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Convergence and stability of Jungck-type iterative procedures in convex b-metric spaces

Fixed Point Theory and Applications20132013:331

https://doi.org/10.1186/1687-1812-2013-331

  • Received: 19 February 2013
  • Accepted: 13 November 2013
  • Published:

Abstract

The purpose of this paper is to investigate some strong convergence as well as stability results of some iterative procedures for a special class of mappings. First, this class of mappings called weak Jungck ( φ , ψ ) -contractive mappings, which is a generalization of some known classes of Jungck-type contractive mappings, is introduced. Then, using an iterative procedure, we prove the existence of coincidence points for such mappings. Finally, we investigate the strong convergence of some iterative Jungck-type procedures and study stability and almost stability of these procedures. Our results improve and extend many known results in other spaces.

MSC:47H06, 47H10, 54H25, 65D15.

Keywords

  • weak Jungck ( φ , ψ ) -contractive mapping
  • iterative procedure
  • coincidence point
  • stability
  • convex b-metric space

1 Introduction

Czerwik [1] initiated the study of multivalued contractions in b-metric spaces.

Definition 1.1 Let X be a set and let s 1 be a given real number. A function d : X × X R + is said to be a b-metric if and only if for all x , y , z X the following conditions are satisfied:
  1. (1)

    d ( x , y ) = 0 if and only if x = y ;

     
  2. (2)

    d ( x , y ) = d ( y , x ) ;

     
  3. (3)

    d ( x , z ) s [ d ( x , y ) + d ( y , z ) ] .

     

Then the pair ( X , d ) is called a b-metric space.

It is clear that normed linear spaces, l p (or L p ) spaces ( p > 0 ), l (or L ) spaces, Hilbert spaces, Banach spaces, hyperbolic spaces, -trees and CAT ( 0 ) spaces are examples of b-metric spaces.

Throughout this paper, R + is the set of nonnegative real numbers and Y is a nonempty arbitrary subset of a b-metric space ( X , d ) . Moreover, F ( T ) = { x Y : T x = x } will be denoted as the set of fixed points of T : Y X . Approximately, all the concepts and results in metric spaces are extended to the setting of b-metric spaces (for more details, see [1]).

The first result on stability of T-stable mappings was introduced by Ostrowski [2] for the Banach contraction principle. Harder and Hicks [3] proved that the sequence { x n } generated by the Picard iterative process in a complete metric space converges strongly to the fixed point of T and is stable with respect to T, provided that T is a Zamfirescu mapping. Rhoades [4] extended the stability results of [3] to more general classes of contractive mappings. Ding [5] constructed the Ishikawa-type iterative process in a convex metric space. He showed that this process converges to the fixed point of T, provided that T belongs in the class which is defined by Rhoades.

A mapping T is said to be a φ-quasinonexpansive if F ( T ) and there exists a function φ : R + R + such that
d ( T x , p ) φ ( d ( x , p ) )

for all x X and p F ( T ) .

Osilike [6] considered a mapping T from a metric space X into itself satisfying the condition d ( T x , T y ) δ d ( x , y ) + L d ( x , T x ) for some δ [ 0 , 1 ) and L 0 for all x , y X . Furthermore, he extended some of the stability results in [4]. Indeed, he proved T-stability for such a mapping with respect to Picard, Kirk, Mann, and Ishikawa iterations. Thereafter, Olatinwo [7] improved this concept to the context of multivalued weak contraction for the Jungck iteration in a complete b-metric space. In [8] this contractive condition was generalized by replacing this condition with d ( T x , T y ) δ d ( x , y ) + φ ( d ( x , T x ) ) , where 0 δ < 1 and φ : R + R + is monotone increasing with φ ( 0 ) = 0 , and some stability results were proved. Recently, Olatinwo [9] extended this condition to d ( T x , T y ) φ ( d ( x , y ) ) + ψ ( d ( x , T x ) ) , where φ : R + R + is a subadditive comparison function and ψ : R + R + is monotone increasing with ψ ( 0 ) = 0 . He studied this contractive condition as a particular case of the class of φ-quasinonexpansive mappings (see [10]). Also, he proved some stability results as well as strong convergence results for the pair of nonself mappings in a complete metric space.

In 1968, Goebel [11] generalized the well-known Banach contraction principle by taking a continuous mapping S in place of the identity mapping, where S commuted with T and T ( X ) S ( X ) . In fact, he used two mappings S , T : Y X for introducing the contractive condition as follows.

A mapping T is called a Jungck contraction if there exists a real number 0 α < 1 such that
(JC) d ( T x , T y ) α d ( S x , S y )

for all x , y Y . In addition, Jungck [12], using a constructive method, proved the existence of a unique common fixed point of S and T, where Y = X .

A mapping T is said to be a Jungck-Zamfirescu contraction (JZ) if there exist real numbers α, β, and γ satisfying 0 α < 1 , 0 β , γ < 1 2 such that for each x , y Y , one has at least one of the following:

  • (z1) d ( T x , T y ) α d ( S x , S y ) ;

  • (z2) d ( T x , T y ) β [ d ( S x , T x ) + d ( S y , T y ) ] ;

  • (z3) d ( T x , T y ) γ [ d ( S x , T y ) + d ( S y , T x ) ] .

