The Study of Fixed Point Theory for Various Multivalued Non-Self-Maps

Abstract

The basic motivation of this paper is to extend, generalize, and improve several fundamental results on the existence (and uniqueness) of coincidence points and fixed points for well-known maps in the literature such as Kannan type, Chatterjea type, Mizoguchi-Takahashi type, Berinde-Berinde type, Du type, and other types from the class of self-maps to the class of non-self-maps in the framework of the metric fixed point theory. We establish some fixed/coincidence point theorems for multivalued non-self-maps in the context of complete metric spaces.

1. Introduction

During the last few decades, the celebrated Banach contraction principle, also known as the Banach fixed point theorem [1], has become one of the core topics of applied mathematical analysis. As a consequence, a number of generalizations, extensions, and improvement of the praiseworthy Banach contraction principle in various direction have been explored and reported by various authors; see, for example, [2–30] and the references therein. In parallel with the Banach contraction principle, Kannan [5] and Chatterjea [6] created, respectively, different type, fixed point theorems as follows.

Theorem 1 (Kannan). Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝑋 is a single-valued map, and 𝛾∈[0,1/2). Assume that 𝑑𝑇𝑥,𝑇𝑦≤𝛾𝑑(𝑥,𝑇𝑥)+𝑑𝑦,𝑇𝑦∀𝑥,𝑦∈𝑋. (1) Then 𝑇 has a unique fixed point in 𝑋.

Theorem 2 (Chatterjea). Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝑋 is a single-valued map, and 𝛾∈[0,1/2). Assume that 𝑑𝑇𝑥,𝑇𝑦≤𝛾𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋. (2) Then 𝑇 has a unique fixed point in 𝑋.

The characterization of the renowned Banach fixed point theorem in the setting of multivalued maps is one of the most outstanding ideas of research in fixed point theory. The remarkable examples in this trend were given by Nadler [2], Mizoguchi and Takahashi [3], and M. Berinde and V. Berinde [4]. On the other hand, investigation of the existence of a fixed point of non-self-maps under certain condition is an interesting research subject of metric fixed point theory, see, for example, [19–27], and references therein.

The following attractive result was reported by M. Berinde and V. Berinde [4] in 2007.

Theorem 3 (M. Berinde and V. Berinde). Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝒞ℬ(𝑋) a multivalued map, 𝜑∶[0,∞)→[0,1) an ℳ𝒯-function (i.e., limsup_𝑠→𝑡+𝜑(𝑠)<1 for all 𝑡∈[0,∞)), and 𝐿≥0. Assume that ℋ𝑇𝑥,𝑇𝑦≤𝜑𝑑𝑥,𝑦𝑑𝑥,𝑦+𝐿𝑑𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋. (3) Then 𝑇 has a fixed point in 𝑋.

If we take 𝐿=0 in Theorem 3, then we conclude the remarkable result of Mizoguchi and Takahashi [3] which is a partial answer of problem 9 in [8].

Theorem 4 (Mizoguchi and Takahashi). Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝒞ℬ(𝑋) a multivalued map, and 𝜑∶[0,∞)→[0,1) an ℳ𝒯-function. Assume that ℋ𝑇𝑥,𝑇𝑦≤𝜑𝑑𝑥,𝑦𝑑𝑥,𝑦∀𝑥,𝑦∈𝑋. (4) Then 𝑇 has a fixed point in 𝑋.

Recently, Du [12] established the following theorem which is an extension of Theorem 3 and hence Theorem 4.

Theorem 5 (Du). Let (𝑋,𝑑) be a complete metric space, 𝑇∶𝑋→𝒞ℬ(𝑋) a multivalued map, 𝜑∶[0,∞)→[0,1) an ℳ𝒯-function and ℎ∶𝑋→[0,∞) a function. Assume that ℋ𝑇𝑥,𝑇𝑦≤𝜑𝑑𝑥,𝑦𝑑𝑥,𝑦+ℎ𝑦𝑑𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋. (5) Then 𝑇 has a fixed point in 𝑋.

The basic objective of this paper is to investigate the existence of coincidence and fixed points of multivalued non-self-maps under the certain conditions in the setting of metric spaces. The presented results generalize, improve, and extend several crucial and notable results that examine the existence of the coincidence/fixed point of well-known maps such as Kannan type, Chatterjea type, Mizoguchi-Takahashi type, Berinde-Berinde type, Du type, and other types in the context of complete metric spaces.

2. Preliminaries

Let (𝑋,𝑑) be a metric space. For each 𝑥∈𝑋 and 𝐴⊆𝑋, let 𝑑(𝑥,𝐴)=inf_𝑦∈𝐴𝑑(𝑥,𝑦). Denote by 𝒩(𝑋) the class of all nonempty subsets of 𝑋 and 𝒞ℬ(𝑋) the family of all nonempty closed and bounded subsets of 𝑋. A function ℋ∶𝒞ℬ(𝑋)×𝒞ℬ(𝑋)→[0,∞) defined by ℋ(𝐴,𝐵)=maxsup_𝑥∈𝐵𝑑(𝑥,𝐴),sup_𝑥∈𝐴𝑑(𝑥,𝐵) (6) is said to be the Hausdorff metric on 𝒞ℬ(𝑋) induced by the metric 𝑑 on 𝑋. It is also known that (𝒞ℬ(𝑋),ℋ) is a complete metric space whenever (𝑋,𝑑) is a complete metric space.

