# Approximation of the Fixed Point of Multivalued Quasi-Nonexpansive Mappings via a Faster Iterative Process with Applications

Authors: Godwin Amechi Okeke; Mujahid Abbas; Manuel de la Sen
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this paper, we approximate the fixed points of multivalued quasi-nonexpansive mappings via a faster iterative process and propose a faster fixed-point iterative method for finding the solution of two-point boundary value problems. We prove analytically and with series of numerical experiments that the Picard–Ishikawa hybrid iterative process has the same rate of convergence as the CR iterative process.


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## Body

## 1. Introduction

If the existence of the solution of a fixed-point equation involving an operatorT is guaranteed, but an exact solution is not possible, then the requirement of approximating the solution becomes very pertinent. This gives rise to the need of different iterative processes [1–3]. In view of theoretical and practical significance of fixed-point iterative schemes, several authors have constructed and applied different fixed-point iteration schemes in approximating the solution of equations which model certain physical problems (e.g., [2–15]). One of the most important criteria in preferring one fixed-point iteration scheme over the other is the rate of convergence of the iteration scheme. Consequently, a faster fixed-point iteration scheme is always preferred in practice.In 2018, Bello et al. [16] developed a Mann-type fixed-point iteration scheme for approximating the solution of two-point boundary value problems. It is worth mentioning that the scheme proposed in [16] is a self-correcting, unlike the variational or weighted residual methods of approximation which depend on the selection of suitable coordinate or basis functions (e.g., [16, 17]). They established that the proposed fixed-point iteration method is more suitable to approximate the exact solution than other existing methods. Moreover, a noticeable advantage of this method is that it solves the boundary value problem without constructing Green’s function which is always difficult to construct for some problems [16].The purpose of this paper is to introduce a new faster iterative scheme to approximate the solution of a fixed-point inclusion which involves a multivalued quasi-nonexpansive mapping. A study of a faster fixed-point iterative method for finding the solution of two-point boundary value problems is also carried out. With a series of numerical experiments and an analytical proof, it is established that the Picard–Ishikawa hybrid iterative process, recently introduced by Okeke [3], has the same rate of convergence as the CR iterative process, introduced by Chugh et al. [8]. Our results improve, extend, and generalize several known results in the literature [18–20].


## 2. Preliminaries

LetX be a Banach space, x ∈ X, and A ⊆ X. Define d(x, A) = inf{d(x, y) : y ∈ A}.Definition 1 (see [20]).
SupposeX is a real Banach space. A subset C is called proximinal if for all x ∈ X there exists an element c ∈ C such that d(x, c) = inf{‖x − y‖ : y ∈ C} = d(x, C).
It is well known that weakly compact convex subsets of a Banach spaceX and closed convex subsets of a uniformly convex Banach space are proximinal. Let N(X), CL(X), CB(X), K(X), and 2X denote the class of all nonvoid subsets, nonvoid closed subsets, nonvoid bounded and closed subsets, nonvoid compact subsets, and subsets of X, respectively. Let PA be the class of all nonvoid proximinal bounded subsets of A. Suppose that H is the generalized Hausdorff metric on CB(X) which is defined as follows:(1)HA,B=maxsupx∈Adx,B,supy∈Bdy,A,for each A, B ∈ CB(X). We say that a point x∗ ∈ X is a fixed point of T : X ⟶ CL(X) if x∗ ∈ Tx∗. We denote by F(T) ≔ {x∗ ∈ X : x∗ ∈ Tx∗} the set of all the fixed points of T and PT(x) = {y ∈ Tx : ‖ x − y‖ = d(x, Tx)} for a multivalued mapping T:K⟶PK. It is known that multivalued fixed-point theory has applications in economics, differential inclusion, optimization, and control theory [20, 21]. The theory of multivalued mappings is harder than the corresponding theory of single-valued mappings. For further studies of multivalued fixed-point theory, interested reader should see, e.g., [20–25] and the references therein.
The following definitions will be needed in the sequel.Definition 2.
