DOI: 10.24193/subbmath.2021.3.12
On a Fredholm-Volterra integral equation
Alexandru-Darius Filip and Ioan A. Rus
Abstract. In this paper we give conditions in which the integral equation x(t) =
Z c
a
K(t, s, x(s))ds+ Z t
a
H(t, s, x(s))ds+g(t), t∈[a, b],
where a < c < b, K ∈ C([a, b]×[a, c]×B,B), H ∈ C([a, b]×[a, b]×B,B), g∈C([a, b],B), withBa (real or complex) Banach space, has a unique solution inC([a, b],B). An iterative algorithm for this equation is also given.
Mathematics Subject Classification (2010):45N05, 47H10, 47H09, 54H25.
Keywords: Fredholm-Volterra integral equation, existence, uniqueness, contrac- tion, fiber contraction, Maia theorem, successive approximation, fixed point, Picard operator.
1. Introduction
The following type of integral equation was studied by several authors (see [11], [2], [3], [6], [1], [5], [10], [7], . . . ),
x(t) = Z c
a
K(t, s, x(s))ds+ Z t
a
H(t, s, x(s))ds+g(t), t∈[a, b], (1.1) wherea < c < b,K∈C([a, b]×[a, c]×B,B),H ∈C([a, b]×[a, b]×B,B),g∈C([a, b],B), with (B,|·|) a (real or complex) Banach space.
The aim of this paper is to give some conditions onKandH in which the equa- tion (1.1) has a unique solution inC([a, b],B). To do this, we shall use the contraction principle, the fiber contraction principle ([9], [13], [10], [11]) and a variant of Maia fixed point theorem given in [8] (see also [4]).
2. Preliminaries
Let us recall some notions, notations and fixed point results which will be used in this paper.
2.1. Picard operators and weakly Picard operators
Let (X,→) be anL-space ((X, d),→; (X, τ),d →; (X,τ k·k),k·k→, *;. . .). An opera- tor A : (X,→)→ (X,→) is called weakly Picard operator (W P O) if the sequence (An(x))n∈Nconverges for all x∈X and the limit (which generally depends onx) is a fixed point ofA.
If an operatorAisW P O and the fixed point set ofAis a singleton, i.e., FA={x∗},
then, by definition,Ais called Picard operator (P O).
For aW P O,A: (X,→)→(X,→), we define the limit operatorA∞: (X,→)→ (X,→), by A∞(x) := lim
n→∞An(x). We remark that,A∞(X) =FA, i.e., A∞ is a set retraction ofX onFA.
2.2. Fiber contraction principle
Regarding this principle, some important results were given in [12] and [13].
Fiber Contraction Theorem. Let (X,→) be an L-space, (Y, d) be a metric space, B :X →X,C :X×Y →Y and A:X×Y →X×Y, A(x, y) := (B(x), C(x, y)).
We suppose that:
(i) (Y, d)is a complete metric space;
(ii) B is a W P O;
(iii) C(x,·) :Y →Y is anl-contraction, for allx∈X; (iv) C:X×Y →Y is continuous.
ThenA is aW P O. Moreover, if B is aP O, thenA is aP O.
Generalized Fiber Contraction Theorem.Let (X,→)be anL-space and(Xi, di),i= 1, m, m ≥ 1 be metric spaces. Let Ai : X0×. . .×Xi → Xi, i = 0, m, be some operators. We suppose that:
(i) (Xi, di),i= 1, m, are complete metric spaces;
(ii) A0 is aW P O;
(iii) Ai(x0, . . . , xi−1,·) :Xi→Xi,i= 1, m, areli-contractions;
(iv) Ai,i= 1, m, are continuous.
Then the operator A:X0×. . .×Xm→X0×. . .×Xm, defined by A(x0, . . . , xm) := (A0(x0), A1(x0, x1), . . . , Am(x0, . . . , xm)) is aW P O. Moreover, ifA0 is aP O, thenA is aP O.
2.3. A variant of Maia fixed point theorem
We recall here the following variant of Maia fixed point theorem, given by I.A.
Rus in [8]:
Theorem 2.1. LetX be a nonempty set,dandρbe two metrics onX andA:X →X be an operator. We suppose that:
(1) there existsc >0such that d(A(x), A(y))≤cρ(x, y), for allx, y∈X; (2) (X, d)is a complete metric space;
(3) A: (X, d)→(X, d)is continuous;
(4) A: (X, ρ)→(X, ρ)is anl-contraction.