A mapping T is said to be a contractive mapping satisfying (JS), (JR) or (JQC) if there exists a constant q [ 0 , 1 ) such that for any x , y Y ,
(JS) d ( T x , T y ) q max { d ( S x , S y ) , 1 2 [ d ( S x , T y ) + d ( S y , T x ) ] , d ( S x , T x ) , d ( S y , T y ) } , (JR) d ( T x , T y ) q max { d ( S x , S y ) , 1 2 [ d ( S x , T x ) + d ( S y , T y ) ] , d ( S x , T y ) , d ( S y , T x ) } , (JQC) d ( T x , T y ) q max { d ( S x , S y ) , d ( S x , T x ) , d ( S y , T y ) , d ( S x , T y ) , d ( S y , T x ) } .
A mapping T is said to be a weak Jungck contraction if there exist two constants a [ 0 , 1 ) and L 0 such that for all x , y Y ,
(WJC) d ( T x , T y ) a d ( S x , S y ) + L d ( S x , T x ) .

It is worth mentioning that a Jungck-Zamfirescu mapping is a (JR) mapping. In [[13], Proposition 3.3], a comparison of the above contractive conditions is established as follows.

Proposition 1.2
  1. (i)

    (JC) (JS) (JQC);

     
  2. (ii)

    (JC) (JR) (JQC);

     
  3. (iii)

    (JS) and (JR) are independent;

     
  4. (iv)

    (JR) (WJC);

     
  5. (v)

    (JS) and (WJC) are independent;

     
  6. (vi)

    (JQC) and (WJC) are independent;

     
  7. (vii)

    reverse implications of (i), (ii), and (iv) are not true.

     

In this paper, a special class of mappings called a weak Jungck ( φ , ψ ) -contraction is introduced, and it is shown that it contains other known classes of Jungck-type contractive mappings. Then, using a Jungck-Picard iterative procedure, we investigate the existence of coincidence points and the uniqueness of the coincidence value of weak Jungck ( φ , ψ ) -contractive mappings. Also, some strong convergence as well as stability results of some Jungck-type iterative procedures (such as Jungck-Ishikawa etc.) are studied. These results play a crucial role in numerical computations for approximation of coincidence values of two nonlinear mappings.

2 Preliminary

In [14], Berinde introduced the concepts of comparison function and ( c ) -comparison function with respect to the function φ : R + R + . A function φ is called a comparison function if it satisfies the following:

  • (i φ ) φ is monotone increasing, i.e., t 1 < t 2 φ ( t 1 ) φ ( t 2 ) ;

  • (ii φ ) The sequence { φ n ( t ) } 0 for all t R + , where φ n stands for the n th iterate of φ.

If φ satisfies (i φ ) and

(iii φ ) n = 0 φ n ( t ) converges for all t R + ,

then φ is said to be a ( c ) -comparison function.

Several results regarding comparison functions can be found in [14] and [15]. Referring to [14] and [15], we have:
  1. 1.

    Any ( c ) -comparison function is a comparison function;

     
  2. 2.

    Any comparison function satisfies φ ( 0 ) = 0 and φ ( t ) < t for all t > 0 ;

     
  3. 3.

    Any subadditive comparison function is continuous;

     
  4. 4.

    Condition (iii φ ) is equivalent to the following one:

     
There exist k 0 N , α ( 0 , 1 ) and a convergent series of nonnegative terms v n such that
φ k + 1 ( t ) α φ k ( t ) + v k

holds for all k k 0 and any t R + .

Berinde [16] expanded the concept of ( c ) -comparison functions in b-metric spaces to s-comparison functions as follows.

Definition 2.1 Let s 1 be a real number. A mapping φ : R + R + is called an s-comparison function if it satisfies (i φ ) and

(iv φ ) There exist k 0 N , α ( 0 , 1 ) , and a convergent series of nonnegative terms v n such that
s k + 1 φ k + 1 ( t ) α s k φ k ( t ) + v k

holds for all k k 0 and any t R + .

Applying results 4 and 1 regarding comparison functions, it is easy to conclude that every s-comparison function is a comparison function.

In the sequel, some lemmas which are useful to obtain our main results are stated.

Lemma 2.2 ([17])

Let φ : R + R + be a comparison function, and let ε n be a sequence of positive numbers such that lim n ε n = 0 . Then
lim n k = 0 n φ n k ( ε k ) = 0 .

Lemma 2.3 ([18])

Let { u n } , { α n } , and { ε n } be sequences of nonnegative real numbers satisfying the inequality
u n + 1 α n u n + ε n , n N .

If α n 1 , n = 1 ( α n 1 ) < and n = 1 ε n < , then lim n u n exists.

Lemma 2.4 Suppose that { u n } and { ε n } are two sequences of nonnegative numbers such that
u n + 1 φ ( u n ) + ε n , n = 0 , 1 , 2 , ,
(2.1)

where φ is a subadditive comparison function. If lim n ε n = 0 , then lim n u n = 0 .

