Multivalued Pseudo-Picard Operators and Fixed Point Results

Abstract

We introduce the concept of multivalued pseudo-Picard (MPP) operator on a metric space. This concept is weaker than multivalued weakly Picard (MWP) operator, which is given by M. Berinde and V. Berinde (2007). Then, we give both fixed point results and examples for MPP operators. Also, we obtain some ordered fixed point results for multivalued maps as application.

1. Introduction and Preliminaries

Let (𝑋,𝑑) be a metric space, and let 𝐶𝐵(𝑋) denote the class of all nonempty, closed, and bounded subsets of 𝑋. It is well known that 𝐻∶𝐶𝐵(𝑋)×𝐶𝐵(𝑋)→ℝ defined by 𝐻(𝐴,𝐵)=maxsup_𝑥∈𝐴𝐷(𝑥,𝐵),sup_𝑦∈𝐵𝐷𝑦,𝐴 (1) is a metric on 𝐶𝐵(𝑋), which is called Hausdorff metric, where 𝐷(𝑥,𝐵)=inf𝑑𝑥,𝑦∶𝑦∈𝐵. (2) Let 𝑇∶𝑋→𝐶𝐵(𝑋) be a map; then, 𝑇 is called multivalued contraction if for all 𝑥,𝑦∈𝑋 there exists 𝛿∈[0,1) such that 𝐻𝑇𝑥,𝑇𝑦≤𝛿𝑑𝑥,𝑦. (3)

In 1969, Nadler [1] proved a fundamental fixed point theorem for multivalued maps: every multivalued contraction on complete metric space has a fixed point.

Then, a lot of generalizations of the result of Nadler were given (see, e.g., [2–4]). Two important generalizations of it were given by M. Berinde and V. Berinde [5] and Mizoguchi and Takahashi [6].

In [5], M. Berinde and V. Berinde introduced the concept of multivalued weakly Picard operator as follows (for single-valued Picard and weakly Picard operators we refer to [7–9]).

Definition 1. Let (𝑋,𝑑) be a metric space, and let 𝑇∶𝑋→𝒫(𝑋) (the family of all nonempty subsets of 𝑋) be a multivalued operator. 𝑇 is said to be multivalued weakly Picard (MWP) operator if and only if for each 𝑥∈𝑋 and any 𝑦∈𝑇𝑥, there exists a sequence {𝑥_𝑛} in 𝑋 such that(i)𝑥₀=𝑥,𝑥₁=𝑦, (ii)𝑥_𝑛+1∈𝑇𝑥_𝑛, (iii)the sequence {𝑥_𝑛} is convergent and its limit is a fixed point of 𝑇.

Then M. Berinde and V. Berinde [5] show that every Nadler [1], Reich [10], Rus [11] and Petruşel [12] type multivalued contractions on complete metric space are MWP operators. Mizoguchi and Takahashi [6], proved the following fixed point theorem. This is also an example of MWP operator.

Theorem 2. Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝐶𝐵(𝑋) be a multivalued map. Assume that 𝐻𝑇𝑥,𝑇𝑦≤𝑘𝑑𝑥,𝑦𝑑𝑥,𝑦 (4) for all 𝑥,𝑦∈𝑋, where 𝑘 is an ℳ𝒯-function (i.e., it satisfies limsup_𝑠→𝑡+𝑘(𝑠)<1 (5) for all 𝑡∈[0,∞)). Then 𝑇 is an MWP operator.

In the same paper, M. Berinde and V. Berinde [5] introduced the concepts of multivalued (𝛿,𝐿)-weak contraction and multivalued (𝑘,𝐿)-weak contraction and proved the following nice fixed point theorems.

Theorem 3. Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝐶𝐵(𝑋) be a multivalued (𝛿,𝐿)-weak contraction; that is, there exist two constants 𝛿∈[0,1) and 𝐿≥0 such that 𝐻𝑇𝑥,𝑇𝑦≤𝛿𝑑𝑥,𝑦+𝐿𝐷𝑦,𝑇𝑥 (6) for all 𝑥,𝑦∈𝑋. Then 𝑇 is an MWP operator.

Theorem 4. Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝐶𝐵(𝑋) be a multivalued (𝑘,𝐿)-weak contraction; that is, there exist an ℳ𝒯-function 𝑘 and a constant 𝐿≥0 such that 𝐻𝑇𝑥,𝑇𝑦≤𝑘𝑑𝑥,𝑦𝑑𝑥,𝑦+𝐿𝐷𝑦,𝑇𝑥 (7) for all 𝑥,𝑦∈𝑋. Then 𝑇 is an MWP operator.

We can find some detailed information about the single-valued case of (𝛿,𝐿)-weak contraction and the nonlinear case of it in [13–15].