A multivalued mappingT : A ⊆ X ⟶ 2X is called(i)
A contraction if there existsa ∈ [0, 1) such that(2)HTx,Ty≤ax−y,holds for allx, y ∈ A. If a > 0, then we say that T is Lipschitzian.(ii)
Nonexpansive if for allx, y ∈ A, we have(3)HTx,Ty≤x−y.(iii)
Quasi-nonexpansive if for eachx ϵ A and x∗ ϵ F(T) ≠ Ø, we have(4)HTx,x∗≤x−x∗.Definition 3 (see [26]).
A Banach spaceX is said to satisfy Opial’s condition if for each sequence {un} in X such that un ⇀ u implies that(5)limsupn⟶∞un−u<limsupn⟶∞un−v,for each v ϵ X with v ≠ u.Definition 4.
A multivalued mappingT : C ⟶ CB(C) is said to be demiclosed at y ϵ C if for each sequence {xn} in C weakly converges to x and yn ϵ Txn strongly converges to y, we have y ϵ Tx.
In 2012, Chugh et al. [8] introduced the so-called CR iterative scheme and proved that the iterative scheme converges faster than all of Picard [15], Mann [11], Ishikawa [9], Agarwal et al. [5], Noor [12], and SP [14] iterative schemes. The CR iterative process is given as follows.
SupposeX is a Banach space, T: X ⟶ X, and u0 ϵ X. Define the sequence unn=0∞ by(6)zn=1−γnun+γnTun,yn=1−βnTun+βnTzn,un+1=1−αnyn+αnTyn,n∈ℕ,where {αn}, {βn}, and {γn} are sequences of positive numbers in [0, 1] satisfying ∑n=0∞αn=∞. Recently, Okeke [3] introduced the Picard–Ishikawa hybrid iterative process. The author proved that this iterative process converges faster than all of Picard [15], Krasnosel'skii [10], Mann [11], Ishikawa [9], Noor [12], Picard-Mann [27], and Picard–Krasnosel'skii [2] iterative processes in the sense of Berinde [1].
Motivated by the investigation of iterative scheme in [3], we introduce the following multivalued fixed-point iterative process. Suppose X is a Banach space, T : X ⟶ 2X, and x0 ϵ X. Define the sequence xnn=0∞ by(7)zn=1−βnxn+βnun,yn=1−αnxn+αnvn,xn+1=wn,n∈ℕ,where {αn} and {βn} are sequences in [0, 1] satisfying appropriate conditions, un ϵ PT(xn), vn ϵ PT(zn), and wn ϵ PT(yn).Remark 1.
We remark that iteration (7) is the multivalued version of the Picard–Ishikawa hybrid iterative process, recently introduced by Okeke [3].
The multivalued version of the CR iterative process (6) is given as follows:(8)zn=1−γnun+γngn,yn=1−βngn+βnhn,un+1=1−αnyn+αnsn,n∈ℕ,where gn ϵ PT(un), hn ϵ PT(zn), and sn ϵ PT(yn).
The aim of this paper is to approximate the fixed point of multivalued quasi-nonexpansive mappings via the newly introduced fixed-point iteration scheme (7). We prove that the Picard–Ishikawa hybrid iterative process (7) has the same rate of convergence as the CR iterative process (6) for a certain class of quasi-nonexpansive and contraction mappings.
Suppose thatunn=0∞ and vnn=0∞ are two fixed-point iteration processes that converge to a certain fixed point x∗FT of a mapping T; we say that unn=0∞ converges faster than vnn=0∞ ([28]) if(9)un−x∗≤vn−x∗,∀n∈ℕ.
The following definitions are due to Berinde [1].Definition 5 (see [1]).
Letann=0∞ and bnn=0∞ be two sequences of positive numbers that converge to a and b, respectively. Assume there exists(10)l=limn⟶∞an−abn−b.(i)
Ifl = 0, then it is said that the sequence ann=0∞ converges to a faster than the sequence bnn=0∞ to b(ii)
If 0 <l < ∞, then we say that the sequences ann=0∞ and bnn=0∞ have the same rate of convergence.Definition 6 (see [1]).
