Abstract
In this paper, based on the very recent work by Nandal et al. (Nandal, A.; Chugh, R.; Postolache, M. Iteration process for fixed point problems and zeros of maximal monotone operators. Symmetry 2019, 11, 655.), we propose a new generalized viscosity implicit rule for finding a common element of the fixed point sets of a finite family of nonexpansive mappings and the sets of zeros of maximal monotone operators. Utilizing the main result, we first propose and investigate a new general system of generalized equilibrium problems, which includes several equilibrium and variational inequality problems as special cases, and then we derive an implicit iterative method to solve constrained multiple-set split convex feasibility problem. We further combine forward-backward splitting method and generalized viscosity implicit rule for solving monotone inclusion problem. Moreover, we apply the main result to solve convex minimization problem.
Keywords:
generalized viscosity implicit rule; zero point; fixed point; system of generalized equilibrium problems; constrained multiple-set split convex feasibility problem; monotone inclusion problem; minimization problem MSC:
47H09; 47H10; 47H05; 47J25; 49J40
1. Introduction
A problem which appears very often in different areas of mathematics and physical sciences consists of finding an element in the intersection of closed and convex subsets of a Hilbert space. This problem is generally named as convex feasibility problem (CFP). The applications of CFP lie in the center of various disciplines such as sensor networking [1], radiation therapy treatment planning [2], computerized tomography [3], and image restoration [4].
The multiple-sets split feasibility problem (MSSFP) is stated as finding a point belonging to a family of closed convex subsets in one space whose image under a bounded linear transformation belongs to another family of closed convex subsets in the image space. It generalizes the CFP and the split feasibility problem (SFP). The MSSFP was firstly introduced by Censor et al. [5] to model the inverse problem of the Intensity-Modulated Radiation Therapy. Recently, Buong [6] studied several iterative algorithms for solving MSSFP, which solves a certain variational inequality. Masad and Reich [7] generalized the MSSFP to the constrained multiple set split convex feasibility problem (CMSSCFP), in which several bounded linear operators are involved. For other new results in this direction, we address the reader to the works of Yao et al. [8,9,10,11,12,13,14]
Equilibrium problem (EP) theory includes many important mathematical problems, for instance, variational inequality problems, optimization problems, saddle point problems, Nash equilibrium problems and fixed point problems [15,16,17]. This problem has been generalized in several interesting and important problems. In 2010, Ceng and Yao [18] introduced and studied a system of generalized Equilibrium problem (SGEP). Several iterative methods have been proposed by a number of authors to solve SGEP (see, e.g., [19,20,21,22,23]).
Monotone inclusion problem with multivalued maximal monotone mapping and inverse-strongly monotone mapping is among the very important and extensively studied problems in recent years. This problem includes various important problems such as convex minimization problem, variational inequality problem, linear inverse problem and split feasibility problem. One of most popular methods for solving this inclusion problem is the forward-backward splitting method [24,25,26,27].
In a very recent paper [28], Nandal, Chugh and Postolache introduced an iterative algorithm to study fixed point problem of nonexpansive mappings and zero point problem of maximal monotone mappings. Then, applicability of the algorithm was shown by discussing and solving different kinds of problems; for instance, general system of variational inequalities, convex feasibility problem, zero point problem of inverse strongly monotone and maximal monotone mapping, split feasibility and its connected problems were solved under weaker control conditions on the parameters.
On the other hand, the implicit midpoint rule, the variational iteration method and the Taylor series method are some powerful methods for solving various important kinds of differential equations. The variational iteration method was first introduced by He [29,30,31]. In [32], Khuri and Sayfy established relation between variational iteration method and the fixed point theory. Very recently, He and Ji [33] suggested a simple approach using Taylor series technology to solve approximately the Lane-Emden equation. The major method for solving ODEs (in particular, stiff equations) is the implicit midpoint rule (see [34,35,36,37,38,39]). For instance, consider the initial value problem for the time dependent ODE with initial condition . It is known that if is Lipschitz continuous and uniformly smooth, then the implicit method which is given by the implicit midpoint rule:
converges to the solution of the above mentioned initial value problem as uniformly over for any fixed . If we take , where T is a nonlinear mapping, then the equilibrium problem associated with the above mentioned initial value problem reduces to the fixed point problem . Therefore, Alghamdi et al. [40] established the implicit midpoint rule for nonexpansive mappings in Hilbert space and also proved that their iterative method can be applied to find periodic solution of a nonlinear time dependent evolution equation. In [41], Xu et al. combined viscosity approximation method and implicit midpoint rule to approximate a fixed point of a nonexpansive mapping. Recently, Ke and Ma [42] established generalized viscosity implicit rule of nonexpansive mappings by replacing the midpoint with any point of the interval .
