Hybrid generalized contractions

Abstract

Sintunavarat and Kumam introduced the notion of hybrid generalized multi-valued contraction mapping and established a common fixed point theorem. We extend their result for four mappings and prove common coincidence and common fixed point theorem.

Introduction and preliminaries

Alber and Guerre-Delabriere [1] introduced the concept of weak contraction in Hilbert spaces. Rhoades [2] has shown that the result concluded by Alber and Guerre-Delabriere in [1] is also valid in complete metric spaces. Berinde and Berinde [3] extended weak contraction for multi-valued mappings and introduced the notion of multi-valued (θ, L)-weak contraction and multi-valued (α, L)-weak contraction. Kamran [4] extended these contractions for hybrid pair of mappings and introduced multi-valued (f, θ, L)-weak contraction and multi-valued (f,α,L)-weak contraction. Sintunavarat and Kumam [5] introduced the notion of generalized (f, α, β)-weak contraction to extend the notion of multi-valued (f, α, L)-weak contraction. They further extended the notion of generalized (f, α, β)-weak contraction by introducing hybrid generalized multi-valued contraction mappings and established a common fixed point theorem [6]. The purpose of this paper is to extend the notion of hybrid generalized multi-valued contraction and to prove common coincidence and common fixed point theorems.

Let (X, d) be a metric space. For x ∈ X and A ⊆ X, d(x, A) = inf{d(x, y): y ∈ A}. We denote by CB(X) the class of all nonempty closed and bounded subsets of X. For every A, B ∈ CB(X)

$$H(A,B)=max\left\{\underset{x\in A}{sup}d\right(x,B),\underset{y\in B}{sup}d(y,A\left)\right\}.$$

Such a map H is called Hausdorff metric induced by d. A point x ∈ X is said to be a common fixed point of f:X → X and T:X → CB(X) if x = f x ∈ T x. The point x ∈ X is said to be a coincidence point of f:X → X and T:X → CB(X) if f x ∈ T x. Some works for hybrid pair are available in [7–10].

The mappings f:X → X and T:X → CB(X) are called R-weakly commuting [11, 12] if for all x ∈ X, f T x ∈ CB(X), and there exists a positive real number R such that H(f T x, T f x) ≤ R d(f x, T x).

Lemma 1

[4] Let (X, d) be a metric space, {A _k } be a sequence in C B(X), and {x _k } be a sequence in X such that x _k  ∈ A_k−1. Let ϕ:[0, ∞) → [0, 1) be a function satisfying ${limsup}_{r\to t^{+}}\varphi \left(r\right)<1$ for every t∈ [0, ∞). Suppose $d(x_{k-1},x_k)$ to be a nonincreasing sequence such that

$$H(A_{k-1},A_k)\le \varphi \left(d\right(x_{k-1},x_k\left)\right)d(x_{k-1},x_k),$$

$$d(x_{k+1},x_k)\le H(A_{k-1},A_k)+{\varphi }^{n_k}\left(d\right(x_{k-1},x_k\left)\right),$$

where n₁ < n₂ < …, $k,n_k\in \mathbb{N}$. Then, {x _k } is a Cauchy sequence in X.

Lemma 2

[13] If A, B ∈ C B(X) and a ∈ A, then for each ϵ > 0, there exists b ∈ B such that

$$d(a,b)\le H(A,B)+\mathit{ϵ}.$$

Definition 1

[6] Let (X, d) be a metric space, f:X → X be a single-valued mapping, and T:X → CB(X) be a multi-valued mapping. T is said to be a hybrid generalized multi-valued contraction mapping if and only if there exist two functions ϕ:[0, ∞) → [0, 1) satisfying ${limsup}_{r\to t^{+}}\varphi \left(r\right)<1$ for every t∈ [0, ∞) and φ:[0, ∞) → [0, ∞) such that

$$\begin{array}{l}H(\mathit{\text{Tx}},\mathit{\text{Ty}})\le \varphi \left(M\right(x,y\left)\right)M(x,y)+\phi \left(N\right(x,y\left)\right)N(x,y),\\ \text{for each}x,y\in X,\end{array}$$

