In 1922, S. Banach [4] established his famous and fundamental result in the metric fixed point theory as follows:

Theorem 1.1.(Banach Contraction Principle) Let (M,d) be a complete metric space and let S:M→M be a Banach contraction, that is, S satisfies that there exists α∈(0,1) such that

d(Sx,Sy)≤αd(x,y)       (z1)

for all x,y∈M. Then, S has a unique fixed point in M.

Notice that Banach's contractions are continuous mappings, so, in the spirit to extend the BCP, in 1968, R. Kannan [11] introduced a new class of contractive mappings admitting discontinuous functions, as follows.

Theorem 1.2.(Kannan, [11]) Let (M,d) be a complete metric space and let S:M→M be a Kannan contraction, that is, S satisfies that there exists β∈[0,1/2) such that

d(Sx,Sy)≤β[d(x,Sx)+d(y,Sy)]      (z2)

for all x,y∈M. Then, S has a unique fixed point in M.

In the same fashion, in 1972, S. K. Chatterjea [6] introduced a similar type of contractive condition.

Theorem 1.3.(Chatterjea, [6]) Let (M,d) be a complete metric space and let S:M→M be a Chatterjea contraction, that is, S satisfies that there exists γ∈[0,1/2) such that

d(Sx,Sy)≤γ[d(x,Ty)+d(y,Tx)]      (z3)

for all x,y∈M. Then, S has a unique fixed points in M.

Remark 1.1. Conditions (z1),(z2) and (z₃) are independent to each other. (See, B.E. Rhoades [19]).

In 1972, T. Zamfirescu [23] combine conditions (z1),(z2) and (z₃) to obtain a general fixed point result that include the Banach, Kannan and Chatterjea contraction mappings.

Theorem 1.4.(Zamfirescu, [23]) Let (M,d) be a complete metric space and let S:M→M be a Zamfirescu operator, that is, S satisfies that there exist real numbers α,β and γ satisfying 0≤α<1 and 0≤β,γ<1/2 such that, for all x,y∈M, at least one of the following inequalities is satisfied:

• (z₁) d(Sx,Sy)≤αd(x,y),
  

• (z₂) d(Sx,Sy)≤β[d(x,Sx)+d(y,Sy)],

• (z₃) d(Sx,Sy)≤γ[d(x,Sy)+d(y,Sx)].

Then, S has a unique fixed point in M.

Notice that conditions (z1),(z2) and (z₃) can be written in the following equivalent form:

d(Sx,Sy)≤hmaxd(x,y),d(x,Sx)+d(y,Sy)2,d(x,Sy)+d(y,Sx)2

for all x,y∈M and 0≤h<1.

On the other hand, in 1976, G. Jungck [9] extend the Banach fixed point principle by using a pair of commuting mappings for which he prove the existence and uniqueness of its common fixed points.

Theorem 1.5.(Jungck, [9]) Let S₁, S₂ be two self-maps on a complete metric space, (M,d) such that,

• (J₁) d(Sx,Sy)≤αd(x,y),

• (J₂) S₂ is continuous,

• (J₃) S1(M)⊂S2(M),

• (J₄) there is a∈[0,1) such that
  d(S1x,S1y)≤ad(S2x,S2y)
  for all x,y∈M.

Then, S₁ and S₂ have a unique common fixed point z₀∈ M.

These classical results have been generalized and extended in several ways, including by defining the contractive maps in more general spaces and controlling the inequality contraction with external functions as well. The purpose of this paper is to define a class of pair of mappings in the fashion of Zamfirescu [23], Jungck [9] and Geraghty [7], in the setting of b-metric spaces, to study conditions for the existence of a point of coincidence, to analyze the convergence and stability of Jungck, Jungck-Mann and Jungck-Ishikawa iteration processes and, finally, to prove the existence of its common fixed points.

In this section we recall some concepts, results and properties of the b-metric spaces that will be useful in the sequel.

Definition 2.1. Let M be a nonempty set and s ≥ 1 be a given real number. A function

ρ:M×M→ℝ+:=[0,+∞)

is said to be a b-metric if and only if for all x,y,z∈M the following conditions hold:

• (ρ₁) ρ(x,y)=0 if only if x=y.

• (ρ₂) ρ(x,y)=ρ(y,x).

• (ρ₃) ρ(x,z)≤s(ρ(x,y)+ρ(y,z)).

The pair (M,ρ) is called a b-metric space (also known as a quasimetric space) and the real number s≥ 1 is called the coefficient of (M,ρ).

From Definition 2.1 it is clear that the class of b-metric spaces is larger than the class of usual metric spaces, thus the b-metric spaces extend the usual metric spaces, since a b-metric is a usual metric space when s=1, but the converse is not true.

Example 2.1.

• (1) Let (M,d) be a metric space. For p>1, the function φ:ℝ+→ℝ+ defined by φ(t)=tp,t∈ℝ+ is convex, then we can define a continuous b-metric with coefficient s=2p−1 as:

  ρ(x,y)=φ(d(x,y))=(d(x,y))p

  for all x,y∈M. It is easy to show that ρ does not hold the triangle inequality, so (M,ρ) is not a metric space. In particular, let M=R and let d(x,y)=|x−y| be the usual metric on R. Then ρ(x,y)=(x−y)2 is a b-metric continuous on R, but it is not a metric on R.

• (2) The set Lp[0,1],(0<p<1) of all real functions x(t),t∈[0,1] is defined by

  Lp[0,1]=x(t) : ∫01 |x(t)| p dt<∞.

  We define a b-metric on Lp[0,1] with coefficient s=21/p−1, by

  ρ(x,y)=∫01 |x(t)−y(t)|pdt1/p,

  for all x,y∈Lp[0,1].

Now, we present the basic concepts concerning to the convergence of sequences, Cauchy sequences and complete b-metric spaces.

Definition 2.2. Let (M,ρ) be a b-metric space with s ≥ 1. A sequence (x_n) in M is called:

• (1) b-convergent if there exists x ∈ M such that limn→∞ρ(xn,x)=0. In this case we write limn→∞xn=x or xn→x as n→∞.

• (2) b-Cauchy sequence in M if limn,m→∞ρ(xn,xm)=0, that is, given ε>0 there exists n0∈ℕ such that for all n,m>n0, we have ρ(xn,xm)<ε. If every b-Cauchy sequence in M is convergent, then (M,ρ) is said to be a complete b-metric space.

