- Research
- Open access
- Published:
Convergence comparison and stability of Jungck-Kirk-type algorithms for common fixed point problems
Fixed Point Theory and Applications volume 2013, Article number: 173 (2013)
Abstract
The aim of this article is to introduce new hybrid iterative schemes, namely Jungck-Kirk-SP and Jungck-Kirk-CR iterative schemes, and prove convergence and stability results for these iterative schemes using certain quasi-contractive operators. Numerical examples showing the comparison of convergence rate and applications of newly introduced iterative schemes are also provided. The obtained results improve, generalize and extend the works of Olatinwo (Acta Math. Univ. Comen. LXXVII(2):299-304, 2008; Fasc. Math. 40:37-43, 2008; Mat. Vesn. 61(4):247-256, 2009; Acta Math. Acad. Paedagog. Nyházi. 25(1):105-118, 2009; Acta Univ. Apulensis 26:225-236, 2011), Chugh and Kumar (Int. J. Contemp. Math. Sci. 7(24):1165-1184, 2012; Int. J. Comput. Appl. 36(12):40-46, 2011), Bosede (Bull. Math. Anal. Appl. 2(3):65-73, 2010), Oleleru and Akewe (Fasc. Math. 47:47-61, 2011) and many others in the literature.
MSC:47H06, 54H25.
1 Introduction
In the recent years, fixed and common points of operators have been approximated by using different iterative schemes (see [1–20]). Let X be a Banach space, Y an arbitrary set and such that . For , consider the following iterative scheme:
This scheme is called Jungck iterative scheme and was essentially introduced by Jungck [21] in 1976. It reduces to the Picard iterative scheme when (identity mapping) and .
For , Singh et al. [18] defined the Jungck-Mann iterative scheme as follows:
For , Olatinwo defined the Jungck-Ishikawa [11] and Jungck-Noor [12] iterative schemes as follows:
and
respectively.
Chugh and Kumar [6] defined the Jungck-SP iterative scheme as
where , and are sequences of positive numbers in .
Remark 1.1 If and (identity mapping), then Jungck-SP (1.5), Jungck-Noor (1.4), Jungck-Ishikawa (1.3) and the Jungck-Mann (1.2) iterative schemes, respectively, become the SP [16], Noor [22], Ishikawa [23] and Mann [24] iterative schemes.
In 2009, Olatinwo [13] introduced the Kirk-Mann and Kirk-Ishikawa iterative schemes as follows.
-
(a)
Kirk-Mann iterative scheme:
(1.6)where , , and k is a fixed integer.
-
(b)
Kirk-Ishikawa iterative scheme:
(1.7)where , , , , , and k, s are fixed integers.
Chugh and Kumar [5] introduced the following Jungck-Kirk-Noor iterative scheme:
, , , , , , , , where r, s and t are fixed integers.
Very recently, Hussain et al. [25] defined the Jungck-CR iterative scheme as follows:
where , and are sequences of positive numbers in .
Putting (identity mapping) and in the Junck-CR iterative scheme, we get the Agarwal et al. iterative scheme [1].
Jungck [21] used iterative scheme (1.1) to approximate the common fixed points of the mappings S and T satisfying the following Jungck-contraction:
Singh et al. [17, 18] established some stability results for Jungck and Jungck-Mann iterative schemes for both contractive conditions (1.10) and (1.11): For some and ,
Olatinwo and Imoru [11] studied the generalized Zamfirescu operators for the pair , satisfying the following condition: For each pair of points x, y in Y, at least one of the following is true:
where a, b, c are nonnegative constants satisfying , , .
Any mapping satisfying (1.12)(ii) is called a Kannan mapping, while the mapping satisfying (1.12)(iii) is called a Chatterjea operator.
The contractive condition (1.12) implies
where (see Berinde [3]).
Ranganathan [26] used the following more general contractive condition than (1.10) and (1.12) to prove some fixed point theorems:
and some .
Note that (1.13) and (1.14) are independent of each other but more general than (1.10).
