# Fixed-Point Theorems for Systems of Operator Equations and Their Applications to the Fractional Differential Equations

Authors: Xinqiu Zhang; Lishan Liu; Yumei Zou
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We study the existence and uniqueness of positive solution for a class of nonlinear binary operator equations systems by means of the cone theory and monotone iterative technique, under more general conditions. Also, we give the iterative sequence of the solution and the error estimation of the system. Moreover, we use this new result to study the existence and uniqueness of the solutions for fractional differential equations systems involving integral boundary value conditions in ordered Banach spaces as an application. The results obtained in this paper are more general than many previous results and complement them.


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## Body

## 1. Introduction

In this paper, using the cone theory and monotone iterative technique, we consider the following nonlinear nonmonotone binary equations systems in real Banach spaces,(1)x=Ax,x,x=Bx,x,where A,B: D×D⟶E, D is a subset of real Banach spaces. There have appeared a series of research results concerning the nonlinear operator equation x=A(x,x), x=Ax [1–7], the sum of several classes of mixed-monotone operator equations, and the nonlinear equations systems (1) [8], in recent years. The techniques they used are cone and semiorder [9, 10], the Granas fixed-point index theory [11], the equivalent classes (which are called components) of a real Banach space [12], the Ishikawa iteration process [13, 14], etc.In [15], Zhang investigated the existence and uniqueness of solutions for a class of nonlinear operator equations x=Ax in ordered Banach space, by using the cone theory and Banach contraction mapping principle. The assumption they used is(2)-Bn0x-y≤Ax-Ay≤Bn0x-y,∀x,y∈E,x≥y,where P is a generating normal cone, A: E⟶E is a nonlinear operator, B: E⟶E is a positive linear bounded operator with r(B)<1 (where r(B) is the spectral radius of B), and n0 is a positive integer.In [16], Zhang investigated the existence and uniqueness theorems of fixed points to a class of mixed-monotone operators with convexity and concavity, in which suppose that there exist v>θ, c>1/2 such that θ<A(v,θ)≤v and (3)Aθ,v≥cAv,θ.But in [15, 16], they did not consider the nonmonotone binary operator equations systems x=A(x,x), x=B(x,x), where A,B: D×D⟶E are two nonlinear operators and D is the order interval in E. In this article, the existence and uniqueness of positive solution for nonlinear nonmonotone binary operator equations systems are established under more general conditions, even not supposing the generating of cone P. Compared with [16], we do not need the assumption of convexity-concavity. Moreover, the operators we consider are Tm0-increasing in x and the different upper and lower bounds in formula (6) are more general; the results here cannot be obtained by using Banach contraction mapping principle. Also, different from the results in [15], the iterative sequence of the solution and the error estimation of the system are obtained.As an application, we study the existence and uniqueness of positive solutions and iterative approximation of the unique solution for the following fractional differential equations involving integral boundary value problems:(4)-D0+αut=f1t,ut,vt,D0+αkus,D0+δkvs,t∈I,-D0+αvt=f2t,vt,ut,D0+αkvs,D0+δkus,t∈I,n-1<α≤n,u0=D0+γ1u0=D0+γ2u0=⋯=D0+γn-2u0=0,v0=D0+γ1v0=D0+γ2v0=⋯=D0+γn-2v0=0,D0+β1u1=∫0ηhsD0+β2usdAs+∫01asD0+β3usdAs,D0+β1v1=∫0ηhsD0+β2vsdAs+∫01asD0+β3vsdAs,where n≥2, D0+αu, D0+αku, D0+δku  (k=1,2,…,n-2), D0+γku, (k=1,2,…,n-2),  D0+βiu  (i=1,2,3) is the standard Riemann-Liouville derivatives and n-1<α≤n, k-1<αk,δk,γk≤k  (k=1,2,…,n-2), n-k-1<α-αk≤n-k, n-k-1<α-δk≤n-k, n-k-1<α-γk≤n-k, (k=1,2,…,n-2), β1≥β2, β1≥β3, α-β1>1, β-β1>1, βi-αn-2≥0, βi-δn-2≥0  (i=1,2,3), f: I×E4→E is continuous and a,h∈C(I,R+), A is a function of bounded variation, ∫0ηD0+β2v(s)dA(s), and ∫01D0+β3v(s)dA(s) denote the Riemann-Stieltjes integral with respect to A.Recently, fractional differential equations, arising in the mathematical modeling of systems and processes, have drawn more and more attention of the research community due to their numerous applications in various fields of science such as engineering, chemistry, physics, and mechanics. Boundary value problems of fractional differential equations have been investigated for many authors [17–22]. Now, there are many papers dealing with the problem for different kinds of boundary value conditions such as multipoint boundary condition [23, 24], integral boundary condition [25–31], and many other boundary conditions [32]. From the application, we can see that the fixed-point theorems in this paper have extensive applied background. The results presented here are more general and complement many previous known results.