Recently, Samet et al. [16] introduced the notion of 𝛼-𝜓-contractive mappings and gave some fixed point results for such mappings. Their results are closely related to some ordered fixed point results. Then, using their idea, some authors presented fixed point results for single and multivalued mappings (see, e.g., [16–20]). First, we recall these results. Denote by Ψ the family of nondecreasing functions 𝜓∶[0,∞)→[0,∞) such that ∑^∞_𝑛=1𝜓^𝑛(𝑡)<∞ for all 𝑡>0.

Definition 5 (see [16]). Let (𝑋,𝑑) be a metric space, 𝑇 be a self map on 𝑋, 𝜓∈Ψ and 𝛼∶𝑋×𝑋→[0,∞) be a function. Then 𝑇 is called 𝛼-𝜓-contractive whenever 𝛼𝑥,𝑦𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑑𝑥,𝑦 (8) for all 𝑥,𝑦∈𝑋.

Note that every Banach contraction mapping is an 𝛼-𝜓-contractive mapping with 𝛼(𝑥,𝑦)=1 and 𝜓(𝑡)=𝛿𝑡 for some 𝛿∈[0,1).

Definition 6 (see [16]). 𝑇 is called 𝛼-admissible whenever 𝛼(𝑥,𝑦)≥1 implies that 𝛼(𝑇𝑥,𝑇𝑦)≥1.

There exist some examples for 𝛼-admissible mappings in [16]. For convenience, we mention in here one of them. Let 𝑋=[0,∞). Define 𝑇∶𝑋→𝑋 and 𝛼∶𝑋×𝑋→[0,∞) by 𝑇𝑥=√𝑥 for all 𝑥∈𝑋 and 𝛼(𝑥,𝑦)=𝑒^𝑥−𝑦 for 𝑥≥𝑦 and 𝛼(𝑥,𝑦)=0 for 𝑥<𝑦. Then 𝑇 is 𝛼-admissible.

Definition 7 (see [17]). 𝛼 is said to have (B) property whenever {𝑥_𝑛} is a sequence in 𝑋 such that 𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1 for all 𝑛∈ℕ and 𝑥_𝑛→𝑥, then 𝛼(𝑥_𝑛,𝑥)≥1 for all 𝑛∈ℕ.

Theorem 8 (see [16]). Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝑋 be an 𝛼-admissible and 𝛼-𝜓-contractive mapping. If there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1 and 𝑇 is continuous, then 𝑇 has a fixed point.

Remark 9. If we assume that 𝛼 has (B) property instead of the continuity of 𝑇, then again 𝑇 has a fixed point (Theorem 2.2 of [16]). If for each 𝑥,𝑦∈𝑋 there exists 𝑧∈𝑋 such that 𝛼(𝑥,𝑧)≥1 and 𝛼(𝑦,𝑧)≥1, then 𝑋 is said to have (H) property. Therefore, if 𝑋 has the (H) property in Theorems 2.1 and 2.2 in [16], then the fixed point of 𝑇 is unique (Theorem 2.3 of [16]).

Then, some generalizations of 𝛼-𝜓-contractive mappings are given as follows.

Definition 10 (see [17]). 𝑇 is called Ćirić type 𝛼-𝜓-generalized contractive mapping whenever 𝛼𝑥,𝑦𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑚𝑥,𝑦 (9) for all 𝑥,𝑦∈𝑋, where 𝑚𝑑(),1𝑥,𝑦=max𝑥,𝑦,𝑑𝑥,𝑇𝑥,𝑑𝑦,𝑇𝑦2𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥. (10)

Note that every Ćirić type generalized contraction mapping is a Ćirić type 𝛼-𝜓-generalized contractive mapping with 𝛼(𝑥,𝑦)=1 and 𝜓(𝑡)=𝜆𝑡 for some 𝜆∈[0,1).

Theorem 11 (see [17]). Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝑋 be an 𝛼-admissible and Ćirić type 𝛼-𝜓-generalized contractive mapping. If there exists 𝑥₀∈𝑋 such that 𝛼(𝑥₀,𝑇𝑥₀)≥1 and 𝑇 is continuous or 𝛼 has (B) property, then 𝑇 has a fixed point. If 𝑋 has the (H) property, then the fixed point of 𝑇 is unique.

We can find some fixed point results for single-valued mappings in these directions in [18, 20]. Now we recall some multivalued cases.

Definition 12 (see [17, 19]). Let (𝑋,𝑑) be a metric space, and let 𝑇∶𝑋→𝐶𝐵(𝑋) be a multivalued mapping. Then 𝑇 is called multivalued 𝛼-𝜓-contractive whenever 𝛼𝑥,𝑦𝐻𝑇𝑥,𝑇𝑦≤𝜓𝑑𝑥,𝑦 (11) for all 𝑥,𝑦∈𝑋, and 𝑇 is called multivalued 𝛼_∗-𝜓-contractive whenever 𝛼_∗𝑇𝑥,𝑇𝑦𝐻𝑇𝑥,𝑇𝑦≤𝜓𝑑𝑥,𝑦, (12) where 𝛼_∗(𝑇𝑥,𝑇𝑦)=inf{𝛼(𝑎,𝑏)∶𝑎∈𝑇𝑥,𝑏∈𝑇𝑦}. Similarly if we replace 𝑑(𝑥,𝑦) with 𝑚(𝑥,𝑦) we can obtain Ćirić type multivalued 𝛼-𝜓-generalized contractive and Ćirić type multivalued 𝛼_∗-𝜓-generalized contractive mappings on 𝑋.

