Some coincidence theorems and stability of iterative procedures

Computers and Mathematics with Applications 55 (2008) 2512–2520 www.elsevier.com/locate/camwa

Some coincidence theorems and stability of iterative procedures S.L. Singh a,∗ , Bhagwati Prasad b a Gurukula Kangri University, Hardwar, India b Department of Mathematics, Jaypee Institute of Information Technology University, A-10, Sector-62, Noida 201307, India

Received 9 July 2007; received in revised form 7 October 2007; accepted 10 October 2007

Abstract The purpose of this paper is to obtain coincidence theorems and study the problem of stability of iterations for solving coincidence equations on a general setting. Several special cases are discussed. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Coincidence point; Fixed point; Stable iteration; Jungck contraction; Jungck iterations or J-iterations; b-metric space

1. Introduction Throughout this paper, let Y be an arbitrary nonempty set and R+ the set of nonnegative real numbers. Let S, T be maps on Y with values in a space X and T (Y ) ⊆ S(Y ). Equations arising in many physical formulations can be expressed as a coincidence point equation (1.1)

Sx = T x.

Notice that (1.1) with Y = X reduces to a fixed point equation T x = x when S is the identity map on X . The usual constructive way to solve the Eq. (1.1) is the following Jungck iterative procedure: Sxn+1 = T xn ,

n = 0, 1, 2, . . . .

(1.2)

For an initial choice x0 ∈ Y (see, for instance, [1–3]), this iteration procedure with Y = X was essentially introduced by Jungck [4,5]. As regards the construction of the sequence {Sxn } and {xn } under the iterative procedure (1.2), we may calculate a1 = T x 0 first, and then may proceed to solve Sx1 = a1 , where x1 ∈ S −1 a1 . However, in actual computations due to discretization or rounding off or other reasons, the choice of x1 is approximative and Sx1 is not exactly equal to a1 . Therefore, in general, we get a sequence {Syn } which is approximately equal to {Sxn }. So, even if {Syn } converges, it need not converge to the desired solution. This is the main problem where the stability of iterative procedures plays a vital role in actual numerical environment. For details, refer to Singh et al. [2]. We remark that the stability of the Picard iterative procedure for a fixed point equation was first studied by Ostrowski [6] ∗ Corresponding address: 21 Govind Nagar, Rishikesh 249201, India.

E-mail addresses: [email protected] (S.L. Singh), [email protected] (B. Prasad). c 2007 Elsevier Ltd. All rights reserved. 0898-1221/$ - see front matter doi:10.1016/j.camwa.2007.10.026

S.L. Singh, B. Prasad / Computers and Mathematics with Applications 55 (2008) 2512–2520

2513

on metric spaces. This subject was formally developed by Harder and Hicks [7,8] and subsequently by Rhoades [9,10], Rhoades and Saliga [11], Osilike [12,13], Berinde [14] (see also [2,9–11,15–20]). Czerwik et al. [15,16] has extended Ostrowski’s classical theorem (cf. Corollary 4.2) for the stability of iterative procedures to the setting of bmetric spaces. In [2], Singh et al. have discussed the stability of Jungck type iterative procedures for the coincidence equation (1.1). As discussed in these papers, the stability problem is both of theoretical and numerical interest. The purpose of this paper is to study the problem of stability for coincidence equations of the type (1.1) on b-metric spaces. Some special cases are discussed. 2. Preliminaries Definition 2.1 (Czerwik [21]). Let X be a (nonempty) set and s ≥ 1 be a given real number. A function d : X × X → R+ is a b-metric iff, for all x, y, z ∈ X , the following conditions are satisfied: (i) d(x, y) = 0 iff x = y, (ii) d(x, y) = d(y, x), (iii) d(x, z) ≤ s[d(x, y) + d(y, z)]. The pair (X, d) is called a b-metric space. Indeed, the class of b-metric spaces is effectively larger than that of metric spaces, since a b-metric is a metric when s = 1. The following example shows that a b-metric on X need not be a metric on X (see also [21, p. 264]): Example 2.1. Let X = {x1 , x2 , x3 , x4 } and d(x1 , x2 ) = k ≥ 2, d(x1 , x3 ) = d(x1 , x4 ) = d(x2 , x3 ) = d(x2 , x4 ) = d(x3 , x4 ) = 1, d(xi , x j ) = d(x j , xi ) for i, j = 1, 2, 3, 4 and d(xi , xi ) = 0, i = 1, 2, 3, 4. Then k [d(xi , xn ) + d(xn , x j )] for n, i, j = 1, 2, 3, 4, 2 and if k > 2, the ordinary triangle inequality does not hold. The following definition of the stability of general iterative procedure (2.1) for the coincidence point equation (1.1) is due to Singh et al. [2] when X is a metric space: d(xi , x j ) ≤

