Cone Metric Space

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Common Fixed Point Theorems for T -Contractive Mappings in D

-Generalized

Cone Metric Space

M. Bousselsal¹ and M.S. Jazmati²

1Laboratoire d’Analyse Nonlineaire et H.M. ENS Department of Mathematics

16050, Vieux-Kouba, Algiers, Algeria E-mail: [email protected]

2Qassim University, College of Science Department of Mathematics P.O. Box 6644, Bouraida, 51452, KSA

E-mail: [email protected] (Received: 26-2-14 / Accepted: 29-3-14)

Abstract

In this note, we prove common fixed point for a Banach pair of mappings on D^∗-Generalized Cone Metric Space.

Keywords: Cone metric space, common fixed point, contractive mappings, sequentially convergent, Banach operator pair.

1 Introduction and Preliminaries

Recently Huang and Zhang [6] generalized the concept of metric spaces re- placing the set of real numbers by an ordered Banach space defining in this way a cone metric space.They have defined convergent, Cauchy sequence in terms of interior points of the underlying Cone. They later proved some fixed point theorems for different contractive mappings. Their results have been generalized and extended by several authors see for instance [11], [7], [3].

Dhage in [4] defined D−metric space as a generalization of metric space and claimed that D−metric defines a Hausdorff topology and D−metric is sequentially continuous with respect to all three variables . He proved some

results on fixed points for a self-map satisfying a contraction for complete and bounded complete metric spaces. In 2003, Zead Mustafa and Brailey Sims [9]

introduced a new structure of generalized metric spaces, which are called G- metric spaces. Recently Shaban Sedghi et al [12] introduced D^∗-metric which is probable modification of the definition ofD−metric space and prove some basic properties in D^∗-metric space and some results on common fixed point theorems. In 2010,C.T Aage and J.N. Salunke [1] introducd generalized D^∗- metric by replacing Rby Banach space in D^∗-metric Spaces.

In [2], A.Beiranvand, S. Moradi et al, introduced a new class of contrac- tive mapping T− contraction and T− Contractive extending the Banach’s contraction principle and the Edelstein’s fixed point theorem.

In the following we always suppose that E is a Banach space, P is a cone inE with intP 6=φ and ≤is a partial ordering with respect to P

In this section we recall some definitions of coneD^∗-metric space and some of their properties see [1] for more details.

Definition 1 let E be a real Banach space andP a subset ofE. The set P is called a cone if and only if

1. P is closed, nonempty andP 6={0}

2. ∀a, b∈R+, x, y ∈P impliesax+by ∈P

3. x∈P and −x∈P then x= 0 that is P ∩(−P) ={0}

Given a cone P ⊂ E, we define a partial ordering ≤ with respect to P by x≤ y ⇐⇒ y−x∈ P. We write x < y and x 6=y. We shall write x y if and only ify−x∈ intP where intP denotes the interior of P.

Definition 2 let E be a real Banach space and P a subset of E.The cone P is called normal if there is a numberk > 0 such that for all

x, y ∈E, 0≤x≤y implies kxk ≤kkyk

The least positive constant k satisfying the inequality above is called the normal constant of P.

The coneP is called regular if every increasing sequence which is bounded from above is convergent. That is , if{x_n}_n≥1is a sequence such thatx₁ ≤x₂ ≤

· · · ≤x_n ≤ · · · ≤y for y∈E, then there is x∈E such that limkx_n−xk= 0 as n −→ ∞.

LetX be a nonempty set. A generalized metric (or D^∗-metric) onX is a function: D^∗ : X³ → E that satisfies the following conditions for each for all (x, y, z, a) ∈X⁴ .

1. D^∗(x, y, z)≥0

2. D^∗(x, y, z) = 0 if and only if x=y=z

3. D^∗(x, y, z) = D^∗(P {x, y, z}), (symmetry) where P is a permutation function

4. D^∗(x, y, z)≤D^∗(x, y, a) +D(a, z, z)

Example 3 Let E = R², P = {(x, y)∈E :x, y ≥0 }, X = R and D^∗ : X×X×X −→E defined by

D^∗(x, y, z) = (|x−y|+|y−z|+|z−x|, α(|x−y|+|y−z|+|z−x|)) where α∈R+. Then (X, D^∗) is a generalized cone D^∗- metric space.

