The Implicit Midpoint Procedures for Asymptotically Nonexpansive Mappings

Aibinu, M. O.; Thakur, S. C.; Moyo, S.

doi:https://doi.org/10.1155/2020/6876385

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Abstract Introduction Preliminaries Numerical Example Conclusion Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

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Volume 2020 | Article ID 6876385 | https://doi.org/10.1155/2020/6876385

The Implicit Midpoint Procedures for Asymptotically Nonexpansive Mappings

M. O. Aibinu,^1,2S. C. Thakur,²and S. Moyo³

Academic Editor: Yongqiang Fu

Received14 Feb 2020

Revised09 Apr 2020

Accepted12 May 2020

Published06 Jun 2020

Abstract

The concept of asymptotically nonexpansive mappings is an important generalization of the class of nonexpansive mappings. Implicit midpoint procedures are extremely fundamental for solving equations involving nonlinear operators. This paper studies the convergence analysis of the class of asymptotically nonexpansive mappings by the implicit midpoint iterative procedures. The necessary conditions for the convergence of the class of asymptotically nonexpansive mappings are established, by using a well-known iterative algorithm which plays important roles in the computation of fixed points of nonlinear mappings. A numerical example is presented to illustrate the convergence result. Under relaxed conditions on the parameters, some algorithms and strong convergence results were derived to obtain some results in the literature as corollaries.

1. Introduction

Let be a bounded subset of a Banach space . A mapping is called asymptotically nonexpansive if there exists a sequence of positive real numbers with as for which

In 1972, Geobel and Kirk [1] introduced this concept of asymptotically nonexpansive mappings as an important generalization of the class of nonexpansive mappings in which for all n [1]. T is said to be uniformly -Lipschitzian, if there exists a constant such that

Notice that every asymptotically nonexpansive is uniformly -Lipschitzian with a constant . The set of fixed point of , will be denoted by . A mapping is called a -contraction if there exists such that

It is well known that a contraction on has a unique fixed point in .

A powerful numerical method for solving ordinary differential equations and differential algebraic equations, which also has a long history, is the implicit midpoint procedure. Akin to the implicit midpoint procedure is the fractal structures of Mandelbrot and Julia sets which have been demonstrated for practical application in quadratic, cubic, and higher degree polynomials. Kang et al. in [2] defined Jungck Noor iteration with -convexity and established the escape criterions for quadratic, cubic, and degree complex polynomials. Auzinger and Frank in [3] studied the structure of the global discretization error for the implicit midpoint and trapezoidal rules applied to nonlinear stiff initial value problems. Bader and Deuflhard in [4] introduced a semi-implicit extrapolation method especially designed for the numerical solution of stiff systems of ordinary differential equations. The implicit midpoint rule is described as a theoretical foundation of the numerical treatment of problems arising in physical and biological sciences (Kastner-Maresch [5]). The implicit midpoint rule is applied to obtain the periodic solution of a nonlinear time-dependent evolution equation and a Fredholm integral equation (Somali and Davulcua [6]). Consider the ordinary differential equation:

A sequence is generated by the implicit midpoint rule via the recursion:where is a stepsize and is the set of positive integers. For a Lipschitz continuous and sufficiently smooth map , it is known that, as uniformly over for any fixed , the sequence generated by (5) converges to the exact solution of (4). By rewriting the function in the form , the differential equation (4) becomes . Consequently, the equilibrium problem associated with the differential equation is transformed to the fixed point problem [7]. Extension of the implicit midpoint rule to nonexpansive mappings by Alghamdi et al. in [8] generates a recursion sequence:where is a nonexpansive mapping and is a closed convex subset of a real Hilbert space . They proved the weak convergence of (6) under certain conditions on . Still, in Hilbert space, Xu et al. in [9] used contractions to regularize the implicit midpoint rule (6) and introduced the implicit procedure:

They proved a strong convergence theorem for the sequence to a fixed point of which also solves the variational inequality:

In 2015, Yao et al. in [7] introducedwhich gives a faster approximation compared with (7), where is a nonexpansive mapping in a Hilbert space and . Aibinu et al. in [10] compared the rate of convergence of the iteration procedures (6), (7), and (9). Aibinu and Kim in [11] recently introduced a new scheme of the viscosity implicit iterative algorithms for nonexpansive mappings in Banach spaces. Suitable conditions were imposed on the control parameters to prove a strong convergence theorem for the considered iterative sequence. Also, Aibinu and Kim in [12] studied the analytical comparison of schemes (7) and (9) to determine the sequence that converges faster in approximating a fixed point of a nonexpansive mapping. Literature review reveals the following problems, which this paper is devoted to address.