(1)

where

$$M(x,y)=max\left\{d\right(\mathit{\text{fx}},\mathit{\text{fy}}),d(\mathit{\text{fy}},\mathit{\text{Tx}}\left)\right\},$$

and

$$\begin{array}{l}N(x,y)=min\left\{d\right(\mathit{\text{fx}},\mathit{\text{fy}}),d(\mathit{\text{fx}},\mathit{\text{Tx}}),d(\mathit{\text{fy}},\mathit{\text{Ty}}),d(\mathit{\text{fx}},\mathit{\text{Ty}}),\\ d(\mathit{\text{fy}},\mathit{\text{Tx}})\}.\end{array}$$

Main results

Definition 2

Let (X, d) be a metric space, f, g:X → X and T:X → CB(X). A mapping S:X → CB(X) is said to be an extended hybrid generalized multi-valued f contraction if and only if there exist two functions ϕ:[0, ∞) → [0, 1) satisfying ${limsup}_{r\to t^{+}}\varphi \left(r\right)<1$ for every t∈ [0, ∞) and φ:[0, ∞) → [0, ∞) such that

$$\begin{array}{l}H(\mathit{\text{Sx}},\mathit{\text{Ty}})\le \varphi \left(M\right(x,y\left)\right)M(x,y)+\phi \left(N\right(x,y\left)\right)N(x,y),\\ \text{for each}x,y\in X,\end{array}$$

(2)

where

$$M(x,y)=max\left\{d\right(\mathit{\text{fx}},\mathit{\text{gy}}),min\{d(\mathit{\text{fx}},\mathit{\text{Ty}}),d(\mathit{\text{gy}},\mathit{\text{Sx}})\left\}\right\},$$

and

$$\begin{array}{l}N(x,y)=min\left\{d\right(\mathit{\text{fx}},\mathit{\text{gy}}),d(\mathit{\text{fx}},\mathit{\text{Tx}}),d(\mathit{\text{gy}},\mathit{\text{Sy}}),d(\mathit{\text{fx}},\mathit{\text{Ty}}),\\ d(\mathit{\text{gy}},\mathit{\text{Sx}})\}.\end{array}$$

Remark 1

If f = g and T = S, then Definition 2 reduces to Definition 1.

Lemma 3

Let (X, d) be a metric space, f, g:X → X and T:X → C B(X). Let S:X → C B(X) be the extended hybrid generalized multi-valued f contraction. Let {g x_2k+1} be a g-orbit of S at x₀ and {f x_2k+2} be an f-orbit of T at x₁ such that

$$d(y_{k+1},y_{k+2})\le H(A_k,A_{k+1})+{\varphi }^{n_{k+1}}\left(M\right(x_k,x_{k+1}\left)\right),$$

(3)

for each k ∈ {0, 2, 4, 6, …}, and

$$d(y_{k+2},y_{k+1})\le H(A_{k+1},A_k)+{\varphi }^{n_{k+1}}\left(M\right(x_{k+1},x_k\left)\right),$$

(4)

for each k ∈ {1, 3, 5, …}, where y_2k+1 = g x_2k+1 ∈ S x_2k = A_2k, y_2k+2 = f x_2k+2 ∈ T x_2k+1 = A_2k+1, for each k ≥ 0. Further, n₁ < n₂ < … and {d(y _k , y_k+1)} is a nonincreasing sequence. Then, {y _k } is a Cauchy sequence in X.

Proof

Let y₀ = x₀. Then, we construct a sequence {y _k } in X, A _k in CB(X) such that y_2k+1 = g x_2k+1 ∈ S x_2k = A_2k and y_2k+2 = f x_2k+1 ∈ T x_2k+1 = A_2k+1.

