# The Solutions of Mixed Monotone Fredholm-Type Integral Equations in Banach Spaces

Authors: Hua Su
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

By introducing new definitions ofϕ convex and -φ concave quasioperator and v0 quasilower and u0 quasiupper, by means of the monotone iterative techniques without any compactness conditions, we obtain the iterative unique solution of nonlinear mixed monotone Fredholm-type integral equations in Banach spaces. Our results are even new to ϕ convex and -φ concave quasi operator, and then we apply these results to the two-point boundary value problem of second-order nonlinear ordinary differential equations in the ordered Banach spaces.


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## Body

## 1. Introduction

In this paper, we will consider the following nonlinear Fredholm integral equation:(1)u(t)=∫I‍H(t,s,u(s))ds,t∈I,
where I=[a,b] and H∈C[I×I×E,E], E is a real Banach space with the norm ∥·∥, and there exists a function G∈C[I×I×E×E,E] such that for any (t,s,x)∈I×I×E(2)H(t,s,x)=G(t,s,x,x).Guo and Lakshmikantham [1] introduced the definition of mixed monotone operator and coupled fixed point; there are many good results (see [1–13]). In the special case where H(t,s,x) is nondecreasing in x for fixed t,s∈I, Guo [2] established an existence theorem of the maximal and minimal solutions for (1) in the ordered Banach spaces by means of monotone iterative techniques. Recently, Jingxian and Lishan [3] and Lishan [4] obtained iterative sequences that converge uniformly to solutions and coupled minimal and maximal quasisolutions of the nonlinear Fredholm integral equations in ordered Banach spaces by using the Möuch fixed point theorem and establishing new comparison results. But these all required the compactness conditions and the monotone conditions in the above papers, and furthermore they did not obtain the unique solutions. In addition, extensive studies have also been carried out to study the global or iterative solutions of initial value problems [8–13].In this paper, by introducing new definitions ofϕ convex and -φ concave quasioperator and v0 quasilower and u0 quasiupper, by means of the monotone iterative techniques without any compactness conditions which are of the essence in [2–4, 7, 8, 14], we obtain the iterative unique solution of nonlinear mixed monotone Fredholm-type integral equations in Banach spaces and then apply these results to the two-point boundary value problem of second-order nonlinear ordinary differential equations.


## 2. Preliminaries and Definitions

LetP be a cone in E, that is, a closed convex subset such that λP⊂P for any λ≥0 and P∩{-P}={θ}. By means of P, a partial order ≤ is defined as x≤y if and only if y-x∈P. A cone P is said to be normal if there exists a constant N>0 such that x,y∈E, θ≤x≤y implies ∥x∥≤N∥y∥, where θ denotes the zero element of E (see [2, 14]), and we call the smallest number N the normal constant of P and denote NP. The cone P is normal if and only if every ordered interval [x,y]={z∈E:x≤z≤y} is bounded.LetPI={u∈C[I,E]:u(t)≥θfor allt∈I}, where C[I,E] denotes the Banach space of all the continuous mapping u:I→E with the norm ∥u∥C=maxt∈I|u(t)|. It is clear that PI is a cone of space C[I,E], and so it defines a partial ordering in C[I,E]. Obviously, the normality of P implies the normality of PI and the normal constants of PI, and P are the same.Letu0,v0∈C[I,E]. Then, u0,v0 are said to be coupled lower and upper quasi-solutions of (1) if
(3)u0(t)≤∫I‍G(t,s,u0(s),v0(s))ds,t∈I,v0(t)≥∫I‍G(t,s,v0(s),u0(s))ds,t∈I.
If the equality in (3) holds, then u0,v0 are said to be coupled quasi-solutions of (1).We will always assume in this paper thatP is a normal cone of E. For any u0,v0∈C[I,E] such that v0≤w0, we define the ordered interval D=[u0,v0]={u∈C[I,E]:u0≤u≤v0}.Next, we will give the new definition ofϕ convex and -φ concave quasi operator and v0 quasi-lower and u0 quasi-upper.Definition 1.
Suppose that,G∈C[I×I×E×E,E]. Then G is called ϕ convex and -φ concave quasi operator, if there exist functions
(4)ϕ:(0,∞)×(0,∞)⟶(0,∞),φ:(0,∞)×(0,∞)⟶(0,∞),
such that(1)
G(t,s,αu,βv)≥ϕ(α,β)G(t,s,u,v),α<β,α,β∈(0,∞), for all u,v∈E,(2)
G(t,s,αu,βv)≤φ(α,β)G(t,s,u,v),α≥β,α,β∈(0,∞), for all  u,v∈E.Definition 2.
Suppose thatG∈C[I×I×E×E,E], u0∈P. Then, G is called u0 quasi-upper, if for any u,v∈E, u,v<u0 such that ∫I‍G(t,s,u,v)ds<u0.Definition 3.
Suppose thatG∈C[I×I×E×E,E], v0∈E. Then, G is called v0 quasi-lower, if for any u,v∈E, u,v>v0 such that ∫I‍G(t,s,u,v)ds>v0.Let us list the following assumption for convenience.(H1)
G is uniformly continuous on I×I×E×E, and G is ϕ convex and -φ concave quasi operator.(H2)
G(t,s,x,y) is nondecreasing in x∈E for fixed (t,s,y)∈I×I×E. G(t,s,x,y) is nonincreasing in y∈E for fixed (t,s,x)∈I×I×E.(H3)
ϕ(α,β),φ(α,β) are all increasing in α, decreasing in β, and ϕ(α0,β0)≥α0,  φ(β0,α0)≤β0 and for α,β∈[α0,β0],  α<β,
(5)φ(β,α)-ϕ(α,β)≤l(β-α),0<l<1.