Proof The monotone increasing and the subadditivity of φ together with inequality (2.1) imply that
u n + 1 φ ( u n ) + ε n φ ( φ ( u n 1 ) + ε n 1 ) + ε n φ 2 ( u n 1 ) + φ ( ε n 1 ) + ε n φ n + 1 ( u 0 ) + i = 0 n φ n i ( ε i ) ,
(2.2)

where φ 0 = I (identity mapping). Moreover, since any comparison function satisfies (ii φ ), hence lim n φ n + 1 ( u 0 ) = 0 . Also, we have lim n i = 0 n φ n i ( ε i ) = 0 from Lemma 2.2. Thus, inequality (2.2) implies that lim n u n = 0 . □

Lemma 2.5 Let { α n } be a real sequence in [ 0 , 1 ] , let { ε n } be a sequence of positive numbers such that n = 0 ε n converges, and let { u n } be a sequence of nonnegative numbers such that
u n + 1 ( 1 α n ) u n + α n φ ( u n ) + ε n , n = 0 , 1 , 2 , ,
(2.3)

where φ is a convex subadditive comparison function. If n = 0 α n = , then lim n u n = 0 .

Proof Since φ ( t ) t for all t 0 , using a straightforward induction and (2.3), one can obtain
u n + p + 1 ( 1 α n + p ) u n + p + α n + p φ ( u n + p ) + ε n + p ( 1 α n + p ) [ ( 1 α n + p 1 ) u n + p 1 + α n + p 1 φ ( u n + p 1 ) + ε n + p 1 ] + α n + p [ ( 1 α n + p 1 ) φ ( u n + p 1 ) + α n + p 1 φ 2 ( u n + p 1 ) + φ ( ε n + p 1 ) ] + ε n + p ( 1 α n + p ) ( 1 α n + p 1 ) u n + p 1 + [ 1 ( 1 α n + p ) ( 1 α n + p 1 ) ] φ ( u n + p 1 ) + ε n + p 1 + ε n + p ( i = n n + p ( 1 α i ) ) u n + ( 1 i = n n + p ( 1 α i ) ) φ ( u n ) + i = n n + p ε i ( i = n n + p ( 1 α i ) ) u n + φ ( u n ) + i = n n + p ε i exp ( i = n n + p α i ) u n + φ ( u n ) + i = n n + p ε i
for all n , p N . Now, n = 0 α n = yields that lim p exp ( i = n n + p α i ) = 0 . Then
lim sup p u p = lim sup p u n + p + 1 φ ( u n ) + i = n ε i , n = 0 , 1 , 2 , ,
(2.4)
which implies that
lim sup p u p lim inf n φ ( u n ) lim inf n u n .

Therefore, there exists u R + such that lim n u n = u . Assume that u > 0 . Since φ is continuous and n = 0 ε n converges, letting n in (2.4), we get that u φ ( u ) < u , which is a contradiction. Hence u = 0 and the desired conclusion follows. □

3 Weak Jungck ( φ , ψ ) -contractive mappings

In this section, the class of weak Jungck ( φ , ψ ) -contractive mappings which contains the class of Jungck φ-quasinonexpansive mappings is studied. Furthermore, it is showed that this class includes the various classes of contractive mappings which is introduced in Section 1.

Definition 3.1 Let Y be an arbitrary subset of a b-metric space ( X , d ) , and let S , T : Y X be such that z is a coincidence point of S and T, i.e., S z = T z = p . We say that T is a Jungck φ-quasinonexpansive mapping with respect to S if there exists a function φ : R + R + such that
d ( T x , p ) φ ( d ( S x , p ) )

for all x Y .

The above definition was used in [19] when S is the identity mapping on Y = X .

Definition 3.2 Let Y be an arbitrary subset of a b-metric space ( X , d ) and S , T : Y X . A mapping T is said to be a weak Jungck ( φ , ψ ) -contractive mapping with respect to S if there exist an s-comparison function φ : R + R + and a monotone increasing function ψ : R + R + with upper semicontinuity from the right at ψ ( 0 ) = 0 such that for all x , y Y ,
d ( T x , T y ) φ ( d ( S x , S y ) ) + ψ ( min { d ( S x , T x ) , d ( S x , T y ) } ) .
(3.1)

It is obvious that any weak Jungck ( φ , ψ ) -contraction is also Jungck φ-quasinonexpansive, but the reverse is not true. The next example illustrates this matter.

Example 3.1 Let S , T : [ 0 , 1 ] [ 0 , 1 ] be given by S x = x and
T x = { 0 , 0 x 1 2 , 1 2 , 1 2 < x 1 ,
where [ 0 , 1 ] is endowed with the usual metric. It is easy to see that T satisfies the following property:
d ( T x , p ) φ ( d ( x , p ) )
for all x [ 0 , 1 ] , p F ( T ) = { 0 } , and φ ( x ) = x . But T is not a weak Jungck ( φ , ψ ) -contractive mapping. Indeed, if there exist a 1-comparison function φ and a monotone increasing function ψ with upper semicontinuity from the right at ψ ( 0 ) = 0 such that for all x , y [ 0 , 1 ] ,
d ( T x , T y ) φ ( d ( x , y ) ) + ψ ( min { d ( x , T x ) , d ( x , T y ) } ) ,

then, taking x = 1 2 , y = 1 , we have 1 2 φ ( 1 2 ) + ψ ( 0 ) . This shows that the class of φ-quasinonexpansive mappings properly includes the class of weak Jungck ( φ , ψ ) -contractive mappings.

In what follows, we prove that all the mappings introduced in Section 1 are in the class of weak Jungck ( φ , ψ ) -contractive mappings. It is clear that every Jungck contractive mapping is a weak Jungck ( φ , ψ ) -contractive mapping with φ ( t ) = α t and ψ ( t ) = 0 , where 0 α < 1 s .

