# Application of Topological Degree Method for Solutions of Coupled Systems of Multipoints Boundary Value Problems of Fractional Order Hybrid Differential Equations

Authors: Muhammad Iqbal; Yongjin Li; Kamal Shah; Rahmat Ali Khan
Journal: Complexity
 (2017)

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## Abstract

We established the theory to coupled systems of multipoints boundary value problems of fractional order hybrid differential equations with nonlinear perturbations of second type involving Caputo fractional derivative. The proposed problem is as follows:Dcαxt-ft,xt=gt,yt,Iαyt, t∈J=[0,1],Dcαyt-ft,yt=gt,xt,Iαxt, t∈J=0,1, Dcpx0=ψxη1, x′0=0,…,xn-20=0, Dcpx1=ψxη2, Dcpy0=ψyη1, y′0=0,…,yn-20=0, Dcpy1=ψyη2, where p,η1,η2∈0,1, ψ is linear, Dcα is Caputo fractional derivative of order α, with n-1<α≤n, n∈N, and Iα is fractional integral of order α. The nonlinear functions f, g are continuous. For obtaining sufficient conditions on existence and uniqueness of positive solutions to the above system, we used the technique of topological degree theory. Finally, we illustrated the main results by a concrete example.


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## Body

## 1. Introduction

Due to a wide range of applications of fractional calculus in various scientific disciplines such as optimization theory, electric networks, signal processing, nonlinear control theory, nonlinear biological systems, controlled thermonuclear fusion, viscoelasticity, chemistry, turbulence, mechanics, oscillation, diffusion, fluid dynamics, stochastic dynamical system, polymer physics, plasma physics, astrophysics, chemical physics, and economics [1–4], the subject area has received much attention among the scientific community. Recently, the theory on existence and uniqueness of solutions to boundary value problems (BVPs) of fractional differential equations (DEs) are well studied and many results are available in literature (see, e.g., [5–10] and the references herein). The perturbed DEs are categorized into various types. Dhage [11] classified different types of perturbations for nonlinear integral and DEs. An important class of DEs which captured great attention in last few decades is the quadratic perturbations of nonlinear differential equations known as hybrid differential equations (HDEs). This class is well studied for BVPs with ordinary DEs. However, existence theory for BVPs with fractional hybrid differential equations (FHDEs) are not well explored and few results are available in the literature (see [12, 13]). This class of DEs includes perturbations of dynamical systems in different ways and hence includes several dynamical systems as special cases. Modern control-command systems often include controllers that perform nonlinear computations to control a physical system, which can typically be described by hybrid automaton containing high dimensional systems of nonlinear DEs. The hybrid systems are dynamical systems that involve the interaction of continuous (real valued) states and discrete (finite valued) states.Recently, existence of solutions to some classes of HDEs is studied with the use of hybrid fixed point theory (see [14–18]). Dhage and Jadhav [17] developed sufficient conditions for existence of maximal and minimal solutions to the following first-order HDEs: (1)ddtrt-Θt,rt=φt,rt,a.et∈I,rt0=r0∈R,where I=t0,t0+a∈R for some t0, a∈R with a>0 and Θ,φ∈CI×R,R. The results of [17] were generalized by Lu et al. [18] to the case of fractional order and developed conditions for existence and uniqueness to the following FHDEs: (2)Dqrt-Θt,rt=φt,rt,a.et∈I,0<q<1,rt0=r0∈R.Bashiri et al. in [13] used coupled fixed point theorem, a Krasnoselskii type generalization of fixed point theorem of Burton [19] in Banach spaces, to the case of coupled systems and developed sufficient conditions for existence of solutions to the following coupled systems of two-point BVPs for FHDEs:(3)Dqrt-Θt,rt=ψt,st,Iαst,a.et∈I,Dqst-Θt,st=ψt,rt,Iαrt,a.et∈I,0<q<1,α>0,r0=0,s0=0.Leray Schauder theory is powerful tool in solving operator equations of the formI-Tu=w, where T is compact. But in many situations, T is not