Proposition 2.1. Let (M,ρ) be a b-metric space with s ≥ 1. The following assertions hold:

• (i) Any b-convergent sequence has a unique limit.

• (ii) The subsequences of a b-convergent sequence are also convergent to the limit of the original sequence.

• (iii) Every sequence which is b-convergent is also a b-Cauchy sequence.

The examples of b-metric spaces given above show that there exist certain b-metric functions that are continuous, but in general a b-metric is not continuous in all its variable (see, [8]).

On the other hand, A. Aghajani, M. Abbas and J.R. Rushan [3] proved the following result about b-convergent sequences

Proposition 2.2. Let (M,ρ) be a b-metric space with s≥ 1, and suppose that (x_n) and (y_n) are sequences such that limn→∞xn=x and limn→∞yn=y. Then, we have,

1s2ρ(x,y)≤ limn→∞infρ(xn,yn)≤limn→∞supρ(xn,yn)≤s2ρ(x,y).

In particular, if x=y, then limn→∞ρ(xn,yn)=0. Moreover, for each z∈ M we have

1sρ(x,z)≤ limn→∞infρ(xn,z)≤ limn→∞supρ(xn,z)≤sρ(x,z).

The following result about b-Cauchy sequences will be useful to establish our convergence results.

Proposition 2.3.([17]) Let (M,ρ) be a b-metric space with s ≥ 1. Let (x_n) be a sequence in M such that

limn→∞ρ(xn,xn+1)=0.

If (x_n) is not a b-Cauchy sequence in M, then there exist ε>0 and sequences of positive integers (n(k)) and (m(k)) with n(k)>m(k)>k>0 such that

ρ(xm(k),xn(k))≥ε and ρ(ym(k),yn(k)−1)<ε,

and

ε≤limk→∞supρ(xm(k),xn(k))≤εs,εs2≤limk→∞supρ(xm(k),xn(k)−1)≤ε,εs2≤limk→∞supρ(xm(k)−1,xn(k)−1)≤εs,εs≤limk→∞supρ(xm(k)−1,xn(k))≤εs2.

Now, we present some definitions and properties for a pair of mappings that are useful to establish our common fixed point results.

Definition 2.3. Let (M,ρ) be a b-metric space with s ≥ 1 and let S,T:M→M be two selfmappings. By C(S,T) we denote the set of coincidence points of S and T, that is,

C(S,T)={x∈M : Sx=Tx},

and by PC(S,T), the set of points of coincidence (POC) of S and T, that is,

PC(S,T)={z∈M : z=Sx=Tx, for some x∈M}.

Definition 2.4. Let (M,ρ) be a b-metric space with s≥ 1. The mappings S,T:M→M are said to

• (1) becompatible [10], if and only if

  limn→∞ρ(STxn,TSxn)=0,

  whenever (x_n) is a sequence in M such that

  limn→∞Sxn=limn→∞Txn=t, for some t∈M.

• (2) be non-compatible, if there exists at least a sequence (x_n) in M such that

  limn→∞Sxn=limn→∞Txn=t, for some t∈M.

  but limn→∞ρ(STxn,TSxn) is either non zero or non-existent.

• (3) be weakly compatible [11], if for all x∈M,Sx=Tx implies that STx=TSx, that is, if for x∈C(S,T) implies that Sx∈C(S,T).

• (4) Satisfy the b-property (EA) [1], if there exists a sequence (x_n) in M such that

  limn→∞Sxn=limn→∞Txn=t, for some t∈M.

• (5) Satisfy the b-limit range property with respect to T, (in short b-CLR_T-property) [21], if there exists (x_n) in M such that

  limn→∞Sxn=limn→∞Txn=Tt, for some t∈M.

Remark 2.1.

• (1) If S and T are compatible selfmappings, then S and T are weakly compatible.

• (2) If S and T are non-compatible, then S and T satisfy the b-property (EA).

• (3) Weak compatibility and the b-property (EA) are independent to each other.

• (4) It may be noted that the b-CLR_T-property avoid the requirement of the condition of closedness of the range of the involved mappings.

In this section, we introduce the class of generalized ψ-Geraghty-Zamfirescu contraction pairs. To do this, we will use altering distance functions and Geraghty-type operators for a pair of mappings. Also, for these pair of maps we study the existence and uniqueness of its points of coincidence.

We recall that in 1973, M. Geraghty [7] introduced a condition (now called Geraghty's property) which essentially replace the constant parameters of any contractive inequality by functions satisfying some properties. This approach had been used to proof fixed point theorems that generalize the BCP. For its use in the setting of $b$-metric spaces, we consider the class of functions Bs, where β∈Bs if β:ℝ+→[0,1/s),(s≥1) and satisfies the Geraghty type property:

(3.1)limn→∞β(tn)=1s, implies that  limn→∞tn=0,

for any (tn)⊂ℝ+. (Cf. [7]).

On the other hand, in 1984, M. S. Khan, M. Swalech and S. Sessa [12] extended the BCP by controlling the contractive inequality with an external function ψ:ℝ+→ℝ+ (called an altering distance function) which satisfies that

• (i) ψ is monotonic increasing function,
  

• (ii) ψ is a continuous mapping,
  

• (iii) ψ(t)=0, if and only if t=0.

By Ψ we denote the set of all altering distance functions.

Definition 3.1. Let (M,ρ) be a b-metric space with s ≥ 1. Two selfmappings S,T:M→M are called generalized ψ-Geraghty-Zamfirescu contraction mappings, if there exist ψ∈Ψ and β∈Bs such that

(3.2)ψ[s2ρ(Sx, Sy)]≤β[ψ(N(x, y))]ψ(N(x, y))

for all x, y∈M, where

N(x, y)=maxρ(Tx, Ty), ρ(Sx, Tx)+ρ(Sy, Ty)2, ρ(Sx, Ty)+ρ(Sy, Tx)2s.

Remark 3.1. Since functions belonging to Bs are strictly smaller than 1/s, for some s≥ 1, then the expression β(ψ(N(x, y))) in (3.2) can be estimated from above as follows:

β[ψ(N(x, y))]<1/s, for all x, y∈M.

Example 3.1. Let us consider the b-metric space ([0,4],ρ), where the b-metric ρ is given by ρ(x,y)=(x−y)2. From Example 2.1, we know that the coefficient in this case is s=2. Let us consider the mappings S,T:[0,4]→[0,4] given by the formulas

Sx=1,x=00,x∈(0,4] and Tx=1,x=04,x∈(0,4].

