# Some Convergence and Stability Results for Two New Kirk Type Hybrid Fixed Point Iterative Algorithms

Authors: Faik Gürsoy; Vatan Karakaya
Journal: Journal of Function Spaces
 (2014)

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## Abstract

We introduce Kirk-multistep-SP and Kirk-S iterative algorithms
and we prove some convergence and stability results for these iterative
algorithms. Since these iterative algorithms are more general than some
other iterative algorithms in the existing literature, our results
generalize and unify some other results in the literature.


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## Body

## 1. Introduction and Preliminaries

Fixed point theory has an important role in the study of nonlinear phenomena. This theory has been applied in a wide range of disciplines in various areas such as science, technology, and economics; see, for example, [1–5]. The importance of this theory has attracted researchers’ interest, and consequently numerous fixed point theorems have been put forward; see, for example, [6–17] and the references included therein. In this highly dynamic area, one of the most celebrated theorems amongst hundreds is Banach fixed point theorem (also known as the contraction mapping theorem or contraction mapping principle) [7]. An important process which is called iteration method arises naturally during proving of this theorem. A fixed point iteration method is given by a general form as follows:
(1)x0∈X,xn+1=f(T,xn),∀n∈ℕ,
where X is an ambient space, x0 is an arbitrary initial point, T:X→X is an operator, and f is some function. For example, if f(T,xn)=Txn in (1), then we obtain well-known Picard iteration [18] as follows:
(2)x0∈X,xn+1=f(T,xn)=Txn,∀n∈ℕ.Iterative methods are important instruments commonly used in the study of fixed point theory. These powerful and useful tools enable us to find solutions for a wide variety of problems that arise in many branches of the above mentioned areas. This is a reason, among a number of reasons, why researchers are seeking new iteration methods or trying to improve existing methods over the years. In this respect, it is not surprising to see a number of papers dealing with the study of iterative methods to investigate various important themes; see, for example, [19–27].The purpose of this paper is to introduce two new Kirk type hybrid iteration methods and to show that these iterative methods can be used to approximate fixed points of certain class of contractive operators. Furthermore, we prove that these iterative methods are stable with respect to the same class of contractive operators.As a background to our exposition, we describe some iteration schemes and contractive type mappings.The following multistep-SP iteration was employed in [20, 28]:
(3)x0∈X,xn+1=(1-αn)yn1+αnTyn1,yni=(1-βni)yni+1+βniTyni+1,ynk-1=(1-βnk-1)xn+βnk-1Txn,n∈ℕ,
where ℕ denotes the set of all nonnegative integers, including zero, and {αn}n=0∞, {βn}n=0∞, {γn}n=0∞, and {βni}n=0∞, i=1,k-2¯, k≥2, are real sequences in [0,1) satisfying certain conditions.By takingk=3 and k=2 in (3) we obtain SP [25] and two-step Mann [27] iterative schemes, respectively. In (3), if we take k=2 with βn1=0 and k=2 with βn1≡0, αn≡λ (const.), then we get the iterative procedures introduced in [23, 29], which are commonly known as the Mann and Krasnoselskij iterations, respectively. The Krasnoselskij iteration [29] reduces to the Picard iteration [18] for λ=1.A sequence{xn}n=0∞ defined by
(4)x0∈X,xn+1=(1-αn)Txn+αnTyn,yn=(1-βn)xn+βnTxn,n∈ℕ,
