contributions to the theory of fredholm-volterra integral equations
TRANSCRIPT
BABES-BOLYAI UNIVERSITYCLUJ NAPOCA
FACULTY OF MATHEMATICS AND COMPUTER SCIENCE
CONTRIBUTIONS TO THETHEORY OF
FREDHOLM-VOLTERRAINTEGRAL EQUATIONS
Resume of ph.d. paper
SCIENTIFIC SUPERVISOR: PH.D. CANDIDATE:
Prof. dr. Ioan A. Rus Szilard Andras
2004
Contents
Introduction 3
1 Preliminaries 6
2 Convex contractions 8
2.1 Subconvex sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Convex contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Convex contractions on generalized metric spaces . . . . . . . . . . . 10
2.3.1 Generalization of Perov’s theorem . . . . . . . . . . . . . . . . 10
2.3.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Gronwall type inequalities . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4.1 An abstract Gronwall type inequality . . . . . . . . . . . . . . 11
2.4.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4.3 A discrete Gronwall inequality . . . . . . . . . . . . . . . . . . 12
2.5 Convex contractions on fibers . . . . . . . . . . . . . . . . . . . . . . 13
2.5.1 The fiber contraction theorem for convex contractions . . . . . 13
2.5.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Fredholm-Volterra integral equations in C[a, b] 15
3.1 Existence theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Existence and uniqueness theorems . . . . . . . . . . . . . . . . . . . 17
3.2.1 The nonlinear case . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.2 The linear case . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.3 Weakly singular Fredholm-Volterra integral equations . . . . . 19
3.3 Differentiability of the solutions with respect to a parameter . . . . . 21
3.4 Fredholm-Volterra integral equations with deviating argument . . . . 22
1
CONTENTS 2
3.5 Comparison theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Fredholm-Volterra integral equations in L2[a, b] 26
4.1 Fredholm-Volterra integral equations on a compact interval . . . . . . 26
4.2 Fredholm-Volterra integral equations on unbounded intervals . . . . . 29
Introduction
The main purpose of this ph.d. thesis is the generalization of some theorems from
the theory of Picard operators to convex contractions both in metric and generalized
metric spaces and the study of the mixed Fredholm-Volterra integral equation
y(x) = f(x) +
x∫a
K1(x, s, y(s); λ)ds +
b∫a
K2(x, s, y(s); λ)ds, (0.0.1)
in the space C([a, b], X) and L2[a, b], where (X, ‖ · ‖) is a Banach space. In the
space C([a, b], X) we study the existence, the uniqueness, the continuity and the
differentiability of the solution with respect to the parameter. In the linear case
we obtain some recurrences for the iterated kernels and we deduce the integral
equations satisfied by the resolvent kernel. Most of the studied properties can be
extended to weakly singular kernels too. In the space L2[a, b] we study the existence
and uniqueness of the solution and the differentiability of the solution operator. In
both spaces we consider equations with modified argument too.
The thesis contains four chapters, an introduction and a bibliography.
In the first chapter we present some basic notations, notions and theorems. Most
of these theorems are well known so we omitted their proof. The original results from
this chapter are contained in lemma 1.4.5, lemma 1.4.6, lemma 1.4.7, theorem 1.4.4
and theorem 1.4.5. These results generalize some theorems and lemmas regarding
ϕ−contractions and were published in the paper [11](the numbers of theorems are
those from the thesis, not from this abstract).
Chapter 2 contains results about convex contractions. The fixed point theorems
were partially obtained by V. Istratescu in [56] and independently by Tascovici in
[112], but our method is different from theirs and permits an estimation of the
distance between the nth iterate and the solution. An other gain of this method is
the adaptability to generalized metric spaces (and even to more general spaces). The
3
CONTENTS 4
main idea is to use the properties of subconvex sequences. In some very specific cases
these sequences were studied by D. Barbosu, M. Andronache in [24], S.M. Soltuz in
[109] and J. van de Lune in [66]. The original results are contained in lemmas 2.1.1-
2.1.3, lemma 2.5.1, theorems 2.1.3, 2.1.4, 2.2.1, 2.3.1, 2.3.2, 2.4.2-2.4.7, 2.5.3, 2.5.4.
The most important theorems in this chapter are theorem 2.3.1, 2.4.3 and theorem
2.5.3. These theorems are generalizations of well known theorems like Perov’s fixed
point theorem, the fiber contractions theorem proved by Ioan A. Rus in [98-99],
and the abstract Gronwall lemma proved by Ioan A. Rus in [102]. Our results from
this chapter generalize results of M. Zima [117], B.G. Pachpatte [78], J.I. Wu and
G. Yang [113], and S.S. Dragomir [39]. The original results from this chapter were
published in [9], [10], [12] and [13].
In the third chapter we use fixed point theorems to study the existence and
uniqueness of solution of mixed Fredholm-Volterra integral equations. In the first
section we establish existence results by the aid of Schauder’s, Leray-Schauder’s and
Krasnoselskii’s fixed point theorem. In the second section we obtain an existence and
uniqueness theorem by applying Perov’s fixed point theorem in a product space. This
technique allows to obtain natural (but not simple) conditions which in special cases
(the absence of one of the two integrals) give us the well known classical conditions.
This theorem is more accurate then those from the papaers of I Narosi ([75]), A.
Petrusel ([84]), B.G. Pachpatte ([77]), D. Gou ([44]), V.M. Mamedov and Ja. D.
Musaev ([68]), I. Bihari ([21]), J. Kwapisz and M. Turo ([62]), R.K. Nohel, J.A. Wong
and J.S.W. Miller ([73]), and C. Corduneanu ([32]). In the next section we study
the linear case and we extend the obtained results to weakly singular equations. In
section 3 we study the differentiability with respect to the parameter by using the
fiber contractions theorem. In the fourth section of this chapter we investigate mixed
Fredholm-Volterra integral equations with deviating (both advanced and retarded)
argument. In the last section we give some comparison theorems using the general
framework of weakly Picard operators developed by Ioan A. Rus.
The original results from this chapter were published in the papers [8] and [14].
In the last chapter we study the continuity and differentiability of the solution
operator S : [λ1, λ2] → L2(I) defined by S(λ)(t) = y∗(t, λ), where y∗(·, λ) ∈ L2(I) is
the unique solution of the mixed Fredholm-Volterra integral equation on the interval
I. In the first section the interval I is considered a bounded interval and in the last
CONTENTS 5
section I is unbounded. The original results from this chapter will be published in
the paper [7].
In what follows we enumerate the original results from each chapter with some
definitions.