Let 𝐾 be a nonempty subset of 𝑋, 𝑔∶𝐾→𝑋 a single-valued map, and 𝑇∶𝐾→𝒩(𝑋) a multivalued map. A point 𝑥 in 𝑋 is a coincidence point of 𝑔 and 𝑇 if 𝑔𝑥∈𝑇𝑥. If 𝑔=𝑖𝑑 is the identity map, then 𝑥=𝑔𝑥∈𝑇𝑥 and call 𝑥 a fixed point of 𝑇. The set of fixed points of 𝑇 and the set of coincidence point of 𝑔 and 𝑇 are denoted by ℱ_𝐾(𝑇) and 𝒞𝒪𝒫_𝐾(𝑔,𝑇), respectively. In particular, if 𝐾≡𝑋, we use ℱ(𝑇) and 𝒞𝒪𝒫(𝑔,𝑇) instead of ℱ_𝐾(𝑇) and 𝒞𝒪𝒫_𝐾(𝑔,𝑇), respectively. Throughout this paper, we denote by ℕ, and ℝ, the set of positive integers and real numbers, respectively.

Let 𝑓 be a real-valued function defined on ℝ. For 𝑐∈ℝ, we recall that limsup_𝑥→𝑐+𝑓(𝑥)=inf_𝜀>0sup_{𝑐<𝑥<𝑐+𝜀}𝑓(𝑥). (7)

Definition 6 (see [9–18]). A function 𝜑∶[0,∞)→[0,1) is said to be an ℳ𝒯-function (or ℛ-function) if limsup_𝑠→𝑡+𝜑(𝑠)<1 for all 𝑡∈[0,∞).

It is evident that if 𝜑∶[0,∞)→[0,1) is a nondecreasing function or a nonincreasing function, then 𝜑 is an ℳ𝒯-function. So the set of ℳ𝒯-functions is a rich class. An example which is not an ℳ𝒯-function is given as follows. Let 𝜑∶[0,∞)→[0,1) be defined by 𝜑(𝑡)∶=sin𝑡𝑡𝜋,if𝑡∈0,20,otherwise. (8) We note that 𝜑 is not an ℳ𝒯-function, since limsup_𝑠→0+𝜑(𝑠)=1.

In what follows that, we recall some characterizations of ℳ𝒯 -functions proved first by Du [12].

Theorem 7 (see [12]). Let 𝜑∶[0,∞)→[0,1) be a function. Then the following statements are equivalent.(a)𝜑is an ℳ𝒯-function.(b)For each 𝑡∈[0,∞), there exist 𝑟_𝑡⁽¹⁾∈[0,1) and 𝜀_𝑡⁽¹⁾>0 such that 𝜑(𝑠)≤𝑟_𝑡⁽¹⁾ for all 𝑠∈(𝑡,𝑡+𝜀_𝑡⁽¹⁾).(c)For each 𝑡∈[0,∞), there exist 𝑟_𝑡⁽²⁾∈[0,1) and 𝜀_𝑡⁽²⁾>0 such that 𝜑(𝑠)≤𝑟_𝑡⁽²⁾ for all 𝑠∈[𝑡,𝑡+𝜀_𝑡⁽²⁾].(d)For each 𝑡∈[0,∞), there exist 𝑟_𝑡⁽³⁾∈[0,1) and 𝜀_𝑡⁽³⁾>0 such that 𝜑(𝑠)≤𝑟_𝑡⁽³⁾ for all 𝑠∈(𝑡,𝑡+𝜀_𝑡⁽³⁾].(e)For each 𝑡∈[0,∞), there exist 𝑟_𝑡⁽⁴⁾∈[0,1) and 𝜀_𝑡⁽⁴⁾>0 such that 𝜑(𝑠)≤𝑟_𝑡⁽⁴⁾ for all 𝑠∈[𝑡,𝑡+𝜀_𝑡⁽⁴⁾).(f)For any nonincreasing sequence {𝑥_𝑛}_𝑛∈ℕ in [0,∞), one has 0≤sup_𝑛∈ℕ𝜑(𝑥_𝑛)<1.(g)𝜑 is a function of contractive factor; that is, for any strictly decreasing sequence {𝑥_𝑛}_𝑛∈ℕ in [0,∞), we have 0≤sup_𝑛∈ℕ𝜑(𝑥_𝑛)<1.

3. Existence Theorems of Coincidence Points and Fixed Points for Multivalued Non-Self-Maps of Kannan Type and Chatterjea Type

In this section, we prove the existence of coincidence points and fixed points of multivalued non-self-maps of Kannan type and Chatterjea type. For this purpose, we first established a new intersection theorem of 𝒞𝒪𝒫_𝐾(𝑔,𝑇) and ℱ_𝐾(𝑇) for multivalued non-self-maps in complete metric spaces.

Theorem 8. Let (𝑋,𝑑) be a complete metric space, 𝐾 a nonempty closed subset of 𝑋, 𝑇∶𝐾→𝒞ℬ(𝑋) a multivalued map and 𝑔∶𝐾→𝑋 a continuous self-map. Suppose that(D1) 𝑇𝑥∩𝐾≠∅ for all 𝑥∈𝐾,(D2) 𝑇𝑥∩𝐾 is 𝑔-invariant (i.e., 𝑔(𝑇𝑥∩𝐾)⊆𝑇𝑥∩𝐾) for each 𝑥∈𝐾,(D3) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑()𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥∩𝐾+𝑑𝑦,𝑇𝑥∩𝐾+𝑑𝑦,𝑇𝑦∩𝐾+ℎ𝑔𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾. (9)

Then 𝒞𝒪𝒫_𝐾(𝑔,𝑇)∩ℱ_𝐾(𝑇)≠∅.