is known as the S iteration process [6, 19].Continuing the above trend, we will introduce and employ the following iterative schemes which are called Kirk-multistep-SP and Kirk-S iterations, respectively:(5)x0∈X,xn+1=∑i1=0s1‍αn,i1Ti1yn1,ynp=∑ip+1=0sp+1‍βn,ip+1pTip+1ynp+1,p=1,k-2¯,ynk-1=∑ik=0sk‍βn,ikk-1Tikxn,k≥2,∀n∈ℕ,(6)x0∈X,xn+1=αn,0Txn+∑i1=1s1‍αn,i1Ti1yn,yn=∑i2=0s2‍βn,i2Ti2xn,∀n∈ℕ,
where ∑i1=0s1αn,i1=1, ∑ip+1=0sp+1βn,ip+1p=1 for p=1,k-1¯; αn,i1, βn,ip+1p are sequences in [0,1] satisfying αn,i1≥0, αn,0≠0, βn,ip+1p≥0, and βn,0p≠0 for p=1,k-1¯; and s1, sp+1 for p=1,k-1¯ are fixed integers with s1≥s2≥⋯≥sk.By takingk=3, k=2, and k=2 with s2=0 in (5) we obtain the Kirk-SP [30], a Kirk-two-step-Mann, and the Kirk-Mann [31] iterative schemes, respectively. Also, (5) gives the usual Kirk iterative process [32] for k=2, with s2=0 and αn,i1=αi1. If we put s1=1 and sp+1=1, p=1,k-1¯ in (5) and (6), then we have the usual multistep-SP iteration (3) and S iteration (4), respectively, with ∑i1=01‍αn,i1=1, αn,1=αn, ∑ip+1=01‍βn,ip+1p=1, βn,1p=βnp, p=1,k-1¯. The SP iteration [25], the two-step Mann iteration [27], the Mann iteration [23], the Krasnoselskij iteration [29], and the Picard iteration [18] schemes are special cases of the multistep-SP iterative scheme (3), as explained above. So we conclude that these are also special cases of the Kirk-multistep-SP iterative scheme (5).We end this section with some definitions and lemmas which will be useful in proving our main results.Definition 1 (see [33]).
LetX be a normed space. A mapping T:X→X is called contractive-like mapping if there exists λ∈[0,1) such that
(7)∥Tx-Ty∥≤φ(∥x-Tx∥)+λ∥x-y∥,∀x,y∈X,
where φ:ℝ+→ℝ+ is a monotone increasing function with φ(0)=0.Remark 2.
By takingφ(t)=Lt in (7), one can get contractive definition due to Osilike and Udomene [34]. Also, by putting φ(t)=2λt in (7), condition (7) reduces to the contractive definition in [35]. In [35] it was shown that the class of these operators is wider than class of Zamfirescu operators given in [17], where λ:=max⁡{a,b/(1-b),c/(1-c)}, λ∈[0,1) and a, b, and c are real numbers satisfying 0<a<1, 0<b, and c<1/2.Remark 3 (see [20, 28]).
A map satisfying (7) need not have a fixed point. However, using (7), it is obvious that if T has a fixed point, then it is unique.Definition 4 (see [36, 37]).
LetX be a normed space, T:X→X a mapping, and {xn}n=0∞⊂X an iterative sequence generated by the iterative process (1) with limit point q∈FT:={q∈X:q=Tq}. Let {yn}n=0∞ be an arbitrary sequence in X and set
(8)ɛn=∥yn+1-f(T,yn)∥forn=0,1,2,….
We will say that the iterative sequence {xn}n=0∞ is T-stable or stable with respect to T if and only if
(9)lim⁡n→∞ɛn=0⇔lim⁡n→∞yn=q.Lemma 5 (see [8]).
Ifσ is a real number such that σ∈[0,1) and {ɛn}n=0∞ is a sequence of nonnegative numbers such that lim⁡n→∞ɛn=0, then, for any sequence of positive numbers {un}n=0∞ satisfying
(10)un+1≤σun+ɛn,∀n∈ℕ,
one has lim⁡n→∞un=0.Lemma 6 (see [31]).
Let(X,∥·∥) be a normed linear space and let T be a self-map of X satisfying (7). Let φ:ℝ+→ℝ+ be a subadditive, monotone increasing function such that φ(0)=0, φ(Lu)≤Lφ(u), L≥0, u∈ℝ+. Then, for all i∈ℕ, L≥0 and for all x,y∈X(11)∥Tix-Tiy∥≤∑j=1i‍(ij)ai-jφj(∥x-Tx∥)+ai∥x-y∥,
where a∈[0,1).


## 2. Main Results

For simplicity we assume in the following four theorems thatX is a normed linear space, T is a self map of X satisfying the contractive condition (7) with some fixed point q∈FT, and φ:ℝ+→ℝ+ is a subadditive monotone increasing function such that φ(0)=0 and φ(Lu)≤Lφ(u), L≥0, u∈ℝ+.Theorem 7.
Let{xn}n∈ℕ be a sequence generated by the Kirk-multistep-SP iterative scheme (5). Then the iterative sequence {xn}n∈ℕ converges strongly to q.Proof.
