International Journal of Analysis and Applications
ISSN 2291-8639
Volume 6, Number 2 (2014), 205-219
http://www.etamaths.com

TOPOLOGICAL VECTOR-SPACE VALUED CONE BANACH

SPACES

NAYYAR MEHMOOD1,∗, AKBAR AZAM1 AND SUZANA ALEKSIĆ2

Abstract. In this paper we introduce the notion of tvs-cone normed spaces,

discuss related topological concepts and characterize the tvs-cone norm in

various directions. We construct generalize locally convex tvs generated by a
family of tvs-cone seminorms. The class of weak contractions properly includes

large classes of highly applicable contractions like Banach, Kannan, Chatterjea

and quasi etc. We prove fixed point results in tvs-cone Banach spaces for
nonexpansive self mappings and self/non-self weak contractive mappings. We

discuss the necessary conditions for T -stability of Picard iteration. To ensure
the novelty of our work we establish an application in homotopy theory without

the assumption of normality on cone and many non-trivial examples.

1. Introduction

Recently Beg et al. [1] introduced and studied topological vector space-valued
cone metric spaces (tvs-cone metric spaces), which generalized the cone metric s-
paces [2]. Many generalizations and extensions have been made by many researcher-
s, (see [3-6] ). For more details about topological vector spaces we refer to [7, 8].
Actually the idea of cone metric space was properly introduced by Huang and Zhang
in [2]. In their setting the set of real numbers was replaced by an ordered Banach
space and a vector valued metric was defined on a nonempty set. Many authors
[9-14] studied the properties of cone metric spaces and generalized important fixed
point results of complete metric spaces. The concept of cone metric space in the
sense of Huang-Zhang was characterized by Al-Rawashdeh et al. in [15].

In [16], the author introduced the notion of cone Banach spaces with normal
cones and proved some results regarding fixed points by using nonexpansive map-
pings. Later on many authors investigated some useful results in fixed and coupled
fixed points, (see [17-19] ).

Weak contractions were considered in [20], to study the fixed point results for
self mappings. It has been shown that the Banach, Kannan, Chatterjea, Zamfires-
cue, quasi and many other contractions are weak contractions. The importance of
non-self mappings is obvious. In fact fixed point theorems for non-self mappings
generalized all the corresponding results presented for self-mappings. A variety of
results on nonself mappings and weak contractions can be found in [21-27].

In this article, we introduce tvs-cone Banach space and investigate some proper-
ties without assumption of normality on cones. We generalize the results of [16] and

2010 Mathematics Subject Classification. 47H10; 54H25.
Key words and phrases. tvs-cone Banach space; non-normal cones; weak contraction; nonself

mappings; iteration processes; T-stability; fixed points; homotopy; generalized locally convex tvs.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

205



206 MEHMOOD, AZAM AND ALEKSIĆ

explore some characteristics of norms in cone normed space. We prove fixed point
results for Picard, Mann, Ishikawa and Krasnoseskij iterations, we also present
results for weak contractive non-self mappings. Many examples have been given
and a homotopy result is established for nonexpansive mappings. We discuss the
necessary conditions for T-stability of Picard iteration.

2. Preliminaries

Let E be a topological vector space with its zero vector θ. A nonempty subset
K of E is called a convex cone if K + K ⊆ K and λK ⊆ K for λ ≥ 0. A convex
cone K is said to be pointed (or proper) if K ∩ (−K) = {θ}, and K is normal
(or saturated) if E has a base of neighborhoods of zero consisting of order-convex
subsets. For a given cone K ⊆ E we define a partial ordering 4 with respect to K
by x 4 y if and only if y −x ∈ K, x ≺ y stands for x 4 y and x 6= y, while x � y
stands for y−x ∈ int K, where int K denotes the interior of K. The cone K is said
to be solid if it has a nonempty interior.

Definition 1. Let V be a vector space over R. A vector-valued function ‖·‖K :
V → E; X → V is called a tvs-cone norm on X if the following conditions are
satisfied:

(N1) ‖x‖K < θ for all x ∈ V ,
(N2) ‖x‖K = θ if and only if x = θ,
(N3) ‖x + y‖K 4 ‖x‖K + ‖y‖K for all x,y ∈ V ,
(N4) ‖kx‖K = |k|‖x‖K for all k ∈ R.
The pair (X,‖·‖K) is called a tvs-cone norm space (in brief tvs-CNS).

Definition 2. Let (V,‖·‖K) be a tvs-cone norm space and {xn} a sequence in V .
(i) {xn} tvs-cone converges to x ∈ V if for every c ∈ E with θ � c there exists

n0 ∈ N such that ‖xn −x‖K � c for all n ≥ n0.
(ii) {xn} is a tvs-cone Cauchy sequence if for every c ∈ E with θ � c there exists

n0 ∈ N such that ‖xn −xm‖K � c for all n,m ≥ n0.
(iii) (V,‖·‖K) is a tvs-cone complete or a tvs-cone Banach space if every tvs-cone

Cauchy sequence in V is tvs-cone convergent.

Using the consequences of Lemma 2.4 from [28], we have the following properties.

