Open access peer-reviewed chapter

Recent Fixed Point Techniques in Fractional Set-Valued Dynamical Systems

By Parin Chaipunya and Poom Kumam

Submitted: May 17th 2016Reviewed: November 25th 2016Published: March 15th 2017

DOI: 10.5772/67069

Downloaded: 896

Abstract

In this chapter, we present a recollection of fixed point theorems and their applications in fractional set-valued dynamical systems. In particular, the fractional systems are used in describing many natural phenomena and also vastly used in engineering. We consider mainly two conditions in approaching the problem. The first condition is about the cyclicity of the involved operator and this one takes place in ordinary metric spaces. In the latter case, we develop a new fundamental theorem in modular metric spaces and apply to show solvability of fractional set-valued dynamical systems.

Keywords

  • fractional set-valued dynamical system
  • fixed point theory
  • contraction
  • modular metric space

1. Introduction

Dynamical system is a wide area that deals with a system that changes over time. The two main characteristics of the time domain here are identified with the discrete and continuous manners. In discrete time domain, major considerations turn to the difference equations and generating functions. While in the latter one, which we shall be considering mainly for this chapter, the system is usually represented by differential equations. It might be more influential to talk about the inclusion problems if a set-valued system is to be analyzed.

The very first and fundamental dynamical system is known nowadays under the term Cauchy problem. It is represented with the following C1initial-valued problem:

{u(t)=f(t,u(t)),u(0)=u0

In this case, we assume that f:[0,T]×RRis continuous and uC1([0,T]). From simple calculus, we may see that this system is equivalent to the following integral equation:

u(t)=u0+[0,t]f(s,u(s))dsE1

This is where Banach got the idea to solve the problem. He proposed his famous fixed point theorem known today as the contraction principle in 1922 [1], mainly to solve this Cauchy problem effectively. Recall that the contraction principle states that if Xis a complete metric space and T:XXis Lipschitz continuous with constant 0<L<1, then Thas a unique fixed point.

Let us consider a map Λ:C1([0,T])C1([0,T])given by

Λ(u)(t):=u0+[0,t]f(s,u(s))ds,uC1([0,T]),t[0,T]

One can notice that uC1([0,T])solves Eq. (1) if and only if it is a fixed point of Λ. With this approach, by considering C1([0,T])with the supremum norm , we end up with the local solvability of the Cauchy problem. To obtain the global solution, we have to apply some techniques to extend the boundary of the local solution.

It is not very obvious that renorming by the L-weighted norm fL:=supt[0,T]eLtf(t), with L>0, will resolve such difficulty. We shall give the short solvability result of the Cauchy problem with the contraction principle here, to illustrate the concept of how we apply fixed point theorem to continuous dynamical systems. Under the assumption that fmust be Lipschitz in the second variable with constant L>0, we have for any x,yC1([0,T])the following:

eLt|Λ(x)(t)Λ(y)(t)|=eLt|[0,t]f(s,x(s))f(s,y(s))ds|eLt[0,t]|f(s,x(s))f(s,y(s))|dseLt[0,t]LeLseLs|x(s)y(s)|dseLtxyL[0,t]LeLseLsdseLt(eLt1)xyL(1eLT)xyL.

Taking supremum over t[0,T]yields the result and the solvability thus follows.

This is the alternative technique to guarantee the solvability of the Cauchy problem, without obtaining the local solution first. It is important to remark that there are many mathematicians that can later adapt different technique and different direction to obtain the solvability of various classes of dynamical systems, under one unifying fact—by applying fixed point theorems.

It is natural to raise the situation of set-valued integral, which proved itself for its importance in practical applications especially in engineering. In 1965, Aumann [2] introduced the concept of definite set-valued integral on real line and Euclidean spaces. Suppose that Ψis an interval [0,T], where T>0. Let F:Ψ2Rbe a set-valued operator. A selection of Fis the function f:ΨR{±}such that f(t)F(t)a.e. tΨ. We write to denote the set containing all integrable selections of F. According to Aumann [2], the set-valued integral is determined by the operator Jin the following:

JΨF(t)dt:={Ψf(t)dt;f}

that is, the set of the integrals of integrable selections of F.

On the other hand, in elementary calculus, one deals with derivatives and integrals, including the higher-integer-order iterations. Here, in fractional integral, one looks at a broader concept where the real-order iteration is taken into account. There are many approaches to study this kind of extensions. In our context, we shall use the classical notion introduced by Riemann and Liouville, the latter of which is the first one to point out the possibility of fractional calculus in 1832. Given a function fL1(Ψ,μ), the fractional integral of order α>0is given by

IΨαf(t)dt:=1Γ(α)Ψ(tτ)α1f(τ)dτ

Naturally, we may further consider the following fractional integral:

JΨαF(t)dt:={IΨαf(t)dt;f}

These two concepts have brought up the studies of new systems, the set-valued dynamical systems and the fractional dynamical systems. Even the combination of the two, the fractional set-valued dynamical systems, is an emerging area in research. We shall be particular with this latter class of systems and give some brief investigations over the problem.

The very concept of set-valued fractional integral operator was first proposed by El-Sayed and Ibrahim [35] and this has opened a new universe of investigation to fractional operator equations. It has been reflected that such theory can better describe nonlinear phenomena, compared to the classical theory of differential and integral equations. The extensive use of this theory lays naturally in automatic control theory, network theory and dynamical systems (see, e.g. [610]).