Definition 2.2. Let (X, d) be a b-metric space and Y ⊆ X . Let S, T : Y → X, T (Y ) ⊆ S(Y ) and z a coincidence point of T and S, that is, Sz = T z = u (say). For any x0 ∈ Y , let the sequence {Sxn } generated by the general iterative procedure Sxn+1 = f (T, xn ),

n = 1, 2, . . . ,

(2.1)

converge to an element u ∈ X . Let {Syn } ⊂ X be an arbitrary sequence, and set εn = d(Syn+1 , f (T, yn )),

n = 0, 1, 2, 3, . . . .

Then the iterative procedure f (T, xn ) is (S, T )-stable or stable with respect to (S, T ) (or simply stable if there is no ambiguity) if and only if limn→∞ εn = 0 implies that limn→∞ Syn = u. We remark that the iterative procedure (2.1) reduces to xn+1 = f (T, xn ),

n = 1, 2, . . . ,

when Y = X and S is the identity map on X . So, the general definition of the stability of iterative procedures for a general fixed point equation due to Harder and Hicks [7,8] is a special case of Definition 2.2. Consider the following conditions for A, B, S, T : Y → X for all x, y ∈ Y , where 0 < q < 1 and X is a metric space: d(T x, T y) ≤ qd(Sx, Sy);   [d(Sx, T y) + d(Sy, T x)] ; d(T x, T y) ≤ q max d(Sx, Sy), d(Sx, T x), d(Sy, T y), 2   [d(Sx, Ay) + d(By, T x)] . d(T x, Ay) ≤ q max d(Sx, By), d(Sx, T x), d(By, Ay), 2

(2.2) (2.3) (2.4)

2514

S.L. Singh, B. Prasad / Computers and Mathematics with Applications 55 (2008) 2512–2520

Notice that (2.2) is included in (2.3), while (2.3) itself is obtained from (2.4) when A = T and B = S. These conditions with Y = X and B = S = id, the identity map on X , are nicely compared in [22]. For example, (2.3) with Y = X and S = id is the condition (210 ) listed by Rhoades [22]. Further, the pair of maps S, T satisfying (2.2) is called Jungck contraction (or simply J-contraction) while q is called the Jungck constant. The examples given below speak of the generality and usefulness of such conditions. Example 2.2. Let X = Y = [0, 1] be endowed with the usual metric. Let T : Y → X be defined as ( 0, 0 ≤ x < 1 Tx = 1 , x = 1. 2 Then T does not satisfy (210 ) (see [22, Th.1(x)]). However, if we take S: Y → X such that Sx = x 2 , then the pair of maps (T , S) satisfies the condition (2.3) for all x, y ∈ Y . Example 2.3. Let X = Y = R+ be endowed with the usual metric and A, B, S and T such that T x = x 3 + 21 , Ax = x 2 + 12 , Sx = 3x 3 and Bx = 3x 2 . Then d(T x, Ay) = |x 3 − y 2 | = 13 d(Sx, By) for all x, y ∈ Y , and the condition (2.4) is satisfied for any q ∈ [ 31 , 1). See also [23, Example 2] and [24]. In the sequel, we shall need the following lemma essentially due to Czerwik et al. [15]: Lemma 2.1 ([15], See Also [10,16]). Let {εn } be a sequence of nonnegative real numbers. Then lim εn = 0 iff

n→∞

lim sn = 0,

n→∞

where sn =

n X

k n−i εi and 0 ≤ k < 1.