Lemma 3. Let (E,K) be a locally convex tvs. The following properties hold.
(a) For a sequence {vn} in E with θ 4 vn → θ, let θ � c then there exists

positive integer n0 such that vn � c for each n > n0.
(b) There exists a sequence {vn} in E such that for some positive integer n0

holds θ 4 vn � c for all n > n0, but vn 6→ θ.
(c) If there exists v in E such that θ 4 v � c for all c ∈ int K, then v = θ.
(d) If a 4 λa, where a ∈ K and 0 ≤ λ < 1, then a = θ.

Remark 4. For a Banach space E with non-normal cone K, with norm ‖·‖ . The
following may hold.
(a) For sequences {vn}, {un} in E with norm ‖·‖ , it may happen that vn → v,
un → u, but ‖vn −un‖ 6→‖v −u‖ (see Example 5). In particular, vn → v, n →∞,
may imply that ‖vn −v‖ 6→ θ, n →∞ (this is impossible in CNS defined in [16] if
the cone is normal).
(b) If vn → v and vn → u, then v = u.



TOPOLOGICAL VECTOR-SPACE VALUED CONE BANACH SPACES 207

Example 5. Let V = R and let E be the set of all real-valued functions on V which
also have continuous derivatives on V. Then E is a vector space over R under the
following operations:

(x + y) (t) = x (t) + y (t) , (αx) (t) = αx (t)

for all x,y ∈ E,α ∈ R. Then E with norm
‖x‖ = ‖x‖∞ + ‖x′‖∞,

has non-normal solid cone, see [5, 8]:

K = {x ∈ E : θ 4 x}, where θ(t) = 0 for all t ∈ X.
Consider the sequences

xn(t) =
1 + sin nt

n + 2
, yn(t) =

1 − sin nt
n + 2

, n ≥ 0.

in E. We have xn → θ, yn → θ, n → +∞, but

‖xn −yn‖ =
∥∥∥∥2 sin ntn + 2

∥∥∥∥ = sup
t∈V

{
2 sin nt

n + 2

}
+ sup
t∈V

{
2n cos nt

n + 2

}
=

2 sin n

n + 2
+ 1 6→ θ, n → +∞.

Also as xn → θ, consider
‖xn −θ‖ = ‖xn‖ = 1 6→ θ.

Definition 6 ([1]). Let X be a nonempty set and (E,K) a tvs. A vector-valued
function d : X ×X → E is said to be a tvs-cone metric if the following conditions
are satisfied:

(C1) θ 4 d(x,y) for all x,y ∈ X and d(x,y) = θ if and only if x = y,
(C2) d(x,y) = d(y,x) for all x,y ∈ X,
(C3) d(x,z) 4 d(x,y) + d(y,z) for all x,y,z ∈ X.
The pair (X,d) is called a tvs-cone metric space.

Note that each tvs-CNS is a tvs-cone metric space with induced tvs-cone metric
d : X ×X → E defined by d(x,y) = ‖x−y‖ for all x,y ∈ X.

Remark 7 ([1]). The concept of cone metric spaces is more general than that of
metric spaces, because each metric space is a cone metric space, and a cone metric
space in the sense of Huang and Zhang is a special case of tvs-cone metric spaces
when (X,d) is a cone metric space with respect to a normal cone K.

If K is a normal cone, then a tvs-CNS (V,‖ · ‖K) becomes a CNS [16] and with
the induced tvs-cone metric [1] this space becomes cone metric space in the sense
of [2].

CNS in the case of [16] gives us generalized induced norm known as b-norm
‖·‖b : V → R defined by ‖·‖b = ‖‖·‖K‖. The triangular property of cone norm

‖x + y‖K 4 ‖x‖K + ‖y‖K ,
gives us the following property of b-norm,

‖x + y‖b ≤ k(‖x‖b + ‖y‖b),
where k is a constant of normality.



208 MEHMOOD, AZAM AND ALEKSIĆ

Obviously every norm is a b-norm, but the contrary is not true, consider the
following example

Example 8. Let X = R and ‖·‖b : X → R defined by ‖x‖b = |x|3. For x,y ∈ X we
have |x + y|3 ≤ (|x|+ |y|)3 ≤ 23(|x|3 + |y|3), but |x + y|3 6≤ (|x|3 + |y|3). Therefore,
‖x‖b is a b-norm, but it is not a norm on X.

Let us recall the following definitions.

Definition 9 ([5, 29]). Let X be a nonempty set. A vector-valued function d :
X ×X → E is said to be cone symmetric if the following conditions are satisfied:

(C1) θ 4 d(x,y) for all x,y ∈ X and d(x,y) = θ if and only if x = y;
(C2) d(x,y) = d(y,x) for all x,y ∈ X.
The pair (X,d) is called a cone symmetric space.

It is clear that the cone symmetric space may not be a cone metric space (see
Example 2.2 in [29]). For a given cone symmetric space (X,d) one can deduce (see
[29]) a symmetric metric space with D : X×X → R defined by D(x,y) = ‖d(x,y)‖
for all x,y ∈ X.

For a cone metric space (X,d) with normal cone K with normal constant k ≥ 1,
we have

D(x,y) = ‖d(x,y)‖≤ k‖d(x,z) + d(z,y)‖≤ k(D(x,z) + D(z,y)).
In this case, the metric D becomes b-metric and, hence, the concept of b-metric
spaces is more general then that of metric spaces and the topology τD generated
by D coincides with τb generated by b-metric on X.