Chapter 1
Preliminaries
Further definitions and a more detailed introduction can be found in V. Berinde
[20] or in M.A. Serban [108] (for metric spaces these results were obtained by these
authors).
Definition 1.0.1 If K is the positive cone of an ordered Banach space, X is an
arbitrary set and d : X ×X → K satisfies the following properties:
1. d(x, y) = 0 ⇔ x = y;
2. d(x, y) = d(y, x), ∀x, y ∈ X;
3. d(x, y) ≤ d(x, z) + d(z, y), ∀x, y, z ∈ X,
then (X, d) is a generalized metric space whose metric takes its values in K.
Remark 1.0.1 If (X, d) is generalized metric space whose metric takes its values in
K, where K is the positive cone of an ordered Banach space with monotone norm,
then (X, d) is called a K−metric space. In many applications we use K = Rm+ .
Lemma 1.0.1 If ϕ : K → K is a (c)-comparison function and (αn)n∈N is a se-
quence in K, with the property limn→∞
αn = 0, then limn→∞
n∑k=0
ϕn−k(αk) = 0.
Lemma 1.0.2 If (X, d) is a K−metric space, ϕ : K → K a subadditive (c)-
comparison function and A, An : X → X some operators with the following proper-
ties:
6
7
a) the sequence (An)n∈N converges to A on each x ∈ X;
b) An and A are generalized ϕ−contractions for all n ∈ N;
then the sequence (An ◦ An−1 ◦ ... ◦ A1 ◦ A0) (x) converges to the unique fixed point
of the operator A.
Lemma 1.0.3 Let (X, d) be a K1−metric space and (Y, ρ) a K−metric space, where
K and K1 are the positive cones of two ordered Banach spaces with monotone norm,
ϕ : K → K be a (c)-comparison function, xn, x∗ ∈ X for all n ∈ N and T : X×Y →
Y an operator. If
a) limn→∞
xn = x∗;
b) ϕ is subadditive;
c) the operator T (·, y) : X → Y is continuous for all y ∈ Y ;
d) the operator T (x, ·) : Y → Y is a generalized ϕ−contraction for all x ∈ X;
e) (Y, ρ) is a complete K−metric space;
then the sequence yn+1 = T (xn, yn) , y1 = y converges to the unique fixed point
of the operator T (x∗, ·) : Y → Y, ∀y ∈ Y .
Theorem 1.0.1 Let (Xj, dj) be complete Kj−metric spaces for j = 1, p, and
(X0, d0) be a K0−metric space, where Kj, j = 0, p are the positive cones in some or-
dered Banach spaces with monotone norm. If the operators Ak : X0×X1× ...×Xk →Xk, k = 0, p satisfy the following conditions:
a) A0 is (weakly) Picard;
b) there exist some subadditive (c)-comparison functions ϕj : Kj → Kj such that
the operators Aj (x0, x1, ..., xj−1, ·) : Xj → Xj are ϕj−contractions for j = 1, p;
c) Aj is continuous with respect to (x0, x1, ..., xj−1) for all xj ∈ Xj and j = 1, p;
then the triangular operator Bp = (A0, A1, ..., Ap−1, Ap) is (weakly) Picard. Moreover
if A0 is a Picard operator and FA0 = {x∗0}, FA1(x∗0,·) = {x∗1}, ... , FAp(x∗0,x∗1,...,x∗p−1,·) ={x∗p
}, then FBp =
{(x∗0, x
∗1, ..., x
∗p−1, x
∗p
)}.
Chapter 2
Convex contractions
2.1 Subconvex sequences
Definition 2.1.1 The sequence (an)n≥1 is subconvex of order p if there exist αi ∈
(0, 1] , i = 0, p− 1 withp−1∑i=0
αi ≤ 1 such that
an+p ≤p−1∑i=0
αi · an+i, for all n ≥ 1.
Definition 2.1.2 The sequence (an)n≥1 is subconvex if there exists p ∈ N\{0} such
that (an)n≥1 is subconvex of order p.
Theorem 2.1.1 Every subconvex sequence of nonnegative real numbers is conver-
gent.
Theorem 2.1.2 If a sequence of nonnegative real numbers (an)n≥1 satisfies the con-
ditions below, then it is convergent.
1. an+p ≤p−1∑j=0
αj(n) · an+j ∀n ≥ 1, where αj(n) ∈ (0, 1] j = 0, p− 1 and
p−1∑j=0
αj(n) ≤ 1;
2. the sequences (αj (n))n≥1 are convergent for j = 0, p− 1;
3. min{
limn→∞
αj(n) |0 ≤ j ≤ p− 1}
> 0.
8
2.2. CONVEX CONTRACTIONS 9
Lemma 2.1.1 If the roots of the characteristic equationp−1∑j=0
βj · xj = 0 are in the
unit disk then any sequence (bn)n≥1 which satisfiesp−1∑j=0
βj · bn+j = 0, ∀n ≥ 1 is
convergent to 0 and∞∑
k=1
|bk| is also convergent.
Lemma 2.1.2 If 1 ≥ βp−1 > βp−2 > βp−3 > ... > β0 > 0, all the solutions of the
equationp−1∑j=0
βj · xj = 0 satisfy |x| < 1.(Kakeya’s theorem)
Lemma 2.1.3 If the sequence (an)n≥1 has positive terms and the sequence (cn)n≥1
with cn =p−1∑j=0
βj · an+j is convergent, where 1 ≥ βp−1 > βp−2 > βp−3 > ... > β0 > 0,
then (an)n≥1 is also convergent.
2.2 Convex contractions
The notion of convex contraction was introduced by V. Istratescu in [56].
Definition 2.2.1 Suppose (X, d) is a metric space and T : X → X is an operator.
T is a convex contraction if there exists p ∈ N\{0} and αj ∈ (0, 1) withp−1∑j=0
αj < 1
such that d(T p(x), T p(y)) ≤p−1∑j=0
αj · d(T j(x), T j(y)), ∀x, y ∈ X.
Theorem 2.2.1 If (X, d) is a complete metric space and T : X → X an operator
which satisfies the condition
d(T px, T py) ≤p−1∑j=0
αj · d(T jx, T jy), ∀x, y ∈ X
wherep−1∑j=0
αj < 1, then a) T has an unique fixed point x∗; b) the sequence (xn)n≥1
defined with
xn+1 = T (xn) is convergent to x∗ for all x0 ∈ X. c) d(x∗, xn) ≤∞∑
j=0
cj, where
cn+p =p−1∑j=0
αj · cn+j ∀n ≥ 1 and cj = d (T j+1x, T jx) , for 0 ≤ j ≤ p− 1.