The central system that we are going to investigate in this chapter is the following delayed system:

u(t)i=1nβi(t)u(tτi)IαF(t,u(t));α(0,1],tJ:=[0,T],T>0E2

where τi[0,t]for all i{1,2,,n}, F:J×RCB(R), IαF(t,u(t))is the definite integral of order αgiven by

IαF(t,u(t)):={1Γ(α)0t(tτ)α1f(τ,u(τ))dτ;fSF(u)}

and

SF(u):={fL1(J,R);f(t)F(t,u(t))a.e.tJ}

denotes the set of selections of Fand βi:JRis continuous for each i{1,2,,n}. Also, set B:=max1insuptJβi(t).

In this chapter, we shall bring up some recent results in fixed point theory in several approaches and then show how these theorems apply to different classes of dynamical systems. Going precise, in Section 2, we investigate the system (2) in standard metric spaces through a newly developed fixed point theorem. The mentioned fixed point theorem deals with an operator that satisfied the so-called implicit contractivity condition only on a portion of a space, where such partial partition is obtained from the cyclicity behavior that we imposed. We also note the relation between this cyclicity behavior and the one that arises from the partial ordering relation approach. The solvability of the dynamical system (2) in this section is naturally obtained via the cyclicity and implicit contractivity assumptions. For further readings related to this topic, consult [1117]. In Section 3, we consider a newly emerged approach of studying fixed point theory, i.e., fixed point theory in modular metric spaces. This theory has only been introduced to researchers only a few years ago and has been investigated reasonably in such a short duration. We bring up one of the fundamental fixed point theorem in this modular metric spaces, give appropriate examples and then apply it to guarantee the solvability of, again, the system (2). Even the studies of modular metric spaces are relatively limited at the time, we suggest that further readings from Refs. [1820] should give some ideas about the theory itself and also how to develop further dynamical systems in this framework.

2. Cyclic operators in metric spaces

In this section, we consider a very general class of operators that satisfy the implicit contractivity condition. Moreover, we also assume the operator to be cyclic over its domain. This cyclicity weakens the contractivity only to a portion of the space. This is a more general case than the contractivity on comparable pairs, as we show later in this chapter. This also allows the coexistence result that is better than the exact solution and the sub-/super-solution.

Note that results in this section are based on our paper [21]. Recall the following notion of cyclic operators.

Definition 2.1. Let Xbe a nonempty set and A1,A2,,Apbe nonempty subsets of X. An operator F:k=1pAk2k=1pAkis called a phset-valued cyclic operator over k=1pAkif F(Ai)Ai+1for all i{1,2,,p1}and F(Ap)A1.

There is a special property about the location of fixed point of this operator, as illustrated in the following.

Proposition 2.2. Let X be a nonempty set and A1,A2,⋯,Ap be nonempty subsets of X. If F is a set-valued cyclic operator over ∪k=1pAk, then we have the inclusion Fix(F)⊆∩k=1pAk, where Fix(F) denotes the fixed point set of F.

Proof. If either Fix(F)=ptysetor k=1pAk=ptyset, the conclusion is clear. Thus, let zk=1pAkbe a fixed point of F. Then, zAqfor some q{1,2,,p}and zFzAq+1. Consequently, we also have zFzAq+2. It is easy to see that zAq+nfor all nN. Therefore, it is enough to conclude that zk=1pAk.

The following classes of functions are necessary to our further contents.

Definition 2.3. Let Φbe the class of functions ϕ:R+R+satisfying the following conditions:

(Φ1) ϕ is right continuous.

(Φ2) ϕ (0) = 0.

(Φ3) ϕ(t) < t for all t>0.

Definition 2.4. Let Ψbe the class of functions ψ:R+6Rsatisfying the following conditions:

  1. (Ψ1)ψis continuous.

  2. (Ψ2)ψis nondecreasing in the first variable and is nonincreasing in the remaining variables.

  3. (Ψ3)There exists a function ϕΦsuch that, for all u,v0, either ψ(u,v,u,v,0,u+v)0orψ(u,v,0,0,u,v)0implies that uϕ(v).

  4. (Ψ4)ψ(u,0,u,0,0,u),ψ(u,u,0,0,u,u)>0for all u>0.

Remark 2.5. If ϕΦ, then ϕn(t)0.

Example 2.6 ([22]). The following functions are contained in the class Ψ:

  1. ψ1(t1,t2,,t6):=t1αmax{t2,t3,t4}(1α)[at5+bt6], where α[0,1)and a,b[0,12).

  2. ψ2(t1,t2,,t6):=t1ϕ(max{t2,t3,t4,12[t5+t6]}), where ϕΦ.

  3. ψ3(t1,t2,,t6):=t12t1(αt2+βt3γt4)δt5t6, where α>0and β,γ,δ0with α+β+γ<1and α+δ<1.

2.1. Fixed point theorem for cyclic operators

Now, we give the main fixed point theorem for cyclic implicit contractive operators.

Theorem 2.7. Let (X,d) be a complete metric space and let A1,A2,…,Ap be nonempty closed subsets of X. Suppose that F is a proximal set-valued cyclic operator over ∪k=1pAk in which there exists some ψ∈Ψ satisfying

ψ(H(Fx,Fy),d(x,y),d(x,Fx),d(y,Fy),d(x,Fy),d(y,Fx))0

whenever either (x,y)Ai×Ai+1or (x,y)Ai+1×Aiholds for some i{1,2,,p}. Then, we have the following:

  1. Fhas at least one fixed point;

  2. Fhas no fixed point outside k=1pAk.

Proof. For (I), let x0be chosen arbitrarily from some Aj. Choose any x1Fx0. Then, we define implicitly a sequence (xn)by choosing xn+1Fxnsatisfying

d(xn,xn+1)=d(xn,Fxn).

