FLETT ’ S MEAN VALUE THEOREM IN TOPOLOGICAL VECTOR SPACES

We prove some generalizations of Flett’s mean value theorem for a class of Gateaux differentiable functions f :X → Y , where X and Y are topological vector spaces. 2000 Mathematics Subject Classification. 58C20, 26E20, 46G05.

. Flett's conclusion implies that the tangent at (η, f (η)) passes through the point (a, f (a)).A recent article by Khan [4], which generalizes the Mean Value Theorem to the context of topological vector spaces, stimulated us to see if there was a similar generalization of Flett's theorem.It turns out that such a generalization does exist.However, even more can be done.This article focuses on three distinct generalizations of Flett's theorem.First, we drop the endpoint condition f (a) = f (b).Second, we consider the case where f is not differentiable at a finite number of points.Third, we drop differentiability.As expected, the conclusion at each step is weaker than the previous conclusion.These generalizations are given in Theorem 2.1.We then place these results in a topological vector space setting replacing ordinary differentiability with Gateaux differentiability.

Results. Let
Following [2] we will say that the graph of f intersects its chord in the extended sense if either there is a number c ∈ (a, b) such that

.2)
We now state some interesting generalizations of Flett's theorem. (2.3) (1) If j = 0, then there exists a point η ∈ (a, b) such that (2) If j ≤ n for some nonnegative integer n and a ∈ J, then there exist n + 1 points η 1 ,η 2 ,...,η n+1 ∈ (a, b) and n + 1 positive numbers α 1 ,α 2 ,...,α n+1 such that (2.5) (3) If j is unbounded and the graph of f intersects its chord in the extended case, then there exist c in (a, b), and two positive numbers δ 1 , δ 2 such that either (2.8) Some remarks are in order.Flett's original result is the case where , that is, the second condition for the graph of f intersecting its chord in the extended sense holds, then (2.9) A proof of item (1) can be found in [1] and a proof of item (3) can be found in [2].In order to prove item (2) we need the following lemma. (2.10) Notice that Lemma 2.2 is a generalization of the mean value theorem.A proof of Lemma 2.2 can be found in [5].

Proof of (2). Consider the function g
(2.11) Note that g is continuous on [a, b] and satisfies the hypotheses of Lemma 2.2.Further where f exists; this implies Applying Lemma 2.2, we get (2.14) After simplifying the previous expression the result follows.
Our main goal is to place Theorem 2.1 in the context of topological vector spaces.However, an exact analogue of Theorem 2.1 for vector-valued functions is not true.For example, to see why item (2.16)Therefore, we have f (0) = (0, 0) = f (2π), that is, the derivatives of f at the endpoints of the closed interval [0, 2π] are equal.Nevertheless, it is not hard to show that the equation has no solution in (0, 2π).
In the sequel, we will let X and Y be Hausdorff topological vector spaces over the field R of real numbers, and A ⊂ X be an open set.Furthermore, we assume that Y has a continuous dual Y * .A function f : A → Y is said to be Gateaux differentiable at x 0 ∈ A if there exists a mapping from X into Y , denoted by f (x 0 ), such that, given any z ∈ X and a balanced neighborhood V of 0 in Y , there exists a δ > 0 satisfying whenever 0 < |t| < δ; f (x 0 ) is called the Gateaux derivative of f at x 0 and we write Note that a Gateaux differentiable function need not be continuous.For example, the function f : R 2 → R, given by → Y is said to intersect its chord in the extended sense if either there is an l in (0, 1) and a u ∈ Y * such that We note that if Y = R, then we may choose u = 1 and • , • is merely multiplication.In this case, the previous condition reduces to the condition where the graph of f intersects its chord in the extended sense.
Finally, if j is unbounded and f intersects its chord in the extended sense for some u ∈ Y * , then the graph of φ intersects its chord in the extended sense and so the third item in Theorem 2.3 follows directly from the third item in Theorem 2.1.This completes the proof.

Theorem 2 . 3 .
Let X, Y be Hausdorff topological vector spaces over the field R of real numbers, let A ⊂ X be an open set and let Y * denote the continuous dual of Y .Let f : A → Y be a function continuous on the line segment [x 0