Problem 1. Can one study the implicit iterative procedure (9) for the class of asymptotically nonexpansive mappings which is more general than nonexpansive mappings?

Problem 2. Under what conditions will the main results of Yao et al. in [7] hold for asymptotically nonexpansive mappings in the general Banach spaces?

Motivated by the previous works, this paper is devoted for the extension of the previous results in the literature, to a more general space and to study the analogue of algorithm (9) for the class of asymptotically nonexpansive mappings. Therefore, for a Banach space with a uniformly Gteaux differentiable norm possessing uniform normal structure and for arbitrary , this paper considers the iterative algorithm given bywhere is an asymptotically nonexpansive mapping and the real sequences and are chosen such that . We study the convergence of the important class of asymptotically nonexpansive mappings, which is a generalization of the class of nonexpansive mappings. The necessary conditions for the convergence of the class of asymptotically nonexpansive mappings are established, by using this well-known iterative algorithm which plays important roles in the computation of fixed points of nonlinear mappings. A numerical example is given to illustrate the convergence result.

2. Preliminaries

Let be a real Banach space with dual and denote the norm on by . The normalized duality mapping is defined aswhere is the duality pairing between and . Let denotes the unit ball of . The modulus of convexity of is defined as follows . is uniformly convex if and only if for every . is said to be smooth (or Gteaux differentiable) if the limit,exists for each . is said to have uniformly Gteaux differentiable norm if, for each , the limit is attained uniformly for and uniformly smooth if it is smooth and the limit is attained uniformly for each . Recall that if is smooth, then is single-valued and onto if is reflexive. Furthermore, the normalized duality mapping is uniformly continuous on bounded subsets of from the strong topology of to the weak-star topology of if is a Banach space with a uniformly Gteaux differentiable norm.

Let be a nonempty bounded closed convex subset of a Banach space , and let the diameter of be defined by . For each , let and let , and the Chebyshev radius of is relative to itself. The normal structure coefficient of (see [13]) is defined as the number:

A space , such that , is said to have uniformly normal structure. Recall that a space with uniformly normal structure is reflexive and that all uniformly convex or uniformly smooth Banach spaces have uniform normal structure [14, 15]. If is a reflexive Banach space with modulus of convexity , then . It was proved that if the space is uniformly convex; then, every asymptotically nonexpansive self-mapping of has a fixed point [1]. Also, it has been proved that if , then , where for [16, 17].

Recall that a function is said to be weakly lower semicontinuous at if whenever is a sequence in such that converges weakly to ; then,

According to Jung and Kim in [18], let be a mean on positive integers . A mean on is a continuous linear functional on satisfying . It is called a Banach limit if for every .

The following lemmas are needed in the sequel.

Lemma 1. (see [19]). Let be a nonempty closed convex subset of a Banach space with a uniformly Gteaux differentiable norm, let be a bounded sequence of , and let be a mean on . Let . Then,if and only if for all , where is the duality mapping of .

Lemma 2. (see [20]). Let be a sequence of nonnegative real numbers satisfying the property:where and such that(i)(ii)Then, converges to zero, as .

Lemma 3. (see [21]). Let be a Banach space with uniform normal structure, a nonempty closed convex and bounded subset of , and an asymptotically nonexpansive mapping. Then, has a fixed point.

Lemma 4. (see [22]). Let be a real Banach space and the normalized duality map on . Then, for any given , the following inequality holds:

Lemma 5. (see [23, 24]). Let be a real number and such that for all Banach limits. If , then .