In the following we explore the concept of tvs-cone seminorm.

Definition 10. Let V be a vector space over scalars F . If a mapping ρK : X →
(E,K) satisfies:
(SN1) ρK(x) < θ for all x ∈ V ,
(SN2) ρK(x + y) 4 ρK(x) + ρK(y) for all x,y ∈ V ,
(SN3) ρK(kx) = |k|ρK(x) for all x ∈ V , k ∈ F .

Then ρK is called a tvs-cone seminorm on X.

Note that a tvs-cone seminorm is a norm if ρK(x) = θ implies x = θ. A tvs-cone
seminorm on X induces a pseudo tvs-cone metric defined by dp(x,y) = ρK(x−y)
which satisfies:

(PC1) θ 4 dp(x,y) for all x,y ∈ X,
(PC2) dp(x,y) = dp(y,x) for all x,y ∈ X,
(PC3) dp(x,z) 4 dp(x,y) + dp(y,z) for all x,y,z ∈ X.

Note that dp(x,y) = θ does not imply x 6= y.
The class of tvs-cone pseudo metric spaces is larger than the class of tvs-cone metric
spaces.

Equivalently, ρK is a tvs-cone seminorm on a vector space V if the following
conditions are satisfied:

(SN(i)) ρK(v + v) 4 ρK(u) + ρK(v) for all u,v ∈ V,
(SN(ii)) ρK(kv) = |k|ρK(v) for all v ∈ V , k ∈ F.

This cone seminorm gives us generalized seminorm, so called b-seminorm ‖·‖bs :
X → R defined by

‖x‖bs = ‖ρK(x)‖ .



TOPOLOGICAL VECTOR-SPACE VALUED CONE BANACH SPACES 209

Using (SN(i)), b-seminorm has the following property

‖x + y‖bs ≤ k(‖x‖bs + ‖y‖bs).
Note that b-seminorm is a seminorm if k = 1 and every seminorm is a b-seminorm.
The next example shows that the contrary is not true, i.e., b-seminorm does not
need to be seminorm.

Example 11. Let X = R and ‖ · ‖bs : X → R is defined by ‖x‖bs = |x|3 + 1. For
x,y ∈ X, we have
|x + y|3 + 1 ≤ (|x| + |y|)3 + 1 ≤ 23(|x|3 + |y|3) + 1 ≤ 23(|x|3 + |y|3) + 16

= 23(|x|3 + 1 + |y|3 + 1),
which implies ‖x+y‖bs ≤ 23(‖x‖bs+‖y‖bs). This shows that ‖x‖bs is a b-seminorm,
but not a seminorm on X.

3. Main Results

Let {ρKi : i ∈ I} be a family of tvs-cone seminorms on a vector space V . For
θ � ε and i ∈ I = {1, 2, 3, . . . ,n}, define
U(u0,ρK1,ρK2,ρK3,...,ρKn,ε) = U(u0,ρKn,ε) = {u ∈ V : ρKi(u−u0) � ε, i ∈ I} .

Note that U(u0,ρKn,ε) = u0 + U(θ,ρKn,ε).

Lemma 12. The set U(θ,ρKn,ε) is balanced and convex in V.

Proof. For any w ∈ U(θ,ρKn,ε) and |k| ≤ 1, we have ρKi(kw) 4 |k|ρKi(w) � ε,
i ∈ I. Thus U(θ,ρKn,ε) is absorbing. Now, for 0 ≤ t ≤ 1 and u,v ∈ U(θ,ρKn,ε), we
obtain

ρKi(tu + (1 − t)v) 4 tρKi(u) + (1 − t)ρKi(v) � tε + (1 − t)ε = ε,
which implies tρKi(u) + (1−t)ρKi(v) ∈U(θ,ρKn,ε). Therefore, U(θ,ρKn,ε) is convex.

Lemma 13. Let {ρKi : i ∈ I} be a family of tvs-cone seminorms on a vector space
(V,F). For each v ∈ V denote with Nv the collection of sets of the form

U(v,ρKn,ε) = {u ∈ V : ρKi(u−v) � ε,i ∈ I}.
Let T be the collection of ∅ and all subsets G of X such that for each u ∈ G there
exists some U ∈ Nv such that U ⊆ G. Then T is topology on V and preserves the
structure of vector space. The sets Nv form an open locally convex neighborhood
base at x. The topological space (V,T ) is Hausdorff iff the family {ρKi : i ∈ I} of
tvs-cone seminorms is separating, i.e. for θ 6= u ∈ V there exists some i0 ∈ I such
that ρKi0 (u) 6= θ.

Proof. It is clear that V and the union of any number of elements of T belong to
T . We will show that A,B ∈T implies A∩B ∈T . The case A∩B = ∅ is obvious.
Suppose that A∩B 6= ∅ and v ∈ A∩B. By definition of T there exist U1, U2 ∈T
such that U1 ⊆ A and U2 ⊆ B. Let for comparable ε,δ ∈ int K, we define

U1 := U(v,ρKn,ε) = {u ∈ V : ρKi(u−v) � ε, 1 ≤ i ≤ n}
and

U2 = U(v,µKm,δ) =
{
u ∈ V : µKj(u−v) � δ, 1 ≤ j ≤ m

}
.

