# Global Existence of Solutions to a System of Integral Equations Related to an Epidemic Model

Authors: Mohamed Jleli; Bessem Samet
Journal: Journal of Function Spaces
 (2020)

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## Abstract

A system of integral equations related to an epidemic model is investigated. Namely, we derive sufficient conditions for the existence and uniqueness of global solutions to the considered system. The proof is based on Perov’s fixed point theorem and some integral inequalities.


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## Body

## 1. Introduction

Many phenomena related to infectious diseases can be modeled as an integral equation (see e.g., [1–4] and the references therein). In [3], Gripenberg investigated the large time behavior of solutions to the integral equation
(1)xt=kpt−∫0tAt−sxsdsft+∫0tat−sxsds,t≥0,which arises in the study of the spread of an infectious disease that does not induce permanent immunity. Namely, sufficient conditions were provided so that (1) admits nonnegative, continuous, and bounded solution. Using the comparison method and some integral estimates, Pachpatte [5] established the convergence of solutions to (1) to 0 as t⟶∞. In [6], Brestovanská studied the integral equation
(2)xt=g1t+∫0tA1t−sF1s,xsds⋯gpt+∫0tApt−sFps,xsds,for all t≥0. Namely, sufficient criteria for the global existence and uniqueness of global solutions to (2) were derived. Moreover, under certain conditions, the convergence of solutions to (2) to 0 as t⟶∞ was proved. In [7], using weakly Picard technique operators in a gauge space, Olaru investigated the qualitative behavior of solutions to the integral equation
(3)xt=g1t+∫0tK1t,s,xsdsg2t+∫0tK2t,s,xsds,t≥0.In this paper, we consider the system of integral equations(4)xt=∏i=12fit+∫0tAit−sFis,xs,ysds,t≥0,yt=∏i=12git+∫0tBit−sGis,xs,ysds,t≥0,where fi,Ai,gi,Bi∈C0,∞ and Fi,Gi∈C0,∞×ℝ×ℝ. Namely, we are concerned with the global existence of solutions to the considered system. Using Perov’s fixed point theorem, sufficient conditions are derived for which the system (4) admits one and only one continuous global solution.The rest of the paper is organized as follows. In Section2, we recall some notions on fixed point theory including Perov’s fixed point theorem. In Section 3, we state and prove our main result.


## 2. Preliminaries

Letn be a positive natural number and define the partial order ≺¯n in ℝn by
(5)y=y1y2⋮yn≺¯nz=z1z2⋮zn⇔yi≤zi,i=1,2,⋯,n,for all y,z∈ℝn. We denote by 0ℝn the zero vector in ℝn, i.e.,
(6)0ℝn=00⋮0.LetS be a nonempty set and d:S×S⟶S be a given mapping. We say that d is a vector-valued metric on S (see, e.g., [8]), if for all x,y,z∈S,
(i)
0ℝn≺¯ndx,y(ii)
dx,y=0ℝn⇔x=y(iii)
dx,y=dy,x(iv)
dx,z≺¯ndx,y+dy,zIn this case, we say thatS,d is a generalized metric space. In such spaces, the notions of convergent sequence, Cauchy sequence, and completeness are similar to those for usual metric spaces.LetMnℝ+ be set of square matrices of size n with nonnegative coefficients. Given K∈Mnℝ+, we denote by ρK its spectral radius.Lemma 1 (Perov’s fixed point theorem, see [9]).
LetS,d be a complete generalized metric space and H:S⟶S be a given mapping. Suppose that there exists K∈Mnℝ+ with ρK<1 such that
(7)dHx,Hy≺¯nKdx,y,for all x,y∈S. Then, the mapping H admits a unique fixed point in S.


