ABSTRACT
Let X be a real Banach space, X its conjugate dual space. Let A be a
monotone angle-bounded continuous linear mapping of X into X with constant
of angle-boundedness c 0. Let N be a hemicontinuous (possibly nonlinear)
mapping of X into X such that for a given constant k 0;
hv1 v2; Nv1 Nv2i kkv1 v2k2
X
for all v1 and v2 in X. Suppose finally that there exists a constant R with
k(1 + c2)R < 1 such that for u 2 X
hAu; ui Rkuk2
X:
Then, there exists exactly one solution w in X of the nonlinear equation
w + ANw = 0:
Existence and uniqueness is also proved using variational methods.
vii
TABLE OF CONTENTS
Dedication iii
Preface iv
Acknowledgement vi
Abstract vii
1 General Introduction
1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Definition and examples of some basic terms . . . . . . . . . . . 2
1.3 Hammerstein Equations . . . . . . . . . . . . . . . . . . . . . . 10
2 Existence and Uniqueness Results Using Factorization of Operators
13
2.1 Existence and uniqueness theorem . . . . . . . . . . . . . . . . . 13
2.2 Result of Minty [5] . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Proof of theorem (2.1.1) . . . . . . . . . . . . . . . . . . . . . . 17
3 Existence and Uniqueness Results Using Variational Methods 20
3.1 G^ateaux derivative and gradient . . . . . . . . . . . . . . . . . . 20
3.2 Maxima and minima of functions . . . . . . . . . . . . . . . . . 22
3.3 Fundamental theorems of optimization . . . . . . . . . . . . . . 23
3.4 Extension of Vainberg’s result to real
Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Bibliography 30
CHAPTER ONE
General Introduction
1.1 Introduction
The contribution of this thesis falls within the general area of nonlinear functional
analysis. Within this area, our attention is focused on the topic: “Existence
and Uniqueness of Solutions of Nonlinear Hammerstein Integral Equations”
in Banach spaces. We study theorems that establish existence and
uniqueness of solutions of these equations using factorization of operators and
variational methods.
Several classical problems in the theory of differential equations lead to
integral equations. In many cases, these equations can be dealt with in a
more satisfactory manner using the integral form than directly with differential
equations.
Interest in Hammerstein equations stem mainly from the fact that several
problems that arise in differential equations, for instance, elliptic boundary
value problems whose linear parts possess Green’s function can, as a rule be
transformed into a nonlinear integral equation of Hammerstein type. Elliptic
boundary value problems are a class of problems which do not involve time
variable but only depend on the space variables. That is, they are class of problems
which are typically associated with steady state behaviour. An example
is a Laplace’s equation:
r2u = 0 e.g @2u
@x2 +
@2u
@y2 = 0 in 2D :
Consequently, solvability of such differential equations is equivalent to the
solvability of the corresponding Hammerstein equation.
1
1.2 Definition and examples of some basic terms
In this section, definitions of basic terms used are given.
Throughout this chapter, X denotes a real Banach space and X denotes
its corresponding dual. We shall denote by the pairing hx; xi or x(x) the
value of the functional x 2 X at x 2 X: The norm in X is denoted by k:k,
while the norm in X is denoted by k:k. If there is no danger of confusion, we
omit the asterisk and denote both norms in X and X by the symbol k:jj. We
shall use the symbol ! to indicate strong and * to indicate weak convergence.
We shall also use w! to indicate the weak-star convergence.
The first term we define is monotone map. The concept of monotonicity
pertains to nonlinear functional analysis, and its use in the theory of functional
equations (ordinary differential equations, integral equations, integrodifferential
equations, delay equations) is probably the most powerful method
in obtaining existence theorems.
Definition 1.2.1 (Monotone Operator): A map A : D(A) X ! 2X is
said to be monotone if 8 x; y 2 D(A); x 2 Ax; y 2 Ay, we have
hx y; x yi 0:
From the definition above, a single-valued map A : D(A) X ! X is monotone
if
hAx Ay; x yi 0; 8 x; y 2 D(A):
Remark 1.2.1 For a linear map A, the above definition reduces to
hAu; ui 0 8 u 2 D(A):
The following are some examples of monotone operators.
