3. Vectorial Maia’s fixed point theorems

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DOI: 10.24193/subbmath.2022.1.14

Some applications of Maia’s fixed point theorem for Fredholm integral equation systems

Alexandru-Darius Filip

Abstract. The aim of this paper is to study the existence and uniqueness of solutions for some Fredholm integral equation systems by applying the vectorial form of Maia’s fixed point theorem. Some abstract Gronwall lemmas and an abstract comparison lemma are also obtained.

Mathematics Subject Classification (2010):47H10, 47H09, 34K05, 34K12, 45D05, 45G10, 54H25.

Keywords:Space of continuous functions, vector-valued metric, matrix convergent to zero,A-contraction, fixed point, Picard operator, weakly Picard operator, integral equation, Fredholm integral equation system, vectorial Maia’s fixed point theorem, abstract Gronwall lemma, abstract comparison lemma.

1. Introduction

Let a, b ∈ R+, with a < b. Let C[a, b] be the set of all real valued functions which are continuous on the interval [a, b]. Using a vectorial form of Maia’s fixed point theorem, we study the existence and uniqueness of solutions (x1, x2)∈(C[a, b])² for the following Fredholm integral equation systems:







x1(t) =g1(t) +Rb

aK1(t, s, x1(s))ds+Rb

aH1(t, s, x1(s))ds x2(t) =g2(t) +Rb

aH2(t, s, x2(s))ds

(1.1) and







x₁(t) =g₁(t) +Rb

aK₁(t, s, x₁(s))ds+Rb

aH₁(t, s, x₂(s))ds x2(t) =g2(t) +Rb

aH2(t, s, x1(s))ds

(1.2) whereg1, g2∈C[a, b],K1, K2, H1, H2∈C([a, b]×[a, b]×R,R) are given functions.

Received 20 October 2019; Revised 28 February 2020; Accepted 08 May 2020.

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2. Preliminaries

We recall here some notions, notations and results which will be used in the sequel of this paper.

2.1. L-space

The notion ofL-space was introduced in 1906 by M. Fr´echet ([4]). It is an abstract space in which works one of the basic tools in the theory of operatorial equations, especially in the fixed point theory: the sequence of successive approximations method.

LetX be a nonempty set. Let s(X) :=

{xn}_n∈N |xn ∈X, n∈N . Letc(X) be a subset of s(X) and Lim : c(X) →X be an operator. By definition, the triple (X, c(X), Lim) is calledL-space (denoted by (X,→)) if the following conditions are satisfied:

(i) ifxn =x, for alln∈N, then{xn}_n∈N∈c(X) andLim{xn}_n∈N=x.

(ii) if{xn}_n∈N∈c(X) andLim{xn}_n∈N=x, then for all subsequences{xn_i}_i∈N of {xn}_n∈N, we have that{xn_i}_i∈N∈c(X) andLim{xn_i}_i∈N =x.

A simple example of anL-space is the pair (X,→), where^d X is a nonempty set and→^d is the convergence structure induced by a metricdonX.

In general, anL-space is any nonempty set endowed with a structure implying a notion of convergence for sequences. Other examples ofL-spaces are: Hausdorff topo- logical spaces, generalized metric spaces in Perov’ sense (i.e.d(x, y)∈R^m+), generalized metric spaces in Luxemburg’ sense (i.e.d(x, y)∈R+∪ {+∞}),K-metric spaces (i.e.

d(x, y)∈K, whereK is a cone in an ordered Banach space), gauge spaces, 2-metric spaces,D-R-spaces, probabilistic metric spaces, syntopogenous spaces.

2.2. Picard operators and weakly Picard operators onL-spaces

Let (X,→) be an L-space. An operator f : X → X is called weakly Picard operator (W P O) if the sequence of successive approximations,{fⁿ(x)}n∈N, converges for allx∈X and its limit (which generally depend onx) is a fixed point off.

If an operatorf isW P O and the fixed point set off is a singleton,F_f ={x^∗}, then by definition,f is called Picard operator (P O).