## 3. The Main Result

The main results of this paper are the following three theorems.Theorem 4.
LetP be a normal cone of E, let u0,v0∈PI be coupled lower and upper quasi-solutions of (1). Assume that conditions (H1),(H2), and (H3) hold and(H4)
There existsw0∈PI such that u0≤w0≤v0, and for α0,β0∈(0,∞) of (H3) such that u0≥α0w0,   β0w0≥v0.
Then, (1) has a unique solution x*(t)∈D=[u0,v0], and for any initial x0,y0∈[u0,v0], one has
(6)xn(t)⟶x*(t),yn(t)⟶x*(t),uniformly  ont∈Iasn⟶∞,
where {xn(t)},{yn(t)} are defined as
(7)xn(t)=∫I‍G(t,s,xn-1(s),yn-1(s))ds,yn(t)=∫I‍G(t,s,yn-1(s),xn-1(s))ds,t∈I.Proof.
We first define the operatorA:[u0,v0]×[u0,v0]→C[I,E] by the formula
(8)A(u,v)=∫I‍G(t,s,u(s),v(s))ds.
It follows from the assumption (H2) that A is a mixed monotone operator, that is, A(u,v) is nondecreasing in u∈[u0,v0] and nonincreasing in v∈[u0,v0], and u0≤A(u0,v0), A(v0,u0)≤v0.
By (7), we have xn(t)=A(xn-1(t),yn-1(t)), yn(t)=A(yn-1(t),xn-1(t)) and set wn(t)=A(wn-1(t),wn-1(t)) for initial w0 in (H4), and we also define that
(9)un(t)=A(un-1(t),vn-1(t)),vn(t)=A(vn-1(t),un-1(t)).
Since A is a mixed monotone operator, it is easy to see that
(10)u0≤u1≤⋯≤un≤⋯≤vn≤⋯≤v1≤v0,≤un≤⋯≤vn≤5⋯≤v1≤v0un≤wn≤vn.
Obviously, by induction, it is easy to see that(11)un≥αnwn,vn≤βnwn,n=0,1,…,(12)α0≤α1≤⋯≤αn≤⋯≤1≤⋯≤βn≤⋯≤β1≤β0,
where αn=ϕ(αn-1,βn-1),  βn=φ(βn-1,αn-1),n=1,2,….
In fact, by the assumption(H4), we have that inequality (11) holds as n=0. Suppose that inequality (11) holds as n=k, that is, uk≥αkwk, vk≤βkwk. Then, as n=k+1, by the assumption (H3), we have
(13)uk+1=A(uk,vk)=∫I‍G(t,s,uk(s),vk(s))ds≥∫I‍G(t,s,αkwk,βkwk)ds≥ϕ(αk,βk)∫I‍G(t,s,wk(s),wk(s))ds=αk+1wk+1,vk+1=A(vk,uk)=∫I‍G(t,s,vk(s),uk(s))ds≤∫I‍G(t,s,βkwk,αkwk)ds≤φ(βk,αk)∫I‍G(t,s,wk(s),wk(s))ds=βk+1wk+1.
Then, it is easy to show by induction that inequality (11) holds.
For inequality (12), by uk+1≤wk+1≤vk+1 and the above discussion, we have 0<αk+1≤1≤βk+1. Obviously, it follows from the assumption (H4) that α0≤α1,  β1≤β0. Suppose that αk-1≤αk,  βk≤βk-1, so it is easy to show by (H3) that
(14)ϕ(αk-1,βk-1)≤ϕ(αk,βk),φ(βk,αk)≤φ(βk-1,αk-1),
that is, αk≤αk+1, βk+1≤βk. Then, it is easy to show by induction that inequality (12) holds.
Then, it follows from the inequality (12) that there exist limits of the sequences {αn},{βn}. Suppose that there exist α,β such that αn→α, βn→β, and  n→∞, and by (H3), we also have
(15)0≤βn-αn=φ(βn-1,αn-1)-ϕ(αn-1,βn-1)≤l(βn-1-αn-1)≤⋯≤ln(β0-α0),