We show that S and T are generalized ψ-Geraghty-Zamfirescu contraction mappings with ψ(t)=t2 and β(t)=t+1t+2. In fact:

ρ(Sx,Sy)=1,x=0, y∈(0,4]1,y=0, x∈(0,4]0,x,y∈(0,4],  ρ(Tx,Ty)=9,x=0, y∈(0,4]9,y=0, x∈(0,4]0,x,y∈(0,4]
ρ(Sx,Tx)=0,x=016,x∈(0,4],  ρ(Sy,Ty)=0,x=016,x∈(0,4]
ρ(Sx,Ty)=0,x=y=09,x=0, y∈(0,4]1,y=0, x∈(0,4]16,x,y∈(0,4],  ρ(Sy,Tx)=0,x=y=01,x=0, y∈(0,4]9,y=0, x∈(0,4]16,x,y∈(0,4].

Thus, we have that

ρ(Sx,Tx)+ρ(Sy,Ty)2=0,x=y=08,x=0, y∈(0,4]8,y=0, x∈(0,4]16,x,y∈(0,4]

and

ρ(Sx,Ty)+ρ(Sy,Tx)4=0,x=y=05/2,x=0, y∈(0,4]5/2,y=0, x∈(0,4]8,x,y∈(0,4].

Hence, we obtain

N(x,y)=9,x=y=09,x=0, y∈(0,4]9,y=0, x∈(0,4]16,x,y∈(0,4].

Now, notice that

ψ(s2ρ(Sx,Sy))=16,x=0, y∈(0,4]16,y=0, x∈(0,4]0,otherwise

and

β(ψ(N(x,y)))ψ(N(x,y))=81⋅8283,x=0, y∈(0,4]81⋅8283,y=0, x∈(0,4]>0,otherwise.

Therefore, S and T are generalized ψ-Geraghty-Zamfirescu contraction mappings.

In the next result we prove that any ψ-Geraghty-Zamfirescu pair cannot has more than one point of coincidence.

Proposition 3.1. Let (M,ρ) be a b-metric space with s≥ 1 and let S,T:M→M be two selfmappings. Assume that S and T are generalized ψ-Geraghty-Zamfirescu contraction mappings. If S and T have a POC in M, then it is unique.

Proof. Let z∈ M be a POC of S and T. Then, there exists u∈ M such that Su=Tu=z. Suppose that for some v∈ M we have Sv=Tv=w and w≠ z. Now, from inequality (3.2) we have

(3.3)ψ[ρ(z,w)]≤ψ[s2ρ(z,w)]=ψ[s2ρ(Su,Sv)]≤β[ψ(N(u,v))]ψ[N(u,v)]

where,

(3.4)N(u,v)=maxρ(Tu,Tv),ρ(Su,Tu)+ρ(Sv,Tv)2,ρ(Su,Tv)+ρ(Sv,Tu)2s   =maxρ(z,w),ρ(z,z)+ρ(w,w)2,ρ(z,w)+ρ(z,w)2s   =maxρ(z,w),0,ρ(z,w)s=ρ(z,w).

Using (3.4) in (3.3), we obtain

ψ[ρ(z,w)]≤ψ[s2ρ(z,w)]≤β[ψ(ρ(z,w))]ψ(ρ(z,w))    <1sψ(ρ(z,w))≤ψ(ρ(z,w))

which is a contradiction. Hence z=w.

Now, we prove that generalized ψ-Geraghty-Zamfirescu contraction mappings have a point of coincidence if they satisfy the b-property (EA).

Proposition 3.2. Let (M,ρ) be a b-metric space with s≥ 1 and let S,T:M→M be two selfmappings. Assume that S and T are generalized ψ-Geraghty-Zamfirescu contraction mappings satisfying the b-property (EA). If S(M)⊂T(M) (or T(M)⊂S(M)) and T(M) (resp. S(M)) is closed, then (S,T) has a unique point of coincidence.

Proof. Let us assume S(M)⊂T(M) and T(M) closed. Since (S,T) satisfy the b-property (EA), there exists a sequence (xn)⊂M such that

limn→∞Sxn=Txn=t∈T(M).

Then, t=Tu for some u∈ M. We will prove that Tu=Su=t. Let us suppose that Su ≠ Tu. From inequality (3.2), we have

(3.5)ψ[ρ(Su,Tu)]≤ψ[s2ρ(Su,Tu)]      ≤limsupn→∞ψ[s2ρ(Su,Txn)]      =limsupn→∞ψ[s2ρ(Su,Sxn)]      ≤limsupn→∞β[ψ(N(u,xn))]limsupn→∞ψ(N(u,xn))

where,

N(u,xn)=maxρ(Tu,Txn),ρ(Su,Tu)+ρ(Sxn,Txn)2,ρ(Su,Txn)+ρ(Sxn,Tu)2s.

Taking upper limit as k→∞ and from Proposition 2.2, we have

limsupn→∞N(u,xn)≤maxsρ(Tu,Tu),ρ(Su,Tu)+s2ρ(Tu,Tu)2,sρ(Su,Tu)+s2ρ(Tu,Tu)2s=max0,ρ(Su,Tu)2,ρ(Su,Tu)2=ρ(Su,Tu)2.

Thus, by taking upper limit as k→∞ and using \eqref{eq3.12} we obtain

ψ[ρ(Su,Tu)]≤ψ[s2ρ(Su,Tu)]≤βψρ(Su,Tu)2ψρ(Su,Tu)2      <1sψρ(Su,Tu)2≤ψρ(Su,Tu)2

which is a contradiction. Hence Su=Tu.

The uniqueness of the POC comes from Propostion 3.1. The case T(M)⊂S(M) and S(M) closed is proved with similarity.

The conditions given in Proposition 3.2 to guarantee the existence of a POC for generalized ψ-Geraghty-Zamfirescu mappings are not necessary, as Example 3.1 shows, since S0=T0=1, but S([0,4])={0,1} and T([0,4])={1,4}. That means, neither S([0,4])⊂T([0,4]) nor T([0,4])⊂S([0,4]). In the next example we show that the b-property (EA) cannot be removed in Proposition 3.2.

Example 3.2. Let us consider the b-metric space ([0,2],ρ), where the b-metric function is given by ρ(x,y)=|x−y|, the usual distance on ℝ. Let us consider the mappings S,T:[0,2]→[0,2] defined as:

Sx=1,x∈[0,1)0,x∈[1,2] and Tx=0,x∈[0,1)2,x=11,x∈(1,2].