i=0

3. Coincidence theorems In this section, we present a few coincidence theorems for maps on an arbitrary nonempty set Y with values in a b-metric space X . First, we present a basic existence theorem in a b-metric space. Theorem 3.1. Let (X, d) be a metric space and S, T : Y → X such that T (Y ) ⊆ S(Y ) and (2.2) holds with qs < 1. If S(Y ) or T (Y ) is a complete subspace of X , then S and T have a coincidence. Indeed, for any x0 ∈ Y , there exists a sequence {xn } in Y such that (1.2) holds and {Sxn } converges to Sz for some z in Y , and Sz = T z, that is, S and T have a coincidence at z. Further, if Y = X , and S and T commute (just) at z, then S and T have a unique common fixed point. We remark that Theorem 3.1 with s = 1 is the Jungck contraction principle in a metric space (cf. [1], [2, Th. JS]). Theorem 3.2. Let (X, d) be a b-metric space and A, B, S, T : Y → X such that T (Y ) ⊆ B(Y ), A(Y ) ⊆ S(Y ) and qs (2.4) holds with qs < 1 and λs < 1, where λ = max{q, 2−qs }. If one of A(Y ), B(Y ), S(Y ) or T (Y ) is a complete subspace of X, then (I) T and S have a coincidence, i.e. there exists a v ∈ Y such that Sv = T v; (II) A and B have a coincidence, i.e. there exists a w ∈ Y such that Bw = Aw. Further, if Y = X , then (III) T and S have a common fixed point provided that T and S commute (just) at the coincidence point v; (IV) A and B have a common fixed point provided that A and B commute (just) at the coincidence point w; (V) S, T , A and B have a common fixed point provided (III) and (IV) both are true. Proof. It may be completed following Singh et al. [25] (see also [24,26]). However, for the sake of completeness, we give a brief sketch of (I) and (II).

S.L. Singh, B. Prasad / Computers and Mathematics with Applications 55 (2008) 2512–2520

2515

Pick x0 ∈ Y . Construct the sequences {xn } and {yn } such that y2n−1 = Bx2n−1 = T x2n−2 and y2n = Sx2n = Ax2n−1 , n = 1, 2, . . . . We can do this since T (Y ) ⊆ B(Y ) and A(Y ) ⊆ S(Y ). Then by (2.4), d(y2n−1 , y2n ) = d(T x 2n−2 , Ax 2n−1 ) ≤ q max {d(Sx2n−2 , Bx2n−1 ), d(Sx2n−2 , T x2n−2 ), d(Bx2n−1 , Ax2n−1 ), [d(Sx2n−2 , Ax2n−1 ) + d(Bx2n−1 , T x2n−2 )]/2} .   d(y2n−2 , y2n ) ≤ q max d(y2n−2 , y2n−1 ), d(y2n−1 , y2n ), . 2 This yields  d(y2n−1 , y2n ) ≤ max q,

 qs d(y2n−2 , y2n−1 ). 2 − qs

Similarly  d(y2n , y2n+1 ) ≤ max q,

 qs d(y2n−1 , y2n ). 2 − qs

qs } < 1. Both together imply d(yn+1 , yn+2 ) ≤ λd(yn , yn+1 ), n = 1, 2, . . . , where λ = max{q, 2−qs Therefore, {yn } is a Cauchy sequence, and its subsequences {y2n } and {y2n+1 } are also Cauchy. Now let S(Y ) be complete. Then the subsequence {y2n } has a limit in S(Y ). Call it u. Let v ∈ S −1 u. Then Sv = u. Notice that the subsequence {y2n+1 } also converges to u. By (2.4),

