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.
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, then
is 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
![]() | (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
![]() | (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
![]() | (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
![]() | (1.4) |
holds;
b) Generalized Hemi contractive in real Hilbert space, if
![]() | (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 let
Ø then, the sequence of Mann-type iterates given by
![]() | (1.6) |
And the sequence of Ishikawa-type iterates
![]() | (1.7) |
such that
![]() | (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.
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 let
Prox(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. Suppose
is 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 sequence
if
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 family
is 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
![]() |
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;
![]() | (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,
![]() | (3.2) |
Suppose, then
![]() | (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
.
![]() | (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;
![]() | (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
![]() | (3.8) |
Which implies
![]() |
cancellation of terms and convergence of with
and hence,
for some
we get
,
Since is continuous, strictly increasing, and convex function
as
.
Also by lemma 2.7 40 as
.
Thus, , which implies that,
.
Thus, by induction and triangle inequality, we have for all
.
Thus,
![]() | (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;
![]() | (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,
![]() | (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;
![]() | (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,
![]() | (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;
![]() | (3.15) |
Thus by lemma 2.15 we have the sequence converges to some
.
Thus, the sequence and hence
are bounded.
Since then we have
for 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
![]() | (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
![]() |
![]() | (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
![]() | (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
![]() | (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
![]() | (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.
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.
Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.
The authors declare that they have no competing interests.
[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. | ||
In article | |||
[2] | R.P.AGARWAL, D.OREGAN, D.R.SAHU. Fixed Point Theory for Lipschitzian type Mappings with Applications. Springer, New York (2009). | ||
In article | |||
[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. | ||
In article | |||
[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, | ||
In article | View Article | ||
[5] | F.E.BROWDER. Nonlinear mappings of nonexpansive and accretive type in Banach Spaces. Bull Am Math Soc 73, (1967) 875-882. | ||
In article | View Article | ||
[6] | F.E.BROWDER. Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. USA 54, (1965)1041-1044. | ||
In article | View Article | ||
[7] | F.E.BROWDER. Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Zeitschr. 100 (1967) 201-225. | ||
In article | View Article | ||
[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). | ||
In article | View Article | ||
[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. | ||
In article | |||
[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). | ||
In article | View Article | ||
[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. | ||
In article | View Article | ||
[12] | V.COLAO, G.MARINO, Krasnoselskii–Mann method for non-self mappings. Fixed Point Theory Appl. (2015). | ||
In article | View Article | ||
[13] | O.P.FERREIRA. P.R.OLIVEIRA. Proximal point algorithm on Riemannian manifolds, Optimization 51(2), (2002) 257-270. | ||
In article | View Article | ||
[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. | ||
In article | View Article | ||
[15] | K.GOEBEL, W.A.KIRK. Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, 28, Cambridge University Press, Cambridge, 1990. | ||
In article | View Article | ||
[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. | ||
In article | |||
[17] | F.O. ISIOGUGU, M.O. OSILIKE, Convergence theorems for new classes of multivalued hemi contractive-type mappings. Fixed Point Theory Appl. (2014). | ||
In article | View Article | ||
[18] | S.H.KHAN, I. YILDIRIM, Fixed points of multivalued nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. (2012). | ||
In article | View Article | ||
[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. | ||
In article | |||
[20] | K. KURATOWSKI, Topology, Academic press, 1, 1966. | ||
In article | |||
[21] | G. MARINO, Fixed points for multivalued mappings defined on unbounded sets in Banach spaces. J. Math. Anal. Appl. 157(2), (1991) 555-567. | ||
In article | View Article | ||
[22] | G. Marino, G.Trombetta, On approximating fixed points for nonexpansive mappings. Indian J. Math. 34, (1992) 91-98. | ||
In article | |||
[23] | J.T.MARKIN, Continuous dependence of fixed point sets. Proc. Am Math Soc. 38(1973)545-547. | ||
In article | View Article | ||
[24] | S.B.JR.NADLER, Multi-valued contraction mappings. Pac. J. Math. 30(2), (1969) 475-488. | ||
In article | View Article | ||
[25] | M. E. OKPALA, An iterative method for multivalued tempered Lipschitz hemi contractive mappings, Afr. Mat, (2017), 28(3-4) 595-604. | ||
In article | View Article | ||
[26] | B.PANYANAK, Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces. Comput. Math. Appl. 54(6), (2007) 872-877 . | ||
In article | View Article | ||
[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. | ||
In article | View Article | ||
[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. | ||
In article | |||
[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. | ||
In article | View Article | ||
[30] | Y.SONG, R. CHEN, Viscosity approximation methods for nonexpansive nonself mappings. J. Math. Anal. Appl. 321(1), (2006) 316-326. | ||
In article | View Article | ||
[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. | ||
