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Journal of Pure and Applied Mathematics

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Oriehi Edisemi Destiny Anyaiwe1* and Chika Moore2
1 Department of Mathematics & Computer Science, Lawrence Technological University, Southfield, Michigan, 480075, USA, Email: [email protected]
2 Mathematics Department, Nnamdi Azikiwe University, Awka, Anambra State, Nigeria, Email: [email protected]
*Correspondence: Oriehi Edisemi Destiny Anyaiwe, Department of Mathematics & Computer Science, Lawrence Technological University, Southfield, Michigan, 480075, USA, Tel: 2482043495, Email: [email protected]

Received Date: Jun 06, 2021 / Accepted Date: Jun 08, 2021 / Published Date: Jun 16, 2021

Citation: Anyaiwe OED, Moore C. Iterative solutions for variational inclusion problems in Banach spaces. J Pur Appl Math. 2019;3(1): 01-04.

This open-access article is distributed under the terms of the Creative Commons Attribution Non-Commercial License (CC BY-NC) (http://creativecommons.org/licenses/by-nc/4.0/), which permits reuse, distribution and reproduction of the article, provided that the original work is properly cited and the reuse is restricted to noncommercial purposes. For commercial reuse, contact [email protected]


Variational inclusion problems have become the apparatus that is generally used to constrain sundry mathematical equations in other to pguarantee the uniqueness and existence of their solutions. The existence of these solutions was earlier studied and proven for uniform Banach Spaces using accretive operators. In this study, we extend the conditions to hold for arbitrary Banach Spaces using uniform accretive operators.


Accretive operators; Banach spaces

In recent years, variational inequalities have been extende in different directions and areas of studies, using novel and innovative techniques. One of such generalization is variational inclusions. Several problems that occur in engineering, optimization and control situations can be modeled by free boundary problems which leads to variational inequality and variational inclusion problems, Eq. 1 in appropriate spaces.


(Variational Inclusion Problem). Let image be two set-valued mappings, image an m-accretive mapping,image a single valued mapping and image a nonlinear mapping. For any given function image and image , we consider the following problem. Find qimage such thatimage (1)

Where E is a real Banach space and E* is its topological dual space. CB (E) is the family of all non-empty convex (closed) and bounded subsets of E.

The duality pairing between E and E*is defined by inner product image if E is a Hilbert space and the Hausdorff metric D (.,.) on CB(E) is defined by


Given that A, B∈CB(E) , the distances d(x,B) or d(A, y) is defined by

image (2)

Also, D (T) denotes the domain of T and the normalized duality map is defined by

image (3)

Variational inequalities were introduced in early nineteen sixties by Hartman and Stampachia [1].

Lemma I.1 (Micheal’s Selection Theorem). Let X and Y be two Banach spaces; image a lower semicontinuous mapping with nonempty closed convex values. Then T admits a continuous selection i.e. there exists a continuous mapping image such thatimage for eachimage

Lemma I.2

Let E be a uniformly smooth Banach space and image be a lower semicontinuous and m-accretive mapping. Then the following conditions hold; (a) T admits a continuous and m-accretive selection

(b) If T is also image strongly accretive, then T admits a continuous m-accretive and image strongly accretive selection.

Lemma I.3

Nadler’s Theorem). Let E be a complete metric space, image be a set-valued mapping then for any given ∈> 0 and for any given image , there exists v∈Ty such that

image (4)

Algorithm I.4

(Iterative Sequence). For any given image compute the sequenceimage by the iterative schemeimage

For image


For image ,


Eq. 5, is called the Mann iterative sequence, it is a direct consequence of invoking Michael’s Selection theorems [2]. Using Nadler’s theorem [3], Chang in [4] proved Lemma I.2, thereby establishing the existence of unique solutions to Variational Inclusion problems using accretive operators in uniform Banach Spaces. In this work we present the extension of Chang’s work to arbitrary Banach Spaces using uniformly accretive operators based on the Lipschitz property of T and F.


Definition I.2.

Let A be a set-valued mapping with domain D(A) and range R(A) in E. A is said to be accretive if for all image there existsimage suchimage (6)

Definition I.3.

Let image be a strictly increasing function with image then the mapping A is strongly accretive if for animage , and image

image (7)

If image then A is said to be k-strongly accretive, and said to be m-accretive if A is accretive and (I + rA)D(A) = E for all r > 0, where I is the identity mapping.

