THEOREMS OF EXISTENCE AND UNIQUENESS FOR
NONLINEAR FREDHOLM AND VOLTERRA
INTEGRAL EQUATIONS FOR FUNCTIONS WITH
VALUES IN L-SPACES
VIRA BABENKO
ABSTRACT. We consider nonlinear integral equations
of Fredholm and Volterra type with respect to functions
having values in L-spaces. Such class of equations includes
set-valued integral equations, fuzzy integral equations and
many others. We prove theorems of existence and uniqueness
of the solutions for such equations and investigate data
dependence of their solutions.
1. Introduction. A wide variety of questions lead to Fredholm and
Volterra integral equations. They have many important applications
in biology, physics, and engineering (see, for example, [5], [14], [10],
[9], and the references therein). Nowdays scientists are more and
more interested in integral equations for functions with values that
are compact and convex sets in finite or infinite dimensional spaces,
or that are fuzzy sets, see [7], [4], [17], [13], [18], [15], [16], [3],
[20]. In this paper we consider a generalized concept, that of an Lspace, that encompasses all of these as special cases. In particular,
we investigate the existence and uniqueness of solutions of nonlinear
Fredholm integral equations and nonlinear Volterra integral equations
of the second kind, for functions with values in L-spaces. Theorems
of existence and uniqueness for linear Fredholm and Volterra integral
equations for functions with values in L-spaces are obtained in [2].
We also investigate the dependence of solutions of such equations on
variations of the data.
The paper is organized as follows. In Section 2 we list some
preliminary results that will be used in the remainder of the paper.
1991 AMS Mathematics subject classification. Primary 45G10; Secondary
28B20, 26E50.
Keywords and phrases. Volterra, Fredholm Integral Equations; L-space; Existence; Uniqueness; Data Dependence.
Received by the editors October 8, 2015.
1
2
VIRA BABENKO
In Section 3 we show existence and uniqueness of the solutions of
these integral equations. Section 4 is devoted to questions of data
dependence. We end the paper with a discussion in Section 5.
2. Preliminary Results.
2.1. L-spaces. The following definition was introduced in [19]:
Definition 2.1. A complete separable metric space X with metric δ is
said to be an L – space if in X operations of addition of elements and
their multiplication with real numbers are defined, and the following
axioms are satisfied:
A1.
∀x, y ∈ X
A2.
∀x, y, z ∈ X
A3.
∃θ ∈ X ∀x ∈ X x + θ = x (where θ is called a zero in X);
A4.
∀x, y ∈ X
A5.
∀x ∈ X
λ, µ ∈ R
A6.
∀x ∈ X
1 · x = x,
A7.
∀x, y ∈ X
A8.
∀x, y, u, v ∈ X
x + y = y + x;
x + (y + z) = (x + y) + z;
λ∈R
λ∈R
λ(x + y) = λx + λy;
λ(µx) = (λµ) x;
0 · x = θ;
δ(λx, λy) = |λ|δ(x, y);
δ(x + y, u + v) ≤ δ(x, u) + δ(y, v).
Let us list some examples of L-spaces:
(1) Any Banach space (Y, k · kY ) over the field of real numbers
endowed with the metric δ(x, y) = kx − ykY is an L-space.
(2) Let K(Rn ) be the set of all nonempty and compact subsets
of Rn and let K c (Rn ) ⊂ K(Rn ) be the subset of convex sets.
We define the required operations and the Hausdorff metric on
K(Rn ) as follows: For A, B ∈ K(Rn ), and α ∈ R:
A + B := {x + y : x ∈ A, y ∈ B}
αA := {αx : x ∈ A}.
NONLINEAR INTEGRAL EQUATIONS IN L-SPACES
3
δ h (A, B) = max sup inf |x − y|, sup inf |x − y| ,
x∈A y∈B
x∈B y∈A
where | · | is the Euclidean norm in Rn . With these operations
and metric, K(Rn ) and its subspace K c (Rn ) are complete,
separable metric spaces (see for example [8]) and since the
axioms A1-A8 hold, these spaces are L-spaces.
