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

()

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

()

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., for all t ∈ [0, ∞)), and L ≥ 0. Assume that

()

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

()

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

()

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

()

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

()

Definition 6 (see [9]–[18].)A function φ : [0, ∞)→[0,1) is said to be an ℳ𝒯-function (or ℛ-function) if 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

()

We note that φ is not an ℳ𝒯-function, since

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)
φ is an ℳ𝒯-function.
(b)
For each t ∈ [0, ∞), there exist and such that for all .
(c)
For each t ∈ [0, ∞), there exist and such that for all .
(d)
For each t ∈ [0, ∞), there exist and such that for all .
(e)
For each t ∈ [0, ∞), there exist and such that for all .
(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
()

Then 𝒞𝒪𝒫_K(g, T)∩ℱ_K(T) ≠ ∅.

Proof. Since K a nonempty closed subset of X and X is 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

()

Inequality (10) shows that

()

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

()

which implies that there exists x₃ ∈ Tx₂∩K such that

()

Next, by (11) again, there exists x₄ ∈ Tx₃∩K such that

()

By induction, we can obtain a sequence {x_n} in K satisfying

()

By (16), we have

()

Let ρ_n = (λⁿ⁻¹/(1 − λ))d(x₁, x₂), n ∈ ℕ. For m, n ∈ ℕ with m > n, we have

()

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

()

Since g is continuous and lim _n→∞x_n = v, we have

()

Since the function x ↦ d(x, Tv) is continuous, by (9), (15), (19), and (20), we get

()

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
()
(K2)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(K3)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(K4)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(K5)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(K6)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(K7)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()

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
()
(P2)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(P3)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(P4)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(P5)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(P6)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(P7)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(P8)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()

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

()

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

()

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

()

which implies that there exists x₃ ∈ Tx₂∩K such that

()

Let γ = β/(1 − β). Then γ ∈ (0,1) and the last inequality implies

()

Continuing in this way, we can construct inductively a sequence {x_n} _n∈ℕ in K satisfying

()

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

()

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)

()
(C2)

()
(C3)

()
(C4)

()
(C5)

()
(C6)

()
(C7)

()
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
()
(Q2)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(Q3)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(Q4)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(Q5)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(Q6)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(Q7)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()
(Q8)
there exist a function h : K → [0, ∞) and γ ∈ [0,1/2) such that
()

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

()

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

()

then 𝒞𝒪𝒫_K(g, T)∩ℱ_K(T) ≠ ∅.

Proof. Since K is a nonempty closed subset of X and X is 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

()

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

()

which implies that there exists x₃ ∈ Tx₂∩K such that

()

Next, by (62) again, there exists x₄ ∈ Tx₃∩K such that

()

Iteratively, we can obtain a sequences {x_n} in K satisfying

()

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

()

Let γ : = sup _n∈ℕκ(d(x_n, x_n+1)). So γ ∈ (0,1). By (67), we have

()

Let α_n = (γⁿ⁻¹/(1 − γ))d(x₁, x₂), n ∈ ℕ. For m, n ∈ ℕ with m > n, we have

()

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

()

Since g is continuous and lim _n→∞x_n = v, we have

()

Since the function x ↦ d(x, Tv) is continuous, by (61), (66), and (72), we get

()

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:

()

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
()

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
()

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
()
(W2)
there exist an ℳ𝒯-function φ : [0, ∞)→[0,1) and a function h : K → [0, ∞) such that
()

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
()

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
()

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
()

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.

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

Abstract

1. Introduction

2. Preliminaries

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

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

Acknowledgment

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

Abstract

1. Introduction

2. Preliminaries

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

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

Acknowledgment

References

Citing Literature

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