F-CONE METRIC SPACES OVER FR ´ECHET ALGEBRA
H. MEHRAVARAN ISLAMIC AZAD UNIVERSITY, MASHHAD, IRAN, R. ALLAHYARI ISLAMIC AZAD UNIVERSITY, MASHHAD, IRAN, AND H. AMIRI KAYVANLOO ISLAMIC AZAD UNIVERSITY, MASHHAD, IRAN
Abstract. The paper deals with the achievements of introducing the notion of F-cone metric spaces over Fr´echet algebra as a generalization of F-cone metric spaces over a Banach algebra. First, we study some of its topological properties. Next, we define a generalized Lipschitz for such spaces. Also, we investigate some fixed points for mappings satisfying such conditions in the new framework. Subsequently, as an application of our results, we provide an example. Our work generalizes some well-known results in the literature.
1. introduction
Malviya and Fisheret [9] introduced the concept ofN-cone metric spaces, which is a new generalization of the generalized G-cone metric [6] and the generalized D∗-metric spaces [2].
Following these ideas, very recently, Fernandez et al.[3] introduced F-cone metric spaces over a Banach algebra, which generalize Np-cone metric spaces over the Banach algebra andNb-cone metric spaces over the Banach algebra.
Now, in this paper, we introduce the notion ofF-cone metric spaces over a Fr´echet algebra as a generalization of F-cone metric spaces over the Banach algebra, Np-cone metric spaces over the Banach algebra, Next, we define a generalized Lipschitz for such spaces. Also, we investigate some fixed points for mappings satisfying such conditions in the new framework. Subsequently, as an application of our results, we provide an example.
2. Preliminaries
Throughout this paper, the notations R,R+, and N denote the set of all real numbers, the set of all nonnegative real numbers, and the set of all positive inte- gers, respectively.
Let A be a real Hausdorff topological vector space (tvs for short) with the zero vector . A proper nonempty and closed subset P of A is called a cone if P+P ⊂P, λP ⊂P forλ≥0 and P ∩(−P) = θ. We will always assume that the cone P has a nonempty interior intP such cones are called solid. Each cone P induces a partial order ⪯ on A by x ⪯y ⇔ y−x ∈ P. x ≺y will stand for
Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz.
∗ Corresponding author.
2010Mathematics Subject Classification. 46B20; 47H10 .
Key words and phrases. F-cone metric spaces over Fr´echet algebra;c-sequence; generalized Lipschitz mapping; fixed point.
1
2 H. MEHRAVARAN, R. ALLAHYARI, AND H. AMIRI KAYVANLOO
x⪯y and x ̸=y, while x≪y will stand for y−x∈intP. The pair (A, p) is an ordered topological vector space (see [7]).
A locally convex algebra is called locally multiplicatively convex if pα(xy) ≤ pα(x)pα(y) for allx, y∈A.A complete metrizable locally multiplicatively convex algebra is called aFr´echet algebra.
The topology of a Fr´echet algebraA can be generated by a sequence (pn)n of separating submultiplicative seminorm, that ispn(xy)≤pn(x)pn(y) for alln∈N and every x, y ∈A,such that pn(x) ≤ pn+1(x) for all x ∈A and n ∈N. If A is unital, then pn can be chosen such that pn(e) = 1. The Fr´echet algebra A with the above generating sequence of seminorm is denoted by (A,(pn)). Note that a sequence (xk) in the Fr´echet (A,(pn)) is convergent to x ∈ A if and only if pn(xk−x)→0,for each n∈N,as k→ ∞ (see [4]).
Example 2.1. [5, pp. 67–77 ] LetC(R) be the space of all continuous complex- valued functions. Then C(R) is a Fr´echet algebra with the seminorms ∥f∥n =
sup
|t|≤n
{|f(t)|}forn≥0.
Definition 2.2. LetXbe a nonempty set. Suppose that a mappingF :X3 →E is a function satisfying the following axioms:
(F1) θ⪯F(x, x, x) ⪯F(x, x, y) ⪯F(x, y, z), for allx, y, z,∈X with x̸=y ̸= z,
(F2) F(x, y, z)⪯s[F(x, x, a) +F(y, y, a) +F(z, z, a)]−F(a, a, a),
for all x, y, z, a ∈ X. Then the pair (X, F) is called an F-cone metric space over Fr´echet algebraE.The numbers≥1 is called the coefficient of (X, F).
Now we give some examples ofF-cone metric spaces over Frchet algebras.