Example 1.2.1 Every nondecreasing function on R is monotone.
Proof.
Let f : R ! R be a nondecreasing function. Then for arbitrary x; y 2 R, both
(f(x) f(y)) and (x y) have the same sign. Thus we see that
hf(x) f(y); x yi = (f(x) f(y))(x y) 0 8 x; y 2 R. Hence, f is
monotone.
Example 1.2.2 Let h : R2 ! R2 be defined as h(x; y) = (2x; 5y);
8 (x; y) 2 R2. Then h is montone.
Proof.
For arbitrary (x1; y1); (x2; y2) 2 R2; we have
hh(x1; y1) h(x2; y2); (x1; y1) (x2; y2)i = 2(x1 x2)2 + 5(y1 y2)2 0:
Thus, h is monotone.
2
Example 1.2.3 Let H be a real Hilbert space, I is the identity map of H and
T : H ! H be a non-expansive map (i:e kTx Tyk kx yk 8 x; y 2 H).
Then the operator I T is monotone.
Proof.
Let x; y 2 H; then
h(I T)x (I T)y; x yi = h(x y) (Tx Ty); x yi
= kx yk2 hTx Ty; x yi
kx yk2 kTx Tyk:kx yk
kx yk2 kx yk2 = 0 (T is nonexpansive).
Thus we have that I T is monotone on H.
Example 1.2.4 Let A = (1 0
0 0) and x = (xy
). Consider the function
g : R2 ! R2 defined by g(x) = Ax: Then g is monotone.
Proof.
Since g is linear, by remark (1.2.1) it suffices to show that hg(x); xi 0. For
arbitrary x = (xy
) 2 R2; we have Ax = (1 0
0 0)(xy
) = (x0
).
Thus hg(x); xi = hAx; xi = x2 + 0 = x2 0. Hence g is monotone.
Example 1.2.5 Let X be a real Banach space. The duality map J : X ! 2X
defined by
Jx := fx 2 X : hx; xi = kxk:kxk; kxk = kxk; x 2 Xg
is monotone.
Proof.
Let x; y 2 X and x 2 Jx; y 2 Jy. Then
hx y; x yi = hx y; xi hx y; yi
= hx; xi hy; xi hx; yi + hy; yi
= kxk2 + kyk2 hy; xi hx; yi
kxk2 + kyk2 kyk:kxk kxk:kyk
= kxk2 + kyk2 2kxk:kyk
= (kxk kyk)2 0:
Thus, J is monotone.
Example 1.2.6 Let f : X ! R[f+1g be convex and proper. The subdifferential
of f; @f : X ! 2X defined as
@f(x) =
fx 2 X : hy x; xi f(y) f(x); y 2 Xg ; if f(x) 6= 1
;; if f(x) = 1;
is monotone.
3
Proof.
Let x; y 2 X; x 2 @f(x) and y 2 @f(y).
x 2 @f(x) ) hy x; xi f(y) f(x) 8 y 2 X: (1.2.1)
y 2 @f(y) ) hx y; yi f(x) f(y) 8 x 2 X
) hy x; yi f(x) f(y) 8 x 2 X: (1.2.2)
Adding inequalities (1.2.1) and (1.2.2), we have
hy x; xi hy x; yi 0:
This implies that hy x; x yi 0, i.e hx y; x yi 0.
Definition 1.2.2 (Hemicontinuity): A mapping A : D(A) X ! X is
said to be hemicontinuous if it is continuous from each line segment of X to
the weak topology of X. That is, 8 u 2 D(A); 8 v 2 X and (tn)n1 R+
such that tn ! 0+ and u + tnv 2 D(A) for n sufficiently large, we have
A(u + tnv) * A(u).
Proposition 1.2.1 Let X denote a Banach space and X its corresponding
dual. Let A : D(A) X ! X be a continuous mapping . Then A is
hemicontinuous.