Hussain et al. [25] and Olatinwo [14], respectively, used the following more general contractive conditions than (1.13) to prove the stability and strong convergence results for various iterative schemes: There exists and a monotone increasing function with such that
Recently, Bosede [4] used the following more general contractive condition than (1.13) to prove the convergence results for the Jungck-Ishikawa iteration process:
Let f and g be two selfmaps on X. A point x in X is called (1) a fixed point of f if ; (2) a coincidence point of a pair if ; (3) a common fixed point of a pair if . If for some x in X, then w is called a point of coincidence of f and g. A pair is said to be weakly compatible if f and g commute at their coincidence points.
The stability theory has extensively been studied by various authors [7, 14, 15, 17, 18, 29, 30] due to its increasing importance in computational mathematics, especially due to revolution in computer programming.
We use the following definition and lemmas to prove our results.
Definition 1.3 [18]
Let be operators such that and , a point of coincidence of S and T. Let be the sequence generated by an iterative procedure
where is the initial approximation and f is some function. Suppose converges to p. Let be an arbitrary sequence and set , . Then the iterative procedure (1.18) is said to be -stable or stable if and only if implies .
Let and be two iteration procedures that converge to the same fixed point p on a normed space X such that the error estimates
and
are available, where and are two sequences of positive numbers (converging to zero). If converges faster than , then we say that converges faster to p than .
Suppose that and are two real convergent sequences with limits a and b, respectively. Then is said to converge faster than if
Definition 1.6 [3]
Any function is called a comparison function if it satisfies the following properties:
-
(1)
ψ is monotonic increasing;
-
(2)
, .
Note that a comparison function always satisfies (i) , , (ii) .
Lemma 1.7 [15]
Let be a subadditive, comparison function and let be a sequence of positive numbers such that , then for any sequence of positive numbers satisfying
where with , we have .
Now, we define the Jungck-Kirk-SP and Jungck-Kirk-CR iterative schemes as follows:
Let , , , , , , , , with where r, s and t are fixed integers. Then we define the Jungck-Kirk-SP iterative scheme as follows:
and the Jungck-Kirk-CR iterative scheme as follows:
Remark 1.8 Putting in Jungck-Kirk-type iterative schemes (i.e., JKCR, JKSP, JKN, JKI, JKM), we get the corresponding Jungck-type iterative schemes (i.e., JCR, JKP, JN, JI, JM).
Also, we shall use the following contractive condition: Let be a comparison function such that
where p is a point of coincidence of S, T, i.e., and , denote the i th iterate of T and ψ, respectively.
The following example shows that (1.21) is more general than Jungck contraction (1.10).
Example 1.9 Let . Define T and S by
It is clear that T and S satisfy (1.21) but not Jungck-contraction (1.10).
If are Kannan operators, then from (1.12)(ii) with , (a coincidence point of S, T), we get
which further implies
Hence every Kannan operator satisfies (1.21) with , .
In a similar manner, it can be shown that Chatterjea operators satisfy (1.21) with , .
Therefore, we conclude that generalized Zamfirescu operators satisfy (1.21).
Also, if , , then (1.21) reduces to (1.13) as well as (1.17), with , , (a coincidence point of S, T).
The condition (1.21) is more general than (1.11) as well as (1.15) and (1.16), with , , , .
Moreover, (1.14) reduces to (1.21) as follows: Let , then from (1.14), we have
Hence every mapping satisfying (1.14) becomes a mapping satisfying (1.21) with ,
2 Main results
Lemma 2.1 Let be a subadditive, comparison function, then for any sequence of positive numbers satisfying
where with , we have .
Proof Let . Now, we know that a linear combination of comparison functions is also a comparison function, hence is a comparison function and it satisfies for all . Hence, . Therefore, from (2.1), we have . So, is a decreasing sequence of positive numbers bounded below by 0. Hence will converge to 0, i.e., . □
Theorem 2.2 Let be a normed linear space, let be operators satisfying (1.21) such that . Assume that or is a complete subspace of X, is a continuous sublinear comparison function and p is a point of coincidence of S and T, i.e., . Then, for , the Jungck-Kirk-SP iteration process defined by (1.19) converges to p and is -stable. Also, p will be the unique common fixed point of S, T provided S and T are weakly compatible.
