On fixed point theorems for mappings with PPF dependence

Abstract

In this paper, we consider two families ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$ of mappings defined on $[0,+\mathrm{\infty })$ satisfy some certain properties. Using the mentioned properties for ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$, we prove the analogous PPF dependent fixed point theorems for mappings as in Drici et al. (Nonlinear Anal. 67:641-647, 2007) in partially ordered Banach spaces where mappings satisfy the weaker contractive conditions without assuming the topological closedness with respect to the norm topology for the Razumikhin class ${\mathcal{R}}_c$.

MSC:54H25, 55M20.

1 Introduction and preliminary results

The fixed point theorems for mappings satisfying certain contractive conditions have been continually studied for decade (see [1–9] and references contained therein). Bernfeld et al. [10] proved the existence of PPF (past, present and future) dependent fixed points in the Razumikhin class for mappings that have different domains and ranges. After that Dhage [11] extended the existence of PPF dependent fixed points to PPF common dependent fixed points for mappings satisfying the weaker contractive conditions. In 2007, Drici et al. [2] proved the fixed point theorems in partially ordered metric spaces for mappings with PPF dependence. In this paper, we consider two families ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$ of mappings defined on $[0,+\mathrm{\infty })$ satisfy some certain properties. Moreover, the PPF dependent fixed point theorems for mappings satisfying some generalized contractive conditions in partially ordered Banach spaces are proven using the mentioned properties for ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$.

Suppose that E is a real Banach space with the norm ${\parallel \cdot \parallel }_E$ and I is a closed interval $[a,b]$ in ℝ. Let $E_0=C(I,E)$ be the set of all continuous E-valued mappings on I equipped with the supremum norm ${\parallel \cdot \parallel }_{E_0}$ defined by

$${\parallel \varphi \parallel }_{E_0}=\underset{t\in I}{sup}{\parallel \varphi (t)\parallel }_E,$$

(1.1)

for all $\varphi \in E_0$. For a fixed element $c\in I$, the Razumikhin class of mappings in $E_0$ is defined by

$${\mathcal{R}}_c=\{\varphi \in E_0:{\parallel \varphi \parallel }_{E_0}={\parallel \varphi (c)\parallel }_E\}.$$

(1.2)

Recall that a point $\varphi \in E_0$ is said to be a PPF dependent fixed point or a fixed point with PPF dependence of $T:E_0\to E$ if $T\varphi =\varphi (c)$ for some $c\in I$.

Example 1.1 Let $T:C([0,1],\mathbb{R})\to \mathbb{R}$ be defined by

$$T\varphi =\frac{1}{2}\left(\underset{t\in [0,1]}{sup}|\varphi (t)|\right)\text{for all }\varphi \in C([0,1],\mathbb{R}).$$

Therefore T is a contraction with a constant $\frac{1}{2}$. Suppose that $\varphi (t)=t^2+1$ for all $t\in [0,1]$. Since $T\varphi =\frac{1}{2}({sup}_{t\in [0,1]}|\varphi (t)|)=1=\varphi (0)$, we find that ϕ is a fixed point with dependence of T.

Definition 1.2 Let A be a subset of E. Then

• (i) A is said to be topologically closed with respect to the norm topology if for each sequence $\{x_n\}$ in A with $x_n\to x$ as $n\to \mathrm{\infty }$ implies $x\in A$.

• (ii) A is said to be algebraically closed with respect to the difference if $x-y\in A$ for all $x,y\in A$.

Recently, Dhage [11] proved the existence of PPF fixed points for mappings satisfying the condition of Cirić type generalized contraction assuming topological closedness with respect to the norm topology for a Razumikhin class.

Definition 1.3 (Dhage, [11])

A mapping $T:E_0\to E$ is said to satisfy the condition of Cirić type generalized contraction if there exists a real number $\lambda \in [0,1)$ satisfying

$$\begin{array}{rcl}\parallel T\varphi -T\alpha \parallel & \le & \lambda max\{{\parallel \varphi -\alpha \parallel }_{E_0},{\parallel \varphi (c)-T\varphi \parallel }_E,{\parallel \alpha (c)-T\alpha \parallel }_E,\\ \frac{1}{2}[{\parallel \varphi (c)-T\alpha \parallel }_E+{\parallel \alpha (c)-T\varphi \parallel }_E]\},\end{array}$$

(1.3)

for all $\varphi ,\alpha \in E_0$ and for some $c\in I$.

Theorem 1.4 (Dhage, [11])

Suppose that $T:E_0\to E$ satisfies the condition of Cirić type generalized contraction. Assume that ${\mathcal{R}}_c$ is topologically closed with respect to the norm topology and is algebraically closed with respect to the difference, then T has a unique PPF dependent fixed point in ${\mathcal{R}}_c$.

It is a natural question that the result of the previous theorem is still valid by omitting the topological closedness of ${\mathcal{R}}_c$. In the next result, we will answer the question.

Proposition 1.5 The Razumikhin class ${\mathcal{R}}_c$ is topologically closed with respect to the norm topology.

Proof Let $\{{\varphi }_n\}$ be a sequence in ${\mathcal{R}}_c$ converging to ϕ. This implies that

$$\underset{n\to \mathrm{\infty }}{lim}{\parallel {\varphi }_n-\varphi \parallel }_{E_0}=0\text{where }{\parallel {\varphi }_n-\varphi \parallel }_{E_0}=\underset{t\in I}{sup}{\parallel {\varphi }_n(t)-\varphi (t)\parallel }_E.$$

Therefore

$$\underset{n\to \mathrm{\infty }}{lim}{\parallel {\varphi }_n\parallel }_{E_0}={\parallel \varphi \parallel }_{E_0}\text{and}\underset{n\to \mathrm{\infty }}{lim}{\parallel {\varphi }_n(t)\parallel }_E={\parallel \varphi (t)\parallel }_E\text{for all }t\in I.$$

Since ${\varphi }_n\in {\mathcal{R}}_c$ for all $n\in \mathbb{N}$, we obtain ${\parallel {\varphi }_n\parallel }_{E_0}={\parallel {\varphi }_n(c)\parallel }_E$. Therefore

$$\underset{n\to \mathrm{\infty }}{lim}{\parallel {\varphi }_n(c)\parallel }_E={\parallel \varphi \parallel }_{E_0}.$$

By the uniqueness of the limit, we have ${\parallel \varphi \parallel }_{E_0}={\parallel \varphi (c)\parallel }_E$. Hence $\varphi \in {\mathcal{R}}_c$ and thus ${\mathcal{R}}_c$ is topologically closed with respect to the norm topology. □

Hence, using Proposition 1.5, we can drop the topological closedness with respect to the norm topology for ${\mathcal{R}}_c$ in Theorem 1.4.