## 2. Preliminaries

Now we present briefly some definitions, lemmas, and basic results that are to be used in the article; for convenience of the reader, we refer the reader to [3, 4, 10, 33, 34] for more details.Suppose that(E,∥·∥) is a real Banach space, and θ is the zero element of E. Recall that a nonempty closed convex set P⊂E is a cone if it satisfies (1) x∈P,λ≥0⟹λx∈P; (2) x∈P,-x∈P⟹x=θ. The real Banach space E can be partially ordered by a cone P⊂E, i.e., x≤y if and only if y-x∈P. If x≤y and x≠y, then we denote x<y or y>x. Let C[I,E]={x(t):I⟶E|x(t)iscontinuous}. Then C[I,E] is a Banach space with the norm xc=maxt∈Ix(t), for x∈C[I,E].The coneP is called normal if there exists a constant N>0 such that for all x,y∈E,θ≤x≤y implies x≤Ny, and the smallest N is called the normality constant of P. If x1,x2∈E, x1≤x2, the set [x1,x2]={x∈E∣x1≤x≤x2} is called the order interval between x1 and x2.Definition 1 ([3, 4]).
LetD be a subset of a real ordered Banach space E; A: D×D⟶E is said to be a mixed-monotone operator if A(x,y) is increasing in x and decreasing in y; i.e., xi,yi∈D(i=1,2), x1≤x2, y1≥y2 imply A(x1,y1)≤A(x2,y2). The element x∈D is called a fixed point of A if A(x,x)=x.


## 3. Main Results

Theorem 2.
LetE be a real Banach space, P be a normal cone in E, and D=[u0,v0]={x∈E∣u0≤x≤v0} be the order interval in E. Assume that A,B: D×D⟶E are two nonlinear operators, T,L: E⟶E are two positive linear bounded operators and satisfy the following conditions:
( J 0 )  u0≤Au0,v0, Bv0,u0≤v0.
( J 1 ) For all x∈D, A(x,y) and B(x,y) are decreasing in y, i.e., for any x∈D, y1,y2∈D, y1≤y2 implies Ax,y1≥Ax,y2, Bx,y1≥Bx,y2; and there exist two positive numbers Mi>0  (i=1,2) such that for all y∈D, x1,x2∈D, x1≤x2, (5)Ax2,y-Ax1,y≥-Tm0x2-x1,Bx2,y-Bx1,y≥-Tm0x2-x1;
( J 2 )  I+Tm0 is reversible (I is the identity operator) and I+Tm0x≥θ⟹x∈P.
( J 3 )  TL=LT and there exist two positive integers m0, n0 such that rI+Tm0-1rLn0+rTm0<1 (where r(·) is the spectral radius of linear bounded operator) and(6)-Tm0y-x≤By,x-Ax,y≤Ln0y-x,∀x,y∈D,x≤y.Then the nonlinear operator equations system (1) has a unique solution x∗,x∗ in D×D. And for any initial values x0,y0∈D, x0≤y0, by constructing successively the sequences as follows:(7)xn=I+Tm0-1Axn-1,yn-1+Tm0xn-1,yn=I+Tm0-1Byn-1,xn-1+Tm0yn-1,n=1,2,…,we have xn⟶x∗, yn⟶x∗ in E, as n⟶∞. Moreover, for any rI+Tm0-1rLn0+rTm0<δ<1, there exists n1 such that(8)xn-x∗≤2Nδnv0-u0,yn-x∗≤2Nδnv0-u0,n≥n1.Proof.
