Inertial Iterative Methods for Fixed Point Problems and Solutions of Variational Inequality Problems in a Hilbert Space

Inertial Iterative Methods for Fixed Point Problems and Solutions of Variational Inequality Problems in a Hilbert Space

C. M. Asanya(1), B. G. Akuchu(2), C. L. Ejikeme(2), M. B. Okofu(2), D. C. Ugo(3), C. Okoye(2), And D. F. Agbebaku(2) *

(1) State University of Medical and Applied Sciences (SUMAS)

(2) Department of Mathematics, University of Nigeria, Nsukka 410001, Nigeria

(3) Department of Mathematical and Statistics Enugu State University of Science and Technology Enugu, Nigeria

Abstract: An inertial Halpern-type iterative method with errors is proposed for finding a common element of the set of common fixed points of an infinite family of demimetric mappings and the set of common solutions of variational inequality problems for a finite family of inverse strongly monotone mappings in a real Hilbert space. Strong convergence theorems are established. Applications to monotone inclusion and fixed-point problems are provided. The results extend and improve several known results in the literature.

2020 Mathematics Subject Classification: 47H05, 47H09, 49J40, 65K15.

Key words and phrases: Halpern-type iteration, demimetric mapping, variational inequality, inertial method, monotone operator.

INTRODUCTION

Let be a real Hilbert space with scalar product and induced norm . Let be a nonempty, closed and convex subset of and be a mapping. The variational inequality problem for is defined as: find

such that

. We shall denote by the solution set of (1). The mapping is called -inverse strongly monotone if there exist a constant such that

Let be a mapping. A point is said to be a fixed point of if

The solution set of (3) shall be denoted by

. If is a nonexpansive map then is closed and convex (see [41]). We have for

that if and only if (see [41]).

Variational inequality theory is an important tool in engineering ,economics, mathematical programming (see for example [6, 22]). If is maximal monotone, then its resolvent is denoted and defined by

, for any . The variational inequality problem of can be transformed to a fixed point problem of its corresponding resolvent(see [12, 25]).

Furthermore, the forward-backward algorithm for approximate common zero of the sum of two monotone operators is a fixed point problem. For more examples of variational inequality problems which are equivalent to fixed point problems (see [7, 3]).

A mapping is said to be:

1. nonexpansive if ;
2. -strict pseudo-contractive if

   . If is -inverse strongly monotone and

   This research was supported by the State University of Medical and Applied Sciences (SUMAS) IBR TETFUND.

   for any ;

3. generalized hybrid if there exist real numbers such that

, then is nonexpansive(see [4]).

. Such a mapping is called ( )-generalized hybrid. For example, a ( 1,0 )-generalized hybrid mappings is nonexpansive, i.e.,

for an arbitrary , let the sequence be iteratively defined by ,

Let be a duality mapping on a smooth Banach space and be a maximal monotone operator on with .

Then we have (see e.g [40]) that, for any and

This implies

Definition 1.1. ([42]) Let be a nonempty, closed and convex subset of and let . A mapping

with is called -demimetric if, for any and ,

Example 1.2. (i) Let be a Hilbert Space, let be a nonempty, closed and convex subset of and let be a real number with . If is a -strictly pseudo contraction and , then is -demimetric; (see 21])

1. Let be a Hilbert Space and let be a nonempty, closed and convex subset of . If is a generalized hybrid and , then is 0-demimetric and also nonexpansive; (see [21])
2. Let be a Hilbert Space and let be a nonempty, closed and convex subset of . Let and let

   be an – inverse strongly monotone mapping with . Then and is a – demimetric mapping; (see 21])

3. Let be a uniformly convex and smooth Banach space and let be a maximal monotone operator with . Let . Then the resolvent is ( -1 )-demimetric; (see [21]).
4. Let denote a smooth Banach space which is strictly convex and reflexive and let denote a closed, convex and non-empty subset of . The metric projection of onto , denoted by is -demimetric. (see [21]).

   The class of -demimeric mappings is more general than the class -strictly pseud-contractions, generalized hybrid mappings and nonexpansive mappings (see [42]). Many authors have developed iterative methods for approximating a solution of problem (5) for demimetric mappings (see [42, 40]). From literature, there are different iterative methods for approximating fixed points of nonexpansive mappings in Hilbert spaces. Notable among such iterative methods is the Halpern iteration method [17], which is given as follows:

   where is a real sequence in [0,1]. This iterative method for approximating the fixed point of nonexpansive mappings both in Hilbert spaces and general Banach spaces has been studied extensively. (see for example, 11, 24, 33, 34, 37, 39, 47, 49, 48]).

