Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

Approximation for a Common Fixed Point for Family of Multivalued Nonself Mappings

Mollalgn Haile Takele*, B. Krishna Reddy

Department of Mathematics, College of Science, Bahir Dar University, Ethiopia

Abstract: In this paper, we introduce Mann type iterative method for finite and infinite family of multivalued nonself and non expansive mappings in real uniformly convex Banach spaces. We extend the result to the class of quasi non expansive mappings in real Uniformity convex Banach spaces. We also extend for approximating a common fixed point for the class of multivalued, strictly pseudo contractive and generalized strictly pseudo contractive nonself mappings in real Hilbert spaces. We prove both weak and strong convergence results of the iterative method.

Keywords: fixed point, nonself mapping, nonexpansive mapping, strictly pseudo contractive, generalized strictly pseudo contractive mappings, multivalued mapping, Mann type iterative method, uniformly convex Banach space

Citation: Mollalgn Haile Takele, B. Krishna Reddy, Approximation for a Common Fixed Point for Family of Multivalued Nonself Mappings. American Journal of Applied Mathematics and Statistics, 5 (6). 175-190. January 2017. doi:10.12691/ajams-5-6-1

1. Introduction

Fixed point theory for multi-valued mappings becomes very interesting for numerous researchers of the field because of its many real world applications in convex optimization, game theory and differential inclusions. Multi-valued mappings are also important in solving critical points in optimal control and other problems (Agarwal et al [2] pp 188). In single valued case, for example in studying the operator equation (when the mapping A is monotone) if K is a subset of a Hilbert space H, thenis monotone mapping if, Browder [5] introduced a new operator T defined by , where I is the identity mapping on the Hilbert space H, the operator is called pseudo contractive operator and the solutions of are the fixed points of the pseudo contractive mapping T and vice versa. Consider a mapping and the Variational inequality , in which the problem is to find satisfying the in equality, this problem is the Variational inequality problem arises in convex optimization, differential inclusions.

Let be convex, continuously differentiable function. Thus, is Variational inequality for, this inequality is optimality condition for minimization problem which appears in many areas. An example of a monotone operator in optimization theory is the multi-valued mapping of the sub differential of the functional and is defined by

[picture] | (1.1)

and satisfies the condition

In particular, if is convex, continuously differentiable function then , the gradient is a sub differential which is single valued mapping and the condition is operator equation and is Variational in equality and both conditions are closely related to optimality conditions. Thus, finding fixed point or common fixed point for Multi valued mapping is important in many practical areas.

Let be a non empty subset of a real normed space , then denotes the set of non empty, closed and bounded subsets of. We say is proximal, if for every there exists some such that We denote the family of nonempty proximal bounded subsets of by Prox(K). We observe that, in Hilbert spaces by projection theorem every non empty, closed and convex subset of H is proximal. Also Agarwal et al [2] presented that every nonempty, closed and convex subset of a uniformly convex Banach space is proximal. For in, we define the Housdorff distance between and in by

where .

Kuratowski [19] presented that is complete if is complete.

A mapping is non self multivalued mapping in general and the set of fixed point of is defined as

As Chidume et al [8] proposed, we give the definition of multi valued version for contractive mappings on a non empty subset of a real Banach spaces which is a generalization of single valued case as follows.

Definition 1.1 The mapping is said to be

a) contraction, if there is such that for all

b) L-Lipschitzian, if for some and for all

c) nonexpansive, if. for all , when .

d) Quasi non expansive mapping if

[picture] | (1.2)

In real Hilbert space H, if K is nonempty subset of H is said to be

e) Pseudo contractive, if

f) Hemi contractive in real Hilbert space, if Ø and for all

g) k-strictly pseudo contractive mapping in Hilbert spaces, if there exists such that

[picture] | (1.3)

holds.

h) Demi contractive Ø and there exists such that holds.

On the other hand, Chidume and Okpala [9] introduced generalized k-strictly pseudo contractive multivalued mapping which is defined as follow.

Definition 1.2 Let, K be a non empty subset of a real Hilbert space, and then the mapping is said to be

a) generalized k –strictly pseudo contractive mapping if there exists such that

[picture] | (1.4)

holds;

b) Generalized Hemi contractive in real Hilbert space, if

[picture] | (1.5)

It can be seen that, the class of generalized k- strictly pseudo contractive mappings includes the class of k-strictly pseudo contractive mappings.