For aW P O,f :X→X, we define the operatorf^∞:X →X, by f^∞(x) := lim

n→∞fⁿ(x).

Notice that,f^∞(X) =F_f, i.e.,f^∞is a set retraction ofX onF_f.

IfX is a nonempty set, then the triple (X,→,≤) is an orderedL-space if (X,→) is an L-space and ≤is a partial order relation onX which is closed with respect to the convergence structure of theL-space.

In the setting of ordered L-spaces, we have some properties concerningW P Os andP Os.

Theorem 2.2.1 (Abstract Gronwall Lemma). Let(X,→,≤)be an orderedL-space and f :X →X be an increasingW P O. Then:

(i) x∈X,x≤f(x)⇒ x≤f^∞(x);

(ii) x∈X,x≥f(x)⇒ x≥f^∞(x).

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In particular, iff is aP O and we denoteFf ={x^∗}, then:

(i⁰) ∀ x∈X,x≤f(x)⇒x≤x^∗; (ii⁰) ∀ x∈X,x≥f(x)⇒x≥x^∗.

Theorem 2.2.2 (Abstract Comparison Lemma). Let(X,→,≤)be an orderedL-space and the operators f, g, h:X →X be such that:

(1) f ≤g≤h;

(2) f, g, hareW P Os;

(3) g is increasing.

Then:

x, y, z∈X, x≤y≤z⇒f^∞(x)≤g^∞(y)≤h^∞(z).

In particular, if f, g, hare P Os and we denote Ff ={x^∗},Fg ={y^∗},Fh = {z^∗}, then

∀x, y, z∈X, x≤y≤z⇒x^∗≤y^∗≤z^∗.

Regarding the theory ofW P OsandP Ossee [12], [13], [15], [16], [18], [11], [17], [3].

2.3. Maia’s fixed point theorem

The following result was proved by M.G. Maia in [5].

Theorem 2.3.1. LetXbe a nonempty set,dandρbe two metrics onXandV :X →X be an operator. We suppose that:

(1) there existsc >0such that, d(x, y)≤cρ(x, y),∀x, y∈X;

(2) (X, d)is a complete metric space;

(3) V : (X, d)→(X, d)is continuous;

(4) V : (X, ρ)→(X, ρ) is anl-contraction, i.e.,

∃ l∈[0,1) such thatρ(V(x), V(y))≤lρ(x, y), ∀ x, y∈X.

Then:

(i) FV ={x^∗};

(ii) V : (X, d)→(X, d)isP O.

Maia’s Theorem 2.3.1 remains true if we replace the condition (1) with the following one:

(1⁰) there existsc >0 such that,d(V(x), V(y))≤cρ(x, y),∀ x, y∈X.

Hence, we obtain the so called Rus’ variant of Maia’s fixed point theorem. More considerations can be found in [11], [9], [10], [14].

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2.4. Matrices which converge to zero

We denote by Mm(R+) the set of allm×msquare matrices with positive real elements, byImthe identitym×mmatrix and by Om the zerom×mmatrix.

A∈Mm(R+) is said to be convergent to zero ifAⁿ→Om asn→ ∞.

Some examples of matrices that converge to zero are the following:

a) A= a a

b b

∈M2(R+), wherea, b∈R+ anda+b <1;

b) A= a b

a b

∈M2(R+), where a, b∈R+ anda+b <1;

c) A= a b

0 c

∈M₂(R+), where a, b, c∈R+ and max{a, c}<1.

A classical result in matrix analysis is the following theorem (see [19], [1]), which characterizes the matrices that converge to zero.

Theorem 2.4.1. Let A∈M_m(R+). The following assertions are equivalent:

(1) A is convergent to zero;

(2) its spectral radius ρ(A) is strictly less than 1; that is, |λ| <1, for any λ ∈ C with det(A−λIm) = 0;

(3) the matrix (Im−A)is nonsingular and

(Im−A)⁻¹=Im+A+A²+. . .+Aⁿ+. . .;

(4) the matrix (Im−A)is nonsingular and (Im−A)⁻¹ has nonnegative elements.