Proof If ψ is sublinear, then (iterate of ψ) is also sublinear (see [15]). Now, first we prove the convergence of the Jungck-Kirk-SP iterative scheme.
Using Jungck-Kirk-SP iterative scheme (1.19) and contractive condition (1.21), we have
Similarly, we have the following estimates:
and
It follows from (2.2), (2.3) and (2.4) that
Using Lemma 2.1, (2.5) yields .
Thus, the Jungck-Kirk-SP iterative scheme converges strongly to p.
Next we prove that the Jungck-Kirk-SP iterative scheme is -stable.
Suppose that is an arbitrary sequence, , , where , .
First, let . Then, we show that as follows:
Using Lemma 1.7, (2.6) yields as .
Conversely, we establish that as follows:
Using again Lemma 1.7, (2.7) yields .
Now, we prove p is the unique common fixed point of S and T provided S, T are weakly compatible. Let there exist another point of coincidence say . Then there exists such that . But from (1.21), we have
which implies .
Now, as S and T are weakly compatible and , so and hence . Therefore Tp is a point of coincidence of S, T and since the point of coincidence is unique, then . Thus and therefore p is the unique common fixed point of S and T. □
Remark 2.3 Since the Jungck-Kirk-Mann iteration scheme is a special case of the Jungck-Kirk-SP iteration scheme, the convergence and stability result similar to Theorem 2.2 also holds for the Jungck-Kirk-Mann scheme.
Theorem 2.4 Let be a normed linear space, let be operators satisfying (1.21) such that . Assume that S(X) or T(X) is a complete subspace of X, is a continuous sublinear comparison function and p is a point of coincidence of S and T. Then, for , the Jungck-Kirk-CR iteration process defined by (1.20) converges to p and is -stable. Also, p will be the unique common fixed point of S, T provided S and T are weakly compatible.
Proof Using Jungck-Kirk-CR iterative scheme (1.20), we have
In a similar manner, we have the following estimates:
and
It follows from (2.8), (2.9) and (2.10) that
Let
Then, obviously, being the linear combination of comparison functions, is also a comparison function and hence (2.11) yields
Since , hence using Lemma 2.1, (2.13) yields .
Thus, the Jungck-Kirk-CR iterative scheme converges strongly to p.
Next we prove that the Jungck-Kirk-CR iterative scheme is -stable.
Suppose that is an arbitrary sequence, , , where , and . We shall establish that as follows:
Then using (2.12) and (2.14), we get
Using Lemma 1.7, from (2.15) we obtain .
Conversely, let . Then we shall show that as follows:
Using (2.12) and (2.16), we get
Lemma 1.7 implies that .
Thus, the Jungck-Kirk-CR iterative scheme is -stable.
The uniqueness of a common fixed point can be proved in the same lines as in Theorem 2.2. □
The following example shows the validity of our Theorems 2.2 and 2.4.
Example 2.5 Let , , , , , , and . It is clear that T and S are weakly compatible operators satisfying (1.21) with a unique common fixed point 0.5. Convergence of the Junck-Kirk-CR iterative scheme as well as the Junck-Kirk-SP iterative scheme to 0.5 is shown in Example 3.2.
3 Results on direct comparison
Various authors [2, 7, 8, 16, 19, 20, 25, 31–34] have worked on convergence speed of iterative schemes. In [2] Berinde showed that Picard iteration is faster than Mann iteration for quasi-contractive operators. In [31], Qing and Rhoades by taking an example showed that Ishikawa iteration is faster than Mann iteration for a certain class of quasi-contractive operators. Chugh and Kumar [7] showed that the SP iterative scheme with error terms converges faster than Ishikawa and Noor iterative schemes for accretive type mappings. Very recently, Hussain et al. [25] showed that Jungck-CR and Jungck-SP iterative schemes have a better convergence rate as compared to other Jungck-type iterative schemes existing in the literature.