d(T v, y2n+2 ) = d(T v, Ax 2n+1 ) ≤ q max {d(Sv, Bx2n+1 ), d(Sv, T v), d(Bx2n+1 , Ax2n+1 ), [d(Sv, Ax2n+1 ) + d(Bx2n+1 , T v)]/2} or d(T v, y 2n+2 ) ≤ q max {d(Sv, y2n+2 ), d(Sv, T v), d(y2n+1 , y2n+2 ), [d(Sv, y2n+2 ) + d(y2n+1 , T v)]/2} . This gives in the limit d(T v, u) ≤ qd(Sv, T v), and Sv = T v. In view of the fact that u ∈ S(Y ) ⊆ B(Y ), there exists a w ∈ Y such that Bw = u. Putting x = x2n and y = w in (2.4) and passing to the limit, we get d(Bw, Aw) ≤ qd(Bw, Aw), yielding Aw = Bw. This proves (I) and (II). Further, (I) and (II) are obviously true when A(Y ) is complete subspace of X , since A(Y ) ⊆ S(Y ). In case B(Y ) or T (Y ) is a complete subspace of X , analogous argument yields (I) and (II).  We remark that in general A, B, S, T :Y → X need not have a common coincidence under the hypotheses of Theorem 3.2 (see, for instance, [24] and [27, Ex. 4]). This justifies the two separate conclusions (I) and (II) regarding the existence of coincidence points. However, if B = S, then we have a slight improvement in Theorem 3.2, which we state below. Corollary 3.1. Let (X, d) be a b-metric space and A, S, T : Y → X such that A(Y ) ∪ T (Y ) ⊆ S(Y ) and the following holds   [d(Sx, Ay) + d(Sy, T x)] (3.1) d(T x, Ay) ≤ q max d(Sx, Sy), d(Sx, T x), d(Sy, Ay), 2 qs with qs < 1 and λs < 1, where λ = max{q, 2−qs }. If one of A(Y ), T (Y ) or S(Y ) is a complete subspace of X , then there exists a z in Y such that Az = T z = Sz. Further, if Y = X , then

(i) T and S have a common fixed point provided that T and S commute (just) at z; (ii) A and S have a common fixed point, provided A and S commute (just) at z; (iii) A, S and T have a common fixed point, provided S commute with each of A and T (just) at z.

2516

S.L. Singh, B. Prasad / Computers and Mathematics with Applications 55 (2008) 2512–2520

Corollary 3.2. Let (X, d) be a b-metric space and S, T : Y → X such that T (Y ) ⊆ S(Y ) and (2.3) holds with qs < 1. If S(Y ) or T (Y ) is a complete subspace of X , then S and T have a coincidence. Indeed, for any x0 ∈ Y , there exists a sequence {xn } in Y such that (1.2) holds and {Sxn } converges to Sz for some z in Y , and Sz = T z, that is, S and T have a coincidence at z. Further, if Y = X , and S and T commute (just) at z, then S and T have a unique common fixed point. Proof. It comes from Theorem 3.2 when T = A and S = B.



4. Stability results First we present a basic result on the stability of J-iterations. Our next theorem is presented on a more general setting for a pair of maps on an arbitrary nonempty set with values in a b-metric space satisfying a condition more general than (2.3). Theorem 4.1. Let (X, d) be a b-metric space and S, T maps on an arbitrary set Y with values in X such that T (Y ) ⊆ S(Y ), and S(Y ) or T (Y ) is a complete subspace of X . Let z be a coincidence point of T and S, that is, T z = Sz = u (say). For any x0 ∈ Y , let the sequence {Sxn }, generated by (1.2) converge to u. Let {Syn } ⊂ X and define εn = d(Syn+1 , T yn ), n = 0, 1, 2, . . . . If the pair (S, T ) is a J -contraction with qs as J -constant, that is, S and T satisfy (2.2) for all x, y ∈ Y and k := sq < 1, then (I) d(u, Syn+1 ) ≤ sd(u, Sxn+1 ) + sk n+1 d(Sx0 , Sy0 ) + s 2

n X

k n−r εr .

r =0

Further, (II) lim Syn = u if and only if lim εn = 0.

n→∞

n→∞

Proof. It closely follows Singh et al. [2, Th. 3.1].