For k ∈ {0, 2, 4, 6, …}, it follows from the extended hybrid generalized multi-valued f contraction that

$$\begin{array}{ll}H& (A_k,A_{k+1})\\ =H(S{x}_k,T{x}_{k+1});\\ \le \varphi \left(M\right(x_k,x_{k+1}\left)\right)M(x_k,x_{k+1})\\ +\phi \left(N\right(x_k,x_{k+1}\left)\right)N(x_k,x_{k+1});\\ =\varphi (max\{d(f{x}_k,g{x}_{k+1}),min\left\{d\right(f{x}_k,T{x}_{k+1}),\\ d(g{x}_{k+1},S{x}_k)\left\}\right\})\\ \times max\left\{d\right(f{x}_k,g{x}_{k+1}),min\{d(f{x}_k,T{x}_{k+1}),\\ d(g{x}_{k+1},S{x}_k)\left\}\right\}\\ +\phi (min\{d(f{x}_k,g{x}_{k+1}),d(f{x}_k,T{x}_k),d(g{x}_{k+1},S{x}_{k+1}),\\ d(f{x}_k,T{x}_{k+1}),d(g{x}_{k+1},S{x}_k)\left\}\right)min\left\{d\right(f{x}_k,g{x}_{k+1}),\\ d(f{x}_k,T{x}_k),d(g{x}_{k+1},S{x}_{k+1}),d(f{x}_k,T{x}_{k+1}),\\ d(g{x}_{k+1},S{x}_k)\};\\ =\varphi \left(d\right(f{x}_k,g{x}_{k+1}\left)\right)d(f{x}_k,g{x}_{k+1});\\ =\varphi \left(d\right(y_k,y_{k+1}\left)\right)d(y_k,y_{k+1}).\end{array}$$

Similarly, we show that for k = {1, 3, 5, …}, we have

$$H(A_{k+1},A_k)\le \varphi \left(d\right(y_{k+1},y_k\left)\right)d(y_{k+1},y_k).$$

By (3), for k ∈ {0, 2, 4, 6, …}, we have

$$\begin{array}{ll}d(y_{k+1},y_{k+2})& =d(g{x}_{k+1},f{x}_{k+2});\\ \le H(S{x}_k,T{x}_{k+1})+{\varphi }^{n_{k+1}}\left(M\right(x_k,x_{k+1}\left)\right);\\ =H(A_k,A_{k+1})+{\varphi }^{n_{k+1}}(max\{d(f{x}_k,g{x}_{k+1}),\\ min\left\{d\right(f{x}_k,T{x}_{k+1}),d(g{x}_{k+1},S{x}_k\left)\right\}\left\}\right);\\ =H(A_k,A_{k+1})+{\varphi }^{n_{k+1}}\left(d\right(f{x}_k,g{x}_{k+1}\left)\right);\\ =H(A_k,A_{k+1})+{\varphi }^{n_{k+1}}\left(d\right(y_k,y_{k+1}\left)\right).\end{array}$$

Similarly, for k = {1, 3, 5, …}, we have

$$d(y_{k+2},y_{k+1})\le H(A_{k+1},A_k)+{\varphi }^{n_{k+1}}\left(d\right(y_{k+1},y_k\left)\right).$$

Given that {d(y _k , y_k+1)} is a nonincreasing sequence, thus, all the conditions of Lemma 1 are satisfied. Hence, {y _k } is a Cauchy sequence in X. □

Theorem 1

Let (X, d) be a complete metric space, f, g:X → X, T:X → C B(X) are continuous mappings, and S:X → C B(X) is a continuous extended hybrid generalized multi-valued f contraction such that S X ⊆ g X and T X ⊆ f X. Then, (i) if g, T and f, S are R-weakly commuting, then g, T and f, S have a common coincidence point say z. (ii) Moreover, if g g z = g z, f g z = g z, then f, g, T, and S have a common fixed point.

