On multivalued L-contractions and an application

The aim of this paper is to present several fixed-point results for L-contractive multivalued mappings involving θ-functions in the class of metric spaces. We also give some examples in support of the related concepts and presented results. A homotopy result is also provided.

Some years later, Vetro [23] introduced multivalued θ -contraction mappings and gave the multivalued version of the main results of Jleli and Samet [24].

Theorem 1.3 ([23])
Let (X, d) be a complete metric space and T : X → K(X) be a weak θ -contraction mapping. Then T admits a fixed point.
In [25], the authors gave the following fixed point theorem, extending the results of Jleli and Samet [24]. In [25] they considered θ ∈ * instead of θ ∈ . Recently, Cho [26] initiated the concept L-simulation functions. Let ξ : [1, ∞) 2 → R verify the following assertions: (ξ 1 ) ξ (1, 1) = 1; (ξ 2 ) ξ (τ , υ) < υ τ for all υ, τ > 1; (ξ 3 ) for all sequences {τ n } and {υ n } in (1, ∞) with τ n < υ n for n = 1, 2, 3, . . . , Any ξ ∈ L is said an L-simulation function. Here, ξ (ι, ι) < 1 for each ι > 1. As examples of L-simulation functions, we state the following: This class of L-simulation functions is used as control functions in order to enrich the fixed point theory when dealing with several types of contraction mappings in variant (generalized) metric spaces. In this paper, based on L-simulation functions, we define a new type of multivalued contraction mappings, called multivalued L-contraction mappings via θ -functions. We give some related fixed point results in the context of complete metric spaces. Some consequences are also derived. Moreover, an example to support our results is given. At the end, as an application, a homotopy result is provided.

Main results
Now, we introduce the definition of multivalued L-contraction mappings via θ -functions. Definition 2.1 Let (X, d) be a metric space. A multivalued mapping T : X → CB(X) is called an L-contraction with respect to ξ whenever there are θ ∈ * and ξ ∈ L such that

From (θ 1), one gets
As n → ∞, we find This proves that T is continuous.
Using (θ 1), we get Since θ is continuous and nondecreasing, one has Hence, there exists σ 2 ∈ Tσ 1 such that If σ 2 ∈ Tσ 2 , then σ 2 is a fixed point. Otherwise, proceeding similarly, there is σ 3 ∈ Tσ 2 such that Similarly, Hence {d(σ n-1 , σ n )} is decreasing, and so there is r ≥ 0 such that Assume that r > 0. Using (θ 1) and the continuity of θ , we get which is a contradiction. Therefore, Now, we will show that {σ i } is bounded in (X, d).
If it is not the case, then there is a subsequence {σ i(q) } of {σ i } such that for i(1) = 1 and for all q = 1, 2, . . . , we have that i(q + 1) is the minimum integer greater than i(q) with Letting q → ∞ and using (2.7), we get By taking the limit as q → ∞, from (2.7) and (2.9), we have Hence, using condition (2.1), we get Tσ i(q)-1 )) .

One then gets
By the continuity of θ , we have By taking the limit as q → ∞, we obtain Due to (ξ 3), we get which is a contradiction. Hence, {σ n } is bounded.
By the triangle inequality, we have As q → ∞, we find that Due to the continuity of θ , we obtain Letting q → ∞, we have This implies that H(Tσ i(q)-1 , Tσ j(q)-1 ) > 0. So by using condition (2.1), we have Tσ j(q)-1 )) .

Corollary 2.5
Let (X, d) be a complete metric space and T : X → K(X) be such that for all ς, τ ∈ X with H(Tς, Tτ ) = 0, where θ ∈ * . Then T possesses a unique fixed point.
Remark 2.7 Corollary 2.4 is the multivalued version of Theorem 3.2 in [24], and improves it by replacing the compact range condition by the closed and bounded range, and by considering that ϕ as lower semicontinuous and not necessary continuous. Corollary 2.5 is the multivalued version of Theorem 1.5. Also, Corollary 2.6 is an extension of Theorem 2.5 in [23] without condition (θ 3 ).
The following example supports Theorem 2.3. Here, the main result of Vetro [23] is not applicable.
Note that for all θ ∈ , we have That is, T is not an θ -contraction of Vetro [23].

Application
Now, we present a homotopy result as an application of Corollary 2.4. Next, we prove that A is closed. Let {u n } be a sequence in A such that lim n→∞ u n = u ∈ [0, 1]. Since {u n } ⊂ A, there is {s n } ⊂ E such that s n ∈ T(s n , u n ). Then, by using condition (iii) and for all m, n ∈ Z + , we have We deduce that s ∈ T(s, u), and by using condition (i), we have s ∈ E. Thus, u ∈ A, and so A is a closed subset of [0, 1]. Similarly, we can deduce the reverse implication.