(3) The set of all closed bounded subsets of a given Banach space,
endowed with the Hausdorff metric is an L-space.
(4) Any quasilinear normed space Y (definition see in [1]) is an
L-space.
(5) Consider (see e.g., [6]) the class of fuzzy sets E n consisting of
functions u : Rn → [0, 1] such that
(a) u is normal, i.e. there exists an x0 ∈ Rn such that
u(x0 ) = 1;
(b) u is fuzzy convex, i.e. for any x, y ∈ Rn and 0 ≤ λ ≤ 1,
u(λx + (1 − λ)y) ≥ min{u(x), u(y)};
(c) u is upper semicontinuous;
(d) the closure of {x ∈ Rn : u(x) > 0}, denoted by [u]0 , is
compact.
For each 0 < α ≤ 1, the α-level set [u]α of a fuzzy set u is
defined as
[u]α = {x ∈ Rn : u(x) ≥ α}.
The addition u + v and scalar multiplication cu, c ∈ R \ {0},
on E n are defined, in terms of α-level sets, by
[u + v]α = [u]α + [v]α , [cu]α = c[u]α for each 0 < α ≤ 1.
Define also 0 · u by the equality [0 · u]α = {θ} (here θ =
(0, ..., 0) ∈ Rn ).
One of the possible metrics in E n is defined in the following
way. For a given 1 ≤ p < ∞
Z
dp (u, v) =
1
α
α p
δ([u] , [v] ) dα
1/p
.
0
Then the space (E n , dp ) is (see [6, Theorem 3]) a complete
separable metric space and therefore an L-space.
4
VIRA BABENKO
2.2. Integrals of functions with values in L-spaces. We need the
following notion of convex elements of L-space:
Definition 2.2. An element x ∈ X is convex if
∀ λ, µ ≥ 0 λx + µx = (λ + µ)x.
Let X c be a set of all convex elements of a given L-space X.
Remark 2.3. X c is a closed subset of X.
We also need the definition of a convexifying operator (see [19])
which we give in a somewhat modified form.
Definition 2.4. Let X be an L-space. The operator P : X → X c is
called a convexifying operator if
(1) ∀x, y ∈ X δ(P (x), P (y)) ≤ δ(x, y);
(2) P ◦ P = P ;
(3) P (αx + βy) = αP (x) + βP (y), ∀x, y ∈ X, α, β ∈ R.
Examples of convexifying operators include
(1) The identity operator in the space K c (Rn ) is a convexifying
operator.
(2) The operator, that on the space K(Rn ) is defined by the
formula P (A) = co (A), is a convexifying operator. Here by
the co (A) we denote the convex hull of a set A.
(3) The identity operator in the space (E n , dp ) is a convexifying
operator.
Below we discuss L-spaces X with some fixed convexifying operator P .
Denote the image of the element x ∈ X for the mapping P : X → X
as x
e i.e. P x = x
e, ∀x ∈ X. If all elements of an L-space X are convex
in the sense of Definition 2.2, then we choose the identity operator as
the convexifying operator.
Next we define the Riemannian integral for a function f : [a, b] → X,
where X is an L-space. We again follow Vahrameev [19] for this
purpose. Let fe(t) := fg
(t).
NONLINEAR INTEGRAL EQUATIONS IN L-SPACES
5
Definition 2.5. Let I be an interval [a, b]. The mapping f : I → X is
called weakly bounded, if δ(θ, fe(t)) ≤ const and weakly continuous,
if fe : I → X is continuous.
Remark 2.6. Note that if a function f : I → X is continuous then f
is weakly continuous. If all elements x ∈ X are convex (i.e. P = Id),
then the concepts of continuity and weak continuity coincide.
We need the notion of a stepwise mapping from I to an L-space X.