For these functions PC(S,T)={∅}, S([0,4])={0,1}⊂T([0,4])={0,1,2} and T([0,4]) is closed. We show that S and T are ψ-Geraghty-Zamfirescu mappings with ψ(t)=t2 and β(t)=t+12t+1. In fact,

ρ(Sx,Sy)=0,x,y∈[0,1)1,x=1, y∈[0,1)0,x=1, y∈(1,2]1,y=1, x∈[0,1)0,y=1, x∈(1,2]0,x,y∈(1,2],  ρ(Tx,Ty)=0,x,y∈[0,1)2,x=1, y∈[0,1)1,x=1, y∈(1,2]2,y=1, x∈[0,1)1,y=1, x∈(1,2]0,x,y∈(1,2],
ρ(Sx,Tx)+ρ(Sy,Ty)2=1,x,y∈[0,1)3/2,x=1, y∈[0,1)3/2,x=1, y∈(1,2]3/2,y=1, x∈[0,1)3/2,y=1, x∈(1,2]1,x,y∈(1,2],
ρ(Sx,Ty)+ρ(Sy,Tx)2=1,x,y∈[0,1)1,x=1, y∈[0,1)3/2,x=1, y∈(1,2]1/2,y=1, x∈[0,1)1,y=1, x∈(1,2]1,x,y∈(1,2].

Thus, we obtain

N(x,y)=1,x,y∈[0,1)2,x=1, y∈[0,1)3/2,x=1, y∈(1,2]2,y=1, x∈[0,1)3/2,y=1, x∈(1,2]1,x,y∈(1,2].

Finally, we compute the following quantities:

ψ(ρ(Sx,Sy))=0,x,y∈[0,1)1,x=1, y∈[0,1)0,x=1, y∈(1,2]1,y=1, x∈[0,1)0,y=1, x∈(1,2]0,x,y∈(1,2].

and

β(ψ(N(x,y)))ψ(N(x,y))=2/3,x,y∈[0,1)20/9,x=1, y∈[0,1)117/88,x=1, y∈(1,2]20/9,y=1, x∈[0,1)117/88,y=1, x∈(1,2]2/3,x,y∈(1,2].

Which allow us to conclude that S and T are generalized ψ-Geraghty-Zamfirescu mappings.

On the other hand, notice that any sequence (xn)⊂[0,1) satisfy that

limn→∞Sxn=1 and limn→∞Txn=0,

whereas, if (xn)⊂(1,2]

limn→∞Sxn=0 and limn→∞Txn=1.

For the case when (x_n) converges to 1 but with their terms (for n sufficiently large) oscillating between the set [0,1) and (1,2], the limits limn→∞Sxn and limn→∞Txn does not exist since we can extract subsequences (xn(k)) and (xn(l)) of (x_n) such that

limk→∞Sxn(k)=0 and liml→∞Sxn(l)=1,

as well as,

limk→∞Txn(k)=1 and liml→∞Txn(l)=0.

Therefore, the mappings S and T does not satisfy the b-property (EA).

Remark 3.2. Notice, from Remark 2.1, that we can drop the condition T(M) (S(M)) be closed in Proposition 3.2, if we assume that (T,S) satisfy the b-CLRT−property (resp. b-CLRS−property).

In this section, we will prove the b-convergence and the (S,T)-stability of the Jungck, Jungck-Mann and Jungck-Ishikawa iterative schemes for ψ-Geraghty-Zamfirescu pairs in the setting of b-metric spaces. To pose the Jungck-Mann and Jungck-Ishikawa iterations we endowed the b-metric space (M, ρ) with a convex structure.

4.1. Convergence of the Jungck Iteration

Let (M, ρ) be a b-metric space and S,T:N→M are two nonself mappings on a subset $N$ of $M$ such that S(N)⊂T(N), where T(N) is a complete subspace of M. For any x0∈N, the Jungck iterative sequence is defined by the formula

(4.1)yn=Sxn=Txn+1, n=0,1,…

We are going to prove that for generalized ψ-Geraghty-Zamfirescu pairs, the Jungck iteration defines a b-Cauchy sequence.

4.1. Convergence of the Jungck Iteration

Proposition 4.1. Let S, T be two selfmaps of a b-metric space (M, ρ) with s≥ 1, such that SM□ TM. Assume that (S, T) satisfies condition 3.2. Then, for each x₀∈ M, the Jungck sequence (4.1) satisfies:

• (i) limn→∞ρ(yn, yn+1)=0, and

• (ii) (y_n+1) is a b-Cauchy sequence in M.

Proof. To prove (i), let x₀ ∈ M be an arbitrary point, using the condition SM⊂TM, we construct the Jungck sequence defined by

yn=Sxn=Txn+1, n=0,1,…

From condition (3.2), we have

(4.2)ψ(s2ρ(yn, yn+1))=ψ[s2ρ(Sxn, Sxn+1)]        ≤β[ψ(ρ(N(xn, xn+1)))]ψ[N(xn, xn+1)],

where:

N(xn, xn+1)=maxρ(Txn, Txn+1), ρ(Sxn, Txn)+ρ(Sxn+1, Txn+1)2,    ρ(Sxn, Txn+1)+ρ(Sxn+1, Txn)2s    =maxρ(yn−1, yn), ρ(yn, yn−1)+ρ(yn+1, yn)2,    ρ(yn, yn)+ρ(yn+1, yn−1)2s.

Since,

ρ(yn−1, yn)+ρ(yn, yn+1)2≤max{ρ(yn−1, yn), ρ(yn, yn+1)}

and

ρ(yn−1, yn+1)2s≤s2s(ρ(yn−1, yn)+ρ(yn, yn+1))≤max{ρ(yn−1, yn), ρ(yn, yn+1)}.

Then, we conclude

N(xn, xn+1)≤max{ρ(yn−1, yn), ρ(yn, yn+1)}.

If N(xn, xn+1)≤ρ(yn, yn+1), using (4.2) we obtain

(4.3)ψ[ρ(yn, yn+1)]≤ψ[s2ρ(yn, yn+1)]≤β[ψ(ρ(yn, yn+1))]ψ(ρ(yn, yn+1))      <1sψ(ρ(yn, yn+1))<ψ(ρ(yn, yn+1)).