Proof

Let x₀ be an arbitrary point in X and y₀ = f x₀. Then, we construct a sequence {y _k } in X, A _k in CB(X) respectively as follows. Since S X ⊆ g X, there exists a point x₁ ∈ X such that y₁ = g x₁ ∈ S x₀ = A₀. We can choose a positive integer n₁ such that

$${\varphi }^{n_1}\left(d\right(y_0,y_1\left)\right)\le [1-\varphi (M(x_0,x_1)\left)\right]M(x_0,x_1).$$

(5)

Since T X ⊆ f X, there exists y₂ = f x₂ ∈ T x₁ = A₁ such that

$$d(y_1,y_2)\le H(S{x}_0,T{x}_1)+{\varphi }^{n_1}\left(d\right(y_0,y_1\left)\right).$$

(6)

Using (5) and the notion of extended hybrid generalized multi-valued f contraction in the above inequality, we have

$$\begin{array}{ll}d(y_1,y_2)& \le H(S{x}_0,T{x}_1)+{\varphi }^{n_1}\left(d\right(y_0,y_1\left)\right);\\ \le \varphi \left(M\right(x_0,x_1\left)\right)M(x_0,x_1)\\ +\phi \left(N\right(x_0,x_1\left)\right)N(x_0,x_1)\\ +[1-\varphi (M(x_0,x_1)\left)\right]M(x_0,x_1);\\ =M(x_0,x_1);\\ =min\left\{d\right(f{x}_k,T{x}_{k+1}),d(g{x}_{k+1},S{x}_k\left)\right\}\left\}\right);\\ =d(f{x}_0,g{x}_1);\\ =d(y_0,y_1).\end{array}$$

Now, we can choose a positive integer n₂ > n₁ such that

$${\varphi }^{n_2}\left(d\right(y_2,y_1\left)\right)\le [1-\varphi (M(x_2,x_1)\left)\right]M(x_2,x_1).$$

(7)

There exists y₃ = g x₃ ∈ S x₂ = A₂ such that

$$d(y_3,y_2)\le H(S{x}_2,T{x}_1)+{\varphi }^{n_2}\left(d\right(y_2,y_1\left)\right).$$

(8)

Using (7) and the notion of extended hybrid generalized multi-valued f contraction in the above inequality, we have

$$\begin{array}{ll}d(y_3,y_2)& \le H(S{x}_2,T{x}_1)+{\varphi }^{n_2}\left(d\right(y_2,y_1\left)\right);\\ \le \varphi \left(M\right(x_2,x_1\left)\right)M(x_2,x_1)\\ +\phi \left(N\right(x_2,x_1\left)\right)N(x_2,x_1)\\ +[1-\varphi (M(x_2,x_1)\left)\right]M(x_2,x_1);\\ =M(x_2,x_1);\\ =max\left\{d\right(f{x}_2,g{x}_1),min\{d(f{x}_2,T{x}_1),\\ d(g{x}_1,S{x}_2)\left\}\right\};\\ =d(f{x}_2,g{x}_1);\\ =d(y_2,y_1).\end{array}$$

Now, we can choose a positive integer n₃ > n₂ such that

$${\varphi }^{n_3}\left(d\right(y_2,y_3\left)\right)\le [1-\varphi (M(x_2,x_3)\left)\right]M(x_2,x_3).$$

(9)

There exists y₄ = f x₄ ∈ T x₃ = A₃ such that

$$d(y_3,y_4)\le H(S{x}_2,T{x}_3)+{\varphi }^{n_3}\left(d\right(y_2,y_3\left)\right).$$

(10)

Using (9) and the notion of extended hybrid generalized multi-valued f contraction in the above inequality, we have

$$\begin{array}{ll}d(y_3,y_4)& \le H(S{x}_2,T{x}_3)+{\varphi }^{n_3}\left(d\right(y_2,y_3\left)\right);\\ \le \varphi \left(M\right(x_2,x_3\left)\right)M(x_2,x_3)\\ +\phi \left(N\right(x_2,x_3\left)\right)N(x_2,x_3)\\ +[1-\varphi (M(x_2,x_3)\left)\right]M(x_2,x_3);\\ =M(x_2,x_3);\\ =max\left\{d\right(f{x}_2,g{x}_3),min\{d(f{x}_2,T{x}_3),\\ d(g{x}_3,S{x}_2)\left\}\right\};\\ =d(f{x}_2,g{x}_3);\\ =d(y_2,y_3).\end{array}$$