Thus, the class of contraction as well as non expansive mappings are subset of the class of Lipschitzian and the class of k strictly Pseudo contractive mappings and hence the generalized k-strictly pseudo contractive mappings. Furthermore, the class of quasi non expansive mappings includes the class of non expansive mappings. Thus, the class of k-generalized strictly pseudo contractive mappings is more general than the class of non expansive mappings and the class of strictly pseudo contractive mappings. The study of fixed points of non expansive and contractive types of Multi valued mappings is very important and more complex in its applications in convex optimization, optimal control theory, differential equations and others.

Example 1.1 Let be given by for all.

Then, for all hence is non expansive and non self mapping.

Example 1.2 Let be given by . Then T is nonself, multivalued, k-strictly pseudo contractive mapping but not non expansive type (see [35]) with.

Example 1.3 Let be defined by. , thus

Then T is nonself which is not nonexpansive mapping.

Markin [23] was the first who presented the work on fixed points for multi-valued (nonexpansive) mappings by the application of Hausdorff metric and following his work, an extensive work was done by Nadler [24], since then existence of fixed points and their approximations for multi-valued contraction and nonexpansive mappings and their generalizations have been studied by several authors [1,3,4,8,10,14,19,20,21,24].

To mention a few, in 2005, Sastry and Babu [27] constructed Mann and Ishikawa-type iterations as given bellow

Let Prox (K) be a multi-valued mapping and letØ then, the sequence of Mann-type iterates given by

[picture] | (1.6)

And the sequence of Ishikawa-type iterates

[picture] | (1.7)

such that

[picture] | (1.8)

and they proved strong convergence of the iterative methods to some points in F(T) assuming that K is compact and a convex subset of a real Hilbert space H, T is nonexpansive mapping with ∅ the parameters satisfying certain nailed conditions.

Panyanak [24], consequently, Song and Wang [32], with additional nailed condition extended the result of Sastry and Babu [27] to more general spaces, uniformly convex Banach spaces, indeed, they proved the convergence results of Ishikawa-type iterative method. Moreover, Shahzad and Zegeye [29] extended the above results to multivalued quasi-nonexpansive mappings and removed the compactness assumption on K. They also constructed a new iterative scheme to relax the strong condition in the Song and Wang [32], consequently, Djitte and Sene [4] constructed the Ishikawa type iterative method for multi-valued and Lipschitz pseudo contractive mapping, they also proved convergence with more restrictions. In addition, Chidume and Okpala[9] constructed iterative method of Mann and Ishikawa type for approximating fixed points for generalized k strictly pseudo contractive Multivalued mapping, later on Okpala[25] modified the iteration for three step Ishikawa iterative method for approximating fixed points for Hemi contractive mappings. However, all the above results were for self mappings, on the other hand, in practical areas, there are cases of which we must consider non self mapping or family of non self mappings.

For approximating fixed points of nonself single-valued mappings, several Mann and Ishikawa-type iterative schemes have been studied via projection for sunny nonexpansive retraction [16,19,22,29,30,31,33,30-40]. However, recently, Colao and Marino [12] presented that the computation for sunny non expansive retraction is costly and they proposed the method with lowering the requirement of metric projection. Motivated by the work of Colao and Marino [12] many authors presented iterative methods for approximating a fixed point and a common fixed point for both finite and infinite family of single valued mappings without the requirement of metric projection [34,35]. More recently, Tufa and Zegeye [37] introduced a Mann-type iterative scheme for approximating fixed points for multi-valued nonexpansive nonself single mapping in real Hilbert space, which generalizes the result of Colao and Marino [12] to the class of multivalued mappings and they proved convergence with the assumption that the mapping satisfies inward condition in the following theorem.

Definition 1.2 Let K be a nonempty subset of a real Banach space E, a mapping is said to be inward if for each

Example 1.3 Considering example 1.1, let . Then , thus we have

Hence, is inward mapping, in fact, .

Thus, T is nonself, nonexpansive inward mapping.

Theorem TZ [37] (Tufa and Zegeye; Theorem 3.2) Let K be a nonempty, closed and convex subset of a real Hilbert H and let Prox(H) be an inward nonexpansive mapping with ∅ and . Let be a sequence of Mann-type given by

such that

Then, weakly converges to a fixed point of T. Moreover, if and K is strictly convex, then the convergence is strong.

It has been observed that, the existence of the sequence satisfying the condition is guaranteed by lemma 2.3 [17] which is stated in our preliminary section.

Authors [37] also extended the result for quasi-nonexpansive type mapping in a real uniformly convex Banach space E with some appropriate restrictions.