Proposition 4 [1] If (X, D^∗) be generalized cone D^∗- metric space, then for all x, y, z∈X,we have D^∗(x, x, y) =D^∗(x, y, y)

Definition 5 Let (X, D^∗) be a generalized cone D^∗- metric space. Let {x_n} be a sequence in X and x ∈ X. If for every c 0 there is N ∈ N such that for allm, n≥N, D^∗(xm, xn, x)c, then {xn}is said to be convergent, {xn} converges to x and x is the limite of {x_n}. We denote this convergence by x_n →x (as n → ∞).

Lemma 6 [1] Let (X, D^∗) be generalized cone D^∗- metric space, P be a nor- mal cone with normal constant k. Let {xn} be a sequence in X. Then {xn} converges to x if and only if D^∗(x_m, x_n, x)→0 (as n, m→ ∞).

Lemma 7 [1] Let (X, D^∗)be a generalized cone D^∗-metric space, P be a nor- mal cone with normal constant k. Let {x_n} be a sequence in X . If {x_n} converges to xand if {xn}converges to y,then x=y, that is the limit of{xn} if there exists, is unique.

Definition 8 Let(X, D^∗)be a generalized coneD^∗-metric space, P be a nor- mal cone with normal constant k. {x_n} be a sequence in X. If for every c0, c∈E there isN ∈Nsuch that for allm, n, l≥N, D^∗(xm, xn, xl)c, then {x_n} is called a Cauchy sequence in X.

Definition 9 Let (X, D^∗) be a generalized cone D^∗- metric space. If every Cauchy sequence in X is convergent in X, then X is called a complete gen- eralized cone D^∗- metric space.

Lemma 10 [1] Let (X, D^∗) be a generalized cone D^∗-metric space, {xn} be a sequence inX. If {x_n} converges to x, then {x_n} is a Cauchy sequence .

Lemma 11 [1] Let (X, D^∗) be generalized cone D^∗- metric space, P be a normal cone with normal constantk. Let{x_n}be a sequence in X. Then{x_n} is a Cauchy sequence if and only if D^∗(xm, xn, xl)→0 (as n, m, l→ ∞) Definition 12 Let (X, D^∗) be a generalized cone D^∗- metric space. Then a functionf :X →X⁰ is said to be D^∗continuous at a pointx∈X if and only if it isD^∗- sequentially continuous atx, that is whenever {x_n} is D^∗-convergent to x, we have {f x_n} is D^∗-convergent to f x.

Definition 13 [1] Let (X, D^∗) be a generalized cone D^∗- metric space, P be a normal cone with normal constantk.Let{x_n},{y_n},{z_n}be three sequences in X and xn→x, yn→y, xn →z (as n → ∞). Then

D^∗(x_n, y_n, z_n)→D^∗(x, y, z) (as n→ ∞)

Definition 14 Let (X, D^∗) be a generalized cone D^∗- metric space, P be a normal cone with normal constant k and T :X →X. Then

1. T is said to be sequentially convergent if we have, for every sequence (y_n),if T(y_n) is convergent, then (y_n) is also convergent.

2. T is said to be subsequentially convergent if we have, for every sequence (y_n),if T(y_n) is convergent, then (y_n) has a convergent subsequence.

Definition 15 LetT be a self mapping of a normed spaceX. ThenT is called a Banach operator of type k if

T²x−T x

≤kkT x−xk for somek ≥0 and all x∈X.

This concept was introduced by Subrahmanyam [10], then Chen and Li [5]

extended this as following:

Definition 16 Let f and T be self mappings of a nonempty subset M of a normed linear space X. Then (f, T) is a Banach operator pair, if one of the following conditions is satisfied:

1. f[F(T)]⊆F(T)

2. T f x =f x for each x∈F(T) 3. T f x =f T x for each x∈F(T)

4. kf T x−T xk ≤kkT x−xk for some k ≥0.

where F(T) denotes the set of fixed points of T.