Now, we can choose a positive integer n₄ > n₃ such that

$${\varphi }^{n_4}\left(d\right(y_4,y_3\left)\right)\le [1-\varphi (M(x_4,x_3)\left)\right]M(x_4,x_3).$$

(11)

There exists y₅ = g x₅ ∈ S x₄ = A₄ such that

$$d(y_5,y_4)\le H(S{x}_4,T{x}_3)+{\varphi }^{n_4}\left(d\right(y_4,y_3\left)\right).$$

(12)

Using (11) and the notion of extended hybrid generalized multi-valued f contraction in the above inequality, we have

$$\begin{array}{ll}d(y_5,y_4)& \le H(S{x}_4,T{x}_3)+{\varphi }^{n_4}\left(d\right(y_4,y_3\left)\right);\\ \le \varphi \left(M\right(x_4,x_3\left)\right)M(x_4,x_3)\\ +\phi \left(N\right(x_4,x_3\left)\right)N(x_4,x_3)\\ +[1-\varphi (M(x_4,x_3)\left)\right]M(x_4,x_3);\\ =M(x_4,x_3);\\ =max\left\{d\right(f{x}_4,g{x}_3),min\{d(f{x}_4,T{x}_3),\\ d(g{x}_3,S{x}_4)\left\}\right\};\\ =d(f{x}_4,g{x}_3);\\ =d(y_4,y_3).\end{array}$$

By repeating this process for all $k\in \mathbb{W}$, we have the following: Case (i). For k ∈ {0, 2, 4, 6, …}, we can choose a positive integer n_k+1 such that

$${\varphi }^{n_{k+1}}\left(d\right(y_k,y_{k+1}\left)\right)\le [1-\varphi (M(x_k,x_{k+1})\left)\right]M(x_k,x_{k+1}).$$

(13)

There exists y_k+2 = f x_k+2 ∈ T x_k+1 = A_k+1 such that

$$d(y_{k+1},y_{k+2})\le H(S{x}_k,T{x}_{k+1})+{\varphi }^{n_{k+1}}\left(d\right(y_k,y_{k+1}\left)\right).$$

(14)

Using (13) and the notion of extended generalized multi-valued f contraction in the above inequality, we have

$$\begin{array}{ll}d(y_{k+1},y_{k+2})& \le H(S{x}_k,T{x}_{k+1})+{\varphi }^{n_{k+1}}\left(d\right(y_k,y_{k+1}\left)\right);\\ \le \varphi \left(M\right(x_k,x_{k+1}\left)\right)M(x_k,x_{k+1})\\ +\phi \left(N\right(x_k,x_{k+1}\left)\right)N(x_k,x_{k+1})\\ +[1-\varphi (M(x_k,x_{k+1})\left)\right]M(x_k,x_{k+1});\\ =M(x_k,x_{k+1});\\ =max\left\{d\right(f{x}_k,g{x}_{k+1}),min\{d(f{x}_k,T{x}_{k+1}),\\ d(g{x}_{k+1},S{x}_k)\left\}\right\};\\ =d(f{x}_k,g{x}_{k+1});\\ =d(y_k,y_{k+1}).\end{array}$$

Case (ii). For k ∈ {1, 3, 5, 7, …}, we can choose a positive integer n _k such that

$${\varphi }^{n_{k+1}}\left(d\right(y_{k+1},y_k\left)\right)\le [1-\varphi (M(x_{k+1},x_k)\left)\right]M(x_{k+1},x_k).$$

(15)

