Some new φ-fixed point and φ-fixed disc results via auxiliary functions

Abstract

In this paper, we first introduce several new contractions by some auxiliary functions. Then we establish some φ-fixed point results for $(\gamma ,\psi ,\varphi ,\phi )$ contractions, rational-$(\gamma , \psi ,\varphi ,\phi )$ contractions and almost-$(\gamma ,\psi , \varphi ,\phi )$ contractions and φ-fixed disc results for $(\psi ,\varphi ,\phi )_{x_{0}}$ type 1 contractions, $(\psi ,\varphi ,\phi )_{x_{0}}$ type 2 contractions, and $(\psi ,\varphi ,\phi )_{x_{0}}$ type 3 contractions in metric spaces. We generalize some existing φ-fixed point and φ-fixed circle results in the literature. Further, we give some examples to demonstrate the usefulness of our results and investigate the existence and uniqueness of solutions of nonlinear integral equations.

1 Introduction

In 2012, Wardowski [1] introduced the family Ϝ and established a fixed point theorem of F-contractions in complete metric spaces, which is an interesting generalization of the Banach contraction principle. Inspired by Wardowski [1], many scholars have obtained many fixed point results by generalizing F-contraction and by changing the conditions on the family Ϝ. In 2014, Wardowski et al. [2] and Cosentino et al. [3] proposed F-weak contractions and F-contractive mappings of Hardy–Rogers-type, respectively, and established some related fixed point results. In 2015, Dung and Hang [4] stated a new generalization of the Wardowski fixed point theorem in [2] by adding four new values $d(T^{2}x,x)$, $d(T^{2}x,Tx)$, $d(T^{2}x,y)$, and $d(T^{2}x,Ty)$ to the right-hand of contraction inequalities. Piri and Kumam [5] replaced condition $(F3)$ in Theorem 2.1 with the continuity of F and obtained relevant fixed point theorems. Moreover, other researchers also obtained some fixed point results for F-contractions in generalized metric spaces. In 2016, Piri and Kumam [6] introduced two classes of modified generalized F-contractions and extended the results of [2, 4, 5]. In 2018, Haitham et al.[7] established the existence of the fixed point for generalized $(\alpha ,\beta ,F)$-Geraghty contractions in metric-like spaces by combining Geraghty contractions with α-admissible mappings.

In 2014, Jleli and Samet [8] introduced θ-contractions, which are different from F-contractions proposed by Wardowski [1]. They denoted the family of all functions θ by Θ and established some fixed point theorems for such contractions in the setting of metric spaces. In the follow-up studies, Jleli and Samet [8] and Liu and Chang et al. [9] proposed the notions of θ-type contractions and θ-type Suzuki contractions, obtained some new fixed point theorems in complete metric spaces, and solved the existence problem of solutions of nonlinear Hammerstein integral equations. Parvaneh et al. [10], as a development of [8], introduced the concept of α-HΘ-contraction with respect to a more general family of functions H in ordered metric spaces, . In 2018, Imdad et al. [11] proved that Theorem 2.2 still holds under $(\theta 2)$ and $(\theta 3)$ only, so that more functions can be used in Theorem 2.2. In 2021, Perveen et al. [12] introduced $\theta ^{*}$-weak contraction mappings satisfying only $(\theta 1)$, obtained some related fixed point results, and investigated the existence and uniqueness of solutions of nonlinear matrix equations and integral equations of Volterra type.

In 2016, combining F-contractions of Piri and Kumam [5] with θ-contractions of Jleli and Samet [8], Liu et al. [13] introduced $(\psi ,\phi )$-type contraction and obtained an important fixed point result. Recently, Proinov [14] extended and unified many existing results. These results generalized those of Wardowski [1], Piri and Kumam [5], and Jleli and Samet [8].

In addition, the new concepts of φ-fixed point and φ-Picard mappings were introduced by Jleli et al. [15], who obtained several φ-fixed point results based on the idea of new control functions. Moreover, they also claimed that some fixed point results in partial metric spaces can be derived from these φ-fixed point results in metric spaces. A lot of well-known results have been published since Jleli et al. [15]. In 2017, Kumrod and Sintunavarat [16] developed the results by Jleli et al. [15] by defining $(F,\varphi ,\theta )$-contractions. Asadi [17] weakened the continuity of the control function γ and proved the fixed point theorem for $(F,\varphi ,\theta )$-contractions. Kumrod and Sintunavarat [18] extended Theorem 2.1 of Jleli et al. [15] to D-generalized metric spaces. Imdad and Khan et al. [19] introduced $(F,\varphi ,\alpha \text{-}\psi )$-contractions and $(F,\varphi ,\alpha \text{-}\psi )$-weak contractions by the control function in [17], which extended the results by Kumrod and Sintunavarat [16], and solved the existence of solutions for boundary value problems of second-order ordinary differential equations. Saleh and Imdad et al. [20] obtained the φ-coincidence point and common φ-fixed point results for two self-mappings via the extended $C_{g}$-simulation functions in metric spaces. Samet [21] proved φ-fixed point results via auxiliary functions. Vetro [22] and Özlem [23] combined control functions with F-contractions to establish some φ-fixed point results. Mohammadi et al. [24] obtained some φ-fixed point results for λ-$(\gamma ,\varphi )$-contractions and studied the existence of solutions of nonlinear integral equations.

Recently, geometric properties of nonunique fixed points have been extensively studied in various aspects, for example, the fixed-circle problem, fixed-disc problem, and so on. Özgür and Taş [25] proposed the notion of a fixed circle and presented the fixed circle problem in a metric space as a new direction of generalization of fixed point theory. We refer some recent studies of fixed circle problem and fixed-disc problem to [26–32]. The concepts of φ-fixed circle (resp., disc) were firstly presented by Özgür and Taş [33]. In 2021, Özgür and Taş [34] established some new results for φ-fixed circles (resp., disc).

Motivated by [14, 15], and [34], in this paper, we establish some φ-fixed point results for $(\gamma ,\psi ,\phi ,\varphi )$ contractions, rational-$(\gamma , \psi ,\phi ,\varphi )$ contractions, and almost-$(\gamma ,\psi ,\phi , \varphi )$ contractions and some fixed disc results for $(\psi ,\varphi ,\phi )_{x_{0}}$ type 1 contractions, $(\psi ,\varphi ,\phi )_{x_{0}}$ type 2 contractions, and $(\psi ,\varphi ,\phi )_{x_{0}}$ type 3 contractions. Our results improve and generalize some existing fixed point results. Further, we give some examples and an application, which show the effectiveness of our results.

2 Preliminaries

In this section, we review some basic concepts and some known results. We denote by $\mathbb{R}$ the set of all real numbers, by ${\mathbb{R}}_{+}$ the set of all nonnegative real numbers, by $\mathbb{N}$ the set of all nonnegative integers, by $F(T)$ the set of all fixed points of T, and by $Z_{\varphi}$ the set of all zero points of φ.

Theorem 2.1

([1])

Let $(X,d)$ be a complete metric space, and let $T:X\rightarrow X$ be an F-contraction, that is, there exist $F\in \digamma$ and $\tau >0$ such that

$$ d(Tx,Ty)>0\quad \Rightarrow\quad \tau +F \bigl(d(Tx,Ty) \bigr)\leqslant F \bigl(d(x,y) \bigr)\quad \textit{for all } x,y \in X, $$

where Ϝ denotes the family of all functions $F:(0,\mathrm{\infty })\to \mathbb{R}$ satisfying the following conditions:

$(F1)$ F is strictly increasing, i.e., $x< y\Rightarrow F(x)< F(y)$ for all $x,y\in (0,+\infty )$;

$(F2)$ for each sequence $\{\alpha _{m}\}$ of positive numbers, $\lim_{m\rightarrow \infty} \alpha _{m}=0$ if and only if $\lim_{m\rightarrow \infty} F(\alpha _{m})=-\infty$;

$(F3)$ there exists $l\in (0,1)$ such that $\lim_{\alpha \rightarrow 0^{+}} \alpha ^{l}F(\alpha )=0$.

Then T has a unique fixed point.