Throughout this paper, we will make an identification between row and column vectors inR^m.

2.5. Vector-valued metric spaces

LetX be a nonempty set. A mappingd:X×X →R^m+ is called a vector-valued metric onX if the following conditions are satisfied:

(1) d(x, y) = 0∈R^m⇔x=y, for allx, y∈X;

(2) d(x, y) =d(y, x), for allx, y∈X;

(3) d(x, y)≤d(x, z) +d(z, y), for allx, y, z∈X.

OnR^m+, the relation≤is defined in the component-wise sense.

Some examples of vector-valued metrics are the following:

Example 2.5.1. LetX := (C[a, b])² andd: (C[a, b])²×(C[a, b])²→R²+, defined by d(x, y) :=

max

t∈[a,b]|x1(t)−y1(t)|, max

t∈[a,b]|x2(t)−y2(t)|

,

for allx= (x1, x2),y= (y1, y2)∈(C[a, b])².

Example 2.5.2. LetX := (C[a, b])² andρ: (C[a, b])²×(C[a, b])²→R²+, defined by ρ(x, y) :=



 Z b

a

|x1(t)−y1(t)|²dt

!¹₂ ,

Z b a

|x2(t)−y2(t)|²dt

!¹₂

,

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A nonempty setX endowed with a vector-valued metricdis called a generalized metric space in Perov’ sense (or a R^m+-metric space) and it is denoted by the pair (X, d). The notions of convergent sequence, Cauchy sequence, completeness, open and closed subset and so forth are similar to those defined for usual metric spaces.

The basic fixed point result which holds in generalized metric spaces in Perov’ sense is the following (see [6], [7]).

Theorem 2.5.3 (Perov’s fixed point theorem). Let (X, d) be a complete generalized metric space, where d:X×X→R^m+. Letf :X→X be an A-contraction, i.e. there exists a matrixA∈Mm(R+)convergent to zero, such that

d(f(x), f(y))≤Ad(x, y),∀ x, y∈X.

Thenf isP O in theL-space (X,→).^d

Remark 2.5.4. It would be of interest to extend the study from [8] and [2] to the case of vector-valued metric spaces.

3. Vectorial Maia’s fixed point theorems

In this section we present the Rus’ variant of Maia’s fixed point theorem in the setting of generalized metric spaces in Perov’s sense.

Theorem 3.1. Let X be a nonempty set, endowed with two vector-valued metrics, d, ρ:X×X →R^m+. Let T :X →X be an operator. We assume that:

(1) there exists a matrixC∈M_m(R+)such that

d(T(x), T(y))≤Cρ(x, y), ∀x, y∈X;

(2) (X, d)is a complete generalized metric space;

(3) T : (X, d)→(X, d)is continuous;

(4) T : (X, ρ)→(X, ρ)is an A-contraction, i.e. there exists a matrixA∈Mm(R+) convergent to zero, such that

ρ(T(x), T(y))≤Aρ(x, y), ∀ x, y∈X.

ThenT isP O in theL-spaces (X,→)^d and(X,→).^ρ

Proof. Letx₀ ∈X. By (4), the sequence of successive approximations {Tⁿ(x₀)}_n∈_N is a Cauchy sequence in (X, ρ). Indeed, forn, p∈Nwe have

ρ(Tⁿ(x0), T^n+p(x0))≤

n+p−1

X

k=n

ρ(T^k(x0), T^k+1(x0))≤

n+p−1

X

k=n

A^kρ(x0, T(x0))

≤Aⁿ(Im−A)⁻¹ρ(x0, T(x0))→0 asn, p→ ∞.

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By (1), we get that{Tⁿ(x0)}_n∈_N is a Cauchy sequence in (X, d). By (2), there exists x^∗∈X, such thatTⁿ(x0)→^d x^∗ asn→ ∞. By (3), it follows that x^∗∈FT, since

d(x^∗, T(x^∗))≤d(x^∗, Tⁿ(x0)) +d(Tⁿ(x0), T(x^∗))

=d(x^∗, Tⁿ(x0)) +d(T(Tⁿ⁻¹(x0)), T(x^∗))

→d(x^∗, x^∗) +d(T(x^∗), T(x^∗)) = 0, asn→ ∞.

