Abstract

In this paper, we study the viscosity iterative algorithms for the implicit double midpoint rule in real Hilbert space and prove strong convergence of the sequence {u_n} to a fixed point of T. As an application we employ our method to obtain an application of it in convex minimization and the solution of Fredholm type of integral equations.

1. Introduction

Let be a Hilbert space, be a nonexpansive mapping and be a contraction. The viscosity approximation method for nonexpansive mapping in Hilbert spaces was introduced by Moudafi ²⁶ by the following iterative method:

Let be a closed convex subset of Hilbert space H. Assume that is a contraction and is a nonexpansive mapping. For given the sequence defined by

(1.1)

converges strongly to a fixed point of under certain conditions, which is a solution to the variational inequality

Moudafi's generalizations are called viscosity approximations. Viscosity approximation methods have been extensively employed in the literature to obtain strong convergence results (cf. ^{11, 21, 28, 30, 34} and references therein).

In 2004, Xu ³² extended the result of Moudafi ²⁶ to uniformly smooth Banach spaces and obtained strong convergence theorem. For related work, see ^{7, 16, 37}.

In 2006, Marino and Xu ³⁸ introduced the following iterative scheme based on viscosity approximation method, for fixed point problem for a nonexpansive mapping on :

(1.2)

where is a contraction mapping on with constant , is a strongly positive self-adjoint bounded linear operator on with constant and . They proved that the sequence generated by 1.2 converge strongly to the unique solution of the variational inequality

(1.3)

which is the optimality condition for the minimization problem

(1.4)

where is the potential function for .

The implicit midpoint rule is one of the powerful numerical methods for solving ordinary differential equations and differential algebraic equations. For related works, we refer to ^{6, 39, 40, 41, 42, 43, 44} and the references cited therein. For instance, consider the initial value problem for the differential equation with the initial condition , where f is a continuous function from to . The implicit midpoint rule in which generates a sequence by the following the recurrence relation

(1.5)

In 2014, implicit midpoint rule has been extended by Alghamdi et al. ⁴⁵ to nonexpansive mappings, which generates a sequence by the following implicit iterative scheme:

(1.6)

In 2015, Xu et al. ³⁴ extended (1.1) and obtained the following Viscosity implicit mid point method:

Theorem 1.1. Let be a Hilbert space, a closed convex subset of a nonexpansive mapping with and a contraction with coefficient . For givin the sequence generated by

(1.7)

satisfying the following conditions:

C1:

C2:

C3: or

Then the sequence converges in norm to a fixed point of which is also the unique solution of the variational inequality

Later, Ke and Ma ²¹ and Cai et al. ¹¹ generalized Theorem 1.1 in the setting of Hilbert space. They proposed the following theorems.

Theorem 1.2. [Ke and Ma] Let be a nonempty closed convex subset of the real Hilbert space . Let be a nonexpansive mapping with and be a contraction with coefficient . Pick any , let be a sequence generated by

(1.8)

where satisfying certain conditions, then the sequence converges strongly to a fixed point of the nonexpansive mapping which is also the unique solution of the variational inequality

Theorem 1.3. [Ke and Ma] Let be a nonempty closed convex subset of the real Hilbert space . Let be a nonexpansive mapping with and be a contraction with coefficient . Pick any , let be a sequence generated by.

(1.9)

where satisfying certain conditions, then the sequence converges strongly to a fixed point of the nonexpansive mapping which is also the unique solution of the variational inequality

Recently, Motivated by Xu et al. ³⁴, Tang and Bao ³⁰ considered the following result:

Theorem 1.4. Let be a nonempty closed uniformly convex and -uniformly smooth Banach space with dual Let be a -Lipschitz continuous monotone mapping such that For given the sequence defined by

where is the normalized duality mapping. Suppose that then the sequence converges strongly to an element

Of recently many work has not yet been done for viscosity implicit double midpoint rule (VIDMR). The recent work done for (VIDMR) was done by Shrijana Dhakal and Wutiphol Sintunavarat in 2019 where they defined the sequence in the following theorem.

Theorem 1.5. Shrijana Dhakal and Wutiphol Sintunavarat ⁴⁶. Let be a nonempty closed convex subset of a real Hilbert space be nonexpansive mapping with , and be contraction mapping with the contractive constant . Define a sequence by the following viscosity method for implicit double midpoint rule (VIDMR) as follows:

(1.10)

where an for all and satisfies the following conditions:

(i):

(ii):

Then, the sequence converges to a fixed point of which is also the unique solution

(1.11)

Motivated by Xu et al. ³⁴, Tang and Bao ³⁰, Shrijana Dhakal and Wutiphol Sintu-navarat ⁴⁶ and others, we consider viscosity iterative algorithms for the implicit double midpoint rule for nonexpansion mapping in real Hilbert space. Applications to convex minimization problem and nonlinear Fredholm integral equations are included. The results presented in the paper extend and improve some recent results announced in the current literature.

2. Preliminary Notes

In the sequel, we always assume that is a real Hilbert space and is a nonempty, closed, and convex subset of . The nearest point projection from onto C, PC, is defined by

(2.1)

Namely, is the only point in that minimizes the objective over and is characterized as follows:

(2.2)

Mapping is said to be nonexpansive if

(2.3)

We use to denote the set of fixed points of A mapping is said to be contractive if there exists a constant a such

(2.4)

for all . In this case, is called contraction.

