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

Okeke, Godwin Amechi; Abbas, Mujahid; de la Sen, Manuel

doi:https://doi.org/10.1155/2020/8634050

Discrete Dynamics in Nature and Society

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 8634050 | https://doi.org/10.1155/2020/8634050

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

Godwin Amechi Okeke,¹Mujahid Abbas,^2,3and Manuel de la Sen⁴

Academic Editor: Douglas R. Anderson

Received24 Jan 2020

Accepted24 Feb 2020

Published20 Mar 2020

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.

1. Introduction

If the existence of the solution of a fixed-point equation involving an operator T 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

Let X be a Banach space, x ∈ X, and A ⊆ X. Define d(x, A) = inf{d(x, y) : y ∈ A}.

Definition 1 (see [20]). Suppose X 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 space X and closed convex subsets of a uniformly convex Banach space are proximinal. Let N(X), CL(X), CB(X), K(X), and 2^X denote the class of all nonvoid subsets, nonvoid closed subsets, nonvoid bounded and closed subsets, nonvoid compact subsets, and subsets of X, respectively. Let 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:for each A, B ∈ CB(X). We say that a point x^∗ ∈ X is a fixed point of T : X ⟶ CL(X) if x^∗ ∈ . We denote by F(T) ≔ {x^∗ ∈ X : x^∗ ∈ Tx^∗} the set of all the fixed points of T and P_T(x) = {y ∈ Tx : ‖ x − y‖ = d(x, Tx)} for a multivalued mapping . 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 mapping T : A ⊆ X ⟶ 2^X is called(i)A contraction if there exists a ∈ [0, 1) such that holds for all x, y ∈ A. If a > 0, then we say that T is Lipschitzian.(ii)Nonexpansive if for all x, y ∈ A, we have(iii)Quasi-nonexpansive if for each x ϵ A and x^∗ ϵ F(T) ≠ Ø, we have

Definition 3 (see [26]). A Banach space X is said to satisfy Opial’s condition if for each sequence {u_n} in X such that u_n ⇀ u implies thatfor each ϵ X with ≠ u.

Definition 4. A multivalued mapping T : C ⟶ CB(C) is said to be demiclosed at y ϵ C if for each sequence {x_n} in C weakly converges to x and y_n ϵ Tx_n 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.
Suppose X is a Banach space, T: X ⟶ X, and u₀ ϵ X. Define the sequence bywhere {α_n}, {β_n}, and {γ_n} are sequences of positive numbers in [0, 1] satisfying . 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 ⟶ 2^X, and x₀ ϵ X. Define the sequence bywhere {α_n} and {β_n} are sequences in [0, 1] satisfying appropriate conditions, u_n ϵ P_T(x_n), ϵ P_T(z_n), and ϵ P_T(y_n).

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:where ϵ P_T(u_n), h_n ϵ P_T(z_n), and s_n ϵ P_T(y_n).
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 that and are two fixed-point iteration processes that converge to a certain fixed point of a mapping T; we say that converges faster than ([28]) ifThe following definitions are due to Berinde [1].

Definition 5 (see [1]). Let and be two sequences of positive numbers that converge to a and b, respectively. Assume there exists(i)If l = 0, then it is said that the sequence converges to a faster than the sequence to b(ii)If 0 < l < ∞, then we say that the sequences and have the same rate of convergence.

Definition 6 (see [1]). Suppose that, for two fixed-point iterative processes and , both converging to the same fixed point , the error estimatesare available, where and are two sequences of positive numbers converging to zero. If converges faster than , then converges faster than to .
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 for each x ϵ C.
The following lemma will be needed in this study.

Lemma 1 (see [30]). Suppose that X is a uniformly convex Banach space and 0 < p ≤ t_n ≤ q < 1 for each . Let {x_n} and {y_n} be two sequences of X such that lim sup_n⟶∞‖x_n‖ ≤ r, lim sup_n⟶∞‖y_n‖ ≤ r, and lim_n⟶∞‖t_nx_n + (1 − t_n)y_n‖ = r hold for some r ≥ 0. Then, lim_n⟶∞‖x_n − y_n‖ = 0.

Lemma 2. (see [31]). If and P_T(x) = {y ϵ Tx:‖x − y‖ = d(x, Tx)}. Then, the following are equivalent:(i)x ϵ F(T)(ii)P_T(x) = {x}(iii)x ϵ F(P_T)Moreover, F(T) = F(P_T).

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

We begin with the following convergence results.

Lemma 3. Suppose that X is a normed linear space and C is a nonvoid closed convex subset of X. Let such that F(T) ≠ Ø and P_T is a quasi-nonexpansive mapping. Suppose that {x_n} is a sequence defined by the iteration scheme (7). Then, lim_n⟶∞‖x_n − ‖ exists for each ϵ F(T) and lim_n⟶∞d(x_n, P_T(x_n)) = 0.