compact. Therefore, it is natural to ask whether the solutions of the above operator equation are possible if T is not compact. Schauder constructed an example and showed that it is impossible. But later on Browder, Sadovski and Vath, and so on proved that it is possible to define a complete analogue of the Leray Schauder theory for condensing type mapping with compactness. They called this method the “topological degree method”; see for detail [20]. In 1970, Mawhin introduced the mentioned degree theory for nonlinear Volterra integral equations and differential equations with boundary conditions. On the other hand, using the classical fixed point theory such as “Schauder fixed point theorem” and “Banach contraction principle” required stronger conditions on the nonlinear functions and thus restrict the applicability of these results to limited classes of applied problems and to some specialized systems of BVPs. Finding the fixed points of the respective operator equations corresponding to fractional integral equations needs strong conditions for the compactness of the operator. To relax the criteria and establish weaker conditions in order to extend tools to more classes of BVPs, researchers need to look for some other refined tools of functional analysis. One of such tools is “topological degree theory.” The topological degree method is a powerful tool for existence of solutions to BVPs of many mathematical models that arise in applied nonlinear analysis. The concerned method is also called the “prior estimate method.” By coincidence degree theory approach, Mawhin [21] studied existence of solutions to the following BVPs:(4)ddtst=ψt,st,t∈0,1,s0=s1,(5)-d2dt2st=ψt,st,ddtst,t∈0,π,s0=sπ=0,under appropriate assumptions. Dinca et al. [22] used this method together with Leray Schauder degree to prove the existence of solutions of the Dirichlet problems with p-Laplacian:(6)-Δpw=ψt,w,inΩw∣∂Ω=0.Isaia [23] used this method along with “the degree for condensing maps” and proved the existence of solutions for the following integral equation by using appropriate assumption on the functions φ and Θ, where φ:a1,a2×R→R and Θ:a1,a2×a1,a2×R→R are continuous functions, a1,a2∈R, (7)st=φt,st+∫a1a2Θt,ξ,sξdξ,t∈a1,a2.Wang et al. [24] used “topological degree method” to a class of “nonlocal Cauchy problems” of the following form to study the existence and uniqueness of solutions:(8)Dcpst=ψt,st,t∈0,T,s0+gs=s0,where Dcp is the Caputo fractional derivative of order p∈0,1, s0∈R, and ψ:0,T×R→R is continuous. For more study of fractional DEs via topological degree method, we refer to [6, 24–26]. Recently, Shah et al. [27] applied the “topological degree method” and established sufficient conditions for the existence of at least one solution to the following coupled system of nonlinear ordinary fractional equations with four-point boundary conditions:(9)Dcpxt=ϕt,xt,yt,t∈0,1,Dcqyt=ψt,xt,yt,t∈0,1,x0=fx,x1=λxη,λ,η∈0,1,y0=gy,y1=δyξ,δ,ξ∈0,1,where p,q∈1,2 and ϕ,ψ:0,1×R×R→R are continuous, also f,g∈0,1,R. The above system (9) contains two nonlinear functions, namely, ϕ, ψ. While the proposed coupled system contains four nonlinearities as follows:(10)Dcαxt-ft,xt=gt,yt,Iαyt,t∈J=0,1,Dcαyt-ft,yt=gt,xt,Iαxt,t∈J=0,1,Dcpx0=ψxη1,x′0=0,…,xn-20=0,Dcpx1=ψxη2,Dcpy0=ψyη1,y′0=0,…,yn-20=0,Dcpy1=ψyη2,which made the considered problem more general and complicated. Furthermore, in the proposed coupled system (10) of FHDEs, the order of fractional differential operator lies in n-1,n. Moreover, the boundary conditions of the proposed problem involve Caputo fractional order derivative as well as ordinary derivative of higher order. Moreover, to the best of our knowledge, the topological degree method has not been applied properly for the systems of nonlinear hybrid fractional differential equations.Motivated by this consideration, our main focus in the present article is to use topological degree approach for condensing mapping to investigate existence of solutions of coupled system (10). We would like to solve the dynamics of the system to determine how the state will evolve in the future, that is, to find a function xt called trajectory or solution of the system.