Lemma 2.1. ²⁹. Let be a sequence of nonnegative real numbers satisfying the following relation:

where and are real sequences such that

(i)

(ii)

Then the sequence converges to 0.

3. Main Results

Theorem 3.1. Let be a closed convex subset of a Hilbert space a nonexpansive mapping with , and a contraction with coefficient a . Let be generated by the following viscosity implicit double midpoint rule (VIDMR):

(3.1)

where is a sequence in such that:

(A1)

(A2)

(A3)

Then converges strongly to a fixed point of which is also the unique solution of the following variational inequality:

Proof. The proof is in five stages.

Step 1: We prove that is bounded.

Fixing any , we have

It then follows that

Therefore

Consequently we have

By induction, it is easy to see that

Hence is bounded for all .

Step 2: We now show that

where , then we have

Therefore we have

Hence from lemma 2.1 . This implies that as

Step 3: We now show that This follows from the argument below

It now follows that

Step 4: Again we prove that

(3.2)

where is the unique fixed point of the contraction , that is . Since the sequence is bounded, then there exist a subsequence of such that converges weakly to . Thus

(3.3)

Since then by 2.2, 3.2 and 3.3, we concludes that

(3.4)

Step 5: We now prove that as

Thus, we have the following

Therefore from lemma 2.1, we can see that and

Hence we can conclude that . This completes the proof.

Theorem 3.2. Let be a closed convex subset of a Hilbert space a nonexpansive mapping with , and a contraction with coefficient a . Let be a constant. Let be generated by the following viscosity implicit double midpoint rule (VIDMR):

where is a sequence in such that:

(A1)

(A2)

(A3)

Then converges strongly to a fixed point of , which is also the unique solution of the following variational inequality:

Corollary 3.3. Let be a closed convex subset of a Hilbert space a nonexpansive mapping with , and a contraction with coefficient a . Let be a constant. Let be generated by the following viscosity implicit double midpoint rule (VIDMR):

(3.5)

where is a sequence in such that: Then converges strongly to a fixed point of , which is also the unique solution of the following variational inequality:

Here we assumed

Corollary 3.4. Let be a closed convex subset of a Hilbert space a nonexpansive mapping with , and a contraction with coefficient a . Let be generated by the following viscosity implicit double midpoint rule (VIDMR):

(3.6)

where is a sequence in such that:

(A1):

(A2):

(A3):

Then converges strongly to a fixed point of , which is also the unique solution of the following variational inequality:

4. Application to Convex Minimization Problems

In this section, we study the problem of finding a minimizer of a convex function defined from Hilbert space

The following basic results are well known.

Remark 4.1. It is well known that if be a real-valued differentiable convex function and , then the point is a minimizer of on if and only if .

Definition 4.2. A function is said to be strongly convex if there exists such that for every with and , the following inequality holds:

(4.1)

Lemma 4.3. Let be normed linear space and a real-valued differentiable convex function. Assume that is strongly convex. Then the differential map is strongly monotone, i.e., there exists a positive constant such that

(4.2)

We now prove the following theorem.

Theorem 4.4. Let be a closed convex subset of a Hilbert space a nonexpansive mapping with , and a contraction with coefficient a . Let be generated by the following viscosity implicit double midpoint rule (VIDMR):

(4.3)

where is a sequence in with conditions A1, A2 and A3, then converges strongly to a fixed point of , which is also the unique solution of the following variational inequality:

Proof. Since is nonempty closed convex, it follows that has a unique minimizer characterized by (Remark 4.1). Finally, from Lemma 4.3 and the fact that the differential map is contraction with a with a contraction coefficient , then the proof follows from Theorem 3.1.

5. Fredholm Integral Equation

Let be space of square integrable function endowed with inner product Now we discuss the solution of following Fredholm integral equation:

(5.1)

and suppose that the following conditions hold: where and To obtain our claim, we consider the followings assumptions:

(A1) The functions are continuous.

(A2) is Lipschitz continuous, that is, for all

(5.2)

(A3) is continuous for all where :

(A4) and

Now, we consider the mapping defined as

(5.3)

It is easy to observe that is a nonexpansive mapping. For this, for every

This implies that

and is a nonexpansive mapping. Define

(5.4)

where is sufficiently large, then is a closed ball of of radius with center at origin. It can be easily seen that . From Theorem in ⁸, operator has a fixed point in and this fixed point of operator is a solution of nonlinear integral equation 5.1.

Theorem 5.1. Let be a Hilbert space defined above and be a operator defined in (5.3). Let be a contraction with coefficient For arbitrary given define the sequence as follows:

(5.5)

where is an identity operator and the sequences is in the interval satisfying the following conditions

(A1)

(A2)

(A3)

then the sequence converges weakly to the solution of nonlinear integral equation (5.1) and the proof is the required conclusion of Theorem 3.1.

Example 5.2. Consider the following integral equation:

(5.6)

The above integral equation is a particular case of 5.1 with

For any and , we have

(5.7)

(5.8)

It can be easily seen that is a continuous function. Thus, integral equation 5.6 has a solution. It can be seen that is a solution of 5.6.

Acknowledgements

The authors thank the referees for their helping comments, which notably improved the presentation of this paper.

The authors also wish to acknowledge the financial support by the university of the Gambia for supporting this project.

Disclosure Statement

No potential conflict of interest was reported by the authors.

References