# On a System of Volterra Type Hadamard Fractional Integral Equations in Fréchet Spaces

Authors: Saïd Abbas; Mouffak Benchohra; Yong Zhou
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

In this article we present some results concerning the existence of solutions for a system of Hadamard integral equations. Our investigation is conducted with an application of an extension of the fixed point theorem of Burton-Kirk in Fréchet spaces.


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## Body

## 1. Introduction

In recent years, fractional differential and integral equations are emerging as a useful tool in modeling of the dynamics of many physical systems and electrical phenomena, which has been demonstrated by many researchers in the fields of mathematics, science, and engineering; see monographs [1–7] and a series of papers [8–18] and the references cited therein. Some properties of the Hadamard fractional calculus were investigated by Butzer et al. [19, 20]. Additional papers can be found in [7, 21] and the references therein. Recently, some interesting results on the existence of the solutions of some classes of integral equations have been obtained by Abbas et al. [22–24], Banaś et al. [25, 26], and Pachpatte [27, 28] and the references therein.This paper deals with the existence of solutions of the following system of Hadamard fractional integral equations of the form(1)u1t,x=μ1t,x+f1t,x,IHσru1t,x,u1t,x+1Γr1Γr2∫1t∫1xlogtsr1-1logxyr2-1×g1t,x,s,y,u1s,y,u2s,y,…,uns,ydydssyu2t,x=μ1t,x+f2t,x,IHσru2t,x,u2t,x+1Γr1Γr2∫1t∫1xlogtsr1-1logxyr2-1×g2t,x,s,y,u1s,y,u2s,y,…,uns,ydydssy…………unt,x=μnt,x+fnt,x,IHσrunt,x,unt,x+1Γr1Γr2∫1t∫1xlogtsr1-1logxyr2-1×gnt,x,s,y,u1s,y,u2s,y,…,uns,ydydssy;t,x∈J≔1,+∞×1,b.That is,(2)uit,x=μit,x+fit,x,IHσruit,x,uit,x+1Γr1Γr2∫1t∫1xlogtsr1-1logxyr2-1×git,x,s,y,u1s,y,u2s,y,…,uns,ydydssy;t,x∈J,i=1,…,n,where b>1, σ=(1,1), r=(r1,r2), r1,r2∈(0,∞), IHσr is the Hadamard integral of order r,μi:J→R, fi:J×R2→R,gi:J′×Rn→R are given continuous functions, J′={(t,x,s,y)∈J2:s≤t,y≤x}, and Γ(·) is the (Euler’s) gamma function defined by (3)Γξ=∫0∞tξ-1e-tdt;ξ>0.


## 2. Preliminaries

LetL1([1,+∞)×[1,b]); for b>1, be the space of Lebesgue-integrable functions u:[1,+∞)×[1,b]→R with the standard norm (4)u1=∫1∞∫1but,xdxdt.Definition 1 (see [4]).
The Hadamard fractional integral of orderq>0 for a function g∈L1([1,a],R) is defined as (5)IH1qgx=1Γq∫1xlogxsq-1gssds.Example 2.
The Hadamard fractional integral of orderq>0 for the function w:[1,e]→R, defined by w(x)=(log⁡x)β-1 with β>0, is (6)IH1qwx=ΓβΓβ+qlog⁡xβ+q-1.Definition 3.
Letr1, r2≥0, σ=(1,1), and r=(r1,r2). For w∈L1(J,R), define the Hadamard partial fractional integral of order r by the expression (7)IHσrwx,y=1Γr1Γr2∫1x∫1ylogxsr1-1logytr2-1ws,tstdtds.LetX be a Fréchet space with a family of seminorms {·n}n∈N∗ such that (8)x1≤x2≤x3≤…foreveryx∈X. A subset Y of X is bounded if, for every n∈N, there exists M¯n>0 such that (9)yn≤M¯nforally∈Y. We associate a sequence of Banach spaces {(Xn,·n)} to X as follows: For every n∈N, we consider the equivalence relation ~n defined by x~ny if and only if x-yn=0 for x,y∈X. We denote by Xn=X~n,·n the quotient space, the completion of Xn with respect to ·n. To every Y⊂X, we associate a sequence {Yn} of subsets Yn⊂Xn as follows: For every x∈X, we denote by [x]n the equivalence class of x of subset Xn and we defined Yn={[x]n:x∈Y}. We denote by Yn¯, intn(Yn), and ∂nYn, respectively, the closure, the interior, and the boundary of Yn with respect to ·n in Xn. For more details, we refer the reader to [29].Definition 4.
LetX be a Fréchet space. A function N:X→X is said to be a contraction if for each n∈N∖{0} there exists kn∈[0,1) such that (10)Nu-Nvn≤knu-vnforallu,v∈X.We need the following extension of the Burton-Kirk fixed point theorem in the case of a Fréchet space.Theorem 5 (see [30]).
Let(X,·n) be a Fréchet space and let A,B:X→X be two operators such that (a)
A is a compact operator(b)
B is a contraction operator with respect to a family of seminorms {·n}(c)
the setx∈X:x=λA(x)+λBx/λ,λ∈(0,1) is bounded
Then the operator equationA(u)+B(u)=u has a solution in X.