they 0<l<1, and taking limits in the above inequality as n→∞, we have α=β.
Next, we will show that the sequences{un},{vn} are all Cauchy sequences on D.
In fact, by (10) and (11), for any natural number p, we know that
(16)θ≤un+p-un≤vn-un≤(βn-αn)u0,θ≤vn-vn+p≤vn-un≤(βn-αn)u0.
By the normality of PI and (15), we have
(17)∥un+p-un∥C≤NPln(β0-α0)∥u0∥C,∥vn-vn+p∥C≤NPln(β0-α0)∥u0∥C,
where NP is a normal constant. So {un},{vn} are all Cauchy sequences on D, and then there exists u*,v*∈[u0,v0] such that limn→∞un=u*,   limn→∞vn=v*.
It is easy to know by (10) and (11) that
(18)θ≤vn-un≤βnwn-αnwn≤(βn-αn)u0≤ln(β0-α0)u0,
so by the normality of PI, we have
(19)∥vn-un∥C≤NPln(β0-α0)∥u0∥C,
and taking limits in the above inequality as n→∞, we have x*=u*=v*∈[u0,v0], and for any natural number n, we also have un≤x*≤vn,  t∈I.
Then, by the mixed monotone quality ofA we have
(20)un+1=A(un,vn)≤A(x*,x*)≤A(vn,un)=vn+1,
and taking limits in above inequality as n→∞, we know that
(21)x*=A(x*,x*),
that is, x*∈[u0,v0] is the fixed point of A; thus, x* is the solution of (1) on D=[u0,v0].
Furthermore, we will show that the solution is unique. Suppose thaty*∈[u0,v0] satisfy y*=A(y*,y*). Then, by the mixed monotone quality of A and induction, for any natural number n, it is easy to have that un≤y*≤vn. Then, by the normality of PI and taking limits in the above inequality as n→∞ and the above discussion, we have y*=x*.
For any initialx0,y0∈[u0,v0], by (7) and (8), the mixed monotone quality of A and induction, for any natural number n, we have un(t)≤xn(t)≤vn(t),  un(t)≤yn(t)≤vn(t),  t∈I. Then, the normality of PI and (19) imply that
(22)∥xn-un∥C≤NPln(β0-α0)∥u0∥C,∥yn-un∥C≤NPNPln(β0-α0)∥u0∥C.
Thus, the sequence {xn(t)},{yn(t)} all converges uniformly to x*(t) on t∈I. This completes the proof of Theorem 4.Theorem 5.
LetP be a normal cone of E, let u0,v0∈PI be coupled lower and upper quasi-solutions of (1). Assume that conditions (H1), (H2), and (H3) hold.(H4′)
G is u0 quasi-upper, and there exists w0∈PI such that w0<u0<v0, and there exist α0=sup{α>0:u0≥αw0}, β0=inf{β>0:v0≤βw0}.
Then, (1) has a unique solution x*(t)∈D=[u0,v0], and for any initial x0,y0∈[u0,v0], one has
(23)xn(t)⟶x*(t),yn(t)⟶x*(t),uniformly  ont∈Iasn⟶∞,
where {xn(t)},{yn(t)} are defined as
(24)xn(t)=∫I‍G(t,s,xn-1(s),yn-1(s))ds,yn(t)=∫I‍G(t,s,yn-1(s),xn-1(s))ds,t∈I.Proof.
We first define the operatorA:[u0,v0]×[u0,v0]→C[I,E] by the formula
(8′)A(u,v)=∫I‍G(t,s,u(s),v(s))ds.