Note that this definition is valid since Fis a proximal operator. Also note that by this definition, we may derive that

d(xn,xn+1)H(Fxn1,Fxn)E3

Now, since (xn+1,xn)Aj+n+1×Aj+n, we have

0ψ(H(Fxn+1,Fxn),d(xn+1,xn),d(xn+1,Fxn+1),d(xn,Fxn),d(xn+1,Fxn),d(xn,Fxn+1))ψ(H(Fxn,Fxn+1),d(xn,xn+1),H(Fxn,Fxn+1),d(xn,xn+1),0,d(xn,xn+1)+H(Fxn,Fxn+1))

Suppose that ϕΦis chosen according to (Ψ3). Thus, we have

H(Fxn,Fxn+1)ϕ(d(xn,xn+1))

At this point, we assume that xnxn+1for all nN, otherwise a fixed point is already obtained. Together with Eq. (3), we may deduce that

d(xn,xn+1)H(Fxn1,Fxn)ϕ(d(xn1,xn))ϕn1(d(x0,x1))

Therefore, we have immediately that d(xn,xn+1)0.

Next, we show that (xn)is Cauchy. Suppose to the contrary. So, we may find ε0>0and two strictly increasing sequences of integers (mk)and (nk)in which

d(xmk,xnk)ε0

We can assume, without loss of generality, that nk>mk>kand nkis minimal in the sense that d(xmk,xr)<ε0for all mkr<nk.

Consequently, d(xmk,xnk1)<ε0. Moreover, we may obtain that ε0d(xmk,xnk)d(xmk,xnk1)+d(xnk1,xnk)<ε0+d(xnk1,xnk).Letting k, we have d(xmk,xnk)ε0.

On the other hand, for each kN, we may find jk{1,2,,p}in which nkmk+jk1(modp). For ksufficiently large, we may see that mkjk>0. Observe that

|d(xmkjk,xnk)d(xnk,xmk)|d(xmkjk,xmk)l=0jk1d(xmkjk+l,xmkjk+l+1)l=0p1d(xmkjk+l,xmkjk+l+1)

Letting k, we have d(xmkjk,xnk)ε0. Also consider that

|d(xnk,xmkjk)d(xmkjk,xnk+1)|d(xnk,xnk+1).

As k, we have d(xmkjk,xnk+1)ε0. Similarly, we have

|d(xmkjk,xnk)d(xnk,xmkjk+1)|d(xmkjk,xmkjk+1).

So, we get d(xnk,xmkjk+1)ε0as k. Also observe that

|d(xnk,xnk+1)d(xnk+1,xmkjk+1)|d(xnk,xmkjk+1).

Again, letting k, we obtain that d(xnk+1,xmkjk+1)ε0. Finally, by the fact that (xmkjk,xnk)Ai×Ai+1for some i{1,2,,p}and Eq. (3), we may obtain that

0ψ(H(Fxmkjk,Fxnk),d(xmkjk,xnk),d(xmkjk,Fxmkjk),d(xnk,Fxnk),d(xmkjk,Fxnk),d(xnk,Fxmkjk))ψ(d(xmkjk+1,xnk+1),d(xmkjk,xnk),d(xmkjk,xmkjk+1),d(xnk,xnk+1),d(xmkjk,xnk+1),d(xnk,xmkjk)+d(xmkjk,Fxmkjk))=ψ(d(xmkjk+1,xnk+1),d(xmkjk,xnk),d(xmkjk,xmkjk+1),d(xnk,xnk+1),d(xmkjk,xnk+1),d(xnk,xmkjk)+d(xmkjk,xmkjk+1))

By the condition (Ψ4)and letting k, we may deduce that

0ψ(ε0,ε0,0,0,ε0,ε0)>0

which is absurd. Hence, the sequence (xn)is Cauchy. Since k=1pAkis closed, it is complete and therefore (xn)converges to some unique point xk=1pAk.

Next, we shall prove that xis, in fact, a fixed point of F. Let us assume now that d(x,Fx)>0. Note that for any nN, (x,xn)Ai×Ai+1for some i{1,2,,p}. So, it is followed that

0ψ(H(Fx,Fxn),d(x,xn),d(x,Fx),d(xn,Fxn),d(x,Fxn),d(xn,Fx))ψ(d(xn+1,Fx),d(x,xn),d(x,Fx),d(xn,xn+1),d(x,xn)+d(xn,Fxn),d(xn,Fx))=ψ(d(xn+1,Fx),d(x,xn),d(x,Fx),d(xn,xn+1),d(x,xn)+d(xn,xn+1),d(xn,Fx))

Passing to the limit as n, we obtain that

0ψ(d(x,Fx),0,d(x,Fx),0,0,d(x,Fx))>0

which is absurd. Therefore, d(x,Fx)=0. Since Fxis closed, we conclude that xFx.

To obtain (II), apply Proposition 2.2.

2.2. Ordered spaces as corollaries

Let Xbe a nonempty set, recall that the binary relation Ôis said to be a ph(partial) ordering on Xif it is reflexive, antisymmetric and transitive. By an phordered set, we shall mean the pair (X,)where Xis nonempty and is an ordering on X. A ph(partially) ordered metric space is the triple (X,,d), where (X,)is an ordered set and (X,d)is a metric space.