Proof. To prove that lim_n⟶∞‖x_n − x^∗‖ exists for each x^∗ ϵ F(T) ≠ Ø, we proceed as follows. Using (4) and (7), we obtain the following estimates:Next, by (4) and (7), we have the following estimates:Next, we obtain thatUsing (14) in (13), we haveUsing (14) in (12), we haveThis implies that {‖x_n − x^∗‖} is decreasing. Therefore, lim_n⟶∞‖x_n − x^∗‖ exists for each x^∗ ϵ F(T).
Next, we show thatLetfor some constant k ≥ 0. If k = 0, then (17) holds trivially. Suppose that k > 0; because d(x_n, P_T(x_n)) ≤ ‖x_n − u_n‖, it suffices to establish that lim_n⟶∞‖x_n − u_n‖ = 0. Clearly, we haveThis implies thatUsing (14) and (15), we obtainObserve thatHence, we haveSimilarly, we haveHence,Clearly,Hence, using (23), (25), (26), and Lemma 1, we obtainFrom (25), we haveHence, we haveUsing (23) and (29), we haveSimilarly, we can show thatThis means thatIt follows from (20), (32), and Lemma 1 thatTherefore, we haveThe proof of Lemma 3 is completed.
Next, we prove the following strong convergence theorem.

Theorem 1. Suppose that C is a nonvoid compact convex subset of a real Banach space X, with F(T) ≠ Ø, and P_T is a quasi-nonexpansive mapping. Let {x_n} be the iterative sequence defined in (7). Then, the sequence {x_n} converges strongly to a fixed point of T.

Proof. We have already established in Lemma 3 that lim_n⟶∞‖x_n − x^∗‖ exists for each x^∗ ϵ F(T) ≠ Ø. Because C is compact, it follows that there exists a subsequence of {x_n} such that for some y^∗ ϵ C. Hence, by Lemma 3 and the triangle inequality, we haveThis means thatTherefore, is a fixed point of P_T. By Lemma 2, the set of fixed points of P_T coincides with the set of fixed points of T, so the sequence {x_n} 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 that C is a nonvoid closed and convex subset of a real Banach space X. Let satisfies Condition (I) such that F(T) ≠ Ø and P_T is a quasi-nonexpansive mapping. Then, the sequence {x_n} defined by (7) converges strongly to a fixed point of T.

Proof. By Lemma 3, lim_n⟶∞‖x_n − x^∗‖ exists for each x^∗ ϵ F(T) = F(P_T)(Lemma 2). If lim_n⟶∞‖x_n − x^∗‖ = 0, then the result trivially holds. We assume that lim_n⟶∞‖x_n − x^∗‖ = k > 0. Using relation (16), we have ‖x_n+1 − x^∗‖ ≤ ‖ x_n − x^∗‖. Therefore, we haveThis implies that lim_n⟶∞d(x_n+1, F(T)) exists. Next, we show that lim_n⟶∞d(x_n+1, F(T)) = 0. Assume on contrary that lim_n⟶∞d(x_n+1, F(T)) = μ > 0. Then, for each , we haveHence, we have ‖p_n‖ = 1 and ‖q_n‖ ≤ 1 because ‖u_n − x^∗‖ ≤ H(P_T(x_n), P_T(x^∗)) ≤ ‖x_n − x^∗‖. Using the fact that f satisfies Condition (I), we haveBecause f is continuous, we obtain that for each . It follows from (7), (18), and (31) thatThis implies thatTherefore, by Lemma 1, we have lim_n⟶∞‖q_n − p_n‖ = 0, a contradiction. Therefore, we have lim_n⟶∞d(x_n+1, F(T)) = 0, and hence,Thus, the sequence {x_n} 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. Suppose X is a uniformly convex Banach space satisfying Opial’s condition and C is a nonvoid closed convex subset of X. Let be a multivalued mapping such that F(T) ≠ Ø and P_T is a quasi-nonexpansive mapping. Suppose that {x_n} is the sequence defined in (7). If I − P_T is demiclosed at zero. Then, the sequence {x_n} converges weakly to a fixed point of T.

Proof. Suppose that x^∗ ϵ F(T) = F(P_T). It follows from Lemma 3 that lim_n⟶∞‖x_n − x^∗‖ exists for each . Next, we show that the sequence {x_n} has a unique weak subsequential limit in F(T). We proceed as follows: let and be weak limits of the subsequences and of {x_n}, respectively. By (33), we have that lim_n⟶∞‖x_n − u_n‖ = 0. Using the fact that the mapping I − P_T is demiclosed at zero, we have . Similarly, we can show that . Next, we prove that the weak limit is unique. Suppose that . Because X satisfies Opial’s condition, we havea contradiction. Hence, the sequence {x_n} 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 that X is a normed linear space and C is a nonvoid closed convex subset of X. Let such that F(T) ≠ Ø and P_T is a quasi-nonexpansive mapping. Suppose x₀ = u₀ ϵ C, {x_n} is a sequence defined by the Picard–Ishikawa hybrid iterative process (7), and {u_n} is the CR iterative process (8) converging to the same fixed point x^∗ ϵ F(T), where {α_n}, {β_n}, and {γ_n} are sequences in [0, 1]. Then, the sequences {x_n} and {u_n} have the same rate of convergence.

