Floquet theory and applications.

Project Abstract

Floquet theory is a powerful tool in the analysis of periodic dynamical systems. It provides a systematic way to study the behavior of solutions to linear and nonlinear differential equations with periodic coefficients. The theory is named after Gaston Floquet, who introduced it in the late 19th century. One of the key advantages of Floquet theory is its ability to reduce the study of time-varying systems to that of time-invariant systems. This simplification allows for the application of well-established techniques from linear algebra and eigenvalue problems to gain insights into the dynamics of periodic systems. By analyzing the Floquet multipliers associated with the solutions of a periodic system, one can determine the stability and bifurcation properties of the system. Floquet theory has found numerous applications in various fields of science and engineering. In physics, it has been used to study the stability of particle motion in periodic potentials, such as in the context of atomic and molecular physics. In electrical engineering, Floquet theory is applied to analyze the stability of power systems with time-varying parameters. The theory also plays a crucial role in the study of nonlinear dynamics and chaos, where it helps in understanding the stability of periodic orbits and the onset of bifurcations. Moreover, Floquet theory has been extended to address more general classes of periodic systems, including systems with time-delayed feedback. The analysis of such systems involves the computation of Floquet exponents, which characterize the stability of periodic solutions in the presence of delays. This extension of Floquet theory has broadened its scope of applicability to include a wider range of dynamical systems with time-dependent properties. Recent advances in computational methods have further enhanced the utility of Floquet theory in practical applications. Numerical algorithms for computing Floquet multipliers and exponents have been developed, enabling researchers to analyze complex periodic systems with high precision. These computational tools have enabled the study of Floquet stability in systems that were previously challenging to analyze using traditional analytical methods. In conclusion, Floquet theory is a versatile mathematical framework that has proven to be indispensable in the study of periodic dynamical systems. Its applications span a wide range of disciplines, from physics to engineering, and its continued development, coupled with advancements in computational techniques, holds great promise for furthering our understanding of complex periodic phenomena.

Project Overview

<p> </p><p>Introduction and Generalities<br>Introduction<br>In physical sciences (e.g, elasticity, astronomy) and natural sciences (e.g, ecology) among others,<br>the consideration of periodic environmental factor in the dynamics of multi-phenomena<br>interractions (or multi-species interractions) leads to the study of dierential systems with<br>periodic data. Therefore, it is worth investigating, the fundamental questions inherent in systems<br>of periodic (ordinary) dierential equations such as: existence, uniqueness and stability.