In this part, we show that contractivity on comparable pairs is particularly a cyclic operator over a single set. The following general assumption on the ordered structure is central in the few forthcoming theorems.

Definition 2.8. Let (X,,d)is said to satisfies the phcondition (Θ)if every convergent sequence (xn)in Xand every point z0Xsuch that z0xnfor all nN, there holds the property z0x, where xXis the limit of (xn).

Theorem 2.9. Let (X,⊑,d) be a complete ordered metric space satisfying the condition (Θ) and let F:X→CB(X) be a nondecreasing proximal operator in the sense that if x,y∈X satisfies x⊑y, then u⊑v for all u∈Fx and v∈Fy. Suppose that there exists ψ∈Ψ such that

ψ(H(Fx,Fy),d(x,y),d(x,Fx),d(y,Fy),d(x,Fy),d(y,Fx))0E4

for all x,yXin which we can find some zXsatisfying both zxand zy. If there exists x0Xsuch that x0wfor all wFx0, then Fhas at least one fixed point.

Proof. By the existence of such a point x0, we shall now construct a set

C(x0):={zX;x0z}

Taking any sequence (xn)in C(x0). By the condition (Θ)with z0:=x0, we may see that if (xn)converges, its limit is also included in C(x0). Hence, C(x0)is closed and therefore it is complete.

On the other hand, we define an operator G:C(x0)CB(X)by

G:=F|C(x0).

For any zC(x0), observe that x0wfor all wGz. Thus, G(C(x0))C(x0)so that Gis cyclic over C(x0). Moreover, for any x,yC(x0), we have by definition that x0xand x0y, so that the inequality (4) holds whenever (x,y)C(x0)×C(x0). Therefore, we can now apply Theorem 2.7 to obtain that Ghas at least one fixed point. Passing this property to F, we have now proved the theorem.

Corollary 2.10. Let (X,⊑,d) be a complete ordered metric space and let F:X→CB(X) be a nondecreasing proximal operator in the sense that if x,y∈X satisfies x⊑y, then u⊑v for all u∈Fx and v∈Fy. Suppose that there exists ψ∈Ψ such that

ψ(H(Fx,Fy),d(x,y),d(x,Fx),d(y,Fy),d(x,Fy),d(y,Fx))0

whenever x,yXsatisfy xy. Also assume that if the sequence (xn)in Xis nondecreasing and converges to xX, then xnxfor all nN. If there exists x0Xsuch that x0⊑w for all wFx0, then Fhas at least one fixed point.

Proof. Note that if x,yXare comparable, then, according to Theorem 2.9, we may choose z:=xXso that zxand zy.

On the other hand, let (yn)be a sequence in Xwhich is both nondecreasing and convergent to yX. According to the condition (Θ), set z0:=y1. We may see easily that, in this case, Xsatisfies the condition (Θ). We next apply Theorem 2.9 to finish the proof.

2.3. An example

We now give a validating example for our fixed point theorem to help the understanding of the content.

Example 2.11. Consider the Euclidean space E2with its standard metric d. For each tR, we define

0:=[0,12]×{0},1:=[0,12]×{12},and2:=[0,12]×{12}.

Suppose that A1and A2are two closed sets defined by

A1:=01andA2:=02.

Let F:A1A22A1A2be an operator defined by

Fx:={{x},ifx0,P11(x)A2,ifx1,P21(x)A1,ifx2.E5

Note that the notation Pas is appeared in Eq. (5) is the metric projection onto the corresponding sets 1and 2, respectively. The cyclicity of Fis apparent.

Claim. The operator Fsatisfies the inequality in Theorem 2.7 with ψdefined as in (c) of Example 2.6 when α=920, β=γ=14and δ=12.

The case x,y0is trivial and so we omit it. For the case x0as y1and x1as y2, we consider the following calculation.

From Table 1(a), we have

[H(Fx,Fy)]2=(x1y1)2+12(920+142+12)((x1y1)2+12)920((x1y1)2+12)+142(x1y1)2+12+12((x1y1)2+12)=(x1y1)2+12(920(x1y1)2+12+142)+12((x1y1)2+12)=H(Fx,Fy)[αd(x,y)+βd(x,Fx)+γd(y,Fy)]+δd(x,Fy)d(y,Fx)
(A) xl0as yl1
H(Fx,Fy)(x1y1)2+1/2
d(x,y)(x1y1)2+1/2
d(x,Tx)0
d(y,Ty)1/2
d(x,Ty)(x1y1)2+1/2
d(y,Tx)(x1y1)2+1/2
(B) xl1as yl2
H(Fx,Fy)(x1y1)2+1/2
d(x,y)(x1y1)2+2
d(x,Tx)1
d(y,Ty)1
d(x,Ty)|x1y1|
d(y,Tx)|x1y1|

Table 1.

Distances.

for all x0and y1. We can similarly obtain from Table 1(b) the following:

[H(Fx,Fy)]2=(x1y1)2+12(92052+2)((x1y1)2+12)(92052)((x1y1)2+12)+2(x1y1)2+1292052((x1y1)2+12)2+2(x1y1)2+12=920((x1y1)2+12)2+32((x1y1)2+12)2+2(x1y1)2+12920((x1y1)2+12)2+32((x1y1)2+12)+2(x1y1)2+12=920((x1y1)2+12)((x1y1)2+12+32)+2(x1y1)2+12=920((x1y1)2+12)((x1y1)2+2)+2(x1y1)2+12=(x1y1)2+12(920(x1y1)2+2+12+12)=H(Fx,Fy)[αd(x,y)+βd(x,Fx)+γd(y,Fy)]H(Fx,Fy)[αd(x,y)+βd(x,Fx)+γd(y,Fy)]+δd(x,Fy)d(y,Fx)

for all x1and y2. Therefore, we have now proved our claim.

