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Integral Equations Chapter 7 INTEGRAL EQUATIONS Chapter 7 INTEGRAL EQUATIONS Chapter 7 Integral Equations 7.1 Normed Vector Spaces 1. Euclidian vector space \n 2. Vector space of continuous functions CG( ) 3. Vector Space LG2 () 4. Cauchy-Bunyakowski inequality 5. Minkowski inequality 7.2 Linear Operators - continuous operators - bounded operators - Lipschitz condition - contraction operator - successive approximations - Banach fixed point theorem 7.3 Integral Operator 7.4 Integral equations - Fredholm integral equations - Volterra integral equations - integro-differential equations - solution of integral equation 7.5 Solution Methods for Integral Equations 1. Method of successive approximations for Fredholm IE (Neumann series) 2. Method of successive substitutions for Fredholm IE (Resolvent method) 3. Method of successive approximations for Volterra IE 7.6 Connection between integral equations and initial and boundary value problems 1. Reduction of IVP to the Volterra IE 2. Reduction of the Volterra IE to IVP 3. Reduction of BVP to the Fredholm IE 7.7 Exercises Future Topics: 7.7 Fixed point theorem (see [Hochstadt “Integral equations”, p.25]) (added in 7.2) Elementary existence theorems 7.8 Practical applications (see [Jerri “Introduction to Integral Equations with Applications”]) 7.9 Inverse problems (see [ Jerri, p.17]) 7.10 Fredholm’s alternatives Chapter 7 INTEGRAL EQUATIONS 7.1 Normed Vector Spaces We will start with some definitions and results from the theory of normed vector spaces which will be needed in this chapter (see more details in Chapter 10). 1. Euclidian vector space \n The n-dimensional Euclidian vector space consists of all points n \\=={x() x12 ,x ,...,x n x k ∈} for which the following operations are defined: n Scalar product ( x,y) =+++ x11 y x 22 y ... x nn y x,y∈ \ 22 2 Norm x ==+++()x,x x12 x ... x n Distance ρ ( x,y) = x− y Convergence lim xk = x if lim x− xk = 0 k →∞ k →∞ \n is a complete vector space (Banach space) relative to defined norm x . 2. Vector space CG() Vector space CG( ) consists of all real valued continuous functions defined on the closed domain G ⊂ \n : C( G) =⊂→{ f() x : D\\n continuous} Norm fmaxfx= ( ) C xG∈ Convergence lim fk = f if lim f− fk = 0 k →∞ k →∞ C CGis a complete vector space (Banach space) relative to defined norm f . ( ) C 3. Vector space LG2 () The space of functions integrable according to Lebesgue (see Section 3.1) 2 \^n LG2 ()= fx:G () ⊂→∫ fx() dx <∞ G Inner product ( f ,g) = ∫ f( x) g( x) dx G 2 Norm f ==f,f f x dx 2 ()∫ () G The following property follows from the definition of the Lebesgue integral ∫ f ()x dx ≤ ∫ f ()x dx G G LGis a complete normed vectors spaces (Banach spaces) relative to f . 2 ( ) 2 4. Cauchy-Bunyakovsky-Schwarz Inequality (see also Theorem 10.1, p.257) f,g≤⋅ f g for all f ,g∈ L G ( ) 22 2 ( ) Proof: If f ,g∈ L2 ( G) , then functions f , g and any combination α f + β g are also integrable and therefore belong to L2 (G). Consider f + λ g ∈ L2 (G) , λ ∈ R for which we have Chapter 7 INTEGRAL EQUATIONS 2 2 2 0 ≤ ∫ ()f + λ g dx = ∫ f dx + 2λ∫ fg dx + λ2 ∫ g dx G G G G The right hand side is a quadratic function of λ . Because this function is non- negative, its discrimenant ( Db=−2 4ac) is non-positive 2 22 4fgdx4fdxgdx0∫∫∫−≤ GGG 2 22 fg dx≤ f dx g dx ∫∫∫ GGG and because ()f ,g = ∫ fgdx ≤ ∫ fg dx ≤ ∫ f g dx , G G G 2 22 f,g≤⋅ f g () 22 from which the claimed inequality yields f ,g≤⋅ f g () 22 ■ 5. Minkowski Inequality (3rd property of the norm “Triangle Inequality”), (see Example 10.7 on p.257) f + g ≤ f + g for all f ,g∈ L G 2 2 2 2 () Proof: 2 Consider f + g = f + g, f + g 2 ( ) = ( f , f )+ ( f ,g)+ (g, f )+ (g, g) 2 2 ≤ f + f ,g + g, f + g 2 ( ) ( ) 2 2 2 ≤ f + 2 f g + g from C-B inequality 2 2 2 2 2 = f + g ( 2 2 ) Then extraction of the square root yields the claimed result. ■ Note that the Minkowski inequality reduces to equality only if functions f and g are equal up to the scalar multiple, f = αg , α ∈ R (why?). Chapter 7 INTEGRAL EQUATIONS 7.2 Linear Operators Let M and N be two complete normed vectors spaces (Banach spaces, see Ch.10) with norms ⋅ and ⋅ , correspondingly. We define an operator L as M N a map (function) from the vector space M to the vector space N : L : M → N Introduce the following definitions concerning the operators in the vector spaces: Operator L : M → N is linear if L(αf + βg) = αLf + βLg for all f , g ∈ M and all