<br>This work is specially devoted to one of the celebrated tools which is crucial in the analysis<br>and control of time-periodic systems, namely Floquet Theory. This basic theory which is<br>about a systematic study of linear systems of ordinary dierential equations with periodic<br>coecients, has had a rich history since the pioneering work of Floquet (1883) followed by the<br>contribution of Lyapunov (1892). Futhermore it gives a change of variables that transforms a<br>linear dierential system with periodic coecients, into a system with constant coecients,<br>and also provides a representation formula of the solutions of (1).<br>The aim of this project is to present Floquet theory and to use it to assess stability of<br>periodic solution of periodic dierential systems (linear or nonlinear).<br>The work is organized as follows:<br>First, we review some basic notions from Analysis and Linear algebra that will be used in<br>the subsequent chapters.<br>In chapter 2, we present the basic notions and theorems of the general theory of Ordinary<br>Dierential Equations (ODE).<br>Finally in chapter 3, we state Floquet’s theorem, prove it and use it to address stability of<br>periodic solutions of linear as well as nonlinear periodic systems. Furthermore, we illustrate<br>our method by studying Hill’s equation which is a generalization of Mathieu equation.<br>1<br>Generalities<br>The purpose of this chapter is to give some concepts and theorems of linear algebra and<br>Mathematical analysis that will be of importance in the subsequent chapters<br>1.1 Basic concepts from linear functional analysis<br>let’s recall some denitions and results from linear functional analysis<br>Denition 1.1.1 Let X be a linear space over a eld K; where K holds either for Rn or C.<br>A mapping k:k: X ô€€€! R is called a norm provided that the following conditions hold:<br>i) kxk 0; for all x 2 X and kxk= 0 , x = 0<br>ii) kxk= jjkxk for all 2 K; x 2 X<br>iii) kx + yk kxk+kyk for arbitrary x; y 2 X<br>If X is a linear space and k:k is a norm on X; then the pair (X; k:k) is called a normed<br>linear space over K.<br>Should no ambiguity arise about the norm, we simply abbreviate this pair by saying that X<br>is a normed linear space over K.<br>Example 1. Each of the following expressions denes on the vector space Rn a norm which<br>is in common use.<br>i) The absolute norm :<br>kxk1 =<br>Xn<br>i=1<br>jxij ; for every x = (x1; : : : ; xn) 2 Rn:<br>ii) The euclidean norm :<br>kxk2 =</p><p>Xn<br>i=1<br>jxij2<br>!1<br>2<br>; for every x = (x1; : : : ; xn) 2 Rn:<br>iii) The maximum norm :<br>kxk1 = max<br>1in<br>jxij; for every x = (x1; : : : ; xn) 2 Rn:<br>2<br>Example 2. Let X = C([0; 1]) be the space of all real-valued continuous functions on [0; 1].<br>Each of the following expressions denes on the vector space C([0; 1]) a norm which is in<br>common use.<br>i) kfk1=<br>R 1<br>0 jf(t)jdt for every f 2 C([0; 1]).<br>ii) kfk2=<br>R 1<br>0 (jf(t)j)<br>1<br>2 dt<br>1<br>2 for every f 2 C([0; 1]).<br>iii) kfk1= max</p><p>jf(t)j: t 2 [0; 1]</p><p>.<br>Denition 1.1.2 (Equivalent norms)<br>Two norms k:k1 and k:k2 dened on a normed linear space X are said to be equivalent if<br>there exists &gt; 0 and &gt; 0 constants such that<br>kxk1 kxk2 kxk1 8x 2 X:<br>Theorem 1.1.3 In a nite dimensional normed linear space, all the norms are equivalent.<br>Denition 1.1.4 Every normed linear space E is canonically endowed with a metric d de-<br>ned on E E by<br>d(x; y) = jjx ô€€€ yjj 8 x; y 2 E:<br>Denition 1.1.5 (Cauchy sequence)<br>A sequence (xn)n1 of elements of a normed vector space X is a Cauchy sequence if<br>lim<br>n;m!1<br>kxn ô€€€ xmk= 0:<br>That is, for any &gt; 0 there is an interger N = N() such that kxn ô€€€ xmk&lt; whenever<br>n N and m N.