Inspired by the above work, we introduce and study a new generalized viscosity implicit iterative rule based on Nandal, Chugh and Postolache’s [28] iterative method for approximating a common element of the fixed point sets of nonexpansive mappings and the sets of zeros of maximal monotone mappings. Then, we introduce and analyze a new general system of generalized equilibrium problems and apply our main result to solve this problem. Moreover, we utilize our main result to solve constrained multiple set split convex feasibility problem, monotone inclusion problem and convex minimization problem.
2. Preliminaries
In this paper, H is assumed a real Hilbert space with the inner product and the norm . Here, is used to denote the fixed point set of a mapping S. The strong and weak convergence of a sequence to x shall be denoted by and , respectively. Assume that is a closed convex set then the nearest point projection (metric projection) from H onto D is denoted by , that is, for each , , . In addition, for given and
The following is a well known result of Hilbert space
for all and .
Next, we recall the definitions of some important operators, which we use below.
Definition 1.
An operator is said to be
- (1)
- Nonexpansive if ∀.
- (2)
- Contraction if there exist a constant such that , ∀.
- (3)
- α-averaged if there exist a constant and a nonexpansive mapping S such that .
- (4)
- θ-inverse strongly monotone (for short, θ-ism) if there exists such that
- (5)
- Firmly nonexpansive if ∀.
Note that metric projection is an example of firmly nonexpansive and, further, every firmly nonexpansive is -averaged in Hilbert space.
A set valued operator is called maximal monotone, if B is monotone, i.e., ∀, and , and there does not exist any other monotone operator whose graph properly includes graph of B. Further, a maximal monotone operator B and generate an operator given as:
which is known as resolvent of B. It is well known [43] that, if is maximal monotone and , then is firmly nonexpansive and .
Next, we consider a sequence of nonexpansive mappings, which is called strongly nonexpansive sequence [44] if
whenever , such that is bounded and . In addition, note that definition of strongly nonexpansive mapping [45] can be obtained by taking .
Next, we collect several lemmas, which we use in our results.
Lemma 1
(Lemma 1, [28]). Let be a β-ism operator on H. Then, is nonexpansive.
Lemma 2
(Lemma 2.5, [46]). Let , and be three real sequences satisfying
Suppose that and . Then, .
Lemma 3
(Corollary 3.13, [44]). Let be a nonempty set and suppose that satisfy . Then, a sequence of mappings of D into H defined by , is a strongly nonexpansive sequence, where is a nonexpansive mapping for each and I is the identity mapping on D.
Lemma 4
(Example 3.2, [44]). In a Hilbert space, every sequence of firmly nonexpansive mappings is a strongly nonexpansive sequence.
Lemma 5
(Theorem 3.4, [44]). Let be nonempty sets. Suppose and are two strongly nonexpansive sequences where and are such that for each . Then, is a strongly nonexpansive sequence.
Lemma 6
(Lemma 2.1, [45]). If are strongly nonexpansive mappings and , then .
Lemma 7
(Propositions 2.1 and 2.2, [47]). The composition of finitely many averaged mappings is averaged. That is, if are averaged mappings, then is also averaged. Furthermore, if , then .
Lemma 8
(Theorem 10.4, [48]). Let be a nonempty closed convex set and be a nonexpansive mapping. Then, is demiclosed at 0, that is, if with and , then .
Lemma 9
(The Resolvent Identity; (Remark 1.3.48, [49])). For each ,
Note that Lemma 9 uses the homotopy technology, which is also widely used in the homotopy perturbation method [50,51].
Lemma 10
(Lemma 3.1, [52]). Let be a sequence of real numbers such that there exists a subsequence of such that for all . Then, there exists a nondecreasing sequence such that and the following properties are satisfied by all (sufficiently large) numbers :
In fact,
3. Main Results
Theorem 1.