The following example shows that the algebraic closedness with respect to the difference of Razumikhin class ${\mathcal{R}}_c$ may fail.

Example 1.6 Let $E_0=C([0,1],\mathbb{R})$ and $c=1$. If we take $\varphi (t)=t^2$ and $\alpha (t)=t$ for all $t\in [0,1]$, then $\varphi ,\alpha \in {\mathcal{R}}_c$ while $\varphi -\alpha \notin {\mathcal{R}}_c$.

Proposition 1.7 If the Razumikhin class ${\mathcal{R}}_c$ is algebraically closed with respect to the difference, then ${\mathcal{R}}_c$ is a convex set.

Proof Since ${\mathcal{R}}_c$ is algebraically closed with respect to the difference, we have ${\mathcal{R}}_c-{\mathcal{R}}_c\subseteq {\mathcal{R}}_c$. Using the fact that $-{\mathcal{R}}_c={\mathcal{R}}_c$, we obtain ${\mathcal{R}}_c+{\mathcal{R}}_c\subseteq {\mathcal{R}}_c$. Since $\lambda {\mathcal{R}}_c\subseteq {\mathcal{R}}_c$ for all $\lambda \in [0,1]$, we get $\lambda {\mathcal{R}}_c+(1-\lambda ){\mathcal{R}}_c\subseteq {\mathcal{R}}_c$. Hence ${\mathcal{R}}_c$ is a convex set. □

One can verify that Razumikhin class ${\mathcal{R}}_c$ is a cone (i.e., $\lambda \varphi \in {\mathcal{R}}_c$, for each $\varphi \in {\mathcal{R}}_c$ and $\lambda \ge 0$). Then by applying the previous theorem Razumikhin class ${\mathcal{R}}_c$ is a convex cone (also closed).

In 2007, Drici et al. [2] proved the following the fixed point theorems in partially ordered complete metric spaces for mappings with PPF dependence.

Theorem 1.8 ([2])

Let $(E,d,\le )$ be a partially ordered complete metric space and $T:E_0\to E$ where $E_0=C(I,E)$ and $I=[a,b]$. Assume that

• (i) T is a nondecreasing mapping;

• (ii) for all $\varphi ,\alpha \in E_0$ with $\varphi \le \alpha$, $d(T\varphi ,T\alpha )\le k{d}_0(\varphi ,\alpha )$ where $d_0(\varphi ,\alpha )={max}_{s\in I}d(\varphi (s),\alpha (s))$ and $k\in [0,1)$;

• (iii) there exists a lower solution ${\varphi }_0$ such that ${\varphi }_0(c)\le T{\varphi }_0$;

• (iv) T is a continuous mapping or if $\{{\varphi }_n\}$ is a nondecreasing sequence in $E_0$ converging to $\varphi \in E_0$, then ${\varphi }_n\le \varphi$ for all $n\in \mathbb{N}$.

Then T has a PPF dependent fixed point in $E_0$.

It is a natural question if one can obtain the result of the aforementioned theorem for a generalized contraction (that is, condition (ii)). One of the aims of this paper is to answer the question by considering a general case of Cirić type generalized contractions.

In this paper, we consider two families ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$ of mappings defined on $[0,+\mathrm{\infty })$ satisfying some certain properties. Using the mentioned properties for ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$, we prove the analogous PPF dependent fixed point theorems for mappings as in [2] in partially ordered real Banach spaces where mappings satisfy the weaker contractive conditions.

2 PPF dependent fixed points in partially ordered Banach spaces

We begin this part with the consideration of the example of a partially ordered real Banach space. Recall that the set $B(X,\mathbb{R})$ of all bounded linear operators from a normed space X into ℝ is a real Banach space with a norm defined by

$$\parallel f\parallel =\underset{x\in X,\parallel x\parallel =1}{sup}|f(x)|\text{for all }f\in B(X,\mathbb{R}).$$

We know that $B(X,\mathbb{R})$ is a partially ordered real Banach space with a partial order defined as follows:

$$f\le g\text{if and only if}f(x)\le g(x)\text{for all }x\in X.$$

From now on, let $(E,\le )$ be a partially ordered real Banach space. In this paper, we use the following notations:

$$\begin{array}{rcl}{\mathrm{\Psi }}_1& =& \{\psi :\psi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })\text{ is nondecreasing with}\\ \sum _{n=1}^{\mathrm{\infty }}{\psi }^n(t)<\mathrm{\infty }\text{ for all }t\in (0,+\mathrm{\infty })\}\text{and}\\ {\mathrm{\Psi }}_2& =& \{\psi :\psi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })\text{ is continuous nondecreasing with}\\ \psi (t)<t\text{ for all }t\in (0,+\mathrm{\infty })\}.\end{array}$$

Lemma 2.1 ([12])

Suppose that $\psi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$. If ψ is nondecreasing, then for each $t\in (0,+\mathrm{\infty })$, ${lim}_{n\to \mathrm{\infty }}{\psi }^n(t)=0$ implies $\psi (t)<t$.

Hence the difference between an element of ${\mathrm{\Psi }}_1$ and an element of ${\mathrm{\Psi }}_2$ is continuity.

Remark 2.2

• (i) It is easily seen that if $\psi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ is nondecreasing and $\psi (t)<t$ for all $t\in (0,+\mathrm{\infty })$, then $\psi (0)=0$.

• (ii) We can see that if $\psi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$, $\psi (t)<t$ for all $t\in (0,+\mathrm{\infty })$ and $\psi (0)=0$, then ψ is continuous at 0.

