ISSN 2581-3463
REVIEW ARTICLE
On Frechet Derivatives with Application to the Inverse Function Theorem of Ordinary Differential Equations Eziokwu C. Emmanuel Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria
Received: 25-10-2019; Revised: 15-11-2019; Accepted: 01-01-2020 ABSTRACT In this paper, the Frechet differentiation of functions on Banach space was reviewed. We also investigated that it is algebraic properties and its relation by applying the concept to the inverse function theorem of the ordinary differential equations. To achieve the feat, some important results were considered which finally concluded that the Frechet derivative can extensively be useful in the study of ordinary differential equations. Key words: Banach space, Frechet differentiability, Lipschitz function, inverse function theorem 2010 Hematics Subject Classification: 46B25
INTRODUCTION The Frechet derivative, a result in mathematical analysis, is derivative usually defined on Banach spaces. It is often used to formalize the functional derivatives commonly used in physics, particularly quantum field theory. The purpose of this work is to review some results obtained on the theory of the derivatives and apply it to the inverse function theorem in ordinary differential equations.[1-5]
DERIVATIVES Definition 1.1 (Kaplon [1958]) Let f be an operator mapping a Banach space X into a Banach space Y. If there exists a bounded linear operator T from X into Y such that
fx ( 00+∆ x) − fx( ) − T() ∆ x lim = 0 (1.1) ∆→x 0 ∆x or, fx( +− thfx) ( ) lim = Thx ( ) t →0 t or fx( +− y) fx( ) − T (y) lim = 0 (1.1) y →0 y
then P is said to be Frechet differentiable at x0, and the bounded linear operator
, Px( 0 ) = T (1.2)
Address for correspondence: Eziokwu C. Emmanuel E-mail: [email protected]
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is called the first Frechet – derivative of f at x0. The limit in (1.1) is supposed to hold independent of the way that ∆x approaches 0. Moreover, the Frechet differential
, fx ( 00 , ∆= x) f( x) ∆ x
is an arbitrary close approximation to the difference fx ( 00+∆ x) − f () x relative to ∆x , for ∆x small.
If f1 and f2 are differentiable at x0, then ′ ′′ ( f1+=+ f 2) ( x 0) fx 10( ) fx 20( )
Moreover, if f2 is an operator from a Banach space X into a Banach space Z, and f1 is an operator from Z into a Banach space Y, their composition ff12 is defined by
( f1 f 2)( x) = f 12( f( x)) , for all x ∈ X
We know that ff12 is differentiable at x0 if f2 is differentiable at x0 and f1 is differentiable at f2 (x0) of Z, with (chain rule): ′ ′′ ( f1 f 2) ( x) = f 1( fx 20( )) f 2 () x 0 To differentiate an operator f we write: fx( 0+∆ x) − fx( 00) = Tx( , ∆ x) ∆ x + ( x 0 , ∆ x),
where Tx(,0 ∆ x ) is a bounded linear operator for given xx0 , ∆ with
limTx (0 , ∆= x) T (1.3) ∆→x 0 and
(xx0 , ∆ ) lim = 0 (1.4) ∆→x 0 ∆x
Estimate (1.3) and (1.4) give
limTx (0 , ∆= x) f′ ∆→x 0
If Tx(0 , ∆ x) is a continuous function of ∆x in some ball ∪>(0, RR),( 0), then
Tx ( 00 , 0) = f′ ( x ) We now present the definition of a mosaic: Higher – orderer derivatives can be defined by induction: Definition 1.2 (Argyros [2005]) If f is (m – 1) – times Frechet – differentiable (m ≥ 2 an integer) and an m – linear operator A from X into Y exists such that (mm−− 1) ( 1) fxxfxAx( 00+∆ ) − ( ) − () ∆ lim = 0 (1.5) ∆→x 0 ∆x then A is called the m – Frechet – derivative of f at x0, and ()m Af= () x0 (1.6)
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Definition 1.3 (Koplan [1958]) nm Suppose fU:�⊆→��RR� where U is an open set, the function f � is classically differentiable at xU0 ��∈ if
∂fi The partial derivatives of f, fori=…=… 1 mandj 1 nexistsatx0 , ∂x j ∂fi mn x The Jacobean matrix Jx( 00) = ( x) ∈ R satisfies ∂x j
fx( ) −− fx( 00) Jx( o )( x − x) lim = 0 . xx → 0 xx− 0
We say that the Jacobean matrix Jx()0 is the derivative of f at x0 that is called total derivative Higher partial derivatives in product spaces can be defined as follows: Define
Xij = TX(j , X i ) (1.7)
where X1, X2,… are Banach spaces and TX(ji , X) is the space of bounded linear operators from X j into
X i . The elements of X ij are denoted by Tij , etc. Similarly,
Xijm = TX( m , X ij ) = TX(jk , X ij12 j …− jm 1), (1.8) which denotes the space of bonded linear operators from X jm into Xij1 j 2 … jm – 1 . The elements
A = Aij12 j … jm are a generalization of m – linear operators.