   In practical applications, iterative methods with high rate of convergence are always desirable see [10, 15, 16, 29] and the references therein. In this regard, Polyak 31 introduced an inertial extrapolation algorithm based on the heavy ball method of the second-order time dynamical system as an acceleration process for solving the smooth convex minimization problem. In order to obtain faster convergence rate, many researchers have proposed inertial extrapolation type algorithms which include inertial proximal method [2, 30, inertial forward-backward algorithm 25, and fast iterative shrinkage thresholding algorithm (FISTA) [8, 9.

   Recently, Shehu et al [36] introduced a Halpern-type iterative method, which is a combination of inertial extrapolation and error terms for approximating fixed point of nonexpansive mapping and proved the following:

   Theorem 1.3. Let be a real Hilbert space and be a nonexpansive mapping with . Let be the sequence generated from arbitrary by

   where are sequences in ( 0,1 ), is a positive sequence and sequence of errors

   satisfying the following conditions:

   1. , where

      means .

   2. for all and .
   3. Either or
   4. choose such that , where

Then the sequence generated by Algorithm (7) converges strongly to .

Research question: Can strong convergence result be obtain when the nonexpansive operator in the algorithm (7) is replace with a demimetric operator? This paper seeks to achieve this result in the affirmative.

Motivated by the work of Shehu et al and other works in literature, we introduced an inertial Halpern-type iterative method with errors for finding a common element of the set

of common fixed points of an infinite family of demimetric mappings and the set of common solutions of variational inequality problems for a finite family of inverse strongly monotone mappings in Hilbert spaces. We prove strong convergence results for monotone inclusion, variational inequality and fixed point problems. The results obtained in this paper generalize some recent results in literature.

1. Preliminaries

   The nearest-point projection of the Hilbert space onto the set , denoted by , is the metric projection of onto . That is, for all and . It can be shown that the metric projection is firmly nonexpansive:

   Moreover, the inequality holds for all and (see [43]).

   In what follows, some well-known lemmas needed for the convergence analysis of our main results are stated.

   Lemma 2.1. [1] The following inequalities hold in a real Hilbert space:

   Lemma 2.2. ([26, 49]) Let be a sequence of nonnegative real numbers satisfying the following relation:

   where the sequence is in ( 0,1 ) and the sequence

   is real. Assume . The following results hold:

   (i) The sequence is a bounded, if for some

(ii) The limit , if and .

Lemma 2.3. ([19]) Let be a nonexpansive mapping, the sequence be in and a point in . Suppose that converges weakly to (i.e., as

) and that . Then, .

Lemma 2.4. (43) Let be a nonempty, closed and convex subset of a real Hilbert space . Let be an – strongly inverse monotone operator with . Then

number such that and be a -demimetric mapping of into . Then is closed and convex.

Lemma 2.6. (38]) Let be a real Hilbert space and a nonempty, closed and convex subset of . Assume that is an infinite family of -demimetric

mapping with sup such that

. Suppose is a positive sequence

such that . Then is a –

demimetric mapping with and

.

Lemma 2.7. [43] Let be a real Hilbert space and a nonempty, closed and convex subset of . Let

and let be a -demimetric mapping of into such that

. Let be a real number with and define . Then is a quasi-nonexpansive mapping of into .

1. Main Results

   In this section, we give a precise statement of our main result. We first state the assumptions of the iterative method that will hold through out the rest of this paper.

   Assumption 3.1. Suppose

   are sequences in is a positive sequence and the sequence of errors satisfy the following conditions:

   (a) ,

   where means .

   1. and for all
   2. Either or .

   Remark 3.2. The imposed conditions on the iterative parameters in Assumption 3.1 are standard and have been used by many related papers in the literature. For example, the conditions and are necessary

   and have been used in [11, 24, 33, 34, 37, 39, 47, 49, 48, Condition (b) has recently been used in [43, 18] and Condition (c) has been used in [36, 45, 46]. Remark 3.3. Some examples satisfying the Assumption (3.1) are:

   and

   where and . In particular if then ( ) is nonexpansive

   Lemma 2.5. (43) Let be a real Hilbert space and a nonempty, closed and convex subset of . Let be a real

   where is any fixed vector. The main result of the paper is the following Theorem 3.4. Let be a real Hilbert space and a nonempty, closed and convex subset of . Let

   be an infinite family of -demimetric

   mappings with for each and such that and each

   is demiclosed at zero. Let be a family of – inverse strongly monotone mappings of into , for each

   . Suppose Assumption 3.1) holds and

   . For , let

   as . Hence, there exists such that

   Set and , then for any

   , by (8), lemma (2.6) and lemma (2.7) we obtain

   be the sequence generated by

   From (8) we have

   and are sequences in satisfy the following conditons:

   • (m1)

     Let , we have

   • (m3) For choose such that

   , where

   From (8) and Remark (3.5) we have

   Then converges strongly to a point where

   Proof. Since is -strongly monotone for all

   and , then is nonexpansive

   and is closed and convex (see

   [43]). Furthermore, we know from lemma (2.5) that is closed and convex, and by lemma (2.6), is – demimetric and , so

   is closed and convex. Therefore, we have that

   is closed and convex.