Proposition 3.3 Let ( X , d ) be a b-metric space with parameter s, let Y be an arbitrary subset of X, and let S , T : Y X . If T is a Jungck-Zamfirescu contraction (JZ), then T is a weak Jungck ( φ , ψ ) -contractive mapping if α < 1 s and β , γ < 1 s ( 1 + s 2 ) . Moreover, it is a weak Jungck ( φ , ψ ) -contraction with φ ( t ) = max { α , β s 2 1 β s , γ s 2 1 γ s } t and ψ ( t ) = max { β ( 1 + s 2 ) 1 β s , γ ( 1 + s 2 ) 1 γ s } t for all t R + .

Proof If min { d ( S x , T x ) , d ( S x , T y ) } = d ( S x , T x ) , then for all x , y Y ,
d ( T x , T y ) β [ d ( S x , T x ) + d ( S y , T y ) ] β d ( S x , T x ) + β s [ d ( S y , T x ) + d ( T x , T y ) ] β d ( S x , T x ) + β s 2 [ d ( S y , S x ) + d ( S x , T x ) ] + β s d ( T x , T y ) ,
which implies that
d ( T x , T y ) β s 2 1 β s d ( S x , S y ) + β ( 1 + s 2 ) 1 β s d ( S x , T x ) .
Also
d ( T x , T y ) γ [ d ( S x , T y ) + d ( S y , T x ) ] γ s [ d ( S x , T x ) + d ( T x , T y ) ] + γ s [ d ( S y , S x ) + d ( S x , T x ) ]
yields that
d ( T x , T y ) γ s 1 γ s d ( S x , S y ) + 2 γ s 1 γ s d ( S x , T x ) .
Similarly, if min { d ( S x , T x ) , d ( S x , T y ) } = d ( S x , T y ) , then for all x , y Y ,
d ( T x , T y ) β [ d ( S x , T x ) + d ( S y , T y ) ] β s [ d ( S x , T y ) + d ( T y , T x ) ] + β s [ d ( S y , S x ) + d ( S x , T y ) ] ,
thus
d ( T x , T y ) β s 1 β s d ( S x , S y ) + 2 β s 1 β s d ( S x , T y ) .
In addition,
d ( T x , T y ) γ [ d ( S x , T y ) + d ( S y , T x ) ] γ d ( S x , T y ) + γ s [ d ( S y , T y ) + d ( T y , T x ) ] γ d ( S x , T y ) + γ s 2 [ d ( S y , S x ) + d ( S x , T y ) ] + γ s d ( T x , T y )
implies that
d ( T x , T y ) γ s 2 1 γ s d ( S x , S y ) + γ ( 1 + s 2 ) 1 γ s d ( S x , T y ) .
Now, let
φ ( t ) : = max { α , β s 1 β s , β s 2 1 β s , γ s 1 γ s , γ s 2 1 γ s } t = max { α , β s 2 1 β s , γ s 2 1 γ s } t
and
ψ ( t ) : = max { 0 , 2 β s 1 β s , β ( 1 + s 2 ) 1 β s , 2 γ s 1 γ s , γ ( 1 + s 2 ) 1 γ s } t = max { β ( 1 + s 2 ) 1 β s , γ ( 1 + s 2 ) 1 γ s } t

for all t R + . It is clear that φ is an s-comparison function, where α < 1 s and β , γ < 1 s ( 1 + s 2 ) and ψ is a monotone increasing function which is continuous from the right at ψ ( 0 ) = 0 . □

The following result shows that this fact is still true for a more general class of mappings.

Proposition 3.4 Let X, Y and S , T : Y X be as in the above proposition. If T satisfies (JS), then T is a weak Jungck ( φ , ψ ) -contractive mapping, provided that q < 1 s ( 1 + s 2 ) . Furthermore, it is a weak Jungck ( φ , ψ ) -contraction with φ ( t ) = q s 2 1 q s t and ψ ( t ) = q s 2 1 q s t for all t R + .

Proof If min { d ( S x , T x ) , d ( S x , T y ) } = d ( S x , T x ) , then according to the inequality
d ( T x , T y ) q d ( S y , T y ) q s [ d ( S y , T x ) + d ( T x , T y ) ] = q s 2 [ d ( S y , S x ) + d ( S x , T x ) ] + q s d ( T x , T y ) ,
we have
d ( T x , T y ) q s 2 1 q s d ( S x , S y ) + q s 2 1 q s d ( S x , T x )
for all x , y Y . Moreover,
d ( T x , T y ) q 2 [ d ( S x , T y ) + d ( T x , S y ) ] q s 2 [ d ( S x , T x ) + d ( T x , T y ) ] + q s 2 [ d ( T x , S x ) + d ( S x , S y ) ]
implies that
d ( T x , T y ) q s 2 q s d ( S x , S y ) + 2 q s 2 q s d ( S x , T x ) .
On the other hand, if min { d ( S x , T x ) , d ( S x , T y ) } = d ( S x , T y ) , then
d ( T x , T y ) q d ( S x , T x ) q s [ d ( S x , T y ) + d ( T y , T x ) ]
yields that
d ( T x , T y ) q s 1 q s d ( S x , T y )
for all x , y Y . Also
d ( T x , T y ) q d ( S y , T y ) q s [ d ( S y , S x ) + d ( S x , T y ) ] .
Moreover,
d ( T x , T y ) q 2 [ d ( S x , T y ) + d ( T x , S y ) ] q 2 d ( S x , T y ) + q s 2 [ d ( T x , T y ) + d ( T y , S y ) ] q 2 d ( S x , T y ) + q s 2 d ( T x , T y ) + q s 2 2 [ d ( T y , S x ) + d ( S x , S y ) ]
yields that
d ( T x , T y ) q s 2 2 q s d ( S x , S y ) + q ( 1 + s 2 ) 2 q s d ( S x , T y ) .
Now, we take
φ ( t ) : = max { 0 , q , q s , q s 2 q s , q s 2 1 q s , q s 2 2 q s } t = q s 2 1 q s t
and
ψ ( t ) : = max { 0 , q , q s , q s 1 q s , 2 q s 2 q s , q s 2 1 q s , q ( 1 + s 2 ) 2 q s } t = q s 2 1 q s t

