# A Sum Operator Method for the Existence and Uniqueness of Positive Solutions to a System of Nonlinear Fractional Integral Equations

Authors: Jing Wu; Tunhua Wu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper is concerned with the existence and uniqueness of positive solutions for
a Volterra nonlinear fractional system of integral equations. Our analysis relies on a fixed point
theorem of a sum operator. The conditions for the existence and uniqueness of a positive solution
to the system are established. Moreover, an iterative scheme is constructed for approximating
the solution. The case of quadratic system of fractional integral equations is also considered.


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## Body

## 1. Introduction

Fractional calculus has been used for the study of problems in various fields of sciences, such as Abel integral equation and viscoelasticity, analysis of feedback amplifiers, capacitor theory, fractances, generalized voltage dividers, and engineering and biological sciences. In [1], Kilbas et al. give a survey of research in fractional calculus and its applications in mathematical analysis such as ODEs, PDEs, convolution integral equations, and theory of generating equations. Particularly, fractional differential equations have successful applications in nonlinear oscillation analysis of earthquakes, seepage flow in porous media [2], and fluid dynamic models for traffic flow [3], as the fractional derivatives can eliminate the deficiency of continuum traffic flow.Open problems in this field are finding easy and effective methods for solving the equations. In recent years, many techniques of functional analysis, such as the fixed point theory, the Banach contraction principle, and the Leray-Schauder theory, are applied for solving the nonlinear fractional differential equations [4–11]. Iterative techniques [12–14] and the upper and lower solution method [15, 16] are also introduced to investigate the existence and uniqueness of the solutions to nonlinear fractional order differential equations with various boundary conditions.Recently, prompted by the applications in physics, the following nonlinear quadratic system of integral equations and its generalizations have provoked some interest:(1)1=φi(t)+λiφi(t)∫0t(t-s)αi-1Γ(αi)φi(s)ds,αi∈(0,1),i=1,2,…,n.
Salem [17] applied Krasnoselskii’s fixed point theorem to obtain the existence of solutions for the system:
(2)xi(t)=φi(t)+λiIαi[fi(x(t))+gi(x(t))],t∈[0,1],αi∈(0,1),1≤i≤n,
under the assumptions that fi:[0,∞)n→[0,∞) is continuous nondecreasing for all variables, and gi:[0,∞)n→[0,∞) is continuous nonincreasing for all variables, where [0,∞)n denotes the n-products [0,∞)×[0,∞)⋯×[0,∞) and x=(x1,x2,…,xn). For the physical point of view, only positive solutions are interesting. A simple form of the system (2):
(3)x(t)=∫0t(t-s)α-1Γ(α)f(x(s))ds,
has been studied in [18, 19].The aim of this paper is to study the existence and uniqueness of positive solutions for the following Volterra nonlinear fractional system of integral equations:(4)xi(t)=∫0t(t-s)αi-1Γ(α)[fi(s,x(s))+gi(s,x(s))]ds,t∈[0,1],αi∈(0,1),1≤i≤n.
Our main interest is to give some alternative answers to the main results of papers [17–19]. By using a fixed point theorem of a sum operator, we not only obtain the existence and uniqueness of positive solutions for the system (4), but also construct some sequences for approximating the unique solution.