Suppose that, for two fixed-point iterative processesunn=0∞ and vnn=0∞, both converging to the same fixed point x∗, the error estimates(11)un−x∗≤an,for alln∈ℕ,vn−x∗≤bn,for alln∈ℕ,are available, where ann=0∞ and bnn=0∞ are two sequences of positive numbers converging to zero. If ann=0∞ converges faster than bnn=0∞, then unn=0∞ converges faster than vnn=0∞ to x∗.
For the rest of this paper, whenever we make reference to the rate of convergence of iterative processes, we mean the rate of convergence in the sense of Berinde [1] as in Definition 5.Definition 7.
(see [29]). A multivalued nonexpansive mapping T : C ⟶ CB(C) is said to satisfy Condition (I) if there exists a continuous nondecreasing function f : [0, ∞) ⟶ [0, ∞) with f(0) = 0 and f(r) > 0 for all r ϵ (0, ∞) such that dx,Tx≥fdx,FT for each x ϵ C.
The following lemma will be needed in this study.Lemma 1 (see [30]).
Suppose thatX is a uniformly convex Banach space and 0 < p ≤ tn ≤ q < 1 for each n∈ℕ. Let {xn} and {yn} be two sequences of X such that lim supn⟶∞‖xn‖ ≤ r, lim supn⟶∞‖yn‖ ≤ r, and limn⟶∞‖tnxn + (1 − tn)yn‖ = r hold for some r ≥ 0. Then, limn⟶∞‖xn − yn‖ = 0.Lemma 2.
(see [31]). If T:C⟶PC and PT(x) = {y ϵ Tx:‖x − y‖ = d(x, Tx)}. Then, the following are equivalent:(i)
x ϵ F(T)(ii)
PT(x) = {x}(iii)
x ϵ F(PT)
Moreover,F(T) = F(PT).


## 3. Convergence Analysis of the Multivalued Picard–Ishikawa Hybrid Iterative Process

We begin with the following convergence results.Lemma 3.
Suppose thatX is a normed linear space and C is a nonvoid closed convex subset of X. Let T:C⟶PC such that F(T) ≠ Ø and PT is a quasi-nonexpansive mapping. Suppose that {xn} is a sequence defined by the iteration scheme (7). Then, limn⟶∞‖xn − x∗ ‖ exists for each x∗ ϵ F(T) and limn⟶∞d(xn, PT(xn)) = 0.Proof.
To prove that limn⟶∞‖xn − x∗‖ exists for each x∗ ϵ F(T) ≠ Ø, we proceed as follows. Using (4) and (7), we obtain the following estimates:(12)xn+1−x∗=wn−x∗≤HPTyn,PTx∗≤yn−x∗.
Next, by (4) and (7), we have the following estimates:(13)yn−x∗=1−αnxn+αnvn−x∗≤1−αnxn−x∗+αnvn−x∗≤1−αnxn−x∗+αnHPTzn,PTx∗≤1−αnxn−x∗+αnzn−x∗.
Next, we obtain that(14)zn−x∗=1−βnxn+βnun−x∗≤1−βnxn−x∗+βnun−x∗≤1−βnxn−x∗+βnHPTxn,PTx∗≤1−βnxn−x∗+βnxn−x∗=xn−x∗.
Using (14) in (13), we have(15)yn−x∗≤1−αnxn−x∗+αnxn−x∗=xn−x∗.
Using (14) in (12), we have(16)xn+1−x∗≤xn−x∗.
This implies that {‖xn − x∗‖} is decreasing. Therefore, limn⟶∞‖xn − x∗‖ exists for each x∗ ϵ F(T).
Next, we show that(17)limn⟶∞dxn,PTxn=0.
Let(18)limn⟶∞xn−x∗=k,for some constant k ≥ 0. If k = 0, then (17) holds trivially. Suppose that k > 0; because d(xn, PT(xn)) ≤ ‖xn − un‖, it suffices to establish that limn⟶∞‖xn − un‖ = 0. Clearly, we have(19)un−x∗≤HPTxn,PTx∗≤xn−x∗.
This implies that(20)lim supn⟶∞un−x∗≤k.
Using (14) and (15), we obtain(21)limsupn⟶∞zn−x∗≤k,limsupn⟶∞yn−x∗≤k.