Proof
Let u 2 D(A); v 2 X, (tn)n1 be a sequence of positive numbers such that
tn ! 0+ as n ! 1 and (u + tnv) 2 D for n large enough. We observe that
(u + tnv) ! u as n ! 1 because tn ! 0+ as n ! 1. By the continuity
of A, we have A(u + tnv) ! A(u) as n ! 1. Since strong convergence
implies weak convergence we have A(u + tnv) * A(u) as n ! 1: Hence A is
hemicontinuous.
Remark 1.2.2 The converse of proposition (1.2.1) is false.
Consider the function f : R2 ! R2 defined by
f(x; y) =
(
( x2+xy2
x2+y4 ; x); if (x; y) 6= (0; 0)
(1; 0); if (x; y) = (0; 0):
Clearly, f is not continuous at (0; 0). For,
f(x; 0) = ( x2
x2 ; x) = (1; x) for all x 6= 0: This implies lim
x!0
f(x; 0) = (1; 0).
f(0; y) = (0; 0); 8y 6= 0. This implies lim
y!0
f(0; y) = (0; 0). Thus, the
limit does not exist at (0; 0). Hence, f is not continuous at (0,0).
4
However, f is hemicontinuous. Indeed, let u = (0; 0); v = (v1; v2) and
ftngn1 be arbitrary such that tn ! 0+ as n ! 1. Then,
f(u + tnv) = f(tnv1; tnv2)) =
v2
1+tnv1v2
2
v2
1+t2
nv4
2
; tnv1
! (1; 0); as n ! 1: Therefore,
lim
n!1
f(u + tnv) = (1; 0) = f(0; 0). Thus, f(u + tnv) ! f(u) as tn ! 0+.
Hence, f is hemicontinuous on R2 since strong and weak convergence are the
same on R2.
Definition 1.2.3 (Coercivity): An operator A : X ! X is said to be
coercive if for any x 2 X; hx;Axi
kxk ! 1 as kxk ! 1:
Example 1.2.7 Let H be a real Hilbert space and f : H ! H be defined by
f(x) = 1
2u. Then, f is coercive.
Proof.
Let x 2 H be arbitrary. Then,
hf(x); xi
kxk
=
1
2 hx; xi
kxk
=
1
2kxk2
kxk
=
1
2
kxk ! +1 as kxk ! 1:
Hence f is coercive.
Definition 1.2.4 (Symmetry): Let A : X ! X be a bounded linear mapping.
A is said be symmetric if for all u and v in X, we have hAu; vi = hAv; ui :
Example 1.2.8 Let A : l2(R) ! l2(R) be a map defined by Au = 1
2u. Then
A is symmetric.
Proof.
For arbitrary u; v 2 l2;
hAu; vi =
1
2
u; v
=
1
2
h(u1; u2; :::); (v1; v2; :::)i =
1
2
X1
i=1
uivi
=
1
2
X1
i=1
viui =
1
2
h(v1; v2; :::); (u1; u2; :::)i
=
1
2
v; u
= hu; Avi :
Hence A is symmetric.
Definition 1.2.5 (Skew-symmetry): Let A : X ! X be a bounded linear
mapping. A is said be skew-symmetric if for all u and v in X, we have
hAu; vi = hAv; ui :
5
Definition 1.2.6 (Angle-boundedness): Let A : X ! X be a bounded
monotone linear mapping . A is said be angle-bounded with constant c 0 if
for all u, v in X, j hAu; vihAv; ui j 2c fhAu; uig
1
2 fhAv; vig
1
2 . (This is well
defined since hAu; ui 0 and hAv; vi 0 by the linearity and monotonicity of
A).
Example 1.2.9 A symmetric map. It follows that every symmetric mapping
A of X into X is angle-bounded with constant of angle-boundedness c = 0:
Definition 1.2.7 (Adjoint Operators): Let X and Y be normed linear
spaces and A 2 B(X; Y ): The adjoint of A, denoted by A, is the operator
A : Y ! X defined by hAy; xi = hy; Axi for all y 2 Y and all
x 2 X.