for all t R + . It shows that φ is an s-comparison function provided that q < 1 s ( 1 + s 2 ) and ψ is a monotone increasing function which is continuous at ψ ( 0 ) = 0 . □

Similar arguments illustrate that every (JR) mapping is a weak Jungck ( φ , ψ ) -contractive mapping, provided that q < 1 s ( 1 + s 2 ) . In fact, it is a weak Jungck ( φ , ψ ) -contraction with φ ( t ) = ψ ( t ) = q s 2 1 q s t for all t R + . Also, every (JQC) mapping is a weak Jungck ( φ , ψ ) -contractive mapping with φ ( t ) = ψ ( t ) = q s 2 1 q s t for all t R + , provided that q < 1 s ( 1 + s 2 ) .

4 Convergence results

In 1970, Takahashi [20] defined a convex structure on metric spaces. In this section a version of the convexity notion in b-metric spaces is stated. Then, using some Jungck-type iterative procedures, we prove the existence of coincidence points as well as the strong convergence theorems for the weak Jungck ( φ , ψ ) -contractive mappings.

Definition 4.1 Let ( X , d ) be a b-metric space. A mapping W : X × X × [ 0 , 1 ] X is said to be a convex structure on X if for each ( x , y , λ ) X × X × [ 0 , 1 ] and z X ,
d ( z , W ( x , y , λ ) ) λ d ( z , x ) + ( 1 λ ) d ( z , y ) .
(4.1)

A b-metric space X equipped with the convex structure W is called a convex b-metric space, which is denoted by ( X , d , W ) .

Example 4.1 The space l p ( p > 1 ) consisting of all the sequences { x n } of real numbers for which n = 1 | x n | p converges, with the function d : l p × l p R given by
d ( x , y ) = n = 1 | x n y n | p ,

for all x , y l p , is a b-metric space with s = 2 p 1 > 1 . Also, regarding the convexity of f ( t ) = t p , we obtain that d ( z , λ x + ( 1 λ ) y ) λ d ( z , x ) + ( 1 λ ) d ( z , y ) for all z l p , that is, l p ( p > 1 ) is a convex b-metric space with W ( x , y , λ ) = λ x + ( 1 λ ) y . (In a similar way, the space L p ( p > 1 ) is a convex b-metric space.)

Now, the iterative procedures in a convex b-metric space are ready to be illustrated. From now on, it is assumed that ( X , d ) is a b-metric space (resp. ( X , d , W ) is a convex b-metric space) with parameter s and that S , T : Y X are two nonself mappings on a subset Y of X such that T ( Y ) S ( Y ) , where S ( Y ) is a complete subspace of X.

Let { x n } be the sequence generated by an iterative procedure involving the mapping T and S, that is,
S x n + 1 = f ( T , x n ) , n = 0 , 1 , 2 , ,
(4.2)

where x 0 Y is the initial approximation and f is a function.

In the sequel, we discuss several special cases of (4.2):
  1. 1.

    The Jungck iteration (or Jungck-Picard iteration) is given from (4.2) for f ( T , x n ) = T x n . This process was essentially introduced by Jungck [12] and it reduces to the Picard iterative process, when S is the identity mapping on Y = X ;

     
  2. 2.
    The Jungck-Krasnoselskij iteration is defined by (4.2) with
    f ( T , x n ) = W ( S x n , T x n , λ ) ,
    (4.3)

    where 0 λ 1 ;

     
  3. 3.
    The Jungck-Mann iteration is stated by (4.2) with
    f ( T , x n ) = W ( S x n , T x n , α n ) ,
    (4.4)

    where { α n } is a sequence of real numbers such that 0 α n 1 ;

     
  4. 4.
    The Jungck-Ishikawa iteration is introduced by (4.2) with
    f ( T , x n ) = W ( S x n , T y n , α n ) , S y n = W ( S x n , T x n , β n ) ,
    (4.5)

    where { α n } and { β n } are two sequences of real numbers such that 0 α n , β n 1 .

     

It is worth noting that Olatinwo and Postolache [21] used the above iterative procedures in the setting of convex metric spaces.

Theorem 4.2 Suppose that ( X , d ) is a b-metric space, and let S , T : Y X be such that T is a weak Jungck ( φ , ψ ) -contractive mapping. Then S and T have a coincidence point. Moreover, for any x 0 Y , the sequence { S x n } generated by the Jungck-Picard iterative process converges strongly to the coincidence value.

