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Stability of kstep fixed point iterative methods for some Prešić type contractive mappings
Journal of Inequalities and Applications volume 2014, Article number: 149 (2014)
Abstract
We introduce the concept of stability of a kstep fixed point iterative method {x}_{n+1}=T({x}_{n},{x}_{n1},\dots ,{x}_{nk+1}), n\ge k1, and study the stability of this equation for mappings T:{X}^{k}\to X satisfying some Prešić type contraction conditions. Our results naturally extend various stability results of fixed point iterative methods in literature, from contractive selfmappings T:X\to X to Prešić type contractive mappings T:{X}^{k}\to X. Illustrative examples of both stable and unstable fixed point iterations are also presented.
MSC:47H09, 47H10, 54H25.
1 Introduction
Let (X,d) be a metric space and T:X\to X a selfmapping. Denote by Fix(T):=\{x\in X:Tx=x\} the set of fixed points of T. If (X,d) is complete and T is a contraction, i.e., there exists a constant \alpha \in [0,1) such that
then, by the wellknown Banach contraction mapping principle, we know that Fix(T)=\{p\} and that, for any {x}_{0}\in X, the Picard iteration, that is, the sequence defined by {x}_{n+1}=T{x}_{n}, n=0,1,\dots , converges to p, as n\to \mathrm{\infty}.
When applying contraction mapping principle for solving concrete nonlinear problems, because of rounding errors, numerical approximations of functions, derivatives or integrals, discretization etc., instead of the theoretical sequence {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}, defined by the given iterative method, we practically work with an approximate sequence {\{{y}_{n}\}}_{n=0}^{\mathrm{\infty}}, satisfying the following approximation bounds:
where the positive quantity {\u03f5}_{n} can be interpreted as the ‘roundoff error’ of {x}_{n}; see [1].
Under these circumstances, the problem of the numerical stability of Picard iteration is whether the approximate sequence {\{{y}_{n}\}}_{n=0}^{\mathrm{\infty}} still converges to the fixed point p of T, provided that {\u03f5}_{n}\to 0 or {\sum}_{n=1}^{\mathrm{\infty}}{\u03f5}_{n}<\mathrm{\infty}.
This question has been answered in the positive, in the case of contraction condition (1), by Ostrowski [2], who established the first stability result for a fixed point iteration procedure, by using the following estimate:
from which easily follows that {y}_{n}\to p, provided {\u03f5}_{n}\to 0.
In 1988, Harder and Hicks [3, 4] introduced the notion of stability for a general fixed point iteration procedure and started a systematic study of this concept, thus obtaining various stability results for Picard iteration that extended Ostrowski’s theorem to mappings satisfying more general contractive conditions and also established some stability results for other fixed point iteration procedures (Mann iteration, Ishikawa iteration, Kirk iteration) in the class of Banach contractions, Zamfirescu operators etc.
Further, some authors, see [1, 5–17], continued the study of the stability of Picard and some other fixed point iterations for various general classes of contractive selfmappings T. Very recently, Rus [18, 19] introduced some alternative concepts of stability for fixed point iterative methods.
On the other hand, the contraction mapping principle has been extended by Prešić [20] to mappings T:{X}^{k}\to X satisfying a contractive condition that include (1) in the particular case k=1. Some other Prešić type fixed point theorems have been obtained in [21–26], in the case T satisfies more general contractive type conditions, while in [27, 28], and [29], some applications to nonlinear cyclic systems of equations and difference equations are obtained.
As for the kstep fixed point iterative methods associated to Prešić type contractive conditions do not exist corresponding stability results, yet, the main aim of this paper is to fill this gap and to introduce an appropriate concept of stability and then establish some stability results. The relationship of our notion of stability to other existing concepts of stability, mainly drawn from the theory of difference equations, is also discussed.
The stability results we shall obtain in this way are extremely general. They unify, extend, generalize, enrich, and complement a multitude of related results from recent literature. Illustrative examples of both stable and unstable fixed point iterative sequences are given.
The paper is organized as follows: in Section 2 we summarize some fixed point theorems for mappings satisfying Prešić type contractive conditions, in Section 3 we present the basic concepts and results concerning the stability of fixed point iteration procedures associated to selfmappings that satisfy explicit contractive conditions. In Section 4, the main stability results of this paper are presented. In Section 5, we end this paper by presenting three detailed examples of stable (Examples 1 and 2) and unstable (Example 3) fixed point iterations.