Definition 1.2 A uniformly convex space E is a normed space E for which for every, there is a such that for every if then .

Hilbert spaces, the sequences space, the Lebsgue space ( are examples of Uniformly convex Banach spaces.

The above results so far discussed were applicable for a single non expansive or quasi non expansive mapping , on the other hand in many practical areas we may face family of mappings and a more general class of mappings the so called the class of strictly pseudo contractive mappings.

Thus, motivated by the ongoing research work, in particular, the result of Tuffa and Zegeye [37], our question is that, is it possible to approximate a common fixed point for the family of nonself, multivalued and non expansive and strictly pseudo contractive mappings in real Hilbert spaces and real uniformly convex Banach spaces?

Thus, it is the purpose of this paper to construct Mann type iterative method for approximating a common fixed point of both finite and infinite family of nonself, multivalued, nonexpansive mappings and quasi nonexpansive mappings as well and to extend the result to the class of strictly pseudo contractive mappings which is a positive answer to our question.

2. Preliminary Concepts

We use the following notations and definitions;

Definition 2.1 Let be a non empty subset of a real Banach space, and let be multivalued mapping, is demi closed at 0, if for any sequence in converges weakly to and, then. Moreover, is demi closed at 0 is strongly demi closed at 0, if for any sequence in converges strongly to and, then.

Lemma 2.1 ([28], lemma 2.6) Let be a nonempty, closed and convex subset of a real Hilbert space and letProx(H) be a nonexpansive multi-valued mapping. Then, is demi closed at zero.

Definition 2.2 A Banach space is said to satisfy Opial’s condition if for any sequence in , converges weakly to some implies

for all.

Definition 2.3 A sequence in is said to be Fejer monotone with respect to a subset of, if.

Lemma 2.2 [24] Let be a real Banach space. Then, if) and, then for every there exists such that .

Lemma 2.3 [17] Let be a real Banach space. Then, if Prox (E) and, then there exists such that .

Lemma 2.4 (Xu [41]). Let be two fixed numbers and is a real Banach space. Then is uniformly convex if and only if there exists a continuous, strictly increasing and convex function with such that

for all and where .

Lemma 2.5 [42] In real Hilbert space, for all and for such that the equality

holds.

Lemma 2.6 (Browder [7], Ferreira-Oliveira [13]) Let be a complete metric space and a nonempty subset. If is Fejer monotone with respect to then is bounded. Furthermore, if a cluster point of belongs to then converges strongly to . In the particular case of a Hilbert space, given the set of all weakly cluster points of

Converges weakly to a point if and only if

Lemma 2.7 (See, for example, Zeidler [43 ]pp 484) Let E be a real uniformly convex Banach space, in be two sequences, if there exists a constant such that

for for some then .

Lemma 2.8 [8]: Let be a nonempty subset of a real Hilbert space and let be a multivalued 𝑘-strictly pseudo contractive mapping. Then, is Lipschitz with Lipchitz constant .

Lemma 2.9 [38] Let be a real Hilbert space. Supposeis a closed, convex, nonempty subset of . Assume that is pseudo contractive multi-valued mapping with F(T) is non empty. Then, F(T) is closed and convex.

Lemma 2.10 [38] Let H be a real Hilbert space. Suppose is a closed, convex, nonempty subset of H. Assume that is Lipschitz pseudo contractive multi-valued mapping. Then is demi closed at zero.

Lemma 2.11 Let be a nonempty subset of a real Hilbert space 𝐻 and let Prox(H) be a multivalued 𝑘-strictly pseudo contractive mapping. Then, is Lipschitzian with Lipschitz constant and hence is demi closed at 0. (Proof can be done with lemma 2.3, lemma 2.8 and lemma 2.10).

Definition 2.4 Let F, K be two closed and convex nonempty sets in a Banach spaces E and. For any sequenceif converges strongly to an element implies that is not Fejer-monotone with respect to the set, we say the pair (F, K) satisfies S-condition.

Example Let. Then the pair satisfies S- condition.

Definition 2.5. Let be sequence of mappings with nonempty common fixed point set Then, the familyis said to be uniformly weakly closed if for any convergent sequence such that, then the weak cluster Points of belong to F.

Lemma 2.12 [9]: Let be a nonempty subset of a real Hilbert space and be a multivalued generalized 𝑘-strictly pseudo contractive mapping. Then, is Lipschitz with Lipschitz constant and F(T) is closed and convex.

