# A Multidimensional Fixed-Point Theorem and Applications to Riemann-Liouville Fractional Differential Equations

Authors: Tamer Nabil; Ahmed H. Soliman
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this work, we introduce a new version of Krasnoselskii fixed-point theorem dealing withN-tupled fixed-point results under certain blended conditions. Herein, we demonstrate that our newly theoretical results are applied to the investigation of Riemann-Liouville fractional differential equations (R-L FDEs for short). Furthermore, an example to illustrate the abstract results is obtained.


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## Body

## 1. Introduction

Fixed-point theory has experienced quick improvement over the most recent quite a few years. The development has been firmly advanced by the vast number of utilizations in the existence theory of functional, fractional, differential, partial differential, and integral equations. Two fundamental theorems concerning fixed points are those of Schauder and of Banach. The Schauder theorem states that ifE is a closed bounded convex subset of a Banach space X and T:E→E is a continuous map such that T(E) is compact, then T has a fixed point. In Banach’s theorem, every contraction operator on a complete metric space has a unique fixed point. In 1964, Krasnoselskii [1] gave a very important fixed-point theorem which is a blend of the nonlinear contraction principle and Schauder’s fixed-point theorem. He gave intriguing applications to differential equations by finding the existence of solutions under some hybrid conditions. In 2013, Dhage [2] proposed an important Krasnoselskii-type fixed-point theorem and used it to study the existence of the solution to the system of nonlinear fractional differential equations. In the same year, Dhage and Jadhav [3] studied the existence of solution for hybrid differential equations. Also, Lu et al. [4] developed the theory of fractional hybrid differential equations including R-L FDEs and gave an existence theorem for fractional hybrid differential equations. In 2016, Bashiri et al. [5] proposed a new version of Kransoselskii-type fixed-point theorem based on coupled fixed-point approach and applied their results to study the existence of solution to a system of two R-L FDEs. Recently, Yang et al. [6] used the concept of fractional derivative due to Caputo to prove Krasnoselskii coupled fixed-point theorem under some certain conditions and applied their results to study the existence of the solution to a nonlinear coupled system for fractional differential equations.In this paper, we give a new version ofN-tupled fixed-point theorem which is utilized to prove the existence of the N-tupled fixed point of sum of more than two operators. Also, we apply our N-tupled fixed-point theorem to prove the existence of solution to the system of R-L FDEs.Our paper is organized as follows. Section2 contains some basic concepts concerning fixed-point theory. Notably, a fractional derivative criterion in normed spaces is stated, which is based on the concept of Riemann-Liouville derivative and Caputo fractional derivative. In Section 3, our main N-tupled fixed-point results are proved. Section 4 is devoted to study the existence of a solution for nonlinear R-L FDEs.


## 2. Preliminaries

Let(X,.) be a normed space and F:X×X→X be a mapping. An element (x,y)∈X×X is called coupled fixed point of F in X×X if F(x,y)=x and F(y,x)=y [7].Definition 1.
A mappingϕ:X→X is said to be D-function if it is upper semicontinuous and monotone nondecreasing such that ϕ(0)=0.Definition 2 (see [8]).
LetQ:∏i=1NX→X. The point (x1,x2,…,xN)∈∏i=1NX is called the N-tupled fixed point of ∏i=1NX if and only if (1)x1=Qx1,x2,…,xN,x2=Qx2,x3,…,xN,x1,⋮xN=QxN,x1,…,xN-1.For more information about coupled, tripled, andN-tupled fixed points, see [9–17].Now, we recall some definitions about the fractional derivative. The fractional derivative is defined via the fractional integral operator. So, we will start by the definition of the fractional integral operator. The fractional integral operator of orderα∈(0,∞) of a function g(t) is given as (2)Iαgt=∫0tt-sα-1Γαgsds.Also, the Riemann-Liouville derivative of order α>0 is defined as(3)Dαgt=1Γm-αdmdtm∫0tgst-sα-m+1ds,ifm-1<α<m,dmgtdtm,ifα=m,wherem is positive integer and m-1<α<m.The Caputo fractional derivative of orderα>0 with m-1<α<m is denoted by D∗α and is given as(4)D∗αgt=1Γm-α∫0tgmst-sα-m+1ds,ifm-1<α<m,dmgtdtm,ifα=m.For more details about the fractional calculus, we refer the reader to [18–23].The following two lemmas are useful in what follows.Lemma 3 (see [21]).
If0<α<1 and g(t)∈L1([0,b],X), then( A 1 )   D α I α g ( t ) = g ( t ),( A 2 )   I α D α g ( t ) = g ( t ) - [ D α - 1 g ( t ) ] t = 0 / Γ ( α ) t α - 1, for all t∈[0,1].Lemma 4 (see [24]).
LetX be a complete normed space and C be a nonempty closed and bounded convex subset of X. Consider P:X→X and Q:C→X to be two mappings, and satisfying the following conditions(1)
P is a contraction with constant ρ∈(0,1)(2)
Q is continuous and compact(3)
the equationPa+Qy=a implies that a∈C, for all y∈C,
then the operator equationPx+Qx=x has a solution in C.


