Iterative Construction of Fixed Points for Functional Equations and Fractional Differential Equations

,


Introduction
Fixed point theory in recent years has suggested very useful techniques for solving nonlinear problems (for details, see the survey article by Karapinar [1,2]).Iterative solutions for functional equations and FDEs are a busy feld of research on their own [3].It is known that a sought solution of a functional equation or a FDE can be expressed as a fxed point of a certain linear or nonlinear operator acting on a subset of a suitable distance space (see, e.g., [4] and others).Te existence as well as the iterative construction of the fxed point of this operator is always desirable.We know that, the existence of a fxed point is possible but to construct a suitable algorithm to approximate the value of the fxed point is not an easy work (see, e.g., [5,6] and others).For example, the Banach Contraction Principle (BCP) [7] suggests a unique fxed point for contractions and suggests the Picard iteration [8], that is, ] i � Ψ] i , to approximate the values of this unique fxed point [9].On the other hand, the Browder-Gohde-Kirk result (see Browder [10], Gohde [11], and Kirk [12]) proved that every nonexpansive mapping on a closed convex bounded subset of a uniformly convex Banach space (UCBS) has a fxed point.Notice that a self-map Ψ on a subset V of a metric space is essentially called a contraction if where α ∈ [0, 1).A fxed point of Ψ in this case is any element, namely, z ∈ V with the property z � Ψz.Te set of all fxed points of the operator Ψ will be denoted simply by F Ψ throughout in this research paper.If (1) holds when we put α � 1 then Ψ is known as nonexpansive.An example of a nonexpansive mapping for which Picard iteration does not converge is the following: Example 1.Let V � [0, 1] and Ψ] � 1 − ] for all ] ∈ V.It follows that Ψ is nonexpansive with F Ψ � 0.5 { } and the Picard iteration is not convergent to 0.5 for all the starting value which is diferent from 0.5.
Example 1 suggests other iterative schemes instead of Picard iteration [8] which are convergent in the setting of nonexpansive mappings (or even generalized nonexpansive mappings).In 2008, Suzuki [13] introduced a condition on mappings, called (C) condition.
Defnition 1 (see [13]).Te self-map Ψ of V is said to satisfy the (C) condition of Suzuki if ( Te (C) condition is essentially weaker than the nonepensiveness property of any operator Ψ.For instance, see an example in [13].
Ullah and Arshad [21] constructed a new iteration called M * -iteration and proved that this iteration is stable and suggests highly accurate results corresponding to other iterations of the literature.Tis iteration generates a sequence ] i   as follows: In the scheme (4), the operator Ψ is a self-map of the set V and the sequences a i   and b i   are in the interval (0,1).Although Ullah and Arshad [21] proved the convergence of the scheme (4) in the case of contractions.We extend here their main outcome to the more general setting of mappings satisfying the KSC condition.Using the same techniques, convergence of the above mentioned iterations can be proved on the same line of proof.Using a nontrivial example, we show that the iteration scheme M * suggests accurate results corresponding to the other iterations in this new setting of mappings.

Preliminaries
Now, we need some basic results of CAT (0) spaces.For more details on CAT (0) spaces, please see the books [22,23].
We now state a result from [24].
Lemma 3. Suppose B is any complete CAT (0) space and ∅ ≠ V⊆B.Ten, (a) If we have ], ξ ∈ B and there is a fxed element θ in the set [0,1], then one has a unique point q ∈ [], ξ], such that Sometimes, we may write (1 − θ)] ⊕ θξ as the unique point q that satisfes (5).
be a bounded sequence in a metric space B and ∅ ≠ V⊆ B closed convex.We denote and set the asymptotic radius of the sequence ] i   in the set V as We denote and set the asymptotic center of the sequence ) contains one and only one point.
Te following is the defnition of a ∆ convergence that can be considered as an analog of the weak convergence in a Banach space.Defnition 5. A bounded sequence, namely, ] i   in a complete CAT (0) space B is said to be ∆-convergent to a point, namely, z ∈ B (and denote it as ∆ − lim i ] i � z) if it is the case that the point z is the unique asymptotic center for each choice of the subsequence s i   of ] i  .
Te CAT (0) version of the Opial's [25] condition holds, that is, if ] i   is any ∆-convergent sequence in a complete CAT (0) space B with the ∆ limit z, then for any y ≠ z ∈ B, one has 2 Journal of Mathematics Lemma 6 (see [26]).Suppose we have complete CAT (0) space B. Ten any bounded sequence ] i   ⊆ B admits a ∆convergent subsequence.
Lemma 7 (see [27]).Suppose we have complete CAT (0) space B. If ∅ ≠ V⊆ B is convex and closed then the asymptotic center of any bounded sequence ] i   is contained in the space B.
Defnition 8 (see [28]).A self-map Ψ on a subset V of a CAT (0) space is said to satisfy condition (I) if one has a function c with c(0) � 0, c(u) > 0 for each u > 0 and Lemma 9 (see [14]).Suppose B is any CAT (0) space and ∅ ≠ V⊆B.Let Ψ be a self-map of V satisfying KSC condition with F Ψ ≠ ∅.Ten for any ] ∈ V and z ∈ F Ψ , one has the following property: Lemma 10 (see [14]).Suppose B is any CAT (0) space and ∅ ≠ V⊆B.Let T be a self-map of V satisfying KSC condition.Ten for any ], ξ ∈ V, one has the following property: Lemma 11 (see [14]).Suppose B is any complete CAT (0) space and ∅ ≠ V⊆B.Let Ψ be a self-map of V satisfying KSC condition.Ten, the following property holds:

Main Results
First, we defne the CAT (0) space version of M * iterative scheme (4) as follows: Now, using (11), we prove our man results.We frst provide the following lemma that will play a key role.Lemma 13.Suppose B is any complete CAT (0) space and ∅ ≠ V ⊆ B is closed and convex.Let Ψ be a self-map of V satisfying the (KSC) condition with F Ψ ≠ ∅.Ten the sequence generated by M * -iteration (11) satisfes Proof.Consider any point z ∈ F Ψ , then applying Lemma 9, one has Hence, we obtained for all

Tis means that d(] i , z)
is essentially bounded as well as nonincreasing and hence it follows that lim i⟶∞ d(] i , z) exists for all z ∈ F Ψ .Now, for the existence of a fxed point, we give the necessary and sufcient condition for mappings with (KSC) condition defned on nonempty closed convex subsets of a UCBS as follows.
□ Theorem 1 .Suppose B is any complete CAT (0) space and ∅ ≠ V⊆B is closed and convex.If Ψ is a self-map of V satisfying KSC condition and ] i   is the sequence of M * -iteration (11).Ten, F Ψ ≠ ∅, if and only if ] i   is bounded and satisfes Proof.First, we assume the case that the set F Ψ ≠ ∅ and prove that ] i   is bounded with lim i⟶∞ d(] i , Ψ] i ) � 0. For this, Lemma 13 suggests that ] i   is bounded and for some r ∈ R + .We assume the nontrivial case, that is, when r > 0. Ten in the view of the proof of Lemma 13, Again, we see that, Tus from ( 14) and ( 16), we have From ( 17), we have Now, applying Lemma 12 on ( 13), (15), and (18), we obtain Finally, we shall assume ] i   is bounded with the property lim i⟶∞ d(Ψ] i , ] i ) � 0 and show that the set F Ψ ≠ ∅.For this, we may assume any point, namely, z in the set A(V, ] i  ).By Lemma 10, we have Tis implies that Ψz ∈ A(V, ] i  ).Since A(V, ] i  ) is singleton, hence, we have Ψz � z and hence F Ψ ≠ ∅.
We frst suggest a ∆ convergence result.

□
Theorem 15.Suppose B is any complete CAT (0) space and ∅ ≠ V⊆B is closed and convex.Let Ψ be a self-map of V satisfying the KSC condition with F Ψ ≠ ∅.Ten the sequence of the M * -iteration ] i   (11) ∆-converges to some fxed point of Ψ provided that the space B has Opial's property.
Proof.Using Teorem 14, we have the sequence of iterates ] i   is bounded in the set V and satisfes the condition Applying Lemmas 6 and 7, one has a subsequence r i   of s i   such that r i  ∆ converges to a point r in B. Now, using Teorem 14, we have lim i⟶∞ d(r i , Ψr i ) � 0. Also, Ψ is endowed with KSC condition, therefore d r i , Ψr  ≤ 5d r i , Ψr i  + d r i , r . ( Applying limit on (21), it follows that r ∈ F Ψ .Hence, using Lemma 13, one has lim i⟶∞ d(r i , r) exists.Te next aim is to obtain that s � r.We prove this by contradiction, that is, we assume that s ≠ r.Keeping   ) � q  .Ten since we proved already that s � r and r ∈ F Ψ , we claim q � r.Because if q ≠ r, then lim i⟶∞ d(] i , r) exists and keeping the uniqueness of asymptotic centers in mind, one has lim sup which is clearly a contradiction.Hence, we conclude that Te following theorem is based on compactness.

□
Theorem 16.Suppose B is any complete CAT (0) space and ∅ ≠ V⊆B is compact and convex.Let Ψ be a self-map of V satisfying the KSC condition with F Ψ ≠ ∅.Ten the sequence of the M * -iteration (11) converges strongly to some fxed point of Ψ.
Proof.As assumed, the set V is convex and compact, so the sequence of iterates ] i   contained in the set V and has a subsequence ] i k   of ] i   that converges strongly to ] ∈ V.Moreover, in the view of Teorem 14, we obtain lim i⟶∞ d(Ψ] i k , ] i k ) � 0. Hence, using these facts together with Lemma 10, we have It follows that Ψ] � ].By Lemma 13, lim i⟶∞ d(] i , ]) exists and hence ] i   is strongly convergent to ].Strong convergence without compactness of the domain is the following.