Remark. The process (1.2) generates the sequence {T x0 , T x1 , T x2 , . . .} and { Sx1 , Sx2 , . . .}. Adjoin Sx0 to the later sequence to get {Sx0 , Sx1 , Sx2 , . . .}. We may assume that, for some y0 ∈ Y , Sy 0 is approximatively close to Sx 0 . If we put Y = X and S = id, the identity map on X in the above result, we get the following theorem of Czerwik et al. [15]: Corollary 4.1 ([15]). Let X be complete b-metric space and T : X → X such that d(T x, T y) ≤ qd(x, y) for all x, y ∈ X , and k := sq < 1. Let x0 be an arbitrary point in X and a sequence {xn } of iterates of T given by (1.1). Let {yn } be a sequence in X , and set εn = d(yn+1 , T yn ), n = 0, 1, 2, . . . , then (i) limn→∞ xn = u = T u, P (ii) d(u, yn+1 ) ≤ sd(u, xn+1 ) + sk n+1 d(x0 , y0 ) + s 2 rn=0 k n−r εr . Further, (iii) limn→∞ yn = u if and only if limn→∞ εn = 0. The following classical theorem of stability due to Ostrowski [6] comes from Corollary 4.1 with s = 1: Corollary 4.2. Let (X, d) be a complete metric space and T : X → X a Banach contraction with contraction constant k, i. e., d(T x, T y) ≤ kd(x, y) for all x, y ∈ X , where 0 ≤ k < 1. Let u ∈ X be a fixed point of T . Let x0 ∈ X and xn+1 = T xn , n = 0, 1, 2, . . . , . Suppose that {yn } is a sequence in X , and εn = d(yn+1 , T yn ). Then d(u, yn+1 ) ≤ d(u, xn+1 ) + k n+1 d(x0 , y0 ) +

n X r =0

Moreover, limn→∞ yn = u iff limn→∞ εn = 0. Corollary 4.3. Theorem 4.1 with s = 1.

k n−r εr .

2517

S.L. Singh, B. Prasad / Computers and Mathematics with Applications 55 (2008) 2512–2520

Theorem 4.2. Let (X, d) be a b-metric space and S, T maps on an arbitrary set Y with values in X such that T (Y ) ⊆ S(Y ) and S(Y ) or T (Y ) is a complete subspace of X . Let z be a coincidence point of T and S, that is, T z = Sz = u (say). For any x0 ∈ Y , let the sequence {Sxn }, generated by (1.2) converge to u. Let {Syn } ⊂ X and define εn = d(Syn+1 , T yn ), n = 0, 1, 2, . . . . If the pair (S, T ) satisfies (4.1)

d(T x, T y) ≤ qd(Sx, Sy) + Ld(Sx, T x) for all x, y ∈ Y where 0 < q <

1, s 2 q

< 1 and L ≥ 0, then

(III) d(u, Syn+1 ) ≤ sd(u, Sxn+1 ) + s(sq)n+1 d(Sx0 , Sy0 ) + s 2 L Further (IV) limn→∞ Syn = u if and only if limn→∞ εn = 0.

Pn

r =0 (sq)

n−r d(Sx

r , T xr ) + s

2

Pn

r =0 (sq)

n−r ε . r

Proof. From (4.1), for any nonnegative integer n, d(Sxn+1 , Syn+1 ) = d(T xn , Syn+1 ) ≤ sd(T xn , T yn ) + sd(T yn , Syn+1 ) ≤ sqd(Sxn , Syn ) + s Ld(Sxn , T xn ) + sεn ≤ (sq)2 d(Sxn−1 , Syn−1 ) + s 2 q Ld(Sxn−1 , T xn−1 ) + s Ld(Sxn , T xn ) + s 2 qεn−1 + sεn . After (n − 1) steps, we obtain n n X X d(Sxn+1 , Syn+1 ) ≤ (sq)n+1 d(Sx0 , Sy0 ) + s L (sq)n−r d(Sxr , T yr ) + s (sq)n−r εr . Therefore r =0 r =0

d(u, Syn+1 ) ≤ sd(u, Sxn+1 ) + sd(Sxn+1 , Syn+1 ) ≤ sd(u, Sxn+1 ) + s(sq)n+1 d(Sx0 , Sy0 ) + s 2 L

n X

(sq)n−r d(Sxr , T xr ) + s 2

r =0

n X

(sq)n−r εr .

r =0

This proves (III). Now assume that limn→∞ Syn = u. Then εn = d(Sy n+1 , T y n ) ≤ sd(Syn+1 , Sxn+1 ) + sd(T xn , T yn ) ≤ sd(Syn+1 , Sxn+1 ) + s[qd(Sxn , Syn ) + Ld(Sxn , T xn )]. Letting n → ∞, we obtain εn → 0. Suppose that limn→∞ εn = 0. Since 0 ≤ sq < 1 and limn→∞ Sxn = u. Applying Lemma 2.1 and passing (III) to the limit, we obtain ) ( n n X X 2 n−r 2 n−r (4.2) lim d(u, Syn+1 ) ≤ lim s L (sq) d(Sxr , T xr ) + s (sq) εr . n→∞

n→∞

r =0

r =0

Let A P denote the lower triangular matrix with entries anr = (sq)n−r . Then, limn→∞ anr = 0 for each r and 1 limn→∞ ( rn=0 anr ) = 1−sq . Therefore, A is multiplicative, i.e. for any convergent sequence {sn }, limn→∞ A(sn ) = 1 1−sq .