## 2. Preliminaries

In the following, we provide some basic definitions and results of fractional calculus and topological degree theory. For detailed study, we refer to [1–4, 10, 28–30].Definition 1 (see [1]).
The fractional integral operator of orderr∈R+ of a function f:0,∞→R is defined as (11)Irft=1Γr∫0tt-ξr-1fξdξ,provided that integral on the right is pointwise defined on 0,∞.Definition 2 (see [1]).
The Caputo fractional order derivative of orderp∈R+of a continuous function f:0,∞→R is defined as (12)Dcpft=1Γm-p∫0tt-ξm-p-1fmξdξ,where m=p+1, provided that integral on the right is pointwise defined on 0,∞.Lemma 3 (see [1, 28]).
The following results hold for fractional integral and Caputo derivative.(i)
Dcpλ1ft+λ2gt=λ1Dcpft+λ2Dcpgt, λ1,λ2∈R.(ii)
DcpIqft=Iq-pft, DcpIpft=ft.(iii)
Dcptq=(Γq+1/Γq+1-p)tq-p, DpC=0, where C is a constant.(iv)
IqDcqft=ft-∑k=0n-1(Dkf0/Γk-q+1)tk=ft-d0-d1t-d2t2-d3t3-⋯-dn-1tn-1, di∈R, for 0≤i≤n-1.LetJ=0,1; the spaces of all continuous functions Y=CJ,R and Z=CJ,R are Banach spaces under the usual norms y=sup⁡yt:t∈J and z=sup⁡zt:t∈J, respectively. Moreover, the product space Y×Z is a Banach space under the norm y,z=y+z and norm y,z=max⁡y+z. In the following, Y is a Banach space and M⊂PY is the family of all its bounded sets.We recall the following notions, which can be found in [29].Definition 4.
“The functionΥ:M→R+ defined as μMk=inf⁡d>0:Mkadmitsafinitecoverbysetsofdiameter≤d, where Mk∈M is called the (Kuratowski-) measure of noncompactness.”Some of the properties of this measure are listed below (without proof).Proposition 5.
The following assertions hold for Kuratowski measureΥ:(i)
ΥMk=0 iff Mk is relatively compact.(ii)
Υ is a seminorm; that is, ΥσMk=σΥMk and ΥMk1+Mk2≤ΥMk1+ΥMk2, where Mk1,Mk2∈M, and σ∈R.(iii)
Mk1⊂Mk2 implies ΥMk1≤ΥMk2 and ΥMk1∪Mk2=max⁡ΥMk1,ΥMk2.(iv)
ΥconvMk=ΥMk.(v)
ΥMk¯=ΥMk.Definition 6.
LetL⊂Y and Θ:L→Y be a continuous bounded mapping. Then Θ is Υ-Lipschitz if ∃k≥0∋(13)ΥΘL≤kΥL∀L⊂Lbounded.
Furthermore, ifk<1, then Θ is a strict Υ-contraction.Definition 7.
The functionΘ is Υ-condensing if(14)ΥΘL<ΥL∀L⊂Lbounded withΥL>0.
IfΥΘL≥ΥL, then ΥL=0.It may be noted that the class of strictΥ-contractions contains the class of Υ-condensing maps. Also every member of Υ-condensing map is Υ-Lipschitz with constant k=1.Definition 8.
Θ
:
L
→
Y is called Lipschitz if ∃k>0∋(15)Θy1-Θy2≤ky1-y2∀y1,y2∈L.If k<1, then Θ is a strict contraction.Proposition 9 (see [23]).
IfΘ:L→Y is compact, then Θ is Υ-Lipschitz with zero constant.Proposition 10 (see [23]).
IfΘ:L→Y is Lipschitz with constant k1, then Θ is Υ-Lipschitz with the same constant k1.Proposition 11 (see [23]).
IfΘ1,Θ2:L→Y are Υ-Lipschitz with constants k1 and k2, respectively, then Θ1+Θ2:L→Y is Υ-Lipschitz with constant k1+k2.The following theorem from [23] plays a vital role for our main result.Theorem 12.
LetΘ:Y→Y be Υ-condensing and(16)S=y∈Y:∃σ∈J∋y=σΘy.
IfS is a bounded set in Y, so there exist r>0∋  S⊂Br0; then the degree(17)DI-σΘ,Br0,0=1∀σ∈J.
Consequently,Θ has at least one fixed point and the set of fixed points of Θ lies in Br0.Definition 13 (see [31]).
An elementy1,y2∈Y×Y is called “a coupled fixed point of a mapping” T:Y×Y→Y if Ty1,y2=y1 and Ty2,y1=y2.