Since ψ∈Ψ, it follows that

ρ(yn, yn+1)<ρ(yn, yn+1),

which is a contradiction. Therefore, ρ(yn, yn+1)<ρ(yN(xn, xn+1)≤ρ(yn−1, yn)n, yn+1),. Using again (4.2), we get

(4.3)ψ[ρ(yn, yn+1)]≤ψ[s2ρ(yn, yn+1)]≤β[ψ(ρ(yn−1, yn))]ψ[ρ(yn−1, yn)]    <1sψ(ρ(yn−1, yn))<ψ(ρ(yn−1, yn)).

Thus, ρ(yn, yn+1)<ρ(yn−1, yn), which means that (ρ(yn, yn+1))n is a nondecreasing sequence of non-negative real numbers. Therefore, there exists L≥0 such that

limn→∞ρ(yn, yn+1)=L.

If we assume that L>0, from (4.3) we have

ψ(L)=limsupn→∞ψ(ρ(yn,yn+1))≤limsupn→∞ψ[s2ρ(yn,yn+1)]≤limsupn→∞β[ψ(ρ(yn−1,yn))]limsupn→∞ψ[ρ(yn−1,yn)].

It follows that

ψ(L)≤ limsupn→∞β[ψ(ρ(yn−1,yn))]ψ(L).

On the other hand, notice that

1s≤1=ψ(L)ψ(L)≤ limsupn→∞β[ψ(ρ(yn−1,yn))]<1/s.

Therefore, limn→∞β[ψ(ρ(yn−1,yn))]=1/s. This implies that limn→∞ψ(ρ(yn−1,yn))=0 and since ψ ∈ Ψ, we conclude that L=limn→∞ρ(yn−1,yn)=0, which is a contradiction. Therefore,

limn→∞ρ(yn,yn+1)=0.

Now, we want to show that (y_n) is a b-Cauchy sequence in M. We assume the contrary, that is, (y_n) is a not a b-Cauchy sequence. By Proposition 2.3, there exist an ε>0 and two sequences n(k) and m(k) with n(k)>m(k)>k>0 such that

ρ(ym(k),yn(k))≥0 and ρ(ym(k),yn(k)−1)<ε.

From (3.2), we have

(4.4)ψ[ρ(ym(k),yn(k))]≤ψ[s2ρ(ym(k),yn(k))]      =ψ[s2ρ(Sxm(k),Sxn(k))]      ≤β[ψ(N(xm(k),xn(k)))]ψ[N(xm(k),xn(k))].

Where,

N(xm(k),xn(k))=max{ρ(Txm(k),Txn(k)),      ρ(Sxm(k),Txm(k))+ρ(Sxn(k),Txn(k))2,      ρ(Sxm(k),Txn(k))+ρ(Sxn(k),Txm(k))2s      =maxρ(ym(k)−1,yn(k)−1),ρ(ym(k),ym(k)−1)+ρ(yn(k),yn(k)−1)2,      ρ(ym(k),yn(k)−1)+ρ(yn(k),ym(k)−1)2s.

Now, taking upper limit as k→∞ and using Proposition 2.3, we have

(4.5)limn→∞N(xm(k),xn(k))≤maxsε,0,ε+εs22s=εs.

Taking upper limit as k→∞ in (4.4) and using (4.5), we get

ψ(sε)≤ψ(s2ε)≤limsupk→∞ψ[s2ρ(ym(k),yn(k))]  ≤limsupk→∞ψ[s2ρ(Sxm(k),Sxn(k))]  ≤limsupk→∞β[ψ[N(xm(k),xn(k))]]limsupk→∞ψ[N(xm(k),xn(k))]  ≤β[ψ(sε)]ψ(sε)<1sψ(sε)≤ψ(sε),

which is a contradiction. Thus, we conclude that (y_n) is a b-Cauchy sequence in M.

Now, we prove that the Jungck iterative process converges to the POC of a ψ-Geraghty-Zamfirescu contraction pair.

Proposition 4.2. Let (M, ρ) be a b-metric space with s≥ 1 and let S,T:M→M be two selfmappings with SM⊂TM. Assume that S and T are generalized ψ-Geraghty-Zamfirescu contraction mappings. If TM□ M is a complete subspace of M. Then, for any x₀ ∈ M, the Jungck sequence defined in (4.1) converges to z∈ M, C(S,T)≠{∅} and PC(S,T)={z}.

Proof. By Proposition 4.1, the Jungck sequence (y_n) is a b-Cauchy sequence in M. Then, (Txn+1)⊂TM is a b-Cauchy sequence and since TM is complete, we have

limn→∞yn=limn→∞Sxn=limn→∞Txn+1=Tu, for some u∈M.

We have been prove that the Jungck sequence (y_n) converges to some z∈ M. The rest of the proof runs as the proof of Proposition 3.2..

4.2. Convergence of the Jungck-Mann and Jungck-Ishikawa Iterations

We recall that in 1970, W. Takahashi [22] introduce the notion of convex structure on metric spaces, which is a natural generalization of convexity in normed linear spaces as A. A. Abdelhakim establishes in his 2016 paper [2]. In 2012, M. O. Olatinwo and M. Postolache [15] proved some stability results in metric space endowed with this structure for Jungck–Mann and Jungck–Ishikawa iterations for nonselfmappings satisfying certain general contractivity condition. Later on, in 2013, A. Razani and M. Bagherboum [18] reintroduce this concept in the framework of b-metric spaces and proved the convergence and stability of Jungck-type interative proceduces in this setting.

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

ρ(z,W(x,y,λ))≤λρ(z,x)+(1−λ)ρ(z,y).

A b-metric space (M,ρ) equipped with the convex structure W is called a convex b-metric space, and it is denoted by (M,ρ,W).

The b-metric space given in Example 2.1(1) is a convex b-metric space with convex structure W(x,y,λ)=λx+(1−λ)y.

From now on, it is assumed that (M,ρ,W) is a convex b-metric space with parameter s and that S,T:N→M are two nonself mappings on a subset N of M such that S(N)⊂T(N), where T(N) is a complete subspace of M.

For any x0∈M, let (x_n) be the sequence generated by the so-called Jungck-Mann iterative procedure:

Txn+1=W(Txn,Sxn,αn),  n=0,1,…

where (α_n) is a sequence of real numbers such that 0≤αn≤1.

Theorem 4.3. Let (M,ρ,W) be a convex b-metric space with s≥1, and let S,T:N→M be generalized ψ-Geraghty-Zamfirescu contraction mappings having a point of coincidence and ψ∈Ψ be a convex function. Let (α_n) be a real sequence in [0,1] such that ∑ k=0∞s+1−(s−1)αks=∞. Then, for any x₀∈ N, the sequence defined by the Jungck-Mann iterative process converges to the point of coincidence of S and T.