If we set
U3 = U(v,ρK1,ρK2,ρK3,...,ρKn,µK1,µK2,µK3,...,µKm,γ),



210 MEHMOOD, AZAM AND ALEKSIĆ

where γ = ε if δ − ε ∈ int K and γ = δ if ε − δ ∈ int K, then U3 ∈ Nv and
U3 ⊆ U1 ∩ U2 ⊆ A ∩ B. Hence T is topology on V . Let U(v,ρKn,ε) ∈ Nv and
w ∈ U(v,ρKn,ε). Then ρKi(w − v) � ε, 1 ≤ i ≤ n. Now choose θ � δ such that
δ � ε − ρKi(w − v) for 1 ≤ i ≤ n. For any 1 ≤ i ≤ n and u ∈ V satisfying
ρKi(w −u) � δ we have

ρKi(u−v) 4 ρKi(u−w) + ρKi(w −v) � δ + ρKi(w −v) � ε.

We see that U(u,ρKn,δ) ⊆ U(v,ρKn,ε), hence U(v,ρKn,ε) is open. Lemma 12 implies
that the elements of Nv are convex. Therefore, Nv is an open locally convex neigh-
borhood base at v consisting of the open sets U(v,ρKn,ε).

Now we will show that the topology T is compatible. Let u,v ∈ V and let
U(u+v,ρKn,ε) be a basic neighborhood of u + v. Let (un,vn) → (u,v) in V × V .
Then there exists an integer n0 such that (un,vn) ∈ U(u,ρKn,ε2 ) ×U(v,ρKn,ε2 ) for all
n ≥ n0. For 1 ≤ i ≤ n and for all n ≥ n0, we have

ρKi(u + v − (un + vn)) 4 ρKi(u−un) + ρKi(v −vn) � ε,

which gives un + vn ∈ U(u+v,ρKn,ε), and, therefore, un + vn → u + v. Now let
(kn,vn) → (k,v) in F ×V. Let U(kv,ρKn,δ) be a basic neighborhood of kv. Choose
t > 0 and θ � γ such that for 1 ≤ i ≤ n there exists an integer m0 such that
(kn,vn) ∈ {ζ ∈ F : |ζ −k| < t}×U(v,ρKn,γ) for all n ≥ m0, with tρKi(v) �

δ
2

and

(|k| + t)γ � δ
2
. For n ≥ m0 we have

ρKi(kv −knvn) 4 ρKi(kv −knv) + ρKi(knv −knvn)
4 |k −kn|ρKi(v) + |kn|ρKi(v −vn)
4 |t|ρKi(v) + |kn|ρKi(v −vn)
� |t|ρKi(v) + (|k| + t)γ

�
δ

2
+
δ

2
= δ,

thus knvn ∈U(kv,ρKn,δ). Therefore (V,T ) is a tvs.
Now, suppose that the family P = {ρKi : i ∈ I} of tvs-cone seminorms is

separating. For any u,v ∈ V with u 6= v there exists some j0 ∈ I such that
θ � δ = ρKj0 (v − u). Thus, the open sets U(u,ρKj0,δ2 ) and U(v,ρKj0,δ2 ) are disjoint
containing u and v and so the space (V,T ) is Hausdorff.

We conclude that the space (V,T ) is locally convex tvs.

Definition 14. [20] Let X be a tvs-cone normed space and T : X → X an operator.
(i) T is an almost weak contraction if for all x,y ∈ E, L ≥ 0 and δ ∈ (0, 1),

we have

(w1) ‖Tu−Tv‖K 4 δ · ‖u−v‖K + L · ‖u−Tu‖K , ∀u,v ∈ X.

(ii) T is a weak contraction if

(w2) ‖Tu−Tv‖K 4 δ · ‖u−v‖K + L · ‖v −Tu‖K .

Definition 15. Let X be a tvs-cone normed space and T : X → X an operator.
(i) T is a Zamfirescue contraction if for all u,v ∈ X and a ∈ [0, 1), b,c ∈

[0, 1
2
), one of the following conditions is satisfied



TOPOLOGICAL VECTOR-SPACE VALUED CONE BANACH SPACES 211

(z1)

‖Tu−Tv‖K 4 a · ‖u−v‖K ,
(z2)

‖Tu−Tv‖K 4 b (‖u−Tu‖K + ‖v −Tv‖K) ,
(z3)

‖Tu−Tv‖K 4 c (‖u−Tv‖K + ‖v −Tu‖K) .
(ii) T is a Quasi contraction if for all u,v ∈ X and α ∈ [0, 1), holds

‖Tu−Tv‖K 4 c ·m,

where

m ∈{‖u−v‖K ,‖u−Tu‖K ,‖v −Tv‖K ,‖u−Tv‖K ,‖v −Tu‖K} .

Remark 16. [20](a) Every Zamfirescue contraction is a weak contraction.
(b) Every Quasi contraction is a weak contraction.