It follows from the assumption (H2) that A is a mixed monotone operator, that is, A(u,v) is nondecreasing in u∈[u0,v0] and nonincreasing in v∈[u0,v0] and u0≤A(u0,v0), A(v0,u0)≤v0. By (7), we have xn(t)=A(xn-1(t), yn-1(t)),yn(t)=A(yn-1(t), xn-1(t)) and set wn(t)=A(wn-1(t), wn-1(t)), and we also define
(25)un(t)=A(un-1(t),vn-1(t)),vn(t)=A(vn-1(t),un-1(t)).
Since A is a mixed monotone operator, it is easy to see that
(10′)u0≤u1≤⋯≤un≤⋯≤vn≤⋯≤v1≤v0.
BecauseG is u0 quasi-upper and w0<u0, we have
(26)w1(t)=A(w0(t),w0(t))=∫I‍G(t,s,w0(s),w0(s))ds<u0.
So for any natural number n, by induction, we know that wn(t)=A(wn-1(t),wn-1(t))<u0.
It is easy to see by induction that(11′)uk≥αkwk,vk≤βkwk,(12′)a0≤a1≤⋯≤ak≤⋯≤bk≤⋯≤b1≤b0,
where αk=ϕ(αk-1,βk-1),  βk=φ(βk-1,αk-1),k=1,2,….
In fact, by the assumptions(H1) and (H3) and the above discussion, as n=0, we have(27)u1=A(u0,v0)=∫I‍G(t,s,u0(s),v0(s))ds≥∫I‍G(t,s,α0w0,β0w0)ds≥ϕ(α0,β0)∫I‍G(t,s,w0(s),w0(s))ds=α1w1,(28)v1=A(v0,u0)=∫I‍G(t,s,v0(s),u0(s))ds≤∫I‍G(t,s,β0w0,α0w0)ds≤φ(β0,α0)∫I‍G(t,s,w0(s),w0(s))ds=β1w1.
By the above two inequalities and assumption (H3), we have
(29)a0≤α1=ϕ(α0,β0)≤φ(β0,α0)=β1≤b0.
Suppose that for k-1 we have uk-1≥αk-1wk-1, vk-1≤βk-1wk-1, and αk-2≤αk-1≤βk-1≤βk-2. Then, for k+1, by the assumption (H3), we have
(30)uk=A(uk-1,vk-1)=∫I‍G(t,s,uk-1(s),vk-1(s))ds≥∫I‍G(t,s,αk-1wk-1,βk-1wk-1)ds≥ϕ(αk-1,βk-1)∫I‍G(t,s,wk-1(s),wk-1(s))ds=αkwk,vk=A(vk-1,uk-1)=∫I‍G(t,s,vk-1(s),uk-1(s))ds≤∫I‍G(t,s,βk-1wk-1,αk-1wk-1)ds≤φ(βk-1,αk-1)∫I‍G(t,s,wk-1(s),wk-1(s))ds=βkwk.
By the above two inequalities and assumption (H3), we have
(31)αk-1=ϕ(αk-2,βk-2)≤ϕ(αk-1,βk-1)=αk≤βk=φ(βk-1,αk-1)≤φ(βk-2,αk-2)=βk-1.
Then, it is easy to show by induction that  inequalities  (11′)  and   (12′) hold.
The following proof is similar to that of Theorem4. This completes the proof of Theorem 5.By a similar argument to that of Theorem5, we obtain the following results.Theorem 6.
LetP be a normal cone of E, and let u0,v0∈PI be coupled lower and upper quasi-solutions of (1). Assume that condition (H1),(H2), and (H3) hold.(H4′′)
G is v0 quasi-lower, and there exists w0∈PI such that u0<v0<w0, and there exist α0=sup{α>0:u0≥αw0}, β0=inf{β>0:v0≤βw0}.
Then, (1) has a unique solution x*(t)∈D=[u0,v0], and for any initial x0,y0∈[u0,v0], one has
(32)xn(t)⟶x*(t),yn(t)⟶x*(t),uniformly  ont∈Iasn⟶∞,
where {xn(t)},{yn(t)} are defined as
(33)xn(t)=∫I‍G(t,s,xn-1(s),yn-1(s))ds,yn(t)=∫I‍G(t,s,yn-1(s),xn-1(s))ds,t∈I.