There exists y_k+2 = g x_k+2 ∈ S x_k+1 = A_k+1 such that

$$d(y_{k+2},y_{k+1})\le H(S{x}_{k+1},T{x}_k)+{\varphi }^{n_{k+1}}\left(d\right(y_{k+1},y_k\left)\right).$$

(16)

Using (15) and the notion of extended generalized multi-valued f contraction in the above inequality, we have

$$\begin{array}{ll}d(y_{k+2},y_{k+1})& \le H(S{x}_{k+1},T{x}_k)+{\varphi }^{n_{k+1}}\left(d\right(y_{k+1},y_k\left)\right);\\ \le \varphi \left(M\right(x_{k+1},x_k\left)\right)M(x_{k+1},x_k)\\ +\phi \left(N\right(x_{k+1},x_k\left)\right)N(x_{k+1},x_k)\\ +[1-\varphi (M(x_{k+1},x_k)\left)\right]M(x_{k+1},x_k);\\ =M(x_{k+1},x_k);\\ =max\left\{d\right(f{x}_{k+1},g{x}_k),min\{d(f{x}_{k+1},T{x}_k),\\ d(g{x}_k,S{x}_{k+1})\left\}\right\};\\ =d(f{x}_{k+1},g{x}_k);\\ =d(y_{k+1},y_k).\end{array}$$

Hence, {d(y _k ,y_k+1)} is a nonincreasing sequence for each k ≥ 0. Thus, by Lemma 3, {y _k } is a Cauchy sequence in X. Then, (2) ensures that {A _k } is a Cauchy sequence in CB(X). As we know that if X is complete, then CB(X) is also complete. Therefore, there exist z ∈ X and A ∈ CB(X) such that y _k  → z and A _k  → A. Moreover, g x_2k+1 → z and f_2k+2 → z, since

$$d(z,A)=\underset{k\to \infty }{\text{lim}}d(y_k,A_k)\le \underset{k\to \infty }{\text{lim}}H(A_{k-1},A_k)=0.$$

(17)

It follows that z ∈ A, since A is closed. Thus, we have

$$\underset{k\to \infty }{\text{lim}}g{x}_{2k+1}=z\in A=\underset{k\to \infty }{\text{lim}}S{x}_{2k}\text{and}$$

$$\underset{k\to \infty }{\text{lim}}f{x}_{2k+2}=z\in A=\underset{k\to \infty }{\text{lim}}T{x}_{2k+1}$$

As g, T and f, S are R-weakly commuting, we have

$$\begin{array}{ll}d(\mathit{\text{gf}}{x}_{2k+2},\mathit{\text{Tg}}{x}_{2k+1})& \le H(\mathit{\text{gT}}{x}_{2k+1},\mathit{\text{Tg}}{x}_{2k+1})\\ \le \mathit{\text{Rd}}(g{x}_{2k+1},T{x}_{2k+1}).\end{array}$$

(18)

$$d(\mathit{\text{fg}}{x}_{2k+1},\mathit{\text{Sf}}{x}_{2k})\le H(\mathit{\text{fS}}{x}_{2k},\mathit{\text{Sf}}{x}_{2k})\le \mathit{\text{Rd}}(f{x}_{2k},S{x}_{2k}).$$

(19)

Letting k → ∞ in (18) and (19) and using (17) and the continuity of f, g, T, and S, we get

$$\mathit{\text{gz}}\in \mathit{\text{Tz}}\text{and}\mathit{\text{fz}}\in \mathrm{Sz.}$$