## 3. Main Result

Assume thatf:J×R→R and g:J×R×R→R are continuous functions. Consider the following hypotheses:(A1)
(∂k/∂tk)ft,xtt=0=0.(A2)
There existk1∈0,1∋  ∀t,x,t,y∈J×R,(18)ft,x-ft,y≤k1x-y.(A3)
There existc1, N1≥0,∋  ∀t,x∈J×R,(19)ft,x≤c1xp1+N1,wherep1∈0,1.(A4)
There existc2, N2≥0,∋  ∀t,x,Iαx∈J×R×R,(20)gt,x,Iαx≤c2xp2+N2,wherep2∈0,1.(A5)
There exist a continuous functionh∈Y∋  gt,xt,yt≤ht, for x,y∈R and t∈J.The following lemma is useful in the existence results.Lemma 14.
IfH:J→R is α-time integrable and assuming that the hypothesis (A1) holds, then the solutions of the multipoints BVPs:(21)Dcαxt-ft,xt=Ht,t∈J,Dcpx0=ψxη1,x′0=0,…,xn-20=0,Dcpx1=ψxη2whereψis linear,0<η1,η2<1,are the following integral equation:(22)xt=ft,xt+IαHt+a1tn-1+a2,wherea1,a2∈R.Proof.
ApplyingIα on Dcαxt-ft,xt=Ht and using Lemma 3, we obtain(23)xt=ft,xt+IαHt+d0+d1t+d2t2+d3t3+d4t4+⋯+dn-2tn-2+dn-1tn-1,wheredi∈R,fori=0,1,2,…,n-1.Now by conditions x′0=0,…,xn-20=0 and hypothesis A1, (23) implies d1=d2=⋯=dn-2=0. Hence,(24)xt=ft,xt+IαHt+d0+dn-1tn-1.Applying Dcp on (24) and using Lemma 3(25)Dcpxt=Dcpft,xt+Iα-pHt+dn-1ΓnΓn-ptn-1-p.By conditions Dcpx0=ψxη1 and Dcpx1=ψxη2 and linearity of ψ,(26)0=ψfη1,xη1+ψIαHη1+ψd0+dn-1ψη1n-1,(27)Dcpf1,x1+Iα-pH1+dn-1ΓnΓn-p=ψfη2,xη2+ψIαHη2+ψd0+dn-1ψη2n-1.Subtracting (26) from (27) and rearranging, we get(28)dn-1=1ψη2n-1-ψη1n-1-Γn/δ2Γn-pDcpf1,x1+Iα-pH1-ψfη2,xη2-ψfη1,xη1+ψIαHη2-ψIαHη1.Using (28) in (26), we get(29)d0=-ψfη1,xη1-ψIαHη1ψ1-ψη1n-1ψ1ψη2n-1-ψη1n-1-Γn/δ2Γn-pDcpf1,x1+Iα-pH1-ψfη2,xη2-ψfη1,xη1+ψIαHη2-ψIαHη1.From (24), (28), and (29), it follows that (30)xt=ft,xt+IαHt+a1tn-1+a2,where a1=κ3ψ1/κ2, a2=-κo-(κ1/κ2)κ3, κo=(-ψfη1,xη1-ψIαHη1)/ψ1, κ1=ψη1n-1, κ2=ψ1ψη2n-1-ψη1n-1-Γn/δ2Γn-p and κ3=Dcpf1,x1+Iα-pH1+ψfη1,xη1-ψfη2,xη2+ψIαHη1-ψIαHη2.In view of Lemma14, system (10) is equivalent to the following coupled systems of integral equations:(31)xt=ft,xt+Iαgt,yt,Iαyt+a1tn-1+a2,yt=ft,yt+Iαgt,xt,Iαxt+b1tn-1+b2,wherea1,b1,a2,b2∈R.Define operators F1,F2,G:Y→Y by(32)F1zt=ft,zt+a1tn-1+a2,F2zt=ft,zt+b1tn-1+b2,Gzt=Iαgt,zt,Iαzt.By virtue of these operators, system (31) can be written as(33)xt=F1xt+Gyt=T1x,y,yt=F2yt+Gxt=T2x,y.Which impliesx,y=T1,T2x,y.IfF~=F1,F2,G~=G,G,T~=T1,T2,thenx,y=T~x,y⟹u=T~u,whereu=x,y;and solutions of system (10) are fixed points of T~.Lemma 15.
Assume that hypothesesA2 and A3 hold, then the operator F~:Y×Y→Y is Υ-Lipschitz with constant k1.Proof.
For=x1,y1, v=x2,y2∈Y×Y, using (A2), it follows that(34)F~ut-F~vt=F1,F2x1,y1t-F1,F2x2,y2t=supt∈J⁡F1x1t,F2y1t-F1x2t,F2y2t=supt∈J⁡F1x1t-F1x2t,F2y1t-F2y2t=supt∈J⁡F1x1t-F1x2t+F2y1t-F2y2t=supt∈J⁡ft,x1t+a1tn-1+a2-ft,x2t+a1tn-1+a2+ft,y1t+b1tn-1+b2-ft,y2t+b1tn-1+b2=supt∈J⁡ft,x1t-ft,x2t+ft,y1t-ft,y2t≤k1x1t-x2t+k1y1t-y2t≤k1x1-x2+y1-y2=k1x1-x2,y1-y2=k1u-v.Thus, F~ is Lipschitz with constant k1. Hence, by Proposition 10, F~ is Υ-Lipschitz with constant k1. Moreover, by using A3, we get the following condition for F~:(35)F~u≤c1up1+N3,whereN3=N1+a1+a2.Lemma 16.
Assume that the hypothesesA4 and A5 hold; then the operator G~:Y×Y→Y is Υ-Lipschitz with zero constant.Proof.
