Existence and uniqueness of solutions of integral equations of hammerstein type

Thesis Overview

<p> </p><p>General Introduction<br>1.1 Introduction<br>The contribution of this thesis falls within the general area of nonlinear functional<br>analysis. Within this area, our attention is focused on the topic: “Existence<br>and Uniqueness of Solutions of Nonlinear Hammerstein Integral Equations”<br>in Banach spaces. We study theorems that establish existence and<br>uniqueness of solutions of these equations using factorization of operators and<br>variational methods.<br>Several classical problems in the theory of differential equations lead to<br>integral equations. In many cases, these equations can be dealt with in a<br>more satisfactory manner using the integral form than directly with differential<br>equations.<br>Interest in Hammerstein equations stem mainly from the fact that several<br>problems that arise in differential equations, for instance, elliptic boundary<br>value problems whose linear parts possess Green’s function can, as a rule be<br>transformed into a nonlinear integral equation of Hammerstein type. Elliptic<br>boundary value problems are a class of problems which do not involve time<br>variable but only depend on the space variables. That is, they are class of problems<br>which are typically associated with steady state behaviour. An example<br>is a Laplace’s equation:<br>r2u = 0 e.g @2u<br>@x2 +<br>@2u<br>@y2 = 0 in 2D :<br>Consequently, solvability of such differential equations is equivalent to the<br>solvability of the corresponding Hammerstein equation.<br>1<br>1.2 Definition and examples of some basic terms<br>In this section, definitions of basic terms used are given.<br>Throughout this chapter, X denotes a real Banach space and X denotes<br>its corresponding dual. We shall denote by the pairing hx; xi or x(x) the<br>value of the functional x 2 X at x 2 X: The norm in X is denoted by k:k,<br>while the norm in X is denoted by k:k. If there is no danger of confusion, we<br>omit the asterisk and denote both norms in X and X by the symbol k:jj. We<br>shall use the symbol ! to indicate strong and * to indicate weak convergence.<br>We shall also use w! to indicate the weak-star convergence.<br>The first term we define is monotone map. The concept of monotonicity<br>pertains to nonlinear functional analysis, and its use in the theory of functional<br>equations (ordinary differential equations, integral equations, integrodifferential<br>equations, delay equations) is probably the most powerful method<br>in obtaining existence theorems.<br>Definition 1.2.1 (Monotone Operator): A map A : D(A) X ! 2X is<br>said to be monotone if 8 x; y 2 D(A); x 2 Ax; y 2 Ay, we have<br>hx ô€€€ y; x ô€€€ yi 0:<br>From the definition above, a single-valued map A : D(A) X ! X is monotone<br>if<br>hAx ô€€€ Ay; x ô€€€ yi 0; 8 x; y 2 D(A):<br>Remark 1.2.1 For a linear map A, the above definition reduces to<br>hAu; ui 0 8 u 2 D(A):<br>The following are some examples of monotone operators.<br>Example 1.2.1 Every nondecreasing function on R is monotone.<br>Proof.<br>Let f : R ! R be a nondecreasing function. Then for arbitrary x; y 2 R, both<br>(f(x) ô€€€ f(y)) and (x ô€€€ y) have the same sign. Thus we see that<br>hf(x) ô€€€ f(y); x ô€€€ yi = (f(x) ô€€€ f(y))(x ô€€€ y) 0 8 x; y 2 R. Hence, f is<br>monotone.