Definition 13 (see [17, 19]). Let (𝑋,𝑑) be a metric space, and let 𝑇∶𝑋→𝐶𝐵(𝑋) be a multivalued mapping.(a)𝑇 is said to be 𝛼-admissible whenever each 𝑥∈𝑋 and 𝑦∈𝑇𝑥 with 𝛼(𝑥,𝑦)≥1 imply that 𝛼(𝑦,𝑧)≥1 for all 𝑧∈𝑇𝑦.(b)𝑇 is said to be 𝛼_∗-admissible whenever each 𝑥∈𝑋 and 𝑦∈𝑇𝑥 with 𝛼(𝑥,𝑦)≥1 imply that 𝛼_∗(𝑇𝑥,𝑇𝑦)≥1.

Remark 14. It is clear that 𝛼_∗-admissible maps are also 𝛼-admissible, but the converse may not be true as shown in the following example.

Example 15. Let 𝑋=[−1,1], and 𝛼∶𝑋×𝑋→[0,∞) is defined by 𝛼(𝑥,𝑥)=0 and 𝛼(𝑥,𝑦)=1 for 𝑥≠𝑦. Define 𝑇∶𝑋→𝐶𝐵(𝑋) by {}{}{}{1}𝑇𝑥=−𝑥,𝑥∉−1,00,1,𝑥=−1,𝑥=0. (13) Leting 𝑥=−1 and 𝑦=0∈𝑇𝑥={0,1}, then 𝛼(𝑥,𝑦)≥1, but 𝛼_∗(𝑇𝑥,𝑇𝑦)=𝛼_∗({0,1},{1})=0. Thus, 𝑇 is not 𝛼_∗-admissible. Now we show that, 𝑇 is 𝛼-admissible with the following cases.
Case 1. If 𝑥=0, then 𝑦=1 and 𝛼(𝑥,𝑦)≥1. Also, 𝛼(𝑦,𝑧)≥1 since 𝑧=−1∈𝑇𝑦={−1}.
Case 2. If 𝑥=−1, then 𝑦∈{0,1} and 𝛼(𝑥,𝑦)≥1. Also, 𝛼(𝑦,𝑧)≥1 for all 𝑧∈𝑇𝑦.
Case 3. If 𝑥∉{−1,0}, then 𝑦=−𝑥 and 𝛼(𝑥,𝑦)≥1. Also, 𝛼(𝑦,𝑧)≥1 since 𝑧=𝑥∈𝑇𝑦={𝑥}.

The multivalued version of the results for 𝛼-𝜓-contractive mappings is given [17, 19] as follows.

Theorem 16 (see [19]). Let (𝑋,𝑑) be a complete metric space, let 𝛼∶𝑋×𝑋→[0,∞) be a function, let 𝜓∈Ψ be a strictly increasing map, and let 𝑇∶𝑋→𝐶𝐵(𝑋), 𝛼-admissible and 𝛼-𝜓-contractive multifunction on 𝑋. Suppose that there exist 𝑥₀∈𝑋 and 𝑥₁∈𝑇𝑥₀ such that 𝛼(𝑥₀,𝑥₁)≥1. Assume that 𝑇 is continuous or 𝛼 has (B) property; then 𝑇 has a fixed point.

Theorem 17 (see [17]). Let (𝑋,𝑑) be a complete metric space, let 𝛼∶𝑋×𝑋→[0,∞) be a function, let 𝜓∈Ψ be a strictly increasing map, and let 𝑇∶𝑋→𝐶𝐵(𝑋), 𝛼_∗-admissible and 𝛼_∗-𝜓-contractive multifunction on 𝑋. Suppose that there exist 𝑥₀∈𝑋 and 𝑥₁∈𝑇𝑥₀ such that 𝛼(𝑥₀,𝑥₁)≥1. Assume that 𝛼 has (B) property. then 𝑇 has a fixed point.

The purpose of this paper is to introduce the concept of multivalued pseudo-Picard (MPP) operators and present fixed point results and examples.

2. Main Results

Definition 18. Let (𝑋,𝑑) be a metric space, and let 𝑇∶𝑋→𝒫(𝑋) be a multivalued operator. 𝑇 is said to be multivalued pseudo-Picard (MPP) operator if and only if there exist 𝑥₀∈𝑋, 𝑥₁∈𝑇𝑥₀ and a sequence {𝑥_𝑛} in 𝑋 such that(i)𝑥_𝑛+1∈𝑇𝑥_𝑛, (ii)the sequence {𝑥_𝑛} is convergent and its limit is a fixed point of 𝑇.