Proof. Let z be the unique coincidence point of (S,T), i.e., Sz=Tz=p. We are going to prove that the Jungck-Mann iterative process converges to p. Notice that

ρ(Txn+1,p)=ρ(W(Txn,Sxn,αn),p)≤αnρ(Txn,p)+(1−αn)ρ(Sxn,p).

Since ψ is increasing and convex, using (3.2), we obtain

(4.6)ψ(sρ(Txn+1,p))≤αnψ(sρ(Txn,p))+(1−αn)ψ(sρ(Sxn,p))      ≤αnψ(sρ(Txn,p))+(1−αn)ψ(s2ρ(Sxn,Sz))      <αnψ(sρ(Txn,p))+(1−αn)sψ(N(xn,z))

where,

N(xn,z)=maxρ(Txn,Tz),ρ(Sxn,Txn)+ρ(Sz,Tz)2,ρ(Sxn,Tz)+ρ(Sz,Txn)2s    =maxρ(Txn,p),ρ(Sxn,Txn)2,ρ(Sxn,p)+ρ(Txn,p)2s.

Note that

ρ(Sxn,Txn)2≤sρ(Sxn,p)+ρ(Txn,p)2≤smax{ρ(Sxn,p),ρ(Txn,p)}

and

ρ(Sxn,p)+ρ(Txn,p)2s≤sρ(Sxn,p)+ρ(Txn,p)2≤smax{ρ(Sxn,p),ρ(Txn,p))}.

Therefore,

N(xn,z)≤smax{ρ(Sxn,p),ρ(Txn,p)}.

Now, if ρ(Sxn,p)≥ρ(Txn,p), from condition (3.2), we get

ψ(sρ(Sxn,Sz))≤ψ(s2ρ(Sxn,Sz))      ≤β(ψ(N(xn,z)))ψ(N(xn,z))      <1sψ(sρ(Sxn,Sz)),

a contradiction. Hence, ψ(N(xn,z))≤ψ(sρ(Txn,p)). In this way, estimate (4.6) is now given by

(4.7)ψ(sρ(Txn+1,p))<αnψ(sρ(Txn,p))+(1−αn)sψ(sρ(Txn,p))      =αn+1−αnsψ(sρ(Txn,p)).

On the other hand, note that

αn+1−αns=1−1−(s−1)αn+1s.

Moreover, since 0≤αn≤1, then (s−1)αn≤s−1, or equivalently, s−1sαn≤1−1s. I.e.,

1−(s−1)αn+1s≥0.

Now, recursively, from (4.7), we obtain

ψ(sρ(Txn+1,p))<1−1−(s−1)αn +1sψ(sρ(Txn,p))      ⋮      <∏ k=0 n1−1−(s−1)αk +1sψ(sρ(Tx0,p)).

Taking lim sup as n→∞, using the fact that ∑ k=0∞s+1−(s−1)αns=∞, we conclude

that

limn→∞ψ(sρ(Txn+1,p))=0.

The condition ψ∈Ψ implies that limn→∞ρ(Txn+1,p)=0. Thus, we obtain the conclusion.

Now, for any x₀∈ M, the sequence (x_n) generated by the following iterative two-step process:

Txn+1=W(Txn,Syn,αn), Tyn=W(Txn,Sxn,βn),  n=0,1,…

where (α_n) and (β_n) are sequences of real numbers such that 0≤αn,βn≤1, is called the Jungck-Ishikawa iterative procedure.

In the next result we prove the convergence of the Jungck-Ishikawa iterative scheme for generalized ψ-Geraghty-Zamfirescu contraction mappings.

Theorem 4.4. Let (M,ρ,W) be a convex b-metric space with s≥1, and let S,T:N→M be generalized ψ-Geraghty-Zamfirescu contraction mappings having a point of coincidence and ψ∈Ψ be a convex function. Let (α_n) and (β_n) be a real sequences in [0,1] such that ∑ k=0∞s+1−(s−1)αks=∞. Then, for any x₀∈ N, the sequence defined by the Jungck-Ishikawa iterative process converges to the point of coincidence of S and T.

Proof. Let z be the unique coincidence point of (S,T), i.e., Sz=Tz=p. Now,

ψ(sρ(Txn+1,p))=ψ(sρ(W(Txn,Syn,αn),p))      ≤αnψ(sρ(Txn,p))+(1−αn)ψ(sρ(Syn,p))      ≤αnψ(sρ(Txn,p))+(1−αn)ψ(s2ρ(Syn,Sz))      <αnψ(sρ(Txn,p))+1−αnsψ(N(yn,z)).

As in the proof of Theorem 4.3, we conclude that

ψ(sρ(Txn+1,p))<αnψ(sρ(Txn,p))+1−αnsψ(sρ(Tyn,p)).

Now, as above, we have

ψ(sρ(Tyn,p))=ψ(sρ(W(Txn,Sxn,βn),p))      ≤βnψ(sρ(Txn,p))+(1−βn)ψ(sρ(Sxn,p))      ≤βnψ(sρ(Txn,p))+(1−βn)ψ(s2ρ(Sxn,Sz))      <βnψ(sρ(Txn,p))+1−βnsψ(N(xn,z))      ≤βnψ(sρ(Txn,p))+1−βnsψ(sρ(Txn,p)).

Thus, from the previous estimates we obtain

ψ(sρ(Txn+1,p))<αn+1−αnsβn+(1−αn)(1−βn)s2ψ(sρ(Txn,p))      ≤αn+1−αnsβn+(1−αn)(1−βn)sψ(sρ(Txn,p))      =αn+1−αnsψ(sρ(Txn,p)),

which is the estimate (4.7). The conclusion then is obtained following the proof of Theorem 4.3.

4.3. Stability Results

A convergent sequence (x_n) generated by an iterative process is said numerically stable iff a sequence (y_n) approximately close to (x_n) converges to the same limit. A. S. Ostrowski [16] appears to be the first to discuss the stability of iterative procedures on metric spaces, since then, the stability theory has extensively been studied. In 2005, S. L. Singh, C. Bhatnagar and S. N. Mishra [20] introduce the notion of the stability of iterative procedures for a pair of self-maps of a metric space (M,d) and develop the theory for this kind of procedures.