Proof First, we prove that S and T have at least one coincidence point in Y. To do this, let { x n } be the Jungck-Picard iterative process defined by S x n + 1 = T x n and x 0 Y . Taking x = x n and y = x n 1 in (3.1), we obtain
d ( T x n , T x n 1 ) φ ( d ( S x n , S x n 1 ) ) + ψ ( min { d ( S x n , T x n ) , d ( S x n , T x n 1 ) } ) ,
which implies that
d ( S x n + 1 , S x n ) φ ( d ( S x n , S x n 1 ) ) ,
and, inductively,
d ( S x n + 1 , S x n ) φ n ( d ( S x 1 , S x 0 ) ) .
Therefore
d ( S x n + p , S x n ) s p 1 d ( S x n + p , S x n + p 1 ) + s p 1 d ( S x n + p 1 , S x n + p 2 ) + + s 2 d ( S x n + 2 , S x n + 1 ) + s d ( S x n + 1 , S x n ) s p φ n + p 1 ( d ( S x 1 , S x 0 ) ) + s p 1 φ n + p 2 ( d ( S x 1 , S x 0 ) ) + + s 2 φ n + 1 ( d ( S x 1 , S x 0 ) ) + s φ n ( d ( S x 1 , S x 0 ) ) = i = 1 p s i φ n + i 1 ( d ( S x 1 , S x 0 ) ) = 1 s n 1 i = n n + p 1 s i φ i ( d ( S x 1 , S x 0 ) ) , n , p N , p 0 .
Since i = 1 s i φ i ( t ) < for all t R + , { S x n } is a Cauchy sequence. Also, S ( Y ) is complete, so { S x n } has a limit in S ( Y ) , that is, there exists z S 1 p such that p = lim n S x n . Hence, S z = p and
d ( S z , T z ) s d ( S z , S x n + 1 ) + s d ( S x n + 1 , T z ) = s d ( S x n + 1 , S z ) + s d ( T z , T x n ) s d ( S x n + 1 , S z ) + s φ ( d ( S z , S x n ) ) + s ψ ( min { d ( S z , T z ) , d ( S z , T x n ) } ) s d ( S x n + 1 , p ) + s d ( S x n , p ) + s ψ ( d ( S x n + 1 , p ) ) .

Taking the upper limit in the above inequality, we obtain d ( S z , T z ) = 0 . Hence, T z = S z = p , i.e., z is a coincidence point.

Now, we show that S and T have a unique coincidence value. Assume that S and T have two coincidence values p , q X such that p q . Then there exist z 1 , z 2 Y such that S z 1 = T z 1 = p and S z 2 = T z 2 = q . Thus, we conclude that
d ( p , q ) = d ( T z 1 , T z 2 ) φ ( d ( S z 1 , S z 2 ) ) + ψ ( min { d ( S z 1 , T z 1 ) , d ( S z 1 , T z 2 ) } ) = φ ( d ( p , q ) ) .

From our assumptions on φ, it is impossible unless d ( p , q ) = 0 , that is, p = q , which is a contradiction. □

Using Proposition 3.3, one can conclude that the above theorem is a significant extension of [[22], Theorem 3.1] and [[23], Theorem 3.1].

Theorem 4.3 Let ( X , d , W ) be a convex b-metric, and let S , T : Y X be such that T is a weak Jungck ( φ , ψ ) -contractive mapping such that φ is a convex subadditive function. Let { α n } be a real sequence in [ 0 , 1 ] such that n = 0 ( 1 α n ) = . Then, for any x 0 Y , the sequence { S x n } defined by the Jungck-Ishikawa iterative process converges strongly to the coincidence value of S and T.

Proof Theorem 4.3 states the existence of coincidence points in Y and one can obtain the uniqueness of coincidence value in a similar way. We now show that the Jungck-Ishikawa iteration given by S x n + 1 = W ( S x n , T y n , α n ) , where S y n = W ( S x n , T x n , β n ) for each x 0 Y , converges to p = S z = T z , where z is a coincidence point of S and T. Using (3.1), we have
d ( S x n + 1 , p ) α n d ( S x n , p ) + ( 1 α n ) d ( T y n , p ) α n d ( S x n , p ) + ( 1 α n ) × [ φ ( d ( S z , S y n ) ) + ψ ( min { d ( S z , T z ) , d ( S z , T y n ) } ) ] = α n d ( S x n , p ) + ( 1 α n ) φ ( d ( S y n , p ) ) ,
(4.6)
and
d ( S y n , p ) β n d ( S x n , p ) + ( 1 β n ) d ( T x n , p ) β n d ( S x n , p ) + ( 1 β n ) × [ φ ( d ( S z , S x n ) ) + ψ ( min { d ( S z , T z ) , d ( S z , T x n ) } ) ] β n d ( S x n , p ) + ( 1 β n ) φ ( d ( S x n , p ) ) β n d ( S x n , p ) + ( 1 β n ) d ( S x n , p ) = d ( S x n , p ) .
(4.7)
Substituting (4.7) in (4.6), it follows that
d ( S x n + 1 , p ) α n d ( S x n , p ) + ( 1 α n ) φ ( d ( S x n , p ) ) , n = 0 , 1 , 2 , .

Since φ is a convex subadditive comparison function, we have the desired result from Lemma 2.5. □

Remark 4.1
  1. (1)

    Based on Theorem 4.3, it is clear that the Jungck-Mann iterative process as well as the Jungck-Krasnoselskij iterative process converge;

     
  2. (2)

    In normed linear spaces, the generalization of this theorem is stated by Olatinwo [9, 24];

     
  3. (3)

    In Hilbert spaces, assuming that q < 1 s ( 1 + s 2 ) in (JQC), Theorem 4.3 is an extension of the results in [25].

