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Mathematics » "Dynamical Systems - Analytical and Computational Techniques", book edited by Mahmut Reyhanoglu, ISBN 978-953-51-3016-1, Print ISBN 978-953-51-3015-4, Published: March 15, 2017 under CC BY 3.0 license. © The Author(s).

Chapter 7

Recent Fixed Point Techniques in Fractional Set-Valued Dynamical Systems

Poom Kumam
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By Parin Chaipunya and Poom Kumam
DOI: 10.5772/67069

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Recent Fixed Point Techniques in Fractional Set-Valued Dynamical Systems

Parin Chaipunya and Poom Kumam
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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 C1 initial-valued problem:


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


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 X is a complete metric space and T:XX is Lipschitz continuous with constant 0<L<1, then T has a unique fixed point.

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


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 f must be Lipschitz in the second variable with constant L>0, we have for any x,yC1([0,T]) the following:


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 Ψ is an interval [0,T], where T>0. Let F:Ψ2R be a set-valued operator. A selection of F is 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 J in the following:


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>0 is given by


Naturally, we may further consider the following fractional integral:


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:


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




denotes the set of selections of F and βi:JR is 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 X be a nonempty set and A1,A2,,Ap be nonempty subsets of X. An operator F:k=1pAk2k=1pAk is called a phset-valued cyclic operator over k=1pAk if F(Ai)Ai+1 for 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)=ptyset or k=1pAk=ptyset, the conclusion is clear. Thus, let zk=1pAk be a fixed point of F. Then, zAq for some q{1,2,,p} and zFzAq+1. Consequently, we also have zFzAq+2. It is easy to see that zAq+n for 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 Φ be 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 Ψ be the class of functions ψ:R+6R satisfying the following conditions:

  • (Ψ1) ψ is continuous.

  • (Ψ2) ψ is nondecreasing in the first variable and is nonincreasing in the remaining variables.

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

  • (Ψ4) ψ(u,0,u,0,0,u),ψ(u,u,0,0,u,u)>0 for 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>0 and β,γ,δ0 with α+β+γ<1 and 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 ψΨ satisfying


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

  1. F has at least one fixed point;

  2. F has no fixed point outside k=1pAk.

Proof. For (I), let x0 be chosen arbitrarily from some Aj. Choose any x1Fx0. Then, we define implicitly a sequence (xn) by choosing xn+1Fxn satisfying


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


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


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


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


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>0 and two strictly increasing sequences of integers (mk) and (nk) in which


We can assume, without loss of generality, that nk>mk>k and nk is minimal in the sense that d(xmk,xr)<ε0 for 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 k sufficiently large, we may see that mkjk>0. Observe that


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


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


So, we get d(xnk,xmkjk+1)ε0 as k. Also observe that


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


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


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

Next, we shall prove that x is, 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+1 for some i{1,2,,p}. So, it is followed that


Passing to the limit as n, we obtain that


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

To obtain (II), apply Proposition 2.2.

2.2. Ordered spaces as corollaries

Let X be a nonempty set, recall that the binary relation Ô is said to be a ph(partial) ordering on X if it is reflexive, antisymmetric and transitive. By an phordered set, we shall mean the pair (X,) where X is 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 X and every point z0X such that z0xn for all nN, there holds the property z0x, where xX is the limit of (xn).

Theorem 2.9. Let (X,,d) be a complete ordered metric space satisfying the condition (Θ) and let F:XCB(X) be a nondecreasing proximal operator in the sense that if x,yX satisfies xy, then uv for all uFx and vFy. Suppose that there exists ψΨ such that


for all x,yX in which we can find some zX satisfying both zx and zy. If there exists x0X such that x0w for all wFx0, then F has at least one fixed point.

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


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


For any zC(x0), observe that x0w for all wGz. Thus, G(C(x0))C(x0) so that G is cyclic over C(x0). Moreover, for any x,yC(x0), we have by definition that x0x and 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 G has 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:XCB(X) be a nondecreasing proximal operator in the sense that if x,yX satisfies xy, then uv for all uFx and vFy. Suppose that there exists ψΨ such that


whenever x,yX satisfy xy. Also assume that if the sequence (xn) in X is nondecreasing and converges to xX, then xnx for all nN. If there exists x0X such that x0w for all wFx0, then F has at least one fixed point.

Proof. Note that if x,yX are comparable, then, according to Theorem 2.9, we may choose z:=xX so that zx and zy.