To show thatG~ is compact it is enough to show that G~ is uniformly bounded and equicontinuous. Let S~=un=xn,yn:xn,yn≤R⊂Y×Y∋xn,yn→x,y as n→∞ in S~. We have to show that G~un-G~u→0. (36)limn→∞⁡G~unt=limn→∞⁡G,Gxn,ynt=limn→∞⁡Gxnt,limn→∞⁡Gynt=limn→∞⁡Iαgt,xnt,Iαxnt,limn→∞⁡Iαgt,ynt,Iαynt=1Γαlimn→∞⁡∫0tt-ξα-1gt,xnt,Iαxntdξ,1Γαlimn→∞⁡∫0tt-ξα-1gt,ynt,Iαyntdξwhich by Lebesgue dominated convergence theorem gives(37)limn→∞⁡G~unt=1Γα∫0tt-ξα-1gt,xt,Iαxtdξ,1Γα∫0tt-ξα-1gt,yt,Iαytdξ=Iαgt,xt,Iαxt,Iαgt,yt,Iαyt=Gxt,Gyt=G~ut.Thus, the image of a convergent sequence is convergent, so G~ is a continuous on S~. Moreover, in view of A4, G~ satisfies the following condition:(38)G~u≤c2up2+N2.Now, using A5, we obtain(39)G~ut=Gxt,Gyt=Gxt+Gyt=Iαgt,xt,Iαxt+Iαgt,yt,Iαyt=1Γα∫0tt-ξα-1gt,xt,Iαxtdξ+1Γα∫0tt-ξα-1gt,yt,Iαytdξ≤1Γα∫0tt-ξα-1gt,xt,Iαxtdξ+1Γα∫0tt-ξα-1gt,yt,Iαytdξ≤2hΓα∫t-ξα-1dξ=2hΓα+1tα.Upon using t≤1, (39) implies that(40)G~u≤2hΓα+1.Hence, G~ is uniformly bounded. Now, for t1,t2∈J, and any u∈S~, consider(41)G~ut1-G~ut2=Gx,yt1-Gx,yt2=Gxt1-Gxt2,Gyt1-Gyt2=Gxt1-Gxt2+Gyt1-Gyt2=Iαgt1,xt1,Iαxt1-Iαgt2,xt2,Iαxt2+Iαgt1,yt1,Iαyt1-Iαgt2,yt2,Iαyt2=1Γα∫0t1t1-ξα-1gξ,xξ,Iαxξdξ-∫0t2t2-ξα-1gξ,xξ,Iαxξdξ+1Γα∫0t1t1-ξα-1gξ,yξ,Iαyξdξ-∫0t2t2-ξα-1gξ,yξ,Iαyξdξ=1Γα∫0t1t1-ξα-1gξ,xξ,Iαxξdξ-∫0t1t2-ξα-1gξ,xξ,Iαxξdξ+∫0t1t2-ξα-1gξ,xξ,Iαxξdξ-∫0t2t2-ξα-1gξ,xξ,Iαxξdξ+1Γα∫0t1t1-ξα-1gξ,yξ,Iαyξdξ-∫0t1t2-ξα-1gξ,yξ,Iαyξdξ+∫0t1t2-ξα-1gξ,yξ,Iαyξdξ-∫0t2t2-ξα-1gξ,yξ,Iαyξdξ=1Γα∫0t1t1-ξα-1-t2-ξα-1gξ,xξ,Iαxξdξ+∫0t1t2-ξα-1gξ,xξ,Iαxξdξ+∫t20t2-ξα-1gξ,xξ,Iαxξdξ+1Γα∫0t1t1-ξα-1-t2-ξα-1gξ,yξ,Iαyξdξ+∫0t1t2-ξα-1gξ,yξ,Iαyξdξ+∫t20t2-ξα-1gξ,yξ,Iαyξdξ≤hΓα∫0t1t1-ξα-1-t2-ξα-1dξ+∫t2t1t2-ξα-1dξ+hΓα∫0t1t1-ξα-1-t2-ξα-1dξ+∫t2t1t2-ξα-1dξ≤2hΓα+1t1α-t2α+t2-t1α-t2-t1α=2hΓα+1t1α-t2α.Since tα is uniformly continuous on J for n-1<α≤n, so for ∈>0  ∃δ>0  ∋ if t1-t2<δ, then t1α-t2α<(Γα+1/2h)∈. Thus, (41) becomes(42)G~ut1-G~ut2<2hΓα+1Γα+12h∈.Thus, G~ut1-G~ut2<∈ if t1-t2<δ; thus, G is equicontinuous. Hence, by Arzela-Ascoli Theorem G~ is compact and by virtue of Proposition 9, G~ is Υ-Lipschitz with zero constant.Theorem 17.
Iff:Y×Y→Y and g:Y×Y→Y satisfying conditions A1–A5, then the integral equation(43)ut=xt,yt=ft,xt+a1tn-1+a2+Iαgt,yt,Iαyt,ft,yt+b1tn-1+b2+Iαgt,xt,Iαxt,t∈J,has at least one solution u∈J and the set of solutions of system (10) is bounded in Y.Proof.
LetF~,G~,T~:Y×Y→Y be the operators defined in (32). These are continuous and bounded. By Lemma 15, it follows that F~ is Υ-Lipschitz with constant k1. Also by Lemma 16, G~ is Υ-Lipschitz with zero constant. Thus, by using Proposition 11, T~ is Υ-Lipschitz with constant k1. Set(44)S~=u∈Y×Y:∃σ∈J∋u=σT~u.We will prove that S~ is bounded in Y×Y. For this let u∈S~; then u=σT~u, where σ∈J. Now by (35) and (38), (45)u=x,y=σT~x,y≤T~x,y=F~x,y+G~x,y≤c1up1+N3+c2up2+N21≤c1u1-p1+N3u+c2u1-p2+N2u.If u→∞, then 1≤0, a contradiction. Thus, u is bounded in Y×Y. Consequently, we deduced by Theorem 12 that T~ has at least one fixed point which is bounded in Y×Y.Remark 18.
Here we remark that the conditions (A3), (A4) hold for p1=p2=1. Therefore, in view of this remark, Theorem 17 is also valid for p1=p2=1.