We note that A is well-defined. Indeed, 8 y 2 Y ; x1; x2 2 X and 2 R,
we have
hAy; x1 + x2i = hy;A(x1 + x2)i = hy; Ax1i + hy; Ax2i
= hy; Ax1i + hy; Ax1i
which shows that Ay is linear.
For boundedness, given y 2 Y and x 2 X;
j hAy; xi j = j hy; Axi j
kyk:kAxk since y 2 Y .
kyk:kAk:kxk since A 2 B(X; Y ).
Therefore, for all y 2 Y ,
j hAy; xi j Kykxk 8 x 2 X; where Ky = kyk:kAk 0:
Hence, for all y 2 Y ;Ay 2 X:
Theorem 1.2.1 Let A : X ! Y be a bounded linear maps with adjoint A.
Then,
(a) A 2 B(Y ;X);
(b) kAk = kAk.
Proof.
(a) Let y; z 2 Y and 2 R. We show that
A (y + z) = Ay + Az;
6
i.e
8 x 2 X; hA (y + z) ; xi = hAy; xi + hAz; xi :
Let x 2 X: Then
hA (y + z) ; xi = hy + z; Axi = hy; Axi + hz; Axi
= hAy; xi + hAz; xi :
So, A is linear.
Furthermore, for any y 2 Y and x 2 X,
j hAy; xi j = j hy; Axi j kyk:kAk:kxk; since A 2 B(X; Y ) :
Thus, kAyk = sup
kxk=1
j hAy; xi j kAk:kyk: Therefore, kAyk
Kkyk; where K = kAk 0: Hence A 2 B(Y ;X).
(b)
kAk = sup
kxk=1
kAxk = sup
kxk=1
sup
kyk=1
hy; Axi
!
= sup
kxk=1
sup
kyk=1
hAy; xi
!
= sup
kyk=1
sup
kxk=1
hAy; xi
!
= sup
kyk=1
kAyk = kAk:
Definition 1.2.8 (Weak Topology): Let (X; !) denote a Banach space endowed
with the weak topology. For an arbitrary sequence fxngn1 X and
x 2 X, we say that fxng converges weakly to x if f(xn) ! f(x) for each
f 2 X. We denote this by xn * x:
Definition 1.2.9 (Weak Star Topology): Let (X; !) denote a Banach
space endowed with the weak star topology. For an arbitrary sequence ffngn1
X and f 2 X we say that ffng converges to f in weak-star topology, denoted
fn
!
! f, if fn(x) ! f(x) for each x 2 X.
Proposition 1.2.2 Let fxng be a sequence and x a point in X. Then the
following hold.
(a) xn ! x ) xn * x;
(b) xn * x ) fxng is bounded and kxk lim inf kxnk;
7
(c) xn * x (in X), fn ! f (in X) ) fn(xn) ! f(x) (in R).
Definition 1.2.10 (Reflexive Space): Let X be a Banach space and let
J : X ! X be the canonical injection from X into X, that is hJ(x); fi =
hf; xi ; 8 x 2 X; f 2 X. Then X is said to be reflexive if J is surjective, i.e
J(X) = X:
Definition 1.2.11 (Uniformly convex Banach spaces): A Banach space
X is called uniformly convex if for any 2 (0; 2], there exists a = () > 0
such that if x; y 2 X, with kxk 1; kyk 1 and kx yk , then
k1
2 (x + y)k 1 .
Hilbert spaces, Lp and lp spaces, 1 < p < 1 are examples of uniformly
convex spaces.
Definition 1.2.12 (Strictly convex spaces): A normed linear space X is
said to be strictly convex if for all x; y 2 X; x 6= y; kxk = kyk = 1, we
have kx + (1 )yk < 1 for all 2 (0; 1).
Theorem 1.2.2 Milman-Pettis Theorem: Every uniformly convex Banach
space X is reflexive.
For the proof of theorem (1.2.2), see, for instance, Chidume [1].
Definition 1.2.13 (algebra): A collection M of subsets of a nonempty
set
is called a algebra if
(a) ;
2M,
(b) A2 M ! Ac 2 M,
(c) [1 n=1An 2 M whenever An 2 M 8 n.