<br>Example 1.2.2 Let h : R2 ! R2 be defined as h(x; y) = (2x; 5y);<br>8 (x; y) 2 R2. Then h is montone.<br>Proof.<br>For arbitrary (x1; y1); (x2; y2) 2 R2; we have<br>hh(x1; y1) ô€€€ h(x2; y2); (x1; y1) ô€€€ (x2; y2)i = 2(x1 ô€€€ x2)2 + 5(y1 ô€€€ y2)2 0:<br>Thus, h is monotone.<br>2<br>Example 1.2.3 Let H be a real Hilbert space, I is the identity map of H and<br>T : H ! H be a non-expansive map (i:e kTx ô€€€ Tyk kx ô€€€ yk 8 x; y 2 H).<br>Then the operator I ô€€€ T is monotone.<br>Proof.<br>Let x; y 2 H; then<br>h(I ô€€€ T)x ô€€€ (I ô€€€ T)y; x ô€€€ yi = h(x ô€€€ y) ô€€€ (Tx ô€€€ Ty); x ô€€€ yi<br>= kx ô€€€ yk2 ô€€€ hTx ô€€€ Ty; x ô€€€ yi<br>kx ô€€€ yk2 ô€€€ kTx ô€€€ Tyk:kx ô€€€ yk<br>kx ô€€€ yk2 ô€€€ kx ô€€€ yk2 = 0 (T is nonexpansive).<br>Thus we have that I ô€€€ T is monotone on H.<br>Example 1.2.4 Let A = (1 0<br>0 0) and x = (xy<br>). Consider the function<br>g : R2 ! R2 defined by g(x) = Ax: Then g is monotone.<br>Proof.<br>Since g is linear, by remark (1.2.1) it suffices to show that hg(x); xi 0. For<br>arbitrary x = (xy<br>) 2 R2; we have Ax = (1 0<br>0 0)(xy<br>) = (x0<br>).<br>Thus hg(x); xi = hAx; xi = x2 + 0 = x2 0. Hence g is monotone.<br>Example 1.2.5 Let X be a real Banach space. The duality map J : X ! 2X<br>defined by<br>Jx := fx 2 X : hx; xi = kxk:kxk; kxk = kxk; x 2 Xg<br>is monotone.<br>Proof.<br>Let x; y 2 X and x 2 Jx; y 2 Jy. Then<br>hx ô€€€ y; x ô€€€ yi = hx ô€€€ y; xi ô€€€ hx ô€€€ y; yi<br>= hx; xi ô€€€ hy; xi ô€€€ hx; yi + hy; yi<br>= kxk2 + kyk2 ô€€€ hy; xi ô€€€ hx; yi<br>kxk2 + kyk2 ô€€€ kyk:kxk ô€€€ kxk:kyk<br>= kxk2 + kyk2 ô€€€ 2kxk:kyk<br>= (kxk ô€€€ kyk)2 0:<br>Thus, J is monotone.<br>Example 1.2.6 Let f : X ! R[f+1g be convex and proper. The subdifferential<br>of f; @f : X ! 2X defined as<br>@f(x) =</p><p>fx 2 X : hy ô€€€ x; xi f(y) ô€€€ f(x); y 2 Xg ; if f(x) 6= 1<br>;; if f(x) = 1;<br>is monotone.<br>3<br>Proof.<br>Let x; y 2 X; x 2 @f(x) and y 2 @f(y).<br>x 2 @f(x) ) hy ô€€€ x; xi f(y) ô€€€ f(x) 8 y 2 X: (1.2.1)<br>y 2 @f(y) ) hx ô€€€ y; yi f(x) ô€€€ f(y) 8 x 2 X<br>) ô€€€hy ô€€€ x; yi f(x) ô€€€ f(y) 8 x 2 X: (1.2.2)<br>Adding inequalities (1.2.1) and (1.2.2), we have<br>hy ô€€€ x; xi ô€€€ hy ô€€€ x; yi 0:<br>This implies that hy ô€€€ x; x ô€€€ yi 0, i.e hx ô€€€ y; x ô€€€ yi 0.<br>Definition 1.2.2 (Hemicontinuity): A mapping A : D(A) X ! X is<br>said to be hemicontinuous if it is continuous from each line segment of X to<br>the weak topology of X. That is, 8 u 2 D(A); 8 v 2 X and (tn)n1 R+<br>such that tn ! 0+ and u + tnv 2 D(A) for n sufficiently large, we have<br>A(u + tnv) * A(u).<br>Proposition 1.2.1 Let X denote a Banach space and X its corresponding<br>dual. Let A : D(A) X ! X be a continuous mapping . Then A is<br>hemicontinuous.<br>Proof<br>Let u 2 D(A); v 2 X, (tn)n1 be a sequence of positive numbers such that<br>tn ! 0+ as n ! 1 and (u + tnv) 2 D for n large enough. We observe that<br>(u + tnv) ! u as n ! 1 because tn ! 0+ as n ! 1. By the continuity<br>of A, we have A(u + tnv) ! A(u) as n ! 1. Since strong convergence<br>implies weak convergence we have A(u + tnv) * A(u) as n ! 1: Hence A is<br>hemicontinuous.<br>Remark 1.2.2 The converse of proposition (1.2.1) is false.