α ,β ∈ R Operator L : M → N is continuous if from f k → f in M follows Lf k → Lf in N (the image of the convergent sequence in M is a convergent sequence in N ) Operator L : M → N is bounded if there exists c > 0 such that Lf ≤ c f for all f ∈ M N M The norm of operator L : M → N can be defined as the greatest lower bound of such constant c Lf L = sup N f ≠0 f M Theorem 7.1 If linear operator L : M → N is bounded then it is continuous Proof: Let operator L : M → N be bounded, then according to the definition there exists c > 0 such that Lf ≤ c f . N M Let f k → f in M . That means that lim f k − f = 0 . From the definition of k →∞ M the limit it follows that for any ε > 0 there exists kNε ∈ such that f − f < ε for all kk≥ . k M ε To prove the theorem, show now that Lf k → Lf in N or that lim Lf k − Lf = 0 . We have to show that for any Ε > 0 there exists K Ε ∈ N k →∞ N such that Lf − Lf < Ε for all k ≥ K . k N Ε Ε Choose ε = , then c Ε Lf − Lf = L()f − f ≤ c f − f < c ⋅ = Ε for all kk≥= K. ■ k N k N k M c Ε c Ε Remark: It is also true that if linear operator is continuous then it is bounded (prove as an exercise). Therefore, for linear operators, properties continuous and bounded are equivalent. Chapter 7 INTEGRAL EQUATIONS Definition Linear operator L:M→ N satisfies the Lipschitz condition with constant k0≥ if Lf− Lg≤− k f g for all f,g∈ M Obviously that if linear operator satisfies the Lipschitz condition (it is called a Lipschitz operator) then it is bounded (take vector g = 0 ) and, therefore, it is continuous. Definition Linear operator L:M→ N is a contraction if it satisfies the Lipschitz condition with constant k1< . distance between images becomes smaller Let S be a closed subset of Banach space M , SM⊂ , and let L:S→ S be an operator. Definition Solution of operator equation fLf= is called a fixed point of operator L . Definition Successive approximations is a sequence {}f012,f ,f ,... constructed in the following way: fS0 ∈ is a starting point fLf10= fLf21= # fLfn1+ = n # Schematic visualization of successive approximations: Chapter 7 INTEGRAL EQUATIONS Successive approximations can be used for solution of operator equation fLf= For example, in this case, the successive approximations converge to the fixed point: But they do not always converge to the fixed point of operator equation. This example shows that even the choice of the starting point close to the fixed point yields the divergent sequence of successive approximations (apparently they are not very successive / ): The following theorem establishes the sufficient condition for convergence of successive approximations to the fixed point of operator equation. Chapter 7 INTEGRAL EQUATIONS Theorem (Banach Fixed Point Theorem, 1922) Let S be a non-empty closed subset of Banach space M , SM⊂ , S ≠ ∅ . And let L:S→ S be a contraction operator with constant k1< . Then the sequence of successive approximations { fnn f=∈ Lf n10− , f S} converges to the unique fixed point fS,∈ f=Lf for any starting point fS0 ∈ fn → f and the following estimate is valid 1kn ff−≤ ff − ≤ ff − nnn101k1k+ − Proof: n • Using mathematical induction, show that ffnn1−≤+ kff 01 − (☼) 0 Verify for n0= ff01−= kff 01 −=− ff 01 true n Assume for n : ffnn1−≤+ kff 01 − n1+ Show for n1+ : ffn1++−≤ n2 kff 0 − 1 Indeed, ffn1++− n2 =−Lfnn1 Lf + definition of s.a. ≤−kfnn1 f+ Lipschitz condition n ≤⋅kk f01 − f assumption n1+ =−kff01 • Show that {}fn is a Cauchy sequence, i.e. lim fmn− f= 0 n,m→∞ Consider fffLfLfLfLffmn−= m − m + m − n + nn − (add and subtract) Apply Minkowski inequality twice: fmn− f ≤−fmmLf + Lf mn − Lf + Lf nn − f ≤−fmmLf + k f mn −+− f Lf nn f Lipschitz condition fLfLff−+− ≤ mm nn 1k− f −+−fff ≤ mm1nn1++ definition of s.a. 1k− kfmn−+ f kf − f ≤ 01 01 equation (☼) 1k− kkmn+ =−f f → 0 when m,n →∞ 1k− 01 Chapter 7 INTEGRAL EQUATIONS • Because vector space M is complete, Cauchy sequence { fn } converges to some fM∈ . And because fSn ∈ and set S is closed (includes all limiting points), fS∈ . Therefore in a limit, equation of successive approximations fLfn1+ = n ⇒ lim fn1+ = lim Lf n nn→∞ →∞ lim fn1+ = L lim f n nn→∞( →∞ ) converges to fLf= And therefore, fS∈ is a fixed point. • (Uniqueness) Let f,g∈ S be two fixed points of operator L : fLf= g = Lg Then from f − gLfLgkfg=−≤ − yields (1k− ) f−≤ g 0 Because 1k− > 0 f − g0≤ That is possible only if f − g0= Therefore, fg= So the fixed point is unique.
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