Lemma 17 If (X, D^∗) be a generalized compact cone metric space, then ev- ery function T : X → X is subsequentially convergent and every continuous function T :X →X is sequentially convergent.

In this section, we introduce the notions of T−contraction and we extend the Banach Contraction principle and Edelstein point theorem given in [2]

Definition 18 Let (X, D^∗) be generalized cone D^∗- metric space and T, S : X→X two functions. A mapping S is said to be a T−contraction if there is α∈[0,1[ constant such that

D^∗(T Sx, T Sy, T Sz)≤αD^∗(T x, T y, T z) for all x, y, z ∈X.

Example 19 Let E = C_[0,1], R

, P = {ϕ∈E :ϕ≥0 } ⊂ E, X = R and D^∗ :X×X×X −→E defined by

D^∗(x, y, z) = (|x−y|+|y−z|+|z−x|)ϕ

where ϕ(t) = e^−t ∈ E. Then (X, D^∗) is a generalized cone D^∗- metric space. We consider the functions T, S : X → X defined by T(x) = exp(−x) and S(x) = 2x+ 1. Then

1. It is clear that S is not a contraction.

2. S is a T−contraction. Indeed

D^∗(T Sx, T Sy, T Sz) =





|T Sx−T Sy|

+|T Sy−T Sz|

+|T Sz−T Sx|



e^−t

= 1 e





|e^−x−e^−y| |e^−x+e^−y| +|e^−y −e^−z| |e^−y+e^−z| +|e^−z−e^−x| |e^−z+e^−x|



e^−t

≤ 2 e





|e^−x−e^−y| +|e^−y −e^−z| +|e^−z−e^−x|



e^−t

= 2

eD^∗(T x, T y, T z)

2 Main Results

The following theorems are the main results of this paper

Theorem 20 Let (X, D^∗) be a generalized complete cone D^∗- metric space , P be a normal cone with constant normalK and letT, S :X →X be two con- tinuous mappings. Assume thatT is injective and subsequentially convergent.

If T and S satisfy

D^∗(T Sx, T Sy, T Sz)≤a₁D^∗(T x, T y, T z) +a₂D^∗(T x, T Sx, T Sx) +

≤a₃D^∗(T x, T Sx, T z) +a₄D^∗(T y, T Sy, T Sy) +

≤a₅D^∗(T x, T Sy, T Sy) +a₆D^∗(T x, T Sz, T Sz) +

≤a₇D^∗(T y, T Sz, T Sz) +a₈D^∗(T x, T Sy, T Sz) +

≤a9D^∗(T x, T y, T Sz) +a10D^∗(T Sx, T Sx, T Sz) for all x, y, z ∈X, where a_i, i∈ {1,2,3,· · ·,10} are all nonnegative constants such that

a₁+ a₂+a₃+a₄+ 3a₅+ 3a₆+a₇+ 3a₈+a₉+a₁₀<1

then S has a unique fixed point in X. Moreover, if (T, S) is a Banach pair, then T and S have a unique common fixed point in X.

Proof. : Let x₀ ∈ X be arbitrary. Define a sequence {x_n} in X such that x_n+1 =Sx_n; for each n = 0,1,2,· · · ,∞. Consider

D^∗(T xn, T xn+1, T xn+1)≤a1D^∗(T xn−1, T xn, T xn) +a₂D^∗(T xn−1, T x_n, T x_n) +a₃D^∗(T xn−1, T x_n, T x_n) +a₄D^∗(T x_n, T x_n+1, T x_n+1) +a₅D^∗(T xn−1, T x_n+1, T x_n+1) +a₆D^∗(T x_n−1, T x_n+1, T x_n+1) +a7D^∗(T xn, T xn+1, T xn+1) +a₈D^∗(T xn−1, T x_n+1, T x_n+1) +a₉D^∗(T xn−1, T x_n, T x_n+1) +a₁₀D^∗(T x_n, T x_n, T x_n+1)

