This document summarizes research on generalized classical Fréchet derivatives in real Banach spaces. It begins by reviewing basic definitions and results for Fréchet derivatives, including the chain rule, mean value theorem, and Taylor's formula. It then proves that Fréchet derivatives exist and are continuous in real Banach spaces. The main results generalize the chain rule, mean value theorem, and Taylor's formula to higher order Fréchet derivatives in real Banach spaces. Proofs are provided for the generalized chain rule and other theorems.

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ISSN 2581-3463
RESEARCH ARTICLE
On Generalized Classical Fréchet Derivatives in the Real Banach Space
Chigozie Emmanuel Eziokwu
Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria
Received: 01-08-2020; Revised: 20-09-2020; Accepted: 10-10-2020
ABSTRACT
In this work, we reviewed the Fréchet derivatives beginning with the basic definitions and touching most
of the important basic results. These results include among others the chain rule, mean value theorem, and
Taylor’s formula for differentiation. Obviously, having clarified that the Fréchet differential operators
exist in the real Banach domain and that the operators are clearly continuous, we then in the last section
for main results developed generalized results for the Fréchet derivatives of the chain rule, mean value
theorem, and Taylor’s formula among others which become highly useful in the analysis of generalized
Banach space problems and their solutions in Rn
.
Key words: Banach space, continuity, Fréchet derivatives, mean value theorem, Taylor’s formula
2010 Mathematics Subject Classification: 46BXX, 46B25
Address for correspondence:
Chigozie Emmanuel Eziokwu
E-mail: okereemm@yahoo.com
THE USUAL FRECHET DERIVATIVES
Given x a fixed point in a Banach space X and Y another Banach space, a continuous linear operator
S: X→Y is called the Frechet derivative of the operator T: X→Y at x if
T x x T x S x x x
,
and
lim
,
x
x x
x
0
0
Or equivalently,
lim
x
T x x T x S x
x
0
0
This derivative is usually denoted by dT (x) or
T x and T is Frechet differentiable on its domain if
T x exists at every point of the domain as in Abdul[1]
and Argyros[2]
.
Remark: If X = R, Y = R, then the classical derivative
f x of real function f: R→R at x
f x
f x x f x
x
x
lim
0
Is a number representing the slope of the graph of the function f at x where the Frechet derivative of f is
not a number but a linear operator on R into R. Existence of
f x implies the existence of the Frechet
derivative[3]
as the two are related by
f x x f x f x x x g x

2. Eziokwu: Generalized classical fréchet derivatives
AJMS/Oct-Dec-2020/Vol 4/Issue 4 2
While S is the operator which multiplies every δx by the number
f x . In elementary calculus the
derivative at x is a local approximation of f in the neighborhood of x while the Frechet derivative is
interpreted as the best local linear approximation. It is clear from definition that if T is linear, then the
Frechet derivative is linear as well, that is,
dT x T x
THEOREM 1.1:[4]
If an operator has the Frechet derivative at a point, then it has the Gateaux derivative at that point and
both derivatives have equal values.
THEOREM 1.2:[5]
Let Ω be an open subset of X and T Y
: have Frechet derivative at an arbitrary point a of Ω. Then
T is continuous at a. This means that every Frechet differentiable operator defined on an open subset of
a Banach space is continuous.
THEOREM 1.3(CHAIN RULE):[1,6]
Let A, B, and C be real Banach spaces. If S: A→B and T: B→C are Frechet differentiable at x and
U x T S x S x . Then, the higher order Frechet derivatives for real U = To
S can successively be
generated iteratively such that
U x T S x S x
n n n
For n ≥ 2 and integer.
THEOREM 1.5 (IMPLICIT FUNCTION THEOREM)[1,7,8]
Suppose that X, Y, and Z are Banach spaces, C an open subset of X×Y and T: C→Z is continuous, suppose
further that for some x y C
1 1
,
i. T x y
1 1 0
,
ii. The Frechet derivative of T (.,.) when x is fixed is denoted by Ty
(x, y) called the partial Frechet derivative
with respect to y, exists at each point in a neighborhood of (x1
, y1
) and is continuous at (x, y).
iii. T x y B z y
y 1 1
1
, ,
then there is an open subset of X containing x and a unique continuous
mapping y: D→Y such that T(x, y (x)) = 0 and y(x1
)=y1
Corollary 1.6: If in addition to theorem 1.5 Tx
(x, y) also exists in the open set, and is continuous at
(x1
, y1
). Then, F: x→y (x) has Frechet derivative at x1
given by
F x T x y T x y
y x
1 1
1
1 1
, ,
THEOREM 1.7 (Taylor’s Formula for differentiation)[1,9,10]
Let T X Y
: and let a a x
,
be any closed segment lying in Ω. If T is Frechet differentiable at
a, then
T a x T a x x x
x
x
lim
0
0
and
T a h T a T a x T a x x x
x
x
1
2
0
2
0
lim
For twice differentiable functions.