{\bf Definition 4.2.} Let (M,ρ,W) be a convex b-metric space, let N be a subset of M, and let S,T:N→M be such that S(N)⊂T(N). For any x₀∈ N, let the sequence (Tx_n) generated by the iterative procedure

(4.8)Txn+1=f(Txn,Sxn,αn), 0≤αn≤1, n=0,1,2,…

converging to p. Also, let (Tyn)⊂M be an arbitrary sequence and let

εn=ρ(Tyn+1,f(Tyn,Syn,αn)).

The iterative process (4.8) will be called (S,T)-stable if limn→∞εn=0 implies that limn→∞Tyn=p.

Notice that f(Txn,Sxn,αn)≡Sxn corresponds to the Jungck iteration, f(Txn,Sxn,αn)≡W(Txn,Sxn,αn) to the Jungck-Mann process and if we consider f(Txn,Sxn,αn)≡W(Txn,Syn,αn), with Tyn=W(Txn,Sxn,βn) then, it the corresponds to the Jungck-Ishikawa iterative scheme.

In order to prove the (S,T)-stability of the Jungck, Jungck-Mann and Jungck-Ishikawa iterative schemes we assume that s>2, and we will use the following fact concerning to recurrent inequalities. See, e.g., Lemma 1.6 of [5].

Lemma 4.5. Let (a_n), (b_n) be sequences of nonnegative numbers and 0 ≤ q <1, so that

an+1≤qan+bn, for all n≥0.

If limn→∞bn=0, then limn→∞an=0.

Theorem 4.6. Let (M,ρ) be a metric space with s>2. Let S,T:M→M be generalized ψ-Geraghty-Zamfirescu mappings and let ψ ∈ Ψ be a convex subadditive function. Let z be a coincidence point of S and T, that is, Sz=Tz=p. Let x₀∈ M and suppose the sequence (Tx_n) generated by the Jungck iteration Txn+1=Sxn, n=0,1,… converges to p. Then, for any arbitrary sequence (Tyn)⊂M and εn as in Definition 4.2,

limn→∞Tyn=p, if and only if limn→∞εn=0.

Proof. Let (Ty_n) be an arbitrary sequence and define

εn=ρ(Tyn+1,Syn), n=0,1,…

Using the s-triangle inequality we get,

ρ(Tyn+1,p)≤s[ρ(Tyn+1,Syn)+ρ(Syn,p)]=sεn+sρ(Syn,Sz).

Since ψ∈Ψ is subadditive, we have

(4.9)ψ(ρ(Tyn+1,p))≤ψ(sεn)+ψ(sρ(Syn,Sz))      ≤ψ(sεn)+ψ(s2ρ(Syn,Sz)).

Now, from condition (3.2) we have

(4.10)ψ(s2ρ(Syn,Sz))≤β(ψ(N(yn,z)))ψ(N(yn,z))      <1sψ(N(yn,z)),

where,

N(yn,z)=maxρ(Tyn,Tz),ρ(Syn,Tyn)+ρ(Sz,Tz)2,ρ(Syn,Tz)+ρ(Sz,Tyn)2s    =maxρ(Tyn,Tz),12ρ(Syn,Tyn),ρ(Syn,Tz)+ρ(Sz,Tyn)2s.

Note that

12ρ(Syn,Tyn)≤s2[ρ(Tyn,Tz)+ρ(Tz,Syn)].

Also,

ρ(Syn,Tz)+ρ(Sz,Tyn)2s≤s2[ρ(Tyn,Tz)+ρ(Tz,Syn)].

Moreover, the condition s>2 implies that

ρ(Tyn,Tz)≤s2[ρ(Tyn,Tz)+ρ(Tz,Syn)].

Thus, we conclude that

N(yn,z)≤s2[ρ(Tyn,Tz)+ρ(Tz,Syn)].

With this upper bound, estimate (4.10) takes the form

ψ(s2ρ(Syn,Sz))<12s[ψ(sρ(Tyn,Tz))+ψ(sρ(Tz,Syn))]      ≤12s[ψ(sρ(Tyn,Tz))+ψ(s2ρ(Tz,Syn))].

Since Tz=Sz=p, we get

ψ(s2ρ(Syn,Sz))<12s−1ψ(sρ(Tyn,Tz)).

Thus, inequality (4.9) is now given by

ψ(ρ(Tyn+1,p))<ψ(sεn)+12s−1ψ(sρ(Tyn,p)).

Notice that a subadditive increasing function ψ satisfies that ψ(sa)≤sψ(a), where s is the smallest integer greater or equal to s. Therefore,

ψ(ρ(Tyn+1,p))<ψ(sεn)+s2s−1ψ(ρ(Tyn,p)).

Taking into account that s>2 and

s≤s,if s∈ℤ+s+1,if s∈ℝ+∖ℤ+

we conclude that s/(2s−1)<1. Therefore, from Lemma 4.5 we conclude that

limn→∞ψ(ρ(Tyn,p))=0, if limn→∞ψ(sεn)=0.

Since ψ∈Ψ, this is equivalent to

limn→∞ρ(Tyn,p)=0, if limn→∞εn=0.

On the other hand,

0≤εn=ρ(Tyn+1,Syn)  ≤sρ(Tyn+1,p)+sρ(Syn,p)  <sρ(Tyn+1,p)+s2s−1ρ(Tyn,p).

Using that ψ∈Ψ is a subadditive function, we get

0≤ψ(εn)<ψ(sρ(Tyn+1,p))+sψ12s−1ρ(Tyn,p).

Taking limits as n→∞, we have

0≤limn→∞ψ(εn)<limn→∞ψ(sρ(Tyn+1,p))+slimn→∞ψ12s−1ρ(Tyn,p).

Therefore, we conclude that

limn→∞ψ(εn)=0, equivalently, limn→∞εn=0

whenever,

limn→∞ψ(sρ(Tyn,p))=0, equivalently, limn→∞Tyn=p,

which completes the proof.

To prove the (S,T)-stability of the Jungck-Mann and Jungck-Ishikawa iterations we will assume, in addition to s>2, that there exists q∈[0,1) such that the sequence (α_n) satisfy that

(4.11)0≤αn≤(2s−1)q−s2s(s−1), for all n=0,1,2,…

where s is the smallest integer greater or equal to s and

(4.12)s2s−1<q<1.

Notice that if (4.12) holds, then

0<(2s−1)q−s2s(s−1)<1.