## 3. Existence Results

Set(11)CnJ=CJ×…×CJn  times.Definition 6.
A function(u1,u2,…,un)∈Cn(J) is said to be a solution of (2) if (u1,u2,…,un) satisfies (2) on J.For eachp∈N∖{0,1} we consider the set Cp=C([1,p]×[1,b]), and we define in C(J) the seminorms by (12)up=supt,x∈1,p×1,b⁡ut,x. Then C is a Fréchet space with the family of seminorms {up}.Also, the product spaceC≔Cn is a Fréchet space with the family of seminorms (13)u1,u2,…,unp=∑i=1nuip.Now, we are concerned with the existence of solutions for system (2). Let us introduce the following hypotheses:(H1)
There exist continuous functionsli,ki:J→R+;i=1,…,n, with(14)limt→∞⁡lit,x=limt→∞⁡kit,x=0;forx∈1,b,such that(15)fit,x,u1,v1-fit,x,u2,v2≤lit,xu1-u2+kit,xv1-v21+u1-u2+v1-v2,for each(t,x)∈J and each u1,u2,v1,v2∈R(H2)
There exist continuous functionsQij,φi:J′→R+; i=1,…,n, j=0,…,n, and nondecreasing functions ψi:[0,∞)→(0,∞) such that (16)git,x,s,y,u1,u2,…,un≤Qi0t,x,s,y+∑j=1nQijt,x,s,yuj1+∑j=1nuj,for(t,x,s,y)∈J′, ui∈R; i=1,…,n, and (17)git1,x1,s,y,u1,…,un-git2,x2,s,y,u1,…,un≤φis,yt1-t2+x1-x2×∑j=1nψiuj;t1,x1,t2,x2,s,y∈J,ui∈R.Moreover, assume that(18)limt→∞∫1tlogtsr1-1Qijt,x,s,ysds=0,for each(x,y)∈J with y≤xFor anyp∈N∖{0,1}, set (19)kp∗=maxi=1,…,n⁡kip=maxi=1,…,n⁡supt,x∈1,p×1,b⁡kit,x, and (20)lp∗=maxi=1,…,n⁡lip=maxi=1,…,n⁡supt,x∈1,p×1,b⁡lit,x.Theorem 7.
Assume that hypotheses(H1) and (H2) hold. If(21)Lp≔kp∗+lp∗log⁡pr1log⁡br2Γ1+r1Γ1+r2<1,then system (2) has at least one solution in the space C.Proof.
Define the operatorsAi,Bi:C→C; i=1,…,n, by (22)Aiuit,x=∫1t∫1xlogtsr1-1logxyr2-1×git,x,s,y,u1s,y,…,uns,ysyΓr1Γr2dyds;t,x∈J,(23)Biuit,x=μit,x+fit,x,HIσruit,x,uit,x;t,x∈J,and consider the continuous operators A,B:C→C defined by(24)Au1,…,un=A1u1,…,Anun,(25)Bu1,…,un=B1u1,…,Bnun.We shall show that operators A and B satisfied all the conditions of Theorem 5.
The proof will be given in several steps.
Step 1 (A is compact). To this aim, we must prove that A is continuous and it transforms every bounded set into a relatively compact set. Recall that M⊂C is bounded if and only if (26)∀p∈N∖0,1,∃lp>0:∀u1,…,un∈M,u1,…,unp≤lp, and M={(u1(t,x),…,un(t,x));(t,x))∈J}⊂C is relatively compact if and only if, ∀p∈N∖{0,1}, the family u1t,x,…,unt,x(t,x)]∈[1,p]×[1,b] is equicontinuous and uniformly bounded on [1,p]×[1,b]. The proof will be given in several claims.
Claim 1 (A is continuous). Let {(u1m,…,unm)}m∈N∖{0,1} be a sequence in M such that (u1m,…,unm)→(u1,…,un) in M. Then, for each (t,x)∈J and i=1,…,n, we have(27)Aiuimt,x-Auit,x≤1Γr1Γr2∫1t∫1xlogtsr1-1logxyr2-1×git,x,s,y,u1ms,y,…,unms,y-git,x,s,y,u1s,y,…,uns,ydyds≤1Γr1Γr2∫1t∫1xlogtsr1-1logxyr2-1×git,x,s,y,u1ms,y,…,unms,y-git,x,s,y,u1s,y,…,uns,ydyds.Since uim→uiasm→∞ and gi is continuous, then for each (t,x)∈[1,p]×[1,b]; p∈N∖{0,1}, and i=1,…,n, (27) gives (28)Au1m,…,unm-Au1,…,unp=∑i=1nAiuim-Aiuip→0asm→∞.
Claim 2 (A maps bounded sets into bounded sets in C). Let M be the bounded set in C as in Claim 1. Then, for each p∈N∖{0,1}, there exists lp>0, such that for all (u1,…,un)∈C we have (u1,…,un)p≤lp.