By (4), we obtain the uniqueness of the fixed pointx^∗. HenceT isP Oin (X,→).^d We show next thatT isP Oin (X,→).^ρ

For anyx0∈X, sincex^∗∈FT, by (4) we have

ρ(x^∗, Tⁿ(x0)) =ρ(Tⁿ(x^∗), Tⁿ(x0))≤Aⁿρ(x^∗, x0)→0 asn→ ∞

which implies thatTⁿ(x₀)→^ρ x^∗asn→ ∞. Sincex^∗is the unique fixed point, we get

thatT isP O in (X,→).^ρ

Remark 3.2. Notice that, in the proof of the above result, Perov’s Theorem cannot be applied for T : (X, ρ)→(X, ρ), because the lack of completeness of the generalized metric space (X, ρ).

Remark 3.3. From the proof of the above result, we can deduce the following weak Perov’s contraction principle:

Theorem 3.4. Let (X, ρ) be a generalized metric space, where ρ:X ×X →R^m+. Let T :X →X be an operator. We assume that:

(i) F_T 6=∅;

(ii) there exists a matrix A ∈ M_m(R+) which converges to zero, such that ρ(T(x), T(y))≤Aρ(x, y), for allx, y∈X.

ThenT isP O in theL-space (X,→).^ρ

Another fixed point result of Maia type in vectorial form is the following.

Theorem 3.5. Let X be a nonempty set, endowed with two vector-valued metrics, d, ρ:X×X →R^m+. Let T :X →X be an operator. We assume that:

(1) FT 6=∅;

(2) there exists a matrixC∈Mm(R+)such that

d(T(x), T(y))≤Cρ(x, y), ∀x, y∈X;

(3) T : (X, ρ)→(X, ρ)is an A-contraction, i.e. there exists a matrixA∈Mm(R⁺) convergent to zero, such that

ρ(T(x), T(y))≤Aρ(x, y), ∀ x, y∈X.

ThenT isP O in theL-spaces (X,→)^d and(X,→).^ρ

Proof. By applying Theorem 3.4,T isP Oin (X,→). So^ρ FT ={x^∗}. For anyx0∈X, d(x^∗, Tⁿ⁺¹(x₀)) =d(Tⁿ⁺¹(x^∗), Tⁿ⁺¹(x₀))

≤Cρ(Tⁿ(x^∗), Tⁿ(x0))

≤CAⁿρ(x^∗, x0)→0

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asn→ ∞. SoT isP O in (X,→).^d

4. Applications of vectorial Maia’s fixed point theorem

In this section we study the existence and uniqueness of solutions for Fredholm integral equations systems (1.1) and (1.2), by applying the vectorial Maia’s fixed point theorem.

First, let us consider the system (1.1)







x1(t) =g1(t) +Rb

aH1(t, s, x1(s))ds x₂(t) =g₂(t) +Rb

aK₂(t, s, x₂(s))ds+Rb

aH₂(t, s, x₂(s))ds

whereg₁, g₂∈C[a, b],K₁, K₂, H₁, H₂∈C([a, b]×[a, b]×R,R), are given functions.

We are searching the conditions in which the system (1.1) has a unique solution (x1, x2)∈(C[a, b])².

We assume that there existLK_j, LH_j >0,j ∈ {1,2}such that:

|Kj(t, s, u)−Kj(t, s, v)| ≤LK_j|u−v|,

|H_j(t, s, u)−H_j(t, s, v)| ≤L_H_j|u−v|, for allt, s∈[a, b],u, v∈R, j∈ {1,2}.