Proof. Since 𝐾 a nonempty closed subset of 𝑋 and 𝑋is complete, (𝐾,𝑑) is also a complete metric space. Let 𝑥∈𝐾. Put 𝑘=𝛾/(1−𝛾) and 𝜆=(1+𝑘)/2. So 0≤𝑘<𝜆<1. Let 𝑦∈𝑇𝑥∩𝐾 be arbitrary. Then 𝑑(𝑦,𝑇𝑥∩𝐾))=0. By (D2), we have 𝑑(𝑔𝑦,𝑇𝑥∩𝐾)=0. Hence (9) implies ℋ𝑑()𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥∩𝐾+ℋ𝑇𝑥,𝑇𝑦∩𝐾∀𝑦∈𝑇𝑥∩𝐾. (10) Inequality (10) shows that 𝑑()𝑦,𝑇𝑦∩𝐾≤ℋ𝑇𝑥,𝑇𝑦∩𝐾≤𝑘𝑑𝑥,𝑇𝑥∩𝐾<𝜆𝑑𝑥,𝑦∀𝑦∈𝑇𝑥∩𝐾. (11) Let 𝑥∈𝐾 be given. Take 𝑥₁=𝑥. By (D1), 𝑇𝑥₁∩𝐾≠∅. Choose 𝑥₂∈𝑇𝑥₁∩𝐾. If 𝑥₂=𝑥₁, then 𝑥₁∈ℱ_𝐾(𝑇) and hence 𝑔𝑥₁∈𝑇𝑥₁ from (D2). Hence 𝑥₁∈𝒞𝒪𝒫_𝐾(𝑔,𝑇)∩ℱ_𝐾(𝑇) and the proof is finished. Otherwise, if 𝑥₂≠𝑥₁, then 𝑑(𝑥₁,𝑥₂)>0. By (11), we have 𝑑𝑥₂,𝑇𝑥₂∩𝐾<𝜆𝑑𝑥₁,𝑥₂, (12) which implies that there exists 𝑥₃∈𝑇𝑥₂∩𝐾 such that 𝑑𝑥₂,𝑥₃<𝜆𝑑𝑥₁,𝑥₂. (13) Next, by (11) again, there exists 𝑥₄∈𝑇𝑥₃∩𝐾 such that 𝑑𝑥₃,𝑥₄<𝜆𝑑𝑥₂,𝑥₃. (14) By induction, we can obtain a sequence {𝑥_𝑛} in 𝐾 satisfying 𝑥_𝑛+1∈𝑇𝑥_𝑛𝑑𝑥∩𝐾,_𝑛+1,𝑥_𝑛+2<𝜆𝑑𝑥_𝑛,𝑥_𝑛+1. (15)(16) By (16), we have 𝑑𝑥_𝑛+1,𝑥_𝑛+2<𝜆𝑑𝑥_𝑛,𝑥_𝑛+1<𝜆²𝑑𝑥_𝑛−1,𝑥_𝑛<⋯<𝜆^𝑛𝑑𝑥₁,𝑥₂,for𝑛∈ℕ. (17) Let 𝜌_𝑛=(𝜆^𝑛−1/(1−𝜆))𝑑(𝑥₁,𝑥₂), 𝑛∈ℕ. For 𝑚, 𝑛∈ℕ with 𝑚>𝑛, we have 𝑑𝑥_𝑛,𝑥_𝑚≤^𝑚−1_𝑗=𝑛𝑑𝑥_𝑗,𝑥_𝑗+1<𝜌_𝑛. (18) Since 0<𝜆<1, lim_𝑛→∞𝜌_𝑛=0 and hence lim_𝑛→∞sup{𝑑(𝑥_𝑛,𝑥_𝑚)∶𝑚>𝑛}=0. This proves that {𝑥_𝑛} is a Cauchy sequence in 𝐾. By the completeness of 𝐾, there exists 𝑣∈𝐾 such that 𝑥_𝑛→𝑣 as 𝑛→∞. By (15) and (D2), we have 𝑔𝑥_𝑛+1∈𝑇𝑥_𝑛∩𝐾foreach𝑛∈ℕ. (19) Since 𝑔 is continuous and lim_𝑛→∞𝑥_𝑛=𝑣, we have lim_𝑛→∞𝑔𝑥_𝑛=𝑔𝑣. (20) Since the function 𝑥↦𝑑(𝑥,𝑇𝑣) is continuous, by (9), (15), (19), and (20), we get 𝑑(𝑣,𝑇𝑣∩𝐾)=lim_𝑛→∞𝑑𝑥_𝑛+1,𝑇𝑣∩𝐾≤lim_𝑛→∞ℋ𝑇𝑥_𝑛,𝑇𝑣∩𝐾≤lim_𝑛→∞𝛾𝑑𝑥_𝑛,𝑇𝑥_𝑛∩𝐾+𝑑𝑣,𝑇𝑥_𝑛((𝑣)𝑑∩𝐾+𝑑𝑣,𝑇𝑣∩𝐾)]+ℎ𝑔𝑣,𝑇𝑥_𝑛∩𝐾≤lim_𝑛→∞𝛾𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑣,𝑥_𝑛+1((𝑣)𝑑+𝑑𝑣,𝑇𝑣∩𝐾)+ℎ𝑔𝑣,𝑔𝑥_𝑛+1=𝛾𝑑(𝑣,𝑇𝑣∩𝐾), (21) which implies 𝑑(𝑣,𝑇𝑣∩𝐾)=0. By the closedness of 𝑇𝑣, we have 𝑣∈𝑇𝑣∩𝐾. From (D2), 𝑔𝑣∈𝑇𝑣∩𝐾⊆𝑇𝑣. Hence we verify 𝑣∈𝒞𝒪𝒫_𝐾(𝑔,𝑇)∩ℱ_𝐾(𝑇). The proof is complete.

