In this subsection, we prove some fixed point results for the mappings satisfying the contractive conditions of integral type by using the fixed point results related with the function ξ in Section 2.

Theorem 3.1: Let (X, d) be a complete metric space and T :X → CL(X) be a multi-valued mapping. Suppose that there exist three functions ψ ∈ Φ, φ : [0, ∞) → [0, ∞) and α : X × X → [0, ∞) satisfying the following conditions:(I nt₁) φ : [0,∞) → [0,∞) is a Lebesgue integrable mapping which is summable on each compact subset of [0,∞);(I nt₂) for each $\epsilon >0,{\displaystyle {\int }_0^{\epsilon }\varphi (t)}dt>0;$;(I nt₃) for each a, b > 0, we have $${\displaystyle \underset{0}{\overset{a+b}{\int }}\varphi \left(t\right)dt\le {\displaystyle \underset{0}{\overset{a}{\int }}\varphi \left(t\right)dt}+{\displaystyle \underset{0}{\overset{b}{\int }}\varphi \left(t\right)dt;}}$$(I nt₄) the following condition holds: $$x\in X,y\in Txwith\alpha (x,y)\ge 1⟹{\displaystyle \underset{0}{\overset{d(y,Ty)}{\int }}\varphi (w)dw\le \psi \left({\displaystyle \underset{0}{\overset{M(x,y)}{\int }}\varphi (w)dw}\right)},$$where $M(x,y)=\text{m}\text{a}\text{x}\left\{d(x,y),d(x,Tx),d(y,Ty),\frac{d(x,Ty)+d(y,Tx)}{2}\right\};$(Int₅) ψ is strictly increasing function.Further, if the following assertions hold:(S₁) T is an α-admissible multi-valued mapping;(S₂) there exist x_o ϵ X and x₁ ϵ T x_o such that α(x₀, x₁) > 1;(S₃) T is a continuous multi-valued mapping,then T has a fixed point in X.

Proof: Define a mapping ξ : [0, ∞) → [0, ∞) by $\xi (1)={\displaystyle {\int }_0^t\varphi (s)}ds$ for each t ∈ [0, ∞). From the conditions (Int₁)-(Int₃), we get ξ ϵ Ξ. By the definition of mapping ξ and the condition (Int₄), it follows that the condition (2) holds. Since ψ is a strictly increasing function, it follows that T is a strictly weakly (α, ψ, ξ)-contractive mapping. Hence the conclusion of Theorem 3.1 follows from Theorem 2.2. □

Corollary 3.2: Let (X, d) be a complete metric space and T:X → CL(X) be a multi-valued mapping. Suppose that there exist three functions ψ ϵ Φ, φ : [0, ∞) → [0, ∞) and α :X χ X → [0, ∞) satisfying the following conditions:(Int₁) φ : [0, ∞) → [0, ∞) is a Lebesgue integrable mapping which is summable on each compact subset of [0, ∞);(Int₂) for each $\epsilon >0,{\displaystyle {\int }_0^{\epsilon }\varphi (t)}dt>0;$(Int₃) for each a, b > 0, we have $${\displaystyle \underset{0}{\overset{a+b}{\int }}\varphi \left(t\right)dt\le {\displaystyle \underset{0}{\overset{a}{\int }}\varphi \left(t\right)dt}+{\displaystyle \underset{0}{\overset{b}{\int }}\varphi \left(t\right)dt;}}$$(Int₄) the following condition holds: $$x,y\in Xwith\alpha (x,y)\ge 1⟹\underset{0}{\overset{H(Tx,Ty)}{\int }}\varphi (w)dw\le \psi \left(\underset{0}{\overset{M(x,y)}{\int }}\varphi (w)dw\right),$$where $M(x,y)=\text{m}\text{a}\text{x}\left\{d(x,y),d(x,Tx),d(y,Ty),\frac{d(x,Ty)+d(y,Tx)}{2}\right\};$(Int₅) ψ is strictly increasing function.Further, if the following assertions hold:(S₁) T is an α-admissible multi-valued mapping;(S₂) there exist x_o ϵ X and x₁ ϵ T x_o such that α(x₀, x₁) > 1;(S₃) T is a continuous multi-valued mapping,then T has a fixed point in X.