3. Eziokwu: Generalized classical fréchet derivatives
AJMS/Oct-Dec-2020/Vol 4/Issue 4 3
MAIN RESULTS ON GENERALIZED FRECHET DERIVATIVES
Let x be a fixed point in the real Banach space. Also let the continuous linear operator S: X→Y be a real
Frechet derivative of the operator T: X→Y such that
lim
x
T x x T x S x
x
0
0
Then, the higher order Frechet derivative successively can be generated in an iterative manner such that
lim
x
i
i
n
i
i
n
i
i
n
T x x T x x S x
0
1 1
1
1
i
i
i
n
i
i
n
x

4. 1
1
n≥2 and an integer.
THEOREM 2.1 (CHAIN RULE): Let A, B, and C be a unitary spaces, if S: A→B and T: B→C are
Frechet differentiable at z and
u x u s x s x
 . Then, the higher order Frechet derivative for
U x
n
can be generated with U S T
=  generating n n
z U z
if and only if
lim
z
i
i
n
i
i
n
i
i
i
i
n
U z z U z z z
1
0
1 1
1
1
n
n
i
i
n
x
1
lim
x
i
i
n
i
i
n
i
i
n
S T x x S T x x
1
0
1 1
1
 
x
x
i
i
n
i
i
n
1
1
THEOREM 2.2 [Generalized Frechet Mean Value theorem]: Let T: A→B where A is an open convex
set containing a, b, and c is a normed space. T n x
exists for each a a b
, and T x
n
1
is continuous
on [a, b], then
T b T a T a T b T a
n n
x a b
n n n
1 1 2 2
,
sup
THEOREM 2.3 [Generalized Implicit function theorem]
Suppose that A, B, and C are real Banach spaces, D is an open subset of A×B and T: D→C is continuous.
Suppose further that for some a b D
,
, then
i. T a b
n
, 0
ii. The nthFrechet derivative of T (.,.) where x is fixed and denoted by T a b
b
n
1 1 1
, called the nth
partial
derivative with respect to b exists at each point in a neighborhood of (a1
, b1
) and is continuous at a1
, b1
iii. T a b B C B
x
n
1 1
1
, , then there is a subset E of A containing a1
and a unique continuous
mapping S: E→C such that T a b a
n
1 1 1 0
, and S a b
n
1 1

5. Eziokwu: Generalized classical fréchet derivatives
AJMS/Oct-Dec-2020/Vol 4/Issue 4 4
Corollary 2.4: If the addition to conditions of theorem 2.3, T a b
a
n
1
, also exists on the open set, and
is continuous at (a1
, b1
), then F: a→b (a) has the nth Frechet derivative at a1
given by
F a T a b T a b
n
n
n
a
n
1
1
1
1
1 1
, ,
THEOREM 2.5 [Taylors formula for nth Frechet differentiable functions]
Let T X Y
: and a a n x
,
be any closed segment lying in Ω. If T is differentiable in Ω and nth
differentiable at a, then
T a n x T a T a x T a x x
n
T x x
n n n n
1
2
1

!
x
where
lim
x
x
0
PROOF OF MAIN RESULTS
Proof of Theorem 2.1 (chain rule)
Let x x X
, and suppose Un
(x) can be generated with U S T
=  such that the generalized Frechet
derivative
n n
i
i
n
i
i
n
x U x U x x U x
T S x