Example 2.3. By using Example2.1,A=C(R) is a Fr´echet algebra with respect to the seminorm (pn)n∈N,given by
pn(f) = sup
|x|≤n
|f(x)|
forn≥0. Also, the constant function 1 acts as an identity. Set {f ∈A:f(t)≥ 0, t∈R} as a cone inA. Suppose that X=R. Define the mappingF :X3 →A by F(x, y, z)(t) =
(|x2 −y2|+|y2−z2|+|x2−z2|)
et for all x, y, z ∈ X. Thus (X, F) withs= 1 is anF-cone metric space over Fr´echet algebraA.
Theorem 2.4. [8]Let (X, F) be an F-cone metric space over a Fr´echet algebra Aand let P be a solid cone inA.Then (X, F) is a Hausdorff space.
Now, we define a θ-Cauchy sequence and a convergent sequence in anF-cone metric space over a Fr´echet algebra A.
Definition 2.5. Let (X, F) be anF-cone metric space over a Fr´echet algebraA. A sequence{xq} in (X, F) converges to a pointx∈X whenever for everyc≫θ there is a natural numberN such thatF(xq, x, x)≪c for all q≥N.We denote this by lim
q→∞xq=x orxq→x asq→ ∞.
F-CONE METRIC SPACES OVER FR ´ECHET ALGEBRA 3
Definition 2.6. The sequence{xq}is aθ-Cauchy sequence in (X, F) if{F(xq, xp, xp)}
is a c-sequence in A, that is, if for every c ≫ θ there exists q0 ∈ N such that F(xq, xp, xp)≪c for all q, p≥q0.
Definition 2.7. The space (X, F) isθ-complete if everyθ-Cauchy sequence con- verges tox∈X such that F(x, x, x) =θ.
Definition 2.8. Let (X, F) be anF-cone metric space with the coefficientsover a Fr´echet algebra Aand letP be a cone inA.A mapT :X→X is said to be a generalized Lipschitz mapping if there exists a vectork ∈P with ρ(k) <1 (the spectral radius) such that
F(T x, T x, T y)⪯kF(x, x, y) for allx, y∈X.
Example 2.9. Let the Fr´echet algebra A, the cone P, and the mapping F : X3 → A be the same ones as those in Example 2.3. Then (X, F) is an F-cone metric space over the Fr´echet algebraA.Now, we define the mappingT :X →X byT(x) = x3.We haveF(T x, T x, T y) = 2|x29−y2|et⪯ 29|x2−y2|et= 19F(x, x, y)(t) fork= 19.Then T is a generalized Lipschitz map inX.
Proposition 2.10. [8]Let A be a Fr´echet algebra with a cone P and k∈P such thatρ(k)<1. Then
( pn(k)
)q
→0 asq → ∞.
Lemma 2.11. [8]Let A be a Fr´echet algebra with a solid cone P. Suppose that {xq} is a sequence inA such thatpn(xq)→0 as q→ ∞;thenxq is ac-sequence.
Lemma 2.12. [10] Let E be a topological vector space with a tvs-cone p. Then the following properties hold:
(1) If a≫θ, then ra≫θ for each r∈R+.
(2) If a1 ≫β1 anda2≥β2, thena1+a2 ≫β1+β2 anda2≥β2⇔a2−β2 ≥ θ⇔a2−β2∈p.
Lemma 2.13. [1] Let (E, P) be an ordered TVS. Then if x∈ P and y ∈intP, thenx+y∈intP. Consequently, ifx≤y and y≪z, then x≪z (x≤y, which we say “x is less then y”, if y−x∈p).
3. Applications to fixed point theory
In this section, we prove fixed point theorems for generalized Lipschitz maps on anF-cone metric space over a Fr´echet algebra.
Theorem 3.1. [8]Let (X, F)be aθ-completeF-cone metric space over a Fr´echet algebraA and let P be a solid cone in A. Let k∈P be a a generalized Lipschitz constant with ρ(k) < 1 and let the mapping T : X → X satisfy the following condition
F(T x, T x, T y)⪯kF(x, x, y)
for allx, y ∈X. Moreover, (e−2s2k)≻θ. Then, T has a unique fixed point in X. For each x∈X, the sequence of iterates {Tqx} converges to the fixed point.
4 H. MEHRAVARAN, R. ALLAHYARI, AND H. AMIRI KAYVANLOO
Example 3.2. Choose Example2.9. Therefore (X, F) is anF-cone metric space over the Fr´echet algebra A and the mapping T : X → X by T(x) = x3 is a a generalized Lipschitz with k = 19. Also, we get k = λe = 19. Therefore λ = 19 < 18 = 2s12. Hence, the conditions of Theorem 3.1 hold. Thus T has a unique fixed point 0.
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hamid mehravaran@mshdiau.ac.ir rezaallahyari@mshdiau.ac.ir amiri.hojjat93@mshdiau.ac.ir
