Abstract

In this paper, authors constructed Mann type of iterative method for the finite family of multi valued, nonself and nonexpansive mappings in a uniformly convex hyperbolic space. Authors proved strong convergence theorems of the iterative method, which approximates a common fixed point for the family single valued and multi valued nonexpansive mappings in a complete uniformly convex hyperbolic space which is more general than a complete CAT(0) space and a uniformly convex Banach space. The results in this work extended many results in the literature.

1. Introduction

Many nonlinear problems are naturally formulated as a fixed point problem for single valued or multi valued mapping. When a fixed point of nonexpansive mapping or contractive mapping exists, approximation technique is required. Following Picard’s iterative method which fails to converge in general for mappings which are not strictly contraction, other approximation techniques were introduced to approximate a fixed point. In the last forty years, numerous researchers have been attracted by this direction, and they developed iterative methods to approximate fixed point for not only nonexpansive mappings but also for some general class of nonexpansive mappings in linear Banach spaces and nonlinear domains too. Fixed point theory and hence approximation techniques have been extended to metric spaces(see, for example, ^{1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and their references).

Let be a non-empty subset of a metric space with metric d. Then we denote the set of non-empty, closed and bounded subsets of by We say is proximal if for every there exists such that

We denote the set of non-empty, proximal and bounded subsets of by We see that in CAT (0) space or uniformly convex Banach space every non-empty, closed and convex subset of is proximal ¹¹. For we define the Hausdorff distance between the two sets A and B by

where. Furthermore, as Kuratowski in ¹² presented that is metric space if is metric space and is complete if is complete.

Definition 1.1. Let be nonself multi valued mapping. Then the set of fixed points of is defined by

For a single valued mapping the set of fixed points is defined by

In particular, investigations have been made on nonlinear hyperbolic spaces.

Definition 1.2. ¹³ A hyperbolic space is a triple where the paira metric space and is a mapping satisfying the following

a)

b) ;

c)

d)

Every normed linear space, R-trees, the Hilbert balls with the hyperbolic metric, the Cartesian products of Hilbert balls, Hadamard manifolds and hence CAT(0) spaces are examples of hyperbolic spaces and the detailed concepts and examples can be found in ^{13, 14, 15, 16, 17}.

The following is found in ⁷.

A metric space E is said to be convex if it satisfies part a) of definition 1.2, hence, even in convex metric space E, for all the following hold;

a)

b)

Definition 1.3 ¹³ A hyperbolic space is uniformly convex if for every and there exists a such that for all

The modulus of uniformly convexity of the hyperbolic space is the mapping

which gives for any and we say that is monotone if it is decreasing with respect to

Authors in ⁹ proved that CAT(0) spaces are uniformly convex hyperbolic spaces. Thus, uniformly convex hyperbolic spaces are generalizations of both uniformly convex Banach spaces and CAT(0) spaces.

Definition 1.4. ^{1, 5, 18} Let K be a non-empty subset of a metric space E. Then the mapping is said to

a) be L-Lipschitzian if for some and for all

b) be nonexpansive if for all, when

c) be Quasi nonexpansive if and for all;

d) satisfy condition(C) if

A single valued mapping is said to

1) satisfy condition(C) if

2) be nonexpansive if

3) be Quasi nonexpansive if and

Thus, we see that every nonexpansive mapping satisfies condition(C), hence, the class of mappings satisfying condition(C) is an intermediate between the class of nonexpansive mappings and that of the class of quasi nonexpansive mappings.

Example 1.1. ¹⁸ Let be defined by

Then the map T satisfies condition(C) but is not nonexpansive mapping.

We may have a more general class of mappings: the class of strictlypseudocontractive mappings and their generalizations.

Definition 1.5. ⁸ Let K be non-empty subset of a hyperbolic space and let be a multi valued mapping. Then T is said to be

a) inward mapping if for any

b) k-strictly pseudocontractive mapping if for all there exists such that

Thus, in particular, if , then is nonexpansive mapping. Moreover, if is single valued mapping we have and .