<br>Consider the function f : R2 ! R2 defined by<br>f(x; y) =<br>(<br>( x2+xy2<br>x2+y4 ; x); if (x; y) 6= (0; 0)<br>(1; 0); if (x; y) = (0; 0):<br>Clearly, f is not continuous at (0; 0). For,<br>f(x; 0) = ( x2<br>x2 ; x) = (1; x) for all x 6= 0: This implies lim<br>x!0<br>f(x; 0) = (1; 0).<br>f(0; y) = (0; 0); 8y 6= 0. This implies lim<br>y!0<br>f(0; y) = (0; 0). Thus, the<br>limit does not exist at (0; 0). Hence, f is not continuous at (0,0).<br>4<br>However, f is hemicontinuous. Indeed, let u = (0; 0); v = (v1; v2) and<br>ftngn1 be arbitrary such that tn ! 0+ as n ! 1. Then,<br>f(u + tnv) = f(tnv1; tnv2)) =</p><p>v2<br>1+tnv1v2<br>2<br>v2<br>1+t2<br>nv4<br>2<br>; tnv1</p><p>! (1; 0); as n ! 1: Therefore,<br>lim<br>n!1<br>f(u + tnv) = (1; 0) = f(0; 0). Thus, f(u + tnv) ! f(u) as tn ! 0+.<br>Hence, f is hemicontinuous on R2 since strong and weak convergence are the<br>same on R2.<br>Definition 1.2.3 (Coercivity): An operator A : X ! X is said to be<br>coercive if for any x 2 X; hx;Axi<br>kxk ! 1 as kxk ! 1:<br>Example 1.2.7 Let H be a real Hilbert space and f : H ! H be defined by<br>f(x) = 1<br>2u. Then, f is coercive.<br>Proof.<br>Let x 2 H be arbitrary. Then,<br>hf(x); xi<br>kxk<br>=<br>1<br>2 hx; xi<br>kxk<br>=<br>1<br>2kxk2<br>kxk<br>=<br>1<br>2<br>kxk ! +1 as kxk ! 1:<br>Hence f is coercive.<br>Definition 1.2.4 (Symmetry): Let A : X ! X be a bounded linear mapping.<br>A is said be symmetric if for all u and v in X, we have hAu; vi = hAv; ui :<br>Example 1.2.8 Let A : l2(R) ! l2(R) be a map defined by Au = 1<br>2u. Then<br>A is symmetric.<br>Proof.<br>For arbitrary u; v 2 l2;<br>hAu; vi =</p><p>1<br>2<br>u; v</p><p>=<br>1<br>2<br>h(u1; u2; :::); (v1; v2; :::)i =<br>1<br>2<br>X1<br>i=1<br>uivi<br>=<br>1<br>2<br>X1<br>i=1<br>viui =<br>1<br>2<br>h(v1; v2; :::); (u1; u2; :::)i<br>=</p><p>1<br>2<br>v; u</p><p>= hu; Avi :<br>Hence A is symmetric.<br>Definition 1.2.5 (Skew-symmetry): Let A : X ! X be a bounded linear<br>mapping. A is said be skew-symmetric if for all u and v in X, we have<br>hAu; vi = ô€€€hAv; ui :<br>5<br>Definition 1.2.6 (Angle-boundedness): Let A : X ! X be a bounded<br>monotone linear mapping . A is said be angle-bounded with constant c 0 if<br>for all u, v in X, j hAu; viô€€€hAv; ui j 2c fhAu; uig<br>1<br>2 fhAv; vig<br>1<br>2 . (This is well<br>defined since hAu; ui 0 and hAv; vi 0 by the linearity and monotonicity of<br>A).<br>Example 1.2.9 A symmetric map. It follows that every symmetric mapping<br>A of X into X is angle-bounded with constant of angle-boundedness c = 0:<br>Definition 1.2.7 (Adjoint Operators): Let X and Y be normed linear<br>spaces and A 2 B(X; Y ): The adjoint of A, denoted by A, is the operator<br>A : Y ! X defined by hAy; xi = hy; Axi for all y 2 Y and all<br>x 2 X.<br>We note that A is well-defined. Indeed, 8 y 2 Y ; x1; x2 2 X and 2 R,<br>we have<br>hAy; x1 + x2i = hy;A(x1 + x2)i = hy; Ax1i + hy; Ax2i<br>= hy; Ax1i + hy; Ax1i<br>which shows that Ay is linear.<br>For boundedness, given y 2 Y and x 2 X;<br>j hAy; xi j = j hy; Axi j<br>kyk:kAxk since y 2 Y .<br>kyk:kAk:kxk since A 2 B(X; Y ).<br>Therefore, for all y 2 Y ,<br>j hAy; xi j Kykxk 8 x 2 X; where Ky = kyk:kAk 0:<br>Hence, for all y 2 Y ;Ay 2 X:<br>Theorem 1.2.1 Let A : X ! Y be a bounded linear maps with adjoint A.