Then:
(i) FA={x∗};
(ii) A: (X, d)→(X, d)isP O.
3. Operatorial point of view on equation (1.1)
LetX :=C([a, b],B) andT :X →X be defined by T(x)(t) :=
Z c a
K(t, s, x(s))ds+ Z t
a
H(t, s, x(s))ds+g(t), t∈[a, b].
Forx∈X, we denote byu:=x
[a,c]andv:=x
[c,b]. Ifxis a solution of the equation (1.1) (i.e. a fixed point ofT), then
u(t) = Z c
a
K(t, s, u(s))ds+ Z t
a
H(t, s, u(s))ds+g(t), t∈[a, c] (3.1) and
v(t) = Z c
a
K(t, s, u(s))ds+ Z c
a
H(t, s, u(s))ds +
Z t c
H(t, s, v(s))ds+g(t), t∈[c, b]. (3.2) LetX1:=C([a, c],B),X2:=C([c, b],B) and
T1:X1→X1,T1(u)(t) :=the second part of (3.1),
T2:X1×X2→X2,T2(u, v)(t) :=the second part of (3.2).
The mappingsT1 andT2 allow us to construct the triangular operator
T˜:X1×X2→X1×X2, T˜(u, v) := (T1(u), T2(u, v)), for all (u, v)∈X1×X2. Remark 3.1. If (u∗, v∗)∈FT˜, thenu∗(c) =v∗(c). So the functionx∗∈X, defined by
x∗(t) :=
(u∗(t), t∈[a, c]
v∗(t), t∈[c, b]
is a fixed point ofT, i.e., a solution of (1.1).
Remark 3.2. For (u0, v0)∈X1×X2 we consider the successive approximations cor- responding to the operator ˜T, (un+1, vn+1) = ˜T(un, vn),n∈N. We observe that, for n∈N∗,un(c) =vn(c). So, the function xn, defined by
xn(t) :=
(un(t), t∈[a, c]
vn(t), t∈[c, b]
is in X.
Remark 3.3. LetY ⊂X1×X2 be defined by
Y :={(u, v)∈X1×X2 |u(c) =v(c)}.
The operator R : X → Y, defined by R(x) := (x [a,c], x
[c,b]) is a bijection. From the above definitions, it is clear thatT(x) = (R−1T R)(x) and the˜ nthiterate of T is Tn=R−1T˜nR.
In conclusion, to study the equation (1.1) (which is equivalent with x=T(x)) it is sufficient to study the fixed point of the operator ˜T. If (u∗, v∗) ∈ FT˜ then R−1(u∗, v∗)∈FT.
4. Existence and uniqueness of solution of equation (1.1)
In what follows, in addition to the continuity ofH,K andg, we suppose onK andH that:
(i) There existsL1∈C([a, b]×[a, c],B) such that:
|K(t, s, ξ)−K(t, s, η)| ≤L1(t, s)|ξ−η|, for allt∈[a, b], s∈[a, c], ξ, η∈B. (ii) There existsL2∈C([a, b]×[a, b],B) such that:
|H(t, s, ξ)−H(t, s, η)| ≤L2(t, s)|ξ−η|, for allt, s∈[a, b], ξ, η∈B. (iii)
Z
[a,c]×[a,c]
L1(t, s) +L2(t, s)2
dtds 12
<1.
The basic result of our paper is the following.
Theorem 4.1. In the above conditions we have that:
(1) The equation (1.1)has inC([a, b],B)a unique solution x∗.
(2) The operatorT˜ is a Picard operator with respect tounif.→ . Let FT˜={(u∗, v∗)}.
(3) The operatorT is a Picard operator with respect tounif.→ andFT ={x∗}. More- over, x∗=R−1(u∗, v∗).
Proof. From the remarks which were given in §3, it is sufficient to prove that the operator ˜T is a Picard operator with respect to the uniform convergence onX1×X2.
In order to apply the Fiber contraction principle, we shall prove that:
(j) T1: (X1,unif.→ )→(X1,unif.→ ) is a Picard operator;
(jj) T2(u,·) : (X2,k·kτ)→(X2,k·kτ) is a contraction.