Definition 17. [19] Let X be a tvs-cone normed space, T : X → X an operator
and u0 ∈ X. A sequence {un} is called:

1) Picard iteration if

(p1) un+1 = Tun;

2) Mann iteration if

(m1) un+1 = (1 −αn) un + αnTun;

3) Ishikawa iteration if

un+1 = (1 −αn) un + αnTvn, (i1)
vn = (1 −βn) xn + βnTxn,

where {αn}⊆ (0, 1) and {βn}⊆ [0, 1).
4) Krasnoselskij iteration if

un+1 = (1 −λ) un + λTun,

where λ ∈ (0, 1).

Denote with F(T) the set of all fixed points of T.

Lemma 18. Let X be a tvs-cone Banach space and {an} and {bn} be sequences
in E satisfying an+1 4 λan + bn, where λ ∈ (0, 1) and bn → θ as n → ∞. Then
lim
n→∞

an = θ.

Proof. On the contrary, suppose that lim
n→∞

an 6= θ and lim
n→∞

an = c, for some θ � c.
Then, by lemma 3 (d), we have an = θ.

In the following theorem we obtain a fixed point result for nonself weak contrac-
tions in a tvs-cone Banach space.

Theorem 19. Let X be a tvs cone Banach space and C be a nonempty closed and
convex subset of X. Suppose that T : C → X is a weak contraction (satisfying
(w2)), such that δ(1 + L) < 1. If T(∂C) ⊆ C, then T has a fixed point.



212 MEHMOOD, AZAM AND ALEKSIĆ

Proof. We construct two sequences {un} and {vn} in the following way. Let us
choose u0 arbitrary in X and set v1 = Tu0. If v1 ∈ C, then set u1 = v1. If not, then
there exists u1 ∈ ∂C such that

‖u1 −u0‖K + ‖u1 −v1‖K = ‖u0 −v1‖K .

Thus u1 ∈ C and let v2 = Tu1. We have

‖v2 −v1‖K = ‖Tu0 −Tu1‖K 4 δ · ‖u1 −u0‖K + L · ‖u1 −Tu0‖ .

If v2 ∈ C, set u2 = v2. Otherwise, there exists u2 ∈ ∂C such that

‖u2 −u1‖K + ‖v2 −u2‖K = ‖v2 −u1‖K .

Thus u2 ∈ C. Let v3 = Tu2 and consider

‖v2 −v3‖K = ‖Tu1 −Tu2‖K 4 δ · ‖u2 −u1‖K + L · ‖u2 −Tu1‖K .

Continuing in the same way, we construct the sequences {un} and {vn} such that
(i) vn+1 = Tun,
(ii) ‖vn −vn+1‖K 4 δ · ‖un−1 −un‖K + L · ‖un −Tun−1‖K,
where
(iii) vn ∈ C implies vn = un.
(iv) If vn 6∈ C, then vn 6= un, and then un ∈ ∂C is such that

‖un−1 −un‖K + ‖vn −un‖K = ‖vn −un−1‖K .

We will show that {un} is a Cauchy sequence. Define

P = {ui ∈{un} : ui = vi} ,
Q = {ui ∈{un} : ui 6= vi} .

It is obvious that if un ∈ Q, then un−1 and un+1 are in P. We have the following
three possibilities.
Case 1. If un, un+1 ∈ P, then

‖un −un+1‖K = ‖vn −vn+1‖K 4 δ · ‖un−1 −un‖K + L · ‖un −Tun−1‖K
4 δ · ‖un−1 −un‖K .

Case 2. If un ∈ P, un+1 ∈ Q, then

‖un −un+1‖K 4 ‖un −un+1‖K + ‖un+1 −vn+1‖K
= ‖un −vn+1‖K
= ‖vn −vn+1‖K
4 δ · ‖un−1 −un‖K + L · ‖un −Tun−1‖K
4 δ · ‖un−1 −un‖K .



TOPOLOGICAL VECTOR-SPACE VALUED CONE BANACH SPACES 213

Case 3. If un ∈ Q, un+1 ∈ P, then

‖un −un+1‖K 4 ‖vn −un‖K + ‖vn −vn+1‖K
4 ‖vn −un‖K + δ · ‖un−1 −un‖K + L · ‖un −Tun−1‖K
4 ‖vn −un‖K + ‖un−1 −un‖K + L · ‖vn −un‖K
= ‖vn −un−1‖K + L · ‖vn −un‖K
= ‖vn −un−1‖K + L · ‖vn −un−1‖K −L · ‖un−1 −un‖K
4 (1 + L)‖vn−1 −vn‖K
4 (1 + L)δ · ‖un−2 −un−1‖K + (1 + L)L · ‖un−1 −Tun−2‖K
4 (1 + L)δ · ‖un−2 −un−1‖K
= h‖un−2 −un−1‖K ,

where h = (1 + L)δ < 1.
Taking α = max{δ,h}, and combining all above three cases we have

‖un −un+1‖K 4
{

α‖un−1 −un‖K
α‖un−2 −un−1‖K

.

By mathematical induction, for all n > 0, we have

‖un −un+1‖K 4 h
(n−1)/2w

for w ∈{‖u1 −u0‖K ,‖u2 −u1‖K} .
Now for n > m, we consider

‖um −un‖K 4 ‖un −un−1‖K + ‖un−1 −un−2‖ + · · · + ‖um−1 −um‖
4 (h(n−1)/2 + h(n−2)/2 + · · · + h(m−1)/2)w

4
h(m−1)/2

1 −h(n−m)/2
w.

