Extrema Constrained by C^{k} Curves

Oltin Dogaru and Vlad Dogaru

Abstract

In this paper we generalize the results obtained in Ref. [17].

§1 raises the following problem: what conection there exists between the local extrema of the function f : D ⊂ R^{p} → R and the local extrema of the functions f ∘ α, α ∈ Γ, where Γ is a given family of parametrized curves ?

§2 proves the existence of a C^{k} curve containing a given sequence of points. §3 solves the problem which was presented in §1 in the case of C^{k} curves.

Mathematics Subject Classification: 49K99 Key words: Minima constrained by C^{k} curves, C^{k} curves by sequences.

1 Introduction

Let us consider the extremum problem

minf(x), subject to x ∈ M,

where M is a subset of R^{p} with a given structure. If M is an open set of R^{p} which coincides to the domain of f, then the extremum problem is called unconstrained; in any other case, the extremum problem is called constrained. Such problems, in which M is a C^{k}, k ≥ 2, finite–dimensional differentiable were developed recently in Refs. [1]–[4], [6]–[9].

The extremum conditions (necessary and sufficient) depend on the fashion of defining the subset M. If M is a differentiable manifold, then they depend also on the geometrical structure of M. Frequently, M is considered as the union of a family of its subsets (a plane as the union of straight lines, an integral manifold of a Pfaff system as the union of some integral curves, and so on, Refs. [10]–[16]). Then the following two problems arise:

1) Let D be an open subset in R^{p} and {A_{i}}_{i∈I} be a family of subsets of D having a common point x_{∗} ∈ A_{i}, ∀i ∈ I. Suppose x_{∗} is a local minimum point for each restriction f |_{A_{i}} of the function f : D → R to the subset A_{i}, i ∈ I. Is x_{∗} a local minimum point of f ?

2) Let f : D ⊂ R^{p} → R and α_{i} : I_{i} ⊂ R → D, i ∈ J a family of parametrized curves. What connection we have between of functions f ∘ α_{i}, the extrema of the restrictions f |_{α_{i}(I_{i})}, and the extrema of the function f ?

Balkan Journal of Geometry and Its Applications, Vol.4, No.1, 1999, pp. 45-52 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

The problem 1) and the problem 2), for the C^{1} or C^{2} curves, were solved in Ref. [17]. In this paper we shall solve the problem 2) for the general case of C^{k} curves and even for the analytic curves. For that reason we shall recall some notions about the curves. Definition 1.1. Let I ⊂ R be an interval. A function α : I → R^{p} of class C^{k}, k ≥ 1, is called parametrized curve of class C^{k} and is denoted by α. We shall say that:

1) α passes (just once) through the point x_{∗} ∈ R^{p} if there exists (only one) t_{0} ∈ IntI such that α(t_{0}) = x_{∗};

2) α is a simple parametrized curve if α is injective;

3) α is regular at the point x_{∗} = α(t_{0}) if α′(t_{0}) ≠ 0;

4) α has a tangent at the point x_{∗} = α(t_{0}) if there exists m ∈ \overline{1, k}, such that α^{(m)}(t_{0}) ≠ 0. Definition 1.2. Two parametrized curves α : I → R^{p}, β : J → R^{p} of class C^{k} are called equivalent if there exists a diffeomorphism h : I → J of class C^{k} such that α = β ∘ h. We shall write α ∼ β. Definition 1.3. The set α̃ of C^{k} parametrized curves equivalent to α : I → R^{p} is called curve of class C^{k}. The curve α̃ has qualities 1) to 4) in the Definition 1.1, if a representative α has these properties.

From now we shall refer to a function f : D → R, where D is an open subset in R^{p}. Definition 1.4. Let f : D → R, let x_{∗} ∈ D, and α : I → D be a parametrized curve passing through x_{∗}. We shall say that

1) x_{∗} is a minimum point for f constrained by α if for any t_{0} ∈ I, with α(t_{0}) = x_{∗}, the point t_{0} is a local minimum point for f ∘α, i.e., there exists a neighborhood I_{t_{0}} ⊂ I of t_{0} such that

f(x_{∗}) = f(α(t_{0})) ≤ f(α(t)), ∀t ∈ I_{t_{0}}.

2) x_{∗} is a minimum point for f constrained by α̃ if there exists a neighborhood V of x_{∗} such that

f(x_{∗}) ≤ f(x), ∀x ∈ V ∩ α(I).