Observe now that Fix(F)=0=A1A2, coincide with the Theorem 2.7.

2.4. Fractional set-valued dynamical systems

For convenience, we shall always consider the nonempty closed and bounded subspace

ΩC(J,R):={u:JR;uiscontinuous},

endowed with the supremum norm ||||given by

||u||:=suptJ|u(t)|.

The solutions for the problem (2) are assumed to be in Ωunder this circumstance. Moreover, we shall need some more notions in order to obtain the existence of solutions for the problem (2).

Definition 2.12. Let (X, d) be a metric space and let Jbe an interval of R. An operator F:J2Xis said to be measurable if for each xXand tJ, the mapping xd(x,F(t))is measurable.

Next, we shall define the set-valued operator Λ:Ω2Ωgiven by

(Λu)(t):={wΩ;w(t)=i=1nβi(t)u(tτi)+Uαf(t,u(t)),fSF(u)},E6

where Uis the ordinary single-valued fractional integral.

We shall next illustrate that the operator Λpossesses closed values.

Lemma 2.13. Suppose that the operator Λ is given as in (2.4), then Λu is closed for all u∈Ω.

Proof. Let uΩand let (uk)be a sequence in Λuwhich converges to some uΩ. We shall prove the statement by showing that limits of convergent sequence in Λuare in Λu. Then, there exists a sequence (fk)in SF(u)in which

uk(t)=i=1nβi(t)u(tτi)+Uαfk(t,u(t)).

Also note that this sequence (fk)converges to some fL1(J,R). Since F(t,u(t))is closed, fSF(u). Actually, we have

u(t)=i=1nβi(t)u(tτi)+Uαf(t,u(t))Λu.

This completes the proof.

Now, we give the solvability of the system (2).

Theorem 2.14. According to Eq. (2), assume that there exist non-empty closed subsets Π1,Π2,⋯,Πp in Ω such that ∪k=1pΠk=Ω and F has the following properties:

  1. tF(t,u(t))is measurable for each uΩ;

  2. there exists a function ξ:R+5R+such that

    H(F(t,u(t)),F(t,v(t)))ξ(uv,d(u,Λu),d(v,Λv),d(u,Λv),d(v,Λu))whenever either(u,v)Πi×Πi+1or (u,v)Πi+1×Πiholds for some i{1,2,,p};

  3. Λis proximal and cyclic over k=1pΠk=Ω.

If the function ψ:R+6R+given by

ψ(t1,t2,,t6):=t1nBt2TαΓ(α+1)ξ(t2,t3,t4,t5,t6)

is in the class Ψ, then the problem (1.2) has at least one solution.

Proof. Let (u,v)Πi×Πi+1for some i{1,2,,p}. By 2, we may choose some f1(t,u(t))F(t,u(t))and f2(t,v(t))F(t,v(t))in which

|f1(t,u(t))f2(t,v(t))|ξ(uv,d(u,Λu),d(v,Λv),d(u,Λv),d(v,Λu))

Consider the two functions

w1(t)=i=1nβi(t)u(tτi)+Uαf1(t,u(t))Λu

and

w2(t)=i=1nβi(t)v(tτi)+Uαf2(t,v(t))Λv.

Next, observe that

|w1(t)w2(t)|i=1nβi(t)|u(tτi)v(tτi)|+|Uαf1(t,u(t))Uαf2(t,v(t))|i=1nβi(t)|u(tτi)v(tτi)|+Uα|f1(t,u(t))f2(t,v(t))|nBuv+TαΓ(α+1)|f1(t,u(t))f2(t,v(t))|nBuv+TαΓ(α+1)ξ(uv,d(u,Λu),d(v,Λv),d(u,Λv),d(v,Λu))

It follows that

H(Λu,Λv)nBuv+TαΓ(α+1)ξ(uv,d(u,Λu),d(v,Λv),d(u,Λv),d(v,Λu)).

Consequently, we have for each (u,v)Πi×Πi+1, i{1,2,,p}, that

ψ(H(Λu,Λv),uv,d(u,Λu),d(v,Λv),d(u,Λv),d(v,Λu))0.

We may deduce similarly that the above inequality holds also in the case (u,v)Πi+1×Πi. Apply Theorem 2.7 to obtain the desired result.

We next consider the existence of solutions to Eq. (2) in the case when an ordering is defined on Ωin such a way that for u,vΩ,

uvu(t)v(t)a.e.tJ

It is easy to see that if (un)is a nondecreasing sequence in Ωwhich converges to some uΩ, then unufor all nN. In the further step, we shall need in the initial state that a weak solution to Eq. (2) exists.

Definition 2.15. Suppose that (Ω,)is a partially ordered set. A phweak solution for the problem (2) (w.r.t. ) is a function uΩsuch that uvfor all vΛu.

Corollary 2.16. According to Eq. (2), assume that there is an ordering ⊑ defined on Ω. Suppose also that we have the following properties:

  1. tF(t,u(t))is measurable for each uΩ;

  2. there exists a function ξ:R+5R+such that

    H(F(t,u(t)),F(t,v(t)))ξ(uv,d(u,Λu),d(v,Λv),d(u,Λv),d(v,Λu))whenever u,vΩare comparable;

  3. Λis proximal and nondecreasing;

  4. a weak solution u0Ωto the problem (2) exists.