Lemma 2.13 [9] Let be a nonempty and closed subset of a real Hilbert space 𝐻 and let CB(K) be a multivalued generalized 𝑘-strictly pseudo contractive mapping. Then, is Lipschitzian with Lipschitz constant and is strongly demi closed at 0.

Definition 2.6 Let be a nonempty and closed subset of a real Hilbert space 𝐻. Then a map CB(H) is said to be Hemi compact, if for any sequence in such , then there exists a sub sequence of such that converges strongly to in K.

Remark: Any mapping on a compact domain is Hemi compact.

Lemma2.15 [36] Let be a sequence of non negative real numbers such that , then converges and if in addition the sequence has a subsequence which converges to 0, then the original sequence converges to 0.

The following lemma can be found in [9].

Lemma 2.16 [9] Let E be a normed linear space, and. Then, the following hold;

a) ;

b)

c)

d)

e)

Consequently, from (d) the following was obtained [9]

Lemma 2.17 [9] Let be a non empty and closed subset of a real Hilbert space H and let be generalized k- strictly pseudo contractive mapping. Then, for any given in K there exists such that.

In particular, if is proximal, there exists

3. Main Results

Let Prox(E) be family of non self and multivalued mappings on a non-empty closed, convex subset of a real uniformly convex Banach space E, our objective is to introduce an iterative method for common fixed point of the family and determine conditions for convergence of the iterative method. We use the condition that mappings to be inward instead of metric projection, which is computationally expensive in many cases, and we prove both weak and strong convergence of the iterative method. Thus, we shall have the following lemma.

Lemma 3.1 Let K be a nonempty, closed and convex subset of a real Banach space E, or Prox(E) be multivalued mappings,.Define by

Then for any, the following hold:

1) and if and only if;

2) If , then ;

3) If is inward mapping;

4) If then where is the boundary of K.

The proof of this lemma follows from lemma 3.1 of Takele and Reddy [32] Calo and Mariao [12] and Tuffa and Zegeye [37].

Theorem 3.2: Let Prox(H) be family of, non self, multi valued, nonexpansive and inward mappings on a non-empty, closed and convex subset K of a real Hilbert space H, with non empty,,for all , . Let be a sequence of Mann type defined by the iterative method given by

is well-defined and if for some , then the sequence converges weakly some element p of . Moreover, if and (F,K) satisfies S-condition, then the convergence is strong.

Proof: By lemma 3.1 is well-defined and is in K, thus, to prove the theorem first we prove is fejer monotone with respect to F, to do so, let, then we have the following inequality;

[picture] | (3.1)

Thus, the sequence is fejer monotone with respect to F.

Since is decreasing and bounded below it converges, and hence and are bounded.

That is,

for some

Also, we have the following inequality,

[picture] | (3.2)

Suppose, then

[picture] | (3.3)

Hence, and

which implies that,

Thus, by induction and triangle inequality, we have

Thus,

Thus, by definition of infimum and we have as .

[picture] | (3.5)

Thus, . Since is bounded, it has a convergent subsequence such that weakly, since K is closed and convex, , and for some and for each there is some such that.

Thus, as . . Since is demi closed, we have, and since is arbitrary, we have.

Since H satisfies opial’s condition and is convergent, we get weakly.

Thus, the sequence converges weakly some element p of .

Moreover, if , then

Hence, the sequence is strongly Cauchy, thus it is Cauchy and converges to some element

Moreover, since is inward, then, hence for every , we have , in particular, since, there is a subsequence of such that , whose limit is . Thus, and since the pair (F, K) satisfies S- condition.

Thus converges strongly to some element.

Theorem 3.3: Let Prox (E) be family of non self, multi valued, nonexpansive and inward mappings on a non-empty, closed and convex subset K of a real Uniformly convex Banach space E, satisfying opial’s condition with non empty, for all for each , , and suppose is demi closed at 0, let be a sequence of Mann type defined by the iterative method,

Then the sequence is well-defined and if for some , and E satisfies opial’s condition, then the sequence converges weakly some element p of Moreover, if and (F,K) satisfies S-condition, then the convergence is strong.

Proof: By lemma 3.1 is well-defined and is in K, thus to prove the theorem, first we prove is fejer monotone with respect to F, to do so, let, then we have the following in equality;

[picture] | (3.6)

Thus, the sequence is fejer monotone with respect to F.

Since is decreasing and bounded below, thus it converges, and hence and are bounded.

That is,

for some

Suppose then

can be shown by Lemma 2.4, Xu [38] since E is uniformly convex Banach space , for real numbers, there exists a continuous, strictly increasing, and convex function with such that for all and where . (3.7)

Since is bounded R can be chosen so that If, we have the inequality.