Theorem 3.1 Let be a normed linear space, let be operators satisfying (1.21) such that . Assume that or is a complete subspace of X, is a continuous sublinear comparison function and p is a point of coincidence of S, T (i.e., ). If then for ,
-
(1)
Jungck-Kirk-Mann (JKM) iterative scheme is faster than Jungck-Mann (JM) iterative scheme;
-
(2)
Jungck-Kirk-Ishikawa (JKI) iterative scheme is faster than Jungck-Ishikawa (JI) iterative scheme;
-
(3)
Jungck-Kirk-Noor (JKN) iterative scheme is faster than Jungck-Noor (JN) iterative scheme;
-
(4)
Jungck-Kirk-SP (JKSP) iterative scheme is faster than Jungck-SP (JSP) iterative scheme;
-
(5)
Jungck-Kirk-CR (JKCR) iterative scheme is faster than Jungck-CR (JCR) iterative scheme.
Proof For a Jungck-type iterative scheme, we have the following estimates:
and
Also, for Jungck-Kirk-type iterative schemes, we have the following estimates:
and
Using the above estimates, we have
Let
Then
But , . Hence
Therefore, . So, by ratio test is convergent.
Hence , which further implies , i.e., the JKM iterative scheme converges faster than the JM iterative scheme in view of Definition 1.5.
Now, since the estimates for JKI, JKN and JKSP are similar to that of JKM, also estimates of JI, JN and JSP are similar to that of JM, therefore using a very similar argument, it can be easily shown that JKI, JKN, JKSP iterative schemes converge faster than JI, JN, JSP iterative schemes.
Now we compare JCR and JKCR iterative schemes.
From estimates of JCR and JKCR iterative schemes, we have
and
where
and
Using , for , it can be easily observed that .
Therefore, in view of Berinde’s Definition 1.4, Jungck-Kirk-CR have better convergence rate as compared to the Jungck-CR iterative scheme. □
Example 3.2 Let S, T and X be the same as in Example 2.5. Then convergence speed comparison of Jungck-Kirk-type iterative schemes with corresponding Jungck-type iterative schemes is shown in Table 1 with initial approximation and .
Remark 3.3 Although direct comparison among Jungck-Kirk-type iterative schemes is not possible in view of Rhoades Definition 1.5, yet the following example shows that newly introduced iterative schemes have better convergence rate.
Example 3.4 Let , , , , , for some and , , . It is clear that T and S are operators satisfying (1.21) with a unique common fixed point 0. Also, it is easy to see that Example 3.4 satisfies all the conditions of Theorems 2.2 and 2.4.
Proof For JM, JI, JN, JCR, JSP, JKM, JKI, JKN, JKCR and JKSP, iterative schemes with initial approximation , we have the following equations:
and
respectively.
First we compare Jungck-Kirk-type iterative schemes with their corresponding Jungck-type iterative schemes.
For , consider
It is easy to see that
Hence, .
Similarly,
with
implies .
Again, similarly,
with
implies .
Again, similarly,
with
implies .
Again, similarly,
with
implies .
Therefore, by Definition 1.5, Jungck-Kirk-type iterative schemes converge faster than corresponding Jungck-type iterative schemes to the common fixed point 0 of T and S.
Now, we compare Jungck-Kirk-type iterative schemes with each other.
For , we have
with
implies .
Similarly, for ,
with
implies .
Also, for ,
with
implies .
Similarly, for ,
with
implies .
Hence, in view of Definition 1.5, we observe that the decreasing order of Jungck-Kirk-type iterative schemes is as follows:
JKSP, JKCR, JKN, JKI and JKM iterative scheme. □
4 Applications
In this section, with the help of computer programs in , we explain how and why the newly introduced Jungck-Kirk-type iterative schemes can be applied to solve different types of problems. The outcome is listed in the form of Tables 2, 3 and 4, by taking , for all iterative schemes.
Goat problem
A farmer has a fenced circular pasture of radius a and wants to tie a goat to the fence with a rope of length b (see Figure 1) so as to allow the goat to graze half the pasture. How long should the rope be to accomplish this?
