#### Abstract

This article is about investigating Krasnoselskii and Mann’s convergence theorems for nonexpansive evolution family with Bochner integral and without convexity of Banach space. We introduced a general form of these results that extends these results to general form of semigroup called evolution families.

#### 1. Introduction

In the 19th century, the fixed point theory was started by Poincaré, followed by many other mathematicians like Brouwer [1], Schauder [2], and Tarski [3] in the 20th century. The fixed point theory has a wide range of applications. It is one of the most important tools of modern mathematical analysis and is useful in various field like mathematics, engineering, physics, statistics, economics, and many more. Fixed point theory can be used as a tool to discuss the uniqueness and existence of solutions of many problems like integral equations, differential equations, numerical equations, and algebraic systems. We refer to [4] for more detailed study of fixed point theory and its applications in metric spaces.

Throughout the paper, we will denote by a Banach space and let be a nonempty, closed, and convex subset of a Banach space . We recall that an operator on is called a nonexpansive mapping if for all . The collection of all fixed points of is denoted by . Let be a compact subset of ; then, Schauder’s fixed point theorem [5] says that is nonempty. For details, we refer to [610] and others.

Now we will again recall an important family which is known to a semigroup of linear operators on . A family of mappings is called a nonexpansive strongly continuous semigroup if it satisfies the following conditions:(i), where is the identity operator on .(ii), for all .(iii)For each , the map is strongly continuous. Such families arise from the solution of autonomous differential system: .

The study of fixed point of a semigroup got attention in the last four decades. For nonexpansive semigroups, some theorems [11] and corollary [12] are given below in details.

Theorem 1 (see [12]). Let be a one-parameter strongly continuous semigroup of nonexpansive mapping on . Let and be two sequences such that and . For , define a sequence in byfor . Then, the sequence converges strongly to a common fixed point of .
Krasnoselskii and Mann’s convergence theorems for nonexpansive mappings have been studied by many researchers (see, e.g., [1321] and the references therein). Recently, the following Krasnoselskii and Mann’s convergence theorem for nonexpansive semigroups has also been proved by Takahashi in [22].

Theorem 2 (see [22]). Let be a one-parameter strongly continuous semigroup of nonexpansive mapping on . Define a sequence in by .For every natural number , where and are two sequences that satisfy and , . Then, the sequence strongly converges to a common fixed point of .
Now we proved the following Browder’s type convergence theorem for nonexpansive evolution family without using the Bochner integral.

Theorem 3 (see [23]). Let be a nonempty and convex subset of complete inner product space . Let be a one-parameter strongly continuous evolution family of nonexpansive mapping on . Let be sequences of real numbers satisfying the conditions , and for natural number . For ordering , prescribe a sequence nearbyfor every natural number . Then, the sequence strongly converges to the member of .
In [13], Suzuki discussed the convergence of Krasnoselskii and Mann type convergence theorems for nonexpansive semigroup except the Bochner integral and without considering the strictly convexity of Banach space. In the current article, we introduced a general form of these results that extends these results to general form of semigroup called evolution family. For other such type of recent results, we refer to [2426].
A family is called a nonexpansive strongly continuous evolution family on if it satisfies the following conditions:(i)For every is a nonexpansive mapping on .(ii).(iii).(iv)For every , the mapping from into is strongly continuous. Such families arise from the solution of nonautonomous differential system: . It is obvious that evolution family is a generalized form of a semigroup.The set represents all common fixed points of and is denoted by . As is compact, then is nonempty (for details, see [8, 2730]).

#### 2. Preliminaries

Throughout this paper, , , , and are used for natural numbers, integers, rational numbers, and real numbers, respectively. For any real numbers , we represent as the greatest integer not greater than . Obviously, for any positive real numbers and ,

It is true for , and we substitute

The following lemma is very necessary for our main work.

Lemma 1. Let be a real sequence and satisfy . Furthermore, let the following be true:(1).(2).Then, let be limit point of sequence . Furthermore, considering that and are any positive integers, there exist for each in as well .
Before the proof of this lemma, we have the following remark.

Remark 1. Statement (1) is comparable to since there is no element in with in the case of . It is the same for statement (2).
Now we will prove our lemma.

Proof. Let the sequence be convergent; then, the conclusion is clearly true. So, on the contrary basis, let the sequence be not convergent, i.e., , and we fix and . Also, we fix with and . Then, there exists in with andBy the hypothesis, for any natural numbers such that and for each , we also consider such that .
PutSince , we get for ,As , then using , we get .
Therefore, for .
So, we havefor .
Now we will discuss Krasnoselskii and Mann’s sequences. Let and be two sequences in a Banach space which satisfiesFor every natural number, a sequence in . We say such a sequences and Krasnoselskii and Mann’s sequence, for more detail, see [31, 32].