By condition (ii) of Theorem 1, we have g g z = g z, f g z = g z. Let v = g z and then we have g v = v = f v. From (2), we have

$$\begin{array}{ll}H(\mathit{\text{Sv}},\mathit{\text{Tz}})& \le \varphi (max\{d(\mathit{\text{fv}},\mathit{\text{gz}}),\\ min\left\{d\right(\mathit{\text{fv}},\mathit{\text{Tz}}),d(\mathit{\text{gz}},\mathit{\text{Sv}}\left)\right\}\left\}\right)\\ \times max\left\{d\right(\mathit{\text{fv}},\mathit{\text{gz}}),\\ min\left\{d\right(\mathit{\text{fv}},\mathit{\text{Tz}}),d(\mathit{\text{gz}},\mathit{\text{Sv}}\left)\right\}\}\\ +\phi (min\{d(\mathit{\text{fv}},\mathit{\text{gz}}),d(\mathit{\text{fv}},\mathit{\text{Tv}}),d(\mathit{\text{gz}},\mathit{\text{Sz}}),\\ d(\mathit{\text{fv}},\mathit{\text{Tz}}),d(\mathit{\text{gz}},\mathit{\text{Sv}})\left\}\right)\\ \times min\left\{d\right(\mathit{\text{fv}},\mathit{\text{gz}}),d(\mathit{\text{fv}},\mathit{\text{Tv}}),d(\mathit{\text{gz}},\mathit{\text{Sz}}),\\ d(\mathit{\text{fv}},\mathit{\text{Tz}}),d(\mathit{\text{gz}},\mathit{\text{Sv}})\}.\end{array}$$

Note that f v = g z and f v ∈ T z. Hence, we have H(S v, T z) = 0, i.e., S v = T z. Again from (2), we have

$$\begin{array}{ll}H(\mathit{\text{Sv}},\mathit{\text{Tv}})& \le \varphi (max\{d(\mathit{\text{fv}},\mathit{\text{gv}}),\\ min\left\{d\right(\mathit{\text{fv}},\mathit{\text{Tv}}),d(\mathit{\text{gv}},\mathit{\text{Sv}}\left)\right\}\left\}\right)\\ \times max\left\{d\right(\mathit{\text{fv}},\mathit{\text{gv}}),\\ min\left\{d\right(\mathit{\text{fv}},\mathit{\text{Tv}}),d(\mathit{\text{gv}},\mathit{\text{Sv}}\left)\right\}\}\\ +\phi (min\{d(\mathit{\text{fv}},\mathit{\text{gv}}),d(\mathit{\text{fv}},\mathit{\text{Tv}}),d(\mathit{\text{gv}},\mathit{\text{Sv}}),\\ d(\mathit{\text{fv}},\mathit{\text{Tv}}),d(\mathit{\text{gv}},\mathit{\text{Sv}})\left\}\right)\\ \times min\left\{d\right(\mathit{\text{fv}},\mathit{\text{gv}}),d(\mathit{\text{fv}},\mathit{\text{Tv}}),d(\mathit{\text{gv}},\mathit{\text{Sv}}),\\ d(\mathit{\text{fv}},\mathit{\text{Tv}}),d(\mathit{\text{gv}},\mathit{\text{Sv}})\}.\end{array}$$

Note that f v = g v and g v ∈ S v. Hence, we have H(S v, T v) = 0, i.e., S v = T v. Therefore, we have v = f v = g v ∈ S v = T v. □

Remark 2

Theorem 1 improves and extends some known results of Kamran [12], Nadler [13], Hu [14], Kaneko [15], and Mizoguchi and Takahashi [16].

Example 1

Let X = [0, ∞) be endowed with the metric

$$d(x,y)=\left\{\begin{array}{c}x+y\text{if}x\ne y,\\ 0\text{if}x=\mathrm{y.}\end{array}\right.$$

(20)

Define T, S:X → C B(X) and f, g:X → X by $\mathit{\text{Tx}}=[0,\frac{x}{3}]$, S x = {0}, $\mathit{\text{fx}}=\frac{3x}{2}$, and $\mathit{\text{gx}}=\frac{x}{2}$ for all x ∈ X. Let $\varphi \left(t\right)=\frac{2t}{3}$ and φ(t) = t for all t ≥ 0. It is easy to show that S is the extended hybrid generalized multi-valued f contraction. It is easy to check that all the conditions of Theorem 1 hold, and 0 is a common fixed point of f, g, T, and S.