## 2. Basic Definitions and Preliminaries

For the convenience of the reader, we present here some definitions, lemmas, and basic results that will be used in the proofs of our main results.Definition 1 (see [1]).
The fractional integral of orderα>0 of a function f:(0,+∞)→R is given by
(5)I0+αf(t)=1Γ(α)∫0t(t-s)α-1f(s)ds,
provided that the right-hand side is defined pointwisely on (0,+∞), and Γ(α) denotes the gamma function.Suppose thatE is a real Banach space which is partially ordered by a cone P⊂E; that is, x≤y if and only if y-x∈P. If x≤y and x≠y, then we denote x<y or y>x. By θ we denote the zero element of E. Recall that a nonempty closed convex set P⊂E is a cone if it satisfies (i) x∈P, λ≥0⇒λx∈P; (ii) x∈P, -x∈P⇒x=θ.LetP∘={x∈P∣xisaninteriorpointofP}, and then a cone P is said to be solid if P∘ is nonempty. Moreover, P is called normal if there exists a constant N>0 such that, for all x,y∈E, θ≤x≤y implies ∥x∥≤N∥y∥; in this case N is called the normality constant of P. If x1,x2∈E, the set [x1,x2]={x∈E∣x1≤x≤x2} is called the order interval between x1 and x2. We say that an operator A:E→E is increasing (decreasing) if x≤y implies Ax≤Ay(Ax≥Ay). For all x,y∈E, the notation x~y means that there exist λ>0 and μ>0 such that λx≤y≤μx. Clearly, ~ is an equivalence relation. Given h>θ (i.e., h≥θ and h≠θ ), we denote by Ph the set Ph={x∈E∣x~h}. It is easy to see that Ph⊂P.Definition 2.
LetD=P or D=P∘ and γ be a real number with 0≤γ<1. An operator A:P→P is said to be γ-concave if it satisfies
(6)A(tx)≥tγAx,∀t∈(0,1),x∈D.Definition 3.
An operatorA:E→E is said to be homogeneous if it satisfies
(7)A(tx)=tAx,∀t>0,x∈E.
An operator A:P→P is said to be subhomogeneous if it satisfies
(8)A(tx)≥tAx,∀t∈(0,1),x∈P.In the recent paper [20], Zhai and Anderson considered the following sum operator equation:
(9)Ax+Bx+Cx=x,
where A is an increasing γ-concave operator, B is an increasing subhomogeneous operator, and C is a homogeneous operator. They established the existence and uniqueness of positive solutions for the above equation, and when C is a null operator, they present the following interesting result.Lemma 4 (see [20]).
LetP be a normal cone in a real Banach space E, let A:P→P be an increasing γ-concave operator, and let B:P→P be an increasing subhomogeneous operator. Assume that(1)
there ish>θ such that Ah∈Ph,Bh∈Ph;(2)
there exists a constantδ>0 such that Ax≥δBx, for all x∈P.
Then the operator equationAx+Bx=x, has a unique solution x* in Ph. Moreover, constructing successively the sequence yn=Ayn-1+Byn-1,n=1,2,…. for any initial value y0∈Ph, we have yn→x*, as n→∞.