Theorem 9. In Theorem 8, if condition (D3) is replaced with one of the following conditions:(K1) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ[𝑑()𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥∩𝐾+𝑑𝑦,𝑇𝑥∩𝐾+𝑑𝑦,𝑇𝑦∩𝐾+ℎ𝑔𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾, (22)(K2) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑()𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥∩𝐾+𝑑𝑦,𝑇𝑥+𝑑𝑦,𝑇𝑦∩𝐾+ℎ𝑔𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾, (23) (K3) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑()𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥∩𝐾+𝑑𝑦,𝑇𝑥+𝑑𝑦,𝑇𝑦∩𝐾+ℎ𝑔𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾, (24) (K4) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑()𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑥∩𝐾+𝑑𝑦,𝑇𝑦∩𝐾+ℎ𝑔𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾, (25)(K5) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑()𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑥∩𝐾+𝑑𝑦,𝑇𝑦∩𝐾+ℎ𝑔𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾, (26)(K6) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑()𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑥+𝑑𝑦,𝑇𝑦∩𝐾+ℎ𝑔𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾, (27)(K7) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑()𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑥+𝑑𝑦,𝑇𝑦∩𝐾+ℎ𝑔𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾. (28)

Then 𝒞𝒪𝒫_𝐾(𝑔,𝑇)∩ℱ_𝐾(𝑇)≠∅.

Proof. It is obvious that any of these conditions (K1)–(K7) implies condition (D3) as in Theorem 8. So the desired conclusion follows from Theorem 8 immediately.

The following fixed point theorem for multivalued non-self-maps of generalized Kannan type can be established immediately from Theorem 9 for 𝑔≡𝑖𝑑 (the identity mapping).

Theorem 10. Let (𝑋,𝑑) be a complete metric space, 𝐾 a nonempty closed subset of 𝑋, and 𝑇∶𝐾→𝒞ℬ(𝑋) a multivalued map. Suppose that 𝑇𝑥∩𝐾≠∅ for all 𝑥∈𝐾 and one of the following conditions holds:(P1) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑()𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥∩𝐾+𝑑𝑦,𝑇𝑥∩𝐾+𝑑𝑦,𝑇𝑦∩𝐾+ℎ𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾, (29) (P2) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑()𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥∩𝐾+𝑑𝑦,𝑇𝑥∩𝐾+𝑑𝑦,𝑇𝑦∩𝐾+ℎ𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾, (30) (P3) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑()𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥∩𝐾+𝑑𝑦,𝑇𝑥+𝑑𝑦,𝑇𝑦∩𝐾+ℎ𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾, (31) (P4) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑()𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥∩𝐾+𝑑𝑦,𝑇𝑥+𝑑𝑦,𝑇𝑦∩𝐾+ℎ𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾, (32)(P5) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑()𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑥∩𝐾+𝑑𝑦,𝑇𝑦∩𝐾+ℎ𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾, (33)(P6) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑()𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑥∩𝐾+𝑑𝑦,𝑇𝑦∩𝐾+ℎ𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾, (34)(P7) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑()𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑥+𝑑𝑦,𝑇𝑦∩𝐾+ℎ𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾, (35)(P8) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑()𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑥+𝑑𝑦,𝑇𝑦∩𝐾+ℎ𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾. (36)

Then ℱ_𝐾(𝑇)≠∅.

As a consequence of Theorem 10, we obtain the following generalized Kannan type fixed point theorems for multivalued maps.

Corollary 11. Let (𝑋,𝑑) be a complete metric space, 𝐾 a nonempty closed subset of 𝑋, and 𝑇∶𝐾→𝒞ℬ(𝑋) a multivalued map. Suppose that 𝑇𝑥∩𝐾≠∅ for all 𝑥∈𝐾 and there exists 𝛾∈[0,1/2) such that ℋ𝑇𝑥,𝑇𝑦∩𝐾≤𝛾𝑑(𝑥,𝑇𝑥∩𝐾)+𝑑𝑦,𝑇𝑦∩𝐾∀𝑥,𝑦∈𝐾. (37) Then ℱ_𝐾(𝑇)≠∅.

Remark 12. (a) If 𝐾=𝑋 in Corollary 11, then we can obtain a multivalued version of Kannan’s fixed point theorem [5].
(b) Theorems 8–10 and Corollary 11 all extend and generalize Kannan’s fixed point theorem.

Theorem 13. Let (𝑋,𝑑) be a complete metric space, 𝐾 a nonempty closed subset of 𝑋, 𝑇∶𝐾→𝒞ℬ(𝑋) a multivalued map, and 𝑔∶𝐾→𝑋 a continuous self-map. Suppose that conditions (D1) and (D2) as in Theorem 8 hold. If there exist ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑥,𝑇𝑦∩𝐾+𝑑𝑦,𝑇𝑥∩𝐾+ℎ𝑔𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾. (38) Then 𝒞𝒪𝒫_𝐾(𝑔,𝑇)∩ℱ_𝐾(𝑇)≠∅.