Theorem 2.2

([8])

Let $(X,d)$ be a complete metric space, and let $T:X\rightarrow X$ be a θ-contraction, that is, there exist $\theta \in \Theta$ and $k\in (0,1)$ such that

$$ d(Tx,Ty)>0\quad \Rightarrow\quad \theta \bigl(d(Tx,Ty) \bigr)\leqslant \theta \bigl(d(x,y) \bigr)^{k} \quad \textit{for all } x,y \in X, $$

where Θ represents the family of all functions $\theta :(0,+\infty )\rightarrow (1,\infty )$ satisfying the following conditions:

$(\theta 1)$ θ is nondecreasing;

$(\theta 2)$ for each sequence $\{\alpha _{m}\}$ of positive numbers, $\lim_{m\rightarrow \infty} \alpha _{m}=0$ if and only if $\lim_{m\rightarrow \infty} \theta (\alpha _{m})=1$;

$(\theta 3)$ there exist $k\in (0,1)$ and $l\in (0,\infty ]$ such that $\lim_{\alpha \rightarrow 0^{+}} \frac{\theta (\alpha )-1}{\alpha ^{k}}=l$,

Then T has a unique fixed point.

By Ψ we denote the family of all functions $\psi :(0,\infty )\rightarrow (0,\infty )$ satisfying the following conditions:

$(\psi 1)$ ψ is nondecreasing;

$(\psi 2)$ for all positive sequence $\{t_{n}\}$, $\lim_{n\rightarrow \infty} \psi (t_{n})=0$ if and only if $\lim_{n\rightarrow \infty} t_{n}=0$;

$(\psi 3)$ ψ is continuous.

By Φ we denote the family of all functions $\phi :(0,\infty )\rightarrow (0,\infty )$ satisfying the following conditions:

$(\phi 1)$ ϕ is nondecreasing;

$(\phi 2)$ $\lim_{n\rightarrow \infty} \phi ^{n}(t)=0$ for all $t>0$, where $\phi ^{n}$ stands for the nth iterate of ϕ.

Remark 2.1

([13])

If $\phi \in \Phi$, then $\phi (t)< t$ for all $t>0$.

Definition 2.1

([13])

Let $(X,d)$ be a metric space. $T:X\rightarrow X$ is said to be a $(\psi ,\phi )$-type contraction if there exist $\psi \in \Psi$, $\phi \in \Phi$, and a continuous function ϕ such that

$$\begin{aligned}& d(Tx,Ty)>0 \\& \quad \Rightarrow \quad \psi \bigl(d(Tx,Ty) \bigr)\leqslant \phi \biggl(\psi \biggl(\max{d(x,y),d(x,Tx),d(y,Ty), \frac{d(x,Ty)}{2}},d(y,Tx) \biggr) \biggr). \end{aligned}$$

Theorem 2.3

([13])

Let $(X,d)$ be a complete metric space, and let $T:X\rightarrow X$ be a $(\psi ,\phi )$-type contraction. Then T has a unique fixed point.

Theorem 2.4

([14])

Let $(X,d)$ be a complete metric space, and let $T:X\rightarrow X$ be a mapping such that

$$ d(Tx,Ty)>0\quad \Rightarrow\quad \psi \bigl(d(Tx,Ty) \bigr)\leqslant \phi \bigl(d(x,y) \bigr) $$

for all $x, y\in X$, where $\psi ,\varphi :(0,\mathrm{\infty })\to \mathbb{R}$ satisfy the following conditions:

(i) ψ is nondecreasing;

(ii) $\phi (t)<\psi (t)$ for all $t>0$;

(iii) $\limsup_{t\rightarrow \varepsilon ^{+}} \phi (t)<\psi ( \varepsilon ^{+})$ for all $\varepsilon >0$.

Then T has a unique fixed point.

Remark 2.2

Let $\mu (t)=\kappa (t)$, $\nu (t)=\iota (\kappa (t))$. If $\iota \in \Phi$, $\kappa \in \Psi$, and ι is continuous, then the following conditions clearly hold:

(i) μ is nondecreasing;

(ii) $\nu (t)<\mu (t)$ for all $t>0$;

(iii) $\limsup_{t\rightarrow \varepsilon ^{+}} \nu (t)<\mu ( \varepsilon ^{+})$ for all $\varepsilon >0$.

Definition 2.2

([15])

Let $\varphi :X\rightarrow [0,\infty )$ and $T:X\rightarrow X$. A point $x\in X$ is called a φ-fixed point of T if $x=Tx$ and $\varphi (x)=0$.

Definition 2.3

([15])

Let $\varphi :X\rightarrow [0,\infty )$. A mapping $T:X\rightarrow X$ is called a φ-Picard mapping if the following conditions hold:

(i) $F(T)\cap Z_{\varphi}=\{x\}$;

(ii) $\lim_{n\rightarrow \infty} T^{n}x_{0}=x$ for all $x_{0}\in X$.

Jleli et al. [15] proposed a new control function $\gamma :[0,\infty )^{3}\rightarrow [0,\infty )$ satisfying the following conditions:

$(\gamma 1)$ $\max \{a,b\}\leqslant \gamma (a,b,c)$;

$(\gamma 2)$ $\gamma (0,0,0)=0$;

$(\gamma 3)$ γ is continuous.

We denote all control functions γ by ϒ.

Example 2.1

([15])

Let $\gamma _{1}(a,b,c)=a+b+c$, $\gamma _{2}(a,b,c)=\max \{a,b\}+c$, and $\gamma _{3}(a,b,c)=a+a^{2}+b+c$ for $a,b,c\in [0,\infty )$. Then $\gamma _{1},\gamma _{2},\gamma _{3}\in \Upsilon$.

Theorem 2.5

([15])

Let $(X,d)$ be a complete metric space, and let a mapping $T:X\rightarrow X$ satisfy

$$ \gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr)\leqslant k\gamma \bigl(d(x,y), \varphi (x),\varphi (y) \bigr), $$

where φ is a lower semicontinuous function, and $k\in (0,1)$. Then T is a φ-Picard mapping.

Definition 2.4

([33])

Let $(X,d)$ be a metric space, $T:X\rightarrow X$, and $\varphi :X\rightarrow [0,\infty )$. Define

$$ r=\inf \bigl\{ d(x,Tx):x\neq Tx \bigr\} . $$

$(1)$ A circle $C_{x_{0},r}=\{x \in X: d(x,x_{0})=r\}$ in X is said to be a φ-fixed circle of T if $C_{x_{0},r}\subset F(T)\cap Z_{\varphi}$.

$(2)$ A disc $D_{x_{0},r}=\{x \in X: d(x,x_{0})\leqslant r\}$ in X is said to be a φ-fixed disc of T if $D_{x_{0},r}\subset F(T)\cap Z_{\varphi}$.

3 φ-fixed point results

In this section, we present some new contractions and some corresponding results.

Definition 3.1

A mapping $T:X\rightarrow X$ is called a $(\gamma ,\psi ,\varphi ,\phi )$ contraction in a metric space $(X,d)$, if there exist ψ, ϕ satisfying conditions (i), (ii), and (iii) in Theorem 2.4 and the inequality

$$ \psi \bigl(\gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr) \bigr)\leqslant \phi \bigl( \gamma \bigl(d(x,y),\varphi (x),\varphi (y) \bigr) \bigr) $$

(1)

for all $x,y\in X$ such that $\gamma (d(Tx,Ty),\varphi (Tx),\varphi (Ty))>0$.

Theorem 3.1

Let $(X,d)$ be a complete metric space, and let $T:X\rightarrow X$ be a $(\gamma ,\psi ,\varphi ,\phi )$ contraction. If φ is lower semicontinuous, then T is a φ-Picard mapping.

Proof

Firstly, we show that $F(T)\subseteq Z_{\varphi}$.

Assume that there exists $x\in F(T)$ such that $\varphi (x)\neq 0$. Set $y=x$ in (1). Then

$$\begin{aligned} \psi \bigl(\gamma \bigl(0,\varphi (x),\varphi (x) \bigr) \bigr)&=\psi \bigl( \gamma \bigl(d(Tx,Tx), \varphi (Tx),\varphi (Tx) \bigr) \bigr) \\ &\leqslant \phi \bigl(\gamma \bigl(d(x,x),\varphi (x),\varphi (x) \bigr) \bigr) \\ &=\phi \bigl(\gamma \bigl(0,\varphi (x),\varphi (x) \bigr) \bigr) \\ &< \psi \bigl(\gamma \bigl(0,\varphi (x),\varphi (x) \bigr) \bigr). \end{aligned}$$

This is a contradiction, so $F(T)\subseteq Z_{\varphi}$.