D^∗(T x_n, T x_n+1, T x_n+1)≤(a₁+a₂+a₃)D^∗(T xn−1, T x_n, T x_n) (1) + (a₄ +a₇ +a₁₀)D^∗(T x_n, T x_n, T x_n+1)

+ (a₅ +a₆ +a₈)D^∗(T xn−1, T x_n+1, T x_n+1) +a₉D^∗(T xn−1, T x_n, T x_n+1)

then by rectangle inequality

D^∗(T xn−1, T xn, T xn+1)≤D^∗(T xn−1, T xn+1, T xn) (2) +D^∗(T x_n, T x_n+1, T x_n+1)

and

D^∗(T xn−1, T xn+1, T xn)≤D^∗(T xn−1, T xn, T xn) +D^∗(T x_n, T x_n+1, T x_n+1) therefore from (2) we have

D^∗(T xn−1, T x_n+1, T x_n+1)≤2D^∗(T x_n, T x_n+1, T x_n+1) +D^∗(T xn−1, T x_n, T x_n) so

D^∗(T xn, T xn+1, T xn+1)≤(a1+a2+a3)D^∗(T xn−1, T xn, T xn) (3) + (a₄+a₇+a₁₀)D^∗(T x_n, T x_n, T x_n+1)

+ (a₅+a₆+a₈)

2D^∗(T x_n, T x_n+1, T x_n+1) +D^∗(T xn−1, T x_n, T x_n)

+a₉(D^∗(T xn−1, T x_n, T x_n) +D^∗(T x_n, T x_n+1, T x_n+1)) hence

D^∗(T xn, T xn+1, T xn+1)≤(a1+a2+a3 +a9 +a5+a6+a8) D^∗(T xn−1, T x_n, T x_n)

+ (a₄+a₇+a₁₀+ 2a₅+ 2a₆+ 2a₈) D^∗(T x_n, T x_n, T x_n+1)

since

D^∗(x, x, y) = D^∗(x, y, y) for all x, y, z ∈X (4) then

(1−(a₄+a₇+a₁₀+ 2a₅+ 2a₆+ 2a₈))D^∗(T x_n, T x_n+1, T x_n+1)

≤(a1+a2 +a3+a9+a5+a6+a8)D^∗(T xn−1, T xn, T xn) hence

D^∗(T x_n, T x_n+1, T x_n+1)≤ a1 +a2 +a3+a9+a5+a6+a8

1−(a₄ +a₇+a₁₀+ 2a₅ + 2a₆+ 2a₈)D^∗(T xn−1, T x_n, T x_n) This implies

D^∗(T x_n, T x_n+1, T x_n+1)≤qD^∗(T xn−1, T x_n, T x_n)

where q= _1−(a^{a1+a2+a3+a9+a5+a6+a8}

4+a7+a10+2a5+2a6+2a8) , then q∈[0,1[.

by repeated the above inequality we obtain,

D^∗(T x_n, T x_n+1, T x_n+1)≤qⁿD^∗(T x₀, T x₁, T x₁) (5) Then, for all n, m ∈ N, n < m we have by repeated use the rectangle inequality and inequality (5) that:

D^∗(T x_n, T x_m, T x_m)≤D^∗(T x_n, T x_n, T x_n+1) +D^∗(T x_n+1, T x_n+1, T x_n+2) +D^∗(T x_n+2, T x_n+2, T x_n+3) +· · ·

+D^∗(T x_m−1, T x_m, T x_m )

≤(qⁿ+· · ·+q^m)D^∗(T x0, T x1, T x1)

≤ qⁿ

1−qD^∗(T x₀, T x₁, T x₁) hence

kD^∗(T xn, T xm, T xm)k ≤K qⁿ

1−q kD^∗(T x0, T x1, T x1)k

SinceK_1−q^qn kD^∗(T x₀, T x₁, T x₁)k →0, as n, m→ ∞ then, it follows that D^∗(T x_n, T x_m, T x_m )→0,as n, m→ ∞.For n, m, l∈N and from