Consider an operator f1 from space n Xx= ∏ jp (1.9) p=1 into X i , and that fi has a partial derivative of orders 1,2,… , m – 1 in some ball ∪(xR0 , ), where R > 0 and
(00) ( ) ( 0) x0=( xxj 12, j, …∈ , x jn ) X
For simplicity and without loss of generality, we (Frechet [1906]) remember the original spaces so that j12 = 1, j =…= 2, , jnn hence, we write
(00) ( ) ( 0) x0 =( xx1, 2 , … , xn )
A partial derivative of order ( m – 1 ) of fi at x0 is an operator
(m− 1) ∂ fxi ()0 Aiq12 q…− qm 1= (1.10) ∂xxq12 ∂ q …∂ x qm− 1
(in X iq12 q…− qm 1) where
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Let PX()qm denote the operator from X qm into X iq1 q2…− qm 1 obtained by letting
(0) xjj= x , jq ≠ m for some qmm, 1 ≤≤ qn . Moreover, if
mm−1 ' (0) ∂∂fxii()00 ∂ fx () Px( qm ) = . = , (1.11) ∂xqm ∂ xx q12 ∂ q …∂ x qm− 1 ∂ x q 1 …∂ x qm exists, it will be called the partial Frechet – derivative of order ��m of fi with respect to xxq1 ,… , qm at x0
Furthermore, if fi is Frechet – differentiable m times at x0 , then
m m ∂ fxi ( 0 ) ∂ fxi ()0 xxq1…= qm xxs1… sm (1.12) ∂xq1 …∂ x qm ∂ xxs12 ∂ q …∂ x sm
For any permutation ss12, ,… , sm of integers qq12, ,… , qm and any choice of point xxq1,,… qm , from
XXq1,,… qm , respectively. Hence, if ff = (1 , … , ft ) is an operator from XX = 12 + X +…+ Xn into
YY = 12 + Y +…+ Yt then
m ()m ∂ fi fx( 0 ) = (1.13) ∂xxj1 …∂ jm xx= 0
(00) ( ) ( 0) i =… 1 , 2, , tj, 12 , j , …=… , jm 1 , 2, n is called the m – Frechet derivative of F � at x 0 = (xx12 , ,… , xn ) .
Integration[6] In this subsection, we due to (Pantryagin [1962]) state results concerning the mean value theorem, Taylor’s theorem and Riemannian integration without the proofs. Hence the mean value theorem for differentiable real functions � f is stated as follows: fb( ) – fa ( ) = f ’( c ) ( b – a ),
Where c ∈(ab , ), does not hold in a Banach space setting. However, if F is a differentiable operator between two Banach spaces XY and , then
fx( ) −≤ fy ( ) sup f′ ( x ) . x − y x ∈ L (,) xy where
T( xy,) = { z : z = y +−( 1 ) x , 0 ≤≤ 1}
Set
zx() = y +− ( 1 ) , 0 ≤≤ 1,
Divide the interval 0≤≤ 1 into n subintervals of lengths ∆=…i , in 1, 2, , , choose points i� inside corresponding subintervals and as in the real Riemann integral consider sums
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n ∑∑ff(ii)∆= ( ii) ∆ i=1 where is the partition of the interval, and set
= max ∆ i (i)
Definition 1.4 (Ince [1956]) If
Sf= lim (ii)∆ →0 ∑ exists, then it is called the Riemann integral from f () on [0,1], denoted by 1 y Sfd= ∫∫() = fd( ) 0 x Definition 1.5[7] A bounded operator P () on [0,1] such that the set of points of discontinuity is of measure zero is said to be integrable on [,01].