As h < 1, we have h(m−1)/2 → 0 when n,m →∞, and this gives us h
(m−1)/2

1−h(n−m)/2 w →
θ, n → ∞, in the locally convex space E. Now, according to Lemma 3-(a), we
conclude that for every c ∈ E with θ � c there is a natural number k1 such that
‖um −un‖K � c for all m,n ≥ k1, so {un} is a tvs-cone Cauchy sequence in C.
As C is closed, thus there exists some u ∈ C, such that un → u as n →∞.
By construction of {un} there exists a subsequence

{
unq
}

such that

vnq = unq = Tunq−1

and unq → u as q → ∞. So, for a given c ∈ E with θ � c, let us choose a natural
number k2 such that

∥∥u−unq∥∥K � c1+L and ∥∥unq−1 −u∥∥K � cδ for all q−1 ≥ k2.
Now, we have

‖u−Tu‖K 4
∥∥u−unq∥∥K + ∥∥unq −Tu∥∥K

4
∥∥u−unq∥∥K + ∥∥Tunq−1 −Tu∥∥K

4
∥∥u−unq∥∥K + δ∥∥unq−1 −u∥∥K + L∥∥u−Tunq−1∥∥K

4 (1 + L)
∥∥u−unq∥∥K + δ∥∥unq−1 −u∥∥K ,

i.e. ‖u−Tu‖K � c(k2) for all q − 1 ≥ k2.
This completes the proof.



214 MEHMOOD, AZAM AND ALEKSIĆ

Theorem 20. Let X be a tvs cone Banach space and C be a nonempty closed and
convex subset of X. Suppose that T : C → X is a weak contraction (satisfying
(w1)), such that δ(1 + L) < 1. If T satisfies the condition: u ∈ ∂C ⇒ Tu ∈ C, then
T has a fixed point.

Corollary 21. Let X be a cone Banach space with normal cone K and C be a
nonempty closed and convex subset of X. Suppose that T : C → X is a weak
contraction/almost weak contraction (satisfying (w1)/(w2)), such that δ(1+L) < 1.
If T satisfies the condition: u ∈ ∂C ⇒ Tu ∈ C, then T has a fixed point.

The following corollaries are due to remark 16.

Corollary 22. Let X be a tvs cone Banach space and C be a nonempty closed and
convex subset of X. Suppose that T : C → X is Zamfirecue operator. If T satisfies
the condition: u ∈ ∂C ⇒ Tu ∈ C, then T has a fixed point.

Corollary 23. Let X be a tvs cone Banach space and C be a nonempty closed and
convex subset of X. Suppose that T : C → X is quasi operator. If T satisfies the
condition: u ∈ ∂C ⇒ Tu ∈ C, then T has a fixed point.

Corollary 24. Let X be a Banach space and C be a nonempty closed and convex
subset of X. Suppose that T : C → X is Zamfirecue operator. If T satisfies the
condition: u ∈ ∂C ⇒ Tu ∈ C, then T has a fixed point.

Corollary 25. Let X be a Banach space and C be a nonempty closed and convex
subset of X. Suppose that T : C → X is a quasi operator. If T satisfies the condition:
u ∈ ∂C ⇒ Tu ∈ C, then T has a fixed point.

Theorem 26. Let E be a tvs-normed space, C be a closed and convex subset of
E. Let T : C → C be an almost weak contractive mapping (satisfying (w1)) with
F(T) 6= ϕ. Let {un} be Ishikawa iteration satisfying

(α)

∞∑
j=0

αj = ∞,

u0 ∈ C is arbitrary chosen. Then {un} converges strongly to a unique fixed point
of T.

Proof. It can be shown that (w1) gives us a unique fixed point. Let p ∈ F(T) be a
unique fixed point of T and {un} be Ishikawa iteration defined in (i1) and u0 ∈ C.
We have

‖un+1 −p‖K = ‖(1 −αn)un + αnTvn −p‖K
= ‖(1 −αn)(un −p) + αn(Tvn −p)‖K
4 (1 −αn)‖un −p‖K + αn‖Tvn −p‖K
4 (1 −αn)‖un −p‖K + αnδ‖vn −p‖K , by (w1),

and

‖vn −p‖K = ‖(1 −βn)un + βnTun −p‖K
= ‖(1 −βn)(un −p) + βn(Tun −p)‖K
4 (1 −βn)‖un −p‖K + βnδ‖un −p‖K , by (w1).



TOPOLOGICAL VECTOR-SPACE VALUED CONE BANACH SPACES 215

So, we obtain

‖un+1 −p‖K 4
(
1 − (1 − δ)2αn

)
‖un −p‖K

4 e−(1−δ)
2αn ‖un −p‖K

4

(
e
−(1−δ)2

n∑
j=0

αj

)
· ‖u0 −p‖K .

Using (α), this implies

(
e
−(1−δ)2

n∑
j=0

αj

)
→ 0, n →∞, which gives us

(
e
−(1−δ)2

n∑
j=0

αj

)
·

‖u0 −p‖K → θ, n → ∞, in the locally convex space E. This completes the proof
of theorem.