We now prove the PPF dependent fixed point theorems for mappings satisfying the generalized contractive conditions concerning with $\psi \in {\mathrm{\Psi }}_1$ without assuming the topological closedness with respect to the norm topology for the Razumikhin class ${\mathcal{R}}_c$.

Theorem 2.3 Suppose that $\psi \in {\mathrm{\Psi }}_1$, $c\in I$ and $T:E_0\to E$ satisfies the following conditions:

• (i) T is a nondecreasing mapping;

• (ii) for all $\varphi ,\alpha \in E_0$ with $\varphi \le \alpha$, we have

  $$\begin{array}{rl}{\parallel T\varphi -T\alpha \parallel }_E\le & \psi (max\{{\parallel \varphi -\alpha \parallel }_{E_0},{\parallel \varphi (c)-T\varphi \parallel }_E,{\parallel \alpha (c)-T\alpha \parallel }_E,\\ \frac{1}{2}[{\parallel \varphi (c)-T\alpha \parallel }_E+{\parallel \alpha (c)-T\varphi \parallel }_E]\left\}\right);\end{array}$$

• (iii) there exists a lower solution ${\varphi }_0\in {\mathcal{R}}_c$ such that ${\varphi }_0(c)\le T{\varphi }_0$;

• (iv) T is a continuous mapping.

Assume that ${\mathcal{R}}_c$ is algebraically closed with respect to the difference. Then T has a PPF dependent fixed point in ${\mathcal{R}}_c$.

Proof Since ${\varphi }_0\in {\mathcal{R}}_c$ and $T{\varphi }_0\in E$, there exists $x_1\in E$ such that $T{\varphi }_0=x_1$. Choose ${\varphi }_1\in {\mathcal{R}}_c$ such that $x_1={\varphi }_1(c)$. Since ${\varphi }_0$ is a lower solution in ${\mathcal{R}}_c$ such that ${\varphi }_0(c)\le T{\varphi }_0$, it follows that ${\varphi }_0\le {\varphi }_1$. Using the algebraic closedness with respect to the difference of ${\mathcal{R}}_c$, this yields

$${\parallel {\varphi }_0-{\varphi }_1\parallel }_{E_0}={\parallel {\varphi }_0(c)-{\varphi }_1(c)\parallel }_E.$$

By the fact that T is nondecreasing, we obtain

$${\varphi }_1(c)=T{\varphi }_0\le T{\varphi }_1.$$

By induction, we can construct the sequence $\{{\varphi }_n\}$ such that

$$\begin{array}{c}T{\varphi }_0={\varphi }_1(c)\le T{\varphi }_1={\varphi }_2(c)\le \cdots \le T{\varphi }_n={\varphi }_{n+1}(c)\le T{\varphi }_{n+1}\cdots ,\hfill \\ {\varphi }_n\le {\varphi }_{n+1}\text{and}{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}={\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E,\hfill \end{array}$$

for all $n\in \mathbb{N}\cup \{0\}$. Assume that ${\varphi }_{n-1}={\varphi }_n$ for some $n\in \mathbb{N}$. It follows that ${\varphi }_{n-1}(c)={\varphi }_n(c)=T{\varphi }_{n-1}$. Therefore T has a fixed point in ${\mathcal{R}}_c$. Suppose that ${\varphi }_{n-1}\ne {\varphi }_n$ for all $n\in \mathbb{N}$. Therefore, for each $n\in \mathbb{N}$, we obtain

$$\begin{array}{rcl}{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}& =& {\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E\\ =& {\parallel T{\varphi }_n-T{\varphi }_{n-1}\parallel }_E\\ \le & \psi (max\{{\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0},{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E,{\parallel {\varphi }_{n-1}(c)-T{\varphi }_{n-1}\parallel }_E,\\ \frac{1}{2}[{\parallel {\varphi }_n(c)-T{\varphi }_{n-1}\parallel }_E+{\parallel {\varphi }_{n-1}(c)-T{\varphi }_n\parallel }_E]\left\}\right)\\ =& \psi (max\{{\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0},{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E,{\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E,\\ \frac{1}{2}[{\parallel {\varphi }_n(c)-{\varphi }_n(c)\parallel }_E+{\parallel {\varphi }_{n-1}(c)-{\varphi }_{n+1}(c)\parallel }_E]\left\}\right)\\ \le & \psi (max\{{\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0},{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0},\frac{1}{2}{\parallel {\varphi }_{n-1}-{\varphi }_{n+1}\parallel }_{E_0}\})\\ \le & \psi (max\{{\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0},{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0},\\ \frac{1}{2}{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}+{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\left\}\right)\\ \le & \psi (max\{{\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0},{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\}).\end{array}$$

If $max\{{\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0},{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\}={\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}$, then

$${\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\le \psi \left({\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\right)<{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0},$$

which leads to a contradiction. Therefore, for each $n\in \mathbb{N}$, we have

$${\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\le \psi \left({\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0}\right).$$

By induction, we obtain

$${\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\le {\psi }^n\left({\parallel {\varphi }_0-{\varphi }_1\parallel }_{E_0}\right).$$

Fix $\epsilon >0$. This implies that there exists $N\in \mathbb{N}$ such that

$$\sum _{n\ge N}{\psi }^n\left({\parallel {\varphi }_0-{\varphi }_1\parallel }_{E_0}\right)<\epsilon .$$

For each $m,n\in \mathbb{N}$ with $m>n>N$, we obtain

$${\parallel {\varphi }_n-{\varphi }_m\parallel }_{E_0}\le \sum _{k=n}^{m-1}{\parallel {\varphi }_k-{\varphi }_{k+1}\parallel }_{E_0}\le \sum _{k=n}^{m-1}{\psi }^k\left({\parallel {\varphi }_0-{\varphi }_1\parallel }_{E_0}\right)\le \sum _{n\ge N}{\psi }^n\left({\parallel {\varphi }_0-{\varphi }_1\parallel }_{E_0}\right)<\epsilon .$$