8. 1 1
0 x
x T S x T x y
i
i
n
i
i
n
i
i
n
1 1 1

10. T yi
i
n

11. 1
where
x S x x S x
i
i
n
i
i
n
i
i
n
1 1 1
Thus
U x x U x T x x z
i i
i
n
n
1
0
since
x S x x x
i
i
n
n
i
i
n
1 1
We get
U x x U x T x S x x
U x x
i
i
n
i
i
n
n n
i
i
n
1 1
0
1
U x T y x T y S x x
i
i
n
n n n
1
In view of the fact that S is continuous at x, we obtain
x x
i
i
n
1

12. Eziokwu: Generalized classical fréchet derivatives
AJMS/Oct-Dec-2020/Vol 4/Issue 4 5
therefore
n n
x
i
i
n
i
i
n
x x
S T x x S T x x
i
i
n
lim
1
0
1 1
 
1
1
1
1

13. x
x
i
i
n
i
i
n
conversely
lim
x
i
i
n
i
i
n
i
i
n
S T x x S T x x x
1
0
1 1
1
  i
i
i
n
i
i
n
x
1
1
implies
T S x x T S x
T S x x x
i
i
n
i
i
n
i i
i
n
 

1
1 1
T S x x
i
i
n
i
i
n

1 1
and
T S x x S T x
T S x x x
i
i
n
i
i
n
i
i
n
 

1
1
1
1
T S x x
i
i
n
i
i
n

1 1
lim
x
i i
i
n
i
i
n
i
T S x x T S x x x
0
1
1
1
 
x
x
x x
i
i
n
x
i
i
n
i
x
i
i n
1
1
0
1
1
0
0
0
lim lim By L'H
Hospital Rule
Hence, U x x
n n
is Frechet differentiable and the proof is complete.
PROOF OF THE GENERALIZED FRECHET MEAN VALUE THEOREM
Let T: K→B where K an open convex set containing is a and b. B is a normal space and T x
n
exists
for each x in [a, b] and
T x is continuous in [a, b] such that
T b T a T x b a
x a b
,
sup
Then by induction for the nth complex iterative Frechet derivative of T, the mean value theorem becomes
T b T a T x T b T a
n n
x a b
n n n
1 1 2 2
,
sup
Proof of Theorem 2.3 [generalized implicit function theorem]
For the sake of convenience, we may take x1
= 0 and x2 0
*
= . let
A T B C B
x
n
2
1
0 0
, ,

14. Eziokwu: Generalized classical fréchet derivatives
AJMS/Oct-Dec-2020/Vol 4/Issue 4 6
Since D is an open set containing 0 0
,
, we find that
D D x C x x D
x
1 2 1 2
,
For all sufficiently small say x1 . For each x1 having this property, we define a function S x x
n
1
1 2
,
D C
x1
→ by S x x x FT x x
n n
1
1 2 2
2
1 2
, , . To prove the theorem, we must prove the existence of a
fixed point for S x x
n
1
1 2
, under the condition that x1 is sufficiently small. Continuity of the mapping
x x x
1 2 1
and x x x
2 1 2
*
. Now,
S x x U U T x x U
x
n
x
2 2
1
1 2 1 2
, ,
and
FF FTx
n
1 1
2
0 0
,
Therefore, assumptions on T(n−1)
guarantees the existence of S(n−1)
(x1
, x2
) for sufficiently small x1 and
x2 and
S x x U F T T x x U
n
x
n
x
n
1
1 2
1 1
1 2
2 2
0 0
, , ,
hence
S x x F T T x x
n
x
n
x
n
1
1 2
1 1
1 2
2 2
0 0
, , ,
Since, Tx
n
2
1
is continuous at (0, 0) there exists a constant L 0 such that
S x x L
n
1
1 2
, (3.3.1)
Or sufficiently small x1 and x2 , we say that x1 1
and x2 2
. Since T n
1
is continuous at
(0, 0), there exists an ε ≤ ε1
such that
S x FT x L
n n
1
1 0 1 0 2 1
, , (3.3.2)
For all x1
with x1 . We now show that S x
n
1
1,. maps the closed ball S x B x
n
1
2 2 2
0
into itself. For this let x1 and x2 2
. Then by the Mean Value theorem and (3.3.1), (3.3.2), we
have
S x x S x x S x S x
n n n n
1
1 2
1
1 2
1
1
1
0 1
0 0
, , , ,
s
sup *
, ,
S x x x S x L L
x
n n
2 2
1 2 2
1
1 2 2
0 1
Therefore, for x S x S
n n
1
1
1
1
2
0
, ,. : . Also for x x S
2 2 2
0
* **
, ;
we obtain by the mean value
theorem of section 2.2 and equation (3.3.1)
S x x S x x S x x x x
n n
x x
n
1
1 2
1
1 2
1
1 2 2
2 2 2
, , ,
* ** sup *
2
2 2 2
** * **
L x x
The Banach contraction mapping theorem guarantees that for each x1
with x1 there exists a unique
x x S n
2 1
1
2
0
such that
x x S x x x x x FT x x x
n n
2 1
1
1 2 1 2 1
1
1 2 1
, ,
That is,
T x x x
n
1
1 2 1 0
,