The following corollaries are due to remark 16.

Corollary 27. Let E be a tvs-normed space, C be a closed and convex subset of
E. Let T : C → C be Zamfirescue operator, with F(T) 6= ϕ. Let {un} be Ishikawa
iteration satisfying

∞∑
j=0

αj = ∞

where u0 ∈ C is arbitrary chosen. Then {un} converges strongly to a unique fixed
point of T.

Corollary 28. Let E be a tvs-normed space, C be a closed and convex subset of E.
Let T : C → C be a quasi operator, with F(T) 6= ϕ. Let {un} be Ishikawa iteration
satisfying

∞∑
j=0

αj = ∞

where u0 ∈ C is arbitrary chosen. Then {un} converges strongly to a unique fixed
point of T.

Corollary 29. [24] Let E be a normed space, C be a closed and convex subset of
E. Let T : C → C be a Zamfirescue operator, with F(T) 6= ϕ. Let {un} be Ishikawa
iteration satisfying

∞∑
j=0

αj = ∞

and u0 ∈ C is arbitrary chosen. Then {un} converges strongly to a unique fixed
point of T.

Corollary 30. [24] Let E be a normed space, C be a closed and convex subset of E.
Let T : C → C be a quasi operator, with F(T) 6= ϕ. Let {un} be Ishikawa iteration
satisfying

∞∑
j=0

αj = ∞

and u0 ∈ C is arbitrary chosen. Then {un} converges strongly to a unique fixed
point of T.

The following theorem is a result for fixed point of non-expansive mappings in
tvs-cone Banach space for Krasnoselskij iteration with λ = 1

2
.



216 MEHMOOD, AZAM AND ALEKSIĆ

Theorem 31. Let C be a closed and convex subset of a tvs-cone Banach space
(X,‖·‖K). Suppose that the mapping F : C → C satisfies
(a) ‖v −Fv‖K + ‖u−Fu‖K 4 η‖v −u‖K
for all u,v ∈ C. Then F has at least one fixed point if 2 ≤ η ≤ 4.

Proof. Let us choose v0 ∈ C arbitrary and define sequence {vn} as follows:

vn+1 =
vn + Fvn

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

Since

vn −Fvn = 2
(
vn −

vn + Fvn
2

)
= 2(vn −vn+1),

we have

(b) ‖vn −Fvn‖K = 2‖vn −vn+1‖K , n = 0, 1, 2, 3, . . . .
Combining (a) and (b), we have

2‖vn−1 −vn‖K + 2‖vn −vn+1‖K 4 η‖vn−1 −vn‖K ,
which gives

‖vn −vn+1‖K 4 λ‖vn−1 −vn‖K , n = 0, 1, 2, 3 . . . ,

for λ = η−2
2

< 1.
According to the previous inequality, for m ≥ n, we obtain

‖vn −vm‖K 4
λn

1 −λ
‖v0 −v1‖K .

Since λn → 0 as n → ∞, then λ
n

1−λ ‖v0 −v1‖K → θ, n → ∞, in the locally convex
space E. Now, according to Lemma 3 part (a), we conclude that for every c ∈ E
with θ � c there exists a natural number n1 such that ‖vn −vm‖K � c for all
m,n ≥ n1. Therefore, {vn} is a tvs-cone Cauchy sequence in C. Since C is closed,
there exists some w ∈ C, such that vn → w as n → ∞. Now, choose a positive
integer m1 such that for every c ∈ E with θ � c we have ‖w −vn‖K �

1
η
c for all

n ≥ m1.
Substituting v = w and u = vn in (a), for all n ≥ m1, we obtain

‖w −Fw‖K + 2‖vn −vn+1‖K 4 η‖w −vn‖K ,
‖w −Fw‖K 4 η‖w −vn‖K − 2‖vn −vn+1‖K � c.

Thus, w = Fw is a fixed point of F.

Corollary 32. [16] Let C be a closed and convex subset of a cone Banach space
(X,‖·‖K). Suppose that the mapping F : C → C satisfies

‖v −Fv‖K + ‖u−Fu‖K 4 η‖v −u‖K
for all u,v ∈ C. Then F has at least one fixed point if 2 ≤ η ≤ 4.

The next theorem is an application of above theorem in topological homotopy
theory.

Theorem 33. Let (X,‖·‖K) be a tvs-cone Banach space, C a closed and convex
subset of X and U an open subset of C. Let K : [0, 1] × Ū → C be a homotopy
mapping with the following conditions:

(a) ξ 6= K(t,ξ), for each ξ ∈ ∂U and each t ∈ [0, 1],



TOPOLOGICAL VECTOR-SPACE VALUED CONE BANACH SPACES 217

(b) K(t, ·) : Ū → C is a mapping satisfying the conditions of Theorem 31,
(c) there exists a continuous increasing function g : (0, 1] → P such that∥∥∥K(s,ξ) −K(t, ξ́)∥∥∥

K
4 g(s) −g(t),

g(s) ∈ g(t) + P,
for all s,t ∈ [0, 1], and each ξ ∈ Ū.

Then K(0, ·) has a fixed point if and only if K(1, ·) has a fixed point.