2.3. CONVEX CONTRACTIONS ON GENERALIZED METRIC SPACES 10
2.3 Convex contractions on generalized metric
spaces
Definition 2.3.1 Suppose (X, d) is a generalized metric space (d : X × X → Rn)
and T : X → X an operator. T is a convex contraction if there exists p ∈ N\{0}and the matrices (Λj)j=0,p−1 ⊂ Mn(R) with the following properties
d(T p(x), T p(y)) ≤p−1∑j=0
Λj · d(T j(x), T j(y)),
∀x, y ∈ X andp−1∑j=0
‖Λj‖m < 1
where ‖·‖m : Mn(R) → R is an operator norm.
2.3.1 Generalization of Perov’s theorem
Theorem 2.3.1 If (X, d) is a generalized complete metric space and the continuous
operator T : X → X is a generalized convex contraction on X, then
1) T has an unique fixed point x∗ ∈ X;
2) the sequence (xn)n≥1 defined by
xn+1 = T (xn), ∀n ∈ N converges to x∗, for all x0 ∈ X;
3) we have the following approximation
‖d(x∗, xn)‖v ≤∞∑
j=0
cn+j,
where cj = ‖d (T j+1(x), T j(x))‖v for 0 ≤ j ≤ p − 1 and cn+p =p−1∑j=0
‖Λj‖m ·
cn+j, ∀n ≥ 1.
2.3.2 Application
Theorem 2.3.2 If Q ∈ Mn(R) is a matrix and α a positive number such that∥∥Q2 − αQ∥∥
m< 1− α,
then the sequence xn+1 = b + Q · xn, ∀n ∈ N converges to the unique solution of the
system (In −Q)x = b for all x0 ∈ Rn.
2.4. GRONWALL TYPE INEQUALITIES 11
2.4 Gronwall type inequalities
2.4.1 An abstract Gronwall type inequality
Theorem 2.4.1 If (X, ‖ · ‖,≤) is an ordered and normed space, T : X → X is an
increasing WPO, then the following implications are true:
1) If x ∈ X and x ≤p−1∑i=0
αi · T i+1(x), then x ≤ T∞(x);
2) If x ∈ X and x ≥p−1∑i=0
αi · T i+1(x), then x ≥ T∞(x),
where αi ∈ (0, 1), i = 0, p− 1 andp−1∑i=0
αi = 1.
Theorem 2.4.2 Suppose (X, +, ·,≤,→) is an ordered linear L-space, αi ∈ (0, 1],
i = 0, p− 1 with∑p−1
i=0 αi = 1, T : X → X is a Picard operator and y ∈ X is a fixed
element. If a) T is continuous, linear and increasing; b) there exist a sequence of
nonnegative real numbers (ck)k≤0 with the following properties (1) ck = 0 for k < 0,
c0 = 1 and
cn+p =
p−1∑k=0
αk · cn+p−1−k, ∀n ≥ −p + 1;
(2) the series∞∑
k=0
ck · T k(y) converges, then we have the following implications: 1)
x ≤p−1∑k=0
αk ·T k(x)+y ⇒ x ≤∞∑
k=0
ck ·T k(y) 2) x ≥p−1∑k=0
αk ·T k(x)+y ⇒ x ≥∞∑
k=0
ck ·T k(y)
If p = 1 and α0 = 1, the above constructed series reduces to the Neumann, series
and we obtain theorem 6.5. proved by I.A. Rus.
2.4.2 Applications
Theorem 2.4.3 If K : [a, b] × R → R is a continuous and positive function,
α, β, α1, α2 are positive constants and α1 + α2 = 1 then the inequality
y(x) ≤ α + α1β
x∫a
K(x, s)y(s)ds + α2β2
x∫a
K2(x, s)y(s)ds + α2αβ
x∫a
K(x, s)ds
2.4. GRONWALL TYPE INEQUALITIES 12
implies y(x) ≤ y∗(x),∀x ∈ [a, b], where K2(x, s) =
x∫s
K(x, t)K(t, s)dt and y∗ is the
unique continuous solution of the equation y(x) = α + β
x∫a
K(x, s)y(s)ds.
Theorem 2.4.4 If K1,2 : [a, b]×R → R are continuous and positive functions, and
they satisfy the conditions of theorem (2) from[1], α, β, α1, α2 are positive constants
and α1 + α2 = 1 then the inequality
y(x) ≤ α + α1β
x∫a
K(x, s)y(s)ds +
b∫a
K2(x, s)y(s)ds
+
+βαα2
x∫a
K(x, s)ds +
b∫a
K2(x, s)ds
+
+α2β2
x∫a
K(2)1 (x, s)y(s)ds +
b∫a
K(2)2 (x, s)ds
implies y(x) ≤ y∗(x), ∀ x ∈ [a, b], where
K(2)1 (x, s) =
x∫s
K1(x, t)K1(t, s)dt +
b∫a
K2(x, t)K1(x, t)dt,
K(2)2 (x, s) =
x∫a
K1(x, t)K2(t, s)dt +
b∫a
K2(x, t)K2(x, t)dt
and y∗(x) is the unique solution of the equation
y(x) = α + β
x∫a
K1(x, s)y(s)ds + β
b∫a
K2(x, s)y(s)ds.
2.4.3 A discrete Gronwall inequality
Theorem 2.4.5 If the terms of the sequences (ak)k≥1 and (bk)k≥1 are positive num-
bers and they satisfy the following inequality:
an ≤ α +1
2
n−1∑j=1
bjaj +a
2
n−1∑j=1
bj +1
2
n−1∑k=1
n−1∑j=k
bjbkak
2.5. CONVEX CONTRACTIONS ON FIBERS 13
then we have an ≤ α
n−1∏k=1
(1 + bk +
b2k
2
).
Theorem 2.4.6 If the sequences of positive real numbers (ak)k≥1 and (bk)k≥1 satisfy
the inequality:
an ≤ α + α1
n−1∑j=1
bjaj + α · α2
n−1∑j=1
bj+ +α2
n−1∑k=1
n−1∑j=k
bjbkak,
where α1,2 ∈ (0, 1), α1 + α2 = 1, then
an ≤ αn−1∏k=1
(1 + bk + α2 · b2k) .