## 3. Main Results

In this section, we apply Lemma4 to study problem (4), and we obtain some new results on the existence and uniqueness of positive solutions.Now byC[0,1], we mean the Banach space of continuous functions on [0,1] with the usual max-norm ∥·∥. Also, recall the Banach space of the cartesian product E=:C[0,1]×C[0,1]×⋯×C[0,1] equipped by the norm ∥x∥=:max1≤i≤n∥xi∥. Notice that this space can be equipped with a partial order:
(10)x,y∈E,x≤y⟺xi(t)≤yi(t),t∈[0,1],i=1,2,…,n.
Set P={x∈E∣x(t)≥0,t∈[0,1]}, the standard cone. It is clear that P is a normal cone in E and the normality constant is 1. Take h(t)=(h1(t),h2(t),…,hn(t)) and hi(t)=tαi,
(11)Ph={x∈P∣x~h}.Theorem 5.
Assume that(S1)
for alli,fi,gi:[0,1]×[0,∞)n→[0,∞) are continuous and increasing with respect to the arguments xi, and gi(t,0,0,…,0)>0 for any t∈[0,1];(S2)
for alli,gi(t,x1,x2,…,τxi,…,xn)≥τgi(t,x1,x2,…,xi,…,xn) for τ∈(0,1),  t∈[0,1], xi∈[0,+∞) and there exist constants γi∈(0,1) such that
(12)fi(t,x1,x2,…,τxi,…,xn)≥τγfi(t,x1,x2,…,xi,…,xn)forτ∈(0,1),  t∈[0,1], xi∈[0,+∞),i=1,2,…,n;(S3)
there existsδi>0 such that
(13)fi(t,x1,x2,…,xi,…,xn)≥δigi(t,x1,x2,…,xi,…,xn),t∈[0,1],xi≥0,i=1,2,…,n.Then problem (4) has a unique positive solution x* in Ph. Moreover, for any initial value x(0)=(x1(0),x2(0),…,xn(0))∈Ph, constructing successively the sequence
(14)xi(m+1)(t)=∫0t(t-s)αi-1Γ(αi)×[fi(s,x1(m)(s),x2(m)(s),…,xi(m)(s),…,xn(m)(s))+gi(s,x1(m)(s),x2(m)(s),…,xi(m)(s),…,xn(m)(s))]ds,m=0,1,…,
then x(m)→x* as m→∞.Proof.
To begin with, we define the following operatorsA,B:P→E by
(15)Ax=(A1x1,A2x2,…,Anxn),Bx=(B1x1,B2x2,…,Bnxn),
where
(16)Aixi=∫0t(t-s)αi-1Γ(αi)fi(s,x(s))ds,Bixi=∫0t(t-s)αi-1Γ(αi)gi(s,x(s))ds.
Thus x is the positive solution of problem (4) if and only if x=Ax+Bx. From (S0) and (S1), we know that A:P→P, B:P→P. In the sequel we check that A,B satisfy all assumptions of Lemma 4.
Firstly, we prove thatA,B are two increasing operators. In fact, by (S0) and (S1), for x,y∈P with x≥y, we know that xi(t)≥yi(t),t∈[0,1],i=1,2,…,n, and obtain
(17)Aixi=∫0t(t-s)αi-1Γ(αi)×fi(s,x1(s),x2(s),…,xi(s),…,xn(s))ds≥∫0t(t-s)αi-1Γ(αi)×fi(s,y1(s),y2(s),…,yi(s),…,yn(s))ds=Aiyi;
that is, Ax≥Ay. Similarly, Bx≥By.
Next we show thatA is a γ-concave operator and B is a subhomogeneous operator. In fact, for any τ∈(0,1) and x∈P, by (S1), we obtain
(18)Ai(τxi)(t)=∫0t(t-s)αi-1Γ(αi)×fi(s,x1(s),x2(s),…,τxi(s),…,xn(s))ds≥τγi∫0t(t-s)αi-1Γ(αi)×fi(s,x1(s),x2(s),…,xi(s),…,xn(s))ds=τγiAixi(t).
Consequently, A(τx)(t)≥τγAx, where γ=max1≤i≤nγi. Also, for any τ∈(0,1) and x∈P, by (S0) and (S1), we obtain
(19)Bi(τxi)(t)=∫0t(t-s)αi-1Γ(αi)×gi(s,x1(s),x2(s),…,τxi(s),…,xn(s))ds≥τ∫0t(t-s)αi-1Γ(αi)×gi(s,x1(s),x2(s),…,xi(s),…,xn(s))ds=τBixi(t);
that is, B(τx)≥τBx for τ∈(0,1), x∈P. So the operator B is a subhomogeneous operator.
Now we show thatAh∈Ph, Bh∈Ph. In fact, by (S3), we have
(20)fi(s,1,1,…,1)≥fi(s,0,0,…,0)≥δigi(s,0,0,…,0)>0,
and thus take