Definition 2.7. The mapping f : I → X is called stepwise, if there
exist a set {xk }nk=0 ⊂ X and a partition a = t0 < t1 < ... < tn = b of
the interval I, such that fe(t) = x
ek for tk−1 < t < tk .
Definition 2.8. The Riemannian integral of a stepwise mapping
f : I → X is an element of the space X that is defined by the following
equality:
Z b
n−1
X
f (t)dt =
(tk+1 − tk )e
xk .
a
k=0
Definition 2.9. We say that a weakly bounded mapping f : I → X is
integrable in the Riemannian sense if there exist a sequence {fk } of
stepwise mappings from I to X, such that
Z ∗ (2.1)
δ fe(t), fek (t) dt → 0, as k → ∞,
where
R∗
is a regular Riemannian integral for real-valued functions.
nR
o
b
It follows from (2.1) that the sequence
f (t)dt is a Cauchy
a k
sequence and thus we can use the following definition.
Definition 2.10. Let f : I → X be integrable in the Riemannian
sense and let {fk } be a sequence of stepwise mappings such that (2.1)
holds. Then the Riemannian integral of f is the limit
Z b
Z b
f (t)dt = lim
fk (t)dt.
a
k→∞
a
6
VIRA BABENKO
As described in [19] the Riemannian integral for a function f : I → X
has the following properties:
(1) If f and g are integrable, then ∀ α, β ∈ R, αf +βg is integrable,
and moreover
Z b
Z b
Z b
(αf (t) + βg(t))dt = α
f (t)dt + β
g(t)dt.
a
a
a
(2) If f is integrable in the Riemannian sense, then fe is also
integrable and
Z b
Z b
f (t)dt =
fe(t)dt.
a
a
(3) If f and g are integrable, then the function t → δ(fe(t), ge(t)) is
integrable in the Riemannian sense and
! Z
Z b
Z b
b δ fe(t), ge(t) dt.
g(t)dt ≤
f (t)dt,
δ
a
a
a
The following theorem (see [19], [1]) guarantees that we can consider
the integrals which arise below as Riemannian integrals.
Theorem 2.11. A weakly bounded mapping f : I → X is integrable
in the Riemannian sense if and only if it is weakly continuous almost
everywhere on I.
3. Theorems of existence and uniqueness. For functions with
values in an L-space we consider integral equations
Z b
x(t) = f (t) + λ
g(t, s, x(s))ds Fredholm Equation
a
and
Z
x(t) = f (t) +
t
g(t, s, x(s))ds
Volterra Equation
a
and prove for these equations theorems of existence and uniqueness of
their solutions. Our results generalize the results of I. Tişe [18] for the
case of X = K c (Rn ) and f (t) = A, A ∈ K c (Rn ).
NONLINEAR INTEGRAL EQUATIONS IN L-SPACES
7
3.1. Fredholm equation. Consider the set Y = [a, b] × [a, b] × X. In
the space Y introduce a metric assuming that for points y = (t, s, x)
and y 0 = (t0 , s0 , x0 ) from Y
d(x, y) = |t − t0 | + |s − s0 | + δ(x, x0 ).
Consider the Fredholm integral equation
Z b
(3.1)
x(t) = f (t) + λ
g(t, s, x(s))ds
a
where f : [a, b] → X and g : Y → X are known functions, λ is fixed
real parameter and x : [a, b] → X is an unknown function.
Theorem 3.1. Suppose the function f is continuous on [a, b] and the
function g(t, s, x) satisfies the following conditions
(1) g is weakly continuous on Y , so the function ge : Y → X c is
continuous;
(2) there exists a constant K such that for ∀(t, s) ∈ [a, b] × [a, b] the
function ge satisfies the Lipschitz condition with constant K > 0
on the variable x, so ∀x0 , x00 ∈ X
(3.2)
Then if |λ| <
C([a, b], X).
δ (e
g (t, s, x0 ), ge(t, s, x00 )) ≤ Kδ(x0 , x00 ).