Thus the right-hand side of (4.2) vanishes (cf. Rhoades [10, p. 692]).



Corollary 4.4 (Rhoades [10]). Let (X, d) be a complete metric space and T a selfmap of X such that   [d(x, T x) + d(y, T y)] , d(x, T y), d(y, T x) d(T x, T y) ≤ q max d(x, y), 2

(4.3)

for all x, y ∈ X , where 0 < q < 1. Let u be a fixed point of T . For any x0 ∈ X , let the sequence {xn }, generated by (1.2) with S identity map on X converge to u. Let {yn } ⊂ X and define εn = d(yn+1 , T yn ), n = 0, 1, 2, . . . . Then (iv) d(u, yn+1 ) ≤ d(u, xn+1 ) + Further,



1 1−q

 X n n X . q n+1−r d(xr , xr +1 ) + q n+1 d(x0 , y0 ) + q n−r εr . r =0

r =0

2518

S.L. Singh, B. Prasad / Computers and Mathematics with Applications 55 (2008) 2512–2520

(v) lim yn = u if and only if lim εn = 0.

n→∞

n→∞

Proof. It closely follows from Theorem 4.2 since the condition (4.3) implies (4.1) with Y = X and S = id.



Theorem 4.3. Let (X, d) be a b-metric space and S, T maps on an arbitrary set Y with values in X such that T (Y ) ⊆ S(Y ), and S(Y ) or T (Y ) is a complete subspace of X . Let z be a coincidence point of T and S, that is, T z = Sz = u (say). For any x0 ∈ Y , let the sequence {Sxn }, generated by Sxn+1 = T xn , n = 0, 1, 2, . . . converges to u. Let {Syn } ⊂ X and define εn = d(Syn+1 , T yn ), n = 0, 1, 2, . . . . If the pair (S, T ) satisfies (2.3) for all x, y ∈ Y and k = sq < 1, then (V) d(u, Syn+1 ) ≤ sd(u, Sxn+1 ) + s(sα)n d(Sx1 , Sy1 ) n n X X + s3α (sα)n−r d(Sxr , T xr ) + s 2 (sα)n−r εr , where α = r =0

r =0

s 2q . 1 − s 2q

Further (VI) lim Sy n = u, if and only if lim εn = 0.

n→∞

n→∞

Proof. From (2.3), for any x, y ∈ Y,one of the following is true: d(T x, T y) ≤ qd(Sx, Sy), d(T x, T y) ≤ qd(Sx, T x), d(T x, T y) ≤ qd(Sy, T y) ≤ qs[d(Sy, Sx) + d(Sx, T y)] k ks ≤ d(Sx, Sy) + d(Sx, T x), and 1 − sk 1 − ks q d(T x, T y) ≤ [d(Sx, T y) + d(Sy, T x)] 2 2k k d(Sx, Sy) + d(Sx, T x). ≤ 2−k 2−k Therefore, in all the cases, we get d(T x, T y) ≤ αd(Sx, Sy) + sαd(Sx, T x). For any nonnegative integer n, d(Sxn+1 , Syn+1 ) = d(T xn , Syn+1 ) ≤ sd(T xn , T yn ) + sd(T yn , Syn+1 ) ≤ sαd(Sxn , Syn ) + s 2 αd(Sxn , T xn ) + sεn ≤ (sα)2 d(Sxn−1 , Syn−1 ) + (sα)2 sd(Sxn−1 , T xn−1 ) + s 2 αd(Sxn , T xn ) + s(αsεn−1 + εn ). After (n − 1) steps, we obtain d(Sxn+1 , Syn+1 ) ≤ (sα)n+1 d(Sx0 , Sy0 ) + s 2 α

n X

(sα)n−r d(Sxr , T yr ) + s

r =0

n X

(sα)n−r εr .

r =0

Therefore d(u, Syn+1 ) ≤ sd(u, Sxn+1 ) + sd(Sxn+1 , Syn+1 ) ≤ d(u, Sxn+1 ) + s(sα)n+1 d(Sx0 , Sy0 ) + s 3 α

n X r =0

This proves (V).