   Therefore, is well defined. Remark 3.5. Observe that from Assumption (3.1) and (8) we

   have that and both exist and

   Using (8), (11) we have

   Substituting (12) in (13), we have

   , which implies that

   Since , then is bounded. Furthermore, and both exists.

   Now taking

   Then (14) becomes

   Since for each n , as easily seen from (8), then

   . It then follows from (8) and (21) that

   By Lemma (2.2), we get that is bounded, hence is bounded.

   Furthermore, considering , then from (14) we obtain

   Taking

   then (16) becomes

   The remaining part of the proof is broken into two cases: Case 1: Put for . Suppose there exists a natural number such that for all

   . Therefore, exists. From (19) and Assumption (3.1) we have

   By Lemma (2.2), we again have that is bounded, hence is bounded. It can also be shown that ,

   and are bounded.

   Let , from (8) and from Lemma (2.1) (ii) we have that

   Thus

   Using (9), (10) and from [80] we have

   Since are bounded, using Assumption 3.1 and noting that

   From (8) and Lemma (2.1), we have that

   we have

   and so we have .

   Since , then . therefore

   Combining (17), (18), (19) we obtain

   Hence from (22) we have that

   Let , then we have that

   Since is demimetric for each , then by (

   ) and Lemma (2.6) we have

   (2.2) (ii) we have that , that is as desired. If , then we derive from (20) that

   where . Observe that

   by (26) and for . Using

   Lemma (2.2) (ii) and Remark (3.5) we have that

   , that is as desired.

   Using and boundedness of we have that

   Case 2: Suppose that there is no such that

   is monotonically decreasing. The method of proof used in this section is adapted from [27]. Let be a mapping defined for all (for some large enough) by

   Since is nonexpansive (hence 0 -demimetric) then for we get

   i.e. is the largest number in such that increases at ; note that, in view of Case 2 , this

   is well-defined for all sufficiently large . Clearly, is a non-decreasing sequence such that as

   and

   After a similar estimate as in (22), it is easy to show that

   with and ( ) we get

   for each .

   Since is bounded, take a subsequence of such that

   From and

   and

   and

   (2.3) that and

   , we have using lemma

   Furthermore, using the boundedness of and

   respectively and . Hence

   . For , we show that

   . To do this, we have

   Assumption 3.1, we get

   If , then from (20) we have that

   where . Since and

   are bounded, and by (25) then from Lemma

   Since is bounded, there exists a subsequence of

   , still denoted by , which converges weakly to some . Similarly, as in Case 1 above, we can show that . We have from (19)

   that

   Let be a real Hilbert space and be a proper convex lower semi-continuous function. Then the subdifferential of , denoted by is defined as:

   which shows

   The subdifferential of is maximal monotone, (see [35]). Let be a closed and convex subset of and be the indication function of , i.e

   We have from (30) that

   which, in turn, implies . Furthermore,

   for , it is easy to see that (observe that for and consider the three cases:

   and . For the first and

   second cases, it is obvious that , for . For the third case , we have from the definition of

   and for any integer that for

   . Thus,

   Since is a proper lower semicontinuous convex function on , the subdifferential of , is a maximal monotone operator. So, we can define the resolvent of by

   .

   Lemma 4.1. [42] Let be a nonempty, closed and convex subset of a real Hilbert space be the metric projection from onto and be the subdifferential of where

   is as above and . Then

   ). As a consequence, we obtain for all sufficiently large that . Hence . Therefore, converges strongly to . This completes the proof.

2. Applications

   4.1. Monotone Inclusion. A set-valued mapping

   is said to be monotone if for all and

   , the inequality holds. It is called a maximal mapping if its graph, , is not properly contained in the grph of any other monotone mapping.