Now we show that $\lim_{n\rightarrow \infty} d(x_{n},x_{n+1})=0$ and $\lim_{n\rightarrow \infty} \varphi (x_{n})=0$. Let $x_{0}\in X$, and let the sequence $\{x_{n}\}$ be defined by $x_{n}=Tx_{n-1}$ for all $n\in \mathbb{N}$. If there exists $n_0\in \mathbb{N}$ such that $x_{n_{0}}=x_{n_{0}+1}$, that is, $x_{n_{0}}=Tx_{n_{0}}$, then $x_{n_{0}}$ is a fixed point of T. In this case, obviously, $\lim_{n\rightarrow \infty} d(x_{n},x_{n+1})=0$, $\lim_{n\rightarrow \infty} \varphi (x_{n})=0$, and the proof is completed. So we suppose that $d(x_{n},x_{n+1})>0$, for all $n\in \mathbb{N}$. Letting $x=x_{n}$, $y=x_{n+1}$ in (1), we get

$$\begin{aligned} \begin{aligned}[b] \psi \bigl(\gamma \bigl(d(x_{n+1},x_{n+2}), \varphi (x_{n+1}),\varphi (x_{n+2}) \bigr) \bigr)&= \psi \bigl(\gamma \bigl(d(Tx_{n},Tx_{n+1}),\varphi (Tx_{n}),\varphi (Tx_{n+1}) \bigr) \bigr) \\ &\leqslant \phi \bigl(\gamma \bigl(d(x_{n},x_{n+1}), \varphi (x_{n}),\varphi (x_{n+1}) \bigr) \bigr) \\ &< \psi \bigl(\gamma \bigl(d(x_{n},x_{n+1}), \varphi (x_{n}),\varphi (x_{n+1}) \bigr) \bigr). \end{aligned} \end{aligned}$$

(2)

By (i) we can deduce that

$$ \gamma \bigl(d(x_{n+1},x_{n+2}),\varphi (x_{n+1}),\varphi (x_{n+2}) \bigr)< \gamma \bigl(d(x_{n},x_{n+1}),\varphi (x_{n}),\varphi (x_{n+1}) \bigr). $$

Therefore $\{\gamma (d(x_{n},x_{n+1}),\varphi (x_{n}),\varphi (x_{n+1}))\}$ is a decreasing sequence with a lower bound. So these exists $\epsilon \geqslant 0$ such that $\lim_{n\rightarrow \infty} \gamma _{n}=\epsilon$, where $\gamma _{n}=\gamma (d(x_{n},x_{n+1}),\varphi (x_{n}),\varphi (x_{n+1}))$. If $\epsilon >0$, then taking the right upper limits on both sides of (2), by (i) and (iii) we have

$$\begin{aligned} \psi \bigl(\epsilon ^{+} \bigr)&= \limsup_{\gamma _{n+1} \rightarrow \epsilon ^{+}} \gamma _{n+1} \\ &\leqslant \limsup_{\gamma _{n}\rightarrow \epsilon ^{+}} \phi (\gamma _{n}) \\ &\leqslant \limsup_{t\rightarrow \epsilon ^{+}} \phi (t) \\ &< \psi \bigl(\epsilon ^{+} \bigr). \end{aligned}$$

This is a contradiction, so $\epsilon =0$. By $(\gamma 1)$ it easily follows that

$$ \max \bigl\{ d(x_{n+1},x_{n+2}),\varphi (x_{n+1}) \bigr\} \leqslant \gamma \bigl(d(x_{n+1},x_{n+2}), \varphi (x_{n+1}),\varphi (x_{n+2}) \bigr). $$

Clearly,

$$ d(x_{n+1},x_{n+2})\leqslant \gamma \bigl(d(x_{n+1},x_{n+2}), \varphi (x_{n+1}), \varphi (x_{n+2}) \bigr)$$

and

$$ \varphi (x_{n+1})\leqslant \gamma \bigl(d(x_{n+1},x_{n+2}), \varphi (x_{n+1}), \varphi (x_{n+2}) \bigr).$$

Taking the limits of these two inequalities, we have

$$\begin{aligned}& 0\leqslant \lim_{n\rightarrow \infty} d(x_{n},x_{n+1}) \leqslant \lim_{n\rightarrow \infty} \gamma \bigl(d(x_{n+1},x_{n+2}), \varphi (x_{n+1}),\varphi (x_{n+2}) \bigr)=0,\\& 0\leqslant \lim_{n\rightarrow \infty} \varphi (x_{n+1}) \leqslant \lim_{n\rightarrow \infty} \gamma \bigl(d(x_{n+1},x_{n+2}), \varphi (x_{n+1}),\varphi (x_{n+2}) \bigr)=0, \end{aligned}$$

that is,

$$\begin{aligned}& \lim_{n\rightarrow \infty} d(x_{n},x_{n+1})=0, \end{aligned}$$

(3)

$$\begin{aligned}& \lim_{n\rightarrow \infty} \varphi (x_{n+1})=0. \end{aligned}$$

(4)

Next, we claim that $\{x_{n}\}$ is a Cauchy sequence.

If $\{x_{n}\}$ were not a Cauchy sequence, then there would exist $\varepsilon >0$ and two sequences $\{x_{n(k)}\}$, $\{x_{m(k)}\}$, where $n(k)$, $m(k)$ are two positive integers and $n(k)>m(k)$, such that $d(x_{m(k)},x_{n(k)})\geqslant \varepsilon$ and $d(x_{m(k)},x_{n(k)-1})<\varepsilon$. By the triangle inequality it follows that

$$\begin{aligned} \varepsilon &\leqslant d(x_{m(k)},x_{n(k)}) \\ &\leqslant d(x_{m(k)},x_{n(k)-1})+d(x_{n(k)-1},x_{n(k)}) \\ &< \varepsilon +d(x_{n(k)-1},x_{n(k)}). \end{aligned}$$

(5)

Taking the limits on the both sides of (5), we obtain

$$ \lim_{k\rightarrow \infty}d(x_{m(k)},x_{n(k)})= \varepsilon ^{+}. $$

(6)

By the triangle inequality we have

$$ d(x_{n(k)-1},x_{m(k)-1})\leqslant d(x_{n(k)-1},x_{m(k)})+d(x_{m(k)},x_{m(k)-1}) $$

(7)

and

$$ d(x_{n(k)},x_{m(k)})\leqslant d(x_{n(k)},x_{n(k)-1})+d(x_{n(k)-1},x_{m(k)-1})+d(x_{m(k)-1},x_{m(k)}). $$

(8)

Letting $k\rightarrow \infty$ in (7) and (8), we have

$$ \lim_{k\rightarrow \infty}d(x_{n(k)-1},x_{m(k)-1})= \varepsilon ^{+}. $$

(9)

Let $x=x_{m(k)-1}$ and $y=x_{n(k)-1}$ in (1). The by (i) it follows that

$$\begin{aligned} \psi (\gamma _{m,n})&= \psi \bigl(\gamma \bigl(d(Tx_{m(k)-1},Tx_{n(k)-1}), \varphi (Tx_{m(k)-1}), \varphi (Tx_{n(k)-1}) \bigr) \bigr) \\ &\leqslant \phi (\gamma _{m-1,n-1}), \end{aligned}$$

(10)

where $\gamma _{m,n}=\gamma (d(x_{m(k)},x_{n(k)}),\varphi (x_{m(k)}), \varphi (x_{n(k)}))$. Taking the right upper limits on both sides of (10), by (4), (6), and (iii) we obtain

$$\begin{aligned} \psi \bigl(\varepsilon ^{+} \bigr)&= \limsup_{\gamma _{n,m}\rightarrow \varepsilon ^{+}} \psi ( \gamma _{n,m}) \\ &\leqslant \limsup_{\gamma _{n-1,m-1}\rightarrow \varepsilon ^{+}} \phi ( \gamma _{n-1,m-1}) \\ &\leqslant \limsup_{t\rightarrow \varepsilon ^{+}} \phi (t) \\ &< \psi \bigl(\varepsilon ^{+} \bigr), \end{aligned}$$

which leads to a contradiction with $\varepsilon ^{+}>0$. So $\{x_{n}\}$ is a Cauchy sequence in the complete metric space $(X,d)$. Thus there exists $x^{*}\in X$ such that $x_{n}\rightarrow x^{*}$ as $n\rightarrow \infty$. Since φ is lower semicontinuous, we have

$$ \varphi \bigl(x^{*} \bigr)\leqslant \liminf_{n\rightarrow \infty} \varphi (x_{n+1})\leqslant \lim_{n\rightarrow \infty} \varphi (x_{n+1}) =0. $$

So $\varphi (x^{*})=0$.

