Recent Fixed Point Techniques in Fractional Set-Valued Dynamical Systems Recent Fixed Point Techniques in Fractional Set-Valued Dynamical Systems

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 con- sider 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.


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 C 1 initial-valued problem: 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 ½0, T, where T > 0. Let F : Ψ ! 2 R be a set-valued operator. A selection of F is the function f : Ψ ! R∪{ AE ∞} such that f ðtÞ ∈ FðtÞ a.e. t ∈ Ψ . We write ℱ 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 f ∈ L 1 ðΨ , μÞ, 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 [3][4][5] 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. [6][7][8][9][10]).
The central system that we are going to investigate in this chapter is the following delayed system: where τ i ∈ ½0, t for all i ∈ {1, 2, ⋯, n}, F : J · R ! CBðRÞ, I α Fðt, uðtÞÞ is the definite integral of order α given by and S F ðuÞ :¼ ff ∈ L 1 ðJ, RÞ ; f ðtÞ ∈ Fðt, uðtÞÞ a:e: t ∈ Jg denotes the set of selections of F and β i : 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 [11][12][13][14][15][16][17]. 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. [18][19][20] should give some ideas about the theory itself and also how to develop further dynamical systems in this framework.

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-/supersolution.
Note that results in this section are based on our paper [21]. Recall the following notion of cyclic operators. A k be a fixed point of F. Then, z ∈ A q for some q ∈ {1, 2, ⋯, p} and z ∈ Fz ⊆ A qþ1 . Consequently, we also have z ∈ Fz ⊆ A qþ2 . It is easy to see that z ∈ A qþn for all n ∈ N. Therefore, it is enough to conclude The following classes of functions are necessary to our further contents.  ðΨ 2Þ ψ is nondecreasing in the first variable and is nonincreasing in the remaining variables.

Fixed point theorem for cyclic operators
Now, we give the main fixed point theorem for cyclic implicit contractive operators.
Next, we show that ðx n Þ is Cauchy. Suppose to the contrary. So, we may find ε 0 > 0 and two strictly increasing sequences of integers ðm k Þ and ðn k Þ in which We can assume, without loss of generality, that n k > m k > k and n k is minimal in the sense that On the other hand, for each k ∈ N, we may find j k ∈ {1, 2, ⋯, p} in which n k −m k þ j k ≡1ðmodpÞ.
For k sufficiently large, we may see that m k −j k > 0. Observe that Again, letting k ! ∞, we obtain that dðx n k þ1 , x m k −j k þ1 Þ ! ε 0 . Finally, by the fact that ðx m k −j k , x n k Þ ∈ A i · A iþ1 for some i ∈ {1, 2, ⋯, p} and Eq. (3), we may obtain that By the condition ðΨ 4Þ and letting k ! ∞, we may deduce that which is absurd. Hence, the sequence ðx n Þ is Cauchy. Since ∪ p k¼1 A k is closed, it is complete and therefore ðx n Þ converges to some unique point x * ∈ ∪ p k¼1 A k .
Next, we shall prove that x * is, in fact, a fixed point of F. Let us assume now that dðx * , Fx * Þ > 0. Note that for any n ∈ N, ðx * , Passing to the limit as n ! ∞, we obtain that To obtain (II), apply Proposition 2.2.