Theorem 1.1 (Day [1973]) ()m If F � is m – times Frechet – differentiable in Ux(0 , R ), R > 0 , and fx() is integrable from x to any
∈U(xR0 , ) then
m−1 1 (n) n f( y) = f( x) + ∑ f( x)( y −+ x) Rm ( xy , ), (1.14) n=1 n!
m−1 m 1 n yx− fy( ) − ∑ f(nm) ( xy)( −≤ x) sup f( ) ( x) , (1.15) n=0 nm!!x ∈ L ( xy, ) where 1 m−1 ()mm(1− ) R xy,= f ( y +− 1 x ) ( y − x ) d (1.16) m ( ) ∫ ( ) 0 (m −1!)
THEOREMS AND PROPERTIES[8,9] Definition 2.1 (Arthanasius [1973]) Assume that XY,� are Banach spaces, S is an open subset of X and f: S → Y , is continuous and Frechet differentiable at every point of S , Moreover, assume that for every ∈> 0 there exists ()∈ > 0 such that
' (i) If x12, x , ∈ S with x1 −≤∈ x 2 ( C ) then [f( x12) − fx ( ) ] h ≤∈ h ∀ h ∈ X (ii) ∝(,xL ) ≤∝ [ h] for every x∈ S , h ∈ X with h ≤∈ b ( )then f is called C − differentiable on S
Theorem 2.1 (Arthanasius [1973]) Let f be a convex function defined on an open convex subset X of a Banach space X that is continuous f ( x++ th) f ( x −− th) 2 f() r at xX ∈ . Then, f is Frechet differentiable at x�iff lim = D t → 0 t uniformly for hS ∈ x
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Remark 2.1 Obviously, we have earlier seen that Frechet differentiability has additive property and the product of two Frechet differentiable functions is Frechet differentiable function (the function that is Frechet differentiable is continuous and therefore locally bounded). Now, we use boundedness of Frechet derivative and triangle inequality. A function which is Frechet differentiable at a point is continuous there.
THE INVERSE FUNCTION THEOREM An inverse function theorem is an important tool in the theory of differential equations. It ensures the existence of the solution of the equation Tx =�� y . Although T is not assumed to be compact and the contraction principle might not be directly applicable, it is shown, in the proof of the inverse function theorem, that the contraction principle can be used indirectly if �T has some appropriate differentiability properties. Definition 3.1 (Arthanasius [1973]) Let S be an open subset of the Banach space X � and let fm� ��ap S into the Banach space �Y . Fix a point x00 ∈ S and let f ( x ) = x0 , then , f is said to be “locally invertible at (xy00, ) if there exist two numbers
∝>0, > 0, with the following property: For every y∈ SS , 00 ( y ) , there exists a unique xSx ∈ 00( ) such that fx ( ) = y .