Remark 19. It is clear that the operators mentioned in Theorems 16 and 17 are MPP operators. Also, note that every MWP operator is an MPP operator, but the converse may not be true as shown in the following examples.

Example 20. Let 𝑋=[−1,1] and 𝑑(𝑥,𝑦)=|𝑥−𝑦|. Define 𝑇∶𝑋→𝐶𝐵(𝑋) by 𝑇𝑥=0,𝑥²(){}{}.,𝑥∈−1,1−𝑥,𝑥∈−1,1 (14) Then 𝑇 is not an MWP operator. Indeed, letting 𝑥=1, then 𝑇𝑥={−1} and so 𝑦=−1. Therefore, 𝑥_𝑛=(−1)^𝑛 for 𝑛≥0, which is not convergent. But 𝑇 is MPP operator. To see this, let 𝑥₀=1/2 and 𝑥₁=(1/4)∈𝑇𝑥₀={0,1/4}. Continuing this way, we can construct a sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛+1∈𝑇𝑥_𝑛 such that it is convergent to 0, which is a fixed point of 𝑇.

Example 21. Let 𝑋=[0,∞) and 𝑑(𝑥,𝑦)=|𝑥−𝑦|. Define 𝑇∶𝑋→𝐶𝐵(𝑋) by 𝑥𝑇𝑥=0,2[],0≤𝑥≤1𝑥+1,𝑥+2,𝑥>1. (15) Then 𝑇 is an MPP but not MWP operator.

Before we give our main results, we recall the following. Let 𝑋 and 𝑌 be two topological spaces. Then a multivalued map 𝑇∶𝑋→𝒫(𝑌) is said to be upper semicontinuous (lower semicontinuous) if the inverse image of a closed set (open set) is closed (open). A multivalued map is continuous if it is upper as well as lower semicontinuous.

Lemma 22 (see [21]). Let (𝑋,𝑑) be a metric space, and let 𝑇∶𝑋→𝒫(𝑋) be an upper semicontinuous map such that 𝑇𝑥 is closed for all 𝑥∈𝑋. If 𝑥_𝑛→𝑥₀,𝑦_𝑛→𝑦₀ and 𝑦_𝑛∈𝑇𝑥_𝑛, then 𝑦₀∈𝑇𝑥₀.

Theorem 23. Let (𝑋,𝑑) be a complete metric space and let 𝑇∶𝑋→𝐶𝐵(𝑋) be an 𝛼-admissible multivalued mapping such that 𝛼𝐻𝑑𝐷𝑥,𝑦𝑇𝑥,𝑇𝑦≤𝜓𝑥,𝑦+𝐿min𝑥,𝑇𝑦,𝐷𝑦,𝑇𝑥 (16) for all 𝑥,𝑦∈𝑋, where 𝜓∈Ψ is strictly increasing and 𝐿≥0. Suppose that there exist 𝑥₀∈𝑋 and 𝑥₁∈𝑇𝑥₀ such that 𝛼(𝑥₀,𝑥₁)≥1. If 𝑇 is upper semicontinuous or 𝛼 has (B) property, then 𝑇 is an MPP operator.