In fact,

s2s−1<q⇔0<(2s−1)q−s    ⇔ 0<(2s−1)q−s2s(s−1).

On the other hand,

q<1⇔ (2s−1)q<2s−1≤s(2s−2)+s  ⇔ (2s−1)q2s(s−1)<1+s2s(s−1)  ⇔ (2s−1)q−s2s(s−1)<1.

Thus, under this condition, (αn)⊂[0,1] for all n=0,1,…

Theorem 4.7. Let (M,ρ,W) be a convex metric space with s>2. Let S,T:M→M be generalized ψ-Geraghty-Zamfirescu mappings and let ψ∈Ψ be a convex subadditive function. Let z be a coincidence point of S and T, that is, Sz=Tz=p. Let x₀∈ M and suppose that the sequence (Tx_n) generated by the Jungck-Mann iteration Txn+1=W(Txn,Sxn,αn), (n=0,1,…) with (α_n) satisfying (4.11) converges to p. Then, for any arbitrary sequence (Tyn)⊂M and εn as in Definition 4.2,

limn→∞Tyn=p, if and only if limn→∞εn=0.

Proof. Let (Tyn)⊂M be an arbitrary sequence. From the s-triangle inequality and the convex structure we have

ρ(Tyn+1,p)≤s[ρ(Tyn+1,W(Tyn,Syn,αn))+ρ(W(Tyn,Syn,αn),p)]    ≤sεn+s[αnρ(Tyn,p)+(1−αn)ρ(Syn,p)].

Since ψ∈Ψ is convex and subadditive, we get

ψ(ρ(Tyn+1,p))≤ψ(sεn)+αnψ(sρ(Tyn,p))+(1−αn)ψ(sρ(Syn,p)).

Now, due to the fact that p=Tz=Sz, for some z∈ M, in Theorem 4.6 we prove that

ψ(sρ(Syn,p))<12s−1ψ(sρ(Tyn,p)).

Therefore,

(4.13)ψ(ρ(Tyn+1,p))<ψ(sεn)+αnψ(sρ(Tyn,p))+1−αn2s−1ψ(sρ(Tyn,p))      ≤ψ(sεn)+αn+1−αn2s−1sψ(ρ(Tyn,p)),

where s is the smallest integer greater or equal to s. Now, since

αn≤(2s−1)q−s2s(s−1)=2s−12(s−1)qs−12(s−1)

then,

12(s−1)+αn≤2s−12(s−1)qs.

Equivalently,

12s−1+2(s−1)2s−1αn=αn+1−αn2s−1≤qs.

Hence, we have

ψ(ρ(Tyn+1,p))<ψ(sεn)+qψ(ρ(Tyn,p)).

Thus, from Lemma 4.5, we conclude that

limn→∞ψ(ρ(Tyn,p))=0, equivalently, limn→∞ρ(Tyn,p)=0

if

limn→∞ψ(sεn)=0, equivalently, limn→∞εn=0.

On the other hand,

0≤εn=ρ(Tyn+1,W(Tyn,Syn,αn))  ≤sρ(Tyn+1,p)+sρ(W(Tyn,Syn,αn),p)  ≤sρ(Tyn+1,p)+s[αnρ(Tyn,p)+(1−αn)ρ(Syn,p)].

Now,

0≤ψ(εn)≤ψ(sρ(Tyn+1,p))+αnψ(sρ(Tyn,p))+(1−αn)ψ(sρ(Syn,Sz))

with

ψ(sρ(Syn,Sz))<12s−1ψ(sρ(Tyn,p)).

Thus,

0≤ψ(εn)<ψ(sρ(Tyn+1,p))+αnψ(sρ(Tyn,p))+1−αn2s−1ψ(sρ(Tyn,p)).

In this way, using the fact that ψ∈Ψ, we conclude that

if limn→∞Tyn=p, then limn→∞εn=0.

This completes the proof.

Theorem 4.8. Let (M,ρ, W) be a convex metric space with s>2. Let S,T:M→M be generalized ψ-Geraghty-Zamfirescu mappings and let ψ∈Ψ be a convex subadditive function. Let z be a coincidence point of S and T, that is, Sz=Tz=p. Let x₀∈ M and suppose that the sequence (Tx_n) generated by the Jungck-Ishkawa iteration

Txn+1=W(Txn,Syn,αn) Tyn=W(Txn,Sxn,βn),  n=0,1,…

with (αn),(βn)⊂[0,1] and (α_n) satisfying (4.11), converges to p. Then, for any arbitrary sequence (Tzn)⊂M and εn as in Definition 4.2,

limn→∞Tzn=p, if and only if limn→∞εn=0.

Proof. Let (Tzn)⊂M be an arbitrary sequence. From the s-triangle inequality we have

ρ(Tzn+1,p)≤s[ρ(Tzn+1,W(Tzn,Swn,αn))+ρ(W(Tzn,Swn,αn),p)],

where Twn=W(Tzn,Szn,βn). Then, from the convex structure we obtain

ρ(Tzn+1,p)≤sε+αnsρ(Tzn,p)+(1−αn)sρ(Swn,p).

Since ψ∈Ψ is convex and subadditive, we get

ψ(ρ(Tzn+1,p))≤ψ(sε)+αnψ(sρ(Tzn,p))+(1−αn)ψ(sρ(Swn,p)).

From the proof of Theorem 4.7 we known that

ψ(sρ(Swn,p))<12s−1ψ(sρ(Twn,p)),

then, we have the following estimate

ψ(sρ(Twn,p))=ψ(sρ(W(Tzn,Szn,βn),p))      ≤βnψ(sρ(Tzn,p))+(1−βn)ψ(sρ(Szn,p))      <βnψ(sρ(Tzn,p))+1−βn2s−1ψ(sρ(Tzn,p))      ≤βnsψ(ρ(Tzn,p))+1−βn2s−1sψ(ρ(Tzn,p)).

Therefore, we obtain

ψ(ρ(Tzn+1,p))<ψ(sε)+αnsψ(ρ(Tzn,p))+1−αn2s−1βns+1−βn2s−1sψ(ρ(Tzn,p))≤ψ(sε)+αnsψ(ρ(Tzn,p))+1−αn2s−1βns+(1−βn)sψ(ρ(Tzn,p))=ψ(sε)+αnsψ(ρ(Tzn,p))+1−αn2s−1sψ(ρ(Tzn,p))

which is the same estimate (4.13). The conclusion is obtained by repeating the arguments of the proof of Theorem 4.7.