Proof. Let 𝑥∈𝐾. Since 𝛼∈[0,1/2), by the denseness of ℝ, we can find 𝛽>0 such that 𝛼<𝛽<1/2. Let 𝑦∈𝑇𝑥∩𝐾 be arbitrary. Then 𝑑(𝑦,𝑇𝑥∩𝐾)=0. By (D2), we have 𝑑(𝑔𝑦,𝑇𝑥∩𝐾)=0. Hence (38) has been reduced to 𝑑𝑦,𝑇𝑦∩𝐾≤ℋ𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑑𝑥,𝑇𝑦∩𝐾<𝛽𝑑𝑥,𝑇𝑦∩𝐾∀𝑦∈𝑇𝑥∩𝐾. (39) Let 𝑥∈𝐾 be given. Take 𝑥₁=𝑥. By (D1), 𝑇𝑥₁∩𝐾≠∅. Choose 𝑥₂∈𝑇𝑥₁∩𝐾. If 𝑥₂=𝑥₁, then 𝑥₁∈ℱ_𝐾(𝑇) and hence 𝑔𝑥₁∈𝑇𝑥₁ from (D2). Hence 𝑥₁∈𝒞𝒪𝒫_𝐾(𝑔,𝑇)∩ℱ_𝐾(𝑇) and the proof is finished. Otherwise, if 𝑥₂≠𝑥₁, then 𝑑(𝑥₁,𝑥₂)>0. By (39), we have 𝑑𝑥₂,𝑇𝑥₂∩𝐾<𝛽𝑑𝑥₁,𝑇𝑥₂∩𝐾, (40) which implies that there exists 𝑥₃∈𝑇𝑥₂∩𝐾 such that 𝑑𝑥₂,𝑥₃<𝛽𝑑𝑥₁,𝑇𝑥₂𝑥∩𝐾≤𝛽𝑑₁,𝑥₃≤𝛽𝑑𝑥₁,𝑥₂+𝑑𝑥₂,𝑥₃. (41) Let 𝛾=𝛽/(1−𝛽). Then 𝛾∈(0,1) and the last inequality implies 𝑑𝑥₂,𝑥₃<𝛾𝑑𝑥₁,𝑥₂. (42) Continuing in this way, we can construct inductively a sequence {𝑥_𝑛}_𝑛∈ℕ in 𝐾 satisfying 𝑥_𝑛+1∈𝑇𝑥_𝑛𝑑𝑥∩𝐾,_𝑛+1,𝑥_𝑛+2<𝛾𝑑𝑥_𝑛,𝑥_𝑛+1 (43) for each 𝑛∈ℕ. Using a similar argument as in the proof of Theorem 8, we have the following: (i)𝑥_𝑛+1∈𝑇𝑥_𝑛∩𝐾; (ii){𝑥_𝑛} is a Cauchy sequence in 𝐾; (iii)there exists 𝑣∈𝐾 such that 𝑥_𝑛→𝑣 as 𝑛→∞; (iv)𝑔𝑥_𝑛+1∈𝑇𝑥_𝑛∩𝐾 for each 𝑛∈ℕ; (v)lim_𝑛→∞𝑔𝑥_𝑛=𝑔𝑣.
By (38), we get 𝑑(𝑣,𝑇𝑣∩𝐾)=lim_𝑛→∞𝑑𝑥_𝑛+1,𝑇𝑣∩𝐾≤lim_𝑛→∞ℋ𝑇𝑥_𝑛,𝑇𝑣∩𝐾≤lim_𝑛→∞𝛼𝑑𝑥_𝑛,𝑇𝑣∩𝐾+𝑑𝑣,𝑇𝑥_𝑛(𝑣)𝑑∩𝐾+ℎ𝑔𝑣,𝑇𝑥_𝑛∩𝐾≤lim_𝑛→∞𝛼𝑑𝑥_𝑛,𝑇𝑣∩𝐾+𝑑𝑣,𝑥_𝑛+1+ℎ(𝑣)𝑑𝑔𝑣,𝑔𝑥_𝑛+1=𝛼𝑑(𝑣,𝑇𝑣∩𝐾), (44) which implies 𝑑(𝑣,𝑇𝑣∩𝐾)=0. By the closedness of 𝑇𝑣, we have 𝑣∈𝑇𝑣∩𝐾. By (D2), 𝑔𝑣∈𝑇𝑣∩𝐾⊆𝑇𝑣 and hence 𝑣∈𝒞𝒪𝒫_𝐾(𝑔,𝑇)∩ℱ_𝐾(𝑇). The proof is complete.

Theorem 14. In Theorem 13, if inequality (38) is replaced with one of the following inequalities: (C1)ℋ𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑥,𝑇𝑦∩𝐾+𝑑𝑦,𝑇𝑥∩𝐾+ℎ𝑔𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾, (45) (C2)ℋ𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥∩𝐾+ℎ𝑔𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾, (46) (C3)ℋ𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥∩𝐾+ℎ𝑔𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾, (47) (C4)ℋ𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑥,𝑇𝑦∩𝐾+𝑑𝑦,𝑇𝑥+ℎ𝑔𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾, (48) (C5)ℋ𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑥,𝑇𝑦∩𝐾+𝑑𝑦,𝑇𝑥+ℎ𝑔𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾, (49) (C6)ℋ𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥+ℎ𝑔𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾, (50) (C7)ℋ𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥+ℎ𝑔𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾, (51) then 𝒞𝒪𝒫_𝐾(𝑔,𝑇)∩ℱ_𝐾(𝑇)≠∅.