<br>Remark. In a normed linear sapce, every Cauchy sequence (xn)n1 is bounded; i.e, there<br>exists a constant M 0 such that jjxnjj M ; 8n 1: (See also Denition 1.1.11 below)<br>Denition 1.1.6 (convergent sequence)<br>A sequence (xn)n1 of elements of a normed vector space X converges to an element x 2 X<br>if<br>lim<br>n!1<br>kxn ô€€€ xk= 0:<br>In such a case, we say that (xn)n1 is a convergent sequence.<br>Remark. In a normed linear space, every convergent sequence is a Cauchy sequence.<br>Denition 1.1.7 A normed linear space is complete if every Cauchy sequence in X has the<br>limit in X. A complete normed linear space is called a Banach space.<br>Remark. The notion of completeness is also dened for metric spaces which need not have<br>any linear structure.<br>Example (Banach space). The normed linear space<br>ô€€€<br>C([0; 1]); k k1</p><p>is a Banach space.<br>3<br>Denition 1.1.8 (Open sets and closed sets)<br>Let X be a normed linear space. We dene open (respectively closed) ball with center at a<br>point x 2 X and radius r &gt; 0 by<br>Br(x) = fx 2 X : kxk&lt; rg (respectively Br(x) = fx 2 X : kxk rg ) :<br>A nonempty subset A of a normed linear space X is said to be open if for all x 2 A; there<br>exists r &gt; 0 such that Br(x) A: And A is said to be closed if XnA is open.<br>Proposition 1.1.9 A subset A of a normed linear space is closed if and only if every con-<br>vergent sequence (an)n1 of elements of A has its limit in A:<br>Denition 1.1.10 (Closure and interior) Let A be a subset of a normed linear space X.<br>The interior of A denoted by intA is dened as the union of all open sets contained in A<br>and the closure of A denoted by cl(A) or A is dened as the intersection of all closed sets<br>containing A:<br>Theorem 1.1.11 Let A be a subset of a normed linear space X and x 2 X then,<br>a) x 2 intA if and only if 9 r &gt; 0 : B(x; r) A:<br>b) x 2 clA if and only if 8 r &gt; 0; B(x; r) A 6= ?:<br>Remark. Given a subset A of a normed linear space X; we have :<br>x 2 A () 9 (an)n A such that lim<br>n!+1<br>an = x:<br>Denition 1.1.12 Let X be a normed linear space, x 2 X and let V be a subset of X<br>containing x: We say that V is a neighbourhood of x if there exists an open set U of X<br>containing x and contained in V: We denote by N(x) the collection of all neighbourhoods of<br>x:<br>Denition 1.1.13 A subset of a normed linear space X is said to be bounded if it can be<br>included in some ball.<br>Theorem 1.1.14 (Riesz/ Heine-Borel) A normed linear space is nite dimensional if<br>and only if its closed unit ball is compact, i.e., every bounded sequence of the closed unit ball,<br>has a convergent subsequence.<br>1.1.1 Linear operators<br>In this section X and Y are normed linear spaces over K.<br>Denition 1.1.15 A K-linear operator T from X into Y is a map T : X ô€€€! Y such that<br>T(x + y) = Tx + Ty<br>for all ; 2 K and all x; y 2 X:<br>When Y = K, such a map is called a linear functional or a linear form.<br>4<br>Proposition 1.1.16 The set of K-linear operators from X into Y has a natural structure<br>of linear space over K and is denoted by L(X; Y ). Note that L(X;X) is simply denoted by<br>L(X).