   Let be a -inverse strongly monotone( -ism) mapping and let be a multi-valued maximal monotone mapping. Consider the following monotone inclusion problem: find such that

   Convex programming and variational inequalities problems are special cases of the monotone inclusion problem (32). Furthermore, some practical problems in image processing, machine learning and linear inverse problems can be modelled mathematically as monotone variational inclusion problems, see for more details. Recall that the resolvent operator associated with and is the mapping defined by

   Theorem 4.2. Let be a real Hilbert space and let be a nonempty, closed and convex subset of . Let be a finite family of -inverse strongly monotone mappings of into and be a proper convex lower semicontinuous function such that be a maximal monotone mapping. Assume that Assumption (3.1) hold and

   . Let . Let be

   a sequence generated by

   where is the resolvent operator of and , , satisfy

   the following conditions:

   • (m1)
   • (m3) choose such that , where

   The resolvent operator is single valued, nonexpansive and 1 -inverse monotone (for example see [5]). Moreover

   if and only if

   (see [23]).

   Then converges strongly to a point where

   Proof. Since is -inverse strongly monotone for all and is

   nonexpansive and by Lemma 4.1) we have that

   , then

   is closed and convex (See [41]). Then we have that is well defined where

   . Then we have the desired

   result from 3.4

   Lemma 4.3. [28] Let be a Hilbert space and let be a nonempty, closed and convex subset of . Let be a real number with and be a -strict pseudo- contraction. If and , then .

   Lemma 4.4. [20] Let be a Hilbert space, let be a nonempty, closed and convex subset of and let

   be generalized hybrid. If and , then

   Lemma 4.5. 44 Let be a Hilbert space, and a closed and convex subset of which is nonempty. Let and be mappings of into and such that and

   . Then, the following are equivalent:

   1. is an -inverse strongly monotone mapping, i.e
      • (m3) choose such that , where

        Then converges strongly to a point where

        Proof. Since is a generalized hybrid mapping of into such that , from example (2) in 21, is 0 – demimetric. Furthermore from Lemma (4.4) is demiclosed. Since is nonexpansive, is a – inverse strongly monotone mapping. We also have from

        that

        Therefore, we have the desired results from Theorem (3.4) Theorem 4.7. Let be a Hilbert space and let be a nonempty, closed and convex subset of . Let

        and let be a infinite family of –

        strictly pseudo-contraction mapping of into . Assume that Assumption (3.1) hold and . Let

        . Let be a sequence generated by

   2. is widely -strictly pseudo-contraction, i.e
   3. is a nonexpansive mapping.

      Theorem 4.6. Let be a Hilbert space and let be a nonempty, closed and convex subset of . Let

      and . Let

      be an infinite family of generalized hybrid mapping of into and let be a finite family of

      the following conditions:

      • (1)
      • (2) ,
      • (3) .

        satisfy

        nonexpansive mappings of into . Assume that Assumption 3.1 hold and is

        nonempty. Let . Let be sequence generated by

      • (m1)

   Then converges strongly to a point where

   .

   Proof. Since is a -strictly pseudo-contraction of into such that , from example in 21, is – demimetric. furthermore, from Lemma (4.4), is

   demiclosed. Then, we have the desired result from (3.4).

3. Numerical Examples

In this section, the numerical examples to demonstrate the convergence of the proposed algorithm (8) discussed in this paper are given in the real Hilbert space setting. Using different examples, we show, graphically, strong convergence results discussed in this paper.

All codes are written in MATLAB, and implemented using an HP Probook computer with Intel COREi5 with 2.00 Hz and 4GB RAM.

Example 5.1. Let with the inner product defined by and the standard norm . Let

be defined as and

be defined as

. The following parameter values are used in plotting the graph.

Then the algorithm (8) converges strongly. The results are shown in figure 1 and Table 1

Example 5.2. Let

equipped with the norm and the inner product . Define an infinite family of

mappings by and

. It can be shown that the . It can also be

shown that the infinite family of mappings is –

demimetric. Taking the pointwise limit as , then the mappings which is also a -1 – demimetric mapping. Let be defined as

Figure 1. Graph showing strong convergence of the iterative algorithm (8) with tolerance of

Table 1. Table showing numerical values of and for the algorithm (8)

. The following parameter values are used in plotting the graph.

Then the algorithm (8) converges strongly. The results are shown in figure 2 and Table 2

Table 2. Table showing numerical values of and for the algorithm (8)

111.0000000 71.2088800 0.0000552 0.0000010

Figure 2. Graph showing strong convergence of the iterative algorithm (8)

ACKNOWLEDGMENT

This research was supported by the State University of Medical and Applied Sciences (SUMAS) IBR TETFUND.

CONFLICT OF INTEREST

The authors declare no Conflict interests.

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