Ordered spaces as corollaries
Let X be a nonempty set, recall that the binary relation Ê a 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 ðX, ⊑Þ where X is nonempty and ⊑ is an ordering on X. A ph(partially) ordered metric space is the triple ðX, ⊑, dÞ, where ðX, ⊑Þ is an ordered set and ðX, dÞ 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 ðX, ⊑, dÞ is said to satisfies the phcondition ðΘÞ if every convergent sequence ðx n Þ in X and every point z 0 ∈ X such that z 0 ⊑ x n for all n ∈ N, there holds the property z 0 ⊑x * , where x * ∈ X is the limit of ðx n Þ. THEOREM 2.9. Let ðX, ⊑, dÞ be a complete ordered metric space satisfying the condition ðΘÞ and let F : X ! CBðXÞ 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 ψðHðFx, FyÞ, dðx, yÞ, dðx, FxÞ, dðy, FyÞ, dðx, FyÞ, dðy, FxÞÞ ≤ 0 for all x, y ∈ X in which we can find some z ∈ X satisfying both z ⊑ x and z ⊑ y. If there exists x 0 ∈ X such that x 0 ⊑ w for all w ∈ Fx 0 , then F has at least one fixed point.
PROOF. By the existence of such a point x 0 , we shall now construct a set Taking any sequence ðx n Þ in Cðx 0 Þ. By the condition ðΘÞ with z 0 :¼ x 0 , we may see that if ðx n Þ converges, its limit is also included in Cðx 0 Þ. Hence, Cðx 0 Þ is closed and therefore it is complete.
On the other hand, we define an operator G : Cðx 0 Þ ! CBðXÞ by For any z ∈ Cðx 0 Þ, observe that x 0 ⊑w for all w ∈ Gz. Thus, GðCðx 0 ÞÞ⊆Cðx 0 Þ so that G is cyclic over Cðx 0 Þ. Moreover, for any x, y ∈ Cðx 0 Þ, we have by definition that x 0 ⊑x and x 0 ⊑y, so that the inequality (4) holds whenever ðx, yÞ ∈ Cðx 0 Þ · Cðx 0 Þ. 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. whenever x, y ∈ X satisfy x⊑y. Also assume that if the sequence ðx n Þ in X is nondecreasing and converges to x * ∈ X, then x n ⊑x * for all n ∈ N. If there exists x 0 ∈ X such that x 0 ⊑w for all w ∈ Fx 0 , then F has at least one fixed point.
PROOF. Note that if x, y ∈ X are comparable, then, according to Theorem 2.9, we may choose z :¼ x ∈ X so that z⊑x and z⊑y.
On the other hand, let ðy n Þ be a sequence in X which is both nondecreasing and convergent to y * ∈ X. According to the condition ðΘÞ, set z 0 :¼ y 1 . We may see easily that, in this case, X satisfies the condition ðΘÞ. We next apply Theorem 2.9 to finish the proof.

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 E 2 with its standard metric d. For each t ∈ R, we define Suppose that A 1 and A 2 are two closed sets defined by Let F : A 1 ∪A 2 ! 2 A1∪A2 be an operator defined by Note that the notation P as is appeared in Eq. (5) is the metric projection onto the corresponding sets ℓ 1 and ℓ 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 ¼ 9 20 , β ¼ γ ¼ 1 4 and δ ¼ 1 2 . The case x, y ∈ ℓ 0 is trivial and so we omit it. For the case x ∈ ℓ 0 as y ∈ ℓ 1 and x ∈ ℓ 1 as y ∈ ℓ 2 , we consider the following calculation. From for all x ∈ ℓ 0 and y ∈ ℓ 1 . We can similarly obtain from HðFx, FyÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffi ffi for all x ∈ ℓ 1 and y ∈ ℓ 2 . Therefore, we have now proved our claim.

Fractional set-valued dynamical systems
For convenience, we shall always consider the nonempty closed and bounded subspace Ω⊂CðJ, RÞ :¼ {u : J ! R ; uiscontinuous}, endowed with the supremum norm ∥ Á ∥ 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 : J ! 2 X is said to be measurable if for each x ∈ X and t ∈ J, the mapping x↦dðx, FðtÞÞ is measurable.
Next, we shall define the set-valued operator Λ : Ω ! 2 Ω 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 u ∈ Ω.
PROOF. Let u ∈ Ω and let ðu k Þ be a sequence in Λu which converges to some u * ∈ Ω. We shall prove the statement by showing that limits of convergent sequence in Λu are in Λu. Then, there exists a sequence ðf k Þ in S F ðuÞ in which Also note that this sequence ðf k Þ converges to some f * ∈ L 1 ðJ, RÞ. Since Fðt, uðtÞÞ is closed, f * ∈ S F ðuÞ. Actually, we have This completes the proof. Now, we give the solvability of the system (2

Λ is proximal and cyclic over
If the function ψ : R 6 þ ! R þ given by Next, observe that We may deduce similarly that the above inequality holds also in the case ðu, vÞ ∈ Π iþ1 · Π i . 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, v ∈ Ω, u⊑v⇔uðtÞ ≤ vðtÞ a:e: t ∈ J It is easy to see that if ðu n Þ is a nondecreasing sequence in Ω which converges to some u * ∈ Ω, then u n ⊑u * for all n ∈ N. In the further step, we shall need in the initial state that a weak solution to Eq. (2)
If the function ψ : R 6 þ ! R þ given by is in the class Ψ , then the problem (2) has at least one solution.