Thus, for we get

[picture] | (3.8)

Which implies

cancellation of terms and convergence of with and hence, for some we get,

Since is continuous, strictly increasing, and convex functionas .

Also by lemma 2.7 [40] as .

Thus, , which implies that, .

Thus, by induction and triangle inequality, we have for all .

Thus,

[picture] | (3.9)

Thus, by definition of infimum and we have as .

Thus, . Since is bounded, it has a convergent subsequence such that weakly, since K is closed and convex, and for some and for each there is some such that, . Thus, as

Thus, as since is demi closed, we have, and since is arbitrary, we have .

Since E satisfies opial’s condition and is convergent, we get weakly.

Thus, the sequence converges weakly to some element p of .

Moreover, if then

Hence, the sequence is strongly Cauchy, thus it is Cauchy and converges to some element

Moreover, since is inward, then, hence for every, we have , in particular, since, there is a subsequence of such that, , whose limit is . Thus, and since the pair (F, K) satisfies S- condition.

Thus converges strongly to some element.

Theorem 3.4 Let K be a convex, closed and nonempty subset of a real Hilbert space H and let be a uniformly weakly closed, countable family of non self, multi valued and nonexpansive mappings with is non empty and for all, .Let be a sequence defined by the Mann type iterative method,

Then, the sequence is well-defined and if for some, then the sequence converges weakly some element p of . Moreover, if, and (F,K) satisfies S-condition, then the convergence is strong.

Proof, let, and by lemma 2.3 [17] there is a sequence, satisfying

thus we have the following in equality;

[picture] | (3.11)

Thus is fejer monotone with respect to F.

Since is decreasing and bounded below, it converges, and hence and are bounded.

That is,

for some

Also, we have the following inequality,

[picture] | (3.12)

Suppose, then

Hence,

Thus, by definition of infimum and we have as.

Since is bounded, it has a convergent subsequence such that weakly, since K is closed and convex, , , since is uniformly weakly closed, , that is, .

Since H satisfies opial’s condition and is convergent, we get weakly.

Thus, the sequence converges weakly to some element p of .

Moreover, if then

Hence, the sequence is strongly Cauchy, thus it is Cauchy and converges to some element

Moreover, since is inward, then, hence for every, we have , in particular, since, there is a subsequence of such that , , whose limit is . Thus, and since the pair (F, K) satisfies S- condition.

Thus converges strongly to some element.

Theorem 3.5 Let K be a convex, closed and nonempty subset of a real Uniformly convex Banach space E satisfying opial’s condition and let be a uniformly weakly closed, countable family of non self, multi valued and nonexpansive(quasi non expansive) mappings with is non empty and for all,. Let be a sequence defined by the Mann type iterative method

Then, the sequence is well-defined and if for some, and E satisfies opial’s condition, then the sequence converges weakly some element p of .

Moreover, if , and (F,K) satisfies S-condition , then the convergence is strong.

Proof can be made in similar way as theorem 3.3 and 3.4.

Theorem 3.6 Let K be a strictly convex, closed and nonempty subset of a real Hilbert space H and let be a non self, multi valued and k-strictly pseudo contractive and inward mapping with is non empty and for each , is closed and for all . Let be a sequence defined by the iterative method,

Then, the sequence is well-defined and the sequence converges weakly to some element p of Moreover, if , then the convergence is strong.

Proof. By lemma 3.1 is well-defined and is in K, thus, to prove the theorem first we prove is fejer monotone with respect to F, to do so, let, then the following holds;

[picture] | (3.13)

Thus is fejer monotone with respect to F.

Since is decreasing and bounded below, it converges, and hence and since T is Lipschitzian by lemma 2.8 [8] are bounded.

That is,

for some

We also have the following inequality;

Thus,

[picture] | (3.14)

Suppose , since there exists, such that, thus, hence, also from the method of proof of Mariano and Trombetta [22] it can be seen is decreasing as and

Letting, we have the following;

solving the inequality we get , which gives is decreasing and hence converges to, thus, as a result, which implies that as.

On the other hand, since the sequence is bounded, it has a weakly convergent subsequence such that weakly, since K is closed and convex since is demi closed at 0,

Since, every Hilbert space satisfies opial’s condition, weakly for some .