2.5 Convex contractions on fibers
2.5.1 The fiber contraction theorem for convex contractions
Theorem 2.5.1 Let (Xk, dk) with k = 0, q and q ≥ 1 be some generalized metric
spaces and Ak : X0 ×X1 × ...×Xk → Xk for k = 0, q be some continuous operators
such that:
a) the spaces (Xk, dk) are generalized complete metric spaces, with dk : Xk → Rnk+ ,
nk ∈ N∗ for k = 1, q;
b) the operator A0 is (weakly) Picard;
c) there exist pk ∈ N∗ and Λ(j)pk ∈ Mnk
(R+) for j = 0, pk − 1 with the propertypk−1∑j=0
||Λ(j)pk ||mk
≤ 1 such that the operators
(Tk)(·) = Ak(x0, ..., xk−1, ·) : Xk → Xk
satisfy the following condition
dk(T(pk)k (xk1), T
(pk)k (xk2)) ≤
pk−1∑j=0
Λ(j)pk· dk(T
(j)k (xk1), T
(j)k (xk2)),
∀ (x0, x1, ..., xk−1) ∈ X0 ×X1 × ...×Xk−1 and xk1, xk2 ∈ Xk, k = 1, q;
d) the operators Ak are continuous with respect to (x0, x1, ..., xk−1) for all xk ∈ Xk
and k = 1, q;
2.5. CONVEX CONTRACTIONS ON FIBERS 14
then the operator Bq = (A0, A1, ..., Aq−1, Aq) is (weakly) Picard operator. Moreover
if A0 is a Picard operator and FA0 = x∗0, FA1(x∗0,·) = x∗1, ..., FAp(x∗0,x∗1,...,x∗q−1,·) = x∗q,
then FBq = (x∗0, x∗1, ..., x
∗q−1, x
∗q).
Lemma 2.5.1 The matrices Λ(j)ipk∈ Mnk
(R+) with i = 1, pk and j = 0, pk − 1 satisfy
the inequalitypk−1∑j=0
||Λ(j)ipk||mk
< 1 for i = 1, pk. If the sequence (xm)m≥0 ⊂ (Rnk+ )pk
satisfies the inequality
xm+1 ≤ A · xm + ym,∀m ∈ N,
where (ym)m≥0 ⊂ (Rnk+ )pk , lim
m→∞ym = 0 and A ∈ Mpk
(Mnk(R+)) such that
A =
Λ
(0)1pk
Λ(1)1pk
... Λ(pk−1)1pk
Λ(0)2pk
Λ(1)2pk
... Λ(pk−1)2pk
... ... ... ...
Λ(0)pkpk Λ
(1)pkpk ... Λ
(pk−1)pkpk
, then the sequence (xm)m≥0 is convergent to 0.
2.5.2 Application{x1(λ) = sin
(56x1(λ) + 1
4x2(λ) + λ
)x2(λ) = cos
(116
x1(λ) + 56x2(λ) + λ2
) (2.5.1)
Theorem 2.5.2 The system 2.5.1 has an unique solution in R2 for every λ ∈[λ1, λ2] and the functions λ → x1(λ) and λ → x2(λ) are continuously differentiable
with respect to λ.
Chapter 3
Fredholm-Volterra integral
equations in C[a, b]
In this chapter we study the equation
y(x) = f(x) +
x∫a
K1(x, s, y(s))ds +
b∫a
K2(x, s, y(s))ds. (3.0.1)
3.1 Existence theorems
Theorem 3.1.1 If
a) K1, K2 ∈ C([a, b]× [a, b]× Rn; Rn) and f ∈ C([a, b], Rn)
b) there exist α, β ∈ R+ such that ‖K1(x, s, u)‖ ≤ α · ‖u‖+ β;
c) there exists M ∈ R with the property M = sup(x,s,u)∈[a,b]×[a,b]×Rn
‖K2(x, s, u)‖;
then the equation (3.0.1) has at least one solution y∗ in the space C([a, b], Rn)
with the property ‖y∗‖c·α ≤ Rc, where Rc is a real number greater then c−1c·
[‖f‖+ (M + β)(b− a)] and c > 1.
For τ > 0 the Bielecki norm of a function y ∈ C([a, b], Rn) is defined by
‖y‖τ = maxx∈[a,b]
‖y(x)‖ · eτ(x−a).
15
3.1. EXISTENCE THEOREMS 16
Theorem 3.1.2 If
a) K1, K2 ∈ C([a, b]× [a, b]× Rn; Rn) and f ∈ C([a, b], Rn)
b) K2 has Lipschitz property with respect to its last variable and there exist
α1, β1 ∈ R+ such that ‖K1(x, s, u)‖ ≤ α1 · ‖u‖+ β1;
c) there exists an integrable function k2 : [a, b]×[a, b] → R+ and a number τ0 > α1
such that
supx∈[a,b]
b∫a
k2(x, s)ds ≤(
1− α1
τ0
)e−τ0(b−a); (3.1.2)
d) there exists β2 ∈ R such that
‖K2(x, s, z)‖ ≤ k2(x, s) · ‖z‖+ β2, ∀(x, s) ∈ [a, b]× [a, b], z ∈ Rn;
then the equation (3.0.1) has at least one solution in C([a, b], Rn).
Theorem 3.1.3 If
a) K1, K2 ∈ C([a, b]× [a, b]× Rn; Rn) and f ∈ C([a, b], Rn)
b) there exist α1, β1 ∈ R+ such that ‖K1(x, s, u)‖ ≤ α1 · ‖u‖+ β1;
c) there exist k2 : [a, b]× [a, b] → R and p > 1 such that
‖K2(x, s, z)‖ ≤ k2(x, s) · ‖z‖+ β2, ∀(x, s) ∈ [a, b]× [a, b], z ∈ Rn;
k2(x, ·) ∈ Lp[a, b] and
‖ ‖k2(x, ·)‖Lp ‖τ0 <1
b− a· e−1−αq2(b−a),
where 1p
+ 1q
= 1, and τ0 = αq + 1q(b−a)
.
then the equation (3.0.1) has at least one solution in C([a, b], Rn).
3.2. EXISTENCE AND UNIQUENESS THEOREMS 17
3.2 Existence and uniqueness theorems
In this section we study the following equations
y(x) = f(x) +
x∫a
K1(x, s, y(s); λ)ds +
b∫a
K2(x, s, y(s); λ)ds, (3.2.3)
and
y(x) = f(x) + λ
x∫a
K1(x, s)y(s)ds + λ
b∫a
K2(x, s)y(s)ds (3.2.4)
where λ is a real parameter.