The uniqueness ofq follows from (7). We will now prove that xn→q.
Using Kirk-multistep-SP iterative process (5), condition (7), and Lemma 6, we get
(12)∥xn+1-q∥=∥∑i1=0s1αn,i1Ti1yn1-q‍∥=∥αn,0(yn1-q)+∑i1=1s1αn,i1(Ti1yn1-Ti1q)‍∥≤αn,0∥yn1-q∥+∑i1=1s1αn,i1∥Ti1yn1-Ti1q∥‍≤αn,0∥yn1-q∥+∑i1=1s1αn,i1ddddd.dd×{∑j=1i1(i1j)ai1-jφj(∥q-Tq∥)+ai1∥yn1-q∥}=(∑i1=0s1αn,i1ai1)∥yn1-q∥,(13)∥yn1-q∥=∥∑i2=0s2βn,i21Ti2yn2-q∥=∥∑i2=0s2βn,i21(Ti2yn2-Ti2q)∥≤βn,01∥yn2-q∥+∑i2=1s2βn,i21∥Ti2yn2-Ti2q∥≤βn,01∥yn2-q∥+∑i2=1s2βn,i21ddddddd×{∑j=1i2(i2j)ai2-jφj(∥q-Tq∥)+ai2∥yn2-q∥}=(∑i2=0s2βn,i21ai2)∥yn2-q∥,(14)∥yn2-q∥≤(∑i3=0s3‍βn,i32ai3)∥yn3-q∥.
By combining (12), (13), and (14) we obtain
(15)∥xn+1-q∥≤(∑i1=0s1αn,i1ai1)(∑i2=0s2βn,i21ai2)(∑i3=0s3βn,i32ai3)∥yn3-q∥.
Continuing the above process we have
(16)∥xn+1-q∥≤(∑i1=0s1‍αn,i1ai1)(∑i2=0s2‍βn,i21ai2)⋯(∑ik-1=0sk-1‍βn,ik-1k-2aik-1)∥ynk-1-q∥.
Using again Kirk-multistep-SP iterative process (5), condition (7), and Lemma 6, we have
(17)∥ynk-1-q∥≤(∑ik=0sk‍βn,ikk-1aik)∥xn-q∥.
Substituting (17) into (16) we derive
(18)∥xn+1-q∥≤(∑i1=0s1‍αn,i1ai1)(∑i2=0s2‍βn,i21ai2)⋯(∑ik=0sk‍βn,ikk-1aik)∥xn-q∥.
Since aik∈[0,1) and ∑i1=0s1αn,i1=1, ∑ip+1=0sp+1βn,ip+1p=1 for p=1,k-1¯, then
(19)(∑i1=0s1‍αn,i1ai1)(∑i2=0s2‍βn,i21ai2)⋯(∑ik=0sk‍βn,ikk-1aik)<(∑i1=0s1‍αn,i1)(∑i2=0s2‍βn,i21)⋯(∑ik=0sk‍βn,ikk-1)=1.
Hence, by an application of Lemma 5 to the inequality (18), we get lim⁡n→∞xn=q.Theorem 8.
Let{xn}n∈ℕ be a sequence generated by the Kirk-multistep-SP iterative scheme (5). Then, the iterative sequence {xn}n∈ℕ is T-stable.Proof.
Let{yn}n∈ℕ⊂X, {unp}n∈ℕ, for p=1,k-1¯, be arbitrary sequences in X. Let ɛn=∥yn+1-∑i1=0s1αn,i1Ti1un1∥, n=0,1,2,…, where unp=∑ip+1=0sp+1βn,ip+1pTip+1unp+1, p=1,k-2¯, unk-1=∑ik=0skβn,ikk-1Tikyn, k≥2, and let lim⁡n→∞ɛn=0. Now we will prove that lim⁡n→∞yn=q.