Moreover, if , then

Hence, the sequence is strongly Cauchy, thus it is Cauchy and converges to some element

Moreover, since is inward, then, hence for every, we have , in particular, since, there is a subsequence of such that,

whose limit is thus,. The continuity of Lipschitz mapping T gives

thus, there is such that as. Since T is continuous and with each is closed the following holds; as , thus, for all we have as a result it can be shown that . Since K is strictly convex, in similar fashion (see [37]) it can be seen that, hence Thus, the sequence converges strongly to some element

Theorem 3.7 Let K be a strictly convex, closed and nonempty subset of a real Hilbert space H and let be a non self, multi valued and generalized k-strictly pseudo contractive and inward mapping with is non empty and for all , for each , is closed. Let be a by the iterative sequence defined method,

Then, the sequence is well-defined and the sequence converges strongly to some element p of .

Proof. By lemma 3.1 is well-defined and is in K.

Let. Since and by lemma 2.6 we have the following inequality;

[picture] | (3.15)

Thus by lemma 2.15 we have the sequence converges to some.

Thus, the sequence and hence are bounded.

Since then we havefor some

Hence, the sequence is strongly Cauchy, thus it is Cauchy and converges to some element

Moreover, since is inward, then, hence for every, we have . Since, there is a subsequence of such that

whose limit is, thus.Since is Lipschitz mapping, hence is Cauchy sequence, thus, there is such that as. Since T Lipchitz continuous we have as , since is closed, hence for all , we have ,as a result it can be shown that . Since K is strictly convex, in similar fashion (see [37]) it can be seen that, . Thus, the sequence converges strongly to some element

Theorem 3.8 Let K be a strictly convex, closed and nonempty subset of a real Hilbert space H and let be a non self, multi valued and generalized k-strictly pseudo contractive and inward mapping with is non empty and for all . Let be a by the iterative sequence defined method,

Then, the sequence is well-defined and the sequence converges strongly to some element p of .

Proof. By lemma 3.1 is well-defined and is in K.

Let. Then, applying lemma 2.5 and lemma 2.16 we have

[picture] | (3.16)

Thus by lemma 2.15 we have the sequence converges to some.

Thus, the sequence and hence are bounded.

Since, then we have.

Hence, the sequence is strongly Cauchy, thus it is Cauchy and converges to some element

Moreover, since is inward, then , hence for every, we have . Since, there is a subsequence of such that,

whose limit is .

Since is Lipschitz mapping

hence is Cauchy sequence, thus, there is such that as. Since T is Lipschitz continuous we have

Since is closed, , hence for all , we have ,as a result it can be shown that . Since K is strictly convex, in similar fashion (see [37]) it can be seen that, .Thus, the sequence converges strongly to some element

Theorem 3.9 Let K be a strictly convex, closed and nonempty subset of a real Hilbert space H and let be a non self, multi valued and generalized k-strictly pseudo contractive and inward mapping with is non empty and for all , . Let

Let such that and . Let be a sequence defined by the iterative method,

Suppose is hemi compact and then converges to some. And if K is strictly convex, and, then converges to some.

Proof. First, we see that, for any , since each is inward, then , indeed, for , we have

Thus, Let.

Thus, applying lemma 2.5 and lemma 2.16 we have

[picture] | (3.17)

Thus, by lemma 2.15 we have converges to some, hence the sequence and are bounded. From (3.17) we have

Since, for some, we have

Case 1 suppose and is hemi compact, since , let by Archimedean property of real numbers we have and

[picture] | (3.18)

Thus, for each , , hence there exists a subsequence of such that , thus as . Since is hemi compact there exists a subsequence of such that Moreover, if we take satisfying, and lipschitz property of we have

[picture] | (3.19)

Thus,, hence , since the result is true for any , .

Since for any , converges, hence the sequence converges strongly to.

Case 2. Suppose K is strictly convex and, then

Hence, the sequence is strongly Cauchy, thus it is Cauchy and converges to some element

Moreover, since is inward, then, hence for every, we have . Since, there is a subsequence of such that,

whose limit is , thus, .

Since is Lipschitz mapping

hence is Cauchy sequence, thus, there is such that as. Since is strongly Cauchy, it converges, hence there exists such that let then we have

[picture] | (3.20)

is Lipschitz continuous we have

as , since is closed, , hence for all , we have , as a result it can be shown that . Since K is strictly convex, in similar fashion (see [37]) it can be seen that, .Thus, the sequence converges strongly to some element

Remark: In the above discussions, if we consider family of strictly pseudo contractive or generalized strictly pseudo contractive mappings we can use in theorem 3.9.

Example 3.1 Now we give an example of sequence of multivalued mappings.

Let be defined by Then .