<br>Proposition 1.1.17 If Z is also a linear space, then<br>f 2 L(X; Y ) and g 2 L(Y;Z) =) gof 2 L(X;Z) :<br>Theorem 1.1.18 Let T 2 L(X; Y ). Then the following are equivalent<br>i) T is continuous at the origin (in the sense that if fxngn is a sequence in X such that<br>xn ! 0 as n ! 1, then T(xn) ! 0 in Y as n ! 1.<br>ii) T is Lipschitz, i.e., there exists a constant K 0 such that for every x 2 X,<br>jjT(x)jj Kjjxjj :<br>iii) The image of the closed unit ball, T<br>ô€€€<br>B1(0)</p><p>, is bounded.<br>Denition 1.1.19 A linear operator T : X ô€€€! Y is said to be bounded if there exist some<br>k 0 such that<br>kT(x)k kkxk<br>for all x 2 X:<br>If T is bounded, then the norm of T is dened by<br>kTk= inffk : kT(x)k kxk; x 2 Xg:<br>The set of bounded linear operators from X into Y is denoted B(X; Y ): If X = Y; one simply<br>writes B(X):<br>Proposition 1.1.20 Suppose X 6= f0g and T 2 B(X), then we have the following:<br>kTk= sup<br>kxk1<br>kT(x)k= sup<br>kxk=1<br>kT(x)k= sup<br>kxk6=0<br>kT(x)k<br>kxk<br>1.2 Matrix calculus and basic Operator theory<br>1.2.1 Matrices, eigenvalues and eigenvectors<br>Denition 1.2.1 An m n matrix A is a rectangular array of numbers, real or complex,<br>with m rows and n columns. We shall write aij for the number that appears in the ith row<br>and the jth column of A; this is called the (i; j) entry of A: We can either write A in the<br>extended form 0<br>[email protected]<br>a11 a12 a1n<br>a21 a22 a2n<br>…<br>…<br>. . .<br>…<br>am1 am2 amn<br>1<br>CCCA<br>5<br>or in the more compact form<br>(aij)m;n:<br>We will denoted A = (aij)m;n with 1 i m; 1 j n:<br>Associated with each matrix A a matrix At; known as the transpose of A, and obtained<br>from A by interchanging the rows and the columns of A. Thus, if A = (aij)m;n then At =<br>(aji)n;m: The trace denoted by tr of A is the sum of a diagonal elements of A. Two m n<br>matrices A and B are said to be equal if all corresponding elements are equal, that is , if<br>aij = dij for each i and j.<br>Let D = (dij)m;n, A = (aij)m;n and B = (bij)n;p be a matrices.<br>1) The sum of two m n matrices A and D is dened as matrix obtained by adding<br>corresponding elements:<br>D + A = (dij + aij)m;n:<br>Similarly, the dierence is<br>D ô€€€ A = (dij ô€€€ aij)m;n:<br>2) The multiplication of a matrix A by a scalar is dened as follows: A = (aij)m;n:<br>3) The product of a m n matrix A and a n p matrix B is<br>AB =</p><p>Xn<br>k=1<br>aikbkj<br>!<br>m;p<br>:<br>Special matrices<br>i) A row-vector or 1 m matrix: A = (a1; a2; :::; an) where the a0<br>is are scalars.<br>ii) A column-vector or n 1 matrix: A =<br>0<br>[email protected]<br>a1<br>…<br>an<br>1<br>CA<br>where the a0<br>is are scalar.<br>iii) A zero matrix is a matrix which all of whose entries are zero. The zero m n matrix<br>is denoted by 0m;n or simply 0:<br>iv) A square matrix is a matrix with the same number of rows and columns.<br>v) The identity matrix of order n has one on the principal diagonal, that is from top<br>left to bottom right, and zeros elsewhere; it is denoted by In = (ij) where ij is the<br>Kronecker’s symbol. From the denition of matrix multiplication we have<br>AI = IA = A<br>for any square matrix A.<br>6<br>iv) A square matrix A is regular or non-singular if its column vectors are linearly independant<br>or equivalently its determinant, det(A), is nonzero. Otherwise A is said to be<br>singular or degenerated when det(A) = 0. (See details/recalls below).<br>v) A square matrix in which all the non-zero elements lie on the principal diagonal is<br>called a diagonal matrix.