SinceI+Tm0 is reversible, let(9)Fx,y=I+Tm0-1Ax,y+Tm0x,Gy,x=I+Tm0-1By,x+Tm0y,x,y∈D.Then (7) can be written as(10)xn=Fxn-1,yn-1,yn=Gyn-1,xn-1,n=1,2,….By (J1), (J2), it is easy to prove that F and G satisfy the following conditions:
( A 1 ) By I+Tm0x≥θ⇒x∈P, we know F,G: D×D→E are two mixed-monotone operators.
( A 2 ) For all x,y∈D,(11)Gy,x-Fx,y=I+Tm0-1By,x+Tm0y-I+Tm0-1Ax,y+Tm0x=I+Tm0-1By,x-Ax,y+Tm0y-x.Combining with (J3), it is easy to prove that(12)θ≤Gy,x-Fx,y≤Hy-x,∀x,y∈D,x≤y,where H≜I+Tm0-1Ln0+Tm0 and I is the identity operator.
( A 3 ) Also, by (J0), we have(13)Fu0,v0=I+Tm0-1Au0,v0+Tm0u0≥I+Tm0-1u0+Tm0u0=u0,(14)Gv0,u0=I+Tm0-1Bv0,u0+Tm0v0≤I+Tm0-1v0+Tm0v0=v0.Thus, combining with (12), we have(15)u0≤Fu0,v0≤Gv0,u0≤v0.Let un=Fun-1,vn-1, vn=Gvn-1,un-1(n=1,2,…). Thus, by (15), we know(16)u0≤u1≤v1≤v0.Therefore, by (A1) and (A2), using mathematical induction, we can easily prove that(17)u0≤u1≤⋯≤un≤⋯≤vn≤⋯v1≤v0.
Firstly, we prove that(18)θ≤vn-un≤Hnv0-u0,n=1,2,….By (A2), we can easily prove that(19)θ≤v1-u1≤Gv0,u0-Fu0,v0≤Hv0-u0,i.e., (18) holds for n=1. Suppose that (18) holds for n=k, i.e.,(20)θ≤vk-uk≤Hkv0-u0.Then, for n=k+1, by (A1) and (A2) we know(21)uk+1=Fuk,vk≤Gvk,uk=vk+1,θ≤vk+1-uk+1=Gvk,uk-Fuk,vk≤Hvk-uk≤Hk+1v0-u0.By (19)-(21), using mathematical induction, we know (18) holds.
Next we prove thatxn is a Cauchy sequence. From TL=LT, we have I+Tm0-1Ln0+Tm0=Ln0+Tm0I+Tm0-1. Then (22)rI+Tm0-1Ln0+Tm0≤rI+Tm0-1rLn0+Tm0≤rI+Tm0-1rLn0+rTm0<1. Consequently, there exists δ>0 such that (23)limn→∞Hn1/n=rH=rI+Tm0-1Ln0+Tm0≤rI+Tm0-1rLn0+rTm0<δ<1. Thus, there exists n1 such that(24)Hn<δn,n≥n1.Then by (17), we have(25)θ≤un≤un+p≤vn+p≤vn,n,p=1,2,….Consequently, by (18) and (25), we have(26)θ≤un+p-un≤vn-un≤Hnv0-u0,θ≤vn-vn+p≤vn-un≤Hnv0-u0,n,p=1,2,…. Therefore, by (24), (26) and the normality of cone P, we have(27)un+p-un≤NHnv0-u0≤Nδnv0-u0,vn-vn+p≤NHnv0-u0≤Nδnv0-u0,n≥n1,p=1,2,…, where N is the normality constant of P. Consequently, un and vn are Cauchy sequences. Since E is complete, thus there exist u∗,v∗∈D, such that(28)limn→∞un=u∗,limn→∞vn=v∗.And by (17), we know(29)un≤u∗≤v∗≤vn,n=0,1,2,….Thus, u∗,v∗∈D. By (18) and (29), we have(30)θ≤v∗-u∗≤vn-un≤Hnv0-u0,n=0,1,2,….Thus, by (26), (30) and the normality of cone P, we have (31)v∗-u∗≤NHnv0-u0≤Nδnv0-u0→0,n→∞. Consequently, u∗=v∗. Let x∗:=u∗=v∗; by (A1) and (A2), we have(32)un+1=Fun,vn≤Fx∗,x∗≤Gx∗,x∗≤Gvn,un=vn+1,n=1,2,….Let n→∞, by (32) and the normality of P we have Fx∗,x∗=Gx∗,x∗=x∗. Therefore, by the definitions of F and G, we have x∗=Ax∗,x∗, x∗=Bx∗,x∗; i.e., x∗,x∗ is the solution of operator equation (1).
