Abstract
Finding the solutions of nonlinear operator equations has been a subject of research for decades but has recently attracted much attention. This paper studies the convergence of a newly introduced viscosity implicit iterative algorithm to a fixed point of a nonexpansive mapping in Banach spaces. Our technique is indispensable in terms of explicitly clarifying the associated concepts and analysis. The scheme is effective for obtaining the solutions of various nonlinear operator equations as it involves the generalized contraction. The results are applied to obtain a fixed point of -strictly pseudocontractive mappings, solution of -inverse-strongly monotone mappings, and solution of integral equations of Fredholm type.
1. Introduction
The Viscosity Approximation Method (VAM) for solving nonlinear operator equations has recently attracted much attention. In 1996, Attouch [1] considered the viscosity solutions of minimization problems. In 2000, Moudafi [2] introduced an explicit viscosity method for nonexpansive mappings. The iterative explicit viscosity sequence is defined by where is a contraction on and the nonexpansive mapping is also defined on , which is a nonempty closed convex subset of a Hilbert space . The sequence (1) converges strongly to a fixed point of a nonexpansive mapping under suitable conditions. Xu et al. [3] proposed the concept of the implicit midpoint rule where , , and remain as defined in (1). Under certain conditions, they established that the implicit midpoint sequence (2) converges to a fixed point of which also solves the variational inequality where is the inner product. Aibinu et al. [4] studied the convergence of the sequence (2) in uniformly smooth Banach spaces. Ke and Ma [5] introduced generalized viscosity implicit rules which extend the results of Xu et al. [3]. The generalized viscosity implicit procedures are given by where with Replacement of strict contractions in (5) by the generalized contractions and extension to uniformly smooth Banach spaces was considered by Yan et al. [6]. Under certain conditions imposed on the parameters involved, the sequence converges strongly to a fixed point of the nonexpansive mapping , which is also the unique solution of the variational inequality where is a normalized duality mapping and is the duality pairing. Aibinu and Kim [7] used an analytical method to compare the rate of convergence of the sequences (4) and (5). Due to the important roles which nonlinear operator equations play in modeling many phenomena in scientific fields, research on the solution and application of nonlinear operator equations are ever green (see e.g., [8–12]).
Inspired by the previous works in this direction, this paper studies an implicit iterative sequence that involves the generalized contraction and which is effective for obtaining the solutions of various nonlinear operator equations. Precisely, for a nonempty closed convex subset of a uniformly smooth Banach space and real sequences and such that , an implicit iterative scheme is defined from an arbitrary by where is a nonexpansive mapping and is a generalized contraction mapping. The sequence is shown to converge strongly to a fixed point of a nonexpansive mapping in Banach spaces. The adopted technique is indispensable in terms of explicitly clarifying the associated concepts and analysis. The results are applied to obtain a fixed point of -strictly pseudocontractive mapping, solution of -inverse-strongly monotone mapping, and solution of the integral equation of Fredholm type. An example of real sequences which satisfy the stated conditions of our iteration is given.
2. Preliminaries
Definition 1. Letbe a real Banach space with dualand denote the norm onby. The normalized duality mappingis defined aswhere is the duality pairing between and
Definition 2. Let. is said to be smooth (or Gâteaux differentiable) if the limitexists for each is said to have uniformly Gâteaux 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 Also, is said to be uniformly smooth if It is well known that is nondecreasing. is said to be -uniformly smooth if there exist a constant and a real number such that , , and are typical examples of such spaces, where 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 Gâteaux differentiable norm.
Definition 3. Let be a metric space and a subset of with a mapping defined on (i) is said to be Lipschitzian if there exists a constant , such that for all is called nonexpansive if , and it is a contraction if . A contraction mapping will be referred to as -contraction mapping(ii) is said to be a Meir-Keeler contraction if for each there exists such that for each with we have (iii)Let be the set of all positive integers and the set of all positive real numbers. A mapping is said to be an -function if for all and for every , there exists such that for each (iv) is called a -contraction if is an -function and , for all
Throughout this paper, the generalized contraction mappings will refer to Meir-Keeler or -contraction contractions. It is assumed that the -function from the definition of -contraction is continuous, strictly increasing, and , where for all . Whenever there is no confusion, and will be written as and , respectively.
We have the following interesting results about the Meir-Keeler contraction.
Proposition 4 Meir and Keeler [13]. Letbe a complete metric space and letbe a Meir-Keeler contraction onThen,has a unique fixed point in.