This implies that $\{{\varphi }_n\}$ is a Cauchy sequence. By the completeness of $E_0$, we have ${lim}_{n\to \mathrm{\infty }}{\varphi }_n=\varphi$ for some $\varphi \in E_0$ and

$$\underset{n\to \mathrm{\infty }}{lim}T{\varphi }_n=\underset{n\to \mathrm{\infty }}{lim}{\varphi }_{n+1}(c)=\varphi (c).$$

Since ${\mathcal{R}}_c$ is algebraically closed with respect to the norm topology, we have $\varphi \in {\mathcal{R}}_c$. We next prove that ϕ is a PPF dependent fixed point of T. Using the continuity of T, we obtain ${lim}_{n\to \mathrm{\infty }}T{\varphi }_n=T\varphi$. By the uniqueness of the limit, we have $T\varphi =\varphi (c)$. □

Remark 2.4

• (i) From the proof of Theorem 2.3, we assume that the Razumikhin class ${\mathcal{R}}_c$ is algebraically closed with respect to difference, that is, $\varphi -\alpha \in {\mathcal{R}}_c$ for all $\varphi ,\alpha \in {\mathcal{R}}_c$, in order to construct the sequence $\{{\varphi }_n\}$ satisfying

  $${\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}={\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E\text{for all }n\in \mathbb{N}\cup \{0\}.$$

• (ii) In the proof of Theorem 2.3, if we choose ${\varphi }_n\in {\mathcal{R}}_c$ to be a constant mapping for each $n\in \mathbb{N}\cup \{0\}$, then

  $${\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}={\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E\text{for all }n\in \mathbb{N}\cup \{0\}.$$

  Therefore the algebraic closedness with respect to the difference of ${\mathcal{R}}_c$ can be dropped.

Example 2.5 Let $E={\mathbb{R}}^2$ with respect to the norm $\parallel (x,y)\parallel =|x|+|y|$ and $E_0=C([0,1],{\mathbb{R}}^2)$. Define a mapping $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ by

$$\psi (t)=\frac{t}{3}\text{for all }t\in [0,\mathrm{\infty }).$$

We see that ψ is a nondecreasing mapping with ${\sum }_{n=1}^{\mathrm{\infty }}{\psi }^n(t)<\mathrm{\infty }$ for all $t\in (0,+\mathrm{\infty })$. Let $\varphi \in E_0$. Therefore, for each $t\in [0,1]$, we obtain $\varphi (t)\in {\mathbb{R}}^2$. Thus we can define mappings $f_{\varphi },g_{\varphi }:[0,1]\to \mathbb{R}$ such that

$$\varphi (t)=(f_{\varphi }(t),g_{\varphi }(t))\text{for all }t\in [0,1].$$

Define a mapping $T:E_0\to E$ by

$$T\varphi =(\frac{1}{4}\parallel f_{\varphi }\parallel ,\frac{1}{4}\parallel g_{\varphi }\parallel )\text{for all }\varphi \in E_0,$$

where $\parallel f_{\varphi }\parallel ={sup}_{t\in [0,1]}|f_{\varphi }(t)|$ and $\parallel g_{\varphi }\parallel ={sup}_{t\in [0,1]}|g_{\varphi }(t)|$. Suppose that $\varphi ,\alpha \in E_0$ with $\varphi \le \alpha$. For each $c\in [0,1]$, we obtain

$$\begin{array}{rcl}{\parallel T\varphi -T\alpha \parallel }_E& =& {\parallel (\frac{1}{4}\parallel f_{\varphi }\parallel ,\frac{1}{4}\parallel g_{\varphi }\parallel )-(\frac{1}{4}\parallel f_{\alpha }\parallel ,\frac{1}{4}\parallel g_{\alpha }\parallel )\parallel }_E\\ =& |\frac{1}{4}\parallel f_{\varphi }\parallel -\frac{1}{4}\parallel f_{\alpha }\parallel |+|\frac{1}{4}\parallel g_{\varphi }\parallel -\frac{1}{4}\parallel g_{\alpha }\parallel |\\ \le & \frac{1}{4}(\parallel f_{\varphi }-f_{\alpha }\parallel +\parallel g_{\varphi }-g_{\alpha }\parallel )\\ =& \frac{1}{4}(\underset{t\in [0,1]}{sup}|f_{\varphi }(t)-f_{\alpha }(t)|+\underset{t\in [0,1]}{sup}|g_{\varphi }(t)-g_{\alpha }(t)|)\\ =& \frac{1}{4}\left(\underset{t\in [0,1]}{sup}(|f_{\varphi }(t)-f_{\alpha }(t)|+|g_{\varphi }(t)-g_{\alpha }(t)|)\right)\\ =& \frac{1}{4}\left(\underset{t\in [0,1]}{sup}{\parallel (f_{\varphi }(t)-f_{\alpha }(t),g_{\varphi }(t)-g_{\alpha }(t))\parallel }_E\right)\\ =& \frac{1}{4}\left(\underset{t\in [0,1]}{sup}{\parallel (f_{\varphi }(t),g_{\varphi }(t))-(f_{\alpha }(t),g_{\alpha }(t))\parallel }_E\right)\\ =& \frac{1}{4}\left(\underset{t\in [0,1]}{sup}{\parallel \varphi (t)-\alpha (t)\parallel }_E\right)\\ =& \frac{1}{4}\left({\parallel \varphi -\alpha \parallel }_{E_0}\right)\\ \le & \psi (max\{{\parallel \varphi -\alpha \parallel }_{E_0},{\parallel \varphi (c)-T\varphi \parallel }_E,{\parallel \alpha (c)-T\alpha \parallel }_E,\\ \frac{1}{2}[{\parallel \varphi (c)-T\alpha \parallel }_E+{\parallel \alpha (c)-T\varphi \parallel }_E]\left\}\right).\end{array}$$

Suppose that ${\varphi }_0=0$. We see that all assumptions in Theorem 2.3 are now satisfied and 0 is the PPF dependent fixed point of T in ${\mathcal{R}}_c$.

By applying Theorem 2.3, we obtain the following corollary.