3.2.1 The nonlinear case
Theorem 3.2.1 If (Xi, di) are complete metric spaces for i = 1, n and the operators
Ti : X1 ×X2 × . . .×Xn → Xi satisfy the condition
di (Ti(x), Ti(y)) ≤n∑
j=1
cij · dj(xj, yj),
where cij are real numbers for i, j = 1, n, x = (x1, x2, . . . , xn), y = (y1, y2, . . . , yn),
and the matrix C = (cij)i,j=1,n converges to 0, then
a) the operator T : X1 ×X2 × . . .×Xn → X1 ×X2 × . . .×Xn defined by
T (x1, x2, ..., xn) = (T1(x1, x2, ..., xn), T2(x1, x2, ..., xn), ..., Tn(x1, x2, ..., xn))
is a Picard operator, so the successive approximation sequence
x(m+1)i = Ti
(x
(m)1 , x
(m)2 , . . . , x
(m)n
)with m ∈ N converges to x∗i , where
T (x∗1, x∗2, . . . , x
∗n) = (x∗1, x
∗2, . . . , x
∗n) ;
b) we have the following estimationd1
(x∗1, x
(m)1
)d2
(x∗2, x
(m)2
). . .
dn
(x∗n, x
(m)n
)
≤ Cm(In − C)−1
d1
(x
(1)1 , x
(0)1
)d2
(x
(1)2 , x
(0)2
). . .
dn
(x
(1)n , x
(0)n
)
, m ≥ 1.
3.2. EXISTENCE AND UNIQUENESS THEOREMS 18
Conditions
C1) L1
2−L2(b−a)+
(e
L1(b−a)2−L2(b−a) − 2
)L1L2(b− a) < 0;
C2) 1b−a
ln 1−L2(b−a)(b−a)2L1L2
+(
1−L2(b−a)(b−a)2
L1L2 − 2)
(b− a)L1L2 > 0 si
1
b− aln
1− L2(b− a)
(b− a)2L1L2
(1− L2(b− a))+
+(b− a)L1L2
(2− 1− L2(b− a)
(b− a)2L1L2
)− L1 > 0.
Theorem 3.2.2 If the functions Ki : [a, b] × [a, b] × R × [λm, λM ] → R are con-
tinuous and they have the Lipschitz property with respect to the third variable (the
corresponding Lipschitz constants are L1 and L2), f ∈ C[a, b] and one of the condi-
tions C1) and C2) holds, then
a) equation (3.0.1) has an unique solution x∗ ın C[a, b];
b) the sequence of successive approximation converges to x∗ for every starting
element;
c) we have the following estimation:[‖x∗ − x
(m)1 ‖B
‖x∗ − x(m)1 ‖C
]≤ Cm(I2 − C)−1
[d1(x
(1)1 , x
(0)1 )
d2(x(1)2 , x
(0)2 )
],
where C =
[L1
τL2(b− a)
L1
τ
[eτ(b−a) − 1
]L2(b− a)
].
3.2.2 The linear case
Theorem 3.2.3 For the integral equation
y(x) = f(x) + λ
x∫a
K1(x, s)y(s)ds + λ
b∫a
K2(x, s)y(s)ds
the iterated kernels are defined by
K(n+1)1 (x, s) =
x∫s
K1(x, t)K(n)1 (t, s)dt
3.2. EXISTENCE AND UNIQUENESS THEOREMS 19
and
K(n+1)2 (x, s) =
x∫a
K1(x, t)K(n)2 (t, s)dt+
b∫a
K2(x, t)K(n)2 (t, s)dt+
b∫s
K2(x, t)K(n)1 (t, s)dt.
The resolvent kernels are
R1(x, s, λ) =∞∑
j=1
λjK(j)1 (x, s) and R2(x, s, λ) =
∞∑j=1
λjK(j)2 (x, s).
The solution can be represented as
y(x) = f(x) +
x∫a
R1(x, s, λ)f(s)ds +
b∫a
R2(x, s, λ)f(s)ds.
The series defining the resolvent kernels are convergent in C[a, b] if L1 =
max |K1(x, s)| and L2 = max |K2(x, s)| satisfy one of the conditions C1 and C2.
The resolvent kernels satisfy the following integral equations:
R1(x, s, λ) = λK1(x, s) + λ
x∫s
K1(x, t)R1(t, s, λ)dt
and
R2(x, s, λ) = λK2(x, s) + λ
x∫a
K1(x, t)R2(t, s, λ)dt +
b∫a
K2(x, t)R2(t, s, λ)ds+
+
b∫s
K2(x, t)R1(t, s, λ)ds.
3.2.3 Weakly singular Fredholm-Volterra integral equations
Theorem 3.2.4 If K(x, s, λ) = L(x,s,λ)(x−s)α with L ∈ C ([a, b]× [a, b]× [λ1, λ2]) and
0 < α < 1, then the equation
u(x) = f(x) +
x∫a
K(x, s, λ)u(s)ds (3.2.5)
with f ∈ C[a, b] and λ ∈ [λ1, λ2] has a unique solution in C([a, b]) and this solution
can be obtained by successive approximation. This solution depends continuously
on λ and if K is continuously differentiable with respect to λ, the solution is also
continuously differentiable with respect to λ.
3.2. EXISTENCE AND UNIQUENESS THEOREMS 20
Remark 3.2.1 We can use a direct proof (without the iterated operators) if we use
the following inequality:
|Tu(x)− Tv(x)| ≤x∫
a
maxx,s∈[a,b],λ∈[λ1,λ2]
|L(x, s, λ)|
|x− s|α· |u(s)− v(s)|ds ≤
≤ L∗||u− v|| ·x∫
a
eτ(s−a)
(x− s)αds ≤
x∫a
ds
(x− s)αp
1p
·
x∫a
eτ(s−a)qds
1q
≤
≤(
(b− a)1−α·p
1− α · p
) 1p
· eτ(x−a)
(τ · q)1q
,
where α · p < 1, 1p
+ 1q
= 1, L∗ = maxx,s∈[a,b],λ∈[λ1,λ2]
|L(x, s, λ)| and
||u− v|| = maxx∈[a,b],λ∈[λ1,λ2]
|u(x, λ)− v(x, λ)| · e−τ(x−a).
So we can choose τ such that the operator T be a contraction with the corresponding
Bielecki metric.