Remark. If x_{∗} is a minimum point of f constrained by the curve α̃, then x_{∗} is a minimum point of f constrained by the parametrized curve α. The converse is not true even so α is a simple parametrized curve. However, in case that α : I → D is a simple and regular parametrzied curve and I is a compact set, both notions coincide. Definition 1.5. Let Γ_{x_{∗}} be a family of parametrized curves (curves) passing through the point x_{∗}. We shall say that x_{∗} is a minimum point of f constrained by the family Γ_{x_{∗}} if x_{∗} is a minimum point of f constrained by every curve of the family Γ_{x_{∗}}.

2 C^{k} curves by given sequences of points

The aim of this paragraph is to show that certain conditions assume the existence of C^{k} curves which contain a given sequence of points. To this we recall shortly the prolongation theorem of Whitney (Ref. [5]).

Let K ⊂ R^{p} be a compact set , k = (k_{1}, …, k_{p}) be a multiindex and |k| = k_{1} + … + k_{p}. A family of continuous functions F = (f^{k})_{|K|≤m}, f^{k} : K → R, is called jet of order m. Denote F (x) = f^{0}(x), x ∈ K and D^{k}F = (f^{k+l})_{|l|≤m−|k|}, |k| ≤ m. Naturally, for any function g ∈ C^{m}(K) one can define the jet

J^{m}(g) = (\frac{∂^{k}g}{∂x^{k}} )_{|k|≤m} .

For any x ∈ R^{p} and a fixed a ∈ K, we introduce the Taylor polynomial function

T_{a}^{m}F (x) = ∑_{|k|≤m} \frac{(x − a)^{k}}{k!} f^{k}(a).

Denote

T̃_{a}^{m}F = J^{m}(T_{a}^{m}F), R_{a}^{m}F = F − T̃_{a}^{m}F.

Prolongation Theorem 2.1. Let F = (f^{k})_{|k|≤m}. There exists a function f ∈ C^{m}(R^{p}) with g^{m}(f) = F if and only if

(R_{x}^{m}F)^{k}(y) = O(|x − y|^{m−|k|},

when |x − y| → 0, for any x, y ∈ K and any |k| ≤ m.

In the sequel we shall apply the preceding theorem in the case p = 1 and K = {0} ∪ {t_{n}|n ∈ N}; where t_{n} ∈ R and t_{n} → 0. Corollary 2.1. Let us consider k ∈ N^{∗}. Given the real sequences t_{n} → 0, x_{n}^{(0)} → 0, x_{n}^{(i)} → a^{(i)}, i = \overline{1, k}, there exists f ∈ C^{k}(R) with f^{(i)}(t_{n}) = x_{n}^{(i)}, ∀i ∈ \overline{0, k}, if and only if

(∗)  \frac{x_{m}^{(p)} − ∑_{i=p}^{k−1} \frac{(t_{m} − t_{n})^{i−p}}{(i − p)!} x_{n}^{(i)}}{(t_{m} − t_{n})^{k−p}} → \frac{a^{(k)}}{(k − p)!}

for m, n → ∞ and for any p ∈ \overline{0, k − 1}. Lemma 2.1. (Ref. [17]) Let (x_{n}), (y_{n}) be two sequences of real numbers such that

1) x_{n} ≠ 0, x_{n} ≠ x_{n+1}, ∀n ∈ N;

2) there exists lim_{n→∞} \frac{y_{n}}{x_{n}} = r;

3) there exists λ > 0 with |\frac{x_{n}}{x_{n+1}} − 1| ≥ λ, ∀n ∈ N.

Then the sequence \frac{y_{n+1} − y_{n}}{x_{n+1} − x_{n}} is convergent towards r.

Lemma 2.2. If (x_{n}) is a sequence of strictly positive real numbers and x_{n+1} ≤ \frac{1}{2^{k}} x_{n},

∀n ∈ N, where k ∈ N^{∗}, then there exists μ > 0 such that |\frac{x_{n} − x_{m}}{(x_{n}^{1/k} − x_{m}^{1/k})^{k}} | ≤

μ, ∀m, n ∈ N, m ≠ n.