Proof. Let 𝑥₀ and 𝑥₁ be as mentioned in the hypotheses. If 𝑥₀=𝑥₁ or 𝑥₁∈𝑇𝑥₁, then the proof is complete. Let 𝑥₀≠𝑥₁ and 𝑥₁∉𝑇𝑥₁, then 0<𝐷𝑥₁,𝑇𝑥₁≤𝛼𝑥₀,𝑥₁𝐻𝑇𝑥₀,𝑇𝑥₁<𝑞𝛼𝑥₀,𝑥₁𝐻𝑇𝑥₀,𝑇𝑥₁, (17) where 𝑞>1 is a constant. Therefore, there exists 𝑥₂∈𝑇𝑥₁ such that 0<𝑑𝑥₁,𝑥₂<𝑞𝛼𝑥₀,𝑥₁𝐻𝑇𝑥₀,𝑇𝑥₁≤𝑞𝜓𝑑𝑥₀,𝑥₁+𝑞𝐿min𝐷𝑥₀,𝑇𝑥₁,𝐷𝑥₁,𝑇𝑥₀=𝑞𝜓𝑑𝑥₀,𝑥₁. (18) Also, since 𝑇 is 𝛼-admissible, 𝑥₁∈𝑇𝑥₀, and 𝛼(𝑥₀,𝑥₁)≥1, then 𝛼(𝑥₁,𝑢)≥1 for all 𝑢∈𝑇𝑥₁ and so 𝛼(𝑥₁,𝑥₂)≥1. Since 𝜓 is strictly increasing, we have 0<𝜓𝑑𝑥₁,𝑥₂<𝜓𝑞𝜓𝑑𝑥₀,𝑥₁. (19) Get 𝑞₁=𝜓(𝑞𝜓(𝑑(𝑥₀,𝑥₁)))/𝜓(𝑑(𝑥₁,𝑥₂)). Then 𝑞₁>1. If 𝑥₂∈𝑇𝑥₂, then the proof is complete. Let 𝑥₂∉𝑇𝑥₂. Then 0<𝐷𝑥₂,𝑇𝑥₂≤𝛼𝑥₁,𝑥₂𝐻𝑇𝑥₁,𝑇𝑥₂<𝑞₁𝛼𝑥₁,𝑥₂𝐻𝑇𝑥₁,𝑇𝑥₂. (20) Therefore, there exists 𝑥₃∈𝑇𝑥₂ such that 0<𝑑𝑥₂,𝑥₃<𝑞₁𝛼𝑥₁,𝑥₂𝐻𝑇𝑥₁,𝑇𝑥₂≤𝑞₁𝜓𝑑𝑥₁,𝑥₂+𝑞₁𝐿min𝐷𝑥₁,𝑇𝑥₂,𝐷𝑥₂,𝑇𝑥₁=𝑞₁𝜓𝑑𝑥₁,𝑥₂=𝜓𝑞𝜓𝑑𝑥₀,𝑥₁. (21) Since 𝑇 is 𝛼-admissible, 𝑥₂∈𝑇𝑥₁, and 𝛼(𝑥₁,𝑥₂)≥1, then 𝛼(𝑥₂,𝑢)≥1 for all 𝑢∈𝑇𝑥₂. Thus, 𝛼(𝑥₂,𝑥₃)≥1 since 𝑥₃∈𝑇𝑥₂. Since 𝜓 is strictly increasing, we have 0<𝜓𝑑𝑥₂,𝑥₃<𝜓²𝑞𝜓𝑑𝑥₀,𝑥₁. (22) Get 𝑞₂=𝜓²(𝑞𝜓(𝑑(𝑥₀,𝑥₁)))/𝜓(𝑑(𝑥₁,𝑥₂)). Then 𝑞₂>1. If 𝑥₃∈𝑇𝑥₃, then the proof is complete. By the way, we can construct a sequence {𝑥_𝑛} in 𝑋 such that 𝑥_𝑛+1∈𝑇𝑥_𝑛,𝑥_𝑛≠𝑥_𝑛+1,𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1, and 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝜓^𝑛𝑞𝜓𝑑𝑥₀,𝑥₁ (23) for all 𝑛∈ℕ. Now, for each 𝑚,𝑛∈ℕ,𝑚>𝑛, we have 𝑑𝑥_𝑛,𝑥_𝑚≤^𝑚−1_𝑘=𝑛𝑑𝑥_𝑘,𝑥_𝑘+1≤^𝑚−1_𝑘=𝑛𝜓^𝑘𝑞𝜓𝑑𝑥₀,𝑥₁. (24) Therefore, {𝑥_𝑛} is a Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists 𝑧∈𝑋 such that lim_𝑛→∞𝑥_𝑛=𝑧. If 𝑇 is upper semicontinuous, then from Lemma 22, we have 𝑧∈𝑇𝑧. Now assume that 𝛼 has (B) property. Then 𝛼(𝑥_𝑛,𝑧)≥1 for all 𝑛∈ℕ. Also, since lim_𝑛→∞𝑑(𝑥_𝑛,𝑧)=0 and 𝜓 is continuous at 0, then 𝐷𝑥_𝑛+1,𝑇𝑧≤𝐻𝑇𝑥_𝑛𝑥,𝑇𝑧≤𝛼_𝑛,𝑧𝐻𝑇𝑥_𝑛𝑑𝑥,𝑇𝑧≤𝜓_𝑛𝐷𝑥,𝑧+𝐿min_𝑛,𝑇𝑧,𝐷𝑧,𝑇𝑥_𝑛≤𝜓𝑑𝑥_𝑛𝐷𝑥,𝑧+𝐿min_𝑛,𝑇𝑧,𝑑𝑧,𝑥_𝑛+1⟶0. (25) Therefore, we have 𝐷(𝑧,𝑇𝑧)=0 and so 𝑧∈𝑇𝑧.

Although 𝛼_∗-admissibility implies 𝛼-admissibility of 𝑇, we will give the following theorem because the contractive condition is slightly different from (16).

Theorem 24. Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝐶𝐵(𝑋) be an 𝛼_∗-admissible multivalued mapping such that 𝛼_∗𝐻𝑑𝐷𝑇𝑥,𝑇𝑦𝑇𝑥,𝑇𝑦≤𝜓𝑥,𝑦+𝐿min𝑥,𝑇𝑦,𝐷𝑦,𝑇𝑥 (26) for all 𝑥,𝑦∈𝑋, where 𝜓∈Ψ is strictly increasing and 𝐿≥0. Suppose that there exist 𝑥₀∈𝑋 and 𝑥₁∈𝑇𝑥₀ such that 𝛼(𝑥₀,𝑥₁)≥1. If 𝑇 is upper semicontinuous or 𝛼 has (B) property, then 𝑇 is an MPP operator.