Now we claim that $x^{*}=Tx^{*}$. If there exists some $n_{0}$ such that for $n\geq n_{0}$, $x_{n}=Tx^{*}$ still holds, then $Tx^{*}=\lim_{n\rightarrow \infty} x_{n}=x^{*}$, that is, $x^{*}=Tx^{*}$. Suppose that $d(x^{*},Tx^{*})>0$. Set $x=x_{n}$ and $y=x^{*}$ in (1). Then

$$ \psi \bigl(\gamma \bigl(d \bigl(Tx_{n},Tx^{*} \bigr), \varphi (Tx_{n}),\varphi \bigl(Tx^{*} \bigr) \bigr) \bigr) \leqslant \phi \bigl(\gamma \bigl(d \bigl(x_{n},x^{*} \bigr),\varphi (x_{n}),0 \bigr) \bigr). $$

By (i) and (ii) it follows that

$$\begin{aligned} d \bigl(x_{n+1},Tx^{*} \bigr)\leqslant \gamma \bigl(d \bigl(Tx_{n},Tx^{*} \bigr),\varphi (Tx_{n}), \varphi \bigl(Tx^{*} \bigr) \bigr)&< \gamma \bigl(d \bigl(x_{n},x^{*} \bigr),\varphi (x_{n}),0 \bigr). \end{aligned}$$

(11)

Taking the limits on the both sides of (11), we get

$$ d \bigl(x^{*},Tx^{*} \bigr)\leqslant 0, $$

which is a contradiction. So $Tx^{*}=x^{*}$.

Finally, we claim that $x^{*}=y$ for all $x^{*},y\in F(T)$. Otherwise, if $x^{*}\neq y$, then letting $x=x^{*}$ in (1), by (ii) and $F(T)\subseteq Z_{\varphi}$ it follows that

$$\begin{aligned} \psi \bigl(\gamma \bigl(d \bigl(x^{*},y \bigr),0,0 \bigr) \bigr)&= \psi \bigl(\gamma \bigl(d \bigl(x^{*},y \bigr),\varphi \bigl(x^{*} \bigr), \varphi (y) \bigr) \bigr) \\ &=\psi \bigl(\gamma \bigl(d \bigl(Tx^{*},Ty \bigr),\varphi \bigl(Tx^{*} \bigr),\varphi (Ty) \bigr) \bigr) \\ &\leqslant \phi \bigl(\gamma \bigl(d \bigl(x^{*},y \bigr),\varphi \bigl(x^{*} \bigr),\varphi (y) \bigr) \bigr) \\ &=\phi \bigl(\gamma \bigl(d \bigl(x^{*},y \bigr),0,0 \bigr) \bigr) \\ &< \psi \bigl(\gamma \bigl(d \bigl(x^{*},y \bigr),0,0 \bigr) \bigr). \end{aligned}$$

This is a contraction, so $x^{*}=y$. Hence T is a φ-Picard mapping. □

Remark 3.1

$(1)$ Take $\gamma =\gamma _{1}$ and $\varphi (x)=0$. Then Theorem 3.1 reduces to Theorem 2.4.

$(2)$ Take $\phi (t)=kt$ and $\psi (t)=t$ for $t\in {\mathbb{R}}_{+}$, where $k\in [0,1)$. Then Theorem 3.1 reduces to Theorem 2.5.

$(3)$ Take $\phi (t)=kt$, $\psi (t)=t$, $\varphi (x)=0$, and $\gamma =\gamma _{1}$ in Theorem 3.1, where $t\in {\mathbb{R}}_{+}$, $x\in X$, $k\in [0,1)$. Then we get the Banach contraction principle.

Now we give an example to show the validity of Theorem 3.1.

Example 3.1

Let $X=[0,2]$ be endowed with the usual metric $d(x,y)=|x-y|$ for $x,y\in X$. Obviously, $(X,d)$ is a complete metric space. Consider the mapping $T:X\rightarrow X$ defined by

$$ T(x)=\textstyle\begin{cases} \frac{1}{4}x& \text{if } x\in [0,2), \\ \frac{1}{8}& \text{otherwise}. \end{cases} $$

Let the functions ϕ, φ, ψ, γ be defined by $\psi (t)=t$, $\phi (t)=\frac{1}{4}t$, $\varphi (x)=x$, $\gamma (a,b,c)=a+b+c$.

Case 1: $x=y\in X-\{0,2\}$. Then

$$ d(Tx,Ty)+\varphi (Tx)+\varphi (Ty)\leqslant \frac{1}{4} \bigl(d(x,y)+ \varphi (x)+\varphi (y) \bigr).$$

Case 2: $x\in [0,2)$, $y=2$. Then

$$\begin{aligned} d(Tx,Ty)+\varphi (Tx)+\varphi (Ty)&= \biggl\vert \frac{1}{8}- \frac{1}{4}x \biggr\vert + \frac{1}{4}x+ \frac{1}{8} \\ &\leqslant \max \biggl\{ \frac{1}{4},\frac{x}{2} \biggr\} \\ &\leqslant 1 \\ &=\frac{1}{4} \bigl(d(x,y)+\varphi (x)+\varphi (y) \bigr). \end{aligned}$$

Case 3: $x\neq y\in [0,2)$. Then

$$\begin{aligned} d(Tx,Ty)+\varphi (Tx)+\varphi (Ty)&= \biggl\vert \frac{1}{4}x- \frac{1}{4}y \biggr\vert + \frac{1}{4}x+ \frac{1}{4}y \\ &\leqslant \max \biggl\{ \frac{x}{2},\frac{y}{2} \biggr\} \\ &=\frac{1}{4} \bigl(d(x,y)+\varphi (x)+\varphi (y) \bigr). \end{aligned}$$

So T is a $(\gamma ,\psi ,\varphi ,\phi )$ contraction and satisfies all conditions of Theorem 3.1. By Theorem 3.1T has unique fixed point $x=0$ such that $T(0)=0$ and $\varphi (0)=0$. In addition, we claim that the Banach contraction principle is useless in this example. Indeed, letting $x=\frac{7}{4}$ and $y=2$, we have $d(T(\frac{7}{4}),T(2))=\frac{5}{16}>\frac{1}{4}$.

Corollary 3.1

Let $(X,d)$ be a complete metric space, and let $T:X\rightarrow X$. Suppose there exists $k\in [0,1)$ such that

$$\begin{aligned}& \gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr)>0 \\& \quad \Rightarrow \quad \psi \bigl( \gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr) \bigr)\leqslant \psi \bigl( \gamma \bigl(d(x,y), \varphi (x),\varphi (y) \bigr) \bigr)^{k}, \end{aligned}$$

where φ is lower semicontinuous, and $\psi :{\mathbb{R}}_{+}\to \mathbb{R}$ is nondecreasing. Then T is a φ-Picard mapping.

Proof

Letting $\phi (t)=(\psi (t))^{k}$, the result follows by Theorem 3.1. □

Remark 3.2

By Corollary 3.1 it is not difficult to find that the result of Theorem 4 in [24] is still valid if the following conditions are deleted:

(i) for each sequence $\{\alpha _{m}\}$ of positive numbers, $\lim_{m\rightarrow \infty} \alpha _{m}=0$ if and only if $\lim_{m\rightarrow \infty} \psi (\alpha _{m})=1$;

(ii) there exist $k\in (0,1)$ and $l\in (0,\infty ]$ such that $\lim_{\alpha \rightarrow 0^{+}} \frac{\theta (\alpha )-1}{\alpha ^{k}}=l$.

Corollary 3.2

Let $(X,d)$ be a complete metric space, and let $T:X\rightarrow X$. Suppose there exists $\tau >0$ such that

$$\begin{aligned} \gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr)>0\quad \Rightarrow \quad &\psi \bigl( \gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr) \bigr) \\ &\quad \leqslant \psi \bigl(\gamma \bigl(d(x,y),\varphi (x),\varphi (y) \bigr) \bigr)- \tau , \end{aligned}$$

where φ is lower semicontinuous, $\psi :{\mathbb{R}}_{+}\to \mathbb{R}$ is nondecreasing. Then T is a φ-Picard mapping.

Proof

Letting $\phi (t)=\psi (t)-\tau$, the result follows by Theorem 3.1. □

Remark 3.3

Corollary 3.2 improves Theorem 3.4 and Theorem 3.5 in [22].

Corollary 3.3

Let $(X,d)$ be a complete metric space, and let $T:X\rightarrow X$. Suppose there exist $(\iota ,\kappa )\in (\Phi ,\Psi )$, with continuous ι, such that for all $x,y\in X$ with $\gamma (d(Tx,Ty),\varphi (Tx),\varphi (Ty))>0$,

$$ \kappa \bigl(\gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr) \bigr) \leqslant \iota \bigl( \kappa \bigl(\gamma \bigl(d(x,y),\varphi (x),\varphi (y) \bigr) \bigr) \bigr), $$

and φ is lower semi-continuous. Then T is a φ-Picard mapping.