1
K(b−a)
the equation (3.1) has a unique solution x ∈
Proof. Denote by C([a, b], X) the space of continuous functions
x : [a, b] → X. Introduce in this space a metric
ρ(x, y) = max δ(x(t), y(t)).
t∈[a,b]
It is known (see for example [1]) that the obtained space is complete
and separable.
We consider an operator A on the space C([a, b], X) defined by
Z b
(3.3)
Ax(t) := f (t) + λ
g(t, s, x(s))ds.
a
Next we show that ∀x ∈ C([a, b], X) Ax ∈ C([a, b], X). For this it is
Rb
enough to prove continuity of the operator Bx(t) = a g(t, s, x(s))ds.
8
VIRA BABENKO
Let the function x be given. Consider a set
M = {(t, s, x(s)) : t, s ∈ [a, b]} ⊂ Y.
Due to the continuity of the function x this set is a compact subset
of the space Y . Constriction of the function ge on M is a continuous
function on M and thus is uniformly continuous.
Take an arbitrary ε > 0 and choose η > 0 such that
(d((t, s, x(s)), (t0 , s0 , x(s0 ))) < η) ⇒
ε
.
δ(e
g (t, s, x(s)), ge(t0 , s0 , x(s0 ))) <
b−a
(3.4)
Estimate
δ(Bx(t0 ), Bx(t00 ))
R
Rb
b
= δ a ge(t0 , s, x(s))ds, a ge(t00 , s, x(s))ds
Rb
≤ a δ(e
g (t0 , s, x(s)), ge(t00 , s, x(s)))ds.
Taking into account (3.4) we have that if |t0 − t00 | < η, then
δ(Bx(t0 ), Bx(t00 )) < ε.
Therefore the function Bx(t) is continuous.
Next we show that with |λ| <
We have
ρ(Ax, Ay)
=
1
K(b−a)
the operator A is contractive.
max δ(Ax(t), Ay(t))
Rb
max δ f (t) + λ a g(t, s, x(s))ds,
t∈[a,b]
Rb
f (t) + λ a g(t, s, y(s))ds
R
Rb
b
max δ λ a ge(t, s, x(s))ds, λ a ge(t, s, y(s))ds
t∈[a,b]
Rb
|λ| max a δ(e
g (t, s, x(s)), ge(t, s, y(s)))ds.
t∈[a,b]
=
≤
≤
t∈[a,b]
Due to the third condition of the Theorem 3.1 ∀t, s:
δ(e
g (t, s, x(s)), ge(t, s, y(s)))
≤ Kδ(x(s), y(s)).
Therefore,
Rb
ρ(Ax, Ay) ≤ |λ| a Kδ(x(s), y(s))ds
Rb
≤ |λ|K max δ(x(s), y(s)) a ds = |λ|K(b − a)ρ(x, y).
s∈[a,b]
NONLINEAR INTEGRAL EQUATIONS IN L-SPACES
9
1
If |λ| < K(b−a)
then this operator is contractive and thus the equation
(3.1) has a unique solution.
Remark 3.2. Note that if the known function f in (3.1) is convexvalued (f : [a, b] → X c ), then the solution of equation (3.1) also is
convex-valued.
3.2. Volterra Equation. Consider the set Y = [a, b]×[a, b]×X. The
Volterra integral equation has the following form
Z t
(3.5)
x(t) = f (t) +
g(t, s, x(s))ds
a
where x : [a, b] → X is an unknown function, g : Y → X is a known
function, and f : [a, b] → X is a known function.
Theorem 3.3. Suppose the function f is continuous on [a, b] and the
function g(t, s, x) satisfies the following conditions
(1) g is weakly continuous on Y , so the function ge : Y → X c is
continuous.
(2) There exist a constant K such that for ∀(t, s) ∈ [a, b] × [a, b] the
function ge satisfies the Lipschitz condition (3.2) with a constant
K on variable x.