On the other hand, let (yn) be a sequence in X which is both nondecreasing and convergent to yX. According to the condition (Θ), set z0:=y1. We may see easily that, in this case, X satisfies 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 E2 with its standard metric d. For each tR, we define


Suppose that A1 and A2 are two closed sets defined by


Let F:A1A22A1A2 be an operator defined by


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

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

The case x,y0 is trivial and so we omit it. For the case x0 as y1 and x1 as y2, we consider the following calculation.

From Table 1(a), we have

(A) xl0 as yl1
d(x,Tx) 0
d(y,Ty) 1/2
d(x,Ty) (x1y1)2+1/2
d(y,Tx) (x1y1)2+1/2
(B) xl1 as 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.


for all x0 and y1. We can similarly obtain from Table 1(b) the following:


for all x1 and 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


endowed with the supremum norm |||| given by


The solutions for the problem (2) are assumed to be in Ω under 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 J be an interval of R. An operator F:J2X is said to be measurable if for each xX and tJ, the mapping xd(x,F(t)) is measurable.

Next, we shall define the set-valued operator Λ:Ω2Ω given by


where U is the ordinary single-valued fractional integral.

We shall next illustrate that the operator Λ possesses closed values.

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

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


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


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+1 or (u,v)Πi+1×Πi holds for some i{1,2,,p};

  3. Λ is proximal and cyclic over k=1pΠk=Ω.

If the function ψ:R+6R+ given by


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

Proof. Let (u,v)Πi×Πi+1 for 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


Consider the two functions




Next, observe that


It follows that


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


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 Ω in such a way that for u,vΩ,


It is easy to see that if (un) is a nondecreasing sequence in Ω which converges to some uΩ, then unu for 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 uΩ such that uv for 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,vΩ are comparable;

  3. Λ is proximal and nondecreasing;

  4. a weak solution u0Ω to the problem (2) exists.

If the function ψ:R+6R+ given by


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


whenever u,vΩ are 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.

Definition 3.1. ([23]). Let X be a nonempty set. A function w:(0,)×X×X[0,+] is said to be a phmetric modular on X if the following conditions are satisfied for any s,t>0 and x,y,zX:

  1. x=y if and only if wt(x,y)=0 for 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 X having the collection of all such open balls as a base. The convergence in this topology can therefore be written by:


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


whenever m,n>n. Naturally, X is said to be phcomplete if Cauchy sequences in X converges.

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 AXw and 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, AC(X) and xX. Then,


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


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., YX a nonempty compact subset. For a point xX, if either infyYsupt>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,ZC(X) are nonempty and zZ. If Y is reachable from z, then for each ε>0, there exists a point yεY 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:XX is said to be a phcontraction if there exists a constant k[0,1) such that


for all t>0 and x,yX.

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


Then, we call F a 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 x0X and x1F(x0) with the following properties:

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

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

Then, F has 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


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


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:


Hence, by the contractivity of F, we have


Thus, by induction, we have


Moreover, it follows that


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


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


which implies that wt(x,F(x))=0 for 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


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


for every xX.

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


Therefore, F is a contraction with contraction constant k=12. Moreover, it is easy to see that the conditions (A) and (B) hold. Finally, we have that 1 is 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 X is defined as in the previous example. Consider the operator G:XX given by


for each xX.

Note that this operator G is also a contraction with constant k=12 and takes compact values on X. Also, the conditions (A) and (B) hold. However, every point in X is 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


Now, observe that


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


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 X with exactly the same properties appearing in the proof of Theorem 3.7.

Now, consider further that


Moreover, we get


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


Thus, we have


Letting n to 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 Ψ is the interval mentioned in the previous section. Let us assume throughout the section that the real line R is equipped with the metric modular


for λ>0 and 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


for λ>0 and ϕ,ψC(Ψ). Note that both ωR and ωC(Ψ) satisfy the Fatou’s property. Also note that the set R is 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×R2R is a set-valued operator with nonempty compact values and uC(Ψ). We shall use the following notation to explain the collection of integrable selections:


It is clear that SF(u) is closed. Next, for each i{0,1,,N}, NN, assume that βi:ΨR is 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


Note here that for any ϕC(Ψ), we have Λ(ϕ) is reachable from ϕ w.r.t. ωC(Ψ). To restrict the operator Λ with 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


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 u is 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 fSF(v) such that


for all tΨ;

(F2) (N+1)BΓ(α)+LTαΓ(α)<1.

Then, Λ has a fixed point.

Proof. For each u,vC(Ψ), we may choose, from the assumption, functions f1,f2 such that


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


Now, consider the following computation:


It follows that


The proof ends here by applying Theorem 3.7.


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