OnX := (C[a, b])² we consider the metricsd, ρ:X×X→R²+, where d(x, y) :=



 max

t∈[a,b]|x1(t)−y1(t)|

max

t∈[a,b]|x2(t)−y2(t)|



 (4.1)

and

ρ(x, y) :=



 Rb

a|x1(t)−y₁(t)|²dt¹₂ Rb

a|x2(t)−y2(t)|²dt¹₂



, (4.2)

We consider the operatorT : (C[a, b])²→(C[a, b])², defined by T(x)(t) = T1(x)(t)

T2(x)(t)

!

:= g₁(t) +Rb

a K₁(t, s, x₁(s))ds+Rb

a H₁(t, s, x₁(s))ds g2(t) +Rb

a K2(t, s, x2(s))ds+Rb

a H2(t, s, x2(s))ds

! (4.3)

for allx= (x1, x2)∈(C[a, b])². We have,

ρ(T(x), T(y)) =



 Rb

a|T1(x)(t)−T1(y)(t)|²dt¹₂ Rb

a|T2(x)(t)−T2(y)(t)|²dt¹₂





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and

|T1(x)(t)−T1(y)(t)|

|T2(x)(t)−T2(y)(t)|

!

≤ Rb

a

K1(t, s, x1(s))−K1(t, s, y1(s)) ds Rb

a

K₂(t, s, x₂(s))−K₂(t, s, y₂(s)) ds

!

+ Rb

a

H1(t, s, x1(s))−H1(t, s, y1(s)) ds Rb

a

H₂(t, s, x₂(s))−H₂(t, s, y₂(s)) ds

!

≤ Rb

aL_K₁|x1(s)−y₁(s)|ds Rb

aLK₂|x2(s)−y2(s)|ds

! +

Rb

aL_H₁|x1(s)−y₁(s)|ds Rb

aLH₂|x2(s)−y2(s)|ds

!

H¨older’s inequality

≤

Rb

a|LK1|²ds¹₂ + Rb

a|LH1|²ds¹₂ Rb

a|x1(s)−y₁(s)|²ds¹₂ Rb

a|LK₂|²ds¹₂ + Rb

a|LH₂|²ds¹₂ Rb

a|x2(s)−y2(s)|²ds¹₂

!

=

L_K₁+L_H₁√

b−a˜ρ(x₁, y₁) LK₂+LH₂√

b−a˜ρ(x2, y2)

, where

˜

ρ(x₁, y₁) :=

Z b a

|x₁(s)−y₁(s)|²ds

!¹₂

, ρ(x˜ ₂, y₂) :=

Z b a

|x₂(s)−y₂(s)|²ds

!¹₂ .

Hence,

ρ(T(x), T(y))≤



 Rb

a[(LK₁+LH₁)√

b−a˜ρ(x1, y1)]²dt¹₂ Rb

a[(L_K₂+L_H₂)√

b−a˜ρ(x₂, y₂)]²dt¹₂





= (LK₁+LH₁)(b−a) ˜ρ(x1, y1) (L_K₂+L_H₂)(b−a) ˜ρ(x₂, y₂)

!

=Aρ(x, y), where

A:=

(LK₁+LH₁)(b−a) 0

0 (LK₂+LH₂)(b−a)

∈M2(R⁺)

is a matrix that converges to zero if (LK1+LH1)(b−a)<1 and (LK2+LH2)(b−a)<1.

So, if we add these two conditions,T becomes anA-contraction with respect toρ.

In addition, for allx, y∈C[a, b], we haved(T(x), T(y))≤Cρ(x, y),where C:=

(LK₁+LH₁)√

b−a 0

0 (L_K₂+L_H₂)√ b−a

∈M2(R+).

By applying Theorem 3.1, the system (1.1) has a unique solution in (C[a, b])². Hence, we have obtained the following result:

Theorem 4.1. Let a, b∈R+ witha < b. We consider the system of Fredholm integral equations







x1(t) =g1(t) +Rb

aH2(t, s, x2(s))ds

(9)

whereg1, g2∈C[a, b],K1, K2, H1, H2∈C([a, b]×[a, b]×R,R)are given functions.

