JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi Publishing Corporation 684191 10.1155/2014/684191 684191 Research Article Some Convergence and Stability Results for Two New Kirk Type Hybrid Fixed Point Iterative Algorithms Gürsoy Faik 1 Karakaya Vatan 2 Mursaleen M. 1 Department of Mathematics Yıldız Technical University Davutpasa Campus Esenler 34220 Istanbul Turkey isbank.com.tr 2 Department of Mathematical Engineering Yıldız Technical University Davutpasa Campus Esenler 34210 Istanbul Turkey isbank.com.tr 2014 2812014 2014 13 09 2013 13 11 2013 28 1 2014 2014 Copyright © 2014 Faik Gürsoy and Vatan Karakaya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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, . The importance of this theory has attracted researchers’ interest, and consequently numerous fixed point theorems have been put forward; see, for example,  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) . 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)x0X,xn+1=f(T,xn),n, where X is an ambient space, x0 is an arbitrary initial point, T:XX is an operator, and f is some function. For example, if f(T,xn)=Txn in (1), then we obtain well-known Picard iteration  as follows: (2)x0X,xn+1=f(T,xn)=Txn,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, .

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)x0X,xn+1=(1-αn)yn1+αnTyn1,yni=(1-βni)yni+1+βniTyni+1,ynk-1=(1-βnk-1)xn+βnk-1Txn,n, where 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¯, k2, are real sequences in [0,1) satisfying certain conditions.

By taking k=3 and k=2 in (3) we obtain SP  and two-step Mann  iterative schemes, respectively. In (3), if we take k=2 with βn1=0 and k=2 with βn10, αnλ (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  reduces to the Picard iteration  for λ=1.

A sequence {xn}n=0 defined by (4)x0X,xn+1=(1-αn)Txn+αnTyn,yn=(1-βn)xn+βnTxn,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)x0X,xn+1=i1=0s1αn,i1Ti1yn1,ynp=ip+1=0sp+1βn,ip+1pTip+1ynp+1,  p=1,k-2¯,ynk-1=ik=0skβn,ikk-1Tikxn,  k2,n,(6)x0X,xn+1=αn,0Txn+i1=1s1αn,i1Ti1yn,yn=i2=0s2βn,i2Ti2xn,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,i10, αn,00, βn,ip+1p0, and βn,0p0 for p=1,k-1¯; and s1, sp+1 for p=1,k-1¯ are fixed integers with s1s2sk.

By taking k=3, k=2, and k=2 with s2=0 in (5) we obtain the Kirk-SP , a Kirk-two-step-Mann, and the Kirk-Mann  iterative schemes, respectively. Also, (5) gives the usual Kirk iterative process  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 , the two-step Mann iteration , the Mann iteration , the Krasnoselskij iteration , and the Picard iteration  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 [<xref ref-type="bibr" rid="B19">33</xref>]).

Let X be a normed space. A mapping T:XX is called contractive-like mapping if there exists λ[0,1) such that (7)Tx-Tyφ(x-Tx)+λx-y,x,yX, where φ:++ is a monotone increasing function with φ(0)=0.

Remark 2.

By taking φ(t)=Lt in (7), one can get contractive definition due to Osilike and Udomene . Also, by putting φ(t)=2λt in (7), condition (7) reduces to the contractive definition in . In  it was shown that the class of these operators is wider than class of Zamfirescu operators given in , 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 [<xref ref-type="bibr" rid="B13">20</xref>, <xref ref-type="bibr" rid="B14">28</xref>]).

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 [<xref ref-type="bibr" rid="B15">36</xref>, <xref ref-type="bibr" rid="B16">37</xref>]).

Let X be a normed space, T:XX a mapping, and {xn}n=0X an iterative sequence generated by the iterative process (1) with limit point qFT:={qX:q=Tq}. Let {yn}n=0 be an arbitrary sequence in X and set (8)ɛn=yn+1-f(T,yn)for  n=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)limnɛn=0limnyn=q.

Lemma 5 (see [<xref ref-type="bibr" rid="B5">8</xref>]).

If σ is a real number such that σ[0,1) and {ɛn}n=0 is a sequence of nonnegative numbers such that limnɛn=0, then, for any sequence of positive numbers {un}n=0 satisfying (10)un+1σun+ɛn,n, one has limnun=0.