## 3. Global Existence

The system (4) is investigated under the following assumptions:
(i)
fi,Ai,gi,Bi∈C0,∞ and Fi,Gi∈C0,∞×ℝ×ℝ, i=1,2(ii)
fi,gi, i=1,2, are bounded functions(iii)
Ai,Bi∈L10,∞∩Lς0,∞, i=1,2, for some ς>1(iv)
For alli=1,2, there exist positive constants L1i and L2i such that(8)Fit,u,v−Fit,u¯,v¯≤L1iu−u¯+L2iv−v¯,for all t≥0 and u,v,u¯,v¯∈ℝ2(v)
For alli=1,2, there exist positive constants Q1i,Q2i,ℓ1i, and ℓ2i such that(9)Fit,u,v≤Q1iu∣ℓ1i+Q2ivℓ2i,for all t≥0 and u,v∈ℝ2(vi)
For alli=1,2, there exist positive constants M1i and M2i such that(10)∣Git,u,v−Git,u¯,v¯∣≤M1i∣u−u¯∣+M2iv−v¯,for all t≥0 and u,v,u¯,v¯∈ℝ2(vii)
For alli=1,2, there exist positive constants P1i,P2i,σ1i, and σ2i such that(11)Git,u,v≤P1iu∣σ1i+P2ivσ2i,for all t≥0 and u,v∈ℝ2(viii)
There exist positive constantsρ1 and ρ2 satisfying(12)μ+Q11ρ1ℓ11+Q21ρ2ℓ21δμ+Q12ρ1ℓ12+Q22ρ2ℓ22δ≤ρ1,(13)μ+P11ρ1σ11+P21ρ2σ21δμ+P12ρ1σ12+P22ρ2σ22δ≤ρ2,where
(14)μ=maxμf,μg,(15)μf=max∣f1t∣,∣f2t∣:t≥0,(16)μg=max∣g1t∣,∣g2t∣:t≥0,(17)δ=maxi=1,2∫0∞∣Ais∣ds,∫0∞∣Bis∣ds.Remark 2.
Notice that from (i) and (ii), one hasμ<∞. Moreover, by (iii), one has δ<∞.Our main result is given by the following theorem.Theorem 3.
Under assumptions (i)–(viii), system (4) admits one and only one solution x∗,y∗∈C0,∞×C0,∞ satisfying∣x∗t∣≤ρ1 and ∣y∗t∣≤ρ2, for all t≥0.Proof.
LetT be an arbitrary positive number and IT=0,T. For i=1,2, let
(18)CT,ρi=x∈CIT:∣xt∣≤ρi,t∈IT.
We introduce the mappingH:CT,ρ1×CT,ρ2⟶CIT×CIT defined by
(19)Hx,yt=H1x,yt,H2x,yt=∏i=12H1ix,yt,∏i=12H2ix,yt,t∈IT,where
(20)H1ix,yt=fit+∫0tAit−sFis,xs,ysds,(21)H2ix,yt=git+∫0tBit−sGis,xs,ysds,for all i=1,2.
Letx,y∈CT,ρ1×CT,ρ2. For all i=1,2 and t∈IT, using (i), (ii), (iii), and (v), and taking in consideration Remark 2, one obtains
(22)H1ix,yt≤∣fit∣+∫0tAit−s∣Fis,xs,ys∣ds≤μ+∫0t∣Ait−s∣Q1ixsℓ1i+Q2iysℓ2ids≤μ+Q1iρ1ℓ1i+Q2iρ2ℓ2i∫0t∣Ait−s∣ds=μ+Q1iρ1ℓ1i+Q2iρ2ℓ2i∫0∞∣Aiτ∣dτ≤μ+Q1iρ1ℓ1i+Q2iρ2ℓ2iδ.
Therefore, using (12), it holds that
(23)H1x,yt=∏i=12H1ix,yt≤μ+Q11ρ1ℓ11+Q21ρ2ℓ21δμ+Q12ρ1ℓ12+Q22ρ2ℓ22δ≤ρ1,which yields