Finally, we prove thatx∗,x∗ is the unique solution of operator equations systems (1) in D×D. In fact, suppose x¯,x¯ is another solution of equations systems (1) in D×D, then by (A1), using mathematical induction, we can easily see that un≤x¯≤vn  (n=1,2,…). Thus, by (28) and the normality of P, we have x¯=x∗. Therefore, the operator equations systems (1) have a unique solution x∗,x∗ in D×D.
Now for any initial pointsx0,y0∈D, x0≤y0, we construct successively the sequences (33)xn=Fxn-1,yn-1,yn=Gyn-1,xn-1,n=1,2,…. Since x0,y0∈D, i.e., (34)u0≤x0≤y0≤v0.Suppose (35)un-1≤xn-1≤yn-1≤vn-1. Then (36)un=Fun-1,vn-1≤xn=Fxn-1,yn-1≤yn=Gyn-1,xn-1≤vn=Gvn-1,un-1. Thus, by mathematical induction, we have(37)un≤xn≤yn≤vn,n=0,1,2,….By (18) and (37), we have(38)θ≤xn-un≤vn-un≤Hnv0-u0,(39)θ≤x∗-un≤vn-un≤Hnv0-u0.Thus,(40)xn-x∗≤xn-un+x∗-un≤2NHnv0-u0≤2Nδnv0-u0,n≥n1.In the same way, we can prove that(41)yn-x∗≤2Nδnv0-u0,n≥n1.Consequently, by (40) and (41) we know that (8) holds.Takingm0=n0=1 in Theorem 2, then we get the following corollary.Corollary 3.
LetE be a real Banach space, P be a normal cone in E, and D=[u0,v0]={x∈E∣u0≤x≤v0} be the order interval in E. Assume that A,B: D×D⟶E are two nonlinear operators, T,L: E→E are two positive linear bounded operators and satisfy the following conditions:
( Q 0 )  u0≤Au0,v0, Bv0,u0≤v0.
( Q 1 ) For all x∈D, A(x,y) and B(x,y) are decreasing in y, i.e., for any x∈D, y1,y2∈D, y1≤y2 implies Ax,y1≥Ax,y2, Bx,y1≥Bx,y2; and there exist two positive numbers Mi>0  (i=1,2) such that for all y∈D, x1,x2∈D, x1≤x2, (42)Ax2,y-Ax1,y≥-Tx2-x1,Bx2,y-Bx1,y≥-Tx2-x1;
( Q 2 )  TL=LT, rI+T-1rL+rT<1 (where r(·) is the spectral radius of linear bounded operator) and(43)-Ty-x≤By,x-Ax,y≤Ly-x,∀x,y∈D,x≤y.Then the nonlinear operator equations system (1) has a unique solution x∗,x∗ in D×D. And for any initial values x0,y0∈D, x0≤y0, by constructing successively the sequences as follows:(44)xn=I+T-1Axn-1,yn-1+Txn-1,yn=I+T-1Byn-1,xn-1+Tyn-1,n=1,2,…, we have xn⟶x∗, yn⟶x∗ in E, as n⟶∞. Moreover, for any rI+T-1rL+rT<δ<1, there exists n1 such that(45)xn-x∗≤2Nδnv0-u0,n≥n1,yn-x∗≤2Nδnv0-u0,n≥n1.