## 4. Applications

Consider the following two-point BVP in the Banach space:(34)-u′′=f(t,u),t∈J=[0,1],u(0)=u(1)=0,
where f∈C[J×P,P], P is a cone in a real Banach space E. Suppose that there exists a mapping g∈C[J×P×P,P] such that f(t,x)=g(t,x,x), and that g satisfies the following conditions:(C1)
g is uniformly continuous on J×P×P, and G is ϕ convex and -φ concave quasi operator,(C2)
g(t,x,y) is nondecreasing in x∈P for fixed (t,y)∈J×P, and g(t,x,y) is nonincreasing in y∈P, for fixed (t,x)∈J×P,(C3)
there exist the bounded nonnegative Lebesgue integrable functionsa(t),  b(t), c(t),  and d(t) satisfying ∫J‍a(s)ds<8,  ∫J‍c(s)ds<8 such that
(35)a(t)x+b(t)≤g(t,x,y)≤c(t)x+d(t),t∈J,x,y∈P.It is well known thatu∈C2[J,P] is a solution of BVP(34) in C2[J,P] if and only if u∈C[J,P] is a solution of the following integral equation:
(36)u(t)=∫J‍h(t,s)g(s,u(s),u(s))ds,t∈J,
where
(37)h(t,s)={t(1-s),t≤s,s(1-t),t>s.Lemma 7.
If assumption(C3) holds, then there exists u0,v0∈C[J,P] such that
(38)u0(t)≤∫J‍h(t,s)g(s,u0(s),v0(s))ds,t∈J,v0(t)≥∫J‍h(t,s)g(s,v0(s),u0(s))ds,t∈J.Proof.
In fact, let(39)L1u(t)=∫J‍h(t,s)a(s)u(s)ds,x0(t)=∫J‍h(t,s)b(s)ds,t∈J,L2v(t)=∫J‍h(t,s)c(s)v(s)ds,y0(t)=∫J‍h(t,s)d(s)ds,t∈J.
Obviously, by assumption (C3), we can get that ∥L1∥≤maxt∈I(t(1-t)/2)∫J‍a(s)ds=(1/8)∫J‍a(s)ds<1, then the equation (I-L1)u=x0 has a unique solution
(40)u0(t)=(I-L1)-1x0=∑n=0∞‍L1nx0∈PI.
Similarly, the equation(I-L2)v=y0 has a unique solution
(41)v0(t)=(I-L2)-1y0=∑n=0∞‍L2ny0∈PI.
Thus, by assumption (C3), for any t∈J, we have
(42)∫J‍h(t,s)g(s,u0(s),v0(s))ds≥∫J‍h(t,s)(a(s)x+b(s))ds=L1u0(t)+x0(t)=u0(t),∫J‍h(t,s)g(s,v0(s),u0(s))ds≤∫J‍h(t,s)(c(s)v0(s)+b(s))ds=L2v0(t)+y0(t)=v0(t),
that is, (38) holds.Theorem 8.
LetP be a normal cone of E. Assume that (C1) and (C3) hold,(C4)
there existsw0∈PI and u0,v0 in (38) of Lemma 7 such that u0<w0<v0, and also there exists α0,β0∈(0,∞) such that u0≥α0w0, β0w0≥v0,(C5)
ϕ(α,β),φ(α,β) are all increasing in α, decreasing in β and ϕ(α0,β0)≥α0,  φ(β0,α0)≤β0, for α,β∈[α0,β0], α<β,