Remark 3.3: Theorem 3.1 and Corollary 3.2 are still valid if we replace condition (S3) by the following condition:

• -

  if {x_n} is a sequence in X with x_n+1 ϵ Tx_n, x_n → x ϵ X as n → 1 and α(x_n, x_n+1) > 1 for all n ϵ N, then we have $${\displaystyle \underset{0}{\overset{d\left(x_{n+1},Tx\right)}{\int }}\varphi \left(w\right)dw\le \psi \left({\displaystyle \underset{0}{\overset{M\left(x_n,x\right)}{\int }}\varphi (w)dw}\right)}$$

In this subsection, we can derive some fixed point results in ordinary metric spaces from the fixed point results related with the contractive condition (2).

Theorem 3.4: Let (X, d) be a complete metric space and T :X → CL(X) be a multi-valued mapping. Suppose that there exist three functions 𝜓 ϵ ѱ ξ ϵ Ξ and α X × X → [0, ∞) such that 𝜓 is strictly increasing and $$\alpha \left(x,y\right)\xi \left(d\left(y,Ty\right)\right)\le \psi \left(\xi \left(M\left(x,y\right)\right)\right)$$(26)for all x ∈ X and y ∈ T x, where $M(x,y)=\text{m}\text{a}\text{x}\left\{d(x,y),d(x,Tx),d(y,Ty),\frac{d(x,Ty)+d(y,Tx)}{2}\right\}.$Further, if the following conditions hold:(S₁) T is an α-admissible multi-valued mapping;(S₂) there exist x₀ and x₁ ϵ T x₀ such that ϵ(x₀; x₁) ≥ 1;(S₃) T is a continuous multi-valued mapping,Then T has a fixed point in X.

Proof: We show that T satisfies the condition (2). For any x ϵ X and y ϵ T x, assume that α(x, y) > 1. Using (26), we get $$\xi \left(d\left(y,Ty\right)\right)\le \alpha \left(x,y\right)\xi \left(d\left(y,.Ty\right)\right)\le \psi \left(\xi \left(M\left(x,y\right)\right)\right).$$This implies that the condition (2) holds. Therefore, the conclusions of this theorem follows from Theorem 2.2. □

Remark 3.5: Theorem 3.4 is still valid if we replace the contractive condition (26) by the following contractive conditions:

• -

  [τ + ξ(d(y, Ty))]^{α(x, y)} ≤ τ + ψ (ξ(M(x, y))) for all x ϵ X and y ϵ Tx, where τ > 1, or

• -

  [τ + α(x, y]^{ξ(d(y, Ty))} ≤ τ^{ψ (ξ(M(x, y)))} for all x ϵ X and y ϵ Tx, where τ > 1.

In this section, we give some fixed point results on metric spaces endowed with the arbitrary binary relation which can be regarded as consequences of the results presented in the previous section.

The following notions and definitions are needed.

Let (X, d) be a metric space and ℛ be a binary relation over X. Denote $$S:=ℛ\cup {ℛ}^{-1}.$$

It is easy to see that, for all x, y ϵ X, $$xSy⟺xℛyoryℛx$$

Definition 3.6: Let X be a nonempty set and ℛ be a binary relation over X. A multi-valued mapping T : X → N(X) is said to be ℛ-weakly comparative if, for each x ϵ X and y ϵ Tx with x𝒮y, we have y𝒮z for all z ϵ Ty.

Definition 3.7: Let (X, d) be a metric space and ℛ be a binary relation over X.

• (1)

  A mapping T :X → CL(X) is said to be weakly (𝒮, ψ, ξ)-contractive if there exist two functions ψ ϵ ψ and ξ ϵ Ξ such that the following condition holds: $$x\in X,y\in TxwithxSy⟹\xi \left(d\left(y,Ty\right)\right)\le \psi \left(\xi \left(M\left(x,y\right)\right)\right),$$(27)

  where $M(x,y)=\text{m}\text{a}\text{x}\left\{d(x,y),d(x,Tx),d(y,Ty),\frac{d(x,Ty)+d(y,Tx)}{2}\right\}.$

• (2)

  If ψ is strictly increasing, then the the weakly (𝒮, ψ, ξ)-contractive mapping is said to be strictly weakly (𝒮, ψ, ξ)-contractive.

Theorem 3.8: Let (X, d) be a complete metric space, ℛ be a binary relation over X and T : X → CL(XM) be a strictly weakly (𝒮, ψ, ξ)-contractive mapping satisfying the following conditions:(S₁)T is a ℛ-weakly comparative mapping;(S₂)there exist x_o and xi ∈ Tx_o such that x_o𝒮x₁;(S₃)T is a continuous multi-valued mapping.

Then T has a fixed point in X.