(21)Mi=max0≤s≤1fi(s,1,1,…,1),mi=min0≤s≤1fi(s,0,0,…,0);
then Mi,mi>0. Let
(22)λ=min1≤i≤n{miαiΓ(αi)},μ=max1≤i≤n{MiαiΓ(αi)}.
It follows from (S1) that
(23)Aihi(t)=∫0t(t-s)αi-1Γ(αi)fi(s,sα1,sα2,…,sαi,…,sαn)ds≥∫0t(t-s)αi-1Γ(αi)fi(s,0,0,…,0)ds≥mi∫0t(t-s)αi-1Γ(αi)ds=miαiΓ(αi)tαi≥λtαi=λhi(t),Aihi(t)=∫0t(t-s)αi-1Γ(αi)fi(s,sα1,sα2,…,sαi,…,sαn)ds≤∫0t(t-s)αi-1Γ(αi)fi(s,1,1,…,1)ds≤Mi∫0t(t-s)αi-1Γ(αi)ds=MiαiΓ(αi)tαi≤μtαi=μhi(t).
So λhi(t)≤Aihi(t)≤μhi(t), and then λh(t)≤Ah(t)≤μh(t), hence Ah∈Ph. Similarly, from gi(s,0,0,…,0)>0 and (S1)–(S3), we easily prove Bh∈Ph. Hence the condition (1) of Lemma 4 is satisfied.
In the following we show that the condition (2) of Lemma4 is satisfied. For x∈P, from (S3), we have
(24)Aixi(t)=∫0t(t-s)αi-1Γ(αi)×fi(s,x1(s),x2(s),…,xi(s),…,xn(s))ds≥δi∫0t(t-s)αi-1Γ(αi)×gi(s,x1(s),x2(s),…,xi(s),…,xn(s))ds=δiBixi(t).
Take
(25)δ=min1≤i≤nδi,
and then we have Ax≥δBx, x∈P. By Lemma 4, the operator equation Ax+Bx=x has a unique solution x*∈Ph; of course, x* is also a unique solution of problem (4). In addition, by (S1) we know that the unique solution is also positive.
Now for any initial valuex(0)=(x1(0),x2(0),…,xn(0))∈Ph, let us construct successively the sequence
(26)xi(m)=Aixi(m-1)+Bixi(m-1),m=1,2,…,
and we have xi(m)→xi* as m→∞, and then problem (4) has a unique positive solution x(m)→x* in Ph; that is, for any initial value x(0)=(x1(0),x2(0),…,xn(0))∈Ph, constructing successively the sequence:
(27)xi(m+1)(t)=∫0t(t-s)αi-1Γ(αi)×[fi(s,x1(m)(s),x2(m)(s),…,xi(m)(s),…,xn(m)(s))+gi(s,x1(m)(s),x2(m)(s),…,xi(m)(s),…,xn(m)(s))]ds,m=0,1,…,
then x(m)→x* as m→∞.Corollary 6.
Assume that(A1)
for alli,fi:[0,1]×[0,∞)n→[0,∞) is continuous and increasing with respect to the arguments xi, and fi(t,0,0,…,0)>0 for any t∈[0,1];(A2)
for alli,i=1,2,…,n, there exists constant γi∈(0,1) such that
(28)fi(t,x1,x2,…,τxi,…,xn)≥τγfi(t,x1,x2,…,xi,…,xn)forτ∈(0,1), t∈[0,1], xi∈[0,+∞). Then the problem
(29)xi(t)=∫0t(t-s)αi-1Γ(α)fi(s,x(s))ds,t∈[0,1],αi∈(0,1),1≤i≤n,
has a unique positive solution x* in Ph. Moreover, for any initial value x(0)=(x1(0),x2(0),…,xn(0))∈Ph, constructing successively the sequence(30)xi(m+1)(t)=∫0t(t-s)αi-1Γ(αi)×fi(s,x1(m)(s),x2(m)(s),…,xi(m)(s),…,xn(m)(s))ds,m=0,1,…,then x(m)→x* as m→∞.In what follows, we establish the existence and uniqueness of positive solutions for the following system of quadratic integral equations of the fractional type:(31)φi(t)=∫0t(t-s)αi-1Γ(α)φi2(s)ds+1,t∈[0,1],αi∈(0,1),1≤i≤n.Corollary 7.
The system (31) has a unique positive solution.Proof.
Letfi(x1,x2,…,xn)=(xi+1)2, and then fi satisfies (A1) and (A2) of Corollary 6. Thus let x=(x1,x2,…,xn) be the unique positive solution of (29), and then we have
(32)xi(t)=∫0t(t-s)αi-1Γ(α)(xi(s)+1)2ds;
that is
(33)xi(t)+1=∫0t(t-s)αi-1Γ(α)(xi(s)+1)2ds+1.
Let φi=xi+1, and the φ=(φ1,φ2,…,φn) is a unique positive solution of (31).



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