Lemma 3.1 (uniqueness property of Frechet derivative) (Kaplan [1958]) Let LS�� → Y � be given with S an open subset of the Banach space XY and another Banach space. Suppose further that f is Frechet differentiable at xS∈ . Then, the Frechet derivative of fa�� tx� is unique. Proof
Suppose that DxDx12( ), ( ) are Frechet derivatives of fa�� tx� with remainders 11(xh, ) , 2( xh 2 ) , respectively. Then, we have
D11( x) h+=+ ( xh , ) D 2( x) h 2 ( xh , ) for every h ∈ X with x +∈ h S11 . Here S is some open subsets of S containing x. It follows that
D12( xh) −=− D( xh) / h 1( xh , ) w 2 ( xh , ) (3.1)
1/≤+ 12(xh , ) h ( xh , ) / h
The last member of (3.1) tends to zero as h → 0. Let
Tx =−∈ Dx12( ) D( x) xx , Xthen T is a linear operator on X such that limTx / x = 0 x− 0 Consequently, given ∈> 0, there exists (∈>) 0 such that Tx/ x > for every xX ∈ with x <∈ ( ) . GivenyX ∈ with y ≠ 0, let x =∈ ( ) yy / 2 . Tx Then ∞< ( ∈) and < or Ty < nn y . Since is arbitrary, we obtain Ty = 0 for every y ∈ X . We x conclude that Dx1 ( ) = Dx2 ( ).
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The existence of a bounded Frechet derivative fu�� is equivalent to the continuity of f at x. This is the content of the next lemma. Lemma 3.2 (Argyros [2005]) Let fS:�→ Y � be given where S is an open subset of a Banach space XY and Another Banach space, Let � f be Frechet differentiable at xS� ∈ . Then, f is continuous at x if fx′() is a bounded linear operator. Proof Let f be continuous at xS�� ∈ . Then, for each > 0 there exists ( ) ∈(0 , 1 ) such that fx( +− h ) f () x <∈ /2, fx( +− h ) fxh ( ) <( /2) h < /2 for all hB ∈ with x +∈ hS and h < ( ∈) .
Therefore. f' ( xh) <(/2) h + f( x +− h) f ( x ) <
For h< ( ), xhS +∈ , it follows that the linear operator fx′() is continuous at the point 0 .
We should note here that the magnitude of the ball S plays no role in the Frechet differentiability of. This means that, to define the Frechet derivative, fx′( ) of at x , we only need to have that certain differentiability conditions hold for all xh+ � in a sufficiently small open neighborhood Sx of the point . We now quote a well-known theorem of functional analysis the “bounded inverse theorem.” Theorem 3.2 (Day [1973]) Let XY,� � be Banach spaces and let TX:�→ Y � be bounded linear, one-to-one and onto. Then, the inverse TT−1 of is a bounded linear operator on Y . We are ready for the inverse function theorem. Theorem 3.3 (inverse function theorem) (Arthanasius [1973]) Let XY,� be Banach spaces and S an open subset of X Let f: S→ Y be C-differentiable on �S . Moreover, assume that the Frechet derivative of the function is one-to-one and onto at some point xS, ∈ . Then, the function f � is locally invertible at the point (x00, fx ( )) .
Proof −1 −1 Let D= fx′( 0 ) . Then, the operator D exists and is defined on YD. Moreover is bounded. Thus, the equation fx()= y� is equivalent to the equation D−−11 f( x) = D y . fix y ∈ Y and define the operator US on as follows.
−1 Ux =+− x D x fx ( ) , x ∈ S (3.2) obviously, the fixed points of the operator X � are solutions to the equation fx()=� � y . We first determine a closed ball inside S with center at x0 , on which X is a contraction operator. To this end fix ∈< (0.1 / 4D−1 )) and let ( ) 0 be such that
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'' fx( 0) − fx () 2( x 12 − x) ≤∈ x 1 − x 2 … (3.3)
(xxx212 ,−≤) xx 12 − … (3.4) ( ) For every xx, ∈ S with x −≤ x x −≤ x ( ) / 2 12 1 0 2 2 0 This is possible by virtue of the C-differentiability of the function � � f . Thus, we have
−1 XxXx1− 2 =−− xxD 12 fx( 1) − fx() 2
−1 −− 11 =Df ' ( x012 ) ( x − xDfx ) '( 212 )( x−− x D ( xxx 212 − )
−−1' ' 1 ≤D fx( 0) − fx( 2) ( x 12) −+ x D() xxx2,12 −
−−11 ≤D xx12 −+ D xx12 −
≤ ()12/ xx12−
For every xx12, ∈ S as above. It follows that X is a contraction operator on the ball Sx00(), where
∝= ( ) / 2 <1/ (4D−1 ) .