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 : ð0, ∞Þ · X · X ! ½0, þ ∞ is said to be a phmetric modular on X if the following conditions are satisfied for any s, t > 0 and x, y, z ∈ X: 1.
x ¼ y if and only if w t ðx, yÞ ¼ 0 for all t > 0.
Here, we use w t ðÁ, ÁÞ :¼ wðt, Á , ÁÞ. In this case, we say that ðX, wÞ 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 ðX, dÞ is a metric space, then w t ðÁ, ÁÞ :¼ dðÁ, ÁÞ 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 ðX, wÞ be a modular metric space. By defining an open ball with B w (x;r):={z∈X; sup t>0 w t (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 ðx n Þ⊂X and x ∈ X. With this characterization, we now have a good hint to define the Cauchy sequence. A sequence ðx n Þ⊂X is said to be phCauchy if for any given ε > 0, there exists n * ∈ N such that sup t>0 w t ðx m , x n Þ < ε 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 CðXÞ to denote the set of all nonempty closed subsets of X. For any subset A⊂X w and point x ∈ X, we denote w t ðx, AÞ :¼ inf y ∈ A w t ðx, yÞ.
Given two subsets A, B ∈ CðXÞ, define w t ðA, BÞ :¼ sup x ∈ A w t ðx, BÞ. Most importantly, the Hausdorff-Pompieu metric modular W t ðA, BÞ :¼ max{w t ðA, BÞ, w t ðB, AÞ}: Let ðX, wÞ be a modular metric space, A ∈ CðXÞ and x ∈ X. Then, This lemma gives a simple criterion of when the reachability holds. LEMMA 3.4. Let ðX, wÞ be a modular metric space with w being l.s.c., Y⊂X a nonempty compact subset. For a point x ∈ X, if either inf y ∈ Y sup t>0 w t ðx, yÞ < ∞ or sup t>0 w t ðx, YÞ < ∞, 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 ∈ CðXÞ are nonempty and z ∈ Z. If Y is reachable from z, then for each ε > 0, there exists a point y ε ∈ Y such that sup t>0 w t ðz, y ε Þ ≤ sup t>0 W t ðX, YÞ þ ε.

