# Krasnoselskii N-Tupled Fixed Point Theorem with Applications to Fractional Nonlinear Dynamical System

Authors: Tamer Nabil
Journal: Advances in Mathematical Physics
 (2019)

---

## Abstract

In this paper, N-tupled fixed point theorems for two monotone nondecreasing mappings in complete normed linear space are established. The extension of Krasnoseskii fixed point theorem for a version of N-tupled fixed point is given. Our theoretical results are applied to prove the existence of a mild solution of the system of N-nonlinear fractional evolution equations. Finally, an example of a nonlinear fractional dynamical system is given to illustrate the results.


---

## Body

## 1. Introduction

It is outstanding that the fixed point theorem of Krasnoselskii [1] might be joined with Banach and Schauder’s fixed point theorems. The sum of operators is obviously found in neutral functional equations and delay integral equations, which have been talked about widely in [2, 3], for instance. In 1964, Krasnoselskii studied the existence of fixed point for a sum of two operators, one being a continuous and compact and the other being a strict contraction. A lot of generalizations of Krasnoselskii’s theorem have been done, mostly by relaxing the continuous and compact conditions and sometimes by withdrawing the requirement of the strict contraction or even both.In 2006, Bhaskar and Lskshmikanthan [4] proposed new approach of the fixed point theorem which was called coupled fixed point. They gave some useful applications for mixed monotone operators.In 2011, Luong and Thuan [5] extend the coupled fixed point approach to study the unique solution of the integral equations. Recently in 2013, Dhage [6] proposed new a version of the Krasnoselskii fixed point theorem and used it to prove the existence of the solution of nonlinear fractional differential equation.More recently, in 2017, Yang et al. [7] proved the Krasnoselskii coupled fixed point theorem under some or certain conditions and used their results to study the existence of the solution of a nonlinear coupled system of fractional differential equations. Imdad et al. [8] in 2013, gave a new approach of the fixed point theory which was called N-tupled fixed point. This approach is useful to study a fixed point of mappings F:X×X×⋯×X︸n-times→X, where X is a nonempty set. In this paper, we extend the results of Yang et al. [7] to prove a generalized version of N-tupled fixed point theorems. We give an application to the study existence of a solution of a system of nonlinear fractional differential evolution equations.We organize the paper as follows: In Section2, some basic definitions are introduced. In Section 3, Krasnoselskii N-tupled fixed point ( KNTFP) is proved. In Section 4, the existence of the solution of a nonlinear N-system of fractional evolution differential equations is considered. In Section 5, we give an example of a nonlinear fractional dynamical system. Finally, in Section 6 the conclusion is presented.