Proof. By (16), we haveLetUsing (4) and (8), we haveNext, we have the following estimate:Using (48) in (47), we obtainUsing (49) in (46), we haveLetHence, using (45), (51), and the condition that x₀ = u₀ ϵ C, we obtainBecause 0 < l = 1 < ∞, it follows that the sequences {x_n} and {u_n} 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 {x_n} is defined as follows:where {α_n} and {β_n} 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. Suppose h(t) is continuous on [a, b] for each t ϵ [a, b]. Functions satisfying x′′ = h(t) are of the following form:

Therefore, the unique solution of the following boundary value problemcould 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 , 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:

Similarly, the solution of the following boundary value problemcan be expressed in the following form:where is the solution of the equation x′′ = 0 satisfying and .

Assume that the function f(t, x, x′) is continuous on . It follows from (59) that if the function is the solution of the equation x′′ = 0 with and , then the function x(t) ϵ C⁽²⁾[a, b] is a solution of the following boundary value problem:if and only if x(t) belongs to C⁽¹⁾[a, b] and it is a solution of the following integral equation:

Hence, suppose T : C⁽¹⁾[a, b] ⟶ C⁽¹⁾[a, b] is defined byfor 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:where λ_i and ν_i are real constants such that , 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:where {α_n} and {β_n} are sequences in [0, 1] satisfying . 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:where G(t, s) is Green’s function of the following associated boundary value problem:and is the solution of (67).

Next, suppose T : C⁽¹⁾[a, b] ⟶ C⁽¹⁾[a, b] is defined as follows: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. Suppose D is a nonvoid convex subset of a real Banach space X and is a multivalued mapping. For arbitrary x₀ ϵ D, let {x_n} be the iterative sequence generated by (53), with sequences {α_n} and {β_n} in [0, 1] satisfying .

Next, we compare our method (65) with (53) to establish their equivalence. We obtain this by first differentiating (68) as follows:

Secondly, differentiating (53), we have

Our third step is to differentiate (69) as follows:

Lastly, substituting (71) in (70), we have

Hence, (72) can be written as follows, which is our proposed fixed-point iteration method:

Next, we give our main results for this section as follows.

Theorem 4. Let T: 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 {x_n} generated by the Picard–Ishikawa hybrid iterative process (53) satisfying x′′ = 0 and the boundary condition defined in (64) with {α_n} and {β_n} sequences in [0, 1] such that . Then, {x_n} converges to a unique solution x^∗ ϵ C⁽¹⁾[a, b] of (63) and (64).

Proof. The existence and uniqueness of x^∗ ϵ C⁽¹⁾[a, b] is guaranteed by the famous Banach contraction mapping principle. We now show that x_n ⟶ x^∗ as n ⟶ ∞. Using (53), we haveNext, we have the following estimates:Next, we haveUsing (76) in (75), we haveUsing (77) in (74), we obtainContinuing this process, we obtain the following inequalities:Using the inequalities in (79), we haveBecause a ϵ (0, 1) and α_n, β_n ϵ [0, 1] for each , [1 − α_n(1 − a(1 − β_n(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 haveThis means that x_n ⟶ x^∗ as n ⟶ ∞. The proof of Theorem 4 is completed.

Remark 2. Theorem 4 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 {u_n} in (6) and the Picard–Ishikawa hybrid iterative scheme {x_n} 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. Suppose is a normed linear space with the usual norm and C = [0, 1]. Define by . Khan et al. [20] proved that T is quasi-nonexpansive with . We consider the following cases for our numerical experiments: Case 1: choose x₀ = u₀ = 0.1, , , and , and the number of iteration for each iterative scheme is n = 100. The graph of this case is presented in Figure 1. Case 2: choose x₀ = u₀ = 1 and , and the number of iteration for each iterative scheme is n = 100. The graph of this case is presented in Figure 2.

Example 2. Let T : X ⟶ X be a mapping such that , with and X = [0, ∞). Suppose the first iteration u₀ = x₀ = 13 and the number of iterations for each iterative scheme is n = 100. Choose , , and . 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.

Remark 3. Clearly, from Figures 1 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.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare no conflict of interests.

Authors’ Contributions

All authors contributed equally to the writing of this paper.

Acknowledgments

The first author’s research was supported by the Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan, through grant number ASSMS/2018-2019/452. This research was funded by Basque Government (IT1207-19).

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Copyright © 2020 Godwin Amechi Okeke et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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