Then the equation (3.5) has a unique solution x ∈ C([a, b], X).
Proof. Consider an operator A : C([a, b], X) → C([a, b], X):
Z t
Ax(t) := f (t) +
g(t, s, x(s))ds
a
(That fact that Ax ∈ C([a, b], X) if x ∈ C([a, b], X) can be derived
analogously to how it was done in the previous section). We have
Rt
δ(Ax(t), Ay(t)) = δ f (t) + a g(t, s, x(s))ds,
Rt
f (t) + a g(t, s, y(s))ds
R
Rt
t
≤ δ a g(t, s, x(s))ds, a g(t, s, y(s))ds
Rt
≤
δ(e
g (t, s, x(s)), ge(t, s, y(s)))ds.
a
10
VIRA BABENKO
Using the second condition of the Theorem 3.3 we obtain
(3.6)
Rt
Rt
δ(Ax(t), Ay(t)) ≤ a Kδ(x(s), y(s))ds ≤ K max δ(x(s), y(s)) a ds
s∈[a,b]
= K(t − a)ρ(x, y).
Therefore,
ρ(Ax, Ay) ≤ K(b − a)ρ(x, y),
1
then this operator is contractive and on any
and thus if b − a < K
1
interval [a, b], where 0 < b − a < K
the equation (3.5) has a unique
solution.
We now prove by induction, that ∀n ≥ 1
δ(An x(t), An y(t)) ≤
(3.7)
(t − a)n n
K ρ(x, y).
n!
Inequality (3.6) is the induction base case. Assume that
δ(An−1 x(t), An−1 y(t)) ≤
(t − a)n−1 n−1
ρ(x, y).
K
(n − 1)!
Then
δ(An x(t), An y(t))
=
≤
≤
≤
≤
=
=
Rt
δ f (t) + a g(t, s, An−1 x(s))ds,
Rt
f (t) + a g(t, s, An−1 y(s))ds
R
t
δ a ge(t, s, An−1 x(s))ds,
Rt
n−1
g
e
(t,
s,
A
y(s))ds
a
Rt
δ
g
e
(t,
s,
An−1 x(s)), ge(t, s, An−1 y(s)) ds
Rat
Kδ An−1 x(s), An−1 y(s) ds
Rat (s−a)n−1 n−1
ρ(x, y)ds
K (n−1)! K
a
Rt
Kn
n−1
ds
(n−1)! ρ(x, y) a (s − a)
(t−a)n
n
n! K ρ(x, y).
Therefore inequality (3.7) is proved. It implies that for any a < b and
any n
ρ(An x, An y) = max δ(An x(t), An y(t)) ≤
a≤t≤b
(b − a)n n
K ρ(x, y).
n!
NONLINEAR INTEGRAL EQUATIONS IN L-SPACES
11
n
n
n
If n is sufficiently large, then (b−a)
n! K < 1, and therefore A is contractive operator. Using the generalized contractive mapping principle
(see [12, ch.2 §14]) we see that the operator A has a unique fixed point.
Thus the equation (3.5) has a unique solution in the space C([a, b], X).
4. Data Dependence. In this section we consider questions about
the dependence of solutions of equations (3.1) and (3.5) on perturbations of the given functions g(t, s, x) and f (t). These questions were
considered in [18] for set-valued functions. We show that the schemes
of the proof of Theorems 3.2 and 3.4 from [18] work in our more general
setting.
Theorem 4.1. Consider the set Y = [a, b] × [a, b] × X and let g1 , g2 :
Y → X be weakly continuous. Consider also the following equations:
Z
(4.1)
b
x(t) = f1 (t) +
g1 (t, s, x(s))ds,
a
Z
(4.2)
y(t) = f2 (t) +
b
g2 (t, s, y(s))ds.
a
Suppose:
(1) For any (t, s) ∈ [a, b] × [a, b] the function ge(t, s, x) satisfies
Lipschitz condition (3.2) on variable x and K(b − a) < 1.