## 2. Preliminaries

LetX≠ϕ and F:X×X→X be a mapping. A point (x,y)∈X×X is said to be a coupled fixed point of F in X×X if F(x,y)=x and F(y,x)=y. Throughout this paper, let E be a nonempty set and (E,≤,.) be a partially ordered normed linear space. If Q:E→E is a mapping. Then Q is said to be monotone nondecreasing if a≤b implies Q(a)≤Q(b), for all a,b∈E. Two elements a,b∈E are said to be comparable if a≤b or b≤a. If C is a nonempty subset of E, C is said to be a chain if each two elements a,b∈C are comparable. We recall the following definitions which are given in [6, 7].Definition 1.
LetQ:E→E be a mapping. Q is called partially compact if for each chain C subset of E, Q(C) is a relatively compact subset of E.Definition 2.
LetQ:E→E be a mapping. Given an element a∈E, define the orbit Υ(a;Q) as(1)Υa;Q=a,Qa,Q2a,Q3a,…,Qna,….If for any sequence {an}∈Υ(a;Q) such that an→a∗ as n→∞ implies Qan→Qa∗, for each a∈E, then Q is called Υa- orbitally continuous in E. Furthermore, (E,≤,.) is said to be Υ- orbitally complete if each sequence {an}∈Υ(a;Q) converges to an element a∗∈E.Definition 3.
A mappingϕ:E→E is said to be D- function if it is upper semicontinuous and monotone nondecreasing such that ϕ(0)=0.Definition 4.
A mappingQ:E→E is called partially nonlinear D- contraction in E, if for each comparable elements a,b∈E, there exists a D- function ϕ:R+→R+ such that the following conditions are satisfied:
(i)Qa-Qb≤ϕ(a-b),
(ii)ϕ(r)<r for all r>0.Definition 5.
Let(E,≤,.) be a Υ- orbitally complete linear space. The positive cone K∈E is defined as K={x∈E:x≥0}.The following theorem was given in [7] and will be used as a tool to prove the main results.Theorem 6.
LetE be a partially ordered Υ- orbitally complete normed linear space and K be the positive cone of E. Let K be normal and C be a nonempty closed subset of E. Consider the two monotone nondecreasing mappings P,Q:C→C such that the following conditions are satisfied:
(1)P is Υ- orbitally continuous and a partially nonlinear D- contraction,
(2)Q is Υ- orbitally continuous and partially compact,
(3) there xists an elementa∈C such that a≤Pa+Qy for all y∈C,
(4) every pair of elements inC has an upper and lower bound.
ThenF(x,y)=Px+Qy has a coupled fixed point in E×E.Now, we recall the definition about the fractional derivative; for more detail about the fractional calculus, please see [9, 10]. 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 order α(>0)∈(0,∞) of a function g∈L1([0,b],E) such that b>0 is defined as(2)Iαgt=∫0tt-sα-1Γαgsds.Also, the Riemann-Liouville derivative of order α(>0) is defined as(3)RDαgt=DmIm-αgt,where m=⌈α⌉ and D=d/dt.The Caputo fractional derivative of orderα(>0) is denoted by D∗α and is defined as(4)D∗αgt=In-αDngt,where n=⌈α⌉, g∈L1([0,b],E) such that b>0 and t>0 [11].