Observe that(22)vn−x∗≤HPTzn,PTx∗≤zn−x∗≤xn−x∗.
Hence, we have(23)limsupn⟶∞vn−x∗≤k.
Similarly, we have(24)wn−x∗≤HPTyn,PTx∗≤yn−x∗≤xn−x∗.
Hence,(25)limsupn⟶∞wn−x∗≤k.
Clearly,(26)limn⟶∞xn+1−x∗=limn⟶∞wn−x∗=k.
Hence, using (23), (25), (26), and Lemma 1, we obtain(27)limn⟶∞vn−wn=0.
From (25), we have(28)liminfn⟶∞xn+1−x∗≤liminfn⟶∞wn−vn+vn−x∗.
Hence, we have(29)k≤liminfn⟶∞vn−x∗.
Using (23) and (29), we have(30)limn⟶∞vn−x∗=k.
Similarly, we can show that(31)limn⟶∞zn−x∗=k.
This means that(32)limn⟶∞zn−x∗=limn⟶∞1−βnxn+βnun−x∗=limn⟶∞1−βnxn−x∗+βnun−x∗=k.
It follows from (20), (32), and Lemma 1 that(33)limn⟶∞xn−un=0.
Therefore, we have(34)limn⟶∞dxn,PTxn=0.
The proof of Lemma3 is completed.
Next, we prove the following strong convergence theorem.Theorem 1.
Suppose thatC is a nonvoid compact convex subset of a real Banach space X, T:C⟶PC with F(T) ≠ Ø, and PT is a quasi-nonexpansive mapping. Let {xn} be the iterative sequence defined in (7). Then, the sequence {xn} converges strongly to a fixed point of T.Proof.
We have already established in Lemma3 that limn⟶∞‖xn − x∗‖ exists for each x∗ ϵ F(T) ≠ Ø. Because C is compact, it follows that there exists a subsequence xnj of {xn} such that limn⟶∞xnj−y∗=0 for some y∗ ϵ C. Hence, by Lemma 3 and the triangle inequality, we have(35)dy∗,PTy∗≤dxnj,y∗+dxnj,PTxnj+HPTxnj,PTy∗≤xnj−y∗+xnj−unj+xnj−y∗⟶0,asn⟶∞.
This means that(36)dy∗,PTy∗=0.
Therefore,y∗ is a fixed point of PT. By Lemma 2, the set of fixed points of PT coincides with the set of fixed points of T, so the sequence {xn} converges strongly to a fixed point y∗ ϵ F(T). The proof of Theorem 1 is completed.
We next prove the following results for multivalued mappings satisfying Condition (I) of Senter and Dotson [29].Theorem 2.
Suppose thatC is a nonvoid closed and convex subset of a real Banach space X. Let T:C⟶PC satisfies Condition (I) such that F(T) ≠ Ø and PT is a quasi-nonexpansive mapping. Then, the sequence {xn} defined by (7) converges strongly to a fixed point x∗ of T.Proof.
By Lemma3, limn⟶∞‖xn − x∗‖ exists for each x∗ ϵ F(T) = F(PT)(Lemma 2). If limn⟶∞‖xn − x∗‖ = 0, then the result trivially holds. We assume that limn⟶∞‖xn − x∗‖ = k > 0. Using relation (16), we have ‖xn+1 − x∗‖ ≤ ‖ xn − x∗‖. Therefore, we have(37)dxn+1,FT≤dxn,FT.
This implies that limn⟶∞d(xn+1, F(T)) exists. Next, we show that limn⟶∞d(xn+1, F(T)) = 0. Assume on contrary that limn⟶∞d(xn+1, F(T)) = μ > 0. Then, for each n∈ℕ, we have(38)pn=un−x∗xn−x∗,qn=xn−x∗xn−x∗.
Hence, we have ‖pn‖ = 1 and ‖qn‖ ≤ 1 because ‖un − x∗‖ ≤ H(PT(xn), PT(x∗)) ≤ ‖xn − x∗‖. Using the fact that f satisfies Condition (I), we have(39)qn−pn=xn−x∗xn−x∗−un−x∗xn−x∗=xn−unxn−x∗≥dxn,Txnxn−x∗≥fdxn,FTxn−x∗.