Proof. For m = n + p, p ≥ 1, it follows x_{n}^{1/k} ≤ \frac{1}{2^{p}} x_{m}^{1/k}. Denote t_{n} = x_{n}^{1/k}. We have

\frac{x_{n} − x_{m}}{(x_{n}^{1/k} − x_{m}^{1/k})^{k}} = \frac{t_{n}^{k} − t_{m}^{k}}{(t_{n} − t_{m})^{k}} = \frac{t_{n}^{k−1} + … + t_{m}^{k−1}}{(t_{n} − t_{m})^{k−1}} ≤

≤ \frac{t_{n}^{k−1} (1 + \frac{1}{2^{p}} + … + (\frac{1}{2^{p}} )^{k−1})}{t_{n}^{k−1} (1 − \frac{1}{2^{p}} )^{k−1}} = \frac{1 − (\frac{1}{2^{p}} )^{k}}{(1 − \frac{1}{2^{p}} )^{k}} < \frac{1}{(1 − \frac{1}{2})^{k}} .

Lemma 2.3. Let (x_{n}), (y_{n}) two sequences of real numbers such that (x_{n}) is strictly monotone,

y_{n} → 0, \frac{y_{n}}{x_{n}} → 0, |x_{n+1}| ≤ \frac{1}{2^{k}} |x_{n}|, ∀n ∈ N

where k ∈ N^{∗}. Then \frac{y_{m} − y_{n}}{x_{m} − x_{n}} → 0 for m, n → ∞.

Proof. Suppose, for example, x_{n} > 0, ∀n ∈ N. Let b_{n} = \frac{y_{n+1} − y_{n}}{x_{n+1} − x_{n}}. From Lemma

2.1 it follows b_{n} → 0. Let m > n and μ_{mn} = max{|b_{n}|, …, |b_{m−1}|}. Then, for any

i ∈ \overline{n, m − 1} we have |\frac{y_{i+1} − y_{i}}{x_{i+1} − x_{i}} | ≤ μ_{mn}, namely

−μ_{mn} ≤ \frac{y_{i+1} − y_{i}}{x_{i} − x_{i+1}} ≤ μ_{mn}

and therefore

−μ_{mn} ∑_{i=n}^{m−1}(x_{i} − x_{i+1}) ≤ ∑_{i=n}^{m−1}(y_{i+1} − y_{i}) ≤ μ_{mn} ∑_{i=n}^{m−1}(x_{i} − x_{i+1}).

It results

|\frac{y_{m} − y_{n}}{x_{m} − x_{n}} | ≤ μ_{mn}.

Now the conclusion is obvious. Lemma 2.4. Let (x_{n}), (y_{n}) be two sequences of real numbers such that (x_{n}) is strictly

monotone, y_{n} → 0, \frac{y_{n}}{x_{n}} → 0, |x_{n+1}| ≤ \frac{1}{2^{k}} |x_{n}|, ∀n ∈ N, where k ∈ N^{∗}. Then,

there exist two functions f, g ∈ C^{k}(R) and a sequence (t_{n}) of real numbers such that t_{n} → 0, f(t_{n}) = x_{n}, g(t_{n}) = y_{n}, f^{(i)}(0) = 0, ∀i ∈ \overline{0, k − 1}, g^{(i)}(0) = 0, ∀i ∈ \overline{0, k} and f^{(k)}(0) ≠ 0.

Moreover, the function f and the sequence (t_{n}) do not depend on the sequence (y_{n}). Proof. Let t_{n} = x_{n}^{1/k}. Then, the function f(x) = x^{1/k} and the sequence (t_{n}) satisfy the required properties. In order to get the function g we should apply the Corollary 2.1 to t_{n} = x_{n}^{1/k}, x_{n}^{(0)} = y_{n}, x_{n}^{(i)} = 0 and a^{(i)} = 0, ∀i ∈ \overline{1, k}. Obviously, the condition (∗) from the statement (Corollary 2.1) is fulfilled, for any p ≥ 1. For p = 0 this

condition becomes \frac{y_{m} − y_{n}}{(t_{m} − t_{n})^{k}} → 0, for m, n → ∞. Taking into account that

\frac{y_{m} − y_{n}}{(t_{m} − t_{n})^{k}} = \frac{y_{m} − y_{n}}{x_{m} − x_{n}} · \frac{x_{m} − x_{n}}{(t_{m} − t_{n})^{k}}

and using the Lemmas 2.2 and 2.3, it results \frac{y_{m} − y_{n}}{(t_{m} − t_{n})^{k}} → 0, for m, n → ∞.