Lemma 6 (see [<xref ref-type="bibr" rid="B31">31</xref>]).

Let (X,·) be a normed linear space and let T be a self-map of X satisfying (7). Let φ:++ be a subadditive, monotone increasing function such that φ(0)=0, φ(Lu)Lφ(u), L0, u+. Then, for all i, L0 and for all x,yX(11)Tix-Tiyj=1i(ij)ai-jφj(x-Tx)+aix-y, where a[0,1).

2. Main Results

For simplicity we assume in the following four theorems that X is a normed linear space, T is a self map of X satisfying the contractive condition (7) with some fixed point qFT, and φ:++ is a subadditive monotone increasing function such that φ(0)=0 and φ(Lu)Lφ(u), L0, u+.

Theorem 7.

Let {xn}n be a sequence generated by the Kirk-multistep-SP iterative scheme (5). Then the iterative sequence {xn}n converges strongly to q.

Proof.

The uniqueness of q follows from (7). We will now prove that xnq.

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,0yn1-q+i1=1s1αn,i1Ti1yn1-Ti1qαn,0yn1-q+i1=1s1αn,i1ddddd.dd×{j=1i1(i1j)ai1-jφj(q-Tq)+ai1yn1-q}=(i1=0s1αn,i1ai1)yn1-q,(13)yn1-q=i2=0s2βn,i21Ti2yn2-q=i2=0s2βn,i21(Ti2yn2-Ti2q)βn,01yn2-q+i2=1s2βn,i21Ti2yn2-Ti2qβn,01yn2-q+i2=1s2βn,i21ddddddd×{j=1i2(i2j)ai2-jφj(q-Tq)+ai2yn2-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 limnxn=q.

Theorem 8.

Let {xn}n be a sequence generated by the Kirk-multistep-SP iterative scheme (5). Then, the iterative sequence {xn}n is T-stable.

Proof.

Let {yn}nX, {unp}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, k2, and let limnɛn=0. Now we will prove that limnyn=q.

It follows from (5) and Lemma 6 that (20)yn+1-q=yn+1-i1=0s1αn,i1Ti1un1+i1=0s1αn,i1Ti1un1-qyn+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,0un1-q+i1=1s1αn,i1Ti1un1-Ti1qɛn+αn,0un1-q+i1=1s1αn,i1ddddddd×{j=1i1(i1j)ai1-jφj(q-Tq)+ai1un1-q}  =ɛn+(i1=0s1αn,i1ai1)un1-q,(21)un1-q=i2=0s2βn,i21Ti2un2-q=i2=0s2βn,i21(Ti2un2-Ti2q)βn,01un2-q+i2=1s2βn,i21Ti2un2-Ti2q  βn,01un2-q+i2=1s2βn,i21ddddddd×{j=1i2(i2j)ai2-jφj(q-Tq)+ai2un2-q}=(i2=0s2βn,i21ai2)un2-q,(22)un2-q(i3=0s3βn,i32ai3)un3-q. Combining (20), (21), and (22) we get (23)yn+1-qɛn+(i1=0s1αn,i1ai1)(i2=0s2βn,i21ai2)×(i3=0s3βn,i32ai3)un3-q. By induction (24)yn+1-qɛn+(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-qɛn+(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 limnyn=q.

Now suppose that limnyn=q. Then, we will show that limnɛn=0.

Using Lemma 6 we have (29)ɛn=yn+1-i1=0s1αn,i1Ti1un1yn+1-q+q-i1=0s1αn,i1Ti1un1=yn+1-q+αn,0(q-un1)+i1=1s1αn,i1(Ti1q-Ti1un1)yn+1-q+αn,0un1-q+i1=1s1αn,i1Ti1q-Ti1un1yn+1-q+αn,0q-un1+i1=1s1αn,i1ddd×{j=1i1(i1j)ai1-jφj(q-Tq)+ai1q-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,01q-un2+i2=1s2βn,i21Ti2q-Ti2un2βn,01q-un2+i2=1s2βn,i21ddddddd×{j=1i2(i2j)ai2-jφj(q-Tq)+ai2q-un2}(i2=0s2βn,i21ai2)q-un2,(31)q-un2(i3=0s3βn,i32ai3)q-un3. Combining (29), (30), and (31) we obtain (32)ɛnyn+1-q+(i1=0s1αn,i1ai1)(i2=0s2βn,i21ai2)×(i3=0s3βn,i32ai3)q-un3. Thus, by induction, we get (33)ɛnyn+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)ɛnyn+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)ɛnyn+1-q+σyn-q. It therefore follows from assumption limnyn=q that ɛn0 as n.