2 Fixed point theorems for Prešić type mappings
All the stability results mentioned in the previous section were basically established in connection with a corresponding fixed point theorem: Banach, Kannan, Chatterjea, Zamfirescu etc.; see for example [15] for more details.
We start by presenting one of the most interesting generalizations of Banach’s contraction mapping principle for mappings T:{X}^{k}\to X, obtained in 1965 by Prešić [20], which will be fundamental in establishing our stability results for kstep fixed point iterative methods in the present paper.
Theorem 1 (Prešić [20], 1965)
Let (X,d) be a complete metric space, k a positive integer, {\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{k}\in {\mathbb{R}}_{+}, \stackrel{k}{\sum _{i=1}}{\alpha}_{i}=\alpha <1 and T:{X}^{k}\to X a mapping satisfying
for all {x}_{0},\dots ,{x}_{k}\in X.
Then:

(1)
T has a unique fixed point {x}^{\ast}, that is, there exists a unique {x}^{\ast}\in X such that f({x}^{\ast},\dots ,{x}^{\ast})={x}^{\ast};

(2)
the sequence {\{{x}_{n}\}}_{n\ge 0} defined by
{x}_{n+1}=T({x}_{n},\dots ,{x}_{nk+1}),\phantom{\rule{1em}{0ex}}n=k1,k,k+1,\dots(4)
converges to {x}^{\ast}, for any {x}_{0},\dots ,{x}_{k1}\in X.
It is easy to see that, in the particular case k=1, from Theorem 1 we get exactly the wellknown Banach contraction mapping principle, while the kstep iterative method (4) reduces to Picard iteration:
Theorem 1 and other similar results, like the ones in [21, 23, 24, 26], have important applications in the iterative solution of nonlinear equations, see [30] and [28, 31, 32], as well as in the study of global asymptotic stability of the equilibrium for nonlinear difference equations; see the very recent paper [29].
An important generalization of Theorem 1 was proved by Rus [26], for operators T fulfilling the more general condition
for any {x}_{0},\dots ,{x}_{k}\in X, where \phi :{\mathbb{R}}_{+}^{k}\to {\mathbb{R}}_{+} satisfies certain appropriate conditions.
Another important generalization of Prešić’s result was recently obtained by Cirić and Prešić in [21], where, instead of (3) and its generalization (6), the following contraction condition is considered:
for any {x}_{0},\dots ,{x}_{k}\in X, where \lambda \in (0,1).
Other general Prešić type fixed point results have been very recently obtained by the second author in [22, 23, 25, 33] based on the contractive condition (18), studied in [23], which is more general than (7), (6) and (3).
3 Some useful lemmas
In order to obtain simple and short proofs for our main results in this paper we shall need some auxiliary lemmas. A proof of the following lemma can be found in [34]; see also [35].
Lemma 1 (Cauchy)
Let {\{{a}_{n}\}}_{n=0}^{\mathrm{\infty}}, {\{{b}_{n}\}}_{n=0}^{\mathrm{\infty}} be sequences of nonnegative numbers satisfying
Then
We note that by Lemma 1 we find that the second term in (2) converges to zero:
Lemma 2 Let k be a positive integer and {\{{a}_{n}\}}_{n=0}^{\mathrm{\infty}}, {\{{b}_{n}\}}_{n=0}^{\mathrm{\infty}} two sequences of nonnegative real numbers satisfying the inequality
where {\alpha}_{1},\dots ,{\alpha}_{k}\in [0,1) and {\alpha}_{1}+\cdots +{\alpha}_{k}<1. If {lim}_{n\to \mathrm{\infty}}{b}_{n}=0, then {lim}_{n\to \mathrm{\infty}}{a}_{n}=0.
Proof For simplicity, we prove lemma for k=2, when we denote {\alpha}_{1}:=\alpha and {\alpha}_{2}:=\beta. All arguments used below work unchanged in the general case, too.
We first show that if the sequence {\{{u}_{n}\}}_{n=0}^{\mathrm{\infty}} is given by {u}_{0}=1, {u}_{1}=\alpha and
then the series {\sum}_{n=0}^{\mathrm{\infty}}{u}_{n} converges.