Fixed point and common fixed point iterative methods are applicable in many areas such as convex optimization, control theory, differential inclusions, economics and physics. Consequently, the existence as well as methods of approximating fixed point and common fixed point for single valued and multi valued, self (nonself), contractive and nonexpansive type of mappings in Banach Spaces and generalizations to general metric spaces have been extensively studied by numerous authors of the field. In particular, fixed point results in a CAT(0) space and generalizations to hyperbolic spaces, which can be applied to graph theory, Biology and computer science have been extensively investigated by several authors.

Lim ¹⁹ was the first to introduce the delta convergence which is analogous to weak convergence in Banach spaces.

Definition 1.6. ⁹ Let E be a metric space and a bound sequence. Then for any point, if we define r by

Then the asymptotic radius of the sequence is given by and the asymptotic centre of is given by

Moreover, a sequence in a metric space E is said to be convergent to the point if for any subsequence of

Let be a non-empty subset of a metric space . Then the infimum of over is the asymptotic radius of the sequencewith respect to and is denoted by

The set of asymptotic centre of with respect to is given by

If the point x in the hyperbolic space is the unique asymptotic centre of every subsequence of a bounded sequences, then the sequence converges to x and we write it as or .

Consequently, fixed point iterative methods for the finite family of single valued and multi valued mappings in uniformly convex Banach spaces as well as in CAT(0) spaces have been studied by various authors (see, ^{20, 21, 22} and their references). Results have also been extended to uniformly convex hyperbolic space which is more general than uniformly convex Banach space and CAT(0) space (see, ² and references).

In particular, approximation techniques for common fixed point of nonself mappings via metric projection have been constructed by numerous researchers of the field ¹⁰. However Colao and Marino in ²³ presented that the computation for metric projection is costly, and they introduced iterative method by using inward condition without metric projection calculation. Consequently, authors in ^{8, 24, 25, 26, 27, 28, 29, 30} constructed iterative methods for approximating a common fixed point for family of nonself and inward mappings for single valued and multi valued mappings in Hilbert spaces, Banach spaces and CAT(0) spaces as well.

We raise an open question that, can we construct iterative methods which approximate common fixed point for the finite family nonself mappings in a uniformly convex Hyperbolic space which is more general than complete CAT(0) spaces and uniformly convex Banach spaces? Thus, it is our purpose in this paper to approximate a common fixed point for the finite family of nonself mappings with inward conditions in uniformly convex hyperbolic spaces, which is a positive answer to our question.

2. Preliminary Concepts

We use the following notations and definitions;

Definition 2.1. ³¹ A sequence in is said to be Fejer monotone with respect to a subset of if

Lemma 2.1. ³² Let be a metric space. Then, if) and , then for every there exists such that

Lemma 2.2. ³³ Let be a metric space. Then if and then there exists such that.

Lemma 2.3. ³⁴ Let be a uniformly convex hyperbolic space with monotone modulus of uniformity convexity and, let in be two sequences, if there exists satisfying the conditions,

and

where and . Then it holds that.

Definition 2.2. ²⁴ Let F and K be two closed and convex non-empty sets in a metric space E and . Then for any sequence, if the sequence converges strongly to an element where implies that is not Fejer-monotone with respect to the subset, and we say the pair (F, K) satisfies condition(S).

Example 2.1. Let . Then, the pair satisfies condition(S) with the metric induced by norm in .

Definition 2.3 ³⁵ The multi valued mapping with non-empty set of fixed pointsis said to satisfy condition(I) if there exists a non decreasing non negative functionsatisfying and such that

Definition 2.4. The mapping is said to be semi compact if every bounded sequencein satisfying

has a convergent subsequence.

Furthermore, the multi valued mapping is semi compact if every bounded sequencein satisfying

has a convergent subsequence.