## 3. Krasnoselshii N-Tupled Fixed Point Theorem

In this section we give the main N-tupled fixed point theorem which is used in the proposed applications. ConsiderEn=E×E×⋯×E︸n-times and define the two operations (sum and scaler multiplication in En) as follows: if W=(w1,w2,…,wn), V=(v1,v2,…,vn)∈En, and λ∈R then, we have that(5)W+V=w1+v1,w2+v2,…,wn+vnλV=λv1,λv2,…,λvnTherefore, the positive cone in En can be defined as(6)KEn=W=w1,w2,…,wn∈En:w1,w2,…,wn≥0.Also the norm in En can be defined as(7)WEn=w1,w2,…,wn=w1+w2+⋯+wn,where . is the norm defined on E. It is easy to prove that (En,.En) is partially ordered normed linear space with order relation ≤ induced by KEn. If D is a nonempty closed subset of E, then D×D×⋯×D︸n-times is also nonempty closed subset of En. On the other hand, if K is normal cone, the KEn is normal.LetQ:En→E. The point (x1,x2,…,xn)∈En is said to be N-tupled fixed point of Q if and only if(8)x1=Qx1,x2,…,xnx2=Qx2,x3,…,xn,x1⋮xn=Qxn,x1,…,xn-1Now, we prove the main fixed point theorem.Theorem 7.
LetE be a partially ordered Υ- orbitally complete normed linear space and K be the positive cone of E. Let K be normal and D be a nonempty closed subset of E. Consider T1,T2,…,Tn:D→D are n monotone nondecreasing mappings such that the following are satisfied:
(1)T1 is Υ- orbitally continuous and a partially nonlinear D- contraction,
(2)T2,…,Tn are Υ- orbitally continuous and a partially compact,
(3) there exists an elementa∈D such that a≤T1a+T2y2+T3y3+⋯+Tnyn for all y2,y3,…,yn∈D,
(4) every pair of elements inD has an upper and lower bounded.
ThenQ(x1,x2,…,xn)=T1x1+T2x2+⋯+Tnxn has a N-tupled fixed point in En.Proof.
According to the fact that(E,≤,.) is a partially ordered Υ- complete normed linear space and K is normal, then (En,≤,.En) is also a partially ordered Υ- complete normed linear space and KEn is normal. Define Dn=D×D×⋯×D︸n-times, then Dn is closed subset of En. Consider P1,P2,…,Pn:Dn→Dn such that, for u=(x1,x2,…,xn)∈Dn,(9)P1u=T1x1,T1x2,…,T1xnP2u=T2x2,T2x3,…,T2xn,T2x1⋮Pnu=Tnxn,Tnx1,Tnx2,…,Tnxn-1Therefore, if we prove that the operator equation u=P1u+P2u+⋯+Pnu has a solution u=(x1,x2,…,xn)∈Dn, then we have that(10)x1,x2,…,xn=T1x1,T1x2,…,T1xn+T2x2,T2x3,…,T2xn,T2x1+⋯+Tnxn,Tnx1,Tnx2,…,Tnxn-1.This implies that (x1,x2,…,xn) is a N-tupled fixed point of Q in En. Our proof of the theorem is divided into 4 steps.Step 1.
P 1 is Υ- orbitally continuous and P1W-P2VEn≤W-VEn, for all comparable W≠V∈Dn. Since T1 is Υ- orbitally continuous then by the definition of P1 we get that P1 is Υ- orbitally continuous.
LetW≠V∈Dn be comparable elements; we have that(11)P1W-P2VEn=T1w1,T1w2,…,T1wn-T1v1,T1v2,…,T1vnEn≤T1w1-T1v1+⋯+T1wn-T1vn≤ϕw1-v1+⋯+ϕwn-vn≤w1-v1+⋯+wn-vn=W-VEn,Step 2.
P 2 , P 3 , … , P n are Υ-orbitally continuous and partially compact.
