X, h = 0 and a number c > 0 such that Consider some examples which illustrate this result.
Example 2.4.Cardioid with one cusp can be described as follows: From the simple calculations we obtain: and we notice that all conditions of Theorem 2.1 are fulfilled, which implies that x * + th + t α h + r + (t) ∈ M (x * ), where The following theorem is the main result of this paper and describes just such a situation for p = 3. (2.12) If there exists some h = 0 such that 4 h and Λ = F (3) [h, h].We will prove that h ∈ T 1 M (x * ).To this end it is sufficient to show that there exists r(α) such that F (x * + αh + α ).For this purpose we construct the following sequence is the right-inverse operator ( [3]).
From the Banach open mapping theorem it follows that (2.17 . We will prove that the operator Ψ(x) described in (2.16) has a fixed point r(α) such that r(α) ∈ Ψ(r(α)) and r(α) = o(α 5 4 ).For this purpose we need to show that Ψ(x) is a contraction in some neighborhood U (0) = {x ∈ X : x ≤ cα ).It means that the Hausdorff distance for Θ ∈ (0, 1) and sufficiently small α > 0. Taking and according to (2.17) the following estimation for distance holds where c(Λ) is Banach's constant depending on α.Using the "mean value" theorem, we get where ∆ ≤ cα and applying Taylor's series, we obtain where γ(α) = o(α 3 ).
Next we will show that Ψ(0) = o(α 4 ).This condition holds if At last we obtain that h ∈ T 1 M (x * ).

Conclusion.
The overall aim of this paper consists in the development and application of p-regularity theory in differential geometry problems related to the theory of curves and spaces.The basic features of p-regularity theory were presented in [5,3,4].In the first part of this paper, we investigated the p-regular methods and procedures by using concepts, models and techniques proposed in [1,2].In the second part, we developed a modified Brezhneva-Tret'yakov theorem in such situation where a tangent cone is not two-sided.Several theoretical examples were studied to illustrate the fundamental notions of p-regularity.

Example 1 . 4 .
As a map F we take(1.20)