## 4. An Application for Fractional Differential Equations Involving Integral Boundary Value Problems

Now we present briefly some definitions, lemmas, and basic results that are to be used in the article for convenience of the reader. We refer the reader to [35–39] for more details.Definition 4 ([35, 36, 38, 39]).
The Riemann-Liouville fractional integral of orderα>0 of a function u: (0,+∞)→R is given by (46)I0+αut=1Γα∫0tt-sα-1usds provided that the right-hand side is pointwise defined on (0,+∞).Definition 5 ([35, 36, 38, 39]).
The Riemann-Liouville fractional derivative of orderα>0 of a continuous function u: (0,+∞)⟶R is given by (47)D0+αut=1Γn-αddtn∫0tust-sα-n+1ds, where n=[α]+1, [α] denotes the integer part of the number α, provided that the right-hand side is pointwise defined on (0,+∞).Lemma 6 ([35, 36, 38, 39]).
(1) Ifu∈L1(0,1) and α>β>0, then(48)I0+αI0+βut=I0+α+βut,D0+βI0+αut=I0+α-βut,D0+βI0+βut=ut.
(2) Ifu∈L1(0,1) and α>0, then D0+αu(t)=0 has unique solution(49)ut=c1tα-1+c2tα-2+⋯+cntα-n,where ci∈R  (i=0,1,2,…,n), n=[α]+1.Lemma 7 ([35, 36, 38, 39]).
Letα>0 and let f(x) be integrable; then(50)I0+αD0+αfu=fu+c1uα-1+c2uα-2+⋯+cnuα-n,where ci∈R(i=1,2,…,n) and n is the smallest integer greater than or equal to α.Lemma 8 ([37]).
Letf∈L1(0,1)∩C(0,1), n-1<α≤n. Assume that the following condition is satisfied:(51)HΓα-β1Γα-β2∫0ηhttα-β2dAt+Γα-β1Γα-β3∫01attα-β3dAt<1;then(52)D0+α-αn-2ut+ft=0,n-1<α≤n,0<t<1,D0+γn-2-αn-2u0=0,D0+β1-αn-2u1=∫0ηhsD0+β2-αn-2usdAs+∫01hsD0+β3-αn-2usdAs has a unique positive solution(53)ut=∫01Gt,sfsds,t∈0,1,where(54)Gt,s=G1t,s+tα-αn-2-1Γα-β2Δ-1∫0ηhtG2t,sdAt+tα-αn-2-1Γα-β3Δ-1∫01atG3t,sdAt,in which(55)δ1≔∫0ηhttα-β2-1dAt,δ2≔∫0ηattα-β3-1dAt,Δ=1Γα-β1-1Γα-β2δ1-1Γα-β3δ2,G1t,s=1Γα-αn-2tα-αn-2-11-sα-β1-1-t-sα-αn-2-1,0≤s≤t≤1,tα-αn-2-11-sα-β1-1,0≤t≤s≤1,G2t,s=1Γα-αn-2tα-β2-11-sα-β1-1-t-sα-β2-1,0≤s≤t≤1,tα-β2-11-sα-β1-1,0≤t≤s≤1,G3t,s=1Γα-αn-2tα-β3-11-sα-β1-1-t-sα-β3-1,0≤s≤t≤1,tα-β2-11-sα-β3-1,0≤t≤s≤1.Lemma 9 ([37]).
If the condition(H) in Lemma 8 is satisfied, the Green function G(t,s) has the following properties:
(1)G(t,s)>0, for all t,s∈(0,1).