Applying Theorem 14, we can prove the following fixed point theorems for multivalued maps of generalized Chatterjea type.

Theorem 15. Let (𝑋,𝑑) a complete metric space, 𝐾 a nonempty closed subset of 𝑋, and 𝑇∶𝐾→𝒞ℬ(𝑋) a multivalued map. Suppose that 𝑇𝑥∩𝐾≠∅ for all 𝑥∈𝐾 and one of the following conditions holds:(Q1) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑥,𝑇𝑦∩𝐾+𝑑𝑦,𝑇𝑥∩𝐾+ℎ𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾, (52)(Q2) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑥,𝑇𝑦∩𝐾+𝑑𝑦,𝑇𝑥∩𝐾+ℎ𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾, (53)(Q3) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥∩𝐾+ℎ𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾, (54)(Q4) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥∩𝐾+ℎ𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾, (55)(Q5) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑥,𝑇𝑦∩𝐾+𝑑𝑦,𝑇𝑥+ℎ𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾, (56)(Q6) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑥,𝑇𝑦∩𝐾+𝑑𝑦,𝑇𝑥+ℎ𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾, (57)(Q7) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥+ℎ𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾, (58)(Q8) there exist a function ℎ∶𝐾→[0,∞) and 𝛾∈[0,1/2) such that ℋ𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥+ℎ𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾. (59)

Then ℱ_𝐾(𝑇)≠∅.

The following result is a generalized Chatterjea’s type fixed point theorem for multivalued maps in complete metric spaces.

Corollary 16. Let (𝑋,𝑑) be a complete metric space, 𝐾 a nonempty closed subset of 𝑋, and 𝑇∶𝐾→ℬ𝒞(𝑋) a multivalued map. Suppose that 𝑇𝑥∩𝐾≠∅ for all 𝑥∈𝐾 and there exists 𝛾∈[0,1/2) such that ℋ𝑇𝑥,𝑇𝑦∩𝐾≤𝛼𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋. (60) Then ℱ_𝐾(𝑇)≠∅.

Remark 17. (a) If 𝐾=𝑋 in Corollary 16, then we can obtain a multivalued version of Chatterjea’s fixed point theorem [6].
(b) Theorems 13–15 and Corollary 16 all improve and generalize Chatterjea’s fixed point theorem.

4. New Coincidence and Fixed Point Results for Various Multivalued Non-Self-Maps: Mizoguchi-Takahashi Type, Berinde-Berinde Type, and Du Type

In this section, we prove some coincidence and fixed point theorems for multivalued non-self-maps of Mizoguchi-Takahashi type, Berinde-Berinde type, and Du type.

Recall first the following auxiliary result.

Lemma 18 (see [9, Lemma 2.1]). Let 𝜑∶[0,∞)→[0,1) be an ℳ𝒯-function. Suppose that 𝜅∶[0,∞)→[0,1) is defined by 𝜅(𝑡)=(1+𝜑(𝑡))/2. Then, 𝜅 is also an ℳ𝒯-function.

Theorem 19. Let (𝑋,𝑑) be a complete metric space, 𝐾 a nonempty closed subset of 𝑋, 𝑇∶𝐾→ℬ𝒞(𝑋) a multivalued map, and 𝑔∶𝐾→𝑋 be a continuous self-map. Suppose that conditions (D1) and (D2) as in Theorem 8 hold. If there exist an ℳ𝒯-function 𝜑∶[0,∞)→[0,1) and a function ℎ∶𝐾→[0,∞) such that ℋ𝑑𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝜑𝑥,𝑦𝑥,𝑦+ℎ𝑔𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾, (61) then 𝒞𝒪𝒫_𝐾(𝑔,𝑇)∩ℱ_𝐾(𝑇)≠∅.