SinceT2,…,Tn are Υ-orbitally continuous, it is easy to prove that P2,…,Pn are Υ- orbitally continuous. Consider C⊂D is bounded chain. According to the fact that T2 is partially compact in D, then we get that T2(C) is equicontinuous and uniformly bounded in D. Let Cn=C×C×⋯×C︸n-times, then Cn⊂Dn and Cn is bounded chain in Dn. Now, we prove that P2(Cn) is equicontinuous and uniformly bounded in Dn. Since P2(C)⊂D is uniformly bounded, there is a constant M2>0 satisfying T2z≤ε for all z∈C. For any Z∈P2(Cn), we can find y1,y2,…,yn∈C such that u=(y1,y2,…,yn)∈Cn and Z=P2u. Therefore, we have that(12)ZEn=P2uEn=T2y2,T2y3,…,T2yn,T2y1En=T2y2+T2y3+⋯+T2yn+T2y1≤nε.Thus, P2(Cn) is uniformly bounded in Dn. By doing the same steps we get that P3,…,Pn are uniformly bounded in Dn. According to the fact that T2(C) is equicontinuous in D, for any z∈C and t2>t1, we have that (T2z)(t2)-(T2z)(t1)→0, as t2-t1→0. Thus, for any Z=P2u∈P2(Cn), where t2-t1→0, we get the following:(13)Zt2-Zt1En=P2ut2-P2ut1)En=T2y2t2-T2y2t1+T2y3t2-T2y3t1+⋯+T2y1t2-T2y1t1→0.Thus, P2(Cn) is equicontinuous in Dn. Therefore, P2(Cn) is relatively compact in Dn. Hence, P2:Dn→Dn is partially compact. By doing the same steps, we get that P3,P3,…,Pn:Dn→Dn are partially compact.Step 3.
There is an elementu0∈D satisfying: u0≤P1u0+P2u+⋯+Pnu for all u∈Dn
Letu0=(a,a,…,a) and u=(x1,x2,…,xn)∈Dn. By condition (3) in the theorem, we get that(14)P1u0+P2u+⋯+Pnu=T1a,T1a,…,T1a+T2x2,T2x3,…,T2x1+⋯+Tnxn,Tnx1,…,Tnxn-1≥a,a,…,a.Step 4.
Every pair of elements inDn has an upper bound and lower bound. For every pair of elements W=(w1,w2,…,wn),V=(v1,v2,…,vn)∈Dn, from condition (4) of the theorem, there exist a1,a2,…,an and b1,b2,…,bn such that, for all i=1,2,…,n, we have that(15)ai≤wi,ai≤vi,bi≥wi,bi≥vi.Hence, we find that(16)a1,a2,…,an≤w1,w2,…,wn≤b1,b2,…,bn,a1,a2,…,an≤v1,v2,…,vn≤b1,b2,…,bn.Thus, we get that every pair W,V∈Dn has an upper and lower bound.
DefineB:Dn→Dn as BU=P2U+P3U+⋯+PnU, for all U∈Dn. Thus B is Υ- orbitally continuous and a partially compact. Also there exists an element u0∈Dn such that u0≤P1u0+BV for all V∈Dn. Hence U=P1U+BU has a solution in En. 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=Qx1,x2,…,xnx2=T1x2+T2x3+⋯+Tnx1=Qx2,x3,…,x1⋮xn=T1xn+T2x1+⋯+Tnxn-1=Qxn,x1,…,xn-1Hence Q(x1,x2,…,xn) has a N-tupled fixed point. By changing condition (3) due to the fact that there exists an element a∈D such that a≥T1a+T2y2+T3y3+⋯+Tnyn for all y2,y3,…,yn∈D, the theoretical results are also correct.Theorem 8.
LetE be a partially ordered Υ- orbitally complete normed linear space and K be the positive cone of E. Let K be normal and D be a nonempty closed subset of E. Consider T1,T2,…,Tn:D→D are n monotone nondecreasing mappings such that the following are satisfied:
(1)T1 is Υ- orbitally continuous and a partially nonlinear D- contraction,
(2)T2,…,Tn are Υ- orbitally continuous and a partially compact,
(3) there exists an elementa∈D such that a≥T1a+T2y2+T3y3+⋯+Tnyn for all y2,y3,…,yn∈D,
(4) every pair of elements inD has an upper and lower bounded.
ThenQ(x1,x2,…,xn)=T1x1+T2x2+⋯+Tnxn has a N-tupled fixed point in En.