(2) For anyt,s∈[0,1], we have(56)θ≤tα-αn-2-1l1s≤Gt,s≤L1tα-αn-2-1≤L1,where(57)l1s=Δ-1Γα-β2∫0ηhtG2t,sdAt+Δ-1Γα-β3∫01atG3t,sdAt,L1=1Γα-αn-2+Δ-1Γα-β21Γα-αn-2∫0ηhtdAt+Δ-1Γα-β31Γα-αn-2∫01atdAt.In the following, we need the following assumptions:
H 0  maxt∈I∫01G¯(t,τ)dτ<1, where (58)G¯t,τ=r1+r2Gt,τ+r3Γαn-2-αk∫τ1Gt,ss-ταn-2-αk-1ds+r4Γδn-2-δk∫τ1Gt,ss-τδn-2-δk-1ds;H1  fi∈C[I×E4,E] and there exists v0∈C[I,E] such that (59)∫01Gt,sf2s,v0s,θ,I0+αn-2-αkv0s,θds≤v0t,∀t∈I;H2 For all xi,yi∈E  (i=1,2,3,4), y1≥x1, y2≤x2, y3≥x3, y4≤x4,(60)fit,y1,y2,y3,y4-fit,x1,x2,x3,x4≥0,∀t∈I,i=1,2;H3 There exist four constants ri>0 (i=1,2,3,4) such that for any t∈I, xi,yi∈E  (i=1,2,3,4) with x1≤y1, x2≥y2, x3≤y3, x4≥y4,(61)0≤f2t,y1,y2,y3,y4-f1t,x1,x2,x3,x4≤r1y1-x1+r2x2-y2+r3y3-x3+r4x4-y4.Theorem 10.
LetE be a real Banach space and P be a normal cone in E. Assume that the conditions (H) and H0–H3 are satisfied. Then the system of nonlinear differential equations (50) has unique positive symmetry solution (w∗,w∗)∈D×D, where D=θ,v0⊂C[I,E]. Moreover, for any initial functions x0,y0∈D, there exist monotone iteration sequences xn, yn, such that xn⟶w∗, yn⟶w∗ in C[I,E], as n⟶∞, where(62)xnt=∫01Gt,sf1s,xn-1s,yn-1s,I0+αn-2-αkxn-1s,I0+δn-2-δkyn-1sds,ynt=∫01Gt,sf2s,yn-1s,xn-1s,I0+αn-2-αkyn-1s,I0+δn-2-δkxn-1sds,t∈I,n=1,2,3,….Proof.
It is well known that(u,v)∈C[I,E]×C[I,E] is a solution of the system (50) if and only if (u,v)∈C[I,E]×C[I,E] is a solution of the system of nonlinear integral equations(63)ut=∫01Gt,sf1s,us,vs,I0+αn-2-αkus,I0+δn-2-δkvsds,vt=∫01Gt,sf2s,vs,us,I0+αn-2-αkvs,I0+δn-2-δkusds,t∈I.
Consider the operatorsA,B: D×D⟶C[I,E] as follows:(64)Au,vt=∫01Gt,sf1s,us,vs,I0+αn-2-αkus,I0+δn-2-δkvsds,Bv,ut=∫01Gt,sf2s,vs,us,I0+αn-2-αkvs,I0+δn-2-δkusds,u,v∈D,t∈I. By H2, for all u1,v1,u2,v2∈D×D, u1≤u2, v1≥v2, t∈I, we obtain(65)Au2,v2t-Au1,v1t=∫01Gt,sf1s,u2s,v2s,I0+αn-2-αku2s,I0+δn-2-δkv2sds-∫01Gt,sf1s,u1s,v1s,I0+αn-2-αku1s,I0+δn-2-δkv1sds≥0,(66)Bu2,v2t-Bu1,v1t=∫01Gt,sf2s,u2s,v2s,I0+αn-2-αku2s,I0+δn-2-δkv2sds-∫01Gt,sf2s,u1s,v1s,I0+αn-2-αku1s,I0+δn-2-δkv1sds≥0.Consequently, A,B: D×D⟶C[I,E] are mixed-monotone. By H1, for all t∈I, we obtain(67)Aθ,v0t=∫01Gt,sf1s,0,v0s,0,I0+δn-2-δkv0sds≥0,(68)Bv0,θt=∫01Gt,sf2s,v0s,0,I0+αn-2-αkv0s,0ds≤v0t.By H3, for any t∈I, u,v∈D with u≤v,(69)