15. Eziokwu: Generalized classical fréchet derivatives
AJMS/Oct-Dec-2020/Vol 4/Issue 4 7
By the uniqueness of x2
, we have that x2
(0) = 0 since
T n
1
0 0 0
,
Finally, we show that x x x
1 2 1
is continuous for if x1
*
and x1
**
, and then selecting
x x x x
2
0
1 2 1
, **
and x S x x
n
2
1
1 2
0
* *
,
. We have by the error bound for fixed point iteration on the
mapping S x
n
1
1,.
x x x x
L
x x
2 1 2 1 2
0
2
1
1
** *
We can write
x x x x S x x x
S x x x
n
n
2
0
2 2 1
1
1 2 1
1
1 2 1
* ** * **
* **
,
,
T x x x
F T x x x T x x x
n
n n
1
1 2 1
1
1 2 1
1
1 2 1
* **
* **
,
, , *
**
Therefore, by continuity of T x x x x
n
1
2 1 2 1
, ** **
can be made arbitrary small for x x
1 1
** *
− sufficiently
small and hence the proof.
Proof of Corollary 2.4
We set x x x
1 1 1
*
and G x F x x
n n
1 1 1
*
. Then G n
0 0 and
G x T x x x
T x x
n
x
n
x
n
1
1
1
1 2
1
1
1
1 2
2
2
* *
* *
,
1
1
1 2 1
1
1
1
1 2 1
2 1
T x x x G x T x x x
x
n n
x
n
, ,
* * *
and
T x x x G x T x x x
T
x
n n
x
n
n
2 1
1
1 2 1
1
1
1
1 2 1
1
, ,
* * *
x x x G x x T x x
T G
n n
x
n
1 1 2
1
1 1
1
1 2
1
2
* * * * * *
, ,
n
n
x
n
x x x x
1
1
1
1 2 1
1
* *
,
If O1
, O2
are numbers in (0,1), then
T x x x G x T x x x
T
x
n n
x
n
x
2 1
1
1
1 2 1
1
1
1
1 2 1
, ,
sup
* * *
n
n n
x
n
x O x x O G x T x x x
1
1 1 1 2 2
1
1
1
1 2 1
1
* * * * * *
, ,
O O x
n n
x
n
T x O x x O G x T x
1 2 1 2
1
1 1 1 2 2
1
1
1
1
sup * * * *
, ,
x
x G x
n
2
2
1
* *
Thus applying continuity od Tx
n
1
1
, Tx
n
2
1
for 0, we find that
such that on x x
1 1
*
, we
have

16. Eziokwu: Generalized classical fréchet derivatives
AJMS/Oct-Dec-2020/Vol 4/Issue 4 8
G x T x x T x x x
T
n
x
n
x
n
x
1
1
1
1 2
1
1
1 2 1
2 1
* * * *
, ,
2
2 2 1
1
1 2
1
1
1 2
1
1
1
1
n
x
n
x
n
x x T x x T x
* * * *
, ,
*
* * *
,
,
x x
T x x
x
n
2 1
1
1 2
1
1 2
The coefficient of
*
1
∆x can be as small as required as x1 0
*
. Thus,
F x F x T x x T x x x
n n
x
n
x
n
1 1
1
1 2
1
1
1 1 1
2 1
** *
, ,
x x x
2 1 1
* *
Hence,
F x T x x T x x
n n
x
n
1
1
1 2
1
1
1 2
2
* * *
, ,
Proof of Taylor’s formula for nth Frechet differentiable function
The proof of this theorem can be generated as in Carton[11]
and Nasheed.[12]
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