Theorem 3.2.5 For the equation
u(x) = f(x) +
x∫a
K1(x, s, λ)y(s)ds +
b∫a
K2(x, s, λ)y(s)ds (3.2.6)
with
L1 = maxx,s∈[a,b],λ∈[λ1,λ2]
|K1(x, s, λ)|
and
L2 =
2 · maxx,s∈[a,b],λ∈[λ1,λ2]
|L(x, s, λ)|
1− α· (b− a)1−α
where K1, L ∈ C([a, b]× [a, b]× [λ1, λ2]) and K2 is a weakly singular kernel
(K2(x, s, λ) = L(x,s,λ)|x−s|α , 0 < α < 1) the iterated kernels are
K(n+1)1 (x, s, λ) =
x∫s
K1(x, t, λ)K(n)1 (t, s, λ)dt +
b∫a
K2(x, t, λ)K(n)1 (x, t, λ)dt (3.2.7)
and
K(n+1)2 (x, s, λ) =
x∫a
K1(x, t, λ)K(n)2 (t, s, λ)dt +
b∫a
K2(x, t, λ)K(n)2 (x, t, λ)dt (3.2.8)
3.3. DIFFERENTIABILITY OF THE SOLUTIONS WITH RESPECT TO A PARAMETER 21
and the resolvent kernels are
R1(x, s, λ) =∞∑
j=1
K(j)1 (x, s, λ), (3.2.9)
R2(x, s, λ) =∞∑
j=1
K(j)2 (x, s, λ). (3.2.10)
If L1 and L2 satisfies condition a) or b), there exist an unique continuous solution
to the equation 3.2.6, this solution depends continuously on λ and if the functions
K1 and L are continuously differentiable with respect to λ, then the solution is also
continuously differentiable with respect to λ. The solution of the equation 3.2.6 can
be represented in the form
u(x) = f(x) +
x∫a
R1(x, s, λ)f(s)ds +
b∫a
R2(x, s, λ)f(s)ds.
The series (3.2.9) and (3.2.10) are convergent if L1 and L2 satisfy the condition C1)
or C2)
3.3 Differentiability of the solutions with respect
to a parameter
Theorem 3.3.1 If
1. the functions Ki : [a, b]× [a, b]×R× [λm, λM ] → R are continuous, f ∈ C[a, b];
2. the functions Ki : [a, b]× [a, b]× R× [λm, λM ] → R are derivable with respect
to the last two variables;
3.∣∣∣∂K1(t,s,x;λ)
∂x
∣∣∣ ≤ L1 si∣∣∣∂K2(t,s,y;λ)
∂y
∣∣∣ ≤ L2 in [a, b]× [a, b]× R× [λm, λM ];
4. the numbers L1 and L2 satisfy one of the conditions C1 and C2,
then
a) the equation (3.2.3) has an unique solution x∗(t, λ) in C([a, b]× [λm, λM ]);
3.4. FREDHOLM-VOLTERRA INTEGRAL EQUATIONS WITH DEVIATING ARGUMENT 22
b) x∗(t, λ) is derivable with respect to λ and its partial derivative satisfies the
integral equation
∂x∗(t, λ)
∂λ=
t∫a
∂K1(t, s, x∗(s, λ); λ)
∂λds +
b∫a
∂K2(t, s, x∗(s, λ); λ)
∂λds+
+
t∫a
∂K1(t, s, x∗(s, λ); λ)
∂x
∂x∗(s, λ)
∂λds +
b∫a
∂K2(t, s, x∗(s, λ); λ)
∂y
∂x∗(s, λ)
∂λds;
c) the sequence of successive approximation for the operator A = (B, C) con-
verges.
3.4 Fredholm-Volterra integral equations with
deviating argument
y(x) = f(x) +
x∫a
K1(x, s, y(g1(s)); λ)ds +
b∫a
K2(x, s, y(g2(s)); λ)ds, (3.4.11)
Theorem 3.4.1 If
1. the functions Ki : [a, b]× [a, b]× Rn × [λm, λM ] → Rn, i = 1, 2 are continuous
and they have the Lipschitz property with respect to the third variable (the
Lipschitz constants are L1, respectively L2);
2. f ∈ C([a, b], Rn) and the relations
f(a) = ϕ1(a)
f(b) = ϕ2(b)
K2(a, s, u) = 0, ∀s ∈ [a, b]u ∈ Rn
K1(b, s, u) = K2(b, s, u) = 0, ∀s ∈ [a, b]u ∈ Rn.
(3.4.12)
are satisfied;
3. g1, g2 : [a, b] → R are continuous and injective with the property Im(g1) =
[a1, a2], Im(g2) = [b2, b1] and a1 ≤ a ≤ a2 ≤ b, respectively a ≤ b2 ≤ b ≤ b1;
4. g1(s) ≤ s, ∀s ∈ [a, b];
3.4. FREDHOLM-VOLTERRA INTEGRAL EQUATIONS WITH DEVIATING ARGUMENT 23
5. the functions ϕ1 : [a1, a] → Rn and ϕ2 : [b, b1] → Rn are continuous;
6. one of the conditions C1) and C2) holds,
then
a) the equation (3.4.11) has an unique solution x∗ in C([a1, b1], Rn);
b) the sequence of successive approximation converges to x∗ for every starting
element;
c) we have the following estimation:[‖x∗ − x
(m)1 ‖B
‖x∗ − x(m)1 ‖C
]≤ Cm(I2 − C)−1
[d1(x
(1)1 , x
(0)1 )
d2(x(1)2 , x
(0)2 )
],
where C =
[L1
τL2(b− a)
L1
τ
[eτ(b−a) − 1
]L2(b− a)
].