Thus, is nonexpansive multivalued nonself mapping. For each let , then , and , thus

hence is inward mapping.

Thus, the sequence of mappings satisfies the condition of the theorem 3.2 thus, the algorithm converges to a unique common fixed point, we also see that and the pair (F,K) satisfies S-condition. We see the first four iterates as;

Let Then taking thus, , thus , and , taking such that

say , we get , , and , in the same fashion taking we get , and .

Remark: Let Prox(H) be non self, multi valued, nonexpansive and inward mapping on a non-empty, closed and convex subset K of a real Hilbert space H, with non empty, for all,. Let be a sequence of Mann type defined by the iterative method

is well-defined and if for some, then the sequence converges weakly to some element p of . Moreover, if , and (F,K) satisfies S-condition, then the convergence is strong.

4. Conclusion

Our theorems extend many results in literature, in particular, our theorems [3.2-3.5] extend the result of Tufa and Zegeye [33] to a common fixed point for the family of non expansive mappings. We also extend the result of [9] and [25] to approximation for a fixed point and a common fixed point for family of more general class of mappings, the so called generalized k-strictly pseudo contractive nonself mappings.

Authors’ Contributions

Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.

Competing Interests

The authors declare that they have no competing interests.

References

[1]   | M.ABBAS.M, YJ. CHO. Fixed point results for multi-valued non-expansive mappings on an unbounded set. Analele Scientific Ale Universitatii Ovidius Constanta 18(2), (2010) 5-14.
[2]   | R.P.AGARWAL, D.OREGAN, D.R.SAHU. Fixed Point Theory for Lipschitzian type Mappings with Applications. Springer, New York (2009).
[3]   | I. BEG, M.ABBAS. Fixed-point theorem for weakly inward multi-valued maps on a convex metric space. Demonstr.Math. 39(1), (2006) 149-160.
[4]   | T.D.BENAVIDES, P.L.RAMREZ. Fixed point theorems for multivalued nonexpansive mappings satisfying inwardness conditions. J. Math. Anal. Appl. 291(1), (2004) 100-108,
[5]   | F.E.BROWDER. Nonlinear mappings of nonexpansive and accretive type in Banach Spaces. Bull Am Math Soc 73, (1967) 875-882.
[6]   | F.E.BROWDER. Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. USA 54, (1965)1041-1044.
[7]   | F.E.BROWDER. Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Zeitschr. 100 (1967) 201-225.
[8]   | C.E.CHIDUME, C.O. CHIDUME, N. DJITTE, M.S.MINJIBIR. Convergence theorems for fixed points of multivalued strictly pseudo contractive mappings in Hilbert spaces, Abstract and Applied Analysis, (2013).
[9]   | C.E.CHIDUME, M.E.OKPALA, On a general class of multi valued strictly pseudo contractive mappings, Journal of Nonlinear Analysis and Optimization, 5(2), (2014), 7-20.
[10]   | C.E.CHIDUME, M. E. OKPALA, Fixed point iteration for a countable family of multi‑valued strictly pseudo‑contractive‑type mappings; Springer Plus (2015).
[11]   | C.E. CHIDUME, H.ZEGEYE, N. SHAHZAD, Convergence theorems for a Common fixed point of a finite family of nonself nonexpansive mappings,: Fixed Point Theory and Applications ;2 (2005) 233-241.
[12]   | V.COLAO, G.MARINO, Krasnoselskii–Mann method for non-self mappings. Fixed Point Theory Appl. (2015).
[13]   | O.P.FERREIRA. P.R.OLIVEIRA. Proximal point algorithm on Riemannian manifolds, Optimization 51(2), (2002) 257-270.
[14]   | J.GARCA-FALSET, E. LLORENS-FUSTER, T.SUZUKI, Fixed point theory for a class of generalized nonexpansive mappings. J. Math. Anal. Appl. 375(1), (2011) 185-195.
[15]   | K.GOEBEL, W.A.KIRK. Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, 28, Cambridge University Press, Cambridge, 1990.
[16]   | K.HUKMI,O.MURAT, A.SEZGIN, On Common Fixed Points of Two Non-self nonexpansive mappings in Banach Spaces, Chiang Mai J. Sci.; 34(3),(2007) 281-288.