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A
u
,
v
t
=
∫
0
1
G
t
,
s
f
2
s
,
v
s
,
u
s
,
I
0
+
α
n
-
2
-
α
k
v
s
,
I
0
+
δ
n
-
2
-
δ
k
u
s
d
s
-
∫
0
1
G
t
,
s
f
1
s
,
u
s
,
v
s
,
I
0
+
α
n
-
2
-
α
k
u
s
,
I
0
+
δ
n
-
2
-
δ
k
v
s
d
s
≤
∫
0
1
G
t
,
s
r
1
v
s
-
u
s
+
r
2
v
s
-
u
s
+
r
3
I
0
+
α
n
-
2
-
α
k
v
s
-
I
0
+
α
n
-
2
-
α
k
u
s
+
r
4
I
0
+
δ
n
-
2
-
δ
k
v
s
-
I
0
+
δ
n
-
2
-
δ
k
u
s
d
s
=
∫
0
1
r
1
+
r
2
G
t
,
s
v
s
-
u
s
d
s
+
∫
0
1
r
3
G
t
,
s
I
0
+
α
n
-
2
-
α
k
v
s
-
u
s
d
s
+
∫
0
1
r
4
G
t
,
s
I
0
+
δ
n
-
2
-
δ
k
v
s
-
u
s
d
s
=
∫
0
1
r
1
+
r
2
G
t
,
s
v
s
-
u
s
d
s
+
∫
0
1
r
3
G
t
,
s
1
Γ
α
n
-
2
-
α
k
∫
0
s
s
-
τ
α
n
-
2
-
α
k
-
1
v
τ
-
u
τ
d
τ
d
s
+
∫
0
1
r
4
G
t
,
s
1
Γ
δ
n
-
2
-
δ
k
∫
0
s
s
-
τ
δ
n
-
2
-
δ
k
-
1
v
τ
-
u
τ
d
τ
d
s
=
∫
0
1
r
1
+
r
2
G
t
,
s
v
s
-
u
s
d
s
+
∫
0
1
r
3
Γ
α
n
-
2
-
α
k
∫
τ
1
G
t
,
s
s
-
τ
α
n
-
2
-
α
k
-
1
d
s
v
τ
-
u
τ
d
τ
+
∫
0
1
r
4
Γ
δ
n
-
2
-
δ
k
∫
τ
1
G
t
,
s
s
-
τ
δ
n
-
2
-
δ
k
-
1
d
s
v
τ
-
u
τ
d
τ
=
L
v
-
u
t
,where (71)Lut=∫01r1+r2Gt,τ+r3Γαn-2-αk∫τ1Gt,ss-ταn-2-αk-1ds+r4Γδn-2-δk∫τ1Gt,ss-τδn-2-δk-1dsuτdτ,∀t∈I.
Set(72)G¯t,τ=r1+r2Gt,τ+r3Γαn-2-αk∫τ1Gt,ss-ταn-2-αk-1ds+r4Γδn-2-δk∫τ1Gt,ss-τδn-2-δk-1ds,∀t∈I; then Lu(t)=∫01G¯(t,τ)u(τ)dτ. Consequently, for any t∈I, for any u,v∈D with u≤v,(73)0≤Bv,ut-Au,vt≤Lv-ut.
In the following, we proverL<1. In fact, by H0, since maxt∈I∫01G¯t,sds<1, there exists a constant m1:0<m1<1 such that ∫01Gt,sds≤m1<1, for any t∈I. Thus, for all t∈I, u∈D,(74)Lut=∫01G¯t,susds≤∫01G¯t,susds≤∫01G¯t,sdsuc≤m1uc,(75)L2ut=∫01G¯t,sLusds≤∫01G¯t,sLusds≤∫01G¯t,sLusds≤∫01G¯t,sdsm1uc≤m12uc. By mathematical induction, we can easily prove that for all natural number n,(76)Lnut≤m1nuc,t∈I;therefore,(77)Lnuc=maxt∈ILnut≤m1nuc,consequently,(78)Ln≤m1n,thus, rL=limn→∞Ln1/n≤m1<1.
Thus, all conditions of Corollary3 are satisfied; therefore, the conclusions of Theorem 10 hold. Consequently, the proof of Theorem 10 is completed.



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*Source: 7469868-2018-08-05.xml*