3. Results and Discussion

Mann Type of iterative method

Let be a finite family of nonself and nonexpansive multi valued mappings on a non-empty, closed and convex subset of a complete uniformly convex hyperbolic space . Then it is our objective to construct Mann type of iterative method for approximating a common fixed point of the family and determine conditions for convergence of the iterative method. We use inward condition instead of the computation for metric projection which is costly, that is computationally expensive in many cases and we prove both delta and strong convergence results of the iterative method.

Lemma 3.1. Let K be a non-empty, closed and convex subset of a complete metric space E and let be a finite family of multi valued mappings, for , define by

Then for any , and if and only if whereas if then . Moreover, if is inward mapping, then, in addition, if , then

where is the boundary of K.

The proof of this lemma follows from, lemma 2.1 and 3.1 of Colao and Mariao and Tuffa and Zegeye in ^{8, 23} respectively.

Theorem 3.2. Let be a family of nonself, nonexpansive and inward mappings on a non-empty, closed and convex subsetof a complete uniformly convex Hyperbolic space with monotone modulus of uniformly convexity, non-empty and for each k, .

Then the sequence which is defined by Mann type of iterative method

(3.1)

is well-defined and

Proof. By lemma 3.1, the sequence is well-defined and in , thus, to prove the theorem first we prove is fejer monotone with respect to F, to do so, let. Then since each is nonexpansive we have

(3.2)

Thus, the sequence is fejer monotone with respect to hence, the sequence is bounded.

Also, the sequence is decreasing, hence it converges for all thus, there exists such that , hence, Moreover, since , taking both sides we have We also see that Thus, by lemma 2.3 we have

(3.3)

Thus,

(3.4)

By induction we have

(3.5)

Thus, from equations (3.3) to (3.5) we have

and vice versa, thus, we have

Corollary 3.3. If then the iterative method in theorem 3.2 is reduced to the following

(3.6)

In this case, the sequence is well-defined and satisfies

Theorem 3.4. In theorem 3.2, if and (F,K) satisfies condition(S), then the sequenceconverges strongly to some elementof

Proof. The sequence is bounded, that is, there exists such that,

for some

thus, we have

(3.7)

Hence, the sequence is strongly Cauchy, hence Cauchy, in a complete metric space it converges. Thus, the sequence converges to some element We need to show

Moreover, as and since for every we have Since there exists a sub sequence of such that . Suppose is a sequence of real numbers such that and the limit , then , hence, we must have and its limit is which is in K, thus, , and assuming that ( F, K) satisfies condition(S) we have .

Thus, the sequence converges strongly to some element.

Theorem 3.5. Let be the family of nonself, multi valued, nonexpansive and inward mappings on a non-empty, closed and convex subset of a complete uniformly convex hyperbolic space E with monotone modulus of uniformly convexity, non-empty, and for every point , . Let be a sequence of Mann type defined by the iterative method

(3.8)

Then the sequence is well-defined, furthermore, if (F,K) satisfies condition (S), then the sequence converges strongly to some of .

Proof. By lemma 3.1 the sequence is well-defined and is in , thus, to prove the theorem first we prove that is fejer monotone with respect to F, to do so, let. Then since each is nonexpansive we have and by lemma 2.1 and lemma 2.2, there exists a sequence satisfying equality;

(3.9)

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

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

Also, from the method of proof of theorem 3.2 we have

Again, since we get

Since and is positive we have

From the proof of theorem 3.2 we have

(3.10)

Thus, the sequence is strongly Cauchy, hence, it is Cauchy and Cauchy sequence converges in the complete space thus, there exists such that as and for each there corresponds. Since the coefficient and we have Also since there exists a subsequence of the sequence such that suppose the sequence is sequence of real numbers and , in particular, hence, the sequence Thus, we have

thus, the limit is x which is in K, thus,, since the pair ( F, K) satisfies condition(S) we have.