## 4. Illustrative Example

Example 1.
Consider the coupled system of FHDEs given by(46)D5/2xt-cos⁡txt410+xt=tyt+I5/2yt10+yt,t∈J=0,1,Dc5/2yt-cos⁡tyt410+yt=txt+I5/2xt10+xt,t∈J=0,1,Dc1/2x0=∑k=010120x12,x′0=0,…,xn-20=0,Dc1/2x1=∑k=010120x12,Dc1/2y0=∑k=010120y12,y′0=0,…,yn-20=0,Dc1/2y1=∑k=010120y12.From system (46) we see that ft,xt=cos⁡txt/410+xt, gt,xt,Iαxt=txt+I5/2xt/(10+xt), η1=η2=1/2, α=5/2, p=1/2, ψxη1=ψxη2=∑k=010(1/20)x1/2. Upon computation, we have c1=1/4, p1=1, N1=0, p2=1, c2=1/4, N2=0, a1=0.0055, a2=1.0852, and N3=1.0907. In view of Theorem 17, S~=u=x,y∈Y×Y:u=(1/2)T~u is the solution set; then (47)u≤T~u=F~u+G~u≤c1up1+N3+c2up2+N2.From which, we have u≤2N3=2.1814. Thus, system (46) has at least one solution and the set of solutions of S~ is bounded in CJ,R×CJ,R.



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