For arbitrarily fixed(t,x)∈[1,p]×[1,b] and each i=1,…,n, we have (29)Aiuit,x≤1Γr1Γr2∫1t∫1xlogtsr1-1logxyr2-1×git,x,s,y,u1s,y,…,uns,ydyds≤1Γr1Γr2∫1t∫1xlogtsr1-1logxyr2-1×Qi0t,x,s,y+∑j=1nQijt,x,s,yus,y1+∑j=1nujs,ydyds≤1Γr1Γr2∫1t∫1xlogtsr1-1logxyr2-1×∑j=0nQijt,x,s,ydyds≤Qip,where(30)Qip=supt,x∈1,p×1,b⁡∫1t∫1xlogtsr1-1logxyr2-1∑j=0nQijt,x,s,yΓr1Γr2dyds.Hence(31)Au1,…,unp≤∑i=0nQip≔Qp∗.
Claim 3 (A maps bounded sets into equicontinuous sets inC). Let (t1,x1),(t2,x2)∈[1,p]×[1,b], t1<t2, x1<x2, and let u∈M; thus for each i=1,…,n, we have (32)Aiuit2,x2-Aiuit1,x1≤1Γr1Γr2∫1t2∫1x2logt2sr1-1logx2yr2-1×git2,x2,s,y,u1s,y,…,uns,y-git1,x1,s,y,u1s,y,…,uns,ydyds+1Γr1Γr2∫1t2∫1x2logt2sr1-1logx2yr2-1git1,x1,s,y,u1s,y,…,uns,ydyds-∫1t2∫1x2logt1sr1-1logx1yr2-1git1,x1,s,y,u1s,y,…,uns,ydyds+1Γr1Γr2∫1t2∫1x2logt1sr1-1logx1yr2-1git1,x1,s,y,u1s,y,…,uns,ydyds-∫1t1∫1x1logt1sr1-1logx1yr2-1git1,x1,s,y,u1s,y,…,uns,ydyds≤1Γr1Γr2∫1t2∫1x2logt2sr1-1logx2yr2-1×git2,x2,s,y,u1s,y,…,uns,y-git1,x1,s,y,u1s,y,…,uns,ydyds+1Γr1Γr2∫1t1∫1x1logt2sr1-1logx2yr2-1-logt1sr1-1logx1yr2-1git1,x1,s,y,u1s,y,…,uns,ydyds+1Γr1Γr2∫1t1∫x1x2logt2sr1-1logx2yr2-1git1,x1,s,y,u1s,y,…,uns,ydyds+1Γr1Γr2∫t1t2∫1x1logt2sr1-1logx2yr2-1git1,x1,s,y,u1s,y,…,uns,ydyds+1Γr1Γr2∫t1t2∫x1x2logt2sr1-1logx2yr2-1git1,x1,s,y,u1s,y,…,uns,ydyds≤1Γr1Γr2∫1t2∫1x2logt2sr1-1logx2yr2-1×φis,yt1-t2+x1-x2ψilpdyds+1Γr1Γr2∫1t1∫1x1logt2sr1-1logx2yr2-1-logt1sr1-1logx1yr2-1∑j=0nQijt1,x1,s,ydyds+1Γr1Γr2∫t2t2∫1x2logt2sr1-1logx2yr2-1∑j=0nQijt1,x1,s,ydyds+1Γr1Γr2∫1t1∫x1x2logt2sr1-1logx2yr2-1∑j=0nQijt1,x1,s,ydyds+1Γr1Γr2∫t1t2∫x1x2logt2sr1-1logx2yr2-1∑j=0nQijt1,x1,s,ydyds.From the continuity of functions Qij,φi and as t1→t2 and x1→x2, the right-hand side of the above inequality tends to zero. As a consequence of Claims 1–3 and from the Arzelá-Ascoli theorem, we can conclude that A is continuous and compact.
Step 2 (B is a contraction). Consider (v1,…,vn),(w1,…,wn)∈C. Then, by (H1), for any p∈N∖{0,1} and each (t,x)∈[1,p]×[1,b] and i=1,…,n, we have (33)Bivit,x-Biwit,x≤lit,xIHσrvi-wit,x+kit,xvi-wit,x≤kip+liplog⁡pr1log⁡br2Γ1+r1Γ1+r2vi-wip≤kp∗+lp∗log⁡pr1log⁡br2Γ1+r1Γ1+r2vi-wip. Thus, (34)∑i=1nBivit,x-Biwit,x≤kp∗+lp∗log⁡pr1log⁡br2Γ1+r1Γ1+r2∑i=1nvi-wip. Hence, (35)Bv1,…,vn-Bw1,…,wnp≤Lpv1,…,vn-w1,…,wnp. Inequality (21) implies that B is a contraction operator.
Step 3 (the set E≔u=(u1,…,un)∈C:u=λA(u)+λBu/λ,λ∈(0,1) is bounded). Let (u1,…,un)∈C, such that (u1,…,un)=λA(u1,…,un)+λBu1/λ,…,un/λ, for some λ∈(0,1). Then, for any p∈N∖{0,1} and each (t,x)∈[1,p]×[1,b] and i=1,…,n, we have (36)uit,x≤λAiui+λBiuiλ≤μit,x+fit,x,0,0+kit,x+lit,x+1Γr1Γr2∫1t∫1xlogtsr1-1logxyr2-1×∑j=0nQijt,x,s,ydydssy≤μip+fip+kip+lip+∑j=0nQijp, where (37)μip=supt,x∈1,p×1,bμit,x,fip=supt,x∈1,p×1,bft,x,0,0;p∈N∖0,1. Thus, (38)u1,…,unp≤μp∗+fp∗+kp∗+lp∗+Qp∗≕lp∗. Hence, the set E is bounded. As a consequence of Steps 1–3 and from Theorem 5, we deduce that N has a fixed point (u1,…,un) which is a solution of system (2).