If the function ψ:R+6R+given by

ψ(t1,t2,,t6):=t1nBt2TαΓ(α+1)ξ(t2,t3,t4,t5,t6)

is in the class Ψ, then the problem (2) has at least one solution.

Proof. As in the proof of the previous theorem, we may similarly derive that

ψ(H(Λu,Λv),uv,d(u,Λu),d(v,Λv),d(u,Λv),d(v,Λu))0

whenever u,vΩare comparable. Therefore, we may apply Corollary 2.10 to obtain the desired result.

3. Fractional set-valued systems in modular metric spaces

In this section, we shall consider on fixed point inclusions that are studied within a modular metric spaces. With certain conditions, we can extend Nadler’s theorem to the context of modular metric spaces successfully. A modular metric space is a relatively new concept. It generalizes and unifies both modular and metric spaces. It is therefore not necessarily equipped with a linear structure.

Before we go further, let us first give basic definitions and related properties of a modular metric space.

Definition3.1. ([23]). Let Xbe a nonempty set. A function w:(0,)×X×X[0,+]is said to be a phmetric modular on Xif the following conditions are satisfied for any s,t>0and x,y,zX:

  1. x=yif and only if wt(x,y)=0for all t>0.

  2. wt(x,y)=wt(y,x).

  3. ws+t(x,y)ws(x,z)+wt(z,y).

Here, we use wt(,):=w(t,,). In this case, we say that (X,w)is a phmodular metric space. Notice that the value of a metric modular can be infinite.

Since we are focusing on the generalized metric space approach, we shall not be discussing about modular space theory here. Suppose that (X,d)is a metric space, then wt(,):=d(,)is a metric modular on X.

Now, we turn to basic definitions we need in this particular space. We start by giving the topology of the space.

Let (X,w)be a modular metric space. By defining an open ball with Bw(x;r):={zX; supt>0wt(x,z)<r}, we can define a Hausdorff topology on Xhaving the collection of all such open balls as a base. The convergence in this topology can therefore be written by:

(xn)x¯supt>0wt(xn,x¯)0,

where (xn)Xand x¯X. With this characterization, we now have a good hint to define the Cauchy sequence. A sequence (xn)Xis said to be phCauchy if for any given ε>0, there exists nNsuch that

supt>0wt(xm,xn)<ε

whenever m,n>n. Naturally, Xis said to be phcomplete if Cauchy sequences in Xconverges.

We next give another route of investigation of fixed point inclusion in modular metric spaces. This time, we shall apply more on analytical assumptions. Briefly said, we shall use the contractivity assumptions.

Before we could stomp into the main exploration, we need the following knowledge of metric modular of sets.

We write C(X)to denote the set of all nonempty closed subsets of X. For any subset AXwand point xX, we denote wt(x,A):=infyAwt(x,y).

Given two subsets A,BC(X), define wt(A,B):=supxAwt(x,B). Most importantly, the Hausdorff-Pompieu metric modular Wt(A,B):=max{wt(A,B),wt(B,A)}.

Lemma 3.2. Let (X,w) be a modular metric space, A∈C(X) and x∈X. Then,

wt(x,A)=0forallt>0xA.

Definition 3.3. Given a modular metric space (X,w)and an arbitrary point xX. A subset YXis said to be phreachable from xif

infyYsupt>0wt(x,y)=supt>0wt(x,Y)<.

This lemma gives a simple criterion of when the reachability holds.

Lemma 3.4. Let (X,w) be a modular metric space with w being l.s.c., Y⊂X a nonempty compact subset. For a point x∈X, if either infy∈Ysupt>0wt(x,y)<∞ or supt>0wt(x,Y)<∞, then Y is reachable from x.

The following lemma is essential in showing the solvability of fixed point inclusion for contractivity condition.

Lemma 3.5. Suppose that Y,Z∈C(X) are nonempty and z∈Z. If Y is reachable from z, then for each ε>0, there exists a point yε∈Y such that supt>0wt(z,yε)≤supt>0Wt(X,Y)+ε.

3.1. Fixed point inclusion in modular metric spaces

Now, we state the notion of the contraction and the Kannan’s contraction. Make note that these two concepts are not generalizations of one another.

Definition 3.6. Let (X,w)be a modular metric space. A set-valued operator F:XXis said to be a phcontraction if there exists a constant k[0,1)such that

Wt(Fx,Fy)kwt(x,y),E7

for all t>0and x,yX.

If kis restricted in [0,12)and Eq. (7) is replaced with the following inequality:

Wt(F(x),F(y))k[wt(x,F(x))+wt(y,F(y))].

Then, we call Fa phKannan’s contraction

Now, we present the main existence theorems.

Theorem 3.7. Let (X,w) be a complete modular metric space with w being l.s.c. and F a contraction on X having compact values with contraction constant k. Suppose that there exists a pair of points x0∈X and x1∈F(x0) with the following properties:

(A) the set {x0,x1}is bounded,

(B) F(x1) is reachable from x1.

Then, Fhas at least one fixed point.

Proof. Since F(x1)is reachable from x1, by using Lemma 3.5, we may choose x2F(x1)such that

supt>0wt(x1,x2)supt>0wt(F(x0),F(x1))+k.