Proof

The result is clearly true by Theorem 3.1 and Remark 2.2. □

Corollary 3.4

Let $(X,d)$ be a complete metric space, and let $T:X\rightarrow X$. Let $\kappa :{\mathbb{R}}_{+}\to \mathbb{R}$ and $\iota :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be functions that satisfy

(a) κ is nondecreasing;

(b) $\limsup_{t\rightarrow \varepsilon ^{+}} \kappa (t)<\kappa ( \varepsilon ^{+})$ for all $\varepsilon >0$;

(c) $\liminf_{t\rightarrow \varepsilon ^{+}} \iota (t) \geqslant 0$ for all $\varepsilon >0$.

Suppose that for all $x,y\in X$ with $\gamma (d(Tx,Ty),\varphi (Tx),\varphi (Ty))>0$, T satisfies

$$ \kappa \bigl(\gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr) \bigr) \leqslant \kappa \bigl( \gamma \bigl(d(x,y),\varphi (x),\varphi (y) \bigr) \bigr)- \iota \bigl(\gamma \bigl(d(x,y), \varphi (x),\varphi (y) \bigr) \bigr), $$

and φ is lower semicontinuous. Then T is a φ-Picard mapping.

Proof

Let $\psi (t)=\kappa (t)$ and $\phi (t)=\kappa (t)-\iota (t)$. Then the result follows by Theorem 3.1. □

Corollary 3.5

Let $(X,d)$ be a complete metric space, and let $T:X\rightarrow X$. Suppose there exists a continuous function $\iota \in \Phi$ such that for all $x,y\in X$, T satisfies

$$ \gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr)\leqslant \iota \bigl( \gamma \bigl(d(x,y), \varphi (x),\varphi (y) \bigr) \bigr), $$

where φ is a lower semicontinuous function. Then T is a φ-Picard mapping.

Proof

The result follows by Corollary 3.3. □

Remark 3.4

Corollary 3.5 shows that the condition $\sum_{i=0}^{\infty}\iota ^{i}(t)<\infty$ of Theorem 2.5 in [16] is redundant.

Corollary 3.6

Let $(X,d)$ be a complete metric space, and let $T:X\rightarrow X$. Suppose there exist a continuous function $\iota \in \Phi$ and a function $\kappa \in \Psi$ such that for all $x,y\in X$ with $d(Tx,Ty)>0$, T satisfies

$$ \kappa \bigl(d(Tx,Ty) \bigr)\leqslant \iota \bigl(\kappa \bigl(d(x,y) \bigr) \bigr). $$

Then T has a unique fixed point.

Proof

The result follows by taking $\gamma =\gamma _{1}$ and $\varphi (x)=0$ for $x\in X$ in Corollary 3.3. □

Definition 3.2

A mapping $T:X\rightarrow X$ is called a rational-$(\gamma ,\psi , \varphi ,\phi )$ contraction in a metric space $(X,d)$ if there exist ψ, ϕ satisfy conditions (i), (ii), and (iii) in Theorem 2.4 such that for all $x,y\in X$ with $\gamma (d(Tx,Ty),\varphi (Tx),\varphi (Ty))>0$, T satisfies

$$ \psi \bigl(\gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr) \bigr)\leqslant \phi ( \gamma \bigl(N(x,y),\varphi (x),\varphi (y) \bigr), $$

(12)

where $N(x,y)=\max \{d(x,y),\frac{d(x,Tx)(1+d(y,Ty))}{1+d(Tx,Ty)}\}$.

Theorem 3.2

Let $(X,d)$ be a complete metric space, and let T be a rational-$( \gamma ,\psi ,\varphi ,\phi )$ contraction. If φ is lower semicontinuous, then T is a φ-Picard mapping.

Proof

Firstly, we show that $F(T)\subseteq Z_{\varphi}$.

Assume that there exists $x\in F(T)$ such that $\varphi (x)\neq 0$. Letting $y=x$ in (12), we get

$$\begin{aligned} \psi \bigl(\gamma \bigl(0,\varphi (x),\varphi (x) \bigr) \bigr)&=\psi \bigl( \gamma \bigl(d(Tx,Tx), \varphi (Tx),\varphi (Tx) \bigr) \bigr) \\ &\leqslant \phi \bigl(\gamma \bigl(N(x,x),\varphi (x),\varphi (x) \bigr) \bigr) \\ &=\phi \bigl(\gamma \bigl(0,\varphi (x),\varphi (x) \bigr) \bigr) \\ &< \psi \bigl(\gamma \bigl(0,\varphi (x),\varphi (x) \bigr) \bigr). \end{aligned}$$

This is a contradiction, and so $\varphi (x)=0$. Hence $F(T)\subseteq Z_{\varphi}$.

Now we show that $\lim_{n\rightarrow \infty} d(x_{n},x_{n+1})=0$ and $\lim_{n\rightarrow \infty} \varphi (x_{n})=0$. Let $x_{0}\in X$. Define the sequence $\{x_{n}\}$ by $x_{n}=Tx_{n-1}$ for all $n\in \mathbb{N}$. If there exists some $n_0\in \mathbb{N}$ such that $x_{n_{0}}=x_{n_{0}+1}$, that is, $x_{n_{0}}=Tx_{n_{0}}$, then $x_{n_{0}}$ is a fixed point of T, and, obviously, $\lim_{n\rightarrow \infty} d(x_{n},x_{n+1})=0$, $\lim_{n\rightarrow \infty} \varphi (x_{n})=0$, and thus the proof is completed. So we suppose that $d(x_{n},x_{n+1})>0$ for all $n\in \mathbb{N}$. Letting $x=x_{n}$ and $y=x_{n+1}$ in (12), we get

$$\begin{aligned} \psi \bigl(\gamma \bigl(d(x_{n+1},x_{n+2}),\varphi (x_{n+1}),\varphi (x_{n+2}) \bigr) \bigr)&= \psi \bigl( \gamma \bigl(d(Tx_{n},Tx_{n+1}),\varphi (Tx_{n}), \varphi (Tx_{n+1}) \bigr) \bigr) \\ &\leqslant \phi \bigl(\gamma \bigl(N(x_{n},x_{n+1}), \varphi (x_{n}),\varphi (x_{n+1}) \bigr) \bigr), \end{aligned}$$

where $N(x_{n},x_{n+1})=\max \{d(x_{n},x_{n+1}), \frac{d(x_{n},x_{n+1})(1+d(x_{n+1},x_{n+2})}{1+d(x_{n+1},x_{n+2})}\}=d(x_{n},x_{n+1})$. Similarly to the proof of Theorem 3.1, we obtain (3) and (4). Then we claim that $\{x_{n}\}$ is a Cauchy sequence. If $\{x_{n}\}$ were not a Cauchy sequence, similarly to the proof of Theorem 3.1, we would deduce (6) and (9). Letting $x=x_{m(k)-1}$ and $y=x_{n(k)-1}$ in (12), it follows that

$$\begin{aligned} \psi \bigl(\gamma \bigl(d(x_{m(k)},x_{n(k)}),\varphi (x_{m(k)}),\varphi (x_{n(k)}) \bigr) \bigr)&= \psi \bigl( \gamma \bigl(d(Tx_{m(k)-1},Tx_{n(k)-1}),\varphi (Tx_{m(k)-1}), \\ &\quad \varphi (Tx_{n(k)}-1) \bigr) \bigr) \\ &\leqslant \phi (\gamma \bigl(N(x_{m(k)-1},x_{n(k)-1}),\varphi (x_{m(k)-1}), \varphi (x_{n(k)-1}) \bigr), \end{aligned}$$

where

$$\begin{aligned}& N(x_{m(k)-1},x_{n(k)-1}) \\& \quad =\max \biggl\{ d(x_{m(k)-1},x_{n(k)-1}), \frac{d(x_{m(k)-1},x_{Tm(k)-1})(1+d(x_{n(k)-1},Tx_{n(k)-1}))}{1+d(x_{Tm(k-1)},Tx_{n(k)-1})} \biggr\} . \end{aligned}$$

By (3) and (9) we have

$$ \lim_{n\rightarrow \infty} N(x_{m(k)},x_{n(k)})= \varepsilon . $$

Similarly to the proof of Theorem 3.1, we attain that $\{x_{n}\}$ is a Cauchy sequence in a complete metric space $(X,d)$, and there exists $x^{*}\in X$ such that $x_{n}\rightarrow x^{*}$ as $n\rightarrow \infty$ and $\varphi (x^{*})=0$.