Theorem 9.

Let {xn}n be a sequence generated by the Kirk-S iterative scheme (6). Then, the iterative sequence {xn}n converges strongly to q.

Proof.

The uniqueness of q follows from (7). We will now prove that xnq.

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,0Txn-Tq+i1=1s1αn,i1Ti1yn-Ti1qaαn,0xn-q+i1=1s1αn,i1{j=1i1(i1j)ai1-jφj(q-Tq)+ai1yn-q}=aαn,0xn-q+(i1=1s1αn,i1ai1)yn-q,(39)yn-q=i2=0s2βn,i2(Ti2xn-Ti2q)βn,0xn-q+i2=1s2βn,i2Ti2xn-Ti2qβn,0xn-q+i2=1s2βn,i2{j=1i2(i2j)ai2-jφj(q-Tq)+ai2xn-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,00, βn,00, (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 limnxn=q.

Theorem 10.

Let {xn}n be a sequence generated by the Kirk-S iterative scheme (6). Then, the iterative sequence {xn}n is T-stable.

Proof.

Let {yn}nX, ɛn=yn+1-αn,0Tyn-i1=1s1αn,i1Ti1un, n=0,1,2,, and un=i2=0s2βn,i2Ti2yn. Assume that limnɛn=0. Now we will prove that limnyn=q.

It follows from (6) and Lemma 6 that (42)yn+1-qyn+1-αn,0Tyn-i1=1s1αn,i1Ti1un+αn,0Tyn+i1=1s1αn,i1Ti1un-q=ɛn+αn,0(Tyn-Tq)+i1=1s1αn,i1(Ti1un-Ti1q)ɛn+aαn,0yn-q+i1=1s1αn,i1Ti1un-Ti1qɛn+aαn,0yn-q+i1=1s1αn,i1{j=1i1(i1j)ai1-jφj(q-Tq)+ai1un-q}=ɛn+aαn,0yn-q+(i1=1s1αn,i1ai1)un-q,(43)un-q(i2=0s2βn,i2ai2)yn-q. Combining (42) and (43) we have (44)yn+1-qɛn+[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-qηyn-q+ɛn. Therefore, an application of Lemma 5 to (47) leads to limnyn=q.

Now suppose that limnyn=q. Then, we will show that limnɛn=0.

Using Lemma 6 we have (48)ɛn=yn+1-αn,0Tyn-i1=1s1αn,i1Ti1unyn+1-q+q-αn,0Tyn-i1=1s1αn,i1Ti1un=yn+1-q+αn,0(Tq-Tyn)+i1=1s1αn,i1(Ti1q-Ti1un)yn+1-q+aαn,0yn-q+i1=1s1αn,i1Ti1q-Ti1unyn+1-q+aαn,0yn-q+i1=1s1αn,i1{j=1i1(i1j)ai1-jφj(q-Tq)+ai1q-un}yn+1-q+aαn,0yn-q+(i1=1s1αn,i1ai1)q-un,(49)q-un(i2=0s2βn,i2ai2)yn-q. Substituting (49) into (48) we get (50)ɛnyn+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)ɛnyn+1-q+ηyn-q. It therefore follows from assumption limnyn=q that ɛn0 as n.

Remark 11.

Theorem 7 is a generalization and extension of Theorem 2.1 of , Theorem 2.1 of , Theorem 1 of , and Theorem 2.4 of . Theorems 8 is a generalization and extension of Theorem 3.6 of  and Theorem 3 of . Theorem 9 is a generalization and extension of Theorem 8 of  and Theorem 3 of .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research has been supported by Yıldız Technical University Scientific Research Projects Coordination Department, Project no. BAPK 2012-07-03-DOP02.

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