Denote by x∗ (t) the unique solution of the equation (4.1).
(2) There exist η1 , η2 > 0 such that δ(e
g1 (t, s, x), ge2 (t, s, x)) ≤ η1 for
all (t, s, x) ∈ [a, b] × [a, b] × X and ρ(f1 (t), f2 (t)) ≤ η2 .
(3) There exists y ∗ (t) a solution of the equation (4.2).
Then
ρ(x∗ , y ∗ ) ≤
η2 + η1 (b − a)
.
1 − K(b − a)
12
VIRA BABENKO
Proof. We have
δ(x∗ (t), y ∗ (t))
Rb
δ f1 (t) + a ge1 (t, s, x∗ (s))ds,
Rb
f2 (t) + a ge2 (t, s, y ∗ (s))ds
R
Rb
b
δ a ge1 (t, s, x∗ (s))ds, a ge2 (t, s, y ∗ (s))ds
+δ(f
R 1 (t), f2 (t))
Rb
b
δ a ge1 (t, s, x∗ (s))ds, a ge1 (t, s, y ∗ (s))ds
R
Rb
b
+δ a ge1 (t, s, y ∗ (s))ds, a ge2 (t, s, y ∗ (s))ds + η2
Rb
δ(e
g1 (t, s, x∗ (s)), ge1 (t, s, y ∗ (s)))ds
aR
b
+ a δ(e
g1 (t, s, y ∗ (s)), ge2 (t, s, y ∗ (s)))ds + η2
Rb
Rb
Kδ(x∗ (s), y ∗ (s))ds + a η1 ds + η2 .
a
=
≤
≤
≤
≤
By taking the maximum for t ∈ [a, b], we have:
R
b
ρ(x∗ , y ∗ ) ≤ max K a δ(x∗ (t), y ∗ (t))dt + η1 (b − a) + η2
t∈[a,b]
Rb
= K max δ(x∗ (t), y ∗ (t)) a dt + η1 (b − a) + η2
t∈[a,b]
= K max δ(x∗ (t), y ∗ (t))(b − a) + η1 (b − a) + η2 .
t∈[a,b]
Therefore,
ρ(x∗ , y ∗ ) ≤
η2 + η1 (b − a)
. 1 − K(b − a)
Consider now the question of data dependence for Volterra integral
equations. We need the following metric on C([a, b], X):
ρ∗ (x, y) := max [δ(x(t), y(t))e−τ (t−a) ],
with arbitrary τ > 0.
t∈[a,b]
The pair (C([a, b], X), ρ∗ ) forms a complete metric space.
It is easily seen that metrics ρ and ρ∗ satisfy the following inequalities
(4.3)
e−τ (b−a) ρ(x, y) ≤ ρ∗ (x, y) ≤ ρ(x, y).
We now prove the following theorem
NONLINEAR INTEGRAL EQUATIONS IN L-SPACES
13
Theorem 4.2. Let Y = [a, b] × [a, b] × X and let g1 , g2 : Y → X be
weakly continuous. Consider the following equations:
Z t
(4.4)
x(t) = f1 (t) +
g1 (t, s, x(s))ds,
a
Z
(4.5)
t
g2 (t, s, y(s))ds.
y(t) = f2 (t) +
a
Suppose:
(1) For any (t, s) ∈ [a, b]×[a, b] function ge(t, s, x) satisfies Lipschitz
condition (3.2) on variable x. (Denote by x∗ (t) the unique
solution of the equation (4.4).)
(2) There exist η1 , η2 > 0 such that δ(e
g1 (t, s, x), ge2 (t, s, x)) ≤ η1 for
all (t, s, x) ∈ [a, b] × [a, b] × X and ρ(f1 (t), f2 (t)) ≤ η2 .
(3) There exists y ∗ (t) a solution of the equation (4.5).