[17]   | F.O. ISIOGUGU, M.O. OSILIKE, Convergence theorems for new classes of multivalued hemi contractive-type mappings. Fixed Point Theory Appl. (2014).
[18]   | S.H.KHAN, I. YILDIRIM, Fixed points of multivalued nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. (2012).
[19]   | H.KIZILTUNC, I.YILDIRIM. On Common Fixed Point of nonself, nonexpansive mappings for Multistep Iteration in Banach Spaces, Thai Journal of Mathematics, 6 ( 2) , (2008)343-349.
[20]   | K. KURATOWSKI, Topology, Academic press, 1, 1966.
[21]   | G. MARINO, Fixed points for multivalued mappings defined on unbounded sets in Banach spaces. J. Math. Anal. Appl. 157(2), (1991) 555-567.
[22]   | G. Marino, G.Trombetta, On approximating fixed points for nonexpansive mappings. Indian J. Math. 34, (1992) 91-98.
[23]   | J.T.MARKIN, Continuous dependence of fixed point sets. Proc. Am Math Soc. 38(1973)545-547.
[24]   | S.B.JR.NADLER, Multi-valued contraction mappings. Pac. J. Math. 30(2), (1969) 475-488.
[25]   | M. E. OKPALA, An iterative method for multivalued tempered Lipschitz hemi contractive mappings, Afr. Mat, (2017), 28(3-4) 595-604.
[26]   | B.PANYANAK, Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces. Comput. Math. Appl. 54(6), (2007) 872-877 .
[27]   | K.P.R.SASTRY, G.V.R. BABU, Convergence of Ishikawa iterates for a multivalued mapping with a fixed point. Czechoslovak Math. J. 55(4), (2005) 817-826.
[28]   | T.W. SEBISEBE, G.S. MENGISTU, Z, HABTU. Strong Convergence Theorems for a Common Fixed Point of a Finite Family of Lipschitz Hemi contractive-type Multivalued Mappings Advances in Fixed Point Theory,5(2) (2015)228-253.
[29]   | N. SHAHZAD, H.ZEGEYE, On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces. Nonlinear Anal. Theory Methods Appl. 71(3), (2009) 838-844.
[30]   | Y.SONG, R. CHEN, Viscosity approximation methods for nonexpansive nonself mappings. J. Math. Anal. Appl. 321(1), (2006) 316-326.
[31]   | Y.S. SONG, Y.J. CHO, Averaged iterates for non-expansive nonself mappings in Banach spaces. J. Comput. Anal. Appl. 11, (2009) 451-460.
[32]   | Y.SONG, H.WANG, Erratum to “Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces”, Comput. Math. Appl. 54(2007) 872-877.
[33]   | W.TAKAHASHI, G.E. KIM, Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces. Nonlinear Anal. Theory Methods Appl. 32(3), (1998). 447-454.
[34]   | M.H. TAKELE AND B. K.REDDY, Approximation of common fixed point of finite family of nonself and nonexpansive mappings in Hilbert space, Indian Journal of Mathematics and mathematical Sciences, 13(1) (2017) 177-201.
[35]   | M.H. TAKELE, B. K.REDDY, Fixed point theorems for approximating a common fixed point for a family of nonself, strictly pseudo contractive and inward mappings in real Hilbert spaces, Global journal of pure and applied Mathematics, 13(7) (2017) 3657-3677.
[36]   | K. K. Tan and H. K. Xu, Approximating Fixed Points of Nonexpansive mappings by the Ishikawa Iteration Process, J. Math. Anal. Appl. 178(2), (1993), 301-308.
[37]   | A.R.TUFA, H.ZEGEYE, Mann and Ishikawa-Type Iterative Schemes for Approximating Fixed Points of Multi-valued Non-Self Mappings Mediterr.J.Math,(2016).
[38]   | S.T.WOLDEAMANUEL, M. G. SANGAGO, H. ZEGEYE, Strong convergence theorems for a fixed point of a Lipchitz pseudo contractive multi-valued mapping, Linear Nonlinear Anal, 2(1) (2016) 87-100.
[39]   | H.K.XU, X.M.YIN, Strong convergence theorems for nonexpansive non-self mappings. Nonlinear Anal. Theory Methods Appl. 24(2), (1995) 223-228.
[40]   | H.K.Xu, Approximating curves of nonexpansive nonself-mappings in Banach spaces. C. R. Acad. Sci. Paris Sr. I Math. 325(2), (1997). 151-156.
[41]   | H.K.Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16, (1991) 1127-1138.
[42]   | H.ZEGEYE, N. SHAHZAD, Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings. Comput. Math. Appl. 62, (2011) 4007-4014.
[43]   | E. ZEIDLER.E. Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems Springer-Verlag New York Berlin Heidelberg Tokyo (1986).