The length of the rope b must be longer than a and shorter than , i.e., . Using polar coordinates, we find the grazing area
We want this to equal half the pasture area, which is , so we get the equation
Multiplying both sides by and integrating, we get
After putting , we get the simplified equation and we are looking for the solution x, with .
Now, we rearrange the above equation as , with S, T defined on as
By taking the initial approximation , , and , the comparison of convergence of Jungck-Kirk-type iterative schemes to the point of coincidence 1.158619 of S, T, is listed in Table 2. So the rope length b should be approximately .
Solution of equation
To solve this equation, we rearrange it as , with S, T defined on defined by and .
With the initial approximation , , and , the comparison of convergence of Jungck-Kirk-type iterative schemes to the common fixed point 0.412391 of S, T, is listed in Table 3.
Oscillating function
In order to solve this function by Jungck-type iterative schemes, we write it in the form , where the functions T, S are defined on as and , respectively. By taking the initial approximation and , the obtained results are listed in Table 4.
For detailed study, these programs are again executed after changing the parameters and some observations are made as given below.
5 Observations
Goat problem
1. Taking initial guess (near coincidence point), Jungck-Kirk-Mann, Jungck-Kirk-Ishikawa, Jungck-Kirk-Noor and Jungck-Kirk-CR iterative schemes converge in 64 iterations, while the Jungck-Kirk-SP iterative scheme converges in 63 iterations.
2. Taking and , we observe that Jungck-Kirk-Mann, Kirk-Ishikawa and Jungck-Kirk-Noor iterative schemes converge in 65 iterations, while the Jungck-Kirk-CR iterative scheme converges in 64 iterations and the Jungck-Kirk-SP iterative scheme converges in 63 iterations.
Equation
1. Taking initial guess (somewhat nearer to the common fixed point), the Jungck-Kirk-Mann iterative scheme converges in 15 iterations, the Jungck-Kirk-Ishikawa iterative scheme converges in 5 iterations, the Jungck-Kirk-Noor iterative scheme converges in 11 iterations and the Jungck-Kirk-CR iterative scheme converges in 5 iterations, while the Jungck-Kirk-SP iterative scheme converges in 11 iterations.
2. Taking and , we observe that the Jungck-Kirk-Mann iterative scheme converges in 8 iterations, the Jungck-Kirk-Ishikawa iterative scheme converges in 7 iterations, the Jungck-Kirk-Noor iterative scheme converges in 8 iterations and Jungck-Kirk-CR as well as Jungck-Kirk-SP iterative schemes converge in 6 iterations.
Oscillating function
1. Taking initial guess (near to the common fixed point), the Jungck-Kirk-Mann iterative scheme converges in 18 iterations, the Jungck-Kirk-Ishikawa iterative scheme converges in 13 iterations, the Jungck-Kirk-Noor iterative scheme converges in 12 iterations and the Jungck-Kirk-CR iterative scheme converges in 6 iterations, while the Jungck-Kirk-SP iterative scheme converges in 5 iterations.
2. Taking and , we observe that Jungck-Kirk-Mann, Jungck-Kirk-Ishikawa and Jungck-Kirk-Noor iterative schemes converge in 10 iterations, while the Jungck-Kirk-CR Noor iterative scheme converges in 6 iterations and the Jungck-Kirk-SP iterative scheme converges in 5 iterations.
6 Conclusions
Goat problem
1. Decreasing order of convergence rate of Jungck-Kirk-type iterative schemes is as follows:
Jungck-Kirk-SP, Jungck-Kirk-CR, and Jungck-Kirk-Noor iterative scheme, where Jungck-Kirk-Noor shows equivalence with Jungck-Kirk-Ishikawa and Jungck-Kirk-Mann iterative schemes.
2. For initial guess somewhat near to the point of coincidence, the number of iterations increases in case of Jungck-Kirk-SP and Jungck-Kirk-CR iterative schemes, while the number of iterations decreases in case of Jungck-Kirk-Noor, Jungck-Kirk-Ishikawa and Jungck-Kirk-Mann iterative schemes.