Corollary 2.6 Suppose that $c\in I$ and $T:E_0\to E$ satisfies the following conditions:

• (i) T is a nondecreasing mapping;

• (ii) for all $\varphi ,\alpha \in E_0$ with $\varphi \le \alpha$, we have

  $$\begin{array}{rcl}{\parallel T\varphi -T\alpha \parallel }_E& \le & k(max\{{\parallel \varphi -\alpha \parallel }_{E_0},{\parallel \varphi (c)-T\varphi \parallel }_E,{\parallel \alpha (c)-T\alpha \parallel }_E,\\ \frac{1}{2}[{\parallel \varphi (c)-T\alpha \parallel }_E+{\parallel \alpha (c)-T\varphi \parallel }_E]\left\}\right),\mathit{\text{where }}k\in [0,1);\end{array}$$

• (iii) there exists a lower solution ${\varphi }_0\in {\mathcal{R}}_c$ such that ${\varphi }_0(c)\le T{\varphi }_0$;

• (iv) T is a continuous mapping.

Assume that ${\mathcal{R}}_c$ is algebraically closed with respect to the difference. Then T has a PPF dependent fixed point in ${\mathcal{R}}_c$.

Proof Define a function $\psi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ by $\psi (t)=kt$ for all $t\in [0,+\mathrm{\infty })$. Therefore ψ is a nondecreasing mapping and

$$\sum _{n=1}^{\mathrm{\infty }}{\psi }^n(t)<\mathrm{\infty }\text{for all }t\in (0,+\mathrm{\infty }).$$

This implies that all assumptions in Theorem 2.3 are satisfied. Hence we obtain the desired result. □

Theorem 2.7 Suppose that $\psi \in {\mathrm{\Psi }}_1$, $c\in I$ and $T:E_0\to E$ satisfies the following conditions:

• (i) T is a nondecreasing mapping;

• (ii) for all $\varphi ,\alpha \in E_0$ with $\varphi \le \alpha$, ${\parallel T\varphi -T\alpha \parallel }_E\le \psi ({\parallel \varphi -\alpha \parallel }_{E_0})$;

• (iii) there exists a lower solution ${\varphi }_0\in {\mathcal{R}}_c$ such that ${\varphi }_0(c)\le T{\varphi }_0$;

• (iv) if $\{{\varphi }_n\}$ is a nondecreasing sequence in $E_0$ converging to $\varphi \in E_0$, then ${\varphi }_n\le \varphi$ for all $n\in \mathbb{N}$.

Assume that ${\mathcal{R}}_c$ is algebraically closed with respect to the difference. Then T has a PPF dependent fixed point in ${\mathcal{R}}_c$.

Proof By the analogous proof as in Theorem 2.3, we can construct a nondecreasing sequence $\{{\varphi }_n\}$ in ${\mathcal{R}}_c$ converging to $\varphi \in {\mathcal{R}}_c$. This implies that ${\varphi }_n\le \varphi$ for all $n\in \mathbb{N}$. Therefore, for each $n\in \mathbb{N}$, we have

$$\begin{array}{rcl}{\parallel T\varphi -\varphi (c)\parallel }_E& \le & {\parallel T\varphi -{\varphi }_{n+1}(c)\parallel }_E+{\parallel {\varphi }_{n+1}(c)-\varphi (c)\parallel }_E\\ \le & {\parallel T\varphi -T{\varphi }_n\parallel }_E+{\parallel {\varphi }_{n+1}-\varphi \parallel }_{E_0}\\ \le & \psi \left({\parallel \varphi -{\varphi }_n\parallel }_{E_0}\right)+{\parallel {\varphi }_{n+1}-\varphi \parallel }_{E_0}.\end{array}$$

Since ψ is continuous at 0, we get ${lim}_{n\to \mathrm{\infty }}\psi ({\parallel \varphi -{\varphi }_n\parallel }_{E_0})=\psi (0)=0$. Taking the limit of the above inequality, this yields ${\parallel T\varphi -\varphi (c)\parallel }_E=0$ and so ϕ is a PPF dependent fixed point of T in ${\mathcal{R}}_c$. □

We next ensure the result on PPF dependent fixed points for mappings concerning with $\psi \in {\mathrm{\Psi }}_2$.

Theorem 2.8 Suppose that $\psi \in {\mathrm{\Psi }}_2$, $c\in I$ and $T:E_0\to E$ satisfies the following conditions:

• (i) T is a nondecreasing mapping;

• (ii) for all $\varphi ,\alpha \in E_0$ with $\varphi \le \alpha$, we have

  $$\begin{array}{rcl}{\parallel T\varphi -T\alpha \parallel }_E& \le & \psi (max\{{\parallel \varphi -\alpha \parallel }_{E_0},{\parallel \varphi (c)-T\varphi \parallel }_E,{\parallel \alpha (c)-T\alpha \parallel }_E,\\ \frac{1}{2}[{\parallel \varphi (c)-T\alpha \parallel }_E+{\parallel \alpha (c)-T\varphi \parallel }_E]\left\}\right);\end{array}$$

• (iii) there exists a lower solution ${\varphi }_0\in {\mathcal{R}}_c$ such that ${\varphi }_0(c)\le T{\varphi }_0$;

• (iv) T is a continuous mapping.

Assume that ${\mathcal{R}}_c$ is algebraically closed with respect to the difference. Then T has a PPF dependent fixed point in ${\mathcal{R}}_c$.