Proof. Since 𝐾 is a nonempty closed subset of 𝑋 and 𝑋is complete, (𝐾,𝑑) is also a complete metric space. Note first that for each 𝑥∈𝐾, by (D2), we have 𝑑(𝑔𝑦,𝑇𝑥∩𝐾)=0 for all 𝑦∈𝑇𝑥∩𝐾. So, for each 𝑥∈𝐾, by (61), we obtain 𝑑𝑦,𝑇𝑦∩𝐾≤𝜑𝑑𝑥,𝑦𝑑𝑥,𝑦∀𝑦∈𝑇𝑥∩𝐾. (62) Define 𝜅∶[0,∞)→[0,1) by 𝜅(𝑡)=(1+𝜑(𝑡))/2. Then, by Lemma 18, 𝜅 is also an ℳ𝒯-function. Let 𝑥∈𝐾 be given. Take 𝑥₁=𝑥. Since 𝑇𝑥₁∩𝐾≠∅ from (D1), we can choose 𝑥₂∈𝑇𝑥₁∩𝐾. If 𝑥₂=𝑥₁, then 𝑥₁∈ℱ_𝐾(𝑇) and hence 𝑔𝑥₁∈𝑇𝑥₁ from (D2). Thus, 𝑥₁∈𝒞𝒪𝒫_𝐾(𝑔,𝑇)∩ℱ_𝐾(𝑇) and hence we achieved the result. Now, suppose that 𝑥₂≠𝑥₁; that is, 𝑑(𝑥₁,𝑥₂)>0. By (62), we have 𝑑𝑥₂,𝑇𝑥₂∩𝐾≤𝜑𝑑𝑥₁,𝑥₂𝑑𝑥₁,𝑥₂<𝜅𝑑𝑥₁,𝑥₂𝑑𝑥₁,𝑥₂, (63) which implies that there exists 𝑥₃∈𝑇𝑥₂∩𝐾 such that 𝑑𝑥₂,𝑥₃<𝜅𝑑𝑥₁,𝑥₂𝑑𝑥₁,𝑥₂. (64) Next, by (62) again, there exists 𝑥₄∈𝑇𝑥₃∩𝐾 such that 𝑑𝑥₃,𝑥₄<𝜅𝑑𝑥₂,𝑥₃𝑑𝑥₂,𝑥₃. (65) Iteratively, we can obtain a sequences {𝑥_𝑛} in 𝐾 satisfying 𝑥_𝑛+1∈𝑇𝑥_𝑛𝑑𝑥∩𝐾,_𝑛+1,𝑥_𝑛+2<𝜅𝑑𝑥_𝑛,𝑥_𝑛+1𝑑𝑥_𝑛,𝑥_𝑛+1 (66)(67) for each 𝑛∈ℕ. Since 𝜅(𝑡)<1 for all 𝑡∈[0,∞), by (ii), we know that {𝑑(𝑥_𝑛,𝑥_𝑛+1)} is strictly decreasing in [0,∞). Since 𝜅 is an ℳ𝒯-function, by (g) of Theorem 7, we obtain 0<𝑑𝑥₁,𝑥₂≤sup_𝑛∈ℕ𝜅𝑑𝑥_𝑛,𝑥_𝑛+1<1. (68) Let 𝛾∶=sup_𝑛∈ℕ𝜅(𝑑(𝑥_𝑛,𝑥_𝑛+1)). So 𝛾∈(0,1). By (67), we have 𝑑𝑥_𝑛+1,𝑥_𝑛+2<𝜅𝑑𝑥_𝑛,𝑥_𝑛+1𝑑𝑥_𝑛,𝑥_𝑛+1≤𝛾𝑑𝑥_𝑛,𝑥_𝑛+1<𝛾²𝑑𝑥_𝑛−1,𝑥_𝑛<⋯<𝛾^𝑛𝑑𝑥₁,𝑥₂,for𝑛∈ℕ. (69) Let 𝛼_𝑛=(𝛾^𝑛−1/(1−𝛾))𝑑(𝑥₁,𝑥₂), 𝑛∈ℕ. For 𝑚, 𝑛∈ℕ with 𝑚>𝑛, we have 𝑑𝑥_𝑛,𝑥_𝑚≤^𝑚−1_𝑗=𝑛𝑑𝑥_𝑗,𝑥_𝑗+1<𝛼_𝑛. (70) Since 0<𝛾<1, lim_𝑛→∞𝛼_𝑛=0 and hence lim_𝑛→∞sup{𝑑(𝑥_𝑛,𝑥_𝑚)∶𝑚>𝑛}=0. This proves that {𝑥_𝑛} is a Cauchy sequence in 𝐾. By the completeness of 𝐾, there exists 𝑣∈𝐾 such that 𝑥_𝑛→𝑣 as 𝑛→∞. Thanks to (66) and (D2), we have 𝑔𝑥_𝑛+1∈𝑇𝑥_𝑛∩𝐾foreach𝑛∈ℕ. (71) Since 𝑔 is continuous and lim_𝑛→∞𝑥_𝑛=𝑣, we have lim_𝑛→∞𝑔𝑥_𝑛=𝑔𝑣. (72) Since the function 𝑥↦𝑑(𝑥,𝑇𝑣) is continuous, by (61), (66), and (72), we get 𝑑(𝑣,𝑇𝑣∩𝐾)=lim_𝑛→∞𝑑𝑥_𝑛+1,𝑇𝑣∩𝐾≤lim_𝑛→∞ℋ𝑇𝑥_𝑛,𝑇𝑣∩𝐾≤lim_𝑛→∞𝜑𝑑𝑥_𝑛,𝑣𝑑𝑥_𝑛,𝑣+ℎ(𝑣)𝑑𝑔𝑣,𝑔𝑥_𝑛+1=0, (73) which implies 𝑑(𝑣,𝑇𝑣∩𝐾)=0. By the closedness of 𝑇𝑣, we have 𝑣∈𝑇𝑣∩𝐾. By (D2), 𝑔𝑣∈𝑇𝑣∩𝐾⊆𝑇𝑣 and hence 𝑣∈𝒞𝒪𝒫_𝐾(𝑔,𝑇)∩ℱ_𝐾(𝑇). The proof is complete.

Theorem 20. In Theorem 19, if inequality (61) is replaced with the following inequality: ℋ𝑑𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝜑𝑥,𝑦𝑥,𝑦+ℎ𝑔𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾. (74) Then 𝒞𝒪𝒫_𝐾(𝑔,𝑇)∩ℱ_𝐾(𝑇)≠∅.