Then
(4.6)
ρ∗ (x∗ , y ∗ ) ≤
η2 + η1 (b − a) −τ (b−a)
e
(where τ > K)
1− K
τ
and moreover
ρ(x∗ , y ∗ ) ≤ η2 + η1 (b − a).
Proof. We estimate
δ(x∗ (t), y ∗ (t))
=
≤
≤
≤
≤
Rt
δ f1 (t) + a ge1 (t, s, x∗ (s))ds,
Rt
f2 (t) + a ge2 (t, s, y ∗ (s))ds
R
Rt
t
δ a ge1 (t, s, x∗ (s))ds, a ge2 (t, s, y ∗ (s))ds
+δ(f
R 1 (t), f2 (t))
Rt
t
δ a ge1 (t, s, x∗ (s))ds, a ge1 (t, s, y ∗ (s))ds
R
Rt
t
+δ a ge1 (t, s, y ∗ (s))ds, a ge2 (t, s, y ∗ (s))ds + η2
Rt
δ(e
g1 (t, s, x∗ (s)), ge1 (t, s, y ∗ (s)))ds
aR
t
+ a δ(e
g1 (t, s, y ∗ (s)), ge2 (t, s, y ∗ (s)))ds + η2
Rt
Kδ(x∗ (s), y ∗ (s))e−τ (s−a) eτ (s−a) ds
aR
t
+ a η1 ds + η2 .
14
VIRA BABENKO
By taking the maximum for t ∈ [a, b], we have:
max (δ(x∗ (t), y ∗ (t))e−τ (s−a) eτ (s−a) )
t∈[a,b]
≤ max
t∈[a,b]
Z t
Z t
K
δ(x∗ (t), y ∗ (t))e−τ (s−a) eτ (s−a) dt +
η1 ds + η2
a
a
and therefore
ρ∗ (x∗ , y ∗ )eτ (b−a)
=
=
≤
Rt
Kρ∗ (x∗ , y ∗ ) a eτ (s−a) ds + η1 (b − a) + η2
K
∗ ∗
τ (t−a)
− 1) + η1 (b − a) + η2
τ ρ∗ (x , y )(e
K
∗ ∗ τ (b−a)
ρ
(x
,
y
)e
+
η1 (b − a) + η2 .
∗
τ
From the derived inequality, for τ > K we obtain
ρ∗ (x∗ , y ∗ ) ≤
η2 + η1 (b − a) −τ (b−a)
e
.
1− K
τ
The inequality (4.6) is proved. Using the last inequality and (4.3) we
have
η2 + η1 (b − a)
ρ(x∗ , y ∗ ) ≤ eτ (b−a) ρ∗ (x∗ , y ∗ ) ≤
.
1− K
τ
Since it is true for any τ > K we obtain
ρ(x∗ , y ∗ ) ≤ η2 + η1 (b − a). 5. Discussion. Along with equations (3.1) and (3.5), equations of
the following form are interesting:
Z b
(5.1)
x(t) + λ
g(t, s, x(s))ds = f (t), t ∈ [a, b],
a
Z
(5.2)
x(t) +
t
g(t, s, x(s))ds = f (t),
t ∈ [a, b].
a
Equations (5.1) and (5.2) are equivalent to equations (3.1) and (3.5)
respectively in the case of real-valued functions, but not for functions
with values in L-spaces.
We say that an element z ∈ X is the Hukuhara type difference of
element x, y ∈ X, if
x = y + z.
NONLINEAR INTEGRAL EQUATIONS IN L-SPACES
15
We denote this difference by
h
z = x − y.
Difference of such type for elements of the space K c (Rn ) is defined in
[11] (see also [16]).