Definition 1.2.14 (Measurable Space): If M is a algebra of
, then
the pair (
; M) is referred to as a measurable space.
Definition 1.2.15 (Measure): A measure on (
; M) is a function
: M! [0; 1] such that
(a) (A) 0 for all A 2M;
(b) () = 0;
(c) if Ai 2M are pairwise disjoint, then ([1i
Ai) =
P1
i=1 (Ai).
Definition 1.2.16 (Measure Space): If M is a algebra of subsets of
, and is a measure on M, then the tripple (
; M; ) is referred to as a
measure space.
8
Definition 1.2.17 (Measurable Functions): Let (
; M) be a measurable
space. A function f :
! R is measurable or Mmeasurable if the set
fx 2
: f(x) > g 2M for all 2 R.
Definition 1.2.18 (finite ) : A measure space (
; M; ) is said to be
finite if there exists a countable family (
n)n1 inMsuch that
= [1 n=1
n
and (
n) < 1; 8 n:
Definition 1.2.19 (Green’s Function): This is a function associated with
a given boundary value problem, which appears as an integrand for an integral
representation of the solution of the problem.
Let L be a differential operator and assume that
L(y) =
Xn
p=0
aP (t)y(p)(t) = an(t)yn(t) + ::: + a(t)y(1)(t) + a0(t)y(t):
Suppose that an(t) is not zero on [0; 1] and that each term of the sequence
ap(t); p = 0; :::; n, has at least n continuous derivatives. Also suppose that
B is the given boundary conditions associated with L and denote by M, the
manifold associated with (L;B). (Manifold simply refers to the differential
equation together with the associated boundary conditions.) We present the
algorithm for constructing the Green’s function, G(t; x) for nth order equations.
For x 2 [0; 1], we denote by x, the values of t 2 [0; x) and by x+, the
values of t 2 (x; 1] .
(a) L(G(:; x)) (t) = 0 for 0 < t < x and for x < t < 1;
(b) G(:; x) is in M;
(c) for 0 p n 2, @pG(t;x)
@tp =t=x+ = @pG(t;x)
@tp =t=x ;
(d) @n1G(t;x)
@tn1 =t=x+ @n1G(t;x)
@tn1 =t=x = 1
an(x) .
Definition 1.2.20 (Caratheodory Condition): Let m and n be positive
integers,
be a nonempty subset of Rm and let f be a function from
Rn
into R. A function f :
Rn ! R is said to satisfy the Caratheodory
conditions if
(i) f(x; 🙂 : Rn ! R is a continuous function for almost all x 2
;
(ii) f(:; u) :
! R is a measurable function for all u 2 Rn.
Definition 1.2.21 (Nemystkii Operators): Let f be a function from
Rn into R. We denote by F(X; Y ), the set of all maps from X to Y . The
Nemystkii operator associated to f is the operator Nf : F(
;Rn) ! F(
;R)
defined by
u 7! Nf (u)
where (Nfu)(x) = f (x; u(x)) 8 u 2 F (
; Rn) ; 8 x 2
: For simplicity, we
shall write Nuf (x) instead of (Nfu)(x).
9
Example 1.2.10 Given a map f : R R ! R defined by
f(x; s) = jsj 8 (x; s) 2 R R;
the Nemystkii operator associated to f is given by the expression Nfu(x) =
ju(x)j for any map u : R ! R and for any x 2 R.
Example 1.2.11 Given a map g : R R ! R defined by
g(x; s) = xes 8 (x; s) 2 R R;
the Nemystkii operator associated to g is given by the expression Nfu(x) =
xeu(x) for any map u : R ! R and for any x 2 R.
Observe that by the continuity of f and g, Nf and Ng map the set of
real-valued continuous function on
; C(
) into itself. Moreover, they map
the set of real-valued measurable function into itself.
1.3 Hammerstein Equations
A nonlinear integral equation of Hammerstein type on
is one of the form
u(x) +
Z
k(x; y)f(y; u(y))dy = h(x) (1.3.1)
where dy stands for
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