<br>vi) An n n matrix N is said to be nilpotent, if there is a positive integer k such that<br>Nk = 0:<br>Denition 1.2.2 Let A and B be two n n matrices. We say that A and B are similar,<br>notation A B; if and only if there exists a nonsingular matrix T such that Tô€€€1AT = B<br>Eigenvalues and Eigenvectors<br>Denition 1.2.3 Let A be an nn square matrix. Then a scalar is called an eigenvalue<br>of A, if there exists a nonzero vector v 2 Rn such that Av = v:<br>In this case, the is called an eigenvalue of A and the v; eigenvector associated with :<br>The eigenvalues of A are also the roots of the characteristic polynomial p() = det(Aô€€€I)<br>with p() of degree n:<br>Theorem 1.2.4 Let A be a square matrix, if the eigenvalues of A are all distincts, A is<br>similar to a diagonal matrix (whose diagonal entries are the eigenvalues of A).<br>Now if the matrix A has repeated eigenvalues, then it is not possible to diagonalize it. In<br>this case, we introduce the concept of generalized eigenvector.<br>Denition 1.2.5 A nonzero vector v is called a generalized eigenvector of rank k of A;<br>associated with an eigenvalue if and only if<br>(A ô€€€ I)kv = 0 and (A ô€€€ I)kô€€€1v 6= 0:<br>Lemma 1.2.6 If v is a generalized eigenvector of rank k; then the vectors<br>v; (A ô€€€ I)v; :::; (A ô€€€ I)kô€€€1v are linearly independent.<br>Recall that a set of vectors v1; v2; : : : ; vk is linearly dependent if there exist scalars c1; c2; : : : ; ck<br>not all zero, such that<br>c1v1 + c2v2 + : : : + ckvk = 0:<br>A set of vectors v1; v2; : : : ; vk is linearly independent if it is not linearly dependent.<br>Jordan form<br>From the lemma above, we construct a new basis for Cn such that the matrix representation<br>of A with respect to this new basis is the one we call Jordan canonical form denoted by J:<br>7<br>Theorem 1.2.7 For every nn complex matrix A with eigenvalues 1; :::; s (not necessarily<br>distinct) of multiplicities n1; :::; ns respectively, there exists a nonsingular n n matrix P<br>such that<br>Pô€€€1AP = J = diag(J1; :::; Js);<br>where each of block matrices J1; :::; Js is of the form<br>Jk =<br>0<br>[email protected]<br>k 1 0 0<br>0 k 1 0</p><p>0 k 1<br>0 0 k<br>1<br>CCCCA<br>; k = 1; :::; s;<br>and<br>Ps<br>k=1 nk = n:<br>For the proof of this theorem, refer to Coddington and Levinson [C/L] or Hirsch and Smale<br>[H/S.]<br>The block matrices J1; :::; Js are called Jordan blocks, and J is called the Jordan canonical<br>form of A: Note, any Jordan block Jk() can be written as Jk = kI + Nk; where Nk is<br>nilpotent of order k and that A is similar to J:<br>1.2.2 Limits of sequences of operators<br>We introduce the concepts of limit in the norm and of strong limit of a sequence of operators.<br>We shall then introduce the derivative and integral of operators depending on a parameter<br>and shall discuss series of operators.<br>Let X be a Banach space and (An) a sequence of operators in L(X):<br>– We say that (An) converges in norm to the operator A 2 L(X); if<br>lim<br>n!1<br>jjAn ô€€€ Ajj = 0: (1.2.1)<br>– If, for each element x 2 X;<br>lim<br>n!<br>jjAnx ô€€€ Axjj = 0; (1.2.2)<br>we shall say that An converges strongly to the operator A 2 L(X):<br>It is immediate that (1.2.1) implies (1.2.2). The converse is not in general true. However, in<br>the case of nite dimensional Banach spaces it is true, for if we put x = e(i); we have from<br>(1.2.2)<br>lim<br>n!1<br>jj(a(1)<br>i1 ; :::; a(k)<br>ik )jj = 0 for i = 1; :::; k:<br>This ensures that all the components tend to 0; and since there are only a nite number of<br>them, the limit is uniform.