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 ðX, wÞ be a modular metric space. A set-valued operator F : X⇉X is said to be a phcontraction if there exists a constant k ∈ ½0, 1Þ such that for all t > 0 and x, y ∈ X.
If k is restricted in ½0, 1 2 Þ and Eq. (7) is replaced with the following inequality: Then, F has at least one fixed point.
PROOF. Since Fðx 1 Þ is reachable from x 1 , by using Lemma 3.5, we may choose From the above evidence and the hypothesis that {x 0 , x 1 } is bounded, it comes to the following inequalities: By the assumptions, we apply Lemma 3.4 to guarantee that Fðx 2 Þ is actually reachable from x 2 .
Inductively, by this procedure, we define a sequence ðx n Þ in X, with the supplement from Lemma 3.5, satisfying the following properties for all n ∈ N: Fðx n Þ is reachable from x n : Hence, by the contractivity of F, we have Thus, by induction, we have sup t>0 w t ðx n , x nþ1 Þ ≤ k n sup t>0 w t ðx 0 , x 1 Þ þ nk n : Moreover, it follows that Without loss of generality, suppose m, n ∈ N and m > n. Observe that for all m > n ≥ n * for some n * ∈ N. Hence, ðx n Þ is a Cauchy sequence so that the completeness of X w implies that ðx n Þ converges to some point x ∈ X w . Consequently, we may conclude from the contractivity of F that the sequence ðFðx n ÞÞ converges to FðxÞ. Since x n ∈ Fðx n−1 Þ, we have for any t > 0, which implies that w t ðx, FðxÞÞ ¼ 0 for all t > 0. Since FðxÞ is closed, it then follows from Lemma 3.2 that x ∈ FðxÞ.
EXAMPLE 3.8. Suppose that X ¼ ½0, 1 and w : ð0, þ ∞Þ · X · X ! ½0, þ ∞ 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 : X⇉X by Observe that F has compact values on X. Note that for each t > 0 and x, y ∈ X, we have Therefore, F is a contraction with contraction constant k ¼ 1 2 . 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 : X⇉X given by for each x ∈ X.
Note that this operator G is also a contraction with constant k ¼ 1 2 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. PROOF. Since Fðx 1 Þ is reachable from x 1 , by using Lemma 3.5, we may choose x 2 ∈ Fðx 1 Þ such that sup t>0 w t ðx 1 , x 2 Þ ≤ sup t>0 W t ðFðx 0 Þ, Fðx 1 ÞÞ þ k: Writing ξ :¼ k 1−k < 1, we obtain, from the boundedness of {x 0 , x 1 } and the reachability of Thus, from the assumptions and Lemma 3.5, we may see that Fðx 2 Þ is reachable from x 2 .
Inductively, we can construct a sequence ðx n Þ in X with exactly the same properties appearing in the proof of Theorem 3.7.
As in the proof of Theorem 3.7, the sequence ðx n Þ converges to some x ∈ X. Observe now that sup t>0 w t ðx, FðxÞÞ Letting n ! ∞ to conclude the theorem.

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 ω R λ ðx, yÞ :¼ for λ > 0 and x, y ∈ R. Thus, for the space CðΨ Þ of all continuous (in ω R -topology) real-valued functions on Ψ , we shall use the metric modular ω CðΨ Þ λ ðϕ, ψÞ :¼ sup t ∈ Ψ ω R λ ðϕðtÞ, ψðtÞÞ, for λ > 0 and ϕ, ψ ∈ CðΨ Þ. Note that both ω R and ω CðΨ Þ satisfy the Fatou's property. Also note that the set R is second countable, i.e., it has a countable base, w.r.t. ω R -topology. Moreover, it is clear that the set {ϕ, ψ} is bounded w.r.t. ω CðΨ Þ , for any ϕ, ψ ∈ CðΨ Þ. Suppose that F : Ψ · R ! 2 R is a set-valued operator with nonempty compact values and u ∈ CðΨ Þ. We shall use the following notation to explain the collection of integrable selections: S F ðuÞ :¼ f ∈ L 1 ðΨ , μÞ ; f ðtÞ ∈ Fðt, uðtÞÞa:e:t ∈ Ψ È É : It is clear that S F ðuÞ is closed. Next, for each i ∈ {0, 1, ⋯, N}, N ∈ N, assume that β i : Ψ ! R is continuous and τ i : Ψ ! R þ is a function with τ i ðtÞ ≤ t. We write B :¼ max 0 ≤ i ≤ N sup t ∈ Ψ β i ðtÞ.
The main aim of this section is to consider the fractional integral inclusion: uðtÞ− X N i¼0 β i ðtÞuðt−τ i ðtÞÞ ∈ J α Ψ Fðt, uðtÞÞdt, α ∈ ð0, 1: (FII) In the above inclusion, the summation here is interpreted to be the delay term.
We shall define a set-valued operator Λ : CðΨ Þ ! 2 CðΨ Þ by Note here that for any ϕ ∈ CðΨ Þ, we have ΛðϕÞ is reachable from ϕ w.r.t. ω CðΨ Þ . To restrict the operator Λ with some nice property, we assume that S F ðuÞ is nonempty. LEMMA 3.11. The operator Λ given above is compact valued if S F ðuÞ is nonempty.
PROOF. For the proof, we shall show the compactness by its sequential characterization. Suppose that u ∈ CðΨ Þ and ðw n Þ is an arbitrary sequence in ΛðuÞ. By definition, there corresponds a convergent sequence ðf n Þ in S F ðuÞ⊂FðÁ, uðÁÞÞ satisfying The conclusion is then followed.