Becausef is continuous, we obtain that liminfnqn−pn≥fμ/k>0 for each n∈ℕ. It follows from (7), (18), and (31) that(40)limn⟶∞1−βnqn+βnpn=limn⟶∞1−βnxn−x∗xn−x∗+βnun−x∗xn−x∗=limn⟶∞1−βnxn+βnun−x∗xn−x∗=limn⟶∞zn−x∗xn−x∗=kk=1.
This implies that(41)limn⟶∞1−βnqn+βnpn=1.
Therefore, by Lemma1, we have limn⟶∞‖qn − pn‖ = 0, a contradiction. Therefore, we have limn⟶∞d(xn+1, F(T)) = 0, and hence,(42)limn⟶∞xn−x∗=0.
Thus, the sequence {xn} converges strongly to a fixed point x∗ ∈ F(T) ≠ ∅. The proof of Theorem 2 is completed.
Next, we prove the following weak convergence result.Theorem 3.
SupposeX is a uniformly convex Banach space satisfying Opial’s condition and C is a nonvoid closed convex subset of X. Let T:C⟶PC be a multivalued mapping such that F(T) ≠ Ø and PT is a quasi-nonexpansive mapping. Suppose that {xn} is the sequence defined in (7). If I − PT is demiclosed at zero. Then, the sequence {xn} converges weakly to a fixed point of T.Proof.
Suppose thatx∗ ϵ F(T) = F(PT). It follows from Lemma 3 that limn⟶∞‖xn − x∗‖ exists for each x∗. Next, we show that the sequence {xn} has a unique weak subsequential limit in F(T). We proceed as follows: let y1∗ and y2∗ be weak limits of the subsequences xnk and xnj of {xn}, respectively. By (33), we have that limn⟶∞‖xn − un‖ = 0. Using the fact that the mapping I − PT is demiclosed at zero, we have y1∗∈FPT=FT. Similarly, we can show that y2∗∈FT. Next, we prove that the weak limit is unique. Suppose that y1∗≠y2∗. Because X satisfies Opial’s condition, we have(43)limn⟶∞xn−y1∗=limnk⟶∞xnk−y1∗<limnk⟶∞xnk−y2∗=limn⟶∞xn−y2∗=limnj⟶∞xnj−y2∗<limnj⟶∞xnj−y1∗=limn⟶∞xn−y1∗,a contradiction. Hence, the sequence {xn} converges weakly to a point of F(T). The proof of Theorem 3 is completed.
Next, we prove that the Picard–Ishikawa hybrid iterative process (7) and the CR iterative process (8) have the same rate of convergence.Proposition 1.
Suppose thatX is a normed linear space and C is a nonvoid closed convex subset of X. Let T:C⟶PC such that F(T) ≠ Ø and PT is a quasi-nonexpansive mapping. Suppose x0 = u0 ϵ C, {xn} is a sequence defined by the Picard–Ishikawa hybrid iterative process (7), and {un} is the CR iterative process (8) converging to the same fixed point x∗ ϵ F(T), where {αn}, {βn}, and {γn} are sequences in [0, 1]. Then, the sequences {xn} and {un} have the same rate of convergence.Proof.
By (16), we have(44)xn+1−x∗≤xn−x∗≤xn−1−x∗≤xn−2−x∗⋮≤x0−x∗.
Let(45)an−x∗=x0−x∗.
Using (4) and (8), we have(46)un+1−x∗=1−αnyn+αnsn−x∗≤1−αnyn−x∗+αnsn−x∗≤1−αnyn−x∗+αnHPTyn,PTx∗≤1−αnyn−x∗+αnyn−x∗=yn−x∗.
Next, we have the following estimate:(47)yn−x∗=1−βngn+βnhn−x∗≤1−βngn−x∗+βnhn−x∗≤1−βnHPTun,PTx∗+βnHPTzn,PTx∗≤1−βnun−x∗+βnzn−x∗,(48)zn−x∗=1−γnun+γngn−x∗≤1−γnun−x∗+γngn−x∗≤1−γnun−x∗+γnHPTun,PTx∗≤1−γnun−x∗+γnun−x∗=un−x∗.