D^∗(T x_n, T x_m, T x_l)≤D^∗(T x_n, T x_m, T x_m) +D^∗(T x_m, T x_l, T x_l) we have

kD^∗(T x_n, T x_m, T x_l)k ≤K(kD^∗(T x_n, T x_m, T x_m)k+kD^∗(T x_m, T x_l, T x_l)k) Taking the limit asn, m, l→ ∞,we get D^∗(T x_n, T x_m, T x_l)→0.So{T x_n} is D^∗- Cauchy sequence, since X isD^∗- complete, there exists a∈X such that

n→∞limT x_n =a (6)

Since T is subsequentially convergent {x_n =Sⁿ⁻¹x₀} has a convergent subse- quence. So there exist b ∈ X and {n_k}^∞_k=1 such that lim

n→∞S^nk−1x₀ = b. Hence, since T is continuous, we have lim

n→∞T S^nk−1x₀ = T b and by (6) we conclude that T b =a. Since S is continuous and lim

n→∞S^nkx₀ =Sb therefore

n→∞limT S^nkx₀ =T Sb

Again by (6), lim

n→∞T S^nkx₀ = a and therefore by the unique limit we have T Sb=a.Since T is one to one and by (6)Sb=b, so S has a fixed point. Now we prove the uniqueness of the fixed point. Ifbis another fixed point ofS,then Sb=b and we have

D^∗(T Sa, T Sa, T Sb) =D^∗(T a, T a, T b)

≤(a₁+a₃+a₇+a₆ +a₈+a₉+a₁₀)D^∗(T Sa, T Sa, T Sb)

< D^∗(T Sa, T Sa, T Sb)

contradiction. HenceD^∗(T Sa, T Sa, T Sb) = 0 which implies thatT a=T Sa= T Sb=T b. As T is injective, a =b is the unique fixed point ofS; As (T, S) is a Banach pair, T and S commutes at the fixed point of S which implies that T Sa=ST afor a∈F(S). that isT a=ST awhich implies that T ais another fixed point of S. The uniqueness of fixed point of S implies that a = T a.

Hence a=Sa=T a is the unique common fixed point ofS and T inX.

Corollary 21 Let (X, D^∗) be a generalized complete cone D^∗- metric space , P be a normal cone with constant normal K and let S : X → X be a continuous mapping. If S satisfies

D^∗(Sx, Sy, Sz)≤a₁D^∗(x, y, z) +a₂D^∗(x, Sx, Sx) +a₃D^∗(x, Sx, z) +a₄D^∗(y, Sy, Sy) +a₅D^∗(x, Sy, Sy) +a₆D^∗(x, Sz, Sz) +a₇D^∗(y, Sz, Sz) +a₈D^∗(x, Sy, Sz) +a₉D^∗(x, y, Sz) +a₁₀D^∗(Sx, Sx, Sz)

for all x, y, z ∈ X,where a_i, i ∈ {1,2,3,· · · ,12} are all nonnegative con- stants such that

a₁+a₂+a₃+a₄+ 3a₅ + 3a₆+a₇+ 3a₈+a₉+a₁₀<1 then S has a unique fixed point in X.

Proof. The proof of this corollary follows by taking T = I, the identity mapping in Theorem 20.

Corollary 22 Let (X, D^∗)be a generalized complete coneD^∗-metric space, P be a normal cone with constant normal K and let T, S^m :X →X be two con- tinuous mappings.Assume that T is injective and subsequentially convergent.