Proof: Define a mapping α :X × X → [0, ∞) by $$\alpha \left(x,y\right)=\{\begin{array}{cc}1,& x,y\in x\mathcal{S}y,\\ 0,& \text{otherwise}.\end{array}$$The conclusion of Theorem 3.8 follows from Theorem 2.2. □

Using Theorem 2.3, we get the following result:

Theorem 3.9: Let (X, d) be a complete metric space, ℛ be a binary relation over X and T : X → CL(X) be a strictly weakly (𝒮, ψ, ξ)-contractive mapping satisfying the following conditions:(S₁)T is a ℛ-weakly comparative mapping;(S₂)there exist x_o and x_i ∈ Tx_o such that x_o𝒮x₁;$({S^{\prime }}_3)$if {x_n} is a sequence in X with x_n+1 ∈ Tx_n, x_n+1 → x ∈ X as n → 1 and x_{n + 1}𝒮x_{n + 1} for all n ∈ ℕ, then we have $$\xi (d(x_{n+1},Tx))\le \psi (\xi (M(x_n,x)))$$

for all n ∈ ℕ.

Then T has a fixed point in X.

Next, we give some fixed point results on partially ordered metric spaces from Theorem 3.8 and Theorem 3.9. We need the following notions and definitions:

Definition 3.10: (X, d,⪯) is called a partially ordered metric space when (X, d) is a metric space and (X, ⪯) is a partially ordered set.Let (X, d, ⪯) be a partially ordered metric space. Denotes ≍ : = ⪯ ∪ ⪯^-1 . This implies that, for all x, y ∈ X, $$x\asymp y\iff x\preccurlyeq y\text{or}y\preccurlyeq x.$$

Definition 3.11: Let (X, ⪯) be a partially ordered set. A multi-valued mapping T :X → N(X) is said to be ⪯ weakly comparative if, for each x ∈ X and y ∈ Tx with x ⪯ y, we have y ⪯ z for all z ∈ Ty.

Definition 3.12: Let (X, d, ⪯) be a partially ordered metric spaces.

• (1)

  A mapping T : X → CL(X) is said to be weakly (≍, ψ, ξ)-contractive if there exist two functions ψ ∈ Φ and ξ ∈ Ξ such that the following condition holds: $$x\in X,y\in Txwithx\asymp y⟹\xi (d(y,Ty))\le \psi (\xi (M(x,y))),$$(28)

  where $M(x,y)=\text{m}\text{a}\text{x}\left\{d(x,y),d(x,Tx),d(y,Ty),\frac{d(x,Ty)+d(y,Tx)}{2}\right\}$

• (2)

  If Ψ ∊ ψ is strictly increasing, then the the weakly (≍, ψ, ξ)-contractive mapping is said to be strictly weakly (≍, ψ, ξ)-contractive.

It is easy to see that ⪯ is a binary relation for a partially ordered metric spaces (X, d, ⪯). Therefore, we obtain the following results from Theorem 3.8 and Theorem 3.9:

Corollary 3.13: Let (X, d, ⪯) be a complete partially ordered metric spaces and T : X → CL(X) be a strictly weakly (≍, ψ, ξ)-contractive mapping satisfying the following conditions:(S₁)T is a ⪯ weakly comparative mapping;(S₂)there exist x_o and x₁ ∈ Tx_o such that x_o ≍ x₁;(S₃)T is a continuous multi-valued mapping.Then T has a fixed point in X.

Corollary 3.14: Let (X, d, ⪯) be a complete partially ordered metric spaces and T : X → CL(X) be a strictly weakly (≍, ψ, ξ)-contractive mapping satisfying the following conditions:(S₁)T is a ⪯ weakly comparative mapping;(S₂)there exist x_o and x₁ ∈ Tx_o such that x_o ∈ x₁;$({S^{\prime }}_3)$if {x_n} is a sequence in X with x_{n + 1} ∈ Tx_n, x_n → x ∈ X as n → 1 and x_n ≍ x_{n + 1} for all n ∈ ℕ, then we have $$\xi (d(x_{n+1},Tx))\le \psi (\xi (M(x_n,x)))$$for all n ∈ ℕ.Then T has a fixed point in X.

Throughout this subsection, let (X, d) be a metric space. A set {(x, x) :x ∈ X} is called a diagonal of the Cartesian product X × X, which is denoted by Δ. Consider a graph G such that the set V(G) of its vertices coincides with X and the set E(G) of its edges contains all loops, i.e., Δ ⊆ E(G). We assume G has no parallel edges and so we can identify G with the pair (V(G), E(G)). Moreover, we may treat G as a weighted graph by assigning to each edge the distance between its vertices. In this subsection, we prove fixed point results on a metric space endowed with graph.