□
Theorem 17. Suppose B is any complete CAT (0) space and ∅ ≠ V ⊆ B is closed and convex.Let Ψ be a self-map of V satisfying the KSC condition with F Ψ ≠ ∅.Ten, the sequence of the M * -iteration (11) converges strongly to some fxed point of Ψ provided that lim inf i⟶∞ dist(] i , F Ψ ) � 0.

Journal of Mathematics
Proof.Te proof of this result is easy and hence we exclude the proof.

□
Theorem 18. Suppose B is any complete CAT (0) space and ∅ ≠ V⊆ B is closed and convex.Let Ψ be a self-map of V satisfying the KSC condition with F Ψ ≠ ∅.Ten, the sequence of the M * -iteration (11) converges strongly to some fxed point of Ψ provided that the Ψ satisfes condition (I).

Numerical Example
In this section, frst we give a numerical example of a mapping with (KSC) condition which does not satisfy (C) condition and then we show that the sequence ] i   generated by M * -iteration process converges faster than some other well-known iteration schemes.
Example 2. Defne a mapping Ψ on [− 1, 1] as follows: Now, we see that the above self-map Ψ is not enriched with (C) condition.For example, if one chose ] � − (1/2) and ξ � − (4/5), then Ψ does not satisfy the (C) condition.Eventually, we shall establish that this map is enriched with KSC condition.To achieve the objective, some elementary cases have been omitted, while nontrivial cases are considered as follows: C2: When ], ξ ∈ [0, 1], we have

Journal of Mathematics
Now, we draw a graph and table which show that the sequence ] i   of the scheme M * -iteration moving faster to the fxed point 0 of Ψ as compared to Tukar, Abbas, Noor, and Ishikawa iterative schemes.Assume that a i � 0.70, b i � 0.65, and c i � 0.90.Te iterative results are shown in Table 1 while the behavior of iterates are given in Figure 1.Te efectiveness of M * iterative scheme is clear in both the table and graph.
We fnish the paper with a nontrivial example.
Here, the space (B, d) is only a CAT (0) space but not a Banach space [22].Also, B is closed and convex.Now, let Ψ be the metric projection on B, then by a well-known result (see, p178 in [22]) that Ψ is nonexpansive and hence it satisfes KSC condition.By our main results, the sequence (11) converges to a fxed point of Ψ.

Application to Differential Equations
In this section, we study the solution of a FDE in our new setting of mappings.Tis problem has been considered by some authors in the class of nonexpansive mappings [30] and other types of spaces [31,32].It is important to note here that our approach is alternative and based on the class of mappings with KSC.Te main diference between our approach and classical approaches to the problems are that mappings with KSC are not necessarily continuous throughout on the their domains.Moreover, our iterative method is more efective and suggests very high accurate numerical results in less step of iterates.To achieve the objective, we follow the idea given by [33].
We consider the following general class of boundary value problems from fractional calculus: where (0 ≤ t ≤ 1), (1 < c < 2), and D c stands for the Caputo fractional derivative with order c and Ω: [0, 1] × R ⟶ R. Consider B � C[0, 1] and Green's function associated with (32) that reads as follows: Te main result is provided in the following way.
Theorem 19.Set a self-map Ψ: B ⟶ B by the following formula: 6 Journal of Mathematics then, theM iterates (11) associated with the Ψ (as defend above) converge to the point of S provided that liminf i ⟶ ∞ dist(] i , S) � 0, where S denotes the set of all solutions of (32).
Proof.Since G is a Green's function to our problem, so by [34], the sought solution can be expressed as an integral form as follows: Now, for every choice of h, g ∈ B and 0 ≤ u ≤ 1, it follows that d(Ψ(h(u)), Ψ(g(u))) ≤ d Hence, Ψ satisfes the (KSC) condition.In the view of Teorem 17, the sequence of the M * iterates converges to a fxed point of Ψ and hence to the solution of the given equation.

Conclusions
Existence as well as iterative constructional for the class of mappings satisfying the KSC condition is established under the iterative scheme M * in a CAT (0) space setting.We proved Δ and strong convergence results for these mappings under certain mild conditions.It has been shown by providing an example that the class of mappings satisfying the KSC condition is diferent than the class of mappings satisfying (C) condition.Eventually, we performed a comparative numerical experiment and proved that the M * iterative scheme in the class of KSC mappings is more efective than the many other iterative scheme.One application is also carried out.Our results refne and improve some main results due to Ullah and Arshad [21] from the case of the (C) condition to the more general case of KSC condition.Similarly, our results extend the results of Abbas and Nazir [19], Agarwal et al. [17], Noor [18], Takur et al. [20], and others.

Figure 1 :
Figure 1: Graphical comparison of diferent iterations in the class of mappings with KSC condition.

Table 1 :
Some iterates of Ψ given in Example 2.Te bold values indicate the frst value tends to zero for every iteration scheme.