<br>It is now possible to dene the meaning of the series<br>X1<br>s=1<br>As;<br>8<br>the series<br>1P<br>s=1<br>As is said to be convergent if the sequence made by the partial sums<br>XN<br>s=1<br>As;<br>converges in L(X).<br>In this case of matrices,<br>NP<br>s=1<br>As corresponds to a matrix of which elements are<br>NP<br>s=1<br>a(s)<br>ij = aij :<br>– We say that the series of operators<br>1P<br>s=1<br>As converges absolutely, if the series<br>X1<br>s=1<br>jjAsjj<br>converges.<br>In the case of matrices, this happens if and only if, for every component,<br>X1<br>s=1<br>ja(s)<br>ij j<br>converges with As = (a(s)<br>ij ):<br>Proposition 1.2.8 Let X be a normed linear space. If A 2 L(X); then the series<br>1P<br>0<br>An<br>n! is<br>absolutely convergent.<br>Proof. If suces to show that the sequence of partial sums fSNg1N<br>=1 for the series<br>1P<br>n=0<br>kAnk<br>n! is a Cauchy sequence. Let us dene<br>NP<br>0<br>An<br>n! : Note that the partial sums of the convergent<br>series of real number<br>1P<br>0<br>kAkn<br>n! = ekAk form a cauchy sequence. Using this fact, it follows<br>that SN is a cauchy sequence in L(X).<br>Dene the exponential map exp : L(X) ! L(X) by exp(A) = eA =<br>1P<br>n=0<br>An<br>n! :<br>The main properties of exponential map are summarized in the following proposition<br>Proposition 1.2.9 Suppose that A;B 2 L(Rn):<br>i) If A 2 L(Rn; ) then eA 2 L(Rn):<br>ii) If B is nonsingular, then Bô€€€1eAB = eBô€€€1AB:<br>iii) eô€€€A = (eA)ô€€€1:<br>9<br>iv) keAk ekAk:<br>– Let A be an operator depending on a real parameter t with a t b: Let t0 2 [a; b]; if<br>h is suciently small, the operator<br>A(t0 + h) ô€€€ A(t0)<br>h<br>can be dened. If its limit as h ! 0 exists, we say that the operator A is dierentiable at t<br>with respect to t: The limiting operator is denoted by d<br>dtA(t): We thus have<br>lim<br>h!0</p><p>A(t0 + h) ô€€€ A(t0)<br>h<br>ô€€€<br>dA<br>dt</p><p>= 0:<br>In the case of matrices, the limits exists if and only if each component aij is dierentiable,<br>and we have<br>d<br>dt<br>A(t) =</p><p>d<br>dt<br>aij(t)</p><p>:<br>In general, if A and B are two dierentiable operators in L(X); the following product rule<br>is valid.<br>d<br>dt<br>(AB) =<br>dA<br>dt<br>B + A<br>dB<br>dt<br>:<br>We say that the series of operators depending on a parameter converges uniformly if, for<br>every &gt; 0; there exists a &gt; 0 such that for every t 2 [a; b] and every &gt;</p><p>X1<br>s=<br>As(t)</p><p>&lt; :<br>If the operator As(t) are dierntiable and the series<br>X1<br>s=0<br>d<br>dt<br>As(t)<br>converges uniformly, then also the operator A =<br>1P<br>s=0<br>As(t) is dierntiable, and we have<br>dA<br>dt<br>=<br>X1<br>s=0<br>d<br>dt<br>As:<br>We illustrate this notion with the following result: If A 2 L(X) then tA 2 L(X) for each<br>t 2 R so that the function t 7ô€€€! etA is dierentiable and<br>d<br>dt<br>(etA) = AetA:<br>If A is a nonsingular matrix, then logarithm of A denoted by ln(A) is well-dened matrix.<br>This important result is stated in the following theorem.