Corollary 21. Let (𝑋,𝑑) be a complete convex metric space, 𝐾 a nonempty closed subset of 𝑋, 𝑇∶𝐾→ℬ𝒞(𝑋) a multivalued map, and 𝑔∶𝐾→𝑋 a continuous self-map. Suppose that(i)𝑇𝑥∩𝐾≠∅ for all 𝑥∈𝐾,(ii)𝑇𝑥∩𝐾 is 𝑔-invariant (i.e., 𝑔(𝑇𝑥∩𝐾)⊆𝑇𝑥∩𝐾) for each 𝑥∈𝐾,(iii) there exist an ℳ𝒯-function 𝜑∶[0,∞)→[0,1) and 𝐿≥0 such that ℋ𝑑𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝜑𝑥,𝑦𝑥,𝑦+𝐿𝑑𝑔𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾. (75)

Then 𝒞𝒪𝒫_𝐾(𝑔,𝑇)∩ℱ_𝐾(𝑇)≠∅.

Corollary 22. Let (𝑋,𝑑) be a complete convex metric space, 𝐾 a nonempty closed subset of 𝑋, 𝑇∶𝐾→ℬ𝒞(𝑋) a multivalued map, and 𝑔∶𝐾→𝑋 a continuous self-map. Suppose that(i)𝑇𝑥∩𝐾≠∅ for all 𝑥∈𝐾,(ii)𝑇𝑥∩𝐾 is 𝑔-invariant (i.e. 𝑔(𝑇𝑥∩𝐾)⊆𝑇𝑥∩𝐾) for each 𝑥∈𝐾,(ii)there exist an ℳ𝒯-function 𝜑∶[0,∞)→[0,1) and 𝐿≥0 such that ℋ𝑑𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝜑𝑥,𝑦𝑥,𝑦+𝐿𝑑𝑔𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾. (76)

Then 𝒞𝒪𝒫_𝐾(𝑔,𝑇)∩ℱ_𝐾(𝑇)≠∅.

As a direct consequence of Theorems 19 and 20, we obtain the following fixed point result for multivalued non-self-maps of Du type in complete metric spaces.

Theorem 23. Let (𝑋,𝑑) be a complete convex metric space, 𝐾 a nonempty closed subset of 𝑋, and 𝑇∶𝐾→ℬ𝒞(𝑋) a multivalued map. Suppose that 𝑇𝑥∩𝐾≠∅ for all 𝑥∈𝐾, and one of the following conditions holds:(W1) there exist an ℳ𝒯-function 𝜑∶[0,∞)→[0,1) and a function ℎ∶𝐾→[0,∞) such that ℋ𝑑𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝜑𝑥,𝑦𝑥,𝑦+ℎ𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾, (77)(W2) there exist an ℳ𝒯-function 𝜑∶[0,∞)→[0,1) and a function ℎ∶𝐾→[0,∞) such that ℋ𝑑𝑑𝑦𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝜑𝑥,𝑦𝑥,𝑦+ℎ𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾. (78)

Then ℱ_𝐾(𝑇)≠∅.

Proof. Let 𝑔=𝑖𝑑 be the identity map. It is easy to verify that all the conditions of Theorem 19 (or Theorem 20) are satisfied. Hence the conclusion follows from Theorem 19 (or Theorem 20).

The following fixed point theorems for multivalued non-self-maps of generalized Berinde-Berinde type and generalized Mizoguchi-Takahashi type are established immediately from Theorem 23.

Corollary 24. Let (𝑋,𝑑) be a complete convex metric space, 𝐾 a nonempty closed subset of 𝑋, and 𝑇∶𝐾→ℬ𝒞(𝑋) a multivalued map. Suppose that(i) 𝑇𝑥∩𝐾≠∅ for all 𝑥∈𝐾,(ii) there exist an ℳ𝒯-function 𝜑∶[0,∞)→[0,1) and 𝐿≥0 such that ℋ𝑑𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝜑𝑥,𝑦𝑥,𝑦+𝐿𝑑𝑦,𝑇𝑥∩𝐾∀𝑥,𝑦∈𝐾. (79)

Then ℱ_𝐾(𝑇)≠∅.

Corollary 25. Let (𝑋,𝑑) be a complete convex metric space, 𝐾 a nonempty closed subset of 𝑋, and 𝑇∶𝐾→ℬ𝒞(𝑋) a multivalued map. Suppose that(i) 𝑇𝑥∩𝐾≠∅ for all 𝑥∈𝐾,(ii) there exist an ℳ𝒯-function 𝜑∶[0,∞)→[0,1) and 𝐿≥0 such that ℋ𝑑𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝜑𝑥,𝑦𝑥,𝑦+𝐿𝑑𝑦,𝑇𝑥∀𝑥,𝑦∈𝐾. (80)

Then ℱ_𝐾(𝑇)≠∅.

Corollary 26. Let (𝑋,𝑑) be a complete convex metric space, 𝐾 a nonempty closed subset of 𝑋, and 𝑇∶𝐾→ℬ𝒞(𝑋) a multivalued map. Suppose that(i)𝑇𝑥∩𝐾≠∅ for all 𝑥∈𝐾,(ii)there exists an ℳ𝒯-function 𝜑∶[0,∞)→[0,1) such that ℋ𝑑𝑑𝑇𝑥,𝑇𝑦∩𝐾≤𝜑𝑥,𝑦𝑥,𝑦∀𝑥,𝑦∈𝐾. (81)

Then ℱ_𝐾(𝑇)≠∅.

Remark 27. (a) If 𝐾=𝑋 in Theorem 23, then we can obtain Du’s fixed point theorem [12, Theorem 2.6].
(b) Theorems 19, 20 and 23, and Corollaries 21–26 all generalize and improve Du’s fixed point theorem, Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem, and Banach’s contraction principle.

Acknowledgment

The first author was supported partially by grant no. NSC 101-2115-M-017-001 of the National Science Council of the Republic of China.