Let H be a real Hilbert space. Let and V be nonexpansive self-mappings on H and be maximal monotone mappings such that
Let be a contraction with coefficient and let be a generalized viscosity implicit rule defined by and
for all , where , for and for some . Suppose that , and are sequences in and and are sequences of positive numbers satisfying the following conditions:
- (i)
- , ;
- (ii)
- ;
- (iii)
- , for all ; and
- (iv)
- for all sufficiently large n, for some .
Then, the sequence converges strongly to where is the unique fixed point of the contraction .
Proof.
First, we show that is bounded.
Let . Set and . Clearly, and are nonexpansive mappings for each . By Lemmas 3 and 4, for each , and are composition of strongly nonexpansive mappings. Therefore, from Lemma 6, we get
Therefore, we obtain
Thus, we have
By induction, we have
This shows that is bounded and so are and .
Since the fixed point set of nonexpansive mapping is closed and convex, the set is nonempty closed and convex subset of H. Hence, the metric projection is well defined. In addition, since is a contraction mapping, there exist such that .
Now, we prove that .
For this purpose, we examine two possible cases.
Case 1.Assume that there exists such that the real sequence is nonincreasing for all . Since is bounded, is convergent. From Equation (4), we have
From the nonexpansiveness of , we have
Since is convergent, and is bounded, we have
Using Lemma 5, is strongly nonexpansive sequence, therefore, we have
From the definition of , we have
Note that
Moreover, we have
Next, we show that
From the definition of , we have
Moreover, we have
In view of the fact that is convergent and using Equation (10), we obtain
In addition, by using Lemma 5, is strongly nonexpansive sequence for each . Therefore, we have
Now, consider
Choose a fixed number s such that and using Lemma 9, for all sufficiently large n, we have
In view of Equation (15), we obtain
Next, we show that
Notice that
It follows from given Condition (iii) and Equation (17) that
Since is convergent, using and Equation (5), we obtain
In addition, by using Lemma 3, is strongly nonexpansive sequence. Therefore, we have
Notice that
It follows from given Condition (ii) and Equation (21) that
Now, consider
Now, following the same way as for Equation (16), we obtain
Put . U being a convex combination of nonexpansive mappings is nonexpansive and
We observe
As is bounded, take a subsequence such that Then, by Equation (24) and Lemma 8, , which results that
where the last inequality follows from Equation (1).
Finally, we show that as . In fact, we have
where .
It turns out that
Solving this quadratic inequality for , we get
Note that
Therefore, we have
This implies that
that is,
which gives the inequality
that is,
It follows that
Note that implies that
Consequently, we have
Let
Note that
Letting ξ satisfy
there exists an integer large enough such that for all and, consequently, we have for all . Hence, we have
for all . Therefore, from Equation (26), we have, for all ,
Note that
Case 2.Assume that there exists a subsequence of such that
for all . Then, by Lemma 10, there exists a nondecreasing sequence of integers such that as and
for all . Now, using Equation (31) in Equation (4), we have
Using given conditions on and with boundedness of and , we obtain
As and is bounded, we obtain
By the same argument as in Case 1, we obtain
From Equation (13), we obtain
Using , we have
Again, using the same argument as in Case 1, we obtain
As , , and are bounded, we obtain
Following similar arguments as in Case 1, we have
Using the fact that and Equation (35), we obtain
This together with Equation (34) implies that as .
However,
for all , which gives that as . □
4. Applications
We apply our main result in this section to solve a number of important problems.
4.1. New General System of Generalized Equilibrium Problems
Let be nonempty closed convex subsets of a Hilbert space H. Let, for each , be a bifunction and be a nonlinear mapping. Consider the following problem of finding such that
where for each . Here, is used to denote the solution set of Equation (36). Next, we discuss some special cases of the problem in Equation (36) as follows:
- (1)
- If , , then the problem in Equation (36) reduces to the following general system of generalized equilibrium problems of finding such thatwhich was introduced and studied by Ceng and Yao [18].
- (2)
- If , , , then the problem in Equation (36) reduces to the following generalized equilibrium problem of finding such thatwhich was introduced and studied by Takahashi and Takahashi [53].
- (3)
- If , , , then the problem in Equation (36) reduces to the following system of equilibrium problems of finding such thatwhich was considered by Combettes and Hirstoaga [54].
- (4)
- (5)
- If , then the problem in Equation (36) reduces to the following general system of variational inequalities of finding such thatwhich was considered and investigated by Nandal, Chugh and Postolache [28].