<br>Then,<br>(a) A 2 B(Y ;X);<br>(b) kAk = kAk.<br>Proof.<br>(a) Let y; z 2 Y and 2 R. We show that<br>A (y + z) = Ay + Az;<br>6<br>i.e<br>8 x 2 X; hA (y + z) ; xi = hAy; xi + hAz; xi :<br>Let x 2 X: Then<br>hA (y + z) ; xi = hy + z; Axi = hy; Axi + hz; Axi<br>= hAy; xi + hAz; xi :<br>So, A is linear.<br>Furthermore, for any y 2 Y and x 2 X,<br>j hAy; xi j = j hy; Axi j kyk:kAk:kxk; since A 2 B(X; Y ) :<br>Thus, kAyk = sup<br>kxk=1<br>j hAy; xi j kAk:kyk: Therefore, kAyk<br>Kkyk; where K = kAk 0: Hence A 2 B(Y ;X).<br>(b)<br>kAk = sup<br>kxk=1<br>kAxk = sup<br>kxk=1</p><p>sup<br>kyk=1<br>hy; Axi<br>!<br>= sup<br>kxk=1</p><p>sup<br>kyk=1<br>hAy; xi<br>!<br>= sup<br>kyk=1</p><p>sup<br>kxk=1<br>hAy; xi<br>!<br>= sup<br>kyk=1<br>kAyk = kAk:<br>Definition 1.2.8 (Weak Topology): Let (X; !) denote a Banach space endowed<br>with the weak topology. For an arbitrary sequence fxngn1 X and<br>x 2 X, we say that fxng converges weakly to x if f(xn) ! f(x) for each<br>f 2 X. We denote this by xn * x:<br>Definition 1.2.9 (Weak Star Topology): Let (X; !) denote a Banach<br>space endowed with the weak star topology. For an arbitrary sequence ffngn1<br>X and f 2 X we say that ffng converges to f in weak-star topology, denoted<br>fn<br>!<br>ô€€€! f, if fn(x) ! f(x) for each x 2 X.<br>Proposition 1.2.2 Let fxng be a sequence and x a point in X. Then the<br>following hold.<br>(a) xn ! x ) xn * x;<br>(b) xn * x ) fxng is bounded and kxk lim inf kxnk;<br>7<br>(c) xn * x (in X), fn ! f (in X) ) fn(xn) ! f(x) (in R).<br>Definition 1.2.10 (Reflexive Space): Let X be a Banach space and let<br>J : X ! X be the canonical injection from X into X, that is hJ(x); fi =<br>hf; xi ; 8 x 2 X; f 2 X. Then X is said to be reflexive if J is surjective, i.e<br>J(X) = X:<br>Definition 1.2.11 (Uniformly convex Banach spaces): A Banach space<br>X is called uniformly convex if for any 2 (0; 2], there exists a = () &gt; 0<br>such that if x; y 2 X, with kxk 1; kyk 1 and kx ô€€€ yk , then<br>k1<br>2 (x + y)k 1 ô€€€ .<br>Hilbert spaces, Lp and lp spaces, 1 &lt; p &lt; 1 are examples of uniformly<br>convex spaces.<br>Definition 1.2.12 (Strictly convex spaces): A normed linear space X is<br>said to be strictly convex if for all x; y 2 X; x 6= y; kxk = kyk = 1, we<br>have kx + (1 ô€€€ )yk &lt; 1 for all 2 (0; 1).<br>Theorem 1.2.2 Milman-Pettis Theorem: Every uniformly convex Banach<br>space X is reflexive.<br>For the proof of theorem (1.2.2), see, for instance, Chidume [1].<br>Definition 1.2.13 (ô€€€algebra): A collection M of subsets of a nonempty<br>set<br>is called a ô€€€algebra if<br>(a) ;<br>2M,<br>(b) A2 M ! Ac 2 M,<br>(c) [1 n=1An 2 M whenever An 2 M 8 n.<br>Definition 1.2.14 (Measurable Space): If M is a ô€€€algebra of<br>, then<br>the pair (<br>; M) is referred to as a measurable space.<br>Definition 1.2.15 (Measure): A measure on (<br>; M) is a function<br>: M! [0; 1] such that<br>(a) (A) 0 for all A 2M;<br>(b) () = 0;<br>(c) if Ai 2M are pairwise disjoint, then ([1i<br>Ai) =<br>P1<br>i=1 (Ai).<br>Definition 1.2.16 (Measure Space): If M is a ô€€€algebra of subsets of</p><p>, and is a measure on M, then the tripple (<br>; M; ) is referred to as a<br>measure space.