We assume that:

(i) there existLK_j, LH_j >0,j∈ {1,2} such that:

|K_j(t, s, u)−K_j(t, s, v)| ≤L_K_j|u−v|,

|Hj(t, s, u)−Hj(t, s, v)| ≤LH_j|u−v|, for allt, s∈[a, b],u, v∈R,j∈ {1,2};

(ii) (LK₁+LH₁)(b−a)<1 and(LK₂+LH₂)(b−a)<1.

Then the system has a unique solutionx^∗= (x^∗₁, x^∗₂)∈(C[a, b])².

Remark 4.2. By Theorem 3.1, the operator T defined in (4.3) is P O. Hence, for all t ∈ [a, b] we have x^∗(t) = lim

n→∞x_n(t), for each x₀ = (x¹₀, x²₀) ∈ (C[a, b])², where {xn}n∈N⊂(C[a, b])² is defined by

x_n+1(t) =

x¹_n+1(t) x²_n+1(t)

= g1(t) +Rb

a K1(t, s, x¹_n(s))ds+Rb

aH1(t, s, x¹_n(s))ds g₂(t) +Rb

a K₂(t, s, x²_n(s))ds+Rb

aH₂(t, s, x²_n(s))ds

! .

Corollary 4.3. Let a, b∈R+ witha < b. We consider the system of Fredholm integral equations







x1(t) =g1(t) +Rb

a K1(t, s, x1(s))ds x₂(t) =g₂(t) +Rb

a K₂(t, s, x₂(s))ds

whereg1, g2∈C[a, b],K1, K2∈C([a, b]×[a, b]×R,R)are given functions.

We assume that:

(i) there existLK_j >0,j∈ {1,2} such that:

|Kj(t, s, u)−Kj(t, s, v)| ≤LK_j|u−v|, for allt, s∈[a, b],u, v∈R,j∈ {1,2};

(ii) L_K₁(b−a)<1 andL_K₂(b−a)<1.

Proof. We apply Theorem 4.1, by considering H1 and H2 as zero functions and by

takingLH1 = 0 andLH2 = 0.

Now, let us consider the system (1.2)







x₁(t) =g₁(t) +Rb

aH₁(t, s, x₂(s))ds x2(t) =g2(t) +Rb

aH2(t, s, x1(s))ds

whereg₁, g₂∈C[a, b],K₁, K₂, H₁, H₂∈C([a, b]×[a, b]×R,R) are given functions.

We assume that there existL_K_j, L_H_j >0,j ∈ {1,2}such that:

|Hj(t, s, u)−H_j(t, s, v)| ≤L_H_j|u−v|, for allt, s∈[a, b],u, v∈R, j∈ {1,2}.

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On X := (C[a, b])² we consider the metrics d, ρ : X×X → R²+ defined as in (4.1) and (4.2). Also, we consider the operatorT : (C[a, b])²→(C[a, b])², defined by

T(x)(t) = T₁(x)(t) T2(x)(t)

!

:= g1(t) +Rb

a H1(t, s, x2(s))ds g₂(t) +Rb

a K₂(t, s, x₂(s))ds+Rb

a H₂(t, s, x₁(s))ds

! (4.4)

for allx= (x1, x2)∈(C[a, b])².

In a similar manner as shown for the system (1.1), we get ρ(T(x), T(y))≤Aρ(x, y), for allx, y∈(C[a, b])², where

A:= LK₁(b−a) LH₁(b−a) LH₂(b−a) LK₂(b−a)

!

∈M2(R+).

The matrixA converges to zero if (LK₁+LK₂)±p

(LK₁+LK₂)²−4(LK₁LK₂−LH₁LH₂)

2 (b−a)<1.

So, if we add this condition,T becomes an A-contraction with respect toρ.

In addition, for allx, y∈(C[a, b])², we obtaind(T(x), T(y))≤Cρ(x, y), where C:= LK₁

√b−a LH₁

√b−a

LH₂

√b−a LK₂

√b−a

!

∈M2(R+).