We claim that $x^{*}=Tx^{*}$. If there exists $n_{0}$ such that for $n\geq n_{0}$, $x_{n}=Tx^{*}$ still holds, then $Tx^{*}=\lim_{n\rightarrow \infty} x_{n}=x^{*}$, that is, $x^{*}=Tx^{*}$. On the contrary, set $x=x_{n}$ and $y=x^{*}$ in (12). Then

$$ \psi \bigl(\gamma \bigl(d \bigl(Tx_{n},Tx^{*} \bigr), \varphi (Tx_{n}),\varphi \bigl(Tx^{*} \bigr) \bigr) \bigr) \leqslant \phi \bigl(\gamma \bigl(N \bigl(x_{n},x^{*} \bigr),\varphi (x_{n}),0 \bigr) \bigr). $$

By (i) and (ii) it follows that

$$\begin{aligned} d \bigl(x_{n+1},Tx^{*} \bigr)\leqslant \gamma \bigl(d \bigl(Tx_{n},Tx^{*} \bigr),\varphi (Tx_{n}), \varphi \bigl(Tx^{*} \bigr) \bigr)&< \gamma \bigl(N \bigl(x_{n},x^{*} \bigr),\varphi (x_{n}),0 \bigr), \end{aligned}$$

(13)

where $N(x_{n},x^{*})=\max \{d(x_{n},x^{*}), \frac{d(x_{n},Tx_{n})(1+d(x^{*},Tx^{*}))}{1+d(Tx_{n},Tx^{*})}\}$.

Taking the limits on the both sides of (13), we get

$$ d \bigl(x^{*},Tx^{*} \bigr)\leqslant 0, $$

that is, $Tx^{*}=x^{*}$.

Finally, we claim that $x^{*}=y$ for all $x^{*},y\in F(T)$. If $x^{*}\neq y$, then letting $x=x^{*}$ in (12), it follows that

$$\begin{aligned} \psi \bigl(\gamma \bigl(d \bigl(x^{*},y \bigr),0,0 \bigr) \bigr)&= \psi \bigl(\gamma \bigl(d \bigl(Tx^{*},Ty \bigr),\varphi \bigl(Tx^{*} \bigr), \varphi (Ty) \bigr) \bigr) \\ &\leqslant \phi \bigl(\gamma \bigl(N \bigl(x^{*},y \bigr),\varphi \bigl(x^{*} \bigr),\varphi (y) \bigr) \bigr) \\ &=\phi \biggl(\gamma \biggl(\max \biggl\{ d \bigl(x^{*},y \bigr), \frac{d(x^{*},Tx^{*})(1+d(y,Ty))}{1+d(Tx^{*},Ty)} \biggr\} ,0,0 \biggr) \biggr) \\ &=\phi \bigl(\gamma \bigl(d \bigl(x^{*},y \bigr),0,0 \bigr) \bigr) \\ &< \psi \bigl(\gamma \bigl(d \bigl(x^{*},y \bigr),0,0 \bigr) \bigr). \end{aligned}$$

This is a contraction, so $x^{*}=y$. Hence T is a φ-Picard mapping. □

Example 3.2

Let $X=[0,1]$ be endowed with the usual metric $d(x,y)=|x-y|$ for $x,y\in X$. Obviously, $(X,d)$ is a complete metric space. Consider the mapping $T:X\rightarrow X$ defined by

$$ T(x)=\textstyle\begin{cases} \frac{x}{2}& \text{if } x\in [0,\frac{1}{2}), \\ \frac{x}{4}& \text{otherwise}. \end{cases} $$

Let the functions ϕ, φ, ψ, γ be defined by $\psi (t)=t$, $\phi (t)=\frac{7}{8}t$, $\varphi (x)=x$, $\gamma (a,b,c)= \max \{a,b,c\}$.

Case 1: for all $x=y\in (0,1]$, it is obvious that

$$\begin{aligned} \max \bigl\{ d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr\} &\leqslant \frac{x}{2} \\ &\leqslant \frac{7}{8}\max \bigl\{ N(x,y),\varphi (x),\varphi (y) \bigr\} . \end{aligned}$$

Case 2: for all $x, y\in [0,1]$ with $x\neq y$,

$$\begin{aligned} \max \bigl\{ d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr\} &\leqslant \max \biggl\{ \frac{x}{2},\frac{y}{2} \biggr\} \\ &\leqslant \frac{7}{8}\max \bigl\{ N(x,y),\varphi (x),\varphi (y) \bigr\} . \end{aligned}$$

So T is a $(\gamma ,\psi ,\varphi ,\phi )$ contraction and satisfies all the conditions of Theorem 3.2. By Theorem 3.2 we get T has a unique fixed point $x=0$ such that $T(0)=0$ and $\varphi (0)=0$.

Corollary 3.7

Let $(X,d)$ be a complete metric space, and let $T:X\rightarrow X$. If T satisfies

$$\begin{aligned} \gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr)>0\quad \Rightarrow \quad &\psi \bigl( \gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr) \bigr) \\ &\quad \leqslant \psi \bigl(\gamma \bigl(N(x,y),\varphi (x),\varphi (y) \bigr) \bigr)- \tau , \end{aligned}$$

where $N(x,y)=\max \{d(x,y),\frac{d(x,Tx)(1+d(y,Ty))}{1+d(Tx,Ty)}\}$, $\tau >0$, φ is lower semi-continuous, and $\psi :{\mathbb{R}}_{+}\to \mathbb{R}$ is nondecreasing, then T is a φ-Picard mapping.

Proof

The result follows by taking $\phi (t)=\psi (t)-\tau$ in Theorem 3.2. □

Remark 3.5

Corollary 3.7 shows that the result holds even if Theorem 11 (resp., Theorem 12) in [23] uses only the following condition:

(i) $\psi :{\mathbb{R}}_{+}\to \mathbb{R}$ is nondecreasing.

Corollary 3.8

Let $(X,d)$ be a complete metric space, and let $T:X\rightarrow X$. If there exists $k\in (0,1)$ such that T satisfies

$$\begin{aligned} \gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr)>0\quad \Rightarrow \quad &\psi \bigl( \gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr) \bigr) \\ &\quad \leqslant \psi \bigl(\gamma \bigl(N(x,y),\varphi (x),\varphi (y) \bigr) \bigr)^{k}, \end{aligned}$$

where $N(x,y)=\max \{d(x,y),\frac{d(x,Tx)(1+d(y,Ty))}{1+d(Tx,Ty)}\}$, φ is lower semi-continuous, and $\psi :{\mathbb{R}}_{+}\to \mathbb{R}$ is nondecreasing, then T is a φ-Picard mapping.

Proof

The result follows by taking $\phi (t)=\psi (t)^{k}$ in Theorem 3.2. □

Definition 3.3

A mapping $T:X\rightarrow X$ is called an almost-$(\gamma ,\psi , \varphi ,\phi )$ contraction in a metric space $(X,d)$ if there exist ψ, ϕ satisfying conditions (i), (ii), and (iii) in Theorem 2.4 such that for all $x,y\in X$ with $\gamma (d(Tx,Ty),\varphi (Tx),\varphi (Ty))>0$, T satisfies

$$ \psi \bigl(\gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr) \bigr)\leqslant \phi ( \gamma \bigl(d(x,y)+LM(x,y),\varphi (x),\varphi (y) \bigr), $$

(14)

where $M(x,y)=\min \{d(y,Ty),d(y,Tx)\}$ and $L\geqslant 0$.

Theorem 3.3

Let $(X,d)$ be a complete metric space, and let T be an almost-$( \gamma ,\psi ,\varphi ,\phi )$ contraction. If φ is lower semicontinuous, then T is a φ-Picard mapping.

Proof

Firstly, we show that $F(T)\subseteq Z_{\varphi}$. Assume that there exists $x\in F(T)$ such that $\varphi (x)\neq 0$. Letting $y=x$ in (14), we get

$$\begin{aligned} \psi \bigl(\gamma \bigl(0,\varphi (x),\varphi (x) \bigr) \bigr)&=\psi \bigl( \gamma \bigl(d(Tx,Tx), \varphi (Tx),\varphi (Tx) \bigr) \bigr) \\ &\leqslant \phi \bigl(\gamma \bigl(d(x,x)+LM(x,x),\varphi (x),\varphi (x) \bigr) \bigr) \\ &=\phi \bigl(\gamma \bigl(0,\varphi (x),\varphi (x) \bigr) \bigr) \\ &< \psi \bigl(\gamma \bigl(0,\varphi (x),\varphi (x) \bigr) \bigr). \end{aligned}$$

This is a contradiction, so $\varphi (x)=0$. Finally, we have $F(T)\subseteq Z_{\varphi}$.