(24)H1CT,ρ1×CT,ρ2⊂CT,ρ1.
Similarly, for alli=1,2 and t∈IT, using (i), (ii), (iii), and (vii), and taking in consideration Remark 2, one obtains
(25)H2ix,yt≤∣git∣+∫0tBit−s∣Gis,xs,ys∣ds≤μ+∫0t∣Bit−s∣P1ixsσ1i+P2iysσ2ids≤μ+P1iρ1σ1i+P2iρ2σ2i∫0t∣Bit−s∣ds≤μ+P1iρ1σ1i+P2iρ2σ2iδ.
Hence, using (13), it holds that
(26)H2x,yt=∏i=12H2ix,yt≤μ+P11ρ1σ11+P21ρ2σ21δμ+P12ρ1σ12+P22ρ2σ22δ≤ρ2,which yields
(27)H2CT,ρ1×CT,ρ2⊂CT,ρ2.
Therefore, it follows from (24) and (27) that the mapping H maps the set CT,ρ1×CT,ρ2 into itself, i.e.,
(28)H:CT,ρ1×CT,ρ2⟶CT,ρ1×CT,ρ2.
Next, let us introduce the metric(29)dr:CIT×CIT⟶ℝ,defined by
(30)drx,y=maxt∈ITe−rtxt−yt,x,y∈CIT×CIT,where r>0 will be specified later. Moreover, we introduce the vector-valued metric
(31)Dr:CT,ρ1×CT,ρ2×CT,ρ1×CT,ρ2⟶ℝ2,defined by
(32)Drx,y,x¯,y¯=drx,x¯dry,y¯,x,y,x¯,y¯∈CT,ρ1×CT,ρ2.
It can be easily seen thatCT,ρ1×CT,ρ2,Dr is a complete generalized metric space. On the other hand, for all x,y,x¯,y¯∈CT,ρ1×CT,ρ2, and t∈IT, using (22), one has
(33)H1x,yt−H1x¯,y¯t=∏i=12H1ix,yt−∏i=12H1ix¯,y¯t=H11x,ytH12x,yt−H11x¯,y¯tH12x¯,y¯t≤H11x,ytH12x,yt−H12x¯,y¯t+H12x¯,y¯tH11x,yt−H11x¯,y¯t≤μ+Q11ρ1ℓ11+Q21ρ2ℓ21δH12x,yt−H12x¯,y¯t+μ+Q12ρ1ℓ12+Q22ρ2ℓ22δH11x,yt−H11x¯,y¯t.
Moreover, using (iii), (iv), (20) and Hölder’s inequality, for all i=1,2, one obtains
(34)H1ix,yt−H1ix¯,y¯t=fit+∫0tAit−sFis,xs,ysds−fit−∫0tAit−sFis,x¯s,y¯sds≤∫0t∣Ait−s∣Fis,xs,ys−Fis,x¯s,y¯sds≤∫0t∣Ait−s∣L1i∣xs−x¯s∣+L2i∣ys−y¯s∣ds=L1i∫0t∣Ait−s∣erse−rs∣xs−x¯s∣ds+L2i∫0t∣Ait−s∣erse−rs∣ys−y¯s∣ds≤L1i∫0t∣Ait−s∣ersdsdrx,x¯+L2i∫0t∣Ait−s∣ersdsdry,y¯=∫0t∣Ait−s∣ersdsL1idrx,x¯+L2idry,y¯≤δς∫0tersς′ds1ς′L1idrx,x¯+L2idry,y¯,where
(35)ς′=ςς−1,(36)δς=maxi=1,2∫0∞Aisςds1/ς,∫0∞Bisςds1/ς.
Notice that by (iii), one hasδς<∞. Hence, it holds that
(37)H1ix,yt−H1ix¯,y¯t≤δςς′−1/ς′r−1/ς′ertς′−11/ς′L1idrx,x¯+L2idry,y¯.
Therefore, by (33), one obtains