     

The following example shows that condition (3.1) in Theorem 4.3 is necessary.

Example 4.2 Let S , T : [ 0 , 1 ] [ 0 , 1 ] be given by S x = x and
T x = { 0 , 0 x 1 2 , 1 2 , 1 2 < x 1 ,

where [ 0 , 1 ] is endowed with the usual metric. Let x 0 ( 1 2 , 1 ] and x n + 1 = λ x n + ( 1 λ ) T x n for n = 0 , 1 , 2 ,  . Then x n + 1 = λ n + 1 x 0 + 1 λ n + 1 2 , which implies that lim n x n = 1 2 if 0 λ < 1 and lim n x n = x 0 0 if λ = 1 . Therefore, the Krasnoselskij iteration associated to T does not converge strongly to the coincidence value.

5 Stability results

This section is devoted entirely to the stability of some various iterative procedures in b-metric spaces. This concept was first proposed by Ostrowski [2] in metric spaces. Then, Czerwik et al. [26, 27] extended Ostrowski’s classical theorem in the setting of b-metric spaces. In addition, Singh et al. [13] introduced the stability and almost stability of Jungck-type iterative procedures in metric spaces. Below, we state these concepts in convex b-metric spaces.

Definition 5.1 Let ( X , d , W ) be a convex b-metric space, let Y be a subset of X, and let S , T : Y Y be such that T ( Y ) S ( Y ) . For any x 0 Y , let the sequence { S x n } , generated by iterative procedure (4.2), converges to p. Also, let { S y n } X be an arbitrary sequence and let ε n = d ( S y n + 1 , f ( T , y n ) ) , n = 0 , 1 , 2 ,  . Then
  1. (i)

    Iterative procedure (4.2) will be called ( S , T ) -stable if lim n ε n = 0 implies that lim n S y n = p .

     
  2. (ii)

    Iterative procedure (4.2) will be called almost ( S , T ) -stable if n = 0 ε n < implies that lim n S y n = p .

     

The above definition reduces to the concept of the stability of iterative procedure due to Harder and Hicks [3] when S is the identity mapping on Y = X .

Example 5.1 Let S , T : [ 0 , 1 ] [ 0 , 3 2 ] be given by S x = x 2 + x 2 and
T x = { 0 , 0 x 1 2 , 1 2 , 1 2 < x 1 ,

where [ 0 , 3 2 ] is endowed with the usual metric. Let x 0 [ 0 , 1 ] and S x n + 1 = T x n for n = 0 , 1 , 2 ,  . If 0 x 0 1 2 , then S x n + 1 = T x n = 0 , and if 1 2 < x 0 1 , we have S x 1 = T x 0 = 1 2 and S x n + 1 = T x n = 0 for all n N . Thus lim n S x n = 0 = S ( 0 ) = T ( 0 ) ; i.e., the Picard iteration converges strongly to the coincidence value. But the Picard iteration is not ( S , T ) -stable. Indeed, take the sequence { y n } given by y n = n + 2 2 n , n N . One can see easily that the sequence { S y n } does not converge to the coincidence value, while ε n = d ( S y n + 1 , T y n ) = 1 ( n + 1 ) 2 + 3 2 ( n + 1 ) 0 as n .

Our next theorem is presented for a pair of mappings on a nonempty subset with values in b-metric spaces under a condition more general than the condition stated by Singh and Prasad [[23], Theorem 4.2]. Further, this theorem reduces the condition s 2 q < 1 to the condition s q < 1 .

Theorem 5.2 Let ( X , d ) be a b-metric space and T be a weak Jungck ( φ , ψ ) -contractive mapping such that φ is subadditive. For x 0 Y , let { S x n } be the Picard iterative process defined by S x n + 1 = T x n . Then the Jungck-Picard iteration is ( S , T ) -stable.

Proof Note that, by Theorem 4.2, there exists a coincidence point z Y such that { S x n } converges to p = S z = T z . Suppose that { S y n } X and define ε n = d ( S y n + 1 , f ( T , y n ) ) , where f ( T , y n ) = T y n . Assume that lim n ε n = 0 . Then we have
d ( S y n + 1 , p ) s [ d ( S y n + 1 , T y n ) + d ( T y n , p ) ] s ε n + s [ φ ( d ( S z , S y n ) ) + ψ ( min { d ( S z , T z ) , d ( S z , T y n ) } ) ] = s ε n + s φ ( d ( S y n , p ) ) .

Since φ is a subadditive s-comparison function, we get that is a subadditive comparison function. Therefore, Lemma 2.4 yields that lim n d ( S y n , p ) = 0 , that is, lim n S y n = p . □

Remark 5.1 Theorem 5.2 is a generalization of Theorem 3.2 of Singh and Alam [22], Theorem 3.4 of Singh et al. [13], Theorems 4.1 and 4.2 of Singh and Prasad [23], Theorem 1 of Osilike [6], Theorem 2 of Berinde [28], Theorem 2.1 of Bosede and Rhoades [29] as well as Corollary 2 of Qing and Rhoades [30].

The following example shows that the Ishikawa iterative process is not ( S , T ) -stable.