In article | |||
[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. | ||
In article | View Article | ||
[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. | ||
In article | View Article | ||
[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. | ||
In article | |||
[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. | ||
In article | |||
[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. | ||
In article | View Article | ||
[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). | ||
In article | View Article | ||
[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. | ||
In article | |||
[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. | ||
In article | View Article | ||
[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. | ||
In article | |||
[41] | H.K.Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16, (1991) 1127-1138. | ||
In article | View Article | ||
[42] | H.ZEGEYE, N. SHAHZAD, Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings. Comput. Math. Appl. 62, (2011) 4007-4014. | ||
In article | View Article | ||
[43] | E. ZEIDLER.E. Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems Springer-Verlag New York Berlin Heidelberg Tokyo (1986). | ||
In article | |||
Published with license by Science and Education Publishing, Copyright © 2017 Mollalgn Haile Takele and B. Krishna Reddy
This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit
https://creativecommons.org/licenses/by/4.0/
[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. | ||
In article | |||
[2] | R.P.AGARWAL, D.OREGAN, D.R.SAHU. Fixed Point Theory for Lipschitzian type Mappings with Applications. Springer, New York (2009). | ||
In article | |||
[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. | ||
In article | |||
[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, | ||
In article | View Article | ||
[5] | F.E.BROWDER. Nonlinear mappings of nonexpansive and accretive type in Banach Spaces. Bull Am Math Soc 73, (1967) 875-882. | ||
In article | View Article | ||
[6] | F.E.BROWDER. Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. USA 54, (1965)1041-1044. | ||
In article | View Article | ||
[7] | F.E.BROWDER. Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Zeitschr. 100 (1967) 201-225. | ||
In article | View Article | ||
[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). | ||
In article | View Article | ||
[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. | ||
In article | |||
[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). | ||
In article | View Article | ||
[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. | ||
In article | View Article | ||
[12] | V.COLAO, G.MARINO, Krasnoselskii–Mann method for non-self mappings. Fixed Point Theory Appl. (2015). | ||
In article | View Article | ||
[13] | O.P.FERREIRA. P.R.OLIVEIRA. Proximal point algorithm on Riemannian manifolds, Optimization 51(2), (2002) 257-270. | ||
In article | View Article | ||
[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. | ||
In article | View Article | ||
[15] | K.GOEBEL, W.A.KIRK. Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, 28, Cambridge University Press, Cambridge, 1990. | ||
In article | View Article | ||
[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. | ||
In article | |||
[17] | F.O. ISIOGUGU, M.O. OSILIKE, Convergence theorems for new classes of multivalued hemi contractive-type mappings. Fixed Point Theory Appl. (2014). | ||
In article | View Article | ||
[18] | S.H.KHAN, I. YILDIRIM, Fixed points of multivalued nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. (2012). | ||
In article | View Article | ||
[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. | ||
In article | |||
[20] | K. KURATOWSKI, Topology, Academic press, 1, 1966. | ||
In article | |||
[21] | G. MARINO, Fixed points for multivalued mappings defined on unbounded sets in Banach spaces. J. Math. Anal. Appl. 157(2), (1991) 555-567. | ||
In article | View Article | ||
[22] | G. Marino, G.Trombetta, On approximating fixed points for nonexpansive mappings. Indian J. Math. 34, (1992) 91-98. | ||
In article | |||
[23] | J.T.MARKIN, Continuous dependence of fixed point sets. Proc. Am Math Soc. 38(1973)545-547. | ||
In article | View Article | ||
[24] | S.B.JR.NADLER, Multi-valued contraction mappings. Pac. J. Math. 30(2), (1969) 475-488. | ||
In article | View Article | ||
[25] | M. E. OKPALA, An iterative method for multivalued tempered Lipschitz hemi contractive mappings, Afr. Mat, (2017), 28(3-4) 595-604. | ||
In article | View Article | ||
[26] | B.PANYANAK, Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces. Comput. Math. Appl. 54(6), (2007) 872-877 . | ||
In article | View Article | ||
[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. | ||
In article | View Article | ||
[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. | ||
In article | |||
[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. | ||
In article | View Article | ||
[30] | Y.SONG, R. CHEN, Viscosity approximation methods for nonexpansive nonself mappings. J. Math. Anal. Appl. 321(1), (2006) 316-326. | ||
In article | View Article | ||
[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. | ||
In article | |||
[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. | ||
In article | View Article | ||
[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. | ||
In article | View Article | ||
[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. | ||
In article | |||
[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. | ||
In article | |||
[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. | ||
In article | View Article | ||
[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). | ||
In article | View Article | ||
[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. | ||
In article | |||
[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. | ||
In article | View Article | ||
[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. | ||
In article | |||
[41] | H.K.Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16, (1991) 1127-1138. | ||
In article | View Article | ||
[42] | H.ZEGEYE, N. SHAHZAD, Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings. Comput. Math. Appl. 62, (2011) 4007-4014. | ||
In article | View Article | ||
[43] | E. ZEIDLER.E. Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems Springer-Verlag New York Berlin Heidelberg Tokyo (1986). | ||
In article | |||