(43)φ(β,α)-ϕ(α,β)≤l(β-α),0<l<1.
Then, (34) has a unique solution x*(t)∈D=[u0,v0], and for any initial x0,y0∈[u0,v0], one has
(44)xn(t)⟶x*(t),yn(t)⟶x*(t),uniformly  ont∈Iasn⟶∞,
where {xn(t)},{yn(t)} are defined as
(45)xn(t)=∫J‍h(t,s)g(s,xn-1(s),yn-1(s))ds,yn(t)=∫J‍h(t,s)g(s,yn-1(s),xn-1(s))ds,t∈J.Proof.
It is easy to see by conditions(C1) and (C2) that G(t,s,x,y)=h(t,s)g(s,x,y) satisfy the conditions (H1) and (H2) of Theorem 4. By (C3) and (38), we have u0,v0∈C[I,P] as coupled lower and upper quasi-solutions of  (34).
Thus, the assumption(H1)–(H4) of Theorem 4 is satisfied from the assumption (C1)–(C5) of Theorem 8. The conclusion of Theorem 8 follows from Theorem 4.Example 9.
In fact, we can construct the functionf(t,x) in Theorem 8.Let(46)f(t,x)=g(t,x,y)=x+1y,t∈[0,1],ϕ(α,β)=sinα+12β,α∈[0,π2],φ(α,β)=3α-5β,
then
(47)G(t,s,αx,βy)=h(t,s)g(s,αx,βy)=h(t,s)(αx+1βy)≥h(t,s)(sinα+12β)(x+1y)=ϕ(α,β)G(t,s,x,y),(48)G(t,s,αx,βy)=h(t,s)g(s,αx,βy)=h(t,s)(αx+1βy)≤h(t,s)(3α-5β)(x+1y)=φ(α,β)G(t,s,x,y).
Thus, G is ϕ convex and -φ concave quasi operator and thus satisfies (C1).It is easy to check thatg(t,x,y) is nondecreasing in x for fixed (t,y) and is nonincreasing in y for fixed (t,x) and thus satisfies (C2).There exista(t)=t/2,  b(t)=t/100, c(t)=2t,  and d(t)=1000t satisfying
(49)∫01‍a(s)ds=12∫01‍tdt=14<8,∫01‍c(s)ds=2∫01‍sds=1<8,
such that
(50)a(t)x+b(t)≤g(t,x,y)≤c(t)x+d(t).
Thus, (C3) holds.There exist(51)u0=∫01‍h(t,s)[u0(s)+1v0(s)]ds,v0=2u0=∫01‍h(t,s)[v0(s)+1u0(s)]ds.
Choose w0=(3/2)u0 such that u0<w0<v0, and also there exist α0=2/3, β0=4/3 such that
(52)u0=α032u0=α0w0,β032u0=2u0=v0.
Thus, (C4) is satisfied.ϕ
(
α
,
β
)
,
φ
(
α
,
β
) are all increasing in α and nondecreasing in β,
(53)ϕ(23,43)=sin23+12×(4/3)≥23=α0,φ(43,23)=3×43-5×23=23≤43=β0,
and for α,β∈[2/3,4/3], α<β, we have
(54)φ(β,α)-ϕ(α,β)=3β-5α-sinα-12β≤99100(β-α).
Thus, (C5) also holds.



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