Lemma 6. (see [15]). Suppose is a Banach space with uniformly normal structure, is a nonempty bounded subset of , and is a uniformly -Lipschitzian mapping with . Suppose also there exists a nonempty, bounded, closed, and convex subset of with the following property:where is the weak -limit set of at , that is, the setThen, has a fixed point in .

3. Main Results

Assumption 1. Let be a real Banach space with a uniformly Gteaux differentiable norm possessing uniform normal structure and a nonempty bounded closed convex subset of . Suppose is an asymptotically nonexpansive mapping with and is a contraction with constant . For arbitrary and real sequences and satisfying , this paper considers the iterative scheme given by (10). Assume that satisfies the following conditions:(i)(ii)(iii)where is a sequence of positive real numbers with as . Then, the sequence is well defined. To show this, for arbitrary , define the mapping :Then, for ,Using condition (iii), for any given positive number , there exists a sufficient large positive integer , such that, for any , we obtain thatTherefore, since is a sequence of positive real numbers with as and for all , it is obvious thatThus, from (21) and (23), it shows that is a contraction. Therefore, (10) is well defined since every contraction in a Banach space has a fixed point.

Firstly, the proof of the following lemmas are given, which are useful in establishing the main result.

Lemma 7. Let be a Banach space with a uniform normal structure, a nonempty bounded closed convex subset of , and an asymptotically nonexpansive mapping. Suppose is a contraction with constant and assume that satisfies conditions . For an arbitrary , the iterative sequence given by (10) is bounded.

Proof. The sequence is shown to be bounded.
By Lemma 3, . Then, for ,Therefore,which is equivalent toObserve thatAlso,Consequently, (26) becomesThis implies that the sequence is bounded and hence and are also bounded.
Obviously,where .

Lemma 8. Let be a real Banach space with a uniformly Gteaux differentiable norm possessing uniform normal structure, a nonempty bounded closed convex subset of an asymptotically nonexpansive mapping with sequence a continuous mapping, and a bounded sequence in such that . Suppose is a path in defined by such that as . Then,

Proof. From , let and define . Clearly,Since is given by , we can writeSince , and are bounded, is bounded. By Lemma 4,Therefore,From (32) and by recalling that as ,Observe thatNoting the hypothesis that as , coupled with the continuity of and uniform continuity of the duality mapping on bounded subsets of , it givesFrom (36)–(38), it can be deduced that

Theorem 1. Let be a real Banach space with a uniformly Gteaux differentiable norm possessing uniform normal structure, a nonempty bounded closed convex subset of , and an asymptotically nonexpansive mapping with the sequence of positive real numbers such that as , and . Suppose is a contraction with constant . Assume that satisfies conditions and in addition (iv)
Then, for an arbitrary , the iterative sequence given by (10) converges in norm to a fixed point of which is the unique solution of variational inequality:

Proof.

Step 1. It is shown in this step that .
Observe thatfor some , where denotes the set of positive real numbers. It is required thatTo evaluate ,Let be a constant such thatThen,Using conditions (i) and (iv),Next, to evaluate ,In a similar manner, one can apply (46) and (47) to obtain thatConsequently, from (46)–(48), it can be deduced that

Step 2. It is claimed thatwhere is a fixed point of .
Simply by applying Lemma 7, it is obtained thatAlternatively, define a function byfor all . Observe that is continuous, convex, and as . Since is reflexive, attains its infimum over . Let such that and letThen, is nonempty because and it is also closed, convex, and bounded (for more details, see e.g., [15, 19, 25]). Moreover, has a fixed point in by Lemma 6. Indeed, let and be the weak -limit set of at , that is, the set . Then, from the weak-lower-semicontinuity of and the fact that (condition (iv)), the following estimate is obtained:This shows that satisfies the property (P). By applying Lemma 6, , it follows that . Suppose that . Then, Lemma 1 gives thatfor all . In particular,Since as (46), and the duality mapping is norm to weak-star uniformly continuous on bounded set (earlier mentioned in Section 2); this givesTherefore, the sequence satisfies the conditions of Lemma 5. Thus,

Step 3. Lastly, it is shown that :Therefore,Observe thatConsequently,By applying Lemma 2 to (58) and (62), it is deduced that as ,

where, respectively, and .