Proposition 5 Suzuki [14]. Letbe a Banach space,a convex subset ofanda Meir-Keeler contraction. Then,there existssuch thatfor all with
Proposition 6 Lim [15]. Letbe a metric space andbe a mapping. The following assertions are equivalent:(i) is a Meir-Keeler type mapping(ii)there exists an -function such that is a -contraction
Proposition 7 Lim [15]. Letbe a nonempty convex subset of a Banach spacea nonexpansive mapping anda Meir-Keeler contraction. Then,andare Meir-Keeler contractions.
The following lemmas are needed in the sequel.
Lemma 8 Suzuki [16]. Letandbe bounded sequences in a Banach spaceandbe a sequence inwithSuppose thatfor allandThen,
Lemma 9 Sunthrayuth and Kumam [17]. Letbe a nonempty closed and convex subset of a uniformly smooth Banach space. Letbe a nonexpansive mapping such thatandbe a generalized contraction mapping. Assume thatdefined byforconverges strongly toasSuppose thatis a bounded sequence such thatasThen,
Lemma 10 Sunthrayuth and Kumam [17]. Letbe a nonempty closed and convex subset of a uniformly smooth Banach spaceLetbe a nonexpansive mapping such thatandbe a generalized contraction mapping. Then,defined byforconverges strongly towhich solves the following variational inequality:
Lemma 11 Xu [18]. Letbe a sequence of nonnegative real numbers satisfying the following relations:where (i), (ii)(iii), Then,
3. Main Results
Assumption 12. Letbe a nonempty closed convex subset of a uniformly smooth Banach spaceanda generalized contraction mapping. Letbe a nonexpansive self-mapping defined onwithThe real sequencesandare assumed to satisfy the following conditions:(i)(ii)(iii)(iv) (v) for all Under the conditions (i)-(v) of Assumption 12 stated above, this study establishes the convergence of the iterative scheme (7).
Firstly, it is shown that for all , the mapping defined by
for all , where , is a contraction with a contractive constant.
Indeed, for all
Therefore, is a contraction. By Banach’s contraction mapping principle, has a fixed point.
The proof of the following lemmas which are useful in establishing the main result are given.
Lemma 13. Letbe a nonempty closed convex subset of a uniformly smooth Banach spaceanda generalized contraction mapping. Letbe a nonexpansive self-mapping defined onwithFor an arbitrary, define the iterative sequenceby (7). Then, the sequenceis bounded under the conditions (i)-(v) of Assumption 12.
Proof. It is needed to show that the sequence is bounded. For Therefore, Then by induction, we have For So, is bounded. Also, Therefore, Hence, is bounded.
Lemma 14. Letbe a nonempty closed convex subset of a uniformly smooth Banach spaceanda generalized contraction mapping. Letbe a nonexpansive self-mapping defined onwithSupposeis a real sequences inandSetthen
Proof.
Theorem 15. Let be a nonempty closed convex subset of a uniformly smooth Banach space and a generalized contraction mapping. Let be a nonexpansive self-mapping defined on with Assume that the conditions (i)-(v) of Assumption 12 are satisfied. Then, the iterative sequence which is defined from an arbitrary by (7) converges strongly to a fixed point of which solves the variational inequality (6), given by
Proof. Setting and , one can obtain that
Let , , and Then,
It is needed to evaluate This leads to
Let Therefore,
Let and substitute (32) into (29) to obtain
It then follows that
and thus,
Invoking Lemma 8 to obtain that
Consequently,
Next is to show that From (7), it is obtained that
Since let then
which goes to zero as by (37) and condition (ii) of Assumption 12.
Let a sequence be defined by for It is known by Lemma 10 that converges strongly to which solves the variational inequality:
which is equivalent to
It can be shown that
where is the unique fixed point of the generalized contraction (Proposition 7), that is,
By (39), it follows from Lemma 9 that
Due to the continuity of the duality map and the fact that as by (37), it is obtained that
Lastly, it is shown that as
Suppose that the sequence does not converge strongly to Then, there exists and a subsequence of such that for all Therefore, for this there exists such that
Observe that
Multiplying (46) by 2 gives
Using Lemma 11, it shows that as A contradiction, hence, converges strongly to
The next result shows that under suitable conditions, the implicit iterative sequences (5) and (7) converge to the same fixed point.
Theorem 16. Let be a nonempty closed convex subset of a uniformly smooth Banach space and a -contraction mapping with Let be a nonexpansive self-mapping defined on with Let and be real sequences such that Given that then defined by (7) converges to if and only if defined by (5) converges to
Proof. Notice that (7) and (5) are, respectively, given by
It is needed to show that as Since and are bounded, let Then,
where From the given condition, it follows that.