Thus, the sequence converges strongly to some element.

Corollary 3.6. Let be a finite family of nonself, single valued, nonexpansive and inward mappings on a non-empty, closed and convex subset of a complete uniformly convex hyperbolic spacewith monotone modulus of uniformly convexity, such that non-empty, and for all. Let be a sequence of Mann type defined by the iterative method

(3.11)

Then the sequence is well-defined, furthermore, if (F,K) satisfies condition(S), then the sequence converges strongly to some of .

Proof. From the method of proof of theorem 3.4, we put hence, the proof can be made in similar fashion.

Furthermore, strong convergence result can be obtained with suitable conditions on the mappings such as condition (I).

Definition 3.1. The finite family of mappingswhere with the intersection of sets of fixed points is said to satisfy condition (I) if there exists a non decreasing non negative function , such that the following holds

(3.12)

Theorem 3.7. Let be a family of nonself, multi valued, nonexpansive and inward mappings satisfying condition (I) on a non-empty, closed and convex subset K of a complete uniformly convex Hyperbolic spacewith monotone modulus of uniformly convexity, non-empty, and for all,. Let be a sequence of Mann type defined by the iterative method

Then the sequence is well-defined and in and if for , then the sequence converges strongly to some fixed point element of

(3.13)

Proof. From the method proof of theorem 3.2 we have , hence, we have Furthermore, since the mappings satisfy condition (I), there exists a non decreasing function satisfying the conditions such that, hence, we have

Thus, the monotonicity gives hence, for and for all we have

Taking infimum over all we get

hence, the sequence is Cauchy sequence, thus, it converges to some Moreover, we have

(3.14)

Since is closed we have which completes the proof.

Theorem 3.8. Let be a family of nonself, multi valued, nonexpansive and inward mappings satisfying condition (I) on a non-empty, closed and convex subset K of a complete uniformly convex hyperbolic space E with monotone modulus of uniformly convexity, non-empty, and for all , . Let be a sequence of Mann type defined by the iterative method

Then the sequence is well-defined and in and if holds for some , then the sequence converges strongly to some element of

(3.15)

Proof. Since lemma 3.1 is applicable if is replaced by Thus, the proof can be made in similar way.

Theorem 3.9. Let be a nonself, single valued and inward mapping satisfying both condition (C) and condition (I) on a non-empty, closed and convex subset K of a complete uniformly convex hyperbolic space E with monotone modulus of uniformly convexity and is non-empty. Letbe a sequence of Mann type defined by the iterative method

Then the sequence is well-defined and in K, and if holds for some, then the sequence converges strongly to some element of

(3.16)

Proof. Let. Then since

We have,

(3.17)

Hence, from method of proof of theorem 3.2 and 3.8 we have the sequence is Cauchy sequence, hence it converges to some point It suffices to show that

But

Since we have which completes the proof.

The results can be extended to the class of quasi nonexpansive mappings too.

4. Conclusion

Authors constructed Mann type of iterative methods to approximate common fixed point for the finite family of nonself and nonexpansive mappings with inward condition by lowering the computation for metric projection, which doesn’t exist in general Banach spaces and more general nonlinear spaces, even in Hilbert spaces, it requires additional computational techniques. Our theorems extended many results in the literature, in particular, we extended the result of ^{8, 25, 26, 27, 28, 29, 30} to a common fixed point for the family of nonexpansive and Suzi type of mappings to uniformly convex hyperbolic space which is more general than uniformly convex Banach spaces and CAT(0) spaces. We also extended many results to nonself single valued and multi valued mappings. Authors proved strong convergence result which is stronger than that of delta and weak convergence results.

Open questions. Finally we propose open questions for

a) the possibility to extend results of this work to more general classes of contractive mappings.

b) the possibility to lower condition (I) and condition(S) by imposing weaker conditions. If so, under what suitable conditions?

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