Proof. Let 𝑥₀ and 𝑥₁ be as mentioned in the hypotheses, then 𝛼_∗(𝑇𝑥₀,𝑇𝑥₁)≥1 since 𝑇 is 𝛼_∗-admissible. If 𝑥₀=𝑥₁ or 𝑥₁∈𝑇𝑥₁, then the proof is complete. Let 𝑥₀≠𝑥₁ and 𝑥₁∉𝑇𝑥₁, then 0<𝐷𝑥₁,𝑇𝑥₁≤𝛼_∗𝑇𝑥₀,𝑇𝑥₁𝐻𝑇𝑥₀,𝑇𝑥₁<𝑞𝛼_∗𝑇𝑥₀,𝑇𝑥₁𝐻𝑇𝑥₀,𝑇𝑥₁, (27) where 𝑞>1 is a constant. Therefore, there exists 𝑥₂∈𝑇𝑥₁ such that 0<𝑑𝑥₁,𝑥₂<𝑞𝛼_∗𝑇𝑥₀,𝑇𝑥₁𝐻𝑇𝑥₀,𝑇𝑥₁≤𝑞𝜓𝑑𝑥₀,𝑥₁+𝑞𝐿min𝐷𝑥₀,𝑇𝑥₁,𝐷𝑥₁,𝑇𝑥₀=𝑞𝜓𝑑𝑥₀,𝑥₁. (28) Since 𝛼(𝑥₁,𝑥₂)≥𝛼_∗(𝑇𝑥₀,𝑇𝑥₁)≥1, then 𝛼_∗(𝑇𝑥₁,𝑇𝑥₂)≥1. Since 𝜓 is strictly increasing, 0<𝜓𝑑𝑥₁,𝑥₂<𝜓𝑞𝜓𝑑𝑥₀,𝑥₁. (29) Get 𝑞₁=𝜓(𝑞𝜓(𝑑(𝑥₀,𝑥₁)))/𝜓(𝑑(𝑥₁,𝑥₂)). Then 𝑞₁>1. If 𝑥₂∈𝑇𝑥₂, then the proof is complete. Let 𝑥₂∉𝑇𝑥₂. Then 0<𝐷𝑥₂,𝑇𝑥₂≤𝛼_∗𝑇𝑥₁,𝑇𝑥₂𝐻𝑇𝑥₁,𝑇𝑥₂<𝑞₁𝛼_∗𝑇𝑥₁,𝑇𝑥₂𝐻𝑇𝑥₁,𝑇𝑥₂. (30) Therefore, there exists 𝑥₃∈𝑇𝑥₂ such that 0<𝑑𝑥₂,𝑥₃<𝑞₁𝛼_∗𝑇𝑥₁,𝑇𝑥₂𝐻𝑇𝑥₁,𝑇𝑥₂≤𝑞₁𝜓𝑑𝑥₁,𝑥₂+𝑞₁𝐿min𝐷𝑥₁,𝑇𝑥₂,𝐷𝑥₂,𝑇𝑥₁=𝑞₁𝜓𝑑𝑥₁,𝑥₂=𝜓𝑞𝜓𝑑𝑥₀,𝑥₁. (31) Since 𝛼(𝑥₂,𝑥₃)≥𝛼_∗(𝑇𝑥₁,𝑇𝑥₂)≥1, then 𝛼_∗(𝑇𝑥₂,𝑇𝑥₃)≥1. By the way, we can construct a sequence {𝑥_𝑛} in 𝑋 such that 𝑥_𝑛+1∈𝑇𝑥_𝑛,𝑥_𝑛≠𝑥_𝑛+1,𝛼(𝑥_𝑛,𝑥_𝑛+1)≥1, and 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝜓^𝑛𝑞𝜓𝑑𝑥₀,𝑥₁ (32) for all 𝑛∈ℕ. Now, for each 𝑚,𝑛∈ℕ,𝑚>𝑛, we have 𝑑𝑥_𝑛,𝑥_𝑚≤^𝑚−1_𝑘=𝑛𝑑𝑥_𝑘,𝑥_𝑘+1≤^𝑚−1_𝑘=𝑛𝜓^𝑘𝑞𝜓𝑑𝑥₀,𝑥₁. (33) Therefore, {𝑥_𝑛} is a Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists 𝑧∈𝑋 such that lim_𝑛→∞𝑥_𝑛=𝑧. If 𝑇 is upper semicontinuous, then from Lemma 22, we have 𝑧∈𝑇𝑧. Now assume that 𝛼 has (B) property. Then 𝛼(𝑥_𝑛,𝑧)≥1 for all 𝑛∈ℕ. Since 𝑇 is 𝛼_∗-admissible, 𝛼_∗(𝑇𝑥_𝑛,𝑇𝑧)≥1. Therefore, 𝐷𝑥_𝑛+1,𝑇𝑧≤𝐻𝑇𝑥_𝑛,𝑇𝑧≤𝛼_∗𝑇𝑥_𝑛,𝑇𝑧𝐻𝑇𝑥_𝑛𝑑𝑥,𝑇𝑧≤𝜓_𝑛𝐷𝑥,𝑧+𝐿min_𝑛,𝑇𝑧,𝐷𝑧,𝑇𝑥_𝑛≤𝜓𝑑𝑥_𝑛𝐷𝑥,𝑧+𝐿min_𝑛,𝑇𝑧,𝑑𝑧,𝑥_𝑛+1, (34) and taking limit 𝑛→∞ we have 𝐷(𝑧,𝑇𝑧)=0 and so 𝑧∈𝑇𝑧.