It follows from (5) and Lemma 6 that
(20)∥yn+1-q∥=∥yn+1-∑i1=0s1αn,i1Ti1un1‍+∑i1=0s1αn,i1Ti1un1-q∥≤∥yn+1-∑i1=0s1αn,i1Ti1un1∥+∥∑i1=0s1αn,i1Ti1un1-q∥=ɛn+∥∑i1=0s1αn,i1(Ti1un1-Ti1q)∥=ɛn+∥αn,0(un1-q)+∑i1=1s1αn,i1(Ti1un1-Ti1q)∥≤ɛn+αn,0∥un1-q∥+∑i1=1s1αn,i1∥Ti1un1-Ti1q∥≤ɛn+αn,0∥un1-q∥+∑i1=1s1αn,i1ddddddd×{∑j=1i1(i1j)ai1-jφj(∥q-Tq∥)+ai1∥un1-q∥}‍=ɛn+(∑i1=0s1αn,i1ai1)∥un1-q∥,(21)∥un1-q∥=∥∑i2=0s2βn,i21Ti2un2-q∥=∥∑i2=0s2βn,i21(Ti2un2-Ti2q)∥≤βn,01∥un2-q∥+∑i2=1s2βn,i21∥Ti2un2-Ti2q∥‍≤βn,01∥un2-q∥+∑i2=1s2βn,i21ddddddd×{∑j=1i2(i2j)ai2-jφj(∥q-Tq∥)+ai2∥un2-q∥}=(∑i2=0s2βn,i21ai2)∥un2-q∥,(22)∥un2-q∥≤(∑i3=0s3‍βn,i32ai3)∥un3-q∥.
Combining (20), (21), and (22) we get
(23)∥yn+1-q∥≤ɛn+(∑i1=0s1‍αn,i1ai1)(∑i2=0s2‍βn,i21ai2)×(∑i3=0s3‍βn,i32ai3)∥un3-q∥.
By induction
(24)∥yn+1-q∥≤ɛn+(∑i1=0s1‍αn,i1ai1)(∑i2=0s2‍βn,i21ai2)⋯(∑ik-1=0sk-1‍βn,ik-1k-2aik-1)∥unk-1-q∥.
Again using (5) and Lemma 6 we have
(25)∥unk-1-q∥≤(∑ik=0sk‍βn,ikk-1aik)∥yn-q∥.
Substituting (25) into (24) we derive
(26)∥yn+1-q∥≤ɛn+(∑i1=0s1‍αn,i1ai1)(∑i2=0s2‍βn,i21ai2)⋯(∑ik=0sk‍βn,ikk-1aik)∥yn-q∥.
Define
(27)σ:=(∑i1=0s1‍αn,i1ai1)(∑i2=0s2‍βn,i21ai2)⋯(∑ik=0sk‍βn,ikk-1aik).
We now show that σ∈(0,1). Since aik∈[0,1), αn,0>0, ∑i1=0s1αn,i1=1, and ∑ip+1=0sp+1βn,ip+1p=1 for p=1,k-1¯, we have
(28)σ<(∑i1=0s1‍αn,i1)(∑i2=0s2‍βn,i21)⋯(∑ik=0sk‍βn,ikk-1)=1.
Therefore, an application of Lemma 5 to (26) yields lim⁡n→∞yn=q.
Now suppose thatlim⁡n→∞yn=q. Then, we will show that lim⁡n→∞ɛn=0.
Using Lemma6 we have
(29)ɛn=∥yn+1-∑i1=0s1‍αn,i1Ti1un1∥≤∥yn+1-q∥+∥q-∑i1=0s1‍αn,i1Ti1un1∥=∥yn+1-q∥+∥αn,0(q-un1)+∑i1=1s1‍αn,i1(Ti1q-Ti1un1)∥≤∥yn+1-q∥+αn,0∥un1-q∥+∑i1=1s1‍αn,i1∥Ti1q-Ti1un1∥≤∥yn+1-q∥+αn,0∥q-un1∥+∑i1=1s1‍αn,i1ddd×{∑j=1i1‍(i1j)ai1-jφj(∥q-Tq∥)+ai1∥q-un1∥}≤∥yn+1-q∥+(∑i1=0s1‍αn,i1ai1)∥q-un1∥,(30)∥q-un1∥=∥q-∑i2=0s2‍βn,i21Ti2un2∥=∥βn,01(q-un2)+∑i2=1s2‍βn,i21(Ti2q-Ti2un2)∥≤βn,01∥q-un2∥+∑i2=1s2‍βn,i21∥Ti2q-Ti2un2∥≤βn,01∥q-un2∥+∑i2=1s2‍βn,i21ddddddd×{∑j=1i2‍(i2j)ai2-jφj(∥q-Tq∥)+ai2∥q-un2∥}≤(∑i2=0s2‍βn,i21ai2)∥q-un2∥,(31)∥q-un2∥≤(∑i3=0s3‍βn,i32ai3)∥q-un3∥.