Indeed, the characteristic equation corresponding to the linear difference equation (9) has two distinct real roots, {r}_{1},{r}_{2}\in (1,1), since, if we denote f(r):={r}^{2}\alpha r\beta, we have f(1)>0, f(1)>0.
Hence, {u}_{n}={c}_{1}{r}_{1}^{n}+{c}_{2}{r}_{2}^{n}, where {c}_{1}, {c}_{2} are two constants, and so the convergence of {\sum}_{n=0}^{\mathrm{\infty}}{u}_{n} follows by the convergence of the geometric progression series {\sum}_{n=0}^{\mathrm{\infty}}{q}^{n}, with 1<q<1.
Note that in the general case, the characteristic equation
has all its roots (not always real) in absolute value in the interval [0,1), by virtue of a consequence of the Rouché theorem, see [36], which reads as follows.
Let p(x)={a}_{0}+{a}_{1}x+\cdots +{a}_{n}{x}^{n} be a polynomial with complex coefficients {a}_{0},{a}_{1},\dots ,{a}_{n} such that {a}_{n}\ne 0. Then there are exactly n (counted with multiplicity) roots of absolute value less than R, where
Now, let’s come back to the inequality (8) with k=2, {\alpha}_{1}:=\alpha and {\alpha}_{2}:=\beta. Take formally n:=i in (8) to obtain
then multiply both sides of this inequality by {u}_{ni} and sum up all the inequalities obtained for i=1,2,\dots ,n to get
Now apply Lemma 1 to get the desired conclusion. □
Remark 1 Note that for k=1, by Lemma 2 we obtain Theorem 2.1 in [37], a result which is fundamental for obtaining short proofs of stability theorems for one step fixed point iterative methods that has been used in [12, 13, 15], and [17].
4 Stability of kstep fixed point iteration procedures
We first recall some concepts of stability for 1step fixed point iteration procedures, before we consider the stability of a kstep fixed point iteration procedure.
Let (X,d) be a metric space, T:X\to X a selfoperator with Fix(T)\ne \mathrm{\varnothing} and let {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}} be a fixed point iteration procedure of the general form
where f(T,{x}_{n}) is given, which converges to a fixed point p of T (for example, in the case of Picard iteration we have f(T,{x}_{n}):=T{x}_{n}).
Definition 1 (Harder and Hicks, [3])
Let {\{{y}_{n}\}}_{n=0}^{\mathrm{\infty}} be an arbitrary sequence in X and set
We shall say that the fixed point iteration procedure (10) is Tstable or stable with respect to T if
As Picard iteration and other fixed point iteration procedures are not stable with respect to some classes of contractive operators, various weak stability concepts have also been introduced; see [9, 11, 12, 15]. For example Osilike [9] introduced the concept of almost stability, while Berinde [12] introduced the concept of summable almost stability, two notions which are presented in the following.
Definition 2 (Osilike, [9])
Let (X,d) be a metric space, T:X\to X a selfoperator with Fix(T)\ne \mathrm{\varnothing} and let {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}} be a fixed point iteration procedure given by (10), supposed to converge to a fixed point p of T. Let {\{{y}_{n}\}}_{n=0}^{\mathrm{\infty}} be an arbitrary sequence in X and let \{{\epsilon}_{n}\} be defined by (11). We shall say that the fixed point iteration procedure (10) is almost Tstable or almost stable with respect to T if
Definition 3 (Berinde, [13])
Let (X,d) be a metric space, T:X\to X a selfoperator with Fix(T)\ne \mathrm{\varnothing} and let {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}} be a fixed point iteration procedure given by (10), supposed to converge to a fixed point p of T. Let {\{{y}_{n}\}}_{n=0}^{\mathrm{\infty}} be an arbitrary sequence in X and let \{{\epsilon}_{n}\} be defined by (11). We shall say that the fixed point iteration procedure (10) is summable almost Tstable or summable almost stable with respect to T if
It is clear from Definitions 13 that:

(1)
any stable iteration procedure is almost stable;

(2)
any summable almost stable procedure is almost stable,
but the reverses of these assertions are not generally true; see Example 1 in [13]. Moreover, in general, the class of stable iteration procedures is independent of the class of summable almost stable procedures.
We introduce now the concept of stability for a kstep fixed point iteration procedure.