3. The speed of iterative schemes depends on , and . On decreasing the value of these parameters, Jungck-Kirk-SP and Jungck-Kirk-CR iterative schemes show an increase, while Jungck-Kirk-Noor, Jungck-Kirk-Ishikawa and Jungck-Kirk-Mann iterative schemes show a decrease in the number of iterations to converge.
Equation
1. Decreasing order of convergence rate of Jungck-Kirk-type iterative schemes is as follows:
Jungck-Kirk-CR, Jungck-Kirk-Ishikawa, Jungck-Kirk-SP and Jungck-Kirk-Mann iterative scheme, while Jungck-Kirk-Noor shows equivalence with the Jungck-Kirk-SP iterative scheme.
2. For initial guess somewhat near to the common fixed point, the number of iterations decreases in case of Jungck-Kirk-Ishikawa and Jungck-Kirk-Mann iterative schemes, while Jungck-Kirk-SP, Jungck-Kirk-CR and Jungck-Kirk-Noor iterative schemes show no change in the number of iterations to converge.
3. The speed of iterative schemes depends on , and . On decreasing the value of these parameters, Jungck-Kirk-Ishikawa and Jungck-Kirk-CR iterative schemes show an increase, while Jungck-Kirk-Noor, Jungck-Kirk-SP and Jungck-Kirk-Mann iterative schemes show a decrease in the number of iterations to converge.
Oscillating function
1. Decreasing order of convergence rate of Jungck-Kirk-type iterative schemes is as follows:
Jungck-Kirk-SP, Jungck-Kirk-CR, Jungck-Kirk-Noor iterative scheme, Jungck-Kirk-Ishikawa and Jungck-Kirk-Mann iterative scheme.
2. For initial guess nearer to the common fixed point, the number of iterations decreases in case of Jungck-Kirk-Noor, Jungck-Kirk-Ishikawa and Jungck-Kirk-Mann iterative schemes, while Jungck-Kirk-SP as well as Jungck-Kirk-CR iterative schemes show no change in the number of iterations to converge.
3. The speed of iterative schemes depends on , and . On decreasing the value of these parameters, Jungck-Kirk-Noor, Jungck-Kirk-Ishikawa and Jungck-Kirk-Mann iterative schemes show a decrease while Jungck-Kirk-SP and Jungck-Kirk-CR iterative schemes show no change in the number of iterations to converge.
Open problem
It is still an open problem to compare Jungck-Kirk-type iterative schemes with each other in view of Rhoades Definition 1.5 and also to study the same using nonself contractive-type operators.
References
Agarwal RP, O’Regan D, Sahu DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 2007, 8(1):61–79.
Berinde V: Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators. Fixed Point Theory Appl. 2004, 2: 97–105.
Berinde V: Iterative Approximation of Fixed Points. Editura Efemeride, Baia Mare; 2007.
Bosede AO: Strong convergence results for the Jungck-Ishikawa and Jungck-Mann iteration processes. Bull. Math. Anal. Appl. 2010, 2(3):65–73.
Chugh R, Kumar V: Stability of hybrid fixed point iterative algorithms of Kirk-Noor type in normed linear space for self and nonself operators. Int. J. Contemp. Math. Sci. 2012, 7(24):1165–1184.
Chugh, R, Kumar, V: Strong convergence and stability results for Jungck-SP iterative scheme. Int. J. Comput. Appl. 36(12), 40–46 (2011)
Chugh R, Kumar V: Convergence of SP iterative scheme with mixed errors for accretive Lipschitzian and strongly accretive Lipschitzian operators in Banach space. Int. J. Comput. Math. 2013. 10.1080/00207160.2013.765558
Hussain N, Chugh R, Kumar V, Rafiq A: On the rate of convergence of Kirk-type iterative schemes. J. Appl. Math. 2012., 2012: Article ID 526503
Khan SH, Rafiq A, Hussain N: A three-step iterative scheme for solving nonlinear ϕ -strongly accretive operator equations in Banach spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 149
Olaleru JO, Akewe H: The equivalence of Jungck-type iterations for generalized contractive-like operators in a Banach space. Fasc. Math. 2011, 47: 47–61.