By applying Theorem 3.1, the system (1.2) has a unique solution in (C[a, b])². Hence, we have obtained the following result:







x1(t) =g1(t) +Rb

aH2(t, s, x1(s))ds

whereg₁, g₂∈C[a, b],K₁, K₂, H₁, H₂∈C([a, b]×[a, b]×R,R)are given functions.

We assume that:

(i) there existLK_j, LH_j >0,j∈ {1,2} such that:

|Hj(t, s, u)−Hj(t, s, v)| ≤LHj|u−v|, for allt, s∈[a, b],u, v∈R,j∈ {1,2};

(ii) ^b−a₂

(LK₁+LK₂)±p

(LK₁+LK₂)²−4(LK₁LK₂−LH₁LH₂) <1.

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Remark 4.5. By Theorem 3.1, the operator T defined in (4.4) is P O. Hence, for all t ∈ [a, b] we have x^∗(t) = lim

n→∞xn(t), for each x0 = (x¹₀, x²₀) ∈ (C[a, b])², where {x_n}_n∈_N⊂(C[a, b])² is defined by

x_n+1(t) =

x¹_n+1(t) x²_n+1(t)

= g1(t) +Rb

a K1(t, s, x¹_n(s))ds+Rb

aH1(t, s, x²_n(s))ds g₂(t) +Rb

a K₂(t, s, x²_n(s))ds+Rb

aH₂(t, s, x¹_n(s))ds

! .

Corollary 4.6. Let a, b∈R+ witha < b. We consider the system of Fredholm integral equations







x1(t) =g1(t) +Rb

a H1(t, s, x2(s))ds x₂(t) =g₂(t) +Rb

a H₂(t, s, x₁(s))ds

whereg1, g2∈C[a, b],H1, H2∈C([a, b]×[a, b]×R,R)are given functions.

We assume that:

(i) there existLH_j >0,j∈ {1,2} such that:

(ii) (b−a)p

LH₁LH₂<1.

Proof. We apply Theorem 4.4, by consideringK₁ and K₂ as zero functions and by

takingL_K₁ = 0 andL_K₂= 0.

5. Abstract Gronwall lemmas

Since the operators T, defined in (4.3) and (4.4), areP Os, by using Theorem 2.2.1 we can establish the following abstract Gronwall lemmas for our systems (1.1) and (1.2).







x₁(t) =g₁(t) +Rb

a K₁(t, s, x₁(s))ds+Rb

aH₁(t, s, x₁(s))ds x2(t) =g2(t) +Rb

aH2(t, s, x2(s))ds

whereg1, g2∈C[a, b],K1, K2, H1, H2∈C([a, b]×[a, b]×R,R)are given functions.

We assume that:

(i) there existL_K_j, L_H_j >0,j∈ {1,2} such that:

|Hj(t, s, u)−H_j(t, s, v)| ≤L_H_j|u−v|, for allt, s∈[a, b],u, v∈R,j∈ {1,2};

(ii) (LK₁+LH₁)(b−a)<1 and(LK₂+LH₂)(b−a)<1;

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(iii) Kj(t, s,·), Hj(t, s,·) : R → R are increasing functions, for all t, s ∈ [a, b] and j∈ {1,2}.

Let x^∗= (x^∗₁, x^∗₂)∈(C[a, b])² be the unique solution of the system.

Then the following implications hold:

(1) for allx= (x1, x2)∈(C[a, b])² with x₁(t)

x₂(t)

≤ g₁(t) +Rb

aH₁(t, s, x₁(s))ds g2(t) +Rb

aH2(t, s, x2(s))ds

! ,

for allt∈[a, b], we have x≤x^∗; (2) for allx= (x1, x2)∈(C[a, b])² with

x1(t) x2(t)

≥ g1(t) +Rb

aH1(t, s, x1(s))ds g₂(t) +Rb

aK₂(t, s, x₂(s))ds+Rb

aH₂(t, s, x₂(s))ds

! ,

for allt∈[a, b], we have x≥x^∗.







x1(t) =g1(t) +Rb

aH1(t, s, x2(s))ds x₂(t) =g₂(t) +Rb

a K₂(t, s, x₂(s))ds+Rb

aH₂(t, s, x₁(s))ds

whereg₁, g₂∈C[a, b],K₁, K₂, H₁, H₂∈C([a, b]×[a, b]×R,R)are given functions.