Theorem I.5

Let E be a uniform smooth Banach space, image and image be three set-valued mappings, q : E−D(A) a single valued mapping satisfying the following conditions;


(iv)The mapping image strongly accretive with respect to the mapping T, and image is a strictly increasing function with φ (0) = 0

(v) The mapping image is accretive with respect to the mapping F Then, for any given image , there existsimage which is a solution to Eq. 1.

Theorem I.6

Let E, T, F, A, g, and N be as in Theorem

I.5 and image , be a sequence in the closed interval [0, 1] satisfying the following conditions


If the ranges image and image are bounded, then for any given image , the iterative sequences {xn}, {un} and {vn} defined by Algorithm I.4 converges strongly to the solution q, u, z of the set-valued vibrational inclusion problem of Eq. 1

An equivalent form of Theorem I.6 is given as lemma [5].

Lemma I.7

(Moore and Nnoli). Let {an}, {n} and {n} be real sequences such that image Also, letimage be a strictly increasing function withimage ifimage.

We are not going to reproduce the proofs of these Theorems and Lemmas, it suffices to indicate their implications, conclusions and some of their rudiments as they are used in the course of this paper. For instance, to prove Theorem I.5, one defines the mapping image that is expressed by image and invoke Morales [6] to establish that S is m-accretive and image strongly accretive and hence use the proof of Lemma I.2(b) to conclude that S admits a continuous and –strongly accretive and m-accretive selection image. Then, the theorem and proof of Theorem 5.3 in Kobayashi [7] can be used to show that image is m-accretive and‑-strongly accretive. Then image can be used to construct a variational inclusion problem that is a subset of Eq. 1 whose solution parses to Eq. 1 by virtue of uniqueness of the element image. The proof of Theorem I.6 is given in [8] and proof of Lemma I.7 is given in [9]. The same assumption that, image as image withimage is made in the proof of both (Theorem I.6 and Lemma I.7) in other to establish that the sequences {un} and {vn} are Cauchy in other to achieve the results presented in appendix A. In this study we present a cheaper way to achieve the same result for arbitrary Banach spaces [10].


We begin by presenting and proving the following lemmas, which extends the algebraic property of φ –strongly accretive operators to uniform accretive operators.


Let E be a real Banach space, image two set valued mappings and image a nonlinear mapping satisfying the following conditions;

(i) The mapping image is uniformly accretive with respect to the mapping T

(ii) The mapping image is accretive with respect to the mapping F

Then the mapping image defined by Sx=N(Tx, Fy) is uniformly accretive.

Proof: For any given image and for anyimage; there existsimage and image such that vi =N (wi, vi). By conditions (i) and (ii) and Definition I.2, we have that


Which implies that the mapping S=N (T(.), F(.)) is uniformly accretive.

Lemma II.2 Let E be a real Banach space and image be a lower semicontinuous, m-accretive mapping, then the following conditions holds;

(i) T admits a continuous and m-accretive selection

(ii) In addition, if T is also uniformly accretive, then it admits continuous, m-accretive and uniform accretive selections

Proof: The proof of (i) follows from the proofs of

Lemma I.1 and Lemma I.2 (b). For any given image and for any image , we have from the result of Lemma II.1 that


Letting image we obtain

image which implies that h is uniformly accretive

Now, since T and F are both Lipschitzian, it follows from Eq. 5, that


In the same vein,


This implies that given any image the iterative sequences un and vn are cauchy sequences. Therefore, there existsimage, image such thatimage By Lemmas II.1 and II.2 and results in results in [5,6,10] we infer that image too is uniform accretive.

Thus, there exist image and image such (8) and establish the results image in Appendix A with ‘less’ continuity restrictions


Appendix A

Consequence of Proof of Theorem I.6 and and Lemma I.7

Claim A.1. To prove Theorem I.6 and Lemma I.7, the claim is made that [11-13]

image (13)

Proof of Claim:There exists n0 such that for any image,


image and there exists image

image asimage. That is, ∀n ≥ 0 Which implies that image as image Now using the fact that T is µ

Lipschitzian and F is ∈Lipschitzian, it follows that from Eq. 5 that





This result implies that the sequences un and vn are Cauchy sequences. Therefore, there exists image such that imageimage. Next, we prove that image In fact, since,


image (14)

Result of Eq. 14 implies that image Similarly, Eq. 15 also implies that [14,15] image

image (15)

It remains to show [16] that image But Eq. 16 and 17 clearly shows this.

image (16)

Which implies that image and since;

image (17)

This implies u*= v. [17] Summing up the above argument we conclude that the sequences xn,un and vn defined by Eq. 5, converges strongly to solution (q, u, v) of problem 1 respectively.