<br>10<br>Theorem 1.2.10 Let A be a nonsingular n n matrix, then there exists an n n matrix<br>B (called logarithm of A) such that A = eB:<br>Let A(t) be a matrix depending on a parameter t; we suppose that the component aij are<br>all integrable functions over the interval [t0; t]: We shall say that the matrix<br>Z t<br>t0<br>A( )d;<br>is the intergral of the matrix A(t) between t0 and t:<br>According to the following denition<br>d<br>dt<br>(aij)m;n =</p><p>daij<br>dt</p><p>m;n<br>;<br>we have that if the functions aij are continuous, then<br>d<br>dt<br>Z<br>A( )d = A(t):<br>If, nally A is an m n matrix, and L is a number greater than or equal to the absolute<br>values of aij ; then<br>Zt<br>t0<br>A( )d</p><p>p<br>m:n:Ljt ô€€€ t0j:<br>1.3 Review of calculus<br>We now give the general denition of dierentiability. Let U be a nonempty open subset of<br>a Banach space X, let Y denote a Banach space, and let the symbol k:k denote the norm in<br>both Banach spaces.<br>Denition 1.3.1 A function f : U ô€€€! Y is dierentiable at x 2 U if there is a map<br>A 2 L(X; Y ) such that<br>lim<br>h!0<br>kf(x + h) + f(x) ô€€€ Ahk<br>khk<br>= 0<br>Remark. If such a linear map exists, then it is unique and we write it as A = f0(x) called<br>the derivative of f at x:<br>Other common notations for the derivative are Df and fx<br>The following is a list of standard facts about the derivative. For the statement in the list,<br>the symbols X; Y; Xi; and Yi denoted Banach spaces.<br>i) If f : X ô€€€! Y is dierentiable at a 2 X, then f is continuous at a.<br>ii) If f : X ô€€€! Y1 ::: Yn is given by f(x) = (f1(x); :::fn(x)); and if fi is dierentiable<br>for each i, then so is f and,<br>Df(x) = (Df1(x); :::;Dfn(x)):<br>11<br>iii) If the function f : X1 X2 ::: Xn ô€€€! Y is given by (x1; x2; :::; xn) 7ô€€€!<br>f(x1; x2; :::; xn), then the ith partial derivative of f at (a1; a2:::; an) 2 X1X2:::Xn<br>is the derivative of the function g : Xi ô€€€! Y dened by g(xi) = f(a1; :::; aiô€€€1; xi; ai+1; :::; an).<br>This derivatives is denoted by Dif(a): If f is dierentiable, then all its partial derivatives<br>exist and, if we dene h = (h1; h2; :::; hn); we have<br>Df(x)h =<br>Xn<br>i=1<br>Dif(x)hi:<br>The converse is not true in general, but if all the partial derivatives of f exist and are<br>continuous in an open set U X1 X2 :::Xn then f is continuously dierentiable<br>in U.<br>1.3.1 The mean value theorem<br>Theorem 1.3.2 Suppose that [a; b] is a closed interval, and f : [a; b] ô€€€! Y is a continuous<br>function. If f is dierentiable on the open interval (a; b) and there is some number M &gt; 0<br>such that kf0(t)k M for all t 2 (a; b); then<br>kf(b) ô€€€ f(a)k M(b ô€€€ a)<br>Theorem 1.3.3 (Mean value theorem) Suppose that f : X ô€€€! Y is dierentiable on an<br>open set U X with a; b 2 U and a + t(b ô€€€ a) 2 U for 0 t 1: If there is some M &gt; 0<br>such that<br>sup<br>0t1<br>kDf(a + t(b ô€€€ a)k M;<br>then<br>kf(b) ô€€€ f(a)k Mkb ô€€€ ak:<br>1.3.2 Integration in Banach space<br>Let X be a Banach space and I = [a; b] R where a &lt; b.<br>Denition 1.3.4 If f : I ! X is continuous on I, we dene its integral in the sense of<br>Rieman by the following formula:<br>Z b<br>a<br>f(t)dt = lim<br>n!1<br>b ô€€€ a<br>n<br>Xnô€€€1<br>k=0<br>f(a + k<br>b ô€€€ a<br>n<br>):<br>It is easily seen that<br>Z b<br>a<br>f(t)dt</p><p>Z b<br>a<br>kf(t)kdt:<br>Theorem 1.3.5 Suppose that U is a nonempty open subset of X: If f : X ! Y is a<br>dierentiable function, and x + ty 2 U for 0 t 1; then<br>f(x + y) ô€€€ f(x) =<br>Z 1<br>0<br>Df(x + ty)ydt:<br>12</p> <br><p></p>