If T and S^m satisfy

D^∗(T S^mx, T S^my, T S^mz)≤a1D^∗(T x, T y, T z) +a2D^∗(T x, T Sx, T S^mx) +a₃D^∗(T x, T S^mx, T z) +a₄D^∗(T y, T S^my, T S^my) +a₅D^∗(T x, T S^my, T Sy) +a₆D^∗(T x, T S^mz, T S^mz) +a₇D^∗(T y, T S^mz, T S^mz) +a8D^∗(T x, T S^my, T S^mz) +a₉D^∗(T x, T y, T S^mz)

+a₁₀D^∗(T S^mx, T S^mx, T S^mz)

for all x, y, z ∈ X,where a_i, i ∈ {1,2,3,· · · ,10} are all nonnegative constants such that

a₁+ a₂+a₃+a₄+ 3a₅+ 3a₆+a₇+ 3a₈+a₉+a₁₀<1

then S has a unique fixed point in X. Moreover, if (T, S) is a Banach pair, then T and S have a unique common fixed point in X.

Proof. By theorem 20 applied by S = S^m, S^m has a unique fixed point u₀ that isS^mu₀ =u₀.Therefore

S(S^mu₀) = Su₀ =S^m(Su₀)

hence Su0 is a fixed point for S^m, so by the uniqueness fixed point of S^m, we have Su₀ = u₀.Hence S has a fixed point.The remainder of the proof is obvious.

Corollary 23 Let (X, D^∗) be a generalized complete cone D^∗- metric space , P be a normal cone with constant normalK and letT, S :X →X be two con- tinuous mappings.Assume that T is injective and subsequentially convergent.

If T and S satisfy

D^∗(T Sx, T Sy, T Sz)≤a₁D^∗(T x, T y, T z)

for all x, y, z ∈ X,where a1 ∈ [0,1[. Then S has a unique fixed point in X.

Moreover, if (T, S) is a Banach pair, then T and S have a unique common fixed point in X.

Proof. It follows from the proof of theorem 1 by taking.

a_i = 0 for i∈ {2,3,4,5,6,7,8,9,10}

Definition 24 Let(X, D^∗) be a generalized cone metric space, P be a normal cone with constant normal K and let T, S : X → X be two functions. A mappingS is said to be a T−contractive if for x, y, z ∈X such that

D^∗(T Sx, T Sy, T Sz)< D^∗(T x, T y, T z)

∀ x, y, z ∈X :T x6=T y orT x6=T z or T z6=T y

Obviously, every T−contraction function is T− contractive but the con- verse is not true.

Example 25 X = [1,∞[, D^∗(x, y, z) =|x−y|+|x−z|+|y−z|, Sx=√ x and T x=x, then S is T− contractive but S is not T− contraction.

Theorem 26 Let (X, D^∗) be a generalized compact cone D^∗- metric space, P be a normal cone with constant normal K and let T, S : X → X be two con- tinuous mappings.Assume that T is injective and subsequentially convergent.

If T and S satisfy

D^∗(T Sx, T Sy, T Sz)≤a₁D^∗(T x, T y, T z) +a₂D^∗(T x, T Sx, T Sx) +a₃D^∗(T x, T Sx, T z) +a₄D^∗(T y, T Sy, T Sy) +a₅D^∗(T x, T Sy, T Sy) +a₆D^∗(T x, T Sz, T Sz) +a₇D^∗(T y, T Sz, T Sz) +a₈D^∗(T x, T Sy, T Sz) +a9D^∗(T x, T y, T Sz) +a10D^∗(T Sx, T Sx, T Sz) for all x, y, z ∈ X,where a_i, i ∈ {1,2,3,· · · ,10} are all nonnegative constants such that

a₁+ a₂+a₃+a₄+ 3a₅+ 3a₆+a₇+ 3a₈+a₉+a₁₀<1

then S has a unique fixed point in X. Moreover, if (T, S) is a Banach pair, then T and S have a unique common fixed point in X.