Olatinwo MO, Imoru CO: Some convergence results for the Jungck-Mann and the Jungck-Ishikawa iteration processes in the class of generalized Zamfirescu operators. Acta Math. Univ. Comen. 2008, LXXVII(2):299–304.
Olatinwo MO: A generalization of some convergence results using the Jungck-Noor three-step iteration process in an arbitrary Banach space. Fasc. Math. 2008, 40: 37–43.
Olatinwo MO: Some stability results for two hybrid fixed point iterative algorithms in normed linear space. Mat. Vesn. 2009, 61(4):247–256.
Olatinwo MO: Some unifying results on stability and strong convergence for some new iteration processes. Acta Math. Acad. Paedagog. Nyházi. 2009, 25(1):105–118.
Olatinwo MO: Convergence and stability results for some iterative schemes. Acta Univ. Apulensis 2011, 26: 225–236.
Phuengrattana W, Suantai S: On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval. J. Comput. Appl. Math. 2011, 235: 3006–3014. 10.1016/j.cam.2010.12.022
Singh SL, Prasad B: Some coincidence theorems and stability of iterative procedures. Comput. Math. Appl. 2008, 55: 2512–2520. 10.1016/j.camwa.2007.10.026
Singh SL, Bhatnagar C, Mishra SN: Stability of Jungck-type iterative procedures. Int. J. Math. Math. Sci. 2005, 19: 3035–3043.
Singh SL: A new approach in numerical praxis. Prog. Math. 1998, 32(2):75–89.
Song Y, Liu X: Convergence comparison of several iteration algorithms for the common fixed point problems. Fixed Point Theory Appl. 2009., 2009: Article ID 824374
Jungck G: Commuting mappings and fixed points. Am. Math. Mon. 1976, 83(4):261–263. 10.2307/2318216
Noor MA: New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 2000, 251(1):217–229. 10.1006/jmaa.2000.7042
Ishikawa S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44(1):147–150. 10.1090/S0002-9939-1974-0336469-5
Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3
Hussain N, Kumar V, Kutbi MA: On the rate of convergence of Jungck-type iterative schemes. Abstr. Appl. Anal. 2013., 2013: Article ID 132626
Ranganathan S: A fixed point theorem for commuting mappings. Math. Semin. Notes Kobe Univ. 1978, 6(2):351–357.
Hussain N, Jungck G, Khamsi MA: Nonexpansive retracts and weak compatible pairs in metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 100
Jungck G, Hussain N: Compatible maps and invariant approximations. J. Math. Anal. Appl. 2007, 325: 1003–1012. 10.1016/j.jmaa.2006.02.058
Hussain N, Rafiq A, Ciric LB, Al-Mezel S: Almost stability of the Mann type iteration method with error term involving strictly hemicontractive mappings in smooth Banach spaces. J. Inequal. Appl. 2012., 2012: Article ID 207
Hussain N, Rafiq A, Ciric LB: Stability of the Ishikawa iteration scheme with errors for two strictly hemicontractive operators in Banach spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 160
Qing Y, Rhoades BE: Comments on the rate of convergence between Mann and Ishikawa iterations applied to Zamfirescu operators. Fixed Point Theory Appl. 2008., 2008: Article ID 387504
Rhoades BE: Comments on two fixed point iteration methods. J. Math. Anal. Appl. 1976, 56: 741–750. 10.1016/0022-247X(76)90038-X
Prasad B, Sahni R: Convergence of general iterative schemes. Int. J. Math. Anal. 2011, 5(25):1237–1242.
Hussain N, Rafiq A, Damjanović B, Lazović R: On rate of convergence of various iterative schemes. Fixed Point Theory Appl. 2011., 2011: Article ID 45
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first and third authors acknowledge with thanks DSR, KAU for financial support.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Alotaibi, A., Kumar, V. & Hussain, N. Convergence comparison and stability of Jungck-Kirk-type algorithms for common fixed point problems. Fixed Point Theory Appl 2013, 173 (2013). https://doi.org/10.1186/1687-1812-2013-173
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-173