Let (𝜀, db, s) be a b-metric space with s > 1. We say that a multivalued mapping is a multivalued almost (F, δb)-contraction if F ∈ 𝔉s (with parameter τ) and there exists λ ≥ 0 such that
(2)
for all x, y ∈ 𝜀 with min{δb,(𝒥x, 𝒥y), db (x, y)} > 0, where
(3)
and
If (2) is satisfied just for (for some x0 ∈ ε), we say that 𝒥 is a multivalued almost orbitally (F, δb)-contraction.
We are equipped now to state our first main result.
Starting from the given point x0, choose a sequence {xn}in ε such that xn+1 ∈ 𝒥xn, for all n ≥ 0. Now, if for some n0, then the proof is finished. Therefore, we assume xn ≠ xn+1for all n ≥ 0. So
db(xn+1, xn+2) > 0 andδb(𝒥xn, 𝒥xn+1) > 0 for all n ≥ 0.
Using the condition (2) for elements x = xn, y = xn+1, for arbitrary n ≥ 0 we have
where
and
As it follows that
Suppose that db(xn, xn+1)≤ db(xn+1, xn+2),for some positive integer n.Then from (4), we have
a contradiction with (Fl). Hence,
and consequently
(5)
It follows by (5) and the property (F4) that
(6)
Denote ϱn = d(xn, xn+1) for n = 0,1,2,.... Then, ϱn > 0 for all n and, using (6), the following holds:
(7)
for all n ∈ℕ. From (7), we get F(sn ϱn)→ —∞as n →∞. Thus, from (F2), we have
(8)
Now, by the property (F3) there exists k ∈(0, 1) such that
(9)
By (7), the following holds for all n ∈ℕ:
(10)
Passing to the limit as n →∞in (10) and using (8) and (9), we obtain
and hence limn →∞ n1/k sn ϱn = 0. Now, the last limit implies that the series is convergent and hence {xn}is a Cauchy sequence in 𝒪(x0;𝒥). Since 𝜀is 𝒥-orbitally complete, there exists a z ∈ 𝜀 such that
Suppose that 𝒥 z is closed.
We observe that if there exists an increasing sequence {nk}⊂ ℕ such that for all k ∈ℕ,since 𝒥 z is closed and , we deduce that z ∈ 𝒥 z and hence the proof is completed. Then we assume that there exists n0∈ℕ such that xn ∉𝒥 z for all n ∈ℕ with n ≥ n0. It follows that δb(𝒥xn,𝒥z) > 0 for all n ≥ n0. Using the condition (2) for x = xn, y = z, we have
where
and
Since F and Db are continuous, if Db(z, 𝒥 z) > 0, passing to the limit as n → ∞ in (11), we obtain
which is impossible since τ > 0, s ≥ 1 and F is strictly increasing. Hence, Db(z, 𝒥 z) = 0 and, since 𝒥 z is closed, we have z ∈ 𝒥 z. Thus, z is a fixed point of 𝒥.
Suppose that G(𝒥) is 𝒥-orbitally closed.
Since (xn,xn+1) ∈ G(𝒥) for all n ∈ ℕ ∪ {0} and limn → ∞ xn = z, we have (z, z) ∈ G(𝒥) by the 𝒥-orbitally closedness. Hence, z ∈ 𝒥 z.
It is proved that z is a fixed point of 𝒥. D
The following corollaries follow from Theorem 3.2 by taking F(α) = In α, resp. F(α) = α + In α in (2).
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