Proof. We first suppose that K(0, ·) has a fixed point z, i.e. z = K(0,z). From (a),
we obtain z ∈ U. Define

Γ := {(t,ξ) ∈ [0, 1] ×C : ξ = K(ξ,t)}.
Clearly Γ 6= φ. We define the partial ordering in Γ as follows:

(t,ξ) - (s, ξ́) ⇔ t ≤ s and
∥∥∥ξ́ − ξ∥∥∥

K
4

2

η − 2
(g(s) −g(t)).

Let B be a totally ordered subset of Γ and t̊ = sup{t : (t,ξ) ∈ B}. Consider
a sequence {(tn,ξn)}n≥0 in B such that, (tn,ξn) - (tn+1,ξn+1) and tn → t̊ as
n →∞. For m > n, we have

‖ξm − ξn‖K 4
2

η − 2
(g(tm) −g(tn)) → θ, as n,m →∞,

and conclude that {ξn} is a tvs-cone Cauchy sequence. There exists ξ̊ ∈ C such
that ξn → ξ̊. Choose n0 ∈ N such that for θ � c we have ‖̊ξ − ξn‖K �

c
η

for

all n ≥ n0. The mapping K(t, ·) satisfies all the conditions of Theorem 31 and
substituting v = ξ̊ and u = ξn into (1), for all n ≥ n0, we obtain∥∥∥̊ξ −K(̊t, ξ̊)∥∥∥

K
+ 2

∥∥ξn − ξn+1∥∥K 4 η∥∥∥̊ξ − ξn∥∥∥K ,∥∥∥̊ξ −K(̊t, ξ̊)∥∥∥
K

4 η
∥∥∥̊ξ − ξn∥∥∥

K
− 2

∥∥ξn − ξn+1∥∥K � c.
We see that ξ̊ = K(̊t, ξ̊) and, hence, ξ̊ ∈ U, which implies (̊t, ξ̊) ∈ Γ. Thus, (t,ξ) -
(̊t, ξ̊) for all (t,ξ) ∈B gives us that (̊t, ξ̊) is an upper bound of B. By Zorn’s lemma,
Γ has maximal element (̊t, ξ̊).

We claim that t̊ = 1. On the contrary, suppose that t̊ ≤ 1. Let us choose θ � r
arbitrary and, for any t ≥ t̊, consider

Br(̊ξ) =
{
ξ :
∥∥ξ − ξ̊∥∥

K
4 r
}
⊂ U,

where r = 2
η−2 (g(t) −g(̊t)).

Using the condition (c), we have∥∥K(t,ξ) −K(̊t, ξ̊)∥∥
K

4 g(t) −g(̊t) =
η − 2

2
r � r.

Hence, for each t ∈ [0, 1], there exists some ξ ∈ Br(̊ξ) ⊂ U such that ξ = K(t,ξ).
Since ∥∥ξ − ξ̊∥∥

K
4 r =

2

η − 2
(g(t) −g(̊t))

implies (̊t, ξ̊) - (t,ξ), we obtain a contradiction. Therefore, t̊ = 1.
From the above it follows that K(1, ·) has a fixed point ξ̊ = K(1, ξ̊).



218 MEHMOOD, AZAM AND ALEKSIĆ

Conversely, if K(1, ·) has a fixed point, then, in the same way, we can prove that
K(0, ·) has a fixed point.

Let X be a tvs cone normed space and T be a self operator of X. Let u0 be any
fixed point and xn+1 = ξ(T,xn) is an iteration process involving T, which computes
the sequence {xn} in X.

Definition 34. (see also [30]) The iteration procedure xn+1 = ξ(T,xn) is said to
be T -stable with respect to T if {xn} converges to a unique fixed point q of T and
whenever {yn} is a sequence in X with

lim
n→∞

‖yn+1 − ξ(T,xn)‖K = θ

we have lim
n→∞

yn = q.

Theorem 35. Let X be a tvs-cone normed space and T be a weak contraction
(satisfying (w1)) with v = q ∈ F(T) 6= ϕ, in addition, whenever {yn} is a sequence
with lim

n→∞
‖yn+1 −Tyn‖K = θ, then the Picard iteration defined in (p1) is T -stable.

Proof. We will show that the sequence {yn} with lim
n→∞

‖yn+1 − ξ(T,xn)‖K = θ,
satisfies lim

n→∞
yn = q.

We have

‖yn+1 −q‖K 4 ‖yn+1 −Tyn‖K + ‖Tyn −q‖K
4 ‖yn+1 −Tyn‖K + δ‖yn −q‖K + L‖yn −Tyn‖K
= δ‖yn −q‖K + (‖yn+1 −Tyn‖K + L‖yn −Tyn‖K)
= δan + bn,

where an = ‖yn −q‖K and bn = (‖yn+1 −Tyn‖K + L‖yn −Tyn‖K).
Using Lemma 18, we have an → θ as n →∞. Thus, lim

n→∞
yn = q.

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1Department of Mathematics, COMSATS Institute of Information Technology, Chak

Shahzad, Islamabad - 44000, Pakistan

2Faculty of Science, University of Kragujevac, Radoja Domanovića 12, 34 000 Kragu-

jevac, Serbia

∗Corresponding author