Theorem 3.4.2 If
1. Ki : [a, b]× [a, b]× Rn × [λm, λM ] → Rn are continuous;
2. f ∈ C([a, b], Rn) and conditions 3.4.12 are satisfied;
3. the components of the functions Ki : [a, b] × [a, b] × Rn × [λm, λM ] → Rn are
derivable with respect to the last n + 1 variables;
4. if x = (x1, x2, ..., xn), y = (y1, y2, ..., yn), K1 = (K11, K12, ..., K1n) si K2 =
(K21, K22, ..., K2n), then∣∣∣∂K1j(t,s,x;λ)
∂xi
∣∣∣ ≤ L1 and∣∣∣∂K2j(t,s,y;λ)
∂yi
∣∣∣ ≤ L2 in [a, b] ×[a, b]× Rn × [λm, λM ], ∀i, j = 1, n;
5. g1, g2 : [a, b] → R are continuous functions, Im(g1) = [a1, a2], Im(g2) = [b2, b1],
a1 ≤ a ≤ a2 ≤ b and a ≤ b2 ≤ b ≤ b1;
4. g1(s) ≤ s, ∀s ∈ [a, b];
6. the functions ϕ1 : [a1, a] → Rn and ϕ2 : [b, b1] → Rn are continuous;
7. the numbers L1 and L2 satisfy one of the conditions C1 and C2,
then
3.4. FREDHOLM-VOLTERRA INTEGRAL EQUATIONS WITH DEVIATING ARGUMENT 24
a) the equation (3.4.11) has an unique solution x∗(t, λ) in C([a, b]×[λm, λM ], Rn);
b) the functions x∗j(t, λ) j = 1, n are derivable with respect to λ and the partial
derivatives satisfy the following system of integral equations:
∂x∗j(t, λ)
∂λ=
t∫a
∂K1j(t, s, x∗(g1(s), λ); λ)
∂λds +
b∫a
∂K2j(t, s, x∗(g2(s), λ); λ)
∂λds+
+n∑
i=1
t∫a
∂K1j(t, s, x∗(g1(s), λ); λ)
∂xi
· ∂x∗i (g1(s), λ)
∂λds
+n∑
i=1
b∫a
∂K2j(t, s, x∗(g2(s), λ); λ)
∂xi
· ∂x∗i (g2(s), λ)
∂λds, j = 1, n;
c) if the operators B and C are defined by
B(x, y) = (T1(x, y), T2(x, y)) (3.4.13)
C((x, y), (x1, y1)) = (x1, y1), (3.4.14)
where T1, T2 are defined by
T1,2(x, y)(t) = f(t) +
t∫a
K1(t, s, λ, x(g1(s)))ds +
b∫a
K2(t, s, λ, y(g2(s)))ds
and
x1j(t, λ) =
t∫a
∂K1j(t, s, x(g1(s), λ); λ)
∂λds +
b∫a
∂K2j(t, s, y(g2(s), λ); λ)
∂λds
y1j (t, λ) =
n∑i=1
t∫a
∂K1j(t, s, x(g1(s), λ); λ)
∂xi
x1i(g1(s), λ)ds+
+n∑
i=1
b∫a
∂K2j(t, s, y(g2(s), λ); λ)
∂yi
y1i(g2(s), λ)ds, j = 1, n,
then the sequence of successive approximation for the operator A = (B, C)
converges.
3.5. COMPARISON THEOREMS 25
3.5 Comparison theorems
Theorem 3.5.1 If the functions Ki : [a, b]× [a, b]× R× [λm, λM ] → R,
Ki : [a, b]× [a, b]×R× [λm, λM ] → R i ∈ 1, 2 satisfy the conditions of theorem 3.2.2,
f1, f2 ∈ C[a, b] and the following implications are true
u ≤ v ⇒ K1(x, s, u) ≤ K1(x, s, v),
u ≤ v ⇒ K2(x, s, u) ≤ K2(x, s, v),
then the solutions y∗ and y∗ of the equations
y(x) = f1(x) +
x∫a
K1(x, s, y(s); λ)ds +
b∫a
K2(x, s, y(s); λ)ds, (3.5.15)
and
y(x) = f2(x) +
x∫a
K1(x, s, y(s); λ)ds +
b∫a
K2(x, s, y(s); λ)ds, (3.5.16)
satisfy the inequality y∗(x) ≤ y∗(x), ∀x ∈ [a, b].
Theorem 3.5.2 If the functions Ki : [a, b]× [a, b]× R× [λm, λM ] → R,
Ki : [a, b]× [a, b]×R× [λm, λM ] → R i ∈ {1, 2}, f1, f2 : [a, b] → R, g1, g2 : [a, b] → Rand ϕ1, ϕ1 : [a1, a] → R, ϕ2, ϕ2 : [b, b1] → R satisfy the conditions of theorem 3.4.1,
ϕ1 ≤ ϕ1, ϕ2 ≤ ϕ2, f1 ≤ f2, and the following implications are true
u ≤ v ⇒ K1(x, s, u) ≤ K1(x, s, v),
u ≤ v ⇒ K2(x, s, u) ≤ K2(x, s, v),
then the unique solutions y∗ and y∗ of the equations
y(x) = f1(x) +
x∫a
K1(x, s, y(g1(s)))ds +
b∫a
K2(x, s, y(g2(s)))ds, (3.5.17)
and
y(x) = f2(x) +
x∫a
K1(x, s, y(g1(s)))ds +
b∫a
K2(x, s, y(g2(s)))ds, (3.5.18)
verifies the inequality y∗(x) ≤ y∗(x), ∀x ∈ [a, b].
Chapter 4
Fredholm-Volterra integral
equations in L2[a, b]
4.1 Fredholm-Volterra integral equations on a
compact interval
Lemma 4.1.1 If I = [a, b] is a bounded interval, k ∈ L2(I2) and the positive valued
u ∈ L2(I) function satisfies the inequality
u(t) ≤ α +
∫ b
a
k(t, s)u(s)ds, a.e. t ∈ I, (4.1.1)
where α > 0 and ‖k‖L2(I2) < 1, then
‖u‖L2(I) ≤α√
2(b− a)
1− ‖k‖L2(I2)
.
Theorem 4.1.1 If
I. (Caratheodory type conditions) the functions Ki : I2 × [λ1, λ2] × R → R,
i ∈ {1, 2} with I = [a, b] satisfy the conditions
a) Ki(·, ·, λ, u) is measurable on I2 = [a, b]× [a, b] for every u ∈ R and every
λ ∈ [λ1, λ2];
b) Ki(x, s, λ, ·) is continuous on R a.e.p. (x, s) ∈ I2 and every λ ∈ [λ1, λ2].
26
4.1. FREDHOLM-VOLTERRA INTEGRAL EQUATIONS ON A COMPACT INTERVAL 27
II. (space invariance) f ∈ L2(I),
Ki(·, ·, λ, 0) ∈ L2(I2) for every λ ∈ [λ1, λ2], i ∈ {1, 2} and there exists M1 > 0
such that ‖Ki(·, ·, λ, 0)‖L2(I2) < M1 for every λ ∈ [λ1, λ2];
III. (Lipschitz type condition) there exist ki ∈ L2(I2), i ∈ {1, 2}, such as
|Ki(t, s, λ, u)−Ki(t, s, λ, v)| ≤ ki(t, s)|u− v|,
∀t, s ∈ I, λ ∈ [λ1, λ2], u, v ∈ R;
IV. (contraction type condition)
L2 :=
b∫a
t∫a
(k1(t, s) + k2(t, s))2dsdt +
b∫a
b∫t
k22(t, s)dsdt < 1 (4.1.2)
then
1. for every λ ∈ [λ1, λ2] exists an unique solution y∗(·, λ) ∈ L2(I) for the equation
y(x) = f(x) +
x∫a
K1(x, s, λ, y(s))ds +
b∫a
K2(x, s, λ, y(s))ds;
2. the sequence of successive approximation
yn+1(x) = f(x) +
x∫a
K1(x, s, λ, yn(s))ds +
b∫a
K2(x, s, λ, yn(s))ds
converges in L2(I) to y∗(·, λ), for every y0(·) ∈ L2(I) and every λ ∈ [λ1, λ2];
3. for every n ∈ N we have
‖yn(·)− y∗(·, λ)‖L2(I) ≤Ln
1− L‖y1(·)− y0(·)‖L2(I).