## 3. Main Fixed-Point Results

In this section, we give the mainN-tupled fixed-point theorem which will play an important role in the proposed applications.The following theorem was given in [5] and will be used with Lemma 4 as a tool to prove the main results.Theorem 5.
LetX be a complete normed space and C be a nonempty closed and bounded convex subset of X. Consider P:X→X and Q:C→X are two mappings and the following are satisfied:(1)
P is a contraction(2)
Q is continuous and compact(3)
The equationPa+Qy=a implies that a∈C for all y∈C.
Then, the equationF(x,y)=Px+Qy has a coupled fixed point in C×C.ConsiderXN=∏i=1nX and define the following two operations (sum and scalar multiplication in XN) as follows:ifW=(w1,w2,…,wN),V=(v1,v2,…,vN)∈XN, and λ∈R, then, we have that (5)W+V=w1+v1,w2+v2,…,wN+vN,λV=λv1,λv2,…,λvN.Therefore, the norm in XN can be defined as (6)WXN=w1,w2,…,wN=w1+w2+…+wN,where . is the norm defined on X. It is easy to prove that (XN,.) is a Banach space.We are now ready to prove our mainN-tupled fixed-point theorem for this section.Theorem 6.
LetX be a complete normed space and C be a nonempty closed and bounded convex subset of X. Consider T1:X→X and T2,T3,…,TN:C→X are mappings and the following are satisfied:
1 ′ For every x1,x2∈X, there exist D-function φT1 which satisfies T1x1-T1x2≤λφT1(x1-x2), where λ∈(0,1) and φT1(r)<r, for all r>0;
2 ′   T 2 , T 3 , … , T N are continuous and compact;
3 ′The equation a=T1a+∑i=2NTiyi implies that a∈C, for all y2,…,yN∈C.
Then,F(x1,x2,…,xN)=∑i=1NTixi has an N-tupled fixed point in ∏i=1nC.Proof.
DefineCN=∏i=1nC. It is easy to prove CN≠∅ is closed convex and bounded subset of Xn. Consider P1,P2,…,PN:CN→CN such that, for u=(x1,x2,…,xN)∈CN, then (7)P1u=T1x1,T1x2,…,T1xN,P2u=T2x2,T2x3,…,T2xN,T2x1,⋮PNu=TNxN,TNx1,TNx2,…,TNxN-1.Therefore, if we prove that the operator equation u=∑i=1NPiu has a solution u=(x1,x2,…,xN)∈CN, then we obtain that (8)x1,x2,…,xN=T1x1,T1x2,…,T1xN+T2x2,T2x3,…,T2xN,T2x1+⋮+TNxN,TNx1,TNx2,…,TNxN-1.Then, we have that (9)x1=T1x1+T2x2+…+TNxN,x2=T1x2+T2x3+…+TNx1,⋮xN=T1xN+T2x1+…+TNxN-1,which implies that (x1,x2,…,xN) is an N-tupled fixed point of F in XN. Now, we prove the theorem by three steps.
Step 1. Prove P1 is a contraction.