From the above evidence and the hypothesis that {x0,x1}is bounded, it comes to the following inequalities:

supw>0wt(x2,F(x2))supt>0wt(F(x1),F(x2))ksupt>0wt(x1,x2)k[supt>0Wt(F(x0),F(x1))+k]k2supt>0wt(x0,x1)+k2<.

By the assumptions, we apply Lemma 3.4 to guarantee that F(x2)is actually reachable from x2.

Inductively, by this procedure, we define a sequence (xn)in X, with the supplement from Lemma 3.5, satisfying the following properties for all nN:

{xnF(xn1),supt>0wt(xn,xn+1)supt>0Wt(F(xn1),F(xn))+kn,F(xn)isreachablefromxn.

Hence, by the contractivity of F, we have

supt>0wt(xn,xn+1)supt>0Wt(F(xn1),F(xn))+knksupt>0wt(xn1,xn)+knk[ksupt>0wt(xn2,xn1)+kn1]+knk2supt>0wt(xn2,xn1)+2kn.

Thus, by induction, we have

supt>0wt(xn,xn+1)knsupt>0wt(x0,x1)+nkn.

Moreover, it follows that

supt>0nNwt(xn,xn+1)supt>0wt(x0,x1)nNkn+nNnkn<.

Without loss of generality, suppose m,nNand m>n. Observe that

supt>0wt(xn,xm)supt>0[wtmn(xn,xn+1)++wtmn(xm1,xm)]supt>0wt(xn,xn+1)++supt>0wt(xm1,xm)n=nsupt>0wt(xn,xn+1)<ε,

for all m>nnfor some nN. Hence, (xn)is a Cauchy sequence so that the completeness of Xwimplies that (xn)converges to some point xXw. Consequently, we may conclude from the contractivity of Fthat the sequence (F(xn))converges to F(x). Since xnF(xn1), we have for any t>0,

0wt(x,F(x))wt2(x,xn)+Wt2(F(xn1),F(x)),

which implies that wt(x,F(x))=0for all t>0. Since F(x)is closed, it then follows from Lemma 3.2 that xF(x).

Example 3.8. Suppose that X=[0,1]and w:(0,+)×X×X[0,+]is defined by

wt(x,y)=1(1+t)|xy|.

Clearly, wis an l.s.c. metric modular on X. Notice that any two-point subset is bounded. Now, we define a set-valued operator F:XXby

F(x):=[x+12,1]

for every xX.

Observe that Fhas compact values on X. Note that for each t>0and x,yX, we have

Wt(Fx,Fy)=12(1+t)|xy|12wt(x,y).

Therefore, Fis a contraction with contraction constant k=12. Moreover, it is easy to see that the conditions (A) and (B) hold. Finally, we have that 1is a fixed point of F(and it is unique).

Next, we shall show that the fixed point in the above theorem needs not be unique, as we shall see in the following example:

Example 3.9. Suppose that Xis defined as in the previous example. Consider the operator G:XXgiven by

G(x):=[0,x+12],

for each xX.

Note that this operator Gis also a contraction with constant k=12and takes compact values on X. Also, the conditions (A) and (B) hold. However, every point in Xis a fixed point of G. This shows the nonuniqueness of fixed points for a set-valued contraction.

Theorem 3.10. Replacing F in Theorem 3.7 with a Kannan’s contraction yields the same existence result.

Proof. Since F(x1)is reachable from x1, by using Lemma 3.5, we may choose x2F(x1)such that

supt>0wt(x1,x2)supt>0Wt(F(x0),F(x1))+k.

Now, observe that

supt>0wt(x2,F(x2))supt>0Wt(F(x1),F(x2))ksupt>0wt(x1,F(x1))+ksupt>0wt(x2,F(x2))ksupt>0Wt(F(x0),F(x1))+ksupt>0wt(x2,F(x2))ksupt>0wt(x0,F(x0))+ksupt>0wt(x1,F(x1))+ksupt>0wt(x2,F(x2))ksupt>0wt(x0,x1)+ksupt>0wt(x1,F(x1))+ksupt>0wt(x2,F(x2)).

Writing ξ:=k1k<1, we obtain, from the boundedness of {x0,x1}and the reachability of F(x1)from x1, that

supt>0wt(x2,F(x2))ξsupt>0wt(x0,x1)+ξsupt>0wt(x1,F(x1))<.

Thus, from the assumptions and Lemma 3.5, we may see that F(x2)is reachable from x2.

Inductively, we can construct a sequence (xn)in Xwith exactly the same properties appearing in the proof of Theorem 3.7.

Now, consider further that

supt>0wt(xn,xn+1)supt>0Wt(F(xn1),F(xn))+knksupt>0wt(xn1,F(xn1))+ksupt>0wt(xn,F(xn))+knksupt>0wt(xn1,F(xn1))+ksupt>0wt(xn,xn+1)+kn.

Moreover, we get

supt>0wt(xn,xn+1)ξsupt>0wt(xn1,xn)+kn1kξ2supt>0wt(xn2,xn1)+kn(1k)2+kn(1k)ξ2supt>0wt(xn2,xn1)+2kn(1k)2ξnsupt>0wt(x0,x1)+nξn.

As in the proof of Theorem 3.7, the sequence (xn)converges to some xX. Observe now that

supt>0wt(x,F(x))=supt>0wt({x},F(x))supt>0wt({x},F(xn))+supt>0wt(F(xn),F(x))=supt>0wt(x,F(xn))+supt>0wt(F(xn),F(x))supt>0wt(x,xn+1)+supt>0Wt(F(xn),F(x))supt>0wt(x,xn+1)+ksupt>0wt(xn,F(xn))+ksupt>0wt(x,F(x))=(1+k)supt>0wt(x,xn+1)+ksupt>0wt(x,F(x)).