Now, we determine a constant > 0 such that XSxSxoo( 0) ⊂( 0) whenever vSY ∈=00( ) Herevfx 0( 0) .in fact we have
−−11 XxXx00− ≤ D yfx −(0 ) = D yy −≤ o ( )/4 whenever
−1 yy−≤ o (n ) / (4 D =
Furthermore
X−= x0 Xx − Xx0 + Xx 00 − x
≤(1/2) xx −+ , ( )/4 ≤ ( )/4 + ( )/4 = ( )/2
For any xSx∈ 00(). We have shown that f is locally invertible at (x00, fx( )). and that for any yS ∈ o ( y0 ) , there exists a unique x ∈= Soo( x) With f ( x) y .
Theorem 3.4 (Ince [1956]) Let the assumption of theorem 3.3 be satisfied with the C-differentiability of the function f replaced by condition ()i of definition 3.1, then the conclusion of theorem 3.3 remains valid.
Example 3.1 (Leighton [1970]) let Ja= [],�b and let
S, =∈< [ xR n : x r ]
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Sr =∈ xC J ; ∞< r n ( ) ∞
Where r is a positive number.
n ' We consider a continuous function FJ:�×−SRo � and the operator XS1 − Cn () J defined as follows: (Xx)( t) = F( t , x ( t)) , t ∈ J , ∞∈ S '
We first note that W is continuous on S� . In fact, since �F is uniformly continuous on the compact set
JS×> o for every 0 there exists ( ) > 0 such that
Ftx( , ) −< f( ty, )
For every xy,� �� ∈So with x−< y ( ) and every tJ . This implies that
Ux−< Uy ∞
' Whenever, xy, ∈< S with x , y∞ ( ) . To compute the Frechet derivative of X , we assume that the Jacobean matrix
F1 ( tx, ) =∂∂ [( F / x )( tx , )] , i =… 1,2 n exists and is continuous on JS × , . then given two function
'1 1 xS01 ∈( 0 < rrhJ <) , ∈, such that xhCJxhS0 +∈ n 0 +∈ ' we have
supFt (+− ht( ) Ftx( , 01( t)) − F( tx , o ( t)) h ( t ) s∈1
≤∂ − sup{ [( F / xFtx ( , 01( t)) , ht ( )) ] F( tx , o ( t)) h ( h )∞ (3.5) s∈1
Where n in=1, 2 … are function of t lying in the interval (,01) . In (3.5), we have used the mean value theorem for real-valued functions on S0 as follows:
F1( tx, 0( t) + ht ( )) − F1( t , ∞ 0( t)) =<∇ Ft11 ( , z 1( tht) ( ) > i = 2 … n
Where z101( t) = x( t) + ht ( ) form the uniform continuity of ∇Ft1( 1 , z 1 ( tht) ( ) >=… i i n on J × S1, it follows that the Frechet derivative Xx()0 exists and is a bounded linear operator given by the formula ' X( x0) h ( t) = F 10( tx ,)( tht) ( )(3.6)
REFERENCES
1. Argyros IK. Approximate Solution of Operator Equations with Application. World Scientific Pub. Co., Inc.; 2005. p. 6-11. 2. Kartsatos AG. Advanced Ordinary Differential Equations. Tampa, USA: Mancorp Publishing; 1973. 3. Day MM. Normed Linear Spaces. 3rd ed. New York: Springer; 1973. 4. Dunford N., Schwartz JT. Linear Operators, 3 Parts. New York: Inter-Science/Wiley; 1958.
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5. Frechet M. Sur quelques points ducalcul functionel rend. Circ Mat Palermo 1906;22:1-74. 6. Ince E. Ordinary Differential Equations. New York: Dover; 1956. 7. Kaplan W. Ordinary Differential Equations. Boston, Massachusetts: Addision-Wesley; 1958. 8. Leighton W. Ordinary Differential Equations. 3rd ed. Belmont, California: Wadsworth; 1970. 9. Pontryagin L. Ordinary Differential Equations. Boston, Massachusetts: Addison-Wesley; 1962.
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