Using (48) in (47), we obtain(49)yn−x∗≤1−βnun−x∗+βnun−x∗=un−x∗.
Using (49) in (46), we have(50)un+1−x∗≤un−x∗≤un−1−x∗≤un−2−x∗⋮≤u0−x∗.
Let(51)bn−x∗=u0−x∗.
Hence, using (45), (51), and the condition that x0 = u0 ϵ C, we obtain(52)limn⟶∞an−x∗bn−x∗=limn⟶∞x0−x∗u0−x∗=x0−x∗u0−x∗=x0−x∗x0−x∗=1.
Because0 < l = 1 < ∞, it follows that the sequences {xn} and {un} have the same rate of convergence.


## 4. Applications to Solution of Two-point Boundary Value Problems

In this section, we apply the Picard–Ishikawa hybrid iterative process recently introduced by Okeke [3] for finding the solution of two-point boundary value problems for a second-order differential equation. Let X be a real Banach space and x0 ϵ X; the Picard–Ishikawa hybrid iterative process {xn} is defined as follows:(53)zn=1−βnxn+βnTxn,yn=1−αnxn+αnTzn,xn+1=Tyn,n∈ℕ,where {αn} and {βn} are appropriate sequences in [0, 1]. It is known (see [3]) that this fixed-point iterative process converges faster than all of Picard [15], Krasnosel'skii [10], Mann [11], Ishikawa [9], Noor [12], Picard-Mann [27], and Picard–Krasnosel'skii [2] iterations.In this section, we propose a Picard–Ishikawa hybrid-type algorithm for finding the solution of two-point boundary value problem.Green’s function is given as follows. Supposeh(t) is continuous on [a, b] for each t ϵ [a, b]. Functions satisfying x′′ = h(t) are of the following form:(54)x=K0+K1t+∫abt−shsds.Therefore, the unique solution of the following boundary value problem(55)x″=ht,(56)xa=0,xb=0,could be expressed in form of equation (54) with appropriate values for constants K0 and K1. Once the values of the constants K0 and K1 are determined, one can easily prove that the solution x(t) of equations (55) and (56) can be expressed in form of xt=∫abGt,shsds, where G(t, s) is called Green’s function of the boundary value problem x′′ = 0, x(a) = 0, and x(b) = 0 and it is defined as follows:(57)Gt,s=t−as−bb−a,fora≤t≤s,t−bs−ab−a,fors≤t≤b.Similarly, the solution of the following boundary value problem(58)x′′=ht,xa=ξ0,xb=ξ1,can be expressed in the following form:(59)xt=∫abGt,shsds+wt,where wt is the solution of the equation x′′ = 0 satisfying wa=ξ0 and wb=ξ1.Assume that the functionf(t, x, x′) is continuous on a,b×ℝ2. It follows from (59) that if the function wt is the solution of the equation x′′ = 0 with wa=ξ0 and wb=ξ1, then the function x(t) ϵ C(2)[a, b] is a solution of the following boundary value problem:(60)x′′=ft,x,x′,xa=ξ0,xb=ξ1,if and only if x(t) belongs to C(1)[a, b] and it is a solution of the following integral equation:(61)xt=∫abGt,sfs,xs,x′sds+wt,ona,b.Hence, supposeT : C(1)[a, b] ⟶ C(1)[a, b] is defined by(62)Txt=∫abGt,sfs,xs,x′sds+wt,for each x ϵ C(1)[a, b] with a ≤ t ≤ b, then a fixed point of the mapping T is a solution of equation (61). Hence, it is also a solution of the boundary value problem (60).Bello et al. [16] considered the following two-point boundary value problem:(63)x″=ft,x,x′,a≤t≤b,(64)λ0xa+ν0x′a=ξ0,λ1xb+ν1x′b=ξ1,where λi and νi