Combining (29), (30), and (31) we obtain
(32)ɛn≤∥yn+1-q∥+(∑i1=0s1‍αn,i1ai1)(∑i2=0s2‍βn,i21ai2)×(∑i3=0s3‍βn,i32ai3)∥q-un3∥.
Thus, by induction, we get
(33)ɛn≤∥yn+1-q∥+(∑i1=0s1‍αn,i1ai1)(∑i2=0s2‍βn,i21ai2)⋯(∑ik-1=0sk-1‍βn,ik-1k-2aik-1)∥q-unk-1∥.
Again using (5) and Lemma 6 we have
(34)∥q-unk-1∥≤(∑ik=0sk‍βn,ikk-1aik)∥yn-q∥.
Substituting (34) into (33) we derive
(35)ɛn≤∥yn+1-q∥+(∑i1=0s1‍αn,i1ai1)(∑i2=0s2‍βn,i21ai2)⋯(∑ik=0sk‍βn,ikk-1aik)∥yn-q∥.
Again define
(36)σ:=(∑i1=0s1‍αn,i1ai1)(∑i2=0s2‍βn,i21ai2)⋯(∑ik=0sk‍βn,ikk-1aik).
Using the same argument as that of the first part of the proof we obtain σ∈(0,1).
Hence (35) becomes
(37)ɛn≤∥yn+1-q∥+σ∥yn-q∥.
It therefore follows from assumption lim⁡n→∞yn=q that ɛn→0 as n→∞.Theorem 9.
Let{xn}n∈ℕ be a sequence generated by the Kirk-S iterative scheme (6). Then, the iterative sequence {xn}n∈ℕ converges strongly to q.Proof.
The uniqueness ofq follows from (7). We will now prove that xn→q.
Using Kirk-S iterative process (6), condition (7), and Lemma 6, we get
(38)∥xn+1-q∥=∥αn,0(Txn-Tq)+∑i1=1s1‍αn,i1(Ti1yn-Ti1q)∥≤αn,0∥Txn-Tq∥+∑i1=1s1‍αn,i1∥Ti1yn-Ti1q∥≤aαn,0∥xn-q∥+∑i1=1s1‍αn,i1{∑j=1i1‍(i1j)ai1-jφj(∥q-Tq∥)+ai1∥yn-q∥}=aαn,0∥xn-q∥+(∑i1=1s1‍αn,i1ai1)∥yn-q∥,(39)∥yn-q∥=∥∑i2=0s2‍βn,i2(Ti2xn-Ti2q)∥≤βn,0∥xn-q∥+∑i2=1s2‍βn,i2∥Ti2xn-Ti2q∥≤βn,0∥xn-q∥+∑i2=1s2‍βn,i2{∑j=1i2‍(i2j)ai2-jφj(∥q-Tq∥)+ai2∥xn-q∥}=(∑i2=0s2‍βn,i2ai2)∥xn-q∥.
Substituting (39) into (38) we obtain
(40)∥xn+1-q∥≤[aαn,0+(∑i1=1s1‍αn,i1ai1)(∑i2=0s2‍βn,i2ai2)]∥xn-q∥.
Since aik∈[0,1) and ∑i1=0s1‍αn,i1=∑i2=0s2‍βn,i2=1 with αn,0≠0, βn,0≠0,
(41)aαn,0+(∑i1=1s1‍αn,i1ai1)(∑i2=0s2‍βn,i2ai2)<αn,0+(∑i1=1s1‍αn,i1)(∑i2=0s2‍βn,i2)=∑i1=0s1‍αn,i1=1.
Utilizing (41) and Lemma 5, (40) yields lim⁡n→∞xn=q.Theorem 10.
Let{xn}n∈ℕ be a sequence generated by the Kirk-S iterative scheme (6). Then, the iterative sequence {xn}n∈ℕ is T-stable.Proof.
Let{yn}n∈ℕ⊂X, ɛn=∥yn+1-αn,0Tyn-∑i1=1s1αn,i1Ti1un∥, n=0,1,2,…, and un=∑i2=0s2βn,i2Ti2yn. Assume that lim⁡n→∞ɛn=0. Now we will prove that lim⁡n→∞yn=q.