Remark 25. If we take 𝛼∶𝑋×𝑋→[0,∞) by 𝛼(𝑥,𝑦)=1, then any multivalued mappings 𝑇∶𝑋→𝐶𝐵(𝑋) are 𝛼-admissible as well as 𝛼_∗-admissible. Therefore, Theorem 3 is a special case of Theorems 23 and 24.

Remark 26. If we take 𝐿=0 in Theorems 23 and 24, then we have Theorems 16 and 17, respectively.

Now we give an example to illustrate our main results. Note that both Theorems 3 and 16 cannot be applied to this example.

Example 27. Let 𝑋=[0,1]∪{2,3} and 𝑑(𝑥,𝑦)=|𝑥−𝑦|. Define 𝑇∶𝑋→𝐶𝐵(𝑋) by 𝑥𝑇𝑥=0,21,𝑥∈0,214,3𝑥−121,𝑥∈2{𝑥}{},1,𝑥∈2,3 (35) and 𝛼∶𝑋×𝑋→[0,∞) by 𝛼=1𝑥,𝑦1,𝑥,𝑦∈0,2∪{2,3}0,otherwise. (36) Then 𝑇 is an 𝛼-admissible, and condition (16) is satisfied for 𝜓(𝑡)=𝑡/2 and 𝐿=1. Indeed, first, we show that 𝑇 is an 𝛼-admissible. Letting 𝑥∈𝑋 and 𝑦∈𝑇𝑥 with 𝛼(𝑥,𝑦)≥1, then it should be 𝑥∈[0,1/2]∪{2,3}. Thus, 𝑦∈𝑇𝑥⊂[0,1/2]∪{2,3}, and hence 𝛼(𝑦,𝑧)≥1 for all 𝑧∈𝑇𝑦⊂[0,1/2]∪{2,3}. Therefore, 𝑇 is an 𝛼-admissible.
Now we consider the following cases.
Case 1. Letting 𝑥,𝑦∈𝑋 with {𝑥,𝑦}∩(1/2,1]≠∅, then 𝛼(𝑥,𝑦)=0. Thus, (16) is satisfied. Also note that if 𝑥=𝑦, then 𝐻(𝑇𝑥,𝑇𝑦)=0 and so (16) is satisfied. Therefore, we will consider 𝑥≠𝑦 in the following.
Case 2. Letting 𝑥,𝑦∈𝑋 with 𝑥,𝑦∈[0,1/2]∪{2,3}, then 𝛼(𝑥,𝑦)=1. There are some subcases as follows.
Subcase 1. Consider that 𝑥,𝑦∈[0,1/2], and suppose that (without loss of generality) 𝑥>𝑦; then 𝐻𝑥𝑇𝑥,𝑇𝑦=𝐻0,2,𝑦0,2=12≤1𝑥−𝑦2𝑑𝑥,𝑦+min𝐷𝑥,𝑇𝑦,𝐷𝑦,𝑇𝑥 (37) and so again (16) is satisfied.
Subcase 2. Consider that 𝑥∈[0,1/2] and 𝑦∈{2,3}; then 𝐻𝑥𝑇𝑥,𝑇𝑦=𝐻0,2,𝑦𝑥=𝑦−2≤32=1𝑦−𝑥2𝑥𝑦−𝑥+min𝑦−𝑥,𝑦−2=12𝑑𝑥,𝑦+min𝐷𝑥,𝑇𝑦,𝐷𝑦,𝑇𝑥. (38)
Subcase 3. If 𝑥=2 and 𝑦=3, then 𝐻≤1𝑇𝑥,𝑇𝑦=1=𝑑𝑥,𝑦2𝑑𝑥,𝑦+min𝐷𝑥,𝑇𝑦,𝐷𝑦,𝑇𝑥. (39) Thus, (16) is satisfied.
Finally, 𝛼 has (B) property, then by Theorem 23, 𝑇 is an MPP operator.
Note that since 𝐻(𝑇1,𝑇(3/4))=3/8, 𝑑(1,3/4)=1/4, and 𝐷(3/4,𝑇1)=0, then condition (6) is not satisfied. Therefore, Theorem 3 cannot be applied to this example.
Also, note that, since 𝛼(2,3)=1, 𝐻(𝑇2,𝑇3)=1, and 𝑑(2,3)=1, 𝑇 is not multivalued 𝛼-𝜓-contractive mapping. Therefore, Theorem 16 cannot be applied to this example.