Remark 1. It is generally known that a Hilbert space is a subclass of uniformly convex Banach spaces. Moreover, it is obvious that every nonexpansive mapping is an asymptotically nonexpansive mapping. Consequently, Theorem 1 is an improvement and generalization of the main results of Zhao et al. [26], Aibinu et al. [10], Luo et al. [27], and Yao et al. [7] as stated below.

Corollary 1. (see [26]). Let be a nonempty closed and convex subset of a Hilbert space , be an asymptotically nonexpansive mapping with a sequence and . Let be a contraction on with coefficient . For an arbitrary initial point , let be the sequence generated byThen, the sequence converges strongly to which solves the following variational inequality (8).

Proof. Take to be a Hilbert space in Theorem 1. Moreover, since , by taking in (10), we have . Therefore, we obtain (63) from (10) by taking . Hence, the desired result follows from Theorem 1.

Corollary 2. (see [10]). Let be a uniformly smooth Banach space and be a nonempty closed convex subset of . Let be a nonexpansive mapping with and be a -contraction. Suppose satisfies and satisfies (C₁) (C₂) For an arbitrary , define the iterative sequence by (9). Then, as , the sequence converges in norm to a fixed point of , where is the unique solution in to the variational inequality (8).

Proof. Every nonexpansive mapping is an asymptotically nonexpansive mapping and all uniformly convex or uniformly smooth Banach spaces have uniform normal structure (see [14, 15]). Therefore, it suffices to show that :Let be a constant such that and since . Then,This completes the proof.

Corollary 3. (see [27]). Let be a closed convex subset of a uniformly smooth Banach space . Let be a nonexpansive mapping with and a contraction with coefficient . Let be a sequence generated by the viscosity implicit midpoint rule (7), where is a sequence in such that it satisfies conditions and either or . Then, converges strongly to a fixed point of , which also solves the variational inequality (8).

Proof. Take in equation (9). The desired result follows from Corollary (28).

Corollary 4. (see [7]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping with . Suppose is a -contraction. For given arbitrarily, let the sequence be generated by (9). Assume that satisfies and and satisfies Then, the sequence generated by (9) converges strongly to .

Proof. Observe that a Hilbert space is a subclass of a uniformly smooth Banach space. By taking to be a Hilbert space in Corollary 2, the result of Yao et al. in [7] is obtained.

4. Numerical Example

Numerical example is given in this section to illustrate the convergence of the sequence of iteration in the main theorem.

Example 1. Let be an orthogonal subspace of with the norm and the inner product for and . Define a mapping byfor each . Then,orfor any (Zhang et al. [28]). Take to be and observe that for any . Thus, is an asymptotically nonexpansive mapping on and . We define by . Let , and . It is obvious that the sequences , and satisfy conditions (i)–(iii) in Assumption 1, and according to (67) or (68), . The table and figures are given to display the numerical results and convergence of the our sequence of iteration to fixed points of . The computations are carried out by using MATLAB. Table 1 exhibits numerical values for the nth iterations as converges to a fixed point. Figure 1 displays the convergence of to the fixed point with . Convergence of to the fixed point with is shown in Figure 2. Graph of as converges to the fixed point with is presented in Figure 3.

5. Conclusion

This paper considered the implicit midpoint rule of asymptotically nonexpansive mappings using the viscosity technique. The necessary conditions for the convergence of the class of asymptotically nonexpansive mappings are established by using this well-known iterative algorithm which plays important roles in the computation of fixed points of nonlinear mappings. Moreover, the previous results have been extended from the Hilbert space to a Banach space with a uniformly Gteaux differentiable norm possessing uniform normal structure. A numerical example is given to show the convergence of the sequence of iteration in the main theorem. The result presented in this paper plays crucial theoretical base of the numerical analysis for nonlinear problems which are often encountered in physical and biological sciences. The future direction for this paper is to determine for the class of asymptotically nonexpansive mappings and the conditions for the convergence of Jungck Noor iteration with -convexity, defined by Kang et al. [2] and to compare the rate of convergence with the implicit midpoint procedure.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

All authors contributed significantly in writing this article. All authors read and approved the final manuscript.

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Copyright © 2020 M. O. Aibinu 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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