are real constants such that λi2+νi2>0, i = 0, 1.In order to solve (63) and (64) using our proposed Picard–Ishikawa hybrid iterative process (53), we first transform (63) and (64) to the following relation:(65)zn″=1−βnxn″+βnxn″=1−βnxn″+βnft,xn,xn′,yn″=1−αnxn″+αnzn″=1−αnxn″+αnft,zn,zn′,xn+1″=yn+1″=ft,yn,yn′,λ0xn+1a+ν0xn+1′a=ξ0,λ1xn+1b+ν1xn+1′b=ξ1,where {αn} and {βn} are sequences in [0, 1] satisfying ∑n=0∞αn=∞. One can prove that x(t) is a solution of (63) and (64) if and only if x(t) is a solution of the following equivalent integral equation:(66)xt=∫abGt,sfs,xs,x′sds+wt,ona,b,where G(t, s) is Green’s function of the following associated boundary value problem:(67)x″=0,λ0xa+ν0x′a=ξ0,λ1x′b+ν1x′b=ξ1,and wt is the solution of (67).Next, supposeT : C(1)[a, b] ⟶ C(1)[a, b] is defined as follows:(68)Txt=∫abGt,sfs,xs,x′sds+wt,Tzt=∫abGt,sfs,zs,z′sds+wt,Tyt=∫abGt,sfs,ys,y′sds+wt,where T is a mapping such that any solution x(t) of (63) and (64) is a fixed point of T [32].We next embark on the derivation of our fixed-point iterative method for solving the proposed boundary value problem. SupposeD is a nonvoid convex subset of a real Banach space X and T:D⟶PD is a multivalued mapping. For arbitrary x0 ϵ D, let {xn} be the iterative sequence generated by (53), with sequences {αn} and {βn} in [0, 1] satisfying ∑n=0∞αn=∞.Next, we compare our method (65) with (53) to establish their equivalence. We obtain this by first differentiating (68) as follows:(69)Txn′=∫ab∂∂tGt,sfs,xns,xn′sds+w′t,Tzn′=∫ab∂∂tGt,sfs,zns,zn′sds+w′t,Tyn′=∫ab∂∂tGt,sfs,yns,yn′sds+w′t.Secondly, differentiating (53), we have(70)zn′′=1−βnxn′′+βnTxn′′,yn′′=1−αnxn′′+αnTzn′′,xn+1′′=Tyn′′,n∈ℕ.Our third step is to differentiate (69) as follows:(71)Txn′′=∫ab∂∂tGt,xns,xn′sds+w′t,Tzn′′=∫ab∂∂tGt,zns,zn′sds+w′t,Tyn′′=∫ab∂∂tGt,yns,yn′sds+w′t.Lastly, substituting (71) in (70), we have(72)zn′′=1−βnxn′′+βn∫ab∂∂tGt,xns,xn′sds+w′t,yn′′=1−αnxn′′+αn∫ab∂∂tGt,zns,zn′sds+w′t,xn+1′′=∫ab∂∂tGt,yns,yn′sds+w′t,n∈ℕ.Hence, (72) can be written as follows, which is our proposed fixed-point iteration method:(73)zn′′=1−βnxn′′+βnxn′′=1−βnxn′′+βnft,xn,xn′,yn′′=1−αnxn′′+αnzn′′=1−αnxn′′+αnft,zn,zn′,xn+1′′=yn′′=ft,yn,yn′,n∈ℕ.Next, we give our main results for this section as follows.Theorem 4.
LetT: C(1)[a, b] ⟶ C(1)[a, b] be a contraction mapping defined in (68) with contractive constant a ϵ (0, 1). Suppose x0 ϵ C(1)[a, b] is an affine function. Construct the sequence {xn} generated by the Picard–Ishikawa hybrid iterative process (53) satisfying x′′ = 0 and the boundary condition defined in (64) with {αn} and {βn} sequences in [0, 1] such that ∑n=0∞αn=∞. Then, {xn} converges to a unique solution x∗ ϵ C(1)[a, b] of (63) and (64).Proof.
The existence and uniqueness ofx∗ ϵ C(1)[a, b] is guaranteed by the famous Banach contraction mapping principle. We now show that xn ⟶ x∗ as n ⟶ ∞. Using (53), we have(74)xn+1−x∗=Tyn−x∗≤ayn−x∗.
Next, we have the following estimates:(75)yn−x∗=1−αnxn+αnTzn−x∗≤1−αnxn−x∗+αnTzn−x∗≤1−αnxn−x∗+aαnzn−x∗.