It follows from (6) and Lemma 6 that
(42)∥yn+1-q∥≤∥yn+1-αn,0Tyn-∑i1=1s1‍αn,i1Ti1un∥+∥αn,0Tyn+∑i1=1s1‍αn,i1Ti1un-q∥=ɛn+∥αn,0(Tyn-Tq)+∑i1=1s1‍αn,i1(Ti1un-Ti1q)∥≤ɛn+aαn,0∥yn-q∥+∑i1=1s1‍αn,i1∥Ti1un-Ti1q∥≤ɛn+aαn,0∥yn-q∥+∑i1=1s1αn,i1{∑j=1i1‍(i1j)ai1-jφj(∥q-Tq∥)+ai1∥un-q∥}=ɛn+aαn,0∥yn-q∥+(∑i1=1s1‍αn,i1ai1)∥un-q∥,(43)∥un-q∥≤(∑i2=0s2‍βn,i2ai2)∥yn-q∥.
Combining (42) and (43) we have
(44)∥yn+1-q∥≤ɛn+[aαn,0+(∑i1=1s1‍αn,i1ai1)(∑i2=0s2‍βn,i2ai2)]×∥yn-q∥.
Define
(45)σ:=aαn,0+(∑i1=1s1‍αn,i1ai1)(∑i2=0s2‍βn,i2ai2).
We now show that σ∈(0,1). Since aik∈[0,1), αn,0>0, ∑i1=0s1αn,i1=1, and ∑i2=0s2βn,i2ai2, we obtain
(46)σ<αn,0+(∑i1=1s1‍αn,i1)(∑i2=0s2‍βn,i2)=∑i1=0s1‍αn,i1=1.
Thus, (44) becomes
(47)∥yn+1-q∥≤η∥yn-q∥+ɛn.
Therefore, an application of Lemma 5 to (47) leads to lim⁡n→∞yn=q.
Now suppose thatlim⁡n→∞yn=q. Then, we will show that lim⁡n→∞ɛn=0.
Using Lemma6 we have
(48)ɛn=∥yn+1-αn,0Tyn-∑i1=1s1‍αn,i1Ti1un∥≤∥yn+1-q∥+∥q-αn,0Tyn-∑i1=1s1‍αn,i1Ti1un∥=∥yn+1-q∥+∥αn,0(Tq-Tyn)+∑i1=1s1‍αn,i1(Ti1q-Ti1un)∥≤∥yn+1-q∥+aαn,0∥yn-q∥+∑i1=1s1‍αn,i1∥Ti1q-Ti1un∥≤∥yn+1-q∥+aαn,0∥yn-q∥+∑i1=1s1‍αn,i1{∑j=1i1‍(i1j)ai1-jφj(∥q-Tq∥)+ai1∥q-un∥}≤∥yn+1-q∥+aαn,0∥yn-q∥+(∑i1=1s1‍αn,i1ai1)∥q-un∥,(49)∥q-un∥≤(∑i2=0s2‍βn,i2ai2)∥yn-q∥.
Substituting (49) into (48) we get
(50)ɛn≤∥yn+1-q∥+[aαn,0+(∑i1=1s1‍αn,i1ai1)(∑i2=0s2‍βn,i2ai2)]×∥yn-q∥.
Again define
(51)σ:=aαn,0+(∑i1=1s1‍αn,i1ai1)(∑i2=0s2‍βn,i2ai2).
Using the same argument as that of the first part of the proof we obtain σ∈(0,1).
Hence (50) becomes
(52)ɛn≤∥yn+1-q∥+η∥yn-q∥.
It therefore follows from assumption lim⁡n→∞yn=q that ɛn→0 as n→∞.Remark 11.
Theorem7 is a generalization and extension of Theorem 2.1 of [38], Theorem 2.1 of [39], Theorem 1 of [20], and Theorem 2.4 of [30]. Theorems 8 is a generalization and extension of Theorem 3.6 of [38] and Theorem 3 of [40]. Theorem 9 is a generalization and extension of Theorem 8 of [41] and Theorem 3 of [20].



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*Source: 684191-2014-01-28.xml*