Proof Since ${\varphi }_0\in {\mathcal{R}}_c$ and $T{\varphi }_0\in E$, there exists $x_1\in E$ such that $T{\varphi }_0=x_1$. As in the proof of Theorem 2.3, we can construct the sequence $\{{\varphi }_n\}$ such that

$$\begin{array}{c}T{\varphi }_0={\varphi }_1(c)\le T{\varphi }_1={\varphi }_2(c)\le \cdots \le T{\varphi }_n={\varphi }_{n+1}(c)\le T{\varphi }_{n+1}\cdots ,\hfill \\ {\varphi }_n\le {\varphi }_{n+1}\text{and}{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}={\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E,\hfill \end{array}$$

for all $n\in \mathbb{N}\cup \{0\}$. Assume that ${\varphi }_{n-1}={\varphi }_n$ for some $n\in \mathbb{N}$. It follows that ${\varphi }_{n-1}(c)={\varphi }_n(c)=T{\varphi }_{n-1}$. Therefore T has a fixed point in ${\mathcal{R}}_c$. Suppose that ${\varphi }_{n-1}\ne {\varphi }_n$ for all $n\in \mathbb{N}$. For each $n\in \mathbb{N}$, we obtain

$$\begin{array}{rcl}{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}& =& {\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E\\ =& {\parallel T{\varphi }_n-T{\varphi }_{n-1}\parallel }_E\\ \le & \psi (max\{{\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0},{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E,{\parallel {\varphi }_{n-1}(c)-T{\varphi }_{n-1}\parallel }_E,\\ \frac{1}{2}[{\parallel {\varphi }_n(c)-T{\varphi }_{n-1}\parallel }_E+{\parallel {\varphi }_{n-1}(c)-T{\varphi }_n\parallel }_E]\left\}\right)\\ =& \psi (max\{{\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0},{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E,{\parallel {\varphi }_{n-1}(c)-{\varphi }_n(c)\parallel }_E,\\ \frac{1}{2}[{\parallel {\varphi }_n(c)-{\varphi }_n(c)\parallel }_E+{\parallel {\varphi }_{n-1}(c)-{\varphi }_{n+1}(c)\parallel }_E]\left\}\right)\\ \le & \psi (max\{{\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0},{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0},\frac{1}{2}{\parallel {\varphi }_{n-1}-{\varphi }_{n+1}\parallel }_{E_0}\})\\ \le & \psi (max\{{\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0},{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0},\\ \frac{1}{2}{\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}+{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\left\}\right)\\ \le & \psi (max\{{\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0},{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\}).\end{array}$$

If $max\{{\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0},{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\}={\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}$, then

$${\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\le \psi \left({\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\right)<{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}.$$

This leads to a contradiction. Therefore

$$\begin{array}{rcl}{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}& \le & \psi \left({\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0}\right)\\ <& {\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0}.\end{array}$$

It follows that ${\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\le {\parallel {\varphi }_{n-1}-{\varphi }_n\parallel }_{E_0}$ for all $n\in \mathbb{N}$. Since the sequence $\{{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\}$ is a nonincreasing sequence of nonnegative real numbers, we see that it is a convergent sequence. Suppose that

$$\underset{n\to \mathrm{\infty }}{lim}{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}=\alpha ,$$

for some nonnegative real number α. We will prove that $\alpha =0$. Suppose that $\alpha >0$. Since

$${\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\le \psi \left({\parallel {\varphi }_n-{\varphi }_{n-1}\parallel }_{E_0}\right),$$

for all $n\in \mathbb{N}$ and the continuity of ψ, we have $\alpha \le \psi (\alpha )<\alpha$ which leads to a contradiction. This implies that $\alpha =0$. We next prove that the sequence $\{{\varphi }_n\}$ is a Cauchy sequence in $E_0$. Assume that $\{{\varphi }_n\}$ is not a Cauchy sequence. It follows that there exist $\epsilon >0$ and two sequences of positive integers $\{m_k\}$ and $\{n_k\}$ satisfying $m_k>n_k>k$ for each $k\in \mathbb{N}$ and

$${\parallel {\varphi }_{m_k}-{\varphi }_{n_k}\parallel }_{E_0}\ge \epsilon .$$

(2.1)

Let $\{m_k\}$ be the sequence of the least positive integers exceeding $\{n_k\}$ which satisfies (2.1) and

$${\parallel {\varphi }_{m_k-1}-{\varphi }_{n_k}\parallel }_{E_0}<\epsilon .$$

(2.2)

We will prove that ${lim}_{k\to \mathrm{\infty }}{\parallel {\varphi }_{m_k}-{\varphi }_{n_k}\parallel }_{E_0}=\epsilon$. Since ${\parallel {\varphi }_{m_k}-{\varphi }_{n_k}\parallel }_{E_0}\ge \epsilon$ for all $k\in \mathbb{N}$, we have

$$\underset{k\to \mathrm{\infty }}{lim}{\parallel {\varphi }_{m_k}-{\varphi }_{n_k}\parallel }_{E_0}\ge \epsilon .$$

For each $k\in \mathbb{N}$, we obtain

$$\begin{array}{rcl}{\parallel {\varphi }_{m_k}-{\varphi }_{n_k}\parallel }_{E_0}& \le & {\parallel {\varphi }_{m_k}-{\varphi }_{m_k-1}\parallel }_{E_0}+{\parallel {\varphi }_{m_k-1}-{\varphi }_{n_k}\parallel }_{E_0}\\ \le & {\parallel {\varphi }_{m_k}-{\varphi }_{m_k-1}\parallel }_{E_0}+\epsilon .\end{array}$$

This implies that ${lim}_{k\to \mathrm{\infty }}{\parallel {\varphi }_{m_k}-{\varphi }_{n_k}\parallel }_{E_0}\le \epsilon$. Therefore

$$\underset{k\to \mathrm{\infty }}{lim}{\parallel {\varphi }_{m_k}-{\varphi }_{n_k}\parallel }_{E_0}=\epsilon .$$

Similarly, we can prove that

$$\underset{k\to \mathrm{\infty }}{lim}{\parallel {\varphi }_{m_k+1}-{\varphi }_{n_k}\parallel }_{E_0}=\epsilon ,\underset{k\to \mathrm{\infty }}{lim}{\parallel {\varphi }_{m_k}-{\varphi }_{n_k-1}\parallel }_{E_0}=\epsilon ,$$

and

$$\underset{k\to \mathrm{\infty }}{lim}{\parallel {\varphi }_{m_k+1}-{\varphi }_{n_k-1}\parallel }_{E_0}=\epsilon .$$