Next, we have(76)zn−x∗=1−βnxn+βnTxn−x∗≤1−βnxn−x∗+βnTxn−x∗≤1−βnxn−x∗+aβnxn−x∗=1−βn1−axn−x∗.
Using (76) in (75), we have(77)yn−x∗≤1−αnxn−x∗+aαn1−βn1−axn−x∗=1−αn1−a1−βn1−axn−x∗.
Using (77) in (74), we obtain(78)xn+1−x∗≤a1−αn1−a1−βn1−axn−x∗.
Continuing this process, we obtain the following inequalities:(79)xn+1−x∗≤a1−αn1−a1−βn1−axn−x∗≤a1−αn−11−a1−βn−11−axn−1−x∗≤a1−αn−21−a1−βn−21−axn−2−x∗⋮≤a1−α01−a1−β01−ax0−x∗.
Using the inequalities in (79), we have(80)xn+1−x∗≤an+1∏k=0n1−αk1−a1−βk1−ax0−x∗.
Becausea ϵ (0, 1) and αn, βn ϵ [0, 1] for each n∈ℕ, [1 − αn(1 − a(1 − βn(1 − a)))] < 1. It is well known in analysis that 1 − y ≤ e−y for each y ϵ [0, 1]. Hence, using these facts in (80), we have(81)xn+1−x∗≤an+1x0−x∗e−1−a1−βn1−a∑k=0nαk⟶0,asn⟶∞.
This means thatxn ⟶ x∗ as n ⟶ ∞. The proof of Theorem 4 is completed.Remark 2.
Theorem4 and the results of this section is an improvement on the results of Bello et al. [16] because it is known (see [3]) that the Picard–Ishikawa hybrid iteration method used in our results converges faster than the Mann-type method used by Bello et al. [16].


## 5. Numerical Examples

In this section, we provide some numerical examples to validate our analytical results. We compare the speed of convergence between the CR iterative scheme {un} in (6) and the Picard–Ishikawa hybrid iterative scheme {xn} in (7). In the following figures, we denote the CR iterative process by CR and the Picard–Ishikawa hybrid iterative process by PI. All the codes were written in MATLAB (R2010a) and run on PC with Intel(R) Core(TM) i3-4030U CPU @ 1.90 GHz.We begin with the following example ([20], Example 2.3).Example 1.
Supposeℝ,⋅ is a normed linear space with the usual norm and C = [0, 1]. Define T:C⟶PK by Tx=0,2x+1/4. Khan et al. [20] proved that T is quasi-nonexpansive with FT=0,1/2. We consider the following cases for our numerical experiments:Case 1: choosex0 = u0 = 0.1, αn=n/5n+1, βn=n/10n+1, and γn=1/7n+2, and the number of iteration for each iterative scheme is n = 100. The graph of this case is presented in Figure 1.Case 2: choosex0 = u0 = 1 and αn=βn=γn=1/8n+7, and the number of iteration for each iterative scheme is n = 100. The graph of this case is presented in Figure 2.Example 2.
LetT : X ⟶ X be a mapping such that Tx=x/3, with a=1/2 and X = [0, ∞). Suppose the first iteration u0 = x0 = 13 and the number of iterations for each iterative scheme is n = 100. Choose αn=n/6n+2, βn=n/10n+1, and γn=1/3n+2. Then, Figure 3 shows the errors versus iteration numbers for this example.
Next, we present the following graphs of errors versus iteration numbers (n) for each case.Figure 1
Errors versus iteration numbers (n): Example 1 (Case 1).Figure 2
Errors versus iteration numbers (n): Example 1 (Case 2).Figure 3
Errors versus iteration numbers (n): Example 2.Remark 3.
Clearly, from Figures1 and 2 of Example 1, we see that the Picard–Ishikawa hybrid iterative process and the CR iterative process have the same rate of convergence for a class of multivalued quasi-nonexpansive mappings. Similarly, from Figure 3 of Example 2, we see that the Picard–Ishikawa hybrid iterative process and the CR iterative process have the same rate of convergence for a class of contraction mappings.



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