- (6)
- If , , then the problem in Equation (39) reduces to the following system of variational inequalities of finding such thatwhich was introduced and considered by Ceng et al. [55].
- (7)
- If , , , then the problem in Equation (39) reduces to the following system of variational inequalities of finding such thatwhich was introduced and studied by Verma [56].
- (8)
It is clear from above mentioned special cases that the problem in Equation (36) is very general and includes a number of equilibrium and variational inequality problems, which shows the special significance of this problem.
To study problem in Equation (36), we need following assumptions for a bifunction .
- (P1) for all ;
- (P2) is monotone, i.e., for all ;
- (P3) for all ; and
- (P4) for each fixed , is a convex and lower semicontinuous function.
Lemma 11
(Lemma 2.12, [54]). Let be a nonempty closed convex set and be a bifunction satisfying (P1)–(P4). Then, for any and , there exists such that
Furthermore, if , then
- (a)
- is a single valued map;
- (b)
- is firmly nonexpansive;
- (c)
- ; and
- (d)
- is closed and convex.
Lemma 12.
Let be nonempty closed convex subsets of a Hilbert space H. Let, for each , be a bifunction satisfying Conditions (P1)–(P4) and be a nonlinear mapping. Then, for given , , is a solution of the problem in Equation (36) if and only if is a fixed point of the mapping defined by
Proof.
Theorem 2.
Let be nonempty closed convex subsets of a Hilbert space H. Let, for each , be a bifunction satisfying Conditions (P1)–(P4) and be a -ism self-mapping on H. Assume that , where T is defined in Lemma 12. Let be a sequence defined by and
where and for some . Suppose satisfying and . Then, the sequence converges strongly to a point .
Proof.
First, we prove that is an averaged map. Observe that , where . Then, applying Lemma 1, is nonexpansive and therefore, is averaged for , . In addition, Lemma 11 implies that is firmly nonexpansive, that is, -averaged for each . Hence, Lemma 7 implies that T is averaged on H and, therefore, , for some and a nonexpansive mapping where . Taking , , and in Theorem 1 yields the conclusion of Theorem 2. □
4.2. Constrained Multiple-Set Split Convex Feasibility Problem (CMSSCFP)
Let and be two real Hilbert spaces. Let and be nonempty closed convex subsets of and , respectively. Let, for each , be a bounded linear operator and let K be another closed convex subset of . The constrained multiple-set split convex feasibility problem (CMSSCFP) [7] is formulated as finding a point such that
This problem extends the multiple set split feasibility problem (MSSFP) [5], which is formulated as finding a point such that
where is a bounded linear operator. It is clearly seen that by taking for each , CMSSCFP reduces to MSSFP.
If then MSSFP reduces to the split feasibility problem (SFP), which is formulated as finding a point such that
Recently, Buong [6] proposed the following algorithm to solve MSSFP
where or and or and F be strongly monotone and Lipschitz continuous map. He proved that this algorithm converges to a solution of the following variational inequality problem: find such that
where is solution set of the MSSFP in Equation (42).
Now, we present an implicit iterative method to solve the CMSSCFP in Equation (41). We use to denote the solution set of the CMSSCFP in Equation (41).
Theorem 3.
Let F be a θ-ism self-mapping on . Assume that is nonempty. Let be a sequence defined by and
where and for some . Suppose , and satisfying
- (i)
- ;
- (ii)
- ; and
- (iii)
- , .
Then, the sequence converges strongly to a point , which is also a solution of
Proof.
Let solve the CMSSCFP in Equation (41), i.e., ; then, and for each
Note that is equivalent to i.e., . Therefore, means for each . Thus,
Now, let , which implies
Choose . Therefore, for each
Therefore, for each .
Thus, . Hence, .
Next, we rewrite as
and for each , as
Using Lemma 15 in [28], is -ism. It follows from Lemma 1, and are nonexpansive. Since the metric projection is -averaged, therefore Lemma 7 implies that is averaged mapping. Now, taking , , , , and in Theorem 1 proves that converges strongly to
It can be easily proven that and
In addition, .
That is,
In addition, note that Thus, is also a solution of □
Remark 1.