Thus we obtain a function g ∈ C^{k}(R) which satisfies the required properties. Theorem 2.2. Let (x_{n}) be a sequence of distinct points of R^{p}, convergent to the point a ∈ R^{p}. Then, for any k ∈ N^{∗}, there exist a subsequence (x_{n_{m}}) and a parametrized curve α : R → R^{P} of class C^{k}, which contains the set of points {x_{n_{m}}, a} such that α has a tangent at the point a. Moreover, if a = α(t_{0}), then α^{(i)}(t_{0}) = 0, ∀i ∈ \overline{1, k − 1}, α^{(k)}(t_{0}) ≠ 0 and there exists a sequence (t_{m}) with t_{m} → t_{0} and α(t_{m}) = x_{n_{m}}. Proof. By a translation, we can suppose a = (0, …, 0) ∈ R^{p}. Since u_{n} = \frac{x_{n}}{||x_{n}||}

is bounded, considering it likely a subsequence, we can asssume u_{n} → u ∈ R^{p}. By a rotation, we can suppose u = (1, 0, …, 0). Consequently, if x_{n} = (x_{u}^{1}, …, x_{n}^{p}), it follows

\frac{x_{n}^{1}}{|x_{n}^{1}|\sqrt{1 + (\frac{x_{n}^{2}}{x_{n}^{1}} ) + … + (\frac{x_{n}^{p}}{x_{n}^{1}} )}} → 1.

Hence x_{n}^{1} > 0 for sufficiently large n and \frac{x_{n}^{i}}{x_{n}^{1}} → 0, ∀i ∈ \overline{2, p}. Obviously, there exists a

subsequence (x_{n_{m}}) such that x_{n_{m+1}}^{1} > 0 and x_{n_{m+1}}^{1} ≤ \frac{1}{2^{k}} x_{n_{m}}^{1}, ∀m ∈ N.

Apllying Lemma 2.4 to the pair of sequences (x_{n_{m}}^{1}) and (x_{n_{m}}^{i}), i ∈ \overline{2, p}, we obtain the functions φ_{i} : R → R, i ∈ \overline{1, p} of class C^{k} and a sequence (t_{m}) of real numbers such that t_{m} → 0, φ_{i}(t_{m}) = x_{n_{m}}^{i}, i ∈ \overline{1, p}, φ_{i}^{(j)}(0) = 0, i ∈ \overline{1, p}, j ∈ \overline{0, k − 1}, φ_{i}^{k}(0) = 0, i ∈ \overline{2, p} and φ_{1}^{(k)}(0) ≠ 0. Then, the parametrized curve α(t) = (φ_{1}(t), …, φ_{p}(t)), t ∈ R, has the required properties.

3 Minimum constrained by C^{k} curves

Let D be an open subset in R^{p} and x_{∗} ∈ D. For any k ∈ N^{∗} we denote by Γ_{x_{∗}}^{k} the family of all C^{k} parametrized curves passing just once through the point x_{∗}, each having a tangent at x_{∗}. Let

A_{x_{∗}}^{k} = \{α ∈ Γ_{x_{∗}}^{k}|α(t_{0}) = x_{∗}, α^{(i)}(t_{0}) = 0, i ∈ \overline{1, k − 1}, α^{(k)}(t_{0}) ≠ 0\}

and

B_{x_{∗}}^{k} = \{α ∈ Γ_{x_{∗}}^{k}|α(t_{0}) = x_{∗}, ∃m ∈ \overline{1, k − 1} such that α^{(m)}(t_{0}) ≠ 0\} , for k ≥ 1.

Theorem 3.1. Let f : D → R. Then, x_{∗} is a local minimum point of f, if and only if, there exists k ∈ N^{∗} such that x_{0} is a minimum point constrained by the family A_{x_{∗}}^{k}. Proof. We can suppose f(x_{∗}) = 0. If x_{∗} would not be a local minimum point of f, then there exists a sequence of distinct points (x_{n}) of R^{p}, with x_{n} → x_{∗} and f(x_{n}) < 0, ∀n ∈ N. Taking into account the Theorem 2.2 we find a curve α ∈ A_{x_{∗}}^{k} such that x_{∗} is not a minimum point of f constrained by α. Contradiction.

It is interesing to remark that Theorem 3.1 does not impose any condition upon the function f. Then, more surprising is the fact that Theorem 3.1 fails for the family B_{x_{∗}}^{k} or for the family of all analytic curves passing through the point x_{∗}, even if f is of class C^{∞}.