We assume that:

(i) there existL_K_j, L_H_j >0,j∈ {1,2} such that:

|Kj(t, s, u)−Kj(t, s, v)| ≤LKj|u−v|,

(ii) ^b−a₂

(L_K₁+L_K₂)±p

(L_K₁+L_K₂)²−4(L_K₁L_K₂−L_H₁L_H₂) <1;

(iii) Kj(t, s,·), Hj(t, s,·) : R → R are increasing functions, for all t, s ∈ [a, b] and j∈ {1,2}.

Let x^∗= (x^∗₁, x^∗₂)∈(C[a, b])² be the unique solution of the system.

Then the following implications hold:

(1) for allx= (x1, x2)∈(C[a, b])² with x1(t)

x2(t)

≤ g1(t) +Rb

aH1(t, s, x2(s))ds g2(t) +Rb

aH2(t, s, x1(s))ds

! ,

for allt∈[a, b], we have x≤x^∗; (2) for allx= (x₁, x₂)∈(C[a, b])² with

x1(t) x2(t)

≥ g₁(t) +Rb

aH₁(t, s, x₂(s))ds g2(t) +Rb

aH2(t, s, x1(s))ds

! ,

for allt∈[a, b], we have x≥x^∗.

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6. Abstract comparison lemmas

We can establish also some abstract comparison results, taking into account Theorem 2.2.2. One of them is the following.

Theorem 6.1. Leta, b∈R+ witha < b. We consider the systems of Fredholm integral equations







x1(t) =g1(t) +Rb

aH2(t, s, x2(s))ds

(6.1)







y1(t) =g3(t) +Rb

aK3(t, s, y1(s))ds+Rb

aH3(t, s, y1(s))ds y2(t) =g4(t) +Rb

aK4(t, s, y2(s))ds+Rb

aH4(t, s, y2(s))ds

(6.2)







z1(t) =g5(t) +Rb

a K5(t, s, z1(s))ds+Rb

aH5(t, s, z1(s))ds z2(t) =g6(t) +Rb

a K6(t, s, z2(s))ds+Rb

aH6(t, s, z2(s))ds

(6.3) whereg_i∈C[a, b], for alli= 1,6andK_j, H_j∈C([a, b]×[a, b]×R,R), for allj= 1,6, are given functions.

We assume that:

(i) there existLK_j, LH_j >0,j= 1,6 such that:

|Hj(t, s, u)−Hj(t, s, v)| ≤LH_j|u−v|, for allt, s∈[a, b],u, v∈R,j= 1,6;

(ii) (L_K_j +L_H_j)(b−a)<1, for all j= 1,6;

(iii) K_j(t, s,·), H_j(t, s,·) : R → R are increasing functions, for all t, s ∈ [a, b] and j= 3,4;

(iv) for allt∈[a, b],

g₁(t) +Rb

a H₁(t, s, x₁(s))ds g2(t) +Rb

a H2(t, s, x2(s))ds

!

≤ g3(t) +Rb

aH3(t, s, x1(s))ds g4(t) +Rb

aH4(t, s, x2(s))ds

!

≤ g5(t) +Rb

aH5(t, s, x1(s))ds g₆(t) +Rb

aK₆(t, s, x₂(s))ds+Rb

aH₆(t, s, x₂(s))ds

! .

Let x^∗ = (x^∗₁, x^∗₂), y^∗ = (y₁^∗, y^∗₂), z^∗ = (z₁^∗, z^∗₂)∈ (C[a, b])² be the unique solutions of the systems (6.1),(6.2)and respectively (6.3).

Then for any x= (x1, x2), y= (y1, y2), z= (z1, z2)∈(C[a, b])² we have x≤y≤z⇒x^∗≤y^∗≤z^∗.

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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]