Thus, we have

supt>0wt(x,F(x))1+k1ksupt>0wt(x,xn+1).

Letting nto conclude the theorem.

3.2. Fractional integral inclusion

In this particular subsection, we shall use notations a bit differently than those of earlier sections. This is due to conventional uses of variables and functions that is common to integral and differential equations.

Suppose that Ψis the interval mentioned in the previous section. Let us assume throughout the section that the real line Ris equipped with the metric modular

ωλR(x,y):=11+λ|xy|,

for λ>0and x,yR. Thus, for the space C(Ψ)of all continuous (in ωR-topology) real-valued functions on Ψ, we shall use the metric modular

ωλC(Ψ)(ϕ,ψ):=suptΨωλR(ϕ(t),ψ(t)),

for λ>0and ϕ,ψC(Ψ). Note that both ωRand ωC(Ψ)satisfy the Fatou’s property. Also note that the set Ris second countable, i.e., it has a countable base, w.r.t. ωR-topology. Moreover, it is clear that the set {ϕ,ψ}is bounded w.r.t. ωC(Ψ), for any ϕ,ψC(Ψ). Suppose that F:Ψ×R2Ris a set-valued operator with nonempty compact values and uC(Ψ). We shall use the following notation to explain the collection of integrable selections:

SF(u):={fL1(Ψ,μ);f(t)F(t,u(t))a.e.tΨ}.

It is clear that SF(u)is closed. Next, for each i{0,1,,N}, NN, assume that βi:ΨRis continuous and τi:ΨR+is a function with τi(t)t. We write B:=max0iNsuptΨβi(t). The main aim of this section is to consider the fractional integral inclusion:

                                        u(t)i=0Nβi(t)u(tτi(t))JΨαF(t,u(t))dt,α(0,1].                                       (FII)

In the above inclusion, the summation here is interpreted to be the delay term.

We shall define a set-valued operator Λ:C(Ψ)2C(Ψ)by

Λ(u):={wC(Ψ);w(t)=i=0Nβi(t)u(tτi(t))+IΨαf(t,u(t))dt,fSF(u)}.

Note here that for any ϕC(Ψ), we have Λ(ϕ)is reachable from ϕw.r.t. ωC(Ψ). To restrict the operator Λwith some nice property, we assume that SF(u)is nonempty.

Lemma 3.11. The operator Λ given above is compact valued if SF(u) is nonempty.

Proof. For the proof, we shall show the compactness by its sequential characterization. Suppose that uC(Ψ)and (wn)is an arbitrary sequence in Λ(u). By definition, there corresponds a convergent sequence (fn)in SF(u)F(,u())satisfying

wn(t)=i=0Nβi(t)u(tτi(t))+IΨαfn(t,u(t))dt.

The conclusion is then followed.

Now, we shall state now the solvability result for the problem (FII). It is clear that uC(Ψ)solves Eq. (FII) if and only if uis a fixed point of Λ.

Theorem 3.12. Suppose that F defined above is compact-valued and SF(u) is nonempty. Assume further that

(F1) for any given u,vC(Ψ)and a selection fSF(u)of F, there corresponds a function f′∈SF(v) such that

{ωλR(f(t,u(t)),f(t,v(t)))=ωλR(f1(t,u(t)),F(t,v(t))),ωλR(f(t,u(t)),f(t,v(t)))LωλC(Ψ)(u,v),

for all tΨ;

(F2) (N+1)BΓ(α)+LTαΓ(α)<1.

Then, Λhas a fixed point.

Proof. For each u,vC(Ψ), we may choose, from the assumption, functions f1,f2such that

{f1SF(u),f2SF(v),ωλR(f1(t,u(t)),f2(t,v(t)))=ωλR(f1(t,u(t)),F(t,v(t))),ωλR(f1(t,u(t)),f2(t,v(t)))LωλC(Ψ)(u,v),

for each tΨ. Consider the two functions w1Λ(u)and w2Λ(v), respectively as follows:

{w1(t):=i=0Nβi(t)u(tτi(t))+IΨαf1(t,u(t))dt,w2(t):=i=0Nβi(t)v(tτi(t))+IΨαf2(t,v(t))dt.

Now, consider the following computation:

ωλR(w1(t),w2(t))i=0Nβi(t)ωλR(u(tτi(t)),v(tτi(t))+ωλC(Ψ)(IΨαf1(t,u(t))dt,IΨαf2(t,u(t))dt)(N+1)BωλC(Ψ)(u,v)+IΨαωλR(f1(t,u(t)),f2(t,v(t)))(N+1)BωλC(Ψ)(u,v)+LTαΓ(α)ωλC(Ψ)(u,v)=[(N+1)BΓ(α)+LTαΓ(α)]ωλC(Ψ)(u,v).

It follows that

ΩλC(Ψ)(Λ(u),Λ(v))[(N+1)BΓ(α)+LTαΓ(α)]ωλC(Ψ)(u,v).

The proof ends here by applying Theorem 3.7.

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Parin Chaipunya and Poom Kumam (March 15th 2017). Recent Fixed Point Techniques in Fractional Set-Valued Dynamical Systems, Dynamical Systems - Analytical and Computational Techniques, Mahmut Reyhanoglu, IntechOpen, DOI: 10.5772/67069. Available from:

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