Moreover if condition I.c) holds, the solution operator S : [λ1, λ2] → L2(I) defined
by S(λ)(x) = y∗(x, λ), ∀x ∈ I, ∀λ ∈ [λ1, λ2] is continuous.
I.c) the functions (Ki(x, s, ·, u))x,s∈I,u∈R are echicontinuous;
If instead of I.b), I.c) and III. we have the following conditions:
4.1. FREDHOLM-VOLTERRA INTEGRAL EQUATIONS ON A COMPACT INTERVAL 28
I.b’) Ki(x, s, λ, ·) ∈ C1(R) for every λ ∈ [λ1, λ2], a.e.p. (x, s) ∈ I2, and there exist
ki ∈ L2(I2), i ∈ {1, 2}, such that∣∣∣∣∂Ki(t, s, λ, u)
∂u
∣∣∣∣ ≤ ki(t, s),
∀t, s ∈ I,∀λ ∈ [λ1, λ2],∀u ∈ R;
I.c’) Ki(x, s, ·, u) ∈ C1[λ1, λ2] for every u ∈ R, a.e.p. (x, s) ∈ I2, the partial deriva-
tives satisfy condition I.,∂Ki
∂λ(·, ·, λ, u) ∈ L2(I2), i ∈ {1, 2} and there exists M2 > 0 such as∥∥∥∥∂Ki
∂λ(·, ·, λ, u)
∥∥∥∥L2(I2)
< M2,
∀λ ∈ [λ1, λ2], ∀u ∈ R,
then the operator S is differentiable with respect to λ.
Theorem 4.1.2 If
a) Ki : I × I × [λ1, λ2] × R → R, i = 1, 2 satisfy conditions I.-IV. from theorem
4.1.1;
b) the injective and measurable functions g1, g2 : [a, b] → R satisfy the conditions
Im(g1) = [a1, a2], Im(g2) = [b2, b1] with a1 ≤ a ≤ a2 ≤ b, a ≤ b2 ≤ b ≤ b1;
c) ϕ1 ∈ L2([a1, a]) and ϕ2 ∈ L2([b, b1]);
then
1) the equation (3.4.11) has an unique solution y∗(·, λ) in L2(I1) for all λ ∈[λ1, λ2], where I1 = [a1, b1];
2) the sequence of successive approximations converges in L2(I1) to y∗(·, λ) for
every admissible starting element y0(·, λ), where the set of admissible functions
is defined by
Ya ={y(·, λ) ∈ L2(I1) | y0(t, λ) = ϕ1(t), ∀t ∈ [a1, a], y0(t, λ) = ϕ2(t), ∀t ∈ [b, b1]
};
4.2. FREDHOLM-VOLTERRA INTEGRAL EQUATIONS ON UNBOUNDED INTERVALS 29
3) we have the following estimation:
‖yn(·)− y∗(·, λ)‖L2(I1) ≤Ln
1− L‖y1(·)− y0(·)‖L2(I1),
where L is given by relation (4.1.2).
Moreover if I.c) is satisfied, the operator S : [λ1, λ2] → L2(I1) defined by S(λ)(x) =
y∗(x, λ), ∀x ∈ [a1, b1], ∀λ ∈ [λ1, λ2] is continuous.
If instead of I.b), I.c) and III. we have I.b’) and I.c’), then the operator S is
differentiable.
4.2 Fredholm-Volterra integral equations on un-
bounded intervals
Theorem 4.2.1 If conditions I.-III. from theorem 4.1.1 are satisfied with I =
[a,∞) and
L2 :=
∫ ∞
a
∫ t
a
(k1(t, s) + k2(t, s))2dsdt +
∫ ∞
a
∫ ∞
t
k22(t, s)dsdt < 1, (4.2.3)
then
1. for each λ ∈ [λ1, λ2] there exists an unique solution y∗(·, λ) ∈ L2(I);
2. the sequence of successive approximation
yn+1(x) = f(x) +
x∫a
K1(x, s, λ, yn(s))ds +
∞∫a
K2(x, s, λ, yn(s))ds
converges in L2(I) to y∗(·, λ), for all y0(·) ∈ L2(I);
3. for all n ∈ N we have the following estimation
‖yn(·)− y∗(·, λ)‖L2(I) ≤Ln
1− L‖y1(·)− y0(·)‖L2(I).
Moreover if
I.c) there exists Λi : [λ1, λ2] × [λ1, λ2] → R, and gi : I2 → R, i ∈ {1, 2} with the
following properties
4.2. FREDHOLM-VOLTERRA INTEGRAL EQUATIONS ON UNBOUNDED INTERVALS 30
i)
|Ki(x, s, λ, u)−Ki(x, s, λ, u)| ≤ Λi(λ, λ) · gi(t, s), (4.2.4)
∀u ∈ R, λ, λ ∈ [λ1, λ2], a.e.p.(t, s) ∈ I2, i ∈ {1, 2};
ii) limλ→λ
Λ(λ, λ) = 0;
iii)∞∫a
[(t∫
a
g1(s, t)ds
)2
+
(∞∫a
g2(s, t)
)2]
dt < +∞
then the solution operator S : [λ1, λ2] → L2(I) defined by S(λ)(x) = y∗(x, λ), ∀x ∈I, ∀λ ∈ [λ1, λ2] is continuous.
If instead of I.b) and III. we have I.b’) from theorem 4.1.1 and
I.c’) Ki(x, s, ·, u) ∈ C1[λ1, λ2] for all u ∈ R, a.e.p. (x, s) ∈ I2, the partial derivatives
satisfy conditions I., and there exists M3 > 0 with the property
∫ ∞
a
(∫ t
a
∂K1
∂λ(t, s, λ, u)ds
)2
dt +
∫ ∞
a
(∫ t
a
∂K1
∂λ(t, s, λ, u)ds
)2
dt,
for all λ ∈ [λ1, λ2] and every u ∈ R,
then the solution operator S is differentiable.
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