Since ${\mathcal{R}}_c$ is algebraically closed with respect to the difference, for each $k\in \mathbb{N}$, we obtain

$$\begin{array}{rcl}{\parallel {\varphi }_{n_k}-{\varphi }_{m_k+1}\parallel }_{E_0}& =& {\parallel {\varphi }_{n_k}(c)-{\varphi }_{m_k+1}(c)\parallel }_E\\ =& {\parallel T{\varphi }_{m_k}-T{\varphi }_{n_k-1}\parallel }_E\\ \le & \psi (max\{{\parallel {\varphi }_{m_k}-{\varphi }_{n_k-1}\parallel }_{E_0},{\parallel {\varphi }_{m_k}(c)-T{\varphi }_{m_k}\parallel }_E,\\ {\parallel {\varphi }_{n_k-1}(c)-T{\varphi }_{n_k-1}\parallel }_E,\\ \frac{1}{2}[{\parallel {\varphi }_{m_k}(c)-T{\varphi }_{n_k-1}\parallel }_E+{\parallel {\varphi }_{n_k-1}(c)-T{\varphi }_{n_k}\parallel }_E]\left\}\right)\\ =& \psi (max\{{\parallel {\varphi }_{m_k}-{\varphi }_{n_k-1}\parallel }_{E_0},{\parallel {\varphi }_{m_k}(c)-{\varphi }_{m_k+1}(c)\parallel }_E,\\ {\parallel {\varphi }_{n_k-1}(c)-{\varphi }_{n_k}(c)\parallel }_E,\\ \frac{1}{2}[{\parallel {\varphi }_{m_k}(c)-{\varphi }_{n_k}(c)\parallel }_E+{\parallel {\varphi }_{n_k-1}(c)-{\varphi }_{n_k+1}(c)\parallel }_E]\left\}\right)\\ \le & \psi (max\{{\parallel {\varphi }_{m_k}-{\varphi }_{n_k-1}\parallel }_{E_0},{\parallel {\varphi }_{m_k}-{\varphi }_{m_k+1}\parallel }_{E_0},{\parallel {\varphi }_{n_k-1}-{\varphi }_{n_k}\parallel }_{E_0},\\ \frac{1}{2}[{\parallel {\varphi }_{m_k}-{\varphi }_{n_k}\parallel }_{E_0}+{\parallel {\varphi }_{n_k-1}-{\varphi }_{n_k+1}\parallel }_{E_0}]\left\}\right)\\ \le & \psi (max\{{\parallel {\varphi }_{m_k}-{\varphi }_{n_k-1}\parallel }_{E_0},{\parallel {\varphi }_{m_k}-{\varphi }_{m_k+1}\parallel }_{E_0},{\parallel {\varphi }_{n_k-1}-{\varphi }_{n_k}\parallel }_{E_0},\\ \frac{1}{2}[{\parallel {\varphi }_{m_k}-{\varphi }_{n_k}\parallel }_{E_0}+{\parallel {\varphi }_{n_k-1}-{\varphi }_{n_k}\parallel }_{E_0}+{\parallel {\varphi }_{n_k}-{\varphi }_{n_k+1}\parallel }_{E_0}]\left\}\right).\end{array}$$

By taking the limit of both sides, we have

$$\epsilon \le \psi (\epsilon )<\epsilon .$$

This leads to a contradiction. It follows that the sequence $\{{\varphi }_n\}$ is a Cauchy sequence. By the completeness of $E_0$, we have ${lim}_{n\to \mathrm{\infty }}{\varphi }_n=\varphi$ for some $\varphi \in E_0$ and

$$\underset{n\to \mathrm{\infty }}{lim}T{\varphi }_n=\underset{n\to \mathrm{\infty }}{lim}{\varphi }_{n+1}(c)=\varphi (c).$$

Since ${\mathcal{R}}_c$ is algebraically closed with respect to the norm topology, we have $\varphi \in {\mathcal{R}}_c$. We will prove that ϕ is a PPF dependent fixed point of T. Using the continuity of T, we obtain ${lim}_{n\to \mathrm{\infty }}T{\varphi }_n=T\varphi$. By the uniqueness of the limit, we can conclude that $T\varphi =\varphi (c)$. □

Example 2.9 Assume that $E=\mathbb{R}$ and $E_0=C([0,1],\mathbb{R})$. Define a mapping $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ by

$$\psi (t)=\frac{t}{2}\text{for all }t\in [0,\mathrm{\infty }).$$

We see that ψ is a continuous nondecreasing mapping with $\psi (t)<t$. Define a mapping $T:E_0\to E$ by

$$T\varphi =\frac{1}{3}\varphi \left(\frac{1}{4}\right)\text{for all }\varphi \in E_0.$$

Suppose that $\varphi ,\alpha \in E_0$ with $\psi \le \alpha$ and $c\in [0,1]$. Therefore

$$\begin{array}{rcl}{\parallel T\varphi -T\alpha \parallel }_E& =& \frac{1}{3}|\varphi \left(\frac{1}{4}\right)-\alpha \left(\frac{1}{4}\right)|\\ \le & \frac{1}{3}{\parallel \varphi -\alpha \parallel }_{E_0}\\ \le & \psi (max\{{\parallel \varphi -\alpha \parallel }_{E_0},{\parallel \varphi (c)-T\varphi \parallel }_E,{\parallel \alpha (c)-T\alpha \parallel }_E,\\ \frac{1}{2}[{\parallel \varphi (c)-T\alpha \parallel }_E+{\parallel \alpha (c)-T\varphi \parallel }_E]\left\}\right).\end{array}$$

Suppose that ${\varphi }_0=0$. We find that all assumptions in Theorem 2.8 are now satisfied and 0 is the PPF dependent fixed point of T in ${\mathcal{R}}_c$.

For the next result, we drop the continuity of T.