Definition 4 Let (X,d) be a metric space, k a positive integer, T:{X}^{k}\to X a mapping with Fix(T)=\{x\in X:T(x,\dots ,x)=x\}\ne \mathrm{\varnothing} and let {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}} be a kstep fixed point iteration procedure of the general form
which converges to a fixed point p of T. Let {\{{y}_{n}\}}_{n=0}^{\mathrm{\infty}} be an arbitrary sequence in X and set
We shall say that the fixed point iteration procedure (15) is Tstable or stable with respect to T if
Remark 2 For k=1, by Definition 4 we get exactly Definition 1.
The main result of this paper is the following theorem, which establish a stability result that corresponds to the fixed point theorem for PrešićKannan contractive mappings ([23], Theorem 2).
Theorem 2 Let (X,d) be a complete metric space, k a positive integer, a\in \mathbb{R} a constant such that 0<ak(k+1)<1 and T:{X}^{k}\to X a mapping satisfying the following contractive type condition:
for any {x}_{0},{x}_{1},\dots ,{x}_{k}\in X.
Then:

(1)
T has a unique fixed point {x}^{\ast}, that is, there exists a unique {x}^{\ast}\in X such that T({x}^{\ast},\dots ,{x}^{\ast})={x}^{\ast};

(2)
The sequence {\{{x}_{n}\}}_{n\ge 0} with {x}_{0},\dots ,{x}_{k1}\in X and
{x}_{n}=T({x}_{nk},{x}_{nk+1},\dots ,{x}_{n1}),\phantom{\rule{1em}{0ex}}n\ge k,(19)
converges to {x}^{\ast};

(3)
The kstep iteration {\{{x}_{n}\}}_{n\ge 0} given by (19) is Tstable.
Proof Items (1) and (2) follow by Theorem 2 in [23]. Now let {\{{y}_{n}\}}_{n=0}^{\mathrm{\infty}} be an arbitrary sequence in X and denote
First, by triangle inequality, we have
In view of the contractive condition (18), we have
If we denote F(x):=T(x,x,\dots ,x), then by the above inequality we get
On the other hand, for any j we have, by the triangle inequality,
Similarly to the way we obtained (22), we get for any x,y\in X,
where, by hypothesis, A=\frac{ak(k+1)}{2}<\frac{1}{2}.
Since {x}^{\ast} is a fixed point of T, by taking x:={x}^{\ast} and y:={x}_{j} in (24) we have
Now, by (23) we get
that is,
Now, by (21), (22), and (25) we find that the sequence of nonnegative real numbers \{d({y}_{n},{x}^{\ast})\} satisfies the recurrence inequality
where {\alpha}_{j}=\frac{ja}{1A}. It is a simple task to show that
since A=\frac{ak(k+1)}{2}<\frac{1}{2}, by hypothesis.
Now, assume {lim}_{n\to \mathrm{\infty}}{\epsilon}_{n}=0. By applying Lemma 2 with {a}_{n}:=d({y}_{n},{x}^{\ast}) and {b}_{n}:={\epsilon}_{n}, by virtue of (26), we conclude that d({y}_{n+1},{x}^{\ast})\to 0 as n\to \mathrm{\infty}, that is, the kstep iteration {\{{x}_{n}\}}_{n\ge 0} defined by (19) is Tstable. □
A similar stability result can be obtained for mappings T satisfying a more general contractive condition than (18).
Denote Fix(T)=\{x\in X:T(x,x,\dots ,x)=x\}.
Theorem 3 Let (X,d) be a metric space, k a positive integer and T:{X}^{k}\to X a mapping with the property that Fix(T)\ne \mathrm{\varnothing} for which there exist {\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{k}\in {\mathbb{R}}_{+}, {\sum}_{i=1}^{k}{\alpha}_{i}=\alpha <1 satisfying
for all {x}_{0},\dots ,{x}_{k1}\in X and some {x}^{\ast}\in Fix(T). Then the kstep iteration {\{{x}_{n}\}}_{n\ge 0} defined by (4) is Tstable.
Proof Let {\{{y}_{n}\}}_{n=0}^{\mathrm{\infty}} be an arbitrary sequence in X and denote
By triangle inequality, we have
and by (27) we obtain
which together with (28) yields
and the rest of the proof is similar to that of Theorem 2. □
Remark 3 It is clear by the proof of Theorem 2 that if T satisfies (18) then T also satisfies (27), but the reverse is not true, as shown by Example 2.