LetW≠V∈C,N we have that(10)P1W-P1VXN=T1w1,T1w2,…,T1wN-T1v1,T1v2,…,T1vNXN≤T1w1-T1v1+…+T1wN-T1vN≤λϕT1w1-v1+…+ϕT1wN-vN≤λw1-v1+⋯+wN-vN=λW-VXN,for some λ∈(0,1). Then, P1 is a contraction.
Step 2. P2, P3, …,PN are continuous and compact.
Let(Un)=(x1n,x2n,…,xNn) be a sequence in CN converging to U=(x1,x2,…,xN)∈CN as n→∞. Since T2 is continuous, we get that(11)limn→∞P2Un=limn→∞T2x2n,limn→∞T2x3n,…,limn→∞T2xNn=T2x2,T2x3,…,T2xN=P2x1,x2,…,xN=P2U.This shows that P2 is continuous. Similarly, one can obtain that P3,…,PN are continuous. Since T2 is bounded, then, for all x∈X, there exist M2>0 such that T2x≤M2. Therefore, we have that(12)P2x1,x2,…,xNXN=T2x2,T2x3,…,T2xN,T2x1XN=T2x2+…+T2x1≤M2N.Thus, P2 is uniformly bounded in CN. By doing the same steps, we get that P3,…,PN are also uniformly bounded in CN.
SinceT2 is compact in C, then we get that T2(C) is equicontinuous and uniformly bounded in X. Now, we prove that P2(Cn) is equicontinuous in Xn. Since T2(C) is equicontinuous in C, for any z∈C and t2>t1, we have that (13)T2zt2-T2zt1→0,ast2-t1→0.Hence, for any Z=P2u∈P2(Cn) and t2-t1→0, we obtain(14)Zt2-Zt1En=P2ut2-P2ut1)En=T2y2t2-T2y2t1+T2y3t2-T2y3t1+⋯+T2y1t2-T2y1t1→0.Thus, P2(Cn) is equicontinuous in Xn. Therefore, P2(Cn) is compact in Xn. Hence, P2:Dn→Xn is compact. By doing the same steps, we get P3,P4,…,Pn:Dn→Xn are compact.
Step 3. The operator equation u0=P1u0+P2u+…+Pnu implies that u0∈CN, for all u∈CN.
Letu0=(a,a,…,a)∈XN and u=(x1,x2,…,xn)∈D.n Consider the operator equation u0=P1u0+P2u+…+Pnu. Hence, we get that(15)P1u0+P2u+…+PNu=T1a,T1a,…,T1a+T2x2,T2x3,…,T2x1+⋯+TNxN,TNx1,…,TNxN-1=a,a,…,a.We get that, for all x2,x3,…,xN∈D, (16)a=T1a+T2x2+…+TNxN.It implies that a∈C. Therefore, we have that (a,…,a)∈CN.
DefineB:CN→XN as BU=P2U+P3U+⋯+PNU, for all U∈C.N Thus, B is continuous and compact. If u0=P1u0+BV holds, then u0∈CN, for all V∈C.N From Lemma 3, the operator equation U=P1U+BU has a solution in C.n Then, if the solution is (x1,x2,…,xn), we get that (17)x1,x2,…,xN=T1x1,T1x2,…,T1xN+T2x2,T2x3,…,T2xn,T2x1+⋯+TNxN,TNx1,TNx2,…,TNxN-1.Therefore, we have that (18)x1=T1x1+T2x2+…+TNxN=Fx1,x2,…,xN,x2=T1x2+T2x3+⋯+T2x1=Fx2,x3,…,x1,⋮xN=T1xN+T2x1+…+TNxN-1=FxN,x1,…,xN-1.Hence, F has an N-tupled fixed point.