Proof. First, we prove that S is continuous. Let lim

n→∞x_n = x,we prove that

n→∞limSx_n =Sx,

D^∗(T Sx_n, T Sx_n, T Sx) = D^∗(T x_n+1, T x_n+1, T Sx) (7) +a1D^∗(T xn, T xn, T x) +a2D^∗(T xn, T xn+1, T xn+1) +a₃D^∗(T x_n, T x_n+1, T x) +a₄D^∗(T x_n, T x_n+1, T x_n+1) +a₅D^∗(T x_n, T x_n+1, T x_n+1) +a₆D^∗(T x_n, T Sx, T Sx) +a₇D^∗(T x_n, T Sx, T Sx) +a₈D^∗(T x_n, T x_n+1, T Sx) +a₉D^∗(T x_n, T x_n, T Sx) +a₁₀D^∗(T x_n+1, T x_n+1, T Sx)

from (7) and from (4) we have

(1−a₁₀)D^∗(T Sx_n, T Sx_n, T Sx)≤a₁D^∗(T x_n, T x_n, T x) (8) + (a₂+a₄+a₅)D^∗(T x_n, T x_n+1, T x_n+1) + (a₆+a₇+a₉)D^∗(T x_n+1, T Sx, T Sx) +a₃D^∗(T x_n, T x_n+1, T x)

+a₈D^∗(T x_n, T x_n+1, T Sx) Since

D^∗(T x_n, T x_n+1, T x_n+1)≤D^∗(T x_n, T x_n, T x) +D^∗(T x_n+1, T x_n+1, T x) (9) D^∗(T xn, T xn+1, T Sx)≤D^∗(T xn, T xn+1, T xn+1) +D^∗(T xn+1, T x, T x)

≤D^∗(T x_n, T x_n, T x) + 2D^∗(T x_n+1, T x_n+1, T x). D^∗(T x_n, T x_n+1, T Sx)≤D^∗(T x_n, T x_n, T x) +D^∗(T x_n+1, T x_n+1, T x)

+D^∗(T x_n+1, T x_n+1, T Sx) by (7), (8) and (9) we obtain

(1−a₆−a₇ −a₈−a₉−a₁₀)D^∗(T Sx_n, T Sx_n, T Sx)≤

(a₁+a₂+a₃ +a₄ +a₅+a₈) D^∗(T x_n, T x_n, T x)

+ (a2+ 2a3+a4+a5+a8) D^∗(T x_n+1, T x_n+1, T x) hence

D^∗(T Sx_n, T Sx_n, T Sx)≤ (a₁+a₂+a₃+a₄+a₅+a₈)

(1−a6−a7−a8−a9−a10)D^∗(T x_n, T x_n, T x) + (a2+ 2a3+a4+a5+a8)

(1−a₆−a₇−a₈−a₉−a₁₀)D^∗(T xn+1, T xn+1, T x) setting

α= a₁+a₂+a₃ +a₄ +a₅+a₈ 1−a6−a7−a8−a9−a10

and β = a₂+ 2a₃+a₄+a₅+a₈ 1−a6−a7−a8−a9−a10

we get

kD^∗(T Sx_n, T Sx_n, T Sx)k ≤KkαD^∗(T x_n, T x_n, T x) +βD^∗(T x_n+1, T x_n+1, T x)k (10) KαkD^∗(T xn, T xn, T x)k+KβkD^∗(T xn+1, T xn+1, T x)k SinceT is continuous, it follows from (10), that

D^∗(T x_n, T x_n, T x)→0

and

D^∗(T x_n+1, T x_n+1, T x)→0 thus

D^∗(T Sxn, T Sxn, T Sx)→0 this shows that

T Sx_n→T Sx

Let{Sx_nk}be arbitrary convergence subsequence of{Sx_n}. there existsy∈X such lim

n→∞Sx_nk =y.Since T is continuous

n→∞limT Sx_nk =T y

By the unique limit T Sx = T y, T is injective, therefore Sx = y. Hence, every convergence subsequence of{Sx_n}converges toSx.SinceXis a compact generalized cone metric space, S is continuous. Now we prove the uniqueness:

we assume that there exist a and b such that a 6= b and Sa = a , Sb = b.