## 4. Existence of Mild Solution of the System of N- Fractional Evolution Equations

LetK~ be the positive convex cone of Υ- orbitally Banach space (X,.). Suppose that K~ is normal. It is clear that X is partially order by the poset ≤ induced by K~. In this section, we prove the existence of mild solution of the system of N- fractional evolution evolution equations:(19)D∗αx1t+Ax1t=f1t,x1t+f2t,x2t+⋯+fnt,xnt,D∗αx2t+Ax2t=f1t,x2t+f2t,x3t+⋯+fnt,x1t,⋮D∗αxnt+Axnt=f1t,xnt+f2t,x1t+⋯+fnt,xn-1t),xi0=ζ0∈X,∀i=1,2,…,n,α∈0,1,t∈0,b,b>0,where -A generates a C0- semigroup {S(t):t≥0} of uniformly linear and bounded operators in X. Also, f1,f2,…,fn are given functions. According to the fact that S(t) are bounded, then there exists d∈(0,∞) such that S(t)≤d for all t≥0.Define two families of operator{Ψα(t):t≥0} and {Φα(t):t≥0}(20)Ψαtx=∫0∞ηαsStαsxds,Φαtx=α∫0∞sηαsStαsxds, where α∈(0,1) and(21)ηαs=1αs-1-1/αγαs-1/α,γαs=1π∑n=1∞-1n-1s-αn-1Γnα+1n!sin⁡nπα,s∈0,∞.According to [12, 13], the family of operators {Ψα(t):t≥0} and {Φα(t):t≥0} has the following properties: (Pr1) For any x∈X and t≥0 fixed, we have that(22)Ψαtx≤dx,Φαtx≤dΓαx,(Pr2){S(t):t≥0} is an equicontinuous semigroup which implies that {Φα(t):t≥0} is equicontinuous in X,(Pr3){S(t):t≥0} is positive C0- semigroup which implies that {Ψα(t):t≥0} and {Φα(t):t≥0} are positive operators.DefineC[0,b](X)={f:[0,b]→X:fis continuous} and K~C={x∈C[0,b](X):x(t)∈K~,t∈[0,b]}. It is easy to show C[0,b](X) is a partially order Υ- orbitally complete normed space with the norm: xC[0,b]=supt≥0x(t), and a poset ≤ induced by K~C. It is easy to show that K~C is normal because K~ is normal.LetE=C[0,b](X) and K=K~C. Define, n- operators T1,T2,…,Tn:E→E as(23)T1xt=Ψαtξ0+∫ott-sα-1Φαt-sf1s,xsds,t∈0,b,T2xt=∫ott-sα-1Φαt-sf2s,xsds,t∈0,b,T3xt=∫ott-sα-1Φαt-sf3s,xsds,t∈0,b,⋮Tnxt=∫ott-sα-1Φαt-sfns,xsds,t∈0,b.Then we can say that an element (x1,x2,…,xn)∈En is a N-tupled mild solution of the system (19) if and only if (x1,x2,…,xn) is the solution of the system of the operator equations:(24)x1t=T1x1t+T2x2t+⋯+Tnxnt,t∈0,b,x2t=T1x2t+T2x3t+⋯+Tnx1t,t∈0,b,⋮xnt=T1xnt+T2x1t+⋯+Tnxn+1t,t∈0,b.We will apply Theorem7 to prove that the system of operator equations (24) has a N-tupled fixed point in En under the following important conditions:(C1) The positive C0- semigroups {S(t):t≥0} is equicontinuous.(C2) The function f1:[0,b]×X→X is continuous in X for all t∈[0,b] and there exist a constant ϱ∈R with 0<ϱ<Γ(α+1)/dbα and a D- function ϕ:[0,∞)→[0,∞) and satisfied the fact that ϕ(t)<r for any r∈(0,∞) and(25)0≤f1t,e1-f1t,e2≤ϱϕe1-e2,for alle1,e2(e1≥e2)∈X and t∈[0,b].(C3) The functions f2(t,x),f3(t,x),…,fn(t,x):[0,b]×X→X are continuous, nondecreasing, and bounded in x.(C4) There exists an element x~∈E which satisfies the fact that(26)D∗αx~t+Ax~t≤f1t,x~)+f2t,y2t+⋯+fnt,ynt,∀y2,…,yn∈E,x~≤ξ0∈X,α∈0,1,t∈0,b,b>0.Theorem 9.
If conditions(C1),(C2),(C3), and (C4) are satisfied, the system of fractional evolution equations (19) has a N-tupled mild solution on [0,b].Proof.
Since condition(C3) holds, then f2(t,x),f3(t,x),…,fn(t,x) are bound for all t∈[0,b], and there are constants N2,N3,…Nn such that, for all t∈[0,b], we get that(27)f2t,x≤N2,f3t,x≤N3,⋮,fnt,x≤Nn,∀x∈X. For all i=2,…,n, we have that(28)r~≥max⁡dNibαΓα+1,1-dϱbαΓα+1-1dξ0+dG~bαΓα+1+1, where G~=sup{f1(t,0):t∈[0,b]}. We can define the following open balls in E as(29)Bx0,r=x∈E:x0-xC0,b<rwhere r=x0C[0,b]+r~. Define D=B¯(x0,r). It is clear that D is bounded and closed subset in E. Now we apply 3 steps to prove the theorem.