## 4. An Example

Consider the following coupled system of Hadamard fractional integral equations of the form(39)ut,x=xe3-2t1+t+x2+xe-t-2cp1+e-2pIHσrut,x+e-put,x+∫1t∫1xlogtsr1-1logxyr2-1g1t,x,s,y,us,y,vs,yΓr1Γr2dydsvt,x=11+et+x+xe-t-2cp1+e-2pIHσrvt,x+e-pvt,x+∫1t∫1xlogtsr1-1logxyr2-1g2t,x,s,y,us,y,vs,yΓr1Γr2dyds;t,x∈1,+∞×1,e,where cp=e-p+e-2ppr1/Γ(1+r1)Γ(1+r2); p∈N∖{0,1}, r=(r1,r2)∈(0,∞)×(0,∞), and (40)g1t,x,s,y,u,v=xs-3/4sin⁡tsin⁡s1+y2+t21+u+v;ift,x,s,y∈J′,andu∈R,g2t,x,s,y,u,v=s-3/4ln⁡t1+t21+u+v;ift,x,s,y∈J′,andu∈R, and (41)J′=t,x,s,y:1≤s≤tand1≤x≤y≤e. Set (42)μ1t,x=xe3-2t1+t+x2,μ2t,x=11+et+x,f1t,x,u,v=xe-t-2cp1+e-2pu+e-pu,f2t,x,u,v=xe-t-2cp1+e-2pv+e-pv;p∈N∖0,1. The functions fi; i=1,2 are continuous and satisfy assumption (H1), with (43)k1t,x=k2t,x=xe-t-2-pcp,l1t,x=l2t,x=xe-t-2-2pcp,kp=e-2-pcp,lp=e-2-2pcp. Also, the functions gi,ı=1,2 are continuous and satisfy assumption (H2), with (44)Qi0t,x,s,y=Qi1t,x,s,y=xs-3/4sin⁡tsin⁡s1+y2+t2;t,x,s,y∈J′;i=1,2. Then, (45)∫1tlogtsr1-1Q01t,x,s,ysds≤∫1tlogtsr1-1xs-1sin⁡tsin⁡sds≤xsin⁡t∫1tlogtsr1-1s-1ds≤elog⁡tr1t→0ast→∞. Finally, we shall show that condition (21) holds with b=e. Indeed, for each p∈N∖{0,1}, we get (46)Lp=kp∗+lp∗log⁡pr1log⁡br2Γ1+r1Γ1+r2=1cpe-2-p+e-2-2ppr1Γ1+r1Γ1+r2=e-2<1. Hence by Theorem 7, system (39) has a solution defined on [1,+∞)×[1,e].



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