Lemma 2 (see [21]). Let and be two bounded sequences in a Banach space and let be a sequence in with the condition that . Letfor every andThen,Using Lemma 2, we have the following lemma for evolution equations.

Lemma 3. Let be a Banach space and be a sequence of nonexpansive mappings on . Defining abe a sequence in byFor every natural numbersuch that along.
Letis bounded or equivalent to is bounded, and that either,Holds. Then.

Proof. We put for , and. Then for a natural numberwhere. For , we haveHence, is bounded. In the case , we haveSimilarly, we can prove that , and we haveUsing Lemma 2, we haveIn both cases, it is proved.

Remark 2. (see [21]). We will prove the boundedness of the sequence and are equivalent. Also, we have shown that the sequence is bounded when the sequence is bounded.
Conversely, if is bounded, put . We have to prove for every natural number . We suppose that ; then, we getSo, by induction, we have and hence is a bounded sequence.

Lemma 4. Let be a sequence of nonexpansive mappings on . Defining a sequence in by ,For every natural number and along,
Let be bounded or be bounded and . Then,

#### 3. Main Result

In this section, we will present our main results.

Theorem 4. Let be a Banach space and be a compact convex subset . Let be a one-parameter strongly continuous evolution family of nonexpansive mapping on . Let be a sequence satisfying the following conditions: and . Introduce a sequence in by with .for every natural number . Then, converges strongly to a common fixed point of .

Proof. First of all, we have to prove thatOn the contrary, let (23) be not true; then, there exist a subsequence of and a sequence in and such thatfor every natural number . Since is compact, there exists a subsequence of which is strongly converged. Without loss of generality, we can assume that is self-strongly converged to some of the points of .
We getwhich is a contradiction. Therefore, (23) is true. So, by using Lemma 4, we haveWe haveWe fix for withWe consider which is a subsequence of with the condition for every natural . From equation (24), there exists such that for every .
Using the result of Lemma 1, we have that is a limit point of sequence . Hence, there exists greater than such that . Again from (24), there exists such that for each . Again using Lemma 1, we note that is a limit point of . Therefore, there exists such that .
To continue this process, we defined a subsequence of which satisfies and for all .
Then, for each and . Also, we haveSince is compact, there exists a subsequence of such that strongly converges to .
Put for and for every natural number . We haveHence, the sequence converges to . This implies that .
Also, we getFix ; then, for every natural number with , we getHence, This implies . This shows that is a common fixed point of . Also, is a limit point of , and we get . But sincefor every natural number , we have to obtain .
The following is another convergence theorem which has been proved in [33]; proof can also be seen in [34].

Theorem 5 (see [33]). Let be a strongly continuous evolution family of nonexpansive mappings on . satisfying which does not belong to . Then,holds.
In [21], the convergence theorem of infinite families for nonexpansive mappings without considering the strictly convexity has been proved by the author. Now we give alternate proof by using arguments to Theorem 4, firstly presenting the corollary related to the main theorem [21].

Theorem 6 (see [33]). Let and be nonexpansive mappings on with . Let be also a sequence satisfying , , and . Introduce a sequence in by andfor all . Then, the sequence strongly converges to a common fixed point of and .

Proof. Let . We haveSince is a nonexpansive mapping on for all , we getBy Lemma 4, for a real number , with , we consider that is a subsequence of which satisfies, and for each . From equation (25), there exists such that for each . Using Lemma 1, we noted that is limit point of . Therefore, there exists such that . Again from (25), such that for each .
Again by Lemma 1, we noted that is a limit point of . Therefore, is greater than such that . By pursuing this method, we describe a subsequence of satisfyingand for each . Obviously, for every andWe haveBut since is compact, there exists a subsequence of such that strongly converges to some .
Let and for each . We get .Hence, the sequence converges to . This implies that . So, we get . We have .for every , and thereforeThis implies that the sequence converges to . So, we get . Hence, is a common fixed point of and . Since is a limit point of sequence , we have .
Sincefor every natural number , we get .
By combining Theorems 5 and 6, we get the following theorem.

Theorem 7. Let be a compact convex subset of a Banach space , and let be a one-parameter strongly continuous evolution family of nonexpansive mappings on . Let , satisfying which does not belong to . Let , and let a sequence that satisfies , , and . Introduce a sequence in by andfor all . Then, the sequence converges strongly to a common fixed point of .

Remark 3. If an evolution family is periodic of every positive real number, then it becomes a semigroup [25]. In such a case, the results given in [13] become special case of our results.

#### 4. An Application

In this section, we are going to present an example which will illustrate the importance of evolution equations as well as our theoretical results.

Example 1. Let be the Hilbert space and let be a semigroup defined by , where . Clearly, it is a strongly continuous and nonexpansive semigroup on , and it is generated by the linear operator given by and the maximal domain of is the set of all such that and are absolutely continuous, , and .
Now, consider the nonautonomous Cauchy problem