3. Applications

Our results can be applied to some ordered fixed point results. First we recall some ordered notions. Let 𝑋 be a nonempty set and ⪯ be a partial order on 𝑋.

Definition 28 (see [22]). Let 𝐴,𝐵 be two nonempty subsets of 𝑋; the relations between 𝐴 and 𝐵 are defined as follows.(r₁) If for every 𝑎∈𝐴, there exists 𝑏∈𝐵 such that 𝑎⪯𝑏, then 𝐴≺₁𝐵.(r₂) If for every 𝑏∈𝐵, there exists 𝑎∈𝐴 such that 𝑎⪯𝑏, then 𝐴≺₂𝐵.(r₃) If 𝐴≺₁𝐵 and 𝐴≺₂𝐵, then 𝐴≺𝐵.

Remark 29 (see [22]). ≺₁ and ≺₂ are different relations between 𝐴 and 𝐵. For example, let 𝑋=ℝ, 𝐴=[1/2,1], 𝐵=[0,1], and ⪯ be a usual order on 𝑋; then 𝐴≺₁𝐵, but 𝐴≺₂̸𝐵; if 𝐴=[0,1], 𝐵=[0,1/2], then 𝐴≺₂𝐵, while 𝐴≺₁̸𝐵.

Remark 30 (see [22]). ≺₁, ≺₂, and ≺ are reflexive and transitive but are not antisymmetric. For instance, let 𝑋=ℝ, 𝐴=[0,3], 𝐵=[0,1]∪2,3], and ⪯ be a usual order on 𝑋; then 𝐴≺𝐵 and 𝐵≺𝐴, but 𝐴≠𝐵. Hence, they are not partial orders.

Theorem 31. Let (𝑋,⪯) be a partially ordered set, and suppose that there exists a metric 𝑑 in 𝑋 such that (𝑋,𝑑) is complete metric space. Let 𝑇∶𝑋→𝐶𝐵(𝑋) be a multivalued mapping such that 𝐻𝑇𝑥,𝑇𝑦≤𝜓𝑑𝑥,𝑦+𝐿min𝐷𝑥,𝑇𝑦,𝐷𝑦,𝑇𝑥 (40) for all 𝑥,𝑦∈𝑋 with 𝑥⪯𝑦, where 𝜓∈Ψ is strictly increasing and 𝐿≥0. Suppose that there exists 𝑥₀∈𝑋 such that {𝑥₀}≺₁𝑇𝑥₀. Assume that for each 𝑥∈𝑋 and 𝑦∈𝑇𝑥 with 𝑥⪯𝑦, are has 𝑦⪯𝑧 for all 𝑧∈𝑇𝑦. If 𝑇 is upper semicontinuous or 𝑋 satisfies the following condition that 𝑥_𝑛⊂𝑋𝑖𝑠𝑎𝑛𝑜𝑛𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑤𝑖𝑡ℎ𝑥_𝑛⟶𝑥𝑖𝑛𝑋,𝑡ℎ𝑒𝑛𝑥_𝑛⪯𝑥∀𝑛, (41) then 𝑇 has a fixed point.

Proof. Define the mapping 𝛼∶𝑋×𝑋→[0,∞) by 𝛼𝑥,𝑦=1,𝑥⪯𝑦0,otherwise. (42) Then we have 𝛼𝐻𝑑𝐷𝑥,𝑦𝑇𝑥,𝑇𝑦≤𝜓𝑥,𝑦+𝐿min𝑥,𝑇𝑦,𝐷𝑦,𝑇𝑥 (43) for all 𝑥,𝑦∈𝑋. Also, since {𝑥₀}≺₁𝑇𝑥₀, then there exists 𝑥₁∈𝑇𝑥₀ such that 𝑥₀⪯𝑥₁ and so 𝛼(𝑥₀,𝑥₁)≥1. Now letting 𝑥∈𝑋 and 𝑦∈𝑇𝑥 with 𝛼(𝑥,𝑦)≥1, then 𝑥⪯𝑦, and so by the hypotheses we have 𝑦⪯𝑧 for all 𝑧∈𝑇𝑦. Therefore, 𝛼(𝑦,𝑧)≥1 for all 𝑧∈𝑇𝑦. This shows that 𝑇 is 𝛼-admissible. Finally, if 𝑇 is upper semicontinuous or 𝑋 satisfies (41), then 𝑇 is upper semicontinuous or 𝛼 has (B) property. Therefore, from Theorem 23, 𝑇 has a fixed point.

Remark 32. We can give a similar result using ≺₂ instead of ≺₁.

Acknowledgment

The authors are grateful to the referees because their suggestions contributed to the improvement of the paper.