<br>8<br>Definition 1.2.17 (Measurable Functions): Let (<br>; M) be a measurable<br>space. A function f :<br>! R is measurable or Mô€€€measurable if the set<br>fx 2<br>: f(x) &gt; g 2M for all 2 R.<br>Definition 1.2.18 (ô€€€finite ) : A measure space (<br>; M; ) is said to be<br>ô€€€finite if there exists a countable family (<br>n)n1 inMsuch that<br>= [1 n=1<br>n<br>and (<br>n) &lt; 1; 8 n:<br>Definition 1.2.19 (Green’s Function): This is a function associated with<br>a given boundary value problem, which appears as an integrand for an integral<br>representation of the solution of the problem.<br>Let L be a differential operator and assume that<br>L(y) =<br>Xn<br>p=0<br>aP (t)y(p)(t) = an(t)yn(t) + ::: + a(t)y(1)(t) + a0(t)y(t):<br>Suppose that an(t) is not zero on [0; 1] and that each term of the sequence<br>ap(t); p = 0; :::; n, has at least n continuous derivatives. Also suppose that<br>B is the given boundary conditions associated with L and denote by M, the<br>manifold associated with (L;B). (Manifold simply refers to the differential<br>equation together with the associated boundary conditions.) We present the<br>algorithm for constructing the Green’s function, G(t; x) for nth order equations.<br>For x 2 [0; 1], we denote by xô€€€, the values of t 2 [0; x) and by x+, the<br>values of t 2 (x; 1] .<br>(a) L(G(:; x)) (t) = 0 for 0 &lt; t &lt; x and for x &lt; t &lt; 1;<br>(b) G(:; x) is in M;<br>(c) for 0 p n ô€€€ 2, @pG(t;x)<br>@tp =t=x+ = @pG(t;x)<br>@tp =t=xô€€€ ;<br>(d) @nô€€€1G(t;x)<br>@tnô€€€1 =t=x+ ô€€€ @nô€€€1G(t;x)<br>@tnô€€€1 =t=xô€€€ = 1<br>an(x) .<br>Definition 1.2.20 (Caratheodory Condition): Let m and n be positive<br>integers,<br>be a nonempty subset of Rm and let f be a function from<br>Rn<br>into R. A function f :<br>Rn ! R is said to satisfy the Caratheodory<br>conditions if<br>(i) f(x; <img alt="" src="https://s.w.org/images/core/emoji/11/svg/1f642.svg">&nbsp;: Rn ! R is a continuous function for almost all x 2<br>;<br>(ii) f(:; u) :<br>! R is a measurable function for all u 2 Rn.<br>Definition 1.2.21 (Nemystkii Operators): Let f be a function from</p><p>Rn into R. We denote by F(X; Y ), the set of all maps from X to Y . The<br>Nemystkii operator associated to f is the operator Nf : F(<br>;Rn) ! F(<br>;R)<br>defined by<br>u 7! Nf (u)<br>where (Nfu)(x) = f (x; u(x)) 8 u 2 F (<br>; Rn) ; 8 x 2<br>: For simplicity, we<br>shall write Nuf (x) instead of (Nfu)(x).<br>9<br>Example 1.2.10 Given a map f : R R ! R defined by<br>f(x; s) = jsj 8 (x; s) 2 R R;<br>the Nemystkii operator associated to f is given by the expression Nfu(x) =<br>ju(x)j for any map u : R ! R and for any x 2 R.<br>Example 1.2.11 Given a map g : R R ! R defined by<br>g(x; s) = xes 8 (x; s) 2 R R;<br>the Nemystkii operator associated to g is given by the expression Nfu(x) =<br>xeu(x) for any map u : R ! R and for any x 2 R.<br>Observe that by the continuity of f and g, Nf and Ng map the set of<br>real-valued continuous function on<br>; C(<br>) into itself. Moreover, they map<br>the set of real-valued measurable function into itself.<br>1.3 Hammerstein Equations<br>A nonlinear integral equation of Hammerstein type on<br>is one of the form<br>u(x) +<br>Z</p><p>k(x; y)f(y; u(y))dy = h(x) (1.3.1)<br>where dy stands for</p> <br><p></p>