Now we show that $\lim_{n\rightarrow \infty} d(x_{n},x_{n+1})=0$ and $\lim_{n\rightarrow \infty} \varphi (x_{n})=0$. Letting $x_{0}\in X$, define the sequence $\{x_{n}\}$ by $x_{n}=Tx_{n-1}$ for all $n\in \mathbb{N}$. If there exists $n_0\in \mathbb{N}$ such that $x_{n_{0}}=x_{n_{0}+1}$, that is, $x_{n_{0}}=Tx_{n_{0}}$, then $x_{n_{0}}$ is a fixed point of T, and, obviously, $\lim_{n\rightarrow \infty} d(x_{n},x_{n+1})=0$, $\lim_{n\rightarrow \infty} \varphi (x_{n})=0$, and thus the proof is completed. So we suppose that $d(x_{n},x_{n+1})>0$ for all $n\in \mathbb{N}$. Letting $x=x_{n}$ and $y=x_{n+1}$ in (14), we obtain

$$\begin{aligned} \psi \bigl(\gamma \bigl(d(x_{n+1},x_{n+2}),\varphi (x_{n+1}),\varphi (x_{n+2}) \bigr) \bigr)&= \psi \bigl( \gamma \bigl(d(Tx_{n},Tx_{n+1}),\varphi (Tx_{n}), \varphi (Tx_{n+1}) \bigr) \bigr) \\ &\leqslant \phi \bigl(\gamma \bigl(d(x_{n},x_{n+1})+LM(x_{n},x_{n+1}), \varphi (x_{n}), \varphi (x_{n+1}) \bigr) \bigr) \\ &=\phi \bigl(\gamma \bigl(d(x_{n},x_{n+1}),\varphi (x_{n}),\varphi (x_{n+1}) \bigr) \bigr). \end{aligned}$$

Similarly to the proof of Theorem 3.1, we get (3) and (4). We claim that $\{x_{n}\}$ is a Cauchy sequence. If $\{x_{n}\}$ were not a Cauchy sequence, then similarly to the proof of Theorem 3.1, we would get (6) and (9). Letting $x=x_{m(k)-1}$ and $y=x_{n(k)-1}$ in (14), it follows that

$$\begin{aligned} \psi \bigl(\gamma \bigl(d(x_{m(k)},x_{n(k)}),\varphi (x_{m(k)}),\varphi (x_{n(k)}) \bigr) \bigr)&= \psi \bigl( \gamma \bigl(d(Tx_{m(k)-1},Tx_{n(k)-1}), \\ &\quad \varphi (Tx_{m(k)-1}),\varphi (Tx_{n(k)}-1) \bigr) \bigr) \\ &\leqslant \phi (\gamma \bigl(d(x_{m(k)-1},x_{n(k)-1})+LM(x_{m(k)-1},x_{n(k)-1}), \\ &\quad \varphi (x_{m(k)-1}),\varphi (x_{n(k)-1}) \bigr). \end{aligned}$$

By (3) and (9) we have

$$ \lim_{n\rightarrow \infty} d(x_{m(k)-1},x_{n(k)-1})+LM(x_{m(k)-1},x_{n(k)-1})= \varepsilon . $$

Similarly to the proof of Theorem 3.1, we attain that $\{x_{n}\}$ is a Cauchy sequence in the complete metric space $(X,d)$ and there exists $x^{*}\in X$ such that $x_{n}\rightarrow x^{*}$ as $n\rightarrow \infty$ and $\varphi (x^{*})=0$.

Finally, we claim that $x^{*}=Tx^{*}$. If there exists $n_{0}$ such that for $n\geq n_{0}$, $x_{n}=Tx^{*}$ still holds, then $Tx^{*}=\lim_{n\rightarrow \infty} x_{n}=x^{*}$, that is, $x^{*}=Tx^{*}$. On the contrary, set $x=x_{n}$ and $y=x^{*}$ in (14). Then

$$ \psi \bigl(\gamma \bigl(d \bigl(Tx_{n},Tx^{*} \bigr), \varphi (Tx_{n}),\varphi \bigl(Tx^{*} \bigr) \bigr) \bigr) \leqslant \phi \bigl(\gamma \bigl(d \bigl(x_{n},x^{*} \bigr)+LM \bigl(x_{n},x^{*} \bigr) \bigr),\varphi (x_{n}),0 \bigr)). $$

By (i) and (ii) it follows that

$$\begin{aligned} d \bigl(x_{n+1},Tx^{*} \bigr)&\leqslant \gamma \bigl(d \bigl(Tx_{n},Tx^{*} \bigr),\varphi (Tx_{n}), \varphi \bigl(Tx^{*} \bigr) \bigr) \\ &< \gamma \bigl(d \bigl(x_{n},x^{*} \bigr)+LM \bigl(x_{n},x^{*} \bigr),\varphi (x_{n}),0 \bigr). \end{aligned}$$

(15)

Taking the limits on both sides of (15), we get

$$ d \bigl(x^{*},Tx^{*} \bigr)\leqslant 0, $$

which is a contradiction. So $Tx^{*}=x^{*}$.

Finally, we claim that $x^{*}=y$ for all $x^{*},y\in F(T)$. If $x^{*}\neq y$, then letting $x=x^{*}$ in (2), by (ii) we get that

$$\begin{aligned} \psi \bigl(\gamma \bigl(d \bigl(x^{*},y \bigr),0,0 \bigr) \bigr)&= \psi \bigl(\gamma \bigl(d \bigl(Tx^{*},Ty \bigr),\varphi \bigl(Tx^{*} \bigr), \varphi (Ty) \bigr) \bigr) \\ &\leqslant \phi \bigl(\gamma \bigl(d \bigl(x^{*},y \bigr)+LM \bigl(x^{*},y \bigr),\varphi \bigl(x^{*} \bigr), \varphi (y) \bigr) \bigr) \\ &=\phi \bigl(\gamma \bigl(d \bigl(x^{*},y \bigr),0,0 \bigr) \bigr) \\ &< \psi \bigl(\gamma \bigl(d \bigl(x^{*},y \bigr),0,0 \bigr) \bigr). \end{aligned}$$

This is a contraction, so $x^{*}=y$. Hence T is a φ-Picard mapping. □

Example 3.3

Let $X=[0,1]$ with the standard metric $d(x,y)=|x-y|$. It is obvious that $(X,d)$ is complete metric space. Define the mapping $T:X\rightarrow X$ by $Tx=\frac{x}{2}$. If $\gamma (x,y,z)=\max \{x,y,z\}$, $\varphi (x)=x$, $\psi (t)=t$, and $\phi (t)=\frac{t}{2}$, then (14) holds. Indeed, $\max \{\frac{|x-y|}{2},\frac{x}{2},\frac{y}{2}\}\leqslant \frac{1}{2}\max \{|x-y|,x,y\}\leqslant \frac{1}{2}\max \{|x-y|+LM(x,y),x,y \}$. So by Theorem 3.3T has a unique fixed point $x=0$, and $\varphi (0)=0$.

Corollary 3.9

Let $(X,d)$ be a complete metric space, and let $T:X\rightarrow X$. If T satisfies

$$\begin{aligned} \gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr)>0\quad \Rightarrow \quad &\psi \bigl( \gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr) \bigr) \\ &\quad \leqslant \psi \bigl(\gamma \bigl(d(x,y)+LM(x,y),\varphi (x),\varphi (y) \bigr) \bigr)- \tau , \end{aligned}$$

where $M(x,y)=\min \{d(y,Ty),d(y,Tx)\}$, $\tau >0$, φ is lower sem-continuous, and $\psi :{\mathbb{R}}_{+}\to \mathbb{R}$ is nondecreasing, then T is a φ-Picard mapping.

Proof

The result follows by letting $\phi (t)=\psi (t)-\tau$ by Theorem 3.3. □

Corollary 3.10

Let $(X,d)$ be a complete metric space, and let $T:X\rightarrow X$. If there exists $k\in (0,1)$ such that T satisfies

$$\begin{aligned} \gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr)>0\quad \Rightarrow \quad &\psi \bigl( \gamma \bigl(d(Tx,Ty),\varphi (Tx),\varphi (Ty) \bigr) \bigr) \\ &\quad \leqslant \psi \bigl(\gamma \bigl(d(x,y)+LM(x,y),\varphi (x),\varphi (y) \bigr) \bigr)^{k}, \end{aligned}$$

where $M(x,y)=\min \{d(y,Ty),d(y,Tx)\}$, φ is lower semicontinuous, and $\psi :{\mathbb{R}}_{+}\to \mathbb{R}$ is nondecreasing, then T is a φ-Picard mapping.

Proof

Take $\phi (t)=\psi (t)^{k}$ in Theorem 3.3. □

4 φ-fixed disc results

In this section, based on the results of Sect. 3, we obtain some new φ-fixed disc results on metric spaces by taking $\gamma (a,b,c)=a+b+c$, $\gamma (a,b,c)=\max \{a,b\}+c$, or $\gamma (a,b,c)=\max \{a,b,c\}$.