Step 1. The operator T1:D→D is Υ-orbitally continues and a partially nonlinear D- contraction in E.
Since condition(C2) holds, we get that, for any x∈E with xC[0,b]≤r~, we have(30)T1xt≤dξ0+dΓα∫0ts-tα-1f1s,xs-f1s,0+f1s,0ds≤dξ0+dΓα∫0ts-tα-1ϱϕxC0,b+G~ds≤dξ0+dϱbαΓα+1r~+dG~bαΓα+1≤r~.By taking the sup, we get that (T1x)tC[0,b]≤r~ for all x∈E, such that (x)C[0,b]≤r~. Therefore, we have the following:(31)x0-T1xC0,b≤x0C0,b+T1xC0,b≤x0C0,b+r~=r1.Hence, T1 is a function from D to D. According to the fact that {S(t):t≥0} is positive C0- semigroup and the properties (Pr1), (Pr2), and (Pr3), it is provided that T1:D→D is nondecreasing. Let {xm}∈Υ(x;T1) for any x∈D such that xm→x^ as m→∞. Thus by condition (C2), we get that T1 is continuous in X for all t∈[0,b], and therefore we have the following:(32)limm→∞T1xmt=Ψαtξ0+∫0tt-sα-1Φαt-slimm→∞f1s,xmsds=Ψαtξ0+∫0tt-sα-1Φαt-sf1s,x^sds=T1x^t,t∈0,b.Thus T1 is Υ-orbitally continuous.
Letx1,x2∈D be comparable such that x1≥x2. We have the following:(33)T1x1t-T1x2t≤∫0ts-tα-1Φαt-sf1s,x1s-f1s,x2sds≤dϱΓα∫0ts-tα-1ϕx1s-x2sds≤dϱbαΓα+1ϕx1s-x2sC0,b.Since 0<ϱ<Γ(α+1)/dbα, we get that (34)T1x1t-T1x2tC0,b≤ϕx1s-x2sC0,b.Hence, T1 is partially nonlinear D- contraction operator in E.
Step 2. The operators T2,T3,…,Tn:D→D are Υ- orbitally continuous, nondecreasing, and partially compact operators in E.
First, we prove thatT2,T3,…,Tn are mappings from D to D. By the condition (C3), for any x∈E such that xC[0,b]≤r~, we get that(35)T2xt≤dΓα∫0tt-sα-1f2s,xsds≤dN2bαΓα+1≤r2.Similarly, we have that(36)T3xt≤r3,⋮Tnxt≤rn.Thus, we get the following:(37)T2xC0,b≤r2,…,TnxC0,b≤rn.Therefore, we get that(38)x0-T2xC0,b≤x0C0,b+T2xC0,b≤x0C0,b+r2=r~2⋮x0-TnxC0,b≤x0C0,b+TnxC0,b≤x0C0,b+rn=r~n.Hence, T2,…,Tn are mappings from D to D. By using the same method as in Step 1, we can prove T2,…,Tn are Υ- orbitally continuous and nondecreasing E. Consider C∈D is any arbitrary chain subset of D. For any Y∈T2(C), there is an element x∈D such that Y(t)=(T2x)(t), for all t∈[0,b]. For any t1,t2∈[0,b] such that t1<t2, we get that(39)Yt2-Yt1≤T2xt2-T2xt1≤H1+H2+H3,where(40)H1=∫0t1t2-sα-1-t1-sα-1Φαt2-sf2s,xsds,H2=∫0t1t1-sα-1Φαt2-s-Φαt1-sf2s,xsds,H3=∫t1t2t2-sα-1Φαt2-sf2s,xsds≤dN2bαΓα+1t2-t1α. Hence, when t2-t1→0, we get that(41)H3→0,and also,(42)H1≤dN2Γα∫0t1t2-sα-1-t1-sα-1ds→0.Let t1=0 and 0<t2<b; the we get that H2→0. For t1>0 and ε∈(0,t1) where ε is very small, then we get that(43)H2≤∫0t1-εt1-sα-1Φαt2-s-Φαt1-sf2s,xsds+∫t1-εt1t1-sα-1Φαt2-s-Φαt1-sf2s,xsds.≤N2t1α-εααsups∈0,t1-εΦαt2-s-Φαt1-s+2dN2εαΓα+1
By using condition(C1), we get that H2→0 as t2-t1→0. Therefore, Y(t2)-Y(t1)→0 as t2-t1→0. Thus, T2(C) is equicontinuous on [0,b]. Also, using condition (C3), we get that(44)Yt=T2xt≤∫0tt-sα-1Φαt-sf2s,xsds≤dN2bαΓα+1,∀t∈0,b.Hence, T2(C) is uniformly bounded operator in E and T2 is partially compact operator from D into itself. Similarly, by doing the same steps we get that T3,…,Tn:D→D are partially compact operators.
Step 3. Prove the existence of the N-tupled fixed point.
By condition(C4), we get that(45)x~=D∗αx~t+Ax~t≤Ψαtξ0+∫0tt-sα-1)Φαt-sf1t,x~tds+∫0tt-sα-1)Φαt-sf2t,y2tds+⋯+∫0tt-sα-1)Φαt-sfnt,yntds=T1x~t+T2y2t+⋯+Tnynt,∀y2,…,yn∈D,t∈0,bb>0. By applying Theorem 7, we get that the system of fractional evolution equations (19) has N-tuple mild solution in En.