Before presenting our results, we give the following notions and definitions:

Definition 3.15: Let X be a nonempty set endowed with a graph G and T : X → N(X) be a multi-valued mapping, where X is a nonempty set X. A mapping T is said to have weakly preserve edge if, for each x ∈ X and y ∈ Tx with (x, y) ∈ E(G), we have (y, z) ∈ E(G) for all z ∈ Ty.

Definition 3.16: Let (X, d) be a metric space endowed with a graph G. A mapping T : X → CL(X) is said to be weakly (E(G), ψ, ξ)-contractive if there exist two functions ψ ∈ Φ and ξ ∈ Ξ such that the following condition holds: $$x\in X,y\in Txwith(x,y)\in E(G)⟹\xi (d(y,Ty))\le \psi (\xi (M(x,y))),$$(29)where $M\left(x,y\right)=\max \left\{d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),\frac{d\left(x,Ty\right)+d\left(y,Tx\right)}{2}\right\}.$

Definition 3.17: Let (X, d) be a metric space endowed with a graph G.

• (1)

  A mapping T : X → CL(X) is said to be (E(G), ψ, ξ) contractive if there exist two functions ψ ∈ Φ and ξ ∈ Ξ such that the following condition holds: $$x\in X,y\in Txwith(x,y)\in E(G)⟹\xi (H(Tx,Ty))\le \psi (\xi (M(x,y))),$$(30)

  where $M\left(x,y\right)=\max \left\{d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),\frac{d\left(x,Ty\right)+d\left(y,Tx\right)}{2}\right\}.$

• (2)

  If ψ ∈ Φ is strictly increasing, then the (weakly) (E(G), ψ, ξ)-contractive mapping is said to be strictly (weakly) (E(G), ψ, ξ)-contractive.

Remark 3.18: The (E(G), ψ, ξ)-contractive condition implies the weakly (E(G), ψ, ξ)-contractive condition.

Theorem 3.19: Let (X, d) be a complete metric space endowed with a graph G and T : X → CL(X) be a strictly weakly (E(G), ψ, ξ)-contractive mapping satisfying the following conditions:(S₁)T has weakly preserve edge;(S₂)there exist x_o and x₁ ∈ Txo such that (x_o, x₁) ∈ E(G);(S₃)T is a continuous multi-valued mapping.Then T has a fixed point in X.

Proof: The conclusion of this theorem can be obtain from Theorem 2.2 if we define a mapping α :X × X → [0, ∞)by $$\alpha \left(x,y\right)=\{\begin{array}{cc}1,& x,y\in \left(x,y\right)\in E\left(G\right),\\ 0,& \text{otherwise}\text{.}\end{array}$$ □

Corollary 3.20: Let (X, d) be a complete metric space endowed with a graph G and T : X → CL(X) be a strictly (E(G), ψ, ξ)-contractive mapping satisfying the following conditions:(S₁)T has weakly preserve edge;(S₂)there exist x_o and x₁ ∈ Tx_o such that (x_o, x₁) ∈ E(G);(S₃)T is a continuous multi-valued mapping.Then T has a fixed point in X.By using Theorem 2.3, we get the following result:

Theorem 3.21: Let (X, d) be a complete metric space endowed with a graph G and T : X → CL(X) be a strictly weakly (E(G), ψ, ξ)-contractive mapping satisfying the following conditions:(S₁)T has weakly preserve edge;(S₂)there exist x_o and x₁ ∈ Txo such that (x_o, x₁) ∈ E(G);$({S^{\prime }}_3)$if {x_n} is a sequence in X with x_{n + 1} ∈ Tx_n, x_{n + 1} → x ∈ X as n → 1 and (x_n, x_{n + 1}) ∈ E(G) for all n ∈ ℕ, then we have $$\xi (d(x_n,Tx))\le \psi (\xi (M(x_n,x)))$$for all n ∈ ℕ.Then T has a fixed point in X.

Corollary 3.22: Let (X, d) be a complete metric space endowed with a graph G and T : X → CL(X) be a strictly (E(G), ψ, ξ)-contractive mapping satisfying the following conditions:(S₁)T has weakly preserve edge;(S₂)there exist xo and xi ∈ Txo such that (xo, xi) ∈ E(G);$({S^{″}}_3)$if {x_n is a sequence in X with x_n+ → x ∈ X as n → 1 and (x_n, x_{n + 1}) ∈ E(G) for all n ∈ ℕ, then we have (x_n, x) ∈ E(G) for all n ∈ ℕ.Then T has a fixed point in X.