Definition 4.1

Let $(X,d)$ be a metric space. A mapping $T:X\rightarrow X$ is called $(\psi ,\varphi ,\phi )_{x_{0}}$ type 1 contraction if there exist ψ and ϕ satisfying (i) and (ii) of Theorem 2.4 such that for all $x\in X$ with $d(x,Tx)+\varphi (x)+\varphi (Tx)>0$, T satisfies

$$ \psi \bigl(d(x,Tx)+\varphi (x)+\varphi (Tx) \bigr)\leqslant \phi \bigl(d(x,x_{0})+ \varphi (x)+\varphi (x_{0}) \bigr), $$

(16)

where $x_{0}\in X$.

Theorem 4.1

Let $(X,d)$ be a metric space, and let $T:X\rightarrow X$ be a $(\psi ,\varphi ,\phi )_{x_{0}}$ type 1 contraction with the point $x_{0}\in X$ and the number r defined in Definition 2.4. If $\varphi (x_{0})=d(x_{0},Tx_{0})=0$ and $\varphi (x)\leqslant d(x,Tx)$ for all $x\in D_{x_{0},r}$, then $D_{x_{0},r}$ is a φ-fixed disc of T.

Proof

Firstly, we show that $x=Tx$ for all $x\in D_{x_{0},r}$.

Case 1: if $r=0$. $D_{x_{0},r}=\{x_{0}\}$, then it is easy to see that $D_{x_{0},r}$ is a φ-fixed disc of T.

Case 2: if $r>0$, then assume that $x\neq Tx$ for all $x\in D_{x_{0},r}$. By (16), the definition of r, (ii), $\varphi (x_{0})=0$, and $x_{0}=Tx_{0}$ we get

$$\begin{aligned} \psi \bigl(d(x,Tx)+\varphi (x)+\varphi (Tx) \bigr)&\leqslant \phi \bigl(d(x,x_{0})+ \varphi (x)+\varphi (x_{0}) \bigr) \\ &=\phi \bigl(d(x,x_{0})+\varphi (x)+\varphi (x_{0}) \bigr) \\ &=\phi \bigl(d(x,x_{0})+\varphi (x) \bigr) \\ &< \psi \bigl(d(x,x_{0})+\varphi (x) \bigr) \\ &\leqslant \psi \bigl(r+\varphi (x) \bigr) \\ &\leqslant \psi \bigl(d(x,Tx)+\varphi (x) \bigr). \end{aligned}$$

By (i) and this inequality we can deduce that

$$ d(x,Tx)+\varphi (x)+\varphi (Tx)< d(x,Tx)+\varphi (x). $$

This is a contradiction. So $x=Tx$. Since $\varphi (x)\leqslant d(x,Tx)$ for all $x\in D_{x_{0},r}$, we have

$$ \varphi (x)\leqslant d(x,Tx)=0. $$

So $\varphi (x)=0$, that i, $D_{x_{0},r}$ is a φ-fixed disc of T. □

Corollary 4.1

Let $(X,d)$ be a metric space, and let $T:X\rightarrow X$ be a $(\psi ,\varphi ,\phi )_{x_{0}}$ type 1 contraction with a point $x_{0}\in X$ and the number r defined in Definition 2.4. If $\varphi (x_{0})=d(x_{0},Tx_{0})=0$ and $\varphi (x)\leqslant d(x,Tx)$ for all $x\in C_{x_{0},r}$, then the circle $C_{x_{0},r}$ is a φ-fixed circle of T.

Proof

Similar to that of Theorem 4.1. □

Remark 4.1

Replace $d(x,x_{0})$ in Theorem 4.1 and Corollary 4.1 with one of the following:

$$\begin{aligned}& Q_{1}(x,x_{0})=\max \biggl\{ d(x,x_{0}),d(x,Tx),d(x_{0},Tx_{0}), \frac{d(x,Tx_{0})+d(Tx,x_{0})}{2} \biggr\} ; \\& Q_{2}(x,x_{0})=a_{1}d(x,x_{0})+a_{2}d(x,Tx)+a_{3}d(x_{0},Tx_{0})+a_{4}d(x,Tx_{0})+a_{5}d(Tx,x_{0}), \\& \quad \text{where }a_{i}\in [0,1]\text{ with }\sum _{i=0}^{5}a_{i}=1; \\& \begin{aligned} Q_{3}(x,x_{0})={}&\max \biggl\{ d(x,x_{0}),vd(x,Tx)+(1-v)d(x_{0},Tx_{0}),(1-v)d(x,Tx)+vd(x_{0},Tx_{0}), \\ &{}\frac{d(x,Tx_{0})+d(Tx,x_{0})}{2} \biggr\} , \end{aligned} \end{aligned}$$

where $x\in X, v,\in [0,1]$. If we remove the condition $d(x_{0},Tx_{0})=0$ and add the condition $d(x_{0},Tx)\leqslant r$ for all $x\in C_{x_{0},r}$ (resp., $D(x_{0},r)$), then the results still hold.

Example 4.1

Let $X=\{-6,-2,0,1,2,5\}$ be endowed with the usual metric $d(x,y)=|x-y|$. Consider the self-mapping $T:X\rightarrow X$ defined by

$$ T(x)=\textstyle\begin{cases} x& \text{if } x\neq 5, \\ 1& \text{otherwise}, \end{cases} $$

and the function $\varphi :X\rightarrow [0,\infty )$ defined by

$$ \varphi (x)=\textstyle\begin{cases} 0& \text{if } x\in \{-6,1,5\}, \\ x^{3}-4x& \text{otherwise}. \end{cases} $$

Then we have $r=4$, $F(T)=X-\{5\}$, $Z_{\varphi}=X$, and $F(T)\cap Z_{\varphi}=\{-6,-2,0,1,2\}$. Now we show that T is a $(\psi ,\varphi ,\phi )_{x_{0}}$ type 1 contraction with $x_{0}=-2$, $\psi (t)=t$, and $\phi (t)=\frac{4}{7}t$. Indeed, when $x=5$, we obtain

$$ d \bigl(5,T(5) \bigr)+\varphi (5)+\varphi \bigl(T(5) \bigr)=4\leqslant \frac{4}{7} \bigl(d(5,-2)+ \varphi (5)+\varphi (-2) \bigr). $$

Clearly, all conditions of Theorem 4.1 and Corollary 4.1 are satisfied by T. We get that $C_{-2,4}=\{-6,2\}$ is a φ-fixed circle of T and $D_{-2,4}=X-\{5\}$ is a φ-fixed disc of T.

Corollary 4.2

Let $(X,d)$ be a metric space, and let the number r be as in Definition 2.4. If there exist ψ and ϕ satisfying (i) and (ii) of Theorem 2.4such that T satisfies

$$ d(x,Tx)>0\quad \Rightarrow\quad \psi \bigl(d(x,Tx) \bigr)\leqslant \phi \bigl(d(x,x_{0}) \bigr) $$

and $d(x_{0},Tx_{0})=0$, where $x_{0}\in X$, then the circle $C_{x_{0},r}$ (resp., the disc $D(x_{0},r)$) is a fixed circle (resp., disc) of T.

Proof

Take $\varphi (t)=0$ in Theorem 4.1 and Corollary 4.1. □

Corollary 4.3

Let $(X,d)$ be a metric space, and let the number r be as in Definition 2.4. If there exist μ with $(\psi 1)$ and $\nu :{\mathbb{R}}_{+}\to \mathbb{R}$ with $\nu (t)< t$ such that T satisfies

$$ d(x,Tx)>0\quad \Rightarrow\quad \mu \bigl(d(x,Tx) \bigr)\leqslant \nu \bigl(\mu \bigl(d(x,x_{0}) \bigr) \bigr) $$

and $d(x_{0},Tx_{0})=0$, where $x_{0}\in X$, then the circle $C_{x_{0},r}$ (resp., the disc $D(x_{0},r)$) is a fixed circle (resp., disc) of T.

Proof

Take $\psi (t)=\mu (t)$ and $\phi (t)=\nu (\mu (t))$ in Corollary 4.2. □

Remark 4.2

Corollary 4.3 shows that the monotonic increasing property of ν in Corollary 3.8 in [32] can be removed, and the condition $d(x_{0},Tx)=r$ for all $x\in C_{x_{0},r}$ can be weakened to $d(x_{0},Tx_{0})=0$.

Corollary 4.4

Let $(X,d)$ be a metric space, and let the number r be as in Definition 2.4. If there exist $\tau >0$ and nondecreasing function $\psi :{\mathbb{R}}_{+}\to \mathbb{R}$ such that $T:X\rightarrow X$ satisfies

$$