Theorem 3 generalizes and extends Buong’s result ([6] (Theorem 3.2)) in many directions. Theorem 3 extends the MSSFP studied in Buong [6] to the related, more general problem, CMSSCFP. In addition, we have considered F an inverse strongly monotone operator in Theorem 3, which is more general from the strongly monotone and Lipschitz continuous operator F taken in Buong’s result ([6] (Theorem 3.2)).
4.3. Monotone Inclusion and Fixed Point Problem
Let and be two operators. Consider the inclusion problem of finding such that
The solution set of Equation (45) is denoted by . A popular method for solving Equation (45) is the forward-backward splitting method, which can be expressed via recursion:
where . Now, we combine forward-backward splitting method and generalized viscosity implicit rule for finding a common element of set of solutions of Equation (45) and fixed point sets of a finite family of nonexpansive mappings.
Theorem 4.
Let H be a real Hilbert space. Let be nonexpansive self-mappings on H. Let S be a θ-ism of H into itself and let be maximal monotone mapping such that . Let be a contraction with coefficient and let be a sequence defined by and
for all , where for and for some . Suppose that , and , satisfying:
- (i)
- (ii)
- ; and
- (iii)
- for all .
Then, the sequence converges strongly to , where is the unique fixed point of the contraction .
Proof.
Firstly, we rewrite as
and . By applying Lemma 1, is nonexpansive and, therefore, for each , is averaged for . Furthermore, it follows from [44] (Lemma 5.8) that for any
By putting , , and in Theorem 1, the conclusion follows immediately. □
Remark 2.
Theorem 4 improves and extends Chang et al.’s result [27] (Theorem 3.2). By taking , in Theorem 4, we obtain Theorem 3.2 of [27] with more relaxed conditions on the parameters. In addition, contraction coefficient k is bounded in in [27] (Theorem 3.2), which we relax to .
4.4. Convex Minimization Problem
Suppose is a convex smooth function and is a proper convex and lower semicontinuous function. In this subsection, we study the following convex minimization problem: find such that
Using Fermat’s rule, the problem in Equation (47) can be transformed into the following equivalent problem:
where is a gradient of f and is a subdifferential of h.
Theorem 5.
Assume that is a convex and differentiable function, and its gradient is -Lipschitz continuous where . In addition, assume that is a proper convex and lower semicontinuous function such that attains a minimizer. Let be a contraction with coefficient and let be a sequence defined by and
where for some . Suppose and satisfy the following conditions:
- (i)
- and
- (ii)
- .
Then, converges strongly to a minimizer of .
Proof.
Since is -Lipschitz continuous, it follows from Corollary 10 of [57] that is -ism. Moreover, is maximal monotone (see [58] (Theorem A)). Taking , and in Theorem 4, we obtain the conclusion of Theorem 5 from Theorem 4. □
Remark 3.
If we take in Theorem 5, we obtain Theorem 4.2 of [27] as special case with improved conditions on the parameters.
5. Concluding Remarks
The base of this paper is the work done by Nandal, Chugh and Postolache [28] and Ke and Ma’s [42] generalized viscosity implicit rule for solving fixed point problems. Under mild conditions, strong convergence of the proposed method is proved. Furthermore, we consider a new general system of generalized equilibrium problems, which is generalization of several equilibrium and variational inequality problems considered in Ceng and Yao [18], Takahashi and Takahashi [53], Combettes and Hirstoaga [54], Nandal et al. [28], Ceng et al. [55] and Verma [56]. Theorem 3 extends the multiple set split feasibility problem (MSSFP) studied by Buong [6] to a related more general problem, the constrained multiple-set split convex feasibility problem (CMSSCFP), which in addition extends F from a strongly monotone and Lipschitz continuous operator to an inverse strongly monotone operator. Then, we combine the forward-backward splitting method and generalized viscosity implicit rule in Theorem 4 to solve monotone inclusion problem. Theorem 4 improves and extends Chang et al.’s result [27] (Theorem 3.2). Finally, we derive Theorem 5 to solve convex minimization problem, which extends [27] (Theorem 4.2). Further, our work can be extended to fractal calculus [59]. We have made an attempt to solve a number of problems of nonlinear analysis as application of the presented algorithm, however, finding real world applications is still an open question.
Author Contributions
All authors participated in the conceptualization, validation, formal analysis, investigation, writing—original draft preparation, and writing—review and editing.
Funding
This research received no external funding.
Conflicts of Interest
The authors declare no conflict of interest.
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