## 4. Existence of the Solution of the System of R-L FDEs

In this section, we prove the existence of a mild solution of the system ofN-fractional evolution equations:(19)Dαx1t-f1t,x1t=f2t,x2t,Iβx2t+…+fNt,xNt,IβxNt,Dαx2t-f1(t,x2t=f2t,x3t,Iβx3t+…+fNt,x1t,Iβx1t,⋮DαxNt-f1(t,xNt=f2t,x1t,Iβx1t+…+fNt,xN-1t,IβxN-1t,xi0=0∈X,∀i=1,2,…,N,α∈0,1,β>0,t∈0,b,b>0.Define C0,b(R)=f:0,b→R:fis  continuous. Recall that C[0,b](R) is a Banach space with respect to the norm xC[0,b]=supt≥0|x(t)|. Also, we recall that L0,b1(R)=f:0,b→R:∫0bxtdt<∞; therefore, L[0,b]1(R) is a Banach space with the norm xL[0,b]1=∫0bxtdt. We study the existence of a solution for system (19) under the following hypotheses:( H 1 ) For all t∈[0,b], the mapping x↦x-f1(t,x) is increasing in R.( H 2 ) There exist two constants M1,d such that M1≥L>0 and (20)f1t,yt-f1t,zt≤dyt-zt2M1+yt-zt,for all t∈[0,b] and y,z∈R.( H 3 ) There exists a constant, F∗=max⁡ft,0:t∈0,b.( H 4 ) There are h2,h3,…,hN∈C[0,b](R), such that (21)f2t,yt,zt≤h2t,f3t,yt,zt≤h3t,⋮fNt,yt,zt≤hNt,for all t∈[0,b] and y,z∈R.Lu et al. [4] proved that if the hypothesis (H1) holds, g∈C[0,b](R), α∈(0,1), f:[0,b]×R→R is continuous, and f(0,0)=0, and the fractional differential equation (22)Dαxt-ft,xt=gt,t∈0,b,x0=0has a unique solution which is the solution of the following integral equation: (23)xt=ft,xt+1Γα∫0tgst-s1-αds,t∈0,b.Theorem 7.
If the conditions(H2), (H3), and (H4) hold, the system of fractional differential equations (19) has a solution on [0,b].Proof.
System (19) has a solution if the following system of the integral equations has a solution:(24)x1t=f1t,x1t+1Γα∫0tf2s,x2s,Iβx2st-s1-αds+…+1Γα∫0tfNs,xNs,IβxNst-s1-αds,t∈0,b,x2t=f1t,x2t+1Γα∫0tf2s,x3s,Iβx3st-s1-αds+…+1Γα∫0tfNs,x1s,Iβx1st-s1-αds,t∈0,b,⋮xnt=f1t,xNt+1Γα∫0tf2s,x1s,Iβx1st-s1-αds+…+1Γα∫0tfNs,xN-1s,IβxN-1st-s1-αds,t∈0,b.Let X=C[0,b](R) and D be subset of X defined as D=x∈X:x≤K, where (25)K≥d+F∗+bαΓα+1h2L0,b1+…+hNL0,b1.It is clear that D is closed bounded convex of X. Define the operators T1:X→X and T2,…,TN:D→X as follows:(26)T1xt=f1t,xt,t∈0,b,T2xt=1Γα∫0tf2s,xs,Iβxst-s1-αds,t∈0,b,T3xt=1Γα∫0tf3s,xs,Iβxst-s1-αds,t∈0,b,⋮TNxt=1Γα∫0tfNs,xs,Iβxst-s1-αds,t∈0,b.Then, we can say that an element (x1,x2,…,xN)∈XN is an N-tupled solution of system (19) if and only if (x1,x2,…,xn) is the solution to the system for the following operator equations:(27)x1t=T1x1t+T2x2t+⋯+TNxNt,t∈0,b,x2t=T1x2t+T2x3t+⋯+TNx1t,t∈0,b,⋮xNt=T1xNt+T2x1t+⋯+TNxN+1t,t∈0,b.
Define the mappingQ:DN→X such that Q(x1,x2,…,xN)=∑i=1NTixi. Thus, system (19) has a solution if and only if Q(x1,x2,…,xN) has an N-tupled fixed point. We apply Theorem 6 to prove the result.
Now we apply three steps to prove the theorem.
Step 1. The operator T1 is a contraction.
Since condition H2 holds, we get that, for anyx,y∈X, we have(28)T1xt-T1yt=ft,xt-ft,yt≤dxt-yt2M1+xt-yt≤dxt-yt2M1+xt-yt.Thus, by defining θ(r)=dr/M1+r, we get that θ(r)<r. Hence, by taking the sup, we get that T1x-T1y≤1/2x-y. Therefore, T1 is contraction.
Step 2. The operators T2,T3,…,TN are continuous and compact on D.
Letx∗∈D and {xn}⊂D such that limn→∞xn=x∗. Using Dominated Convergence Theorem of Lebesgue integral, we get that(29)limn→∞T2xnt=1Γαlimn→∞∫0tf2s,x3s,Iβx3st-s1-αds≤1Γα∫0tt-sα-1limn→∞f2s,xns,Iβxnsds≤1Γα∫0tt-sα-1f2s,x∗s,Iβx∗sds=T2x∗t,∀t∈0,b.Hence, T2 is a continuous operator on D. Similarly, by doing the same steps, we get that (30)limn→∞T3xnt=T3x∗t,…,limn→∞TNxnt=TNx∗t.Thus, T3,…,TN are continuous operators on D. From (H3), for all x∈D, we get that(31)T2xt=1Γα∫0tf2s,xs,Iβx3st-s1-αds≤1Γα∫0tt-sα-1f2s,xs,Iβxsds≤1Γα∫0tt-sα-1h2sds≤bαΓα+1h2L0,b1,∀t∈0,b.Therefore, if we take the supremum, we have (32)T2x≤bαΓα+1h2L0,b1.Doing the same steps for T3,…,TN, we obtain (33)T3x≤bαΓα+1h3L0,b1,…,TNx≤bαΓα+1hNL0,b1.Hence, T2,…,TN are uniformly bounded functions on D. For any x∈D and for any t1,t2∈[0,b] such that t1<t2, we get that(34)T2xt1-T2xt2=1Γα∫0t1f2s,xs,Iβxst1-s1-αds-∫0t2f2s,xs,Iβxst2-s1-αds≤1Γα∫0t1f2s,xs,Iβxst1-s1-αds-∫0t1f2s,xs,Iβxst2-s1-αds+1Γα∫0t1f2s,xs,Iβxst2-s1-αds-∫0t2f2s,xs,Iβxst2-s1-αds≤h2L0,b1Γα∫0t1t1-sα-1-t2-sα-1ds+∫t1t2t2-sα-1ds≤h2L0,b1Γα+1t1α-t2α-t2-t1α,∀t∈0,b.Hence, when t2-t1→0, we have that (35)T2xt1-T2xt2→0.Thus, T2(D) is equicontinuous on [0,b]. Hence, T2(D) is uniformly bounded operator in D and T2 is a compact operator. Similarly, by doing the same steps, we get T3,…,Tn:D→D are compact operators.
Step 3. Prove that the equation a=T1a+T2y2+T3y3+…+TNyN implies that a∈D, for all y2,…,yN∈D.
Letx∈D and y2,…,yN∈D. Let x=T1x+T2y2+T3y3+⋯+TNyN. Using condition (H4), we get that(36)xt≤T1xt+T2y2t+…+TNyNt≤ft,xt-ft,0+ft,0+1Γα∫0tf2s,y2s,Iβy2st-s1-αds+…+1Γα∫0tfNs,yNs,IβyNst-s1-αds≤d+F∗+bαΓα+1h2L0,b1+…+hNL0,b1,∀y2,…,yn∈D,t∈0,bb>0,which implies (37)x≤d+F∗+bαΓα+1h2L0,b1+…+hNL0,b1.Thus, x∈D. By applying Theorem 6, we get that the system of fractional equations (19) has a solution in [0,b].Example 8.
Consider the following system of R-L FDEs:(38)D1/2x1t-sin⁡tx1t22+x1t=tx2t1+x2t+f3t,x3t,t∈0,π,D1/2x2t-sin⁡tx2t22+x2t=tx3t1+x3t+f3t,x1t,t∈0,π,,D1/2x3t-sin⁡tx3t22+x3t=tx1t1+x1t+f3t,x2t,t∈0,π,x10=x20=x30=0.In system (38), let f1t,xt=sin⁡(t)|x(t)|/2(2+|x(t)|, such that f2(t,x(t),Iβ(x(t)))=tx(t)/1+|x(t)| is defined as(39)f3t,xt,Iβxt=5,ifx≤0,5+5x1+2x,if0<x<2,7,ifx≥2.
Letx,y∈X. Then, for all t∈[0,π], we have that (40)f1t,xt-f1t,yt≤xt-yt22+xt-yt.
Therefore,f2(t,x(t),Iβ(x(t)))≤h2(t), where h2(t)=t. Also f3(t,x(t),Iβ(x(t)))≤h3 (t), where h3(t)=7. Let F∗=0, d=1, M1=2, b=π, and h2(t)=t, h3(t)=7 in Theorem 7; we obtain K≥22. Then, system (38) has a solution.


## 5. Conclusions

This paper introduced a new version of Kransnoselskii fixed-point theorem. This generalization is more general because we can study theN-tupled fixed point by it. Another advantage of the proposed N-tupled fixed point is to study the existence of systems of more than two operator equations. For that, we applied the abstract proposed fixed-point theorem to prove the existence of solution of the system of N-RL- FDEs. We gave an example to illustrate the abstract proposed fixed-point theorem.



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