(sα)n−r d(Sxr , T xr ) + s 2

n X r =0

(sα)n−r εr .

S.L. Singh, B. Prasad / Computers and Mathematics with Applications 55 (2008) 2512–2520

2519

Now assume that limn→∞ Syn = u. Then εn = d(Sy n+1 , T y n ) ≤ sd(Syn+1 , Sxn+1 ) + sd(T xn , T yn ) ≤ sd(Syn+1 , Sxn+1 ) + s[αd(Sxn , Syn ) + sαd(Sxn , T xn )]. Letting n → ∞, we obtain εn → 0. Suppose that limn→∞ εn = 0. Since 0 ≤ sα < 1 and limn→∞ Sxn = u, applying Lemma 2.1, the inequality (V) of Theorem 4.3 yields lim d(u, Syn+1 ) ≤ lim s 3 α

n→∞

n→∞

n X

(sα)n−r d(Sxr , T xr ).

(4.4)

r =0

Let A P denote the lower triangular matrix with entries anr = (sα)n−r . Then, limn→∞ anr = 0 for each r and 1 . Therefore, A is multiplicative, i.e. for any convergent sequence {sn }, limn→∞ A(sn ) = limn→∞ ( rn=0 anr ) = 1−sα 1 . Thus right hand side of (4.4) vanishes (cf. Rhoades [10, p. 692]).  1−sα We remark that Theorem 4.3 with s = 1, reduces to the following result of Singh et al. [2]: Corollary 4.5 ([2]). Let (X, d) be a metric space and S, T be maps on an arbitrary set Y with values in X such that T (Y ) ⊆ S(Y ), and S(Y ) or T (Y ) is a complete subspace of X . Let z be a coincidence point of T and S, that is, T z = Sz = u (say). For any x0 ∈ Y , let the sequence {Sxn }, generated by Sxn+1 = T xn , n = 0, 1, 2, . . . converge to u. Let {Syn } ⊂ X and define εn = d(Syn+1 , T yn ), n = 0, 1, 2, . . . . If the pair (S, T ) satisfies (4.1) for all x, y ∈ Y , then (viii) d(u, Syn+1 ) ≤ d(u, Sxn+1 ) + q n+1 d(Sx0 , Sy0 ) + L

n X

q n−r d(Sxr , T xr ) +

r =0

n X

q n−r εr .

r =0

Further, (ix) lim Sy n = u if and only if lim εn = 0.

n→∞

n→∞

When we put s = 1 and S = id, the identity map, in the Theorem 4.3 we obtain the following result of Osilike [13]. Corollary 4.6 ([13]). Let (X, d) be a complete metric space and T : X → X such that (4.1) holds with S = id, the identity map. Let u be a fixed point of T . For any x0 ∈ X , let the sequence {xn }, generated by xn+1 = T xn , n = 0, 1, 2, . . . converges to u. Let {yn } ⊂ X and define εn = d(yn+1 , T yn ), n = 0, 1, 2, . . . . Then (x) d(u, yn+1 ) ≤ d(u, xn+1 ) + q n+1 d(x0 , y0 ) + L

n X r =0

q n−r d(xr , T xr ) +

n X

q n−r εr .

r =0

(xi) lim yn = u, if and only if lim εn = 0.

n→∞

n→∞

Acknowledgements The authors thank the referees for their appreciation and useful suggestions to improve upon the original typescript. References [1] S.L. Singh, A new approach in numerical praxis, Progr. Math. (Varanasi) 32 (2) (1998) 75–89. MR1820099. [2] S.L. Singh, Charu Bhatnagar, S.N. Mishra, Stability of Jungck-type iterative procedures, Int. J. Math. Math. Sci. 19 (2005) 3035–3043. MR2206082.