Then

D^∗(T Sa, T Sb, T Sa)≤(a₁+a₃+a₅+a₆ +a₇+a₈+a₉+a₁₀)D^∗(T a, T b, T a)

< D^∗(T a, T b, T a)

then necessarly T a = T b, Hence, since T is one to one , we have a = b contradiction. For the existence, we consider the function ϕ:X →E defined by ϕ(y) = D^∗(T Sy, T Sy, T y), ϕ is continuous and hence by compacteness attains its minimum say at x. If Sx6=x, therefore since T is injective T Sx6=

T x. Since

a₁+a₃

1−a₂−a₃−a₄−a₅−a₈−a₁₀ <1 it is easy to see that

ϕ(Sx) = D^∗ T S²x, T S²x, T Sx

≤ a₁+a₃

1−a₂−a₃−a₄−a₅−a₈−a₁₀D^∗(T Sx, T Sx, T x)

< D^∗(T Sx, T Sx, T x) =ϕ(x)

which is a contradiction to the definition ofx, henceSx=x.Now, letx₀ ∈X, and setα_n =D^∗(T Sⁿx₀, T Sⁿx₀, T x).

αn+1 =D^∗ T Sⁿ⁺¹x0, T Sⁿ⁺¹x0, T x

=D^∗ T Sⁿ⁺¹x0, T Sⁿ⁺¹x0, T x

(11)

≤a₁D^∗(T Sⁿx₀, T Sⁿx₀, T x) +a₂D^∗ T Sⁿx₀, T Sⁿ⁺¹x₀, T Sⁿ⁺¹x₀ +a₃D^∗ T Sⁿx₀, T Sⁿ⁺¹x₀, T x

+a₄D^∗ T Sⁿx₀, T Sⁿ⁺¹x₀, T Sⁿ⁺¹x₀ +a₅D^∗ T Sⁿx₀, T Sⁿ⁺¹x₀, T Sⁿ⁺¹x₀

+a₆D^∗(T Sⁿx₀, T x, T x) +a₇D^∗(T Sⁿx₀, T x, T x) +a₈D^∗ T Sⁿx₀, T Sⁿ⁺¹x₀, T x +a₉D^∗(T Sⁿx₀, T Sⁿx₀, T x) +a₁₀D^∗ T Sⁿ⁺¹x₀, T Sⁿ⁺¹x₀, T x

by using rectangular inequality we get

a₂D^∗ T Sⁿx₀, T Sⁿ⁺¹x₀, T Sⁿ⁺¹x₀

≤a₂(α_n+α_n+1) (12) a₃D^∗ T Sⁿx₀, T Sⁿ⁺¹x₀, T x

≤a₃(α_n+α_n+1) a₄D^∗ T Sⁿx₀, T Sⁿ⁺¹x₀, T Sⁿ⁺¹x₀

≤a₄(α_n+α_n+1) a₅D^∗ T Sⁿx₀, T Sⁿ⁺¹x₀, T Sⁿ⁺¹x₀

≤a₅(α_n+α_n+1) a8D^∗ T Sⁿx0, T Sⁿ⁺¹x0, T x

≤a8(αn+αn+1) from 11 and 12 , it follows that

αn+1 ≤ a₁+a₂ +a₃+a₄+a₅+a₆+a₇+a₈+a₉ 1−a₂−a₃−a₄−a₅−a₈−a₁₀ αn

Since ^a1+a_1−a^{2+a3+a4+a5+a6+a7+a8+a9}

2−a3−a4−a5−a8−a10 <1, then α_n+1 ≤α_n

this implies that{αn} is a nonincreasing sequence of nonegative real numbers and so has a limit denoted by α. By compactness {T Sⁿx₀} has a convergent subsequence {T S^nkx₀}say

n→∞limT S^nkx₀ =z (13)

SinceT is sequentially convergent for a β ∈X, we have

n→∞limS^nkx₀ =β (14)

By (13) and (14) we have T β = z. So D^∗(T β, T β, T x) = a, Next we show that Sβ=x, if Sβ6=x then

a= lim

n→∞D^∗(T Sⁿx₀, T Sⁿx₀, T x)

= lim

n→∞D^∗(T S^nkx₀, T S^nkx₀, T x)

=D^∗(T Sβ, T Sβ, T x)

=D^∗(T Sβ, T Sβ, T Sx)< D^∗(T β, T β, T x) = a contradiction. HenceSβ =x.

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