Apply Lemma 11 to (52) and take to get that , as Next, suppose as It follows that
Similarly, suppose as Then,
Hence, the implicit iterative sequences (5) and (7) converge to the same fixed point under suitable conditions.
Remark 17. One can deduce the following results from Theorem 15.
Corollary 18. Let be a nonempty closed convex subset of a uniformly smooth Banach space and a nonexpansive self-mapping defined on with Assume that the real sequences and satisfy the conditions: (i)(ii)(iii)(iv)Then, the iterative sequence which is defined from an arbitrary by converges strongly to a fixed point of which solves the variational inequality (6).
Proof. The result follows from Theorem 15 by simply taking to be the identity mapping on
Corollary 19. Let be a nonempty closed convex subset of a uniformly smooth Banach space and a nonexpansive self-mapping defined on with Assume that the real sequence satisfies the following conditions: (i)(ii)(iii)Then, the iterative sequence which is defined from an arbitrary by converges strongly to a fixed point of which solves the variational inequality (6).
Proof. The result follows from Theorem 15 by simply taking to be the identity mapping on and for all Consequently, this improves and extend the results of Alghamdi et al. [19].
4. Applications
4.1. Application to Fixed Points of -Strictly Pseudocontractive Mappings
Let be a closed convex subset of a real Banach space A mapping is said to be -strictly pseudocontractive mapping if there exists such that where denotes the identity operator on
Zhou [20] established the following lemma which gives a relationship between -strictly pseudocontractive mappings and nonexpansive mappings.
Lemma 20. Letbe a nonempty subset of a 2-uniformly smooth Banach spaceLetbe a-strictly pseudocontractive mapping. Fordefine
Then, as (where is the -uniformly smooth constant of a 2-uniformly smooth Banach space), is nonexpansive such that
The following result is obtained by using Lemma 20 and Theorem 15.
Corollary 21. Let be a nonempty closed convex subset of a -uniformly smooth Banach space and a generalized contraction mapping. Let a-strictly pseudocontractive mapping with Suppose that the conditions (i)-(v) of Assumption 12 are satisfied and is a mapping from into itself, defined by Then, for an arbitrary the iterative sequence defined by converges strongly to a fixed point of which solves the variational inequality
4.2. Application to Solution of -Inverse-Strongly Monotone Mappings
Let be a nonempty closed convex subset of a Hilbert space . The metric projection is defined from onto by and characterized by
is known as the only point in that minimizes the objective over A mapping of into is called monotone if for all The classical variational inequality (VI) problem is to find such that where is a (single-valued) monotone operator in Hilbert space [21, 22]. In this work, the solution set of (63) is denoted by In the context of the variational inequality problem, (62) implies that
is said to be -inverse-strongly monotone if there exists a positive real number such that for all If is an -inverse-strongly monotone mapping of to , it is known that is -Lipschitz continuous. Also, we have that for all and
Therefore, if then is a nonexpansive mapping of into Consequently, one can apply Theorem 15 to deduce the following result:
Corollary 22. Let be a nonempty closed convex subset of a real Hilbert space and a generalized contractions. Let be an -inverse-strongly monotone mapping of to with . Assume that the conditions (i)-(v) of Assumption 12 are satisfied. Then, the iterative sequence which is defined from an arbitrary by converges strongly to a solution in which solves the variational inequality
4.3. Application to Fredholm Integral Equation in Hilbert Spaces
Consider a Fredholm integral equation of the form where is a continuous function on and is continuous. The existence of solutions of (69) has been studied (see [23] and the references therein). If satisfies the Lipschitz continuity condition
then equation (69) has at least one solution in the Hilbert space ([23], Theorem 3.3). Precisely, define a mapping by
It is known that is nonexpansive. Indeed, for
Thus, finding a solution of integral equation (69) is reduced to finding a fixed point of the nonexpansive mapping in the Hilbert space Consequently, the following result is obtainable.
Corollary 23. Let be a nonempty closed convex subset of a Hilbert space defined by (71) with and is a generalized contraction. Suppose that the conditions (i)-(v) of Assumption 12 are satisfied. Then, for an arbitrary the iterative sequence defined by converges strongly to a fixed point of which solves the variational inequality Examples of real sequences which satisfy the conditions of Assumption 12 are
Data Availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
All authors contributed significantly in writing this article. All authors read and approved the final manuscript.