Examples. 1) Let f : R^{3} → R,

f(x, y) = (y^{k} − x^{k+1})(y^{k} − 2^{k}x^{k+1}),

where k ∈ N, k ≥ 1. It is obvious that f is of class C^{∞} and that the critical point x_{∗} = (0, 0) is not a local minimum point of f. Let us show that x_{∗} = (0, 0) is a minimum point of f constrained by the family B_{x_{∗}}^{k}.

Let D^{−} = {(x, y) ∈ R^{2}|f(x, y) < 0}. For any (x, y) ∈ D^{−} it results

(∗)  |\frac{y}{x}| < 2|x|^{1/k}

and

(∗∗)  \frac{|y|}{|x|^{(m+1)/m}} > |x|^{−1/m(m+1)},

with |x| < 1 and m ∈ \overline{1, k − 1}.

Let α ∈ B_{x_{∗}}^{k}. We can suppose x_{∗} = α(0). If, by reductio ad absurdum, the point x_{∗} would not be a minimum point of f constrained by the parametrized curve α, then would exist a sequence (t_{n}) with t_{n} → 0 and α(t_{n}) = (x_{n}, y_{n}) ∈ D^{−}, ∀n ∈ N. Consequently, the numbers x_{n} and y_{n} satisfy the above conditions (∗) and (∗∗). Obviously, if α(t) = (x(t), y(t)), then x(t) = t^{m}(a + tf(t)) and y(t) = t^{m}(b + tg(t)) where a^{2} + b^{2} > 0, m ∈ \overline{1, k − 1} and f, g are continuous functions. We assume that

a ≠ 0. Hence, |\frac{y_{n}}{x_{n}} | = \frac{|b + t_{n}g(t_{n})|}{|a + t_{n}f(t_{n})|}. Using the relation (∗), we get that |\frac{y_{n}}{x_{n}} | → 0 and

therefore b = 0. Hence,

\frac{|y_{n}|}{|x_{n}|^{(m+1)/m}} = \frac{g(t_{n})}{|a + t_{n}f(t_{n})|^{(m+1)/m}} → |g(0)|.

On the other hand, using the relation (∗∗), it results that \frac{|y_{n}|}{|x_{n}|^{(m+1)/m}} → ∞. Con-

tradiction. Now, we assume that a = 0. Then,

|\frac{x_{n}}{y_{n}} | = \frac{|t_{n}||f(t_{n})|}{|b + t_{n}g(t_{n})|} → 0,

which is a contradiction to the relation (∗).

2) Let g : R → R, g(x) = e^{−1/x^{2}}, for any x > 0 and g(x) = 0 for any x ≤ 0. Let f : R^{2} → R, f(x, y) = y(y − g(x)) which is of class C^{∞}. Also, it is obvious that the critical point x_{∗} = (0, 0). Let us show that x_{∗} = (0, 0) is a minimum point of f constrained by the family Γ_{x_{∗}}^{ω}, where Γ_{x_{∗}}^{ω} is the family of all analytic parametrized curves passing through the point x_{∗}.

Let D^{−} = {(x, y)|f(x, y) < 0}. It follows that for any (x, y) ∈ D^{−} we have x > 0 and

(∗)  0 < ye^{1/x^{2}} < 1

Let α ∈ Γ_{x_{∗}}^{ω}. We can suppose x_{∗} = α(0). If, by reductio ad absurdum, the point x_{∗} would not be a minimum point of f constrained by α, then would exist a sequence (t_{n}) with t_{n} → 0 and α(t_{n}) = (x_{n}, y_{n}) ∈ D^{−}, ∀n ∈ N. Hence, the numbers x_{n} and y_{n} satisfy the above condition (∗) and x_{n} → 0, y_{n} → 0. Obviously, if α(t) = (x(t), y(t)), then x(t) = t^{p}(a + …) and y(t) = t^{q}(b + …), with ab ≠ 0. It follows that

y_{n}e^{1/x_{n}^{2}} = [(x_{n}^{2})^{q/2p}e^{1/x_{n}^{2}}]y_{n}(x_{n}^{2})^{−q/2p} → ∞ · \frac{|b|}{|a|^{q/p}} = ∞,

which is in contradiction to the relation (∗).