(38)H1x,yt−H1x¯,y¯t≤μ+Q11ρ1ℓ11+Q21ρ2ℓ21δδςς′−1/ς′r−1ς′ertς′−11/ς′L12drx,x¯+L22dry,y¯+μ+Q12ρ1ℓ12+Q22ρ2ℓ22δδςς′−1/ς′r−1/ς′ertς′−11/ς′L11drx,x¯+L21dry,y¯=δςς′−1/ς′r−1/ς′ertς′−11/ς′L12μ+Q11ρ1ℓ11+Q21ρ2ℓ21δ+L11μ+Q12ρ1ℓ12+Q22ρ2ℓ22δdrx,x¯+δςς′−1/ς′r−1/ς′ertς′−11/ς′L22μ+Q11ρ1ℓ11+Q21ρ2ℓ21δ+L21μ+Q12ρ1ℓ12+Q22ρ2ℓ22δdry,y¯,which yields
(39)drH1x,y,H1x¯,y¯≤1−e−rTς′1/ς′δςς′−1/ς′r−1/ς′α11drx,x¯+α12dry,y¯,where
(40)α11=L12μ+Q11ρ1ℓ11+Q21ρ2ℓ21δ+L11μ+Q12ρ1ℓ12+Q22ρ2ℓ22δ,(41)α12=L22μ+Q11ρ1ℓ11+Q21ρ2ℓ21δ+L21μ+Q12ρ1ℓ12+Q22ρ2ℓ22δ.
Similarly, using (iii), (vi), (21), and (25), one obtains
(42)drH2x,y,H2x¯,y¯≤1−e−rTς′1/ς′δςς′−1/ς′r−1/ς′α21drx,x¯+α22dry,y¯,where
(43)α21=M12μ+P11ρ1σ11+P21ρ2σ21δ+M11μ+P12ρ1σ12+P22ρ2σ22δ,(44)α22=M22μ+P11ρ1σ11+P21ρ2σ21δ+M21μ+P12ρ1σ12+P22ρ2σ22δ.
Therefore, it follows from (19), (39), and (42) that
(45)DrHx,y,Hx¯,y¯°ℝ2KDrx,y,x¯,y¯,x,y,x¯,y¯∈CT,ρ1×CT,ρ2,where K is the square matrix of size 2 defined by
(46)K=1−e−rTς′1/ς′δςς′−1/ς′r−1/ς′α11α12α21α22.
On the other hand, one has(47)ρK=1−e−rTς′1/ς′δςς′−1/ς′r−1/ς′ρα11α12α21α22.
Therefore, taking(48)r1/ς′≥δςς′−1/ς′ρα11α12α21α22,one obtains
(49)ρK≤1−e−rTς′1/ς′<1.
Then, by Lemma1, one deduces that the mapping H defined by (19) admits a unique fixed point in CT,ρ1×CT,ρ2, which is the unique solution to (4) in CT,ρ1×CT,ρ2. On the other hand, since the real number T>0 is arbitrary, it holds that (4) admits a unique continuous global solution x∗,y∗ satisfying ∣x∗t∣≤ρ1 and ∣y∗t∣≤ρ2, for all t≥0. The proof is completed.We end the paper with the following example.Example 4.
Consider the system of integral equations(50)xt=2t+1+∫0te−2t−s8sinxs4+3yss+16ds∫0t3e−t−s216πxss2+2+ys2ds,t≥0,yt=tt+1+∫0te−t−s16arctan2xs+ys5ds1+∫0te−t−s216πxs6+2yss2+3ds,t≥0.
System (50) is a special case of System (4), where
(51)f1t=2t+1,f2t=0,A1t=e−2t8,A2t=3e−t216π,(52)F1t,u,v=sinu4+3vt+16,F2t,u,v=ut2+2+v2,(53)g1t=tt+1,g2t=1,B1t=e−t16,B2t=e−t216π,(54)G1t,u,v=arctan2u+v5,G2t,u,v=u6+2vt2+3.
Let us check the validity of assumptions (i)–(viii). It can be easily seen that(55)fi,Ai,gi,Bi∈C0,∞,Fi,Gi∈C0,∞×ℝ×ℝ,i=1,2,(56)μf=max∣f1t∣,∣f2t∣:t≥0=2,(57)μg=max∣g1t∣,∣g2t∣:t≥0=1,(58)μ=maxμf,μg=2.