Example 5.2 Let S , T : [ 0 , 1 ] R be given by S x = x and T x = x 2 , where is again endowed with the usual metric. Then T is a weak Jungck ( I 2 , 0 ) -contraction. Let { x n } be a sequence generated by the Ishikawa iterative process with α n = β n = 1 1 n + 1 and x 0 [ 0 , 1 ] . Then
{ z n = S z n = β n S x n + ( 1 β n ) T x n = ( 1 1 n + 1 ) x n + 1 n + 1 T x n = ( 1 3 2 ( n + 1 ) ) x n , x n + 1 = S x n + 1 = α n S x n + ( 1 α n ) T z n = ( 1 1 n + 1 ) x n + 1 n + 1 T z n = ( 1 3 2 ( n + 1 ) + 3 4 ( n + 1 ) 2 ) x n .

Suppose that t n = 3 2 ( n + 1 ) 3 4 ( n + 1 ) 2 . As t n ( 0 , 1 ) and n = 0 t n = , Lemma 2 of [31] implies that lim n x n = 0 = S ( 0 ) = T ( 0 ) (the unique coincidence value of S and T).

To prove the fact that the Ishikawa iteration is not ( S , T ) -stable, we use the sequence { y n } given by y n = n + 1 n + 2 . Then
ε n = | y n + 1 f ( T , y n ) | = | y n + 1 ( 1 3 2 ( n + 1 ) + 3 4 ( n + 1 ) 2 ) y n | = | n + 2 n + 3 ( 1 3 2 ( n + 1 ) + 3 4 ( n + 1 ) 2 ) n + 1 n + 2 | = 6 n 2 + 25 n + 13 4 ( n + 1 ) ( n + 2 ) ( n + 3 ) .

It is clear that lim n ε n = 0 and n = 0 ε n = , while lim n y n = 1 . Therefore, the Ishikawa iterative procedure is not ( S , T ) -stable, but it is almost ( S , T ) -stable. (The almost ( S , T ) -stability is shown in the following.)

The following theorem states that Jungck-Mann iterative and Jungck-Ishikawa iterative process are almost ( S , T ) -stable provided that n = 0 α n < .

Theorem 5.3 Let ( X , d , W ) be a convex b-metric space and let T be a weak Jungck ( φ , ψ ) -contractive mapping such that φ is a convex subadditive function. Let { α n } be a real sequence in [ 0 , 1 ] such that n = 0 α n < . For x 0 Y , let { S x n } be the Ishikawa iterative process given by (4.5). Then the Jungck-Ishikawa iteration is almost ( S , T ) -stable.

Proof In view of Theorem 4.3, there exists a coincidence point z Y such that { S x n } converges to p = S z = T z . Suppose that { S y n } X , ε n = d ( S y n + 1 , W ( S y n , T u n , α n ) ) , n = 0 , 1 , 2 ,  , where S u n = W ( S y n , T y n , β n ) . Assume that n = 0 ε n < . Then
d ( S y n + 1 , p ) s [ d ( S y n + 1 , W ( S y n , T u n , α n ) ) + d ( W ( S y n , T u n , α n ) , p ) ] s ε n + s [ α n d ( S y n , p ) + ( 1 α n ) d ( T u n , p ) ] s ε n + s α n d ( S y n , p ) + s ( 1 α n ) × [ φ ( d ( S z , S u n ) ) + ψ ( min { d ( S z , T z ) , d ( S z , T u n ) } ) ] s ε n + s α n d ( S y n , p ) + s ( 1 α n ) φ ( d ( S u n , p ) ) ,
(5.1)
and
d ( S u n , p ) β n d ( S y n , p ) + ( 1 β n ) d ( T y n , p ) β n d ( S y n , p ) + ( 1 β n ) × [ φ ( d ( S z , S y n ) ) + ψ ( min { d ( S z , T z ) , d ( S z , T y n ) } ) ] β n d ( S y n , p ) + ( 1 β n ) φ ( d ( S y n , p ) ) β n d ( S y n , p ) + ( 1 β n ) d ( S y n , p ) = d ( S y n , p ) .
(5.2)
From (5.1) and (5.2), we conclude that
d ( S y n + 1 , p ) s ε n + s α n d ( S y n , p ) + s ( 1 α n ) φ ( d ( S y n , p ) ) .
(5.3)
Since φ is an s-comparison function, is a comparison function. Thus, inequality (5.3) implies that
d ( S y n + 1 , p ) s ε n + s α n d ( S y n , p ) + ( 1 α n ) d ( S y n , p ) = ( 1 + ( s 1 ) α n ) d ( S y n , p ) + s ε n .

Now, according to Lemma 2.3, lim n d ( S y n , p ) exists. Therefore, there exists u R + such that lim n d ( S y n , p ) = u . Assume that u > 0 . Since is a subadditive comparison function, φ is continuous and s φ ( t ) < t for all t > 0 . Then, letting n in (5.3), we get u s φ ( u ) < u , which is a contradiction. Hence, u = 0 and this completes the proof. □

In a similar way, using Lemma 1 of [32] in place of Lemma 2.3 in the previous proof, by omitting the condition α n < , one can prove that Theorem 5.3 holds in convex metric spaces. This indicates that the Ishikawa iterative process given Example 5.2 is almost ( S , T ) -stable.

Declarations

Acknowledgements

The authors express their gratitude to the referees for reading this paper carefully, providing valuable suggestions and comments, which improved the contents of this paper.

Authors’ Affiliations

(1)
Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

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