In the following, we shall denote by Γ̃_{x_{∗}}^{k} the family of all C^{k} curves passing through the point x_{∗}, regular at x_{∗}. Theorem 3.2. Let f : D ⊂ R^{p} → R. If there exists k ∈ N^{∗} such that for any α̃ ∈ Γ̃_{x_{∗}}^{k} the point x_{∗} is an extrema point of f constrained by α̃, then x_{∗} is a local extrema point of f. Proof. Let us suppose that x_{∗} is not a local extrema point for f and f(x_{∗}) = 0. Then, there exist two sequences (x_{n}) and (y_{n}) of distinct points of D with x_{n} → x_{∗}, y_{n} → x_{∗}, f(x_{n}) < 0 and f(y_{n}) > 0, ∀n ∈ N. By the Theorem 2.2 there exist two subsequences (x_{n_{m}}) and (y_{n_{r}}), two C^{k} parametrized curves α and β, and two sequences of real numbers (t_{m}) and (t′_{r}) with t_{m} → 0, t′_{r} → 0, t_{m} > 0 t′_{r} > 0 such that α(t_{m}) = x_{n_{m}} and β(t′_{r}) = y_{n_{r}}, ∀m, r ∈ N. Then, it is easy to show that there exists a parametrized curve γ : R → R^{p} of class C^{k} such that γ(t) = α(t), ∀t ≤ 1, γ(t) = β(1/t), ∀t ≥ 3, γ(2) = x_{∗} and γ′(2) ≠ 0. It follows that γ̃ ∈ Γ̃_{x_{∗}}^{k} and γ̃ contains the points x_{n_{m}} and y_{n_{r}}, ∀m, r ∈ N. Hence, the point x_{∗} is not a local extrema point of f.

References

[1] R.L. Bishop, R.I. Crittenden, Geometry of Manifolds, Academic Press, New York, New York, 1964.

[2] D. Gabay, Minimizing a Differentiable Function over a Differentiable Manifold, Journal of Optimization Theory and Applications, Vol.37, pp.177-219, 1982.

[3] N.I. Hicks, Notes of Differential Geometry, Van Nostrand Publishing Company, Princeton, New Jersey, 1965.

[4] D.G. Luenberger, Introduction to Linear and Nonlinear Programming, Addison-Wesley Publishing Company, Reading, Massachusetts, 1973.

[5] B. Malgrange, Ideals of Differenatiable Functions, Oxford University Press, Oxford, England, 1964.

[6] J.N. Milnor, Morse Theory, Princeton University Press, Princeton, New Jersey, 1969.

[7] T. Rapcsak, Minimum Problems on Differentiable Manifolds, Optimization Vol.20, pp. 3-13, 1989.

[8] M. Spivak, A Comprehensive Introduction to Differential Geometry I-V, Vols. 1-5, Publish or Perish, Berkeley, California, 1979.

[9] C. Udrişte, Convex Functions and Optimization Methods on Riemannian Manifolds, Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, Holland, Vol.297, 1994.

[10] C. Udrişte, O.Dogaru, Extrema with Nonholonomic Constraints, Buletinul Institutului Politehnic Bucuresti, Seria Energetica, Vol. 50, pp.3-8, 1988.

[11] C. Udrişte, O.Dogaru, Mathematical Programming Problems with Nonholononic Constraints, Seminarul de Mecanică , Universitatea din Timişoara, Vol. 14, 1988.

[12] C. Udrişte, O.Dogaru, Extreme Condiţionate pe Orbite, Buletinul Institutului Politehnic Bucureşti, Seria Mecanică, Vol.51, pp. 3-9, 1989.

[13] C. Udrişte, O.Dogaru, Convex Nonholonomic Hypersurfaces, The Mathematical Heritage of C.F.Gauss, Edited by G.M.Rassias, World Scientific Publishers, Singapore, pp. 769-784. 1990.

[14] C. Udrişte, O.Dogaru,, I. Ţevy, Sufficient Conditions for Extremum on Differentiable Manifolds, Proceedings of the 22nd Conference on Differential Geometry and Topology Applications in Physics and Technics, Bucharest, Romania, 1991.

[15] C. Udrişte, O.Dogaru, I. Ţevy, Extrema Points Associated to a Pfaff Form, Proceedings of the International Conference on Differential Geometry and Its Applications, Bucharest, Romania, 1992.

[16] C. Udrişte, O.Dogaru, I. Ţevy, Open Problems in Extrema Theory, Workshop on Differetial Geometry and Applications, Bucharest, Romania, 1993.

[17] O.Dogaru, I. Ţevy, C.Udrişte, Extrema Constained by a Family of Curves and Local Extrema, Journal of Optimization Theory and Its Applications, vol.97, nr. 3(1998), 605-621.

University Politehnica of Bucharest Department of Mathematics I Splaiul Independenţei 313 77206 Bucharest, Romania