Let us prove (j).
We consider on X1, the norms k·k∞ and k·kL2. By using the assumptions (i) and (ii), we have the following estimations:
|T1(u1)(t)−T1(u2)(t)| ≤ Z c
a
|K(t, s, u1(s))−K(t, s, u2(s))|ds +
Z t a
|H(t, s, u1(s))−H(t, s, u2(s))|ds
≤ Z c
a
L1(t, s)|u1(s)−u2(s)|ds+ Z c
a
L2(t, s)|u1(s)−u2(s)|ds
H¨older’s inequality
≤
Z c a
L1(t, s)2ds
12 Z c a
|u1(s)−u2(s)|2ds 12
+ Z c
a
L2(t, s)2ds
12 Z c a
|u1(s)−u2(s)|2ds 12
. By taking the max
t∈[a,c]in the above inequalities, there exists a real positive constant c:= max
t∈[a,c]
Z c a
L1(t, s)2ds 12
+ max
t∈[a,c]
Z c a
L2(t, s)2ds 12
such that
kT1(u1)−T1(u2)k∞≤cku1−u2kL2, for allu1, u2∈X1. On the other hand, we have that
kT1(u1)−T1(u2)kL2 = Z c
a
|T1(u1)(t)−T1(u2)(t)|2dt 12
≤ Z c
a
Z c a
(L1(t, s)ds+L2(t, s))2ds
ku1−u2k2L2dt 12
= Z c
a
Z c a
(L1(t, s) +L2(t, s))2dsdt 12
ku1−u2kL2, for allu1, u2∈X1.
By using the assumption (iii), it follows that the operator T1 is a contraction with respect tok·kL2 onX1.
The conclusion follows from the variant of Maia theorem.
Let us prove (jj).
Fort∈[c, b] andML2 := max
t,s∈[c,b]L2(t, s), we have that
|T2(u, v1)(t)−T2(u, v2)(t)| ≤ Z t
c
|H(t, s, v1(s))−H(t, s, v2(s))|ds
≤ Z t
c
L2(t, s)|v1(s)−v2(s)|ds
≤ML2
Z t c
|v1(s)−v2(s)|e−τ(s−c)eτ(s−c)ds
≤ML2kv1−v2kτ
Z t c
eτ(s−c)ds≤ML2kv1−v2kτ
eτ(t−c) τ . It follows that
|T2(u, v1)(t)−T2(u, v2)(t)|e−τ(t−c)≤ ML2
τ kv1−v2kτ.
By taking max
t∈[c,b]and by choosingτ > ML2, there exists a real positive constant l:=ML2
τ <1 such that
kT2(u, v1)−T2(u, v2)kτ ≤lkv1−v2kτ, for allv1, v2∈X2. Remark 4.2. Let K := R or C, |·| be a norm on B := Km (|·|1, |·|2, |·|∞, . . .), a < c < b,K= (K1, . . . , Km)∈C([a, b],Km) andH = (H1, . . . , Hm)∈C([a, b],Rm).
In this case, the equation (1.1) takes the following form
x1(t) = Z c
a
K1(t, s, x1(s), . . . , xm(s))ds +
Z t a
H1(t, s, x1(s), . . . , xm(s))ds, t∈[a, b]
... xm(t) =
Z c a
Km(t, s, x1(s), . . . , xm(s))ds Z t
a
Hm(t, s, x1(s), . . . , xm(s))ds, t∈[a, b].
(4.1)
From Theorem 4.1 we have an existence and uniqueness result for the system (4.1).
In the case whenB is a Banach space of infinite sequences with elements inK (c(K),Cp(K),m(K),lp(K),. . .) we have from Theorem 4.1 an existence and unique- ness result for an infinite system of Fredholm-Volterra integral equations.
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Alexandru-Darius Filip Babe¸s-Bolyai University,
Faculty of Economics and Business Administration, Department of Statistics-Forecasts-Mathematics, Teodor Mihali Street, No. 58-60,
400591 Cluj-Napoca, Romania
e-mail:[email protected] Ioan A. Rus
Babe¸s-Bolyai University,
Faculty of Mathematics and Computer Sciences, 1, Kog˘alniceanu Street,
400084 Cluj-Napoca, Romania e-mail:[email protected]