2520

S.L. Singh, B. Prasad / Computers and Mathematics with Applications 55 (2008) 2512–2520

[3] S.L. Singh, S.N. Mishra, Fixed point theorems in a locally convex space, Quaest. Math. 19 (3–4) (1996) 505–515. MR1415107 (97k:47049). [4] G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly 83 (1976) 261–263. MR0400196 (53 #4031). [5] G. Jungck, Common fixed points for commuting and compatible maps on compacta, Proc. Amer. Math. Soc. 103 (1988) 977–983. MR0947693 (89h:54030). [6] A.M. Ostrowski, The round-off stability of iterations, Z. Angew. Math. Mech. 47 (1967) 77–81. MR0216731 (35 #7560). [7] A.M. Harder, T.L. Hicks, A stable iteration procedure for non-expansive mappings, Math. Japon. 33 (1988) 687–692. MR0972378 (90a:54109b). [8] A.M. Harder, T.L. Hicks, Stability results for fixed point iteration procedures, Math. Japon. 33 (1988) 693–706. MR0972379 (90a:54109a). [9] B.E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure Appl. Math. 21 (1990) 1–9. MR1048010 (91e:54101). [10] B.E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures-II, Indian J. Pure Appl. Math. 24 (1993) 697–703. MR1251180 (95a:47064). [11] B.E. Rhoades, L. Saliga, Some fixed point iteration procedures II, Nonlinear Anal. Forum 6 (1) (2001) 193–217. MR1844482 (2002f:47115). [12] M.O. Osilike, A stable iteration procedure for quasi-contractive maps, Indian J. Pure Appl. Math. 27 (1) (1996) 25–34. MR1374885 (96m: 47104). [13] M.O. Osilike, Stability results for fixed point iteration procedure, J. Nigerian Math. Soc. 14 (1995) 17–27. MR1775011 (2001d: 47091). [14] Vasile Berinde, Iterative Approximation of Fixed Points, Efemeride Publishing House, Romania, 2002. MR1995230 (2004i:47102). [15] Stefan Czerwik, Krzysztof Dlutek, S.L. Singh, Round-off stability of iteration procedures for operators in b-metric spaces, J. Natur. Phys. Sci. 11 (1997) 87–94. MR1659318. [16] Stefan Czerwik, Krzysztof Dlutek, S.L. Singh, Round-off stability of iteration procedures for set-valued operators in b-metric spaces, J. Natur. Phys. Sci. 15 (1–2) (2001) 1–8. MR1932188 (2004b:47083). [17] Vasile I. Istrˇa¸tescu, Fixed Point Theory; An Introduction, D. Reidel Publishing Co., Dordrecht, Holland, 1981. MR0620639 (83c:54065). [18] J. Matkowski, S.L. Singh, Round-off stability of functional iterations on product spaces, Indian J. Math. 39 (3) (1997) 275–286. MR1632880 (99f:65081). [19] Ioan A. Rus, A. Petrusel, G. Petrusel, Fixed Point Theory 1950-2000: Romanian Contribution, House of the Book of Science, Cluj-Napoca, ISBN: 973-686-311-5, 2002. MR1947195 (2003h: 47104). [20] S.L. Singh, Charu Bhatnagar, S.N. Mishra, Stability of iterative procedures for multivalued maps in metric spaces, Demonstratio. Math. 38 (4) (2005) 905–916. MR2199144 (2006j:49034). [21] Stefan Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena 46 (2) (1998) 263–276. MR1665883 (99j:54043). [22] B.E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977) 257–290. MR0433430 (55 #6406). [23] S.L. Singh, S.N. Mishra, On general hybrid contractions, J. Austral. Math. Soc. Ser. A 66 (1999) 244–254. MR1671885 (2000b:54057). [24] S.L. Singh, S.N. Mishra, Remarks on recent fixed point theorems and applications to integral equations, Demonstratio. Math. 34 (4) (2001) 847–857. MR2002g:47127. [25] S.L. Singh, V. Chadha, S.N. Mishra, Remarks on recent fixed point theorems for compatible maps, Int. J. Math. Math. Sci. 19 (4) (1996) 801–804. MR1397849. [26] S.L. Singh, Anita Tomar, Weaker forms of commuting maps and existence of fixed points, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 10 (3) (2003) 145–161. MR2011365 (2004h: 54039). [27] S.L. Singh, S.N. Mishra, Coincidences and fixed points of nonself hybrid contractions, J. Math. Anal. Appl. 256 (2) (2001) 486–497. MR1821752.