Another general stability result is given by the next theorem.
Theorem 4 Let (X,d) be a metric space, k a positive integer and T:{X}^{k}\to X a mapping with the property that Fix(T)\ne \mathrm{\varnothing} and for which there exist {\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{k}\in {\mathbb{R}}_{+}, {\sum}_{i=1}^{k}{\alpha}_{i}=\alpha <1 and a constant L>0 such that
for all {x}_{0},\dots ,{x}_{k},x\in X. Then the kstep iteration {\{{x}_{n}\}}_{n\ge 0} defined by (4) is Tstable.
Proof Let {\{{y}_{n}\}}_{n=0}^{\mathrm{\infty}} be an arbitrary sequence in X and denote
If x:={x}^{\ast}\in Fix(T) then by triangle inequality we get (28) and by taking {x}_{0}:={y}_{n}, …, {x}_{k1}:={y}_{nk+1} and x:={x}^{\ast}\in Fix(T) in (30) we obtain
since {x}^{\ast}=T({x}^{\ast},\dots ,{x}^{\ast}). The rest of the proof follows similarly to the proof of Theorem 2. □
5 Examples and concluding remarks
Example 1 ([23], Example 1)
Let X=[0,1] with the usual metric and T:{X}^{2}\to X be defined by
Then, see the detailed proof in [23], T satisfies (18) but does not satisfy (3), (6) and (7). Hence by Theorem 2 it follows that the kstep fixed point iteration (19) is Tstable.
To show that (18) holds, while (3), (6), (7) do not hold; see the detailed proof in [23].
Example 2 Let X=\mathbb{R} with the usual metric and T:{X}^{2}\to X be defined by
It is easy to check that T satisfies condition (3) (with {\alpha}_{1}=\frac{1}{4}, {\alpha}_{2}=\frac{1}{2}), condition (6) (with \phi ({t}_{1},{t}_{2})=\frac{1}{4}{t}_{1}+\frac{1}{2}{t}_{2}), condition (7) (with \lambda =\frac{3}{4}) and condition (27) (with {\alpha}_{1}=\frac{1}{4}, {\alpha}_{2}=\frac{1}{2}) but does not satisfy (18).
Assume condition (18) does hold and take {x}_{0}:=\frac{1}{2}, {x}_{1}:=\frac{1}{2} and {x}_{2}:=0. We get
a contradiction, since, by hypothesis, a<\frac{1}{6}. Hence by Theorem 3 or Theorem 4 (but not by Theorem 2) it follows that the kstep fixed point iteration (19) is Tstable.
The next example illustrates unstable fixed point iterations, in the case neither (18) nor (27) is satisfied.
Example 3 Consider the case k=1 (in order to graph the orbits), X=[\frac{3}{2},\frac{1}{2}] with the Euclidean norm and let T:X\to X be given by Tx=2{x}^{2}+2x1, for all x\in X.
Then T does not satisfy (3), (18), (27), and (30), Fix(T)=\{1,\frac{1}{2}\} and the orbits of {x}_{0}=0.49 under T behave chaotic; see Figure 1.
Indeed, assume that (3) holds, i.e.,
and take x=\frac{3}{2}, y=1 to get the contradiction 3\le {\alpha}_{1}<1. For the same values of x and y, (18) and (30) are not satisfied. Now, assume that (27) holds for {x}^{\ast}=1, i.e.,
and take x=0, to get the contradiction 1\le {\alpha}_{1}<1.
Note that for {x}_{0}=0.5 we obtain a convergent (and constant) sequence, {x}_{n}=0.5, n\ge 0, while for {x}_{0} as close as possible to 0.5 but different of 0.5 we have always unstable orbits \{{x}_{n}\}.
Similar approaches to those in the present paper can be done for mappings defined on product spaces but adapted from the ones in the usual case; see the recent related fixed point results [38–42], and [43].
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Acknowledgements
The authors research was supported by the Grant PNIIRUTE20113239 of the Romanian Ministry of Education and Research. The first author was also supported by the Grant PNIIIDPCE201130087 of the Romanian Ministry of Education and Research.
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Berinde, V., Păcurar, M. Stability of kstep fixed point iterative methods for some Prešić type contractive mappings. J Inequal Appl 2014, 149 (2014). https://doi.org/10.1186/1029242X2014149
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DOI: https://doi.org/10.1186/1029242X2014149