Equations (5.1) and (5.2) are equivalent to equations
Z b
h
x(t) = f (t) − λ
g(t, s, x(s))ds, t ∈ [a, b],
a
h
Z
x(t) = f (t) −
t
g(t, s, x(s))ds,
t ∈ [a, b].
a
That fact that the Hukuhara type difference is not defined for all x
and y brings significant difficulties into the investigation of existence
and uniqueness of the solutions of these equations. We know only two
references [15] and [16] in which theorems of existence and uniqueness
are proved for equations of the form (5.2) in the space K c (Rn ) for the
special case the f (t) ≡ a, where a is fixed element of K c (Rn ). We will
study the existence and uniqueness of solutions of (5.1) and (5.2) in
future work.
REFERENCES
1. Aseev, S.M. Quasilinear operators and their application in the theory of multivalued mappings. Proceedings of the Steklov Institute of Mathematics 167, (1985)
25-52 (Russian) Zbl 0582.46048.
2. Babenko, V. Numerical methods for solution of Volterra and Fredholm integral
equations for functions with values in L-spaces. Manuscript sumbitted for publication, (2015).
3. Balachandran, K., Prakash, P.. On fuzzy Volterra integral equations with deviating arguments. J. of Appl. Math. and Stochastic Anal. (2) (2004), 169-176.
4. Bica, A.M. One-sided fuzzy number and applications to integral equations from
epidemiology. Fuzzy Sets and Systems, 219(2013) 27-48.
5. Corduneanu, C. Integral equations and applications. Cambridge university press,
Cambridge. 1991.
6. Diamond, P., Kloeden, P. Metric spaces of fuzzy sets. Fuzzy sets and systems,
35, (1990), 241-249.
7. Diamond, P. Theory and applications of fuzzy Volterra integral equations. IEEE
Transactions on Fuzzy Systems. V.10 I.1 (2002) 97-102.
16
VIRA BABENKO
8. Dyn, N., Farkhi, E., Mokhov, A. Approximation of set-valued functions: Adaptation of classical approximation operators. Hackensack: Imperial College Press
2014.
9. Gripenberg, G., Londen, S.-O., Staffans, O. J. Volterra Integral and Functional
Equations, Cambridge University Press, Cambridge, 1990.
10. Guo, D., Lakshmikantham, V., Liu, X. Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic Publishers. In Math. and Its Appl., V. 373 (1996).
11. Hukuhara, M. Integration des Applicaitons Mesurables dont la Valeur est un
Compact Convexe. Funkcialaj Ekvacioj, 10 (1967), 205-223.
12. Kolmogorov, A.N., Fomin, S.V. Elements of the Theory of Functions and
Functional Analysis. Dover Books on Math., Dover Publications. 1999.
13. Lakshmikantham, V., Mohapatra R.N. Theory of fuzzy differential equations
and inclusions. Series in Math. Anal. and Appl. Taylor and Francis Inc. 2003.
14. Linz, P. Analytical and Numerical methods for Volterra equations. SIAM,
Philadelphia. 1985.
15. Plotnikov, A. V., Skripnik, N. V. Existence and Uniqueness Theorem for SetValued Volterra Integral Equations. American J. of Applied Math. and Stat., Vol.
1, No. 3 (2013), 41-45.
16. Plotnikov, A. V., Skripnik, N. V. Existence and Uniqueness Theorem for Set
Volterra Integral Equations. JARDCS. Vol. 6, 3 (2014), 1-7.
17. Da Prato, G., Iannelli, M. Volterra integrodifferential equations in Banach
spaces and applications. Pitman research notes in mathematics series. Longman
Scientific and Technical. 1989
18. Tişe, I. Set integral equations in metric spaces. Math. Morav. 13 (1) (2009),
95-102.
19. Vahrameev, S.A. Integration in L-spaces. Book: Applied Math. and Math.
Software of Computers, M.: MSU Publisher, (1980) 45-47 (Russian).
20. Wang, H., Liu, Y. Existence results for fuzzy integral equations of fractional
order. Int. J. of Math. Analysis. 5(17), (2011) 811-818.
Department of Mathematics, The University of Utah, 155 S 1400 E Room
233, Salt Lake City, UT 84112, USA
Email address: vera.babenko@gmail.com