Theorem 2.10 Suppose that $\psi \in {\mathrm{\Psi }}_2$, $c\in I$, and that $T:E_0\to E$ satisfies the following conditions:

• (i) T is a nondecreasing mapping;

• (ii) for all $\varphi ,\alpha \in E_0$ with $\varphi \le \alpha$, we have

  $$\begin{array}{rcl}{\parallel T\varphi -T\alpha \parallel }_E& \le & \psi (max\{{\parallel \varphi -\alpha \parallel }_{E_0},{\parallel \varphi (c)-T\varphi \parallel }_E,{\parallel \alpha (c)-T\alpha \parallel }_E,\\ \frac{1}{2}[{\parallel \varphi (c)-T\alpha \parallel }_E+{\parallel \alpha (c)-T\varphi \parallel }_E]\left\}\right);\end{array}$$

• (iii) there exists a lower solution ${\varphi }_0\in {\mathcal{R}}_c$ such that ${\varphi }_0(c)\le T{\varphi }_0$;

• (iv) if $\{{\varphi }_n\}$ is a nondecreasing sequence in $E_0$ converging to $\varphi \in E_0$, then ${\varphi }_n\le \varphi$ for all $n\in \mathbb{N}$.

Assume that ${\mathcal{R}}_c$ is algebraically closed with respect to the difference. Then T has a PPF dependent fixed point in ${\mathcal{R}}_c$.

Proof As in the proof of Theorem 2.8, we can construct a nondecreasing sequence $\{{\varphi }_n\}$ converging to $\varphi \in {\mathcal{R}}_c$ and this yields

$$\underset{n\to \mathrm{\infty }}{lim}T{\varphi }_n=\underset{n\to \mathrm{\infty }}{lim}{\varphi }_{n+1}(c)=\varphi (c).$$

Using (iv), we have ${\varphi }_n\le \varphi$ for all $n\in \mathbb{N}$. Therefore, for each $n\in \mathbb{N}$, we obtain

$$\begin{array}{rcl}{\parallel T\varphi -\varphi (c)\parallel }_E& \le & {\parallel T\varphi -{\varphi }_{n+1}(c)\parallel }_E+{\parallel {\varphi }_{n+1}(c)-\varphi (c)\parallel }_E\\ \le & {\parallel T\varphi -T{\varphi }_n\parallel }_E+{\parallel {\varphi }_{n+1}-\varphi \parallel }_{E_0}\\ \le & \psi (max\{{\parallel \varphi -{\varphi }_n\parallel }_{E_0},{\parallel \varphi (c)-T\varphi \parallel }_E,{\parallel {\varphi }_n(c)-T{\varphi }_n\parallel }_E,\\ \frac{1}{2}[{\parallel \varphi (c)-T{\varphi }_n\parallel }_E+{\parallel {\varphi }_n(c)-T\varphi \parallel }_E]\left\}\right)+{\parallel {\varphi }_{n+1}-\varphi \parallel }_{E_0}\\ =& \psi (max\{{\parallel \varphi -{\varphi }_n\parallel }_{E_0},{\parallel \varphi (c)-T\varphi \parallel }_E,{\parallel {\varphi }_n(c)-{\varphi }_{n+1}(c)\parallel }_E,\\ \frac{1}{2}[{\parallel \varphi (c)-{\varphi }_{n+1}(c)\parallel }_E+{\parallel {\varphi }_n(c)-T\varphi \parallel }_E]\left\}\right)+{\parallel {\varphi }_{n+1}-\varphi \parallel }_{E_0}\\ \le & \psi (max\{{\parallel \varphi -{\varphi }_n\parallel }_{E_0},{\parallel \varphi (c)-T\varphi \parallel }_E,{\parallel {\varphi }_n-{\varphi }_{n+1}\parallel }_{E_0}\\ \frac{1}{2}[{\parallel \varphi -{\varphi }_{n+1}\parallel }_{E_0}+{\parallel {\varphi }_n(c)-T\varphi \parallel }_E]\left\}\right)+{\parallel {\varphi }_{n+1}-\varphi \parallel }_{E_0}.\end{array}$$

Letting $n\to \mathrm{\infty }$, we obtain ${\parallel T\varphi -\varphi (c)\parallel }_E\le \psi ({\parallel \varphi (c)-T\varphi \parallel }_E)$. If $\varphi (c)\ne T\varphi$, then

$${\parallel T\varphi -\varphi (c)\parallel }_E\le \psi \left({\parallel \varphi (c)-T\varphi \parallel }_E\right)<{\parallel \varphi (c)-T\varphi \parallel }_E.$$

This leads to a contradiction. Therefore $T\varphi =\varphi (c)$. This implies that ϕ is a PPF dependent fixed point of T. □

By applying Theorem 2.8 and Theorem 2.10, we obtain the following corollary.

Corollary 2.11 Suppose that $c\in I$ and that $T:E_0\to E$ satisfies the following conditions:

• (i) T is a nondecreasing mapping;

• (ii) for all $\varphi ,\alpha \in E_0$ with $\varphi \le \alpha$, we have

  $$\begin{array}{rcl}{\parallel T\varphi -T\alpha \parallel }_E& \le & k(max\{{\parallel \varphi -\alpha \parallel }_{E_0},{\parallel \varphi (c)-T\varphi \parallel }_E,{\parallel \alpha (c)-T\alpha \parallel }_E,\\ \frac{1}{2}[{\parallel \varphi (c)-T\alpha \parallel }_E+{\parallel \alpha (c)-T\varphi \parallel }_E]\left\}\right),\mathit{\text{where }}k\in [0,1);\end{array}$$

• (iii) there exists a lower solution ${\varphi }_0\in {\mathcal{R}}_c$ such that ${\varphi }_0(c)\le T{\varphi }_0$;

• (iv) T is a continuous mapping or if $\{{\varphi }_n\}$ is a nondecreasing sequence in $E_0$ converging to $\varphi \in E_0$, then ${\varphi }_n\le \varphi$ for all $n\in \mathbb{N}$.

Assume that ${\mathcal{R}}_c$ is algebraically closed with respect to the difference. Then T has a PPF dependent fixed point in ${\mathcal{R}}_c$.

Proof Define a function $\psi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ by $\psi (t)=kt$ for all $t\in [0,+\mathrm{\infty })$. Therefore ψ is a continuous nondecreasing mapping and

$$\psi (t)<t\text{for all }t\in (0,+\mathrm{\infty }).$$

This implies that all assumptions in Theorem 2.8 or Theorem 2.10 are satisfied. Hence the proof is complete. □