Research Article | Open Access
Mathew Aibinu, Surendra Thakur, Sibusiso Moyo, "Solutions of Nonlinear Operator Equations by Viscosity Iterative Methods", Journal of Applied Mathematics, vol. 2020, Article ID 5198520, 12 pages, 2020. https://doi.org/10.1155/2020/5198520
Solutions of Nonlinear Operator Equations by Viscosity Iterative Methods
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.
The Viscosity Approximation Method (VAM) for solving nonlinear operator equations has recently attracted much attention. In 1996, Attouch  considered the viscosity solutions of minimization problems. In 2000, Moudafi  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.  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.  studied the convergence of the sequence (2) in uniformly smooth Banach spaces. Ke and Ma  introduced generalized viscosity implicit rules which extend the results of Xu et al. . 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. . 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  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.
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 . Letbe a complete metric space and letbe a Meir-Keeler contraction onThen,has a unique fixed point in.
Proposition 5 Suzuki . Letbe a Banach space,a convex subset ofanda Meir-Keeler contraction. Then,there existssuch thatfor all with
Proposition 6 Lim . 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 . 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 . Letandbe bounded sequences in a Banach spaceandbe a sequence inwithSuppose thatfor allandThen,
Lemma 9 Sunthrayuth and Kumam . 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 . 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 . 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
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