Moreover, one has(59)∫0∞∣A1t∣dt=116,∫0∞A1tςdt=12ς8ςforallς>1,(60)∫0∞∣A2t∣dt=332,∫0∞A2tςdt=316πςπ2ςforallς>1,(61)∫0∞∣B1t∣dt=116,∫0∞B1tςdt=1ς16ςforallς>1,(62)∫0∞∣B2t∣dt=132,∫0∞B2tςdt=π2ς16πςforallς>1,(63)δ=maxi=1,2∫0∞∣Ais∣ds,∫0∞∣Bis∣ds=332.
Therefore, assumptions (i)–(iii) of Theorem3 are satisfied. For all t≥0 and u,v,u¯,v¯∈ℝ2, one has
(64)∣F1t,u,v−F1t,u¯,v¯∣=sinu4+3vt+16−sinu¯4+3v¯t+16≤14∣u−u¯∣+34∣v−v¯∣(65)∣F2t,u,v−F2t,u¯,v¯∣=ut2+2+v2−u¯t2+2−v¯2≤12∣u−u¯∣+12v−v¯,which shows that assumption (iv) of Theorem 3 is satisfied with
(66)L11=14,L21=34,L12=L22=12.
Similarly, one has(67)∣G1t,u,v−G1t,u¯,v¯∣=arctan2u+v5−arctan2u¯+v¯5≤25∣u−u¯∣+15∣v−v¯∣(68)∣G2t,u,v−G2t,u¯,v¯∣=u6+2vt2+3−u¯6−2v¯t2+3≤16∣u−u¯∣+23v−v¯,which shows that assumption (vi) of Theorem 3 is satisfied with
(69)M11=25,M21=15,M12=16,M22=23.
For allt≥0 and u,v∈ℝ2, one has
(70)F1t,u,v=sinu4+3vt+16≤14u+34v,(71)∣F2t,u,v∣=ut2+2+v2≤12∣u∣+12v,which shows that assumption (v) of Theorem 3 is satisfied with
(72)Q11=14,Q21=34,Q12=Q22=12,ℓ11=ℓ21=ℓ12=ℓ22=1.
Similarly, one has(73)G1t,u,v=arctan2u+v5≤25u+15v,(74)∣G2t,u,v∣=u6+2vt2+3≤16∣u∣+23v,which shows that assumption (VII) is satisfied with
(75)P11=25,P21=15,P12=16,P22=23,σ11=σ21=σ12=σ22=1.
From the above estimates, one deduces that the system of inequalities (12) and (13) is equivalent to
(76)2+33214ρ1+34ρ22+364ρ1+ρ2≤ρ1,2+31602ρ1+ρ22+164ρ1+4ρ2≤ρ2.
Takingρ1=ρ2=ρ>0, (76) reduces to
(77)2+332ρ2≤ρ,2+9160ρ2+564ρ≤ρ.
On the other hand, one observes easily that(78)2+9160ρ2+564ρ≤2+332ρ2,ρ>0.
Therefore, anyρ>0 satisfying
(79)2+332ρ2≤ρ,is a solution to (77). In particular, for ρ=36, one has
(80)2+332ρ2=28.890625<36=ρ.Therefore,ρ=36 is a solution to (77), which shows that assumption (viii) of Theorem 3 is satisfied with ρ1,ρ2=36,36.Finally, by Theorem3, one deduces that system (50) admits one and only one solution x∗,y∗∈C0,∞×C0,∞ satisfying
(81)supt≥0∣x∗t∣,∣y∗t∣≤36.



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