## 5. An Application to Fractional Dynamical System

Consider the following fractional dynamical system:(46)D∗1/2x1z,t+∂x1z,t∂z=f1z,t,x1z,t+f2z,t,x2z,t+f3z,t,x3z,t,z,t∈0,1×0,1,D∗1/2x2z,t+∂x2z,t∂z=f1z,t,x2z,t+f2z,t,x3z,t+f1z,t,x3z,t,z,t∈0,1×0,1,D∗1/2x3z,t+∂x3z,t∂z=f1z,t,x3z,t+f2z,t,x1z,t+f3z,t,x2z,t,z,t∈0,1×0,1,x10,t=x11,t=x20,t=x21,t=x30,t=x31,t=0,t∈0,1,x1z,0=x2z,0=x3z,0=ξ0z,z∈0,1, such that f2:[0,1]×[0,1]×R is defined as(47)f2z,t,x=5,ifx≤05+5x1+2x,if0<x<2,7,ifx≥2, and f3:[0,1]×[0,1]×R is defined as(48)f3z,t,x=1,ifx≤01+2x,if0<x<5,11,ifx≥5,Thus f2,f3 are continuous and nondecreasing, |f1(z,t,x)|≤7 and |f3(z,t,x)|≤11. Hence, condition (C3) is satisfied.Theorem 10.
Let the following two conditions hold:(C1)′ The function f1:[0,1]×[0,1]×R→R is continuous and there exists a constant ϱ∈R with 0<ϱ<π/2 and a D- function ϕ:[0,∞)→[0,∞) and satisfied the fact that ϕ(t)<r for any r∈(0,∞) and(49)0≤f1z,t,y1z,t-f1z,t,y2z,t≤ϱϕy1-y2,for all(z,t)∈[0,1]×[0,1] and y1,y2∈C[0,1](R) such that y1≥y2.(C2)′ There exists an element x~∈C[0,1](R)∈ which satisfies the fact that(50)D∗αx~z,t+∂x~z,t∂z≤f1z,t,x~+f2t,y2t+f3t,y3t,∀y2,y3∈C0,1R,x~≤ξ0z,0,z∈0,1,α∈0,1,t∈0,1.Then the fractional dynamical system (46) has a N-tupled mild solution.Proof.
It is clear that ifX=C[0,1](R), X is Υ- orbitally complete normed linear space with x(t)X=maxt∈[0,1]|x(t)|. Let K={x∈X:x≥0}; the K is closed convex positive cone in X. Thus K is normal. Let F:D(F)→X, which D(F)⊂X, be defined as(51)Fy=y′,y∈DF=y=X:y′∈X,y0=y1=0.Thus we can find A- generates a C0- semigroup {S(t):t≥0} which is defined as(52)Styz=yt+z,t≥0,y∈X.Hence, {S(t):t≥0} is an equicontinuous C0- semigroup. Therefore, it is clear that supt∈[0,1]S(t)≤1. Thus, condition (C1)′ holds. Define,(53)x1tz=x1z,t,x2tz=x2z,t,x3tz=x3z,t,f1t,x1tz=f1z,t,x1z,t;f2t,x1tz=f2z,t,x1z,t;f3t,x1tz=f3z,t,x1z,t;Thus, the fractional dynamical system (46) can be written as the fractional system (19). From Theorem 9 system (46) has a N-tupled mild solution.


## 6. Conclusions

This paper introduced a new version of Kransnoselskii fixed point theorem. This version is more general because by it we can study the N-tupled fixed point. We applied the abstract proposed fixed point theorem to prove the existence of mild solution of the system of N-fractional evolution equations. We gave an example of the fractional dynamical system to illustrate the abstract fixed point proposed theorem. The results are interesting and can be directly applied to some other interesting problems such as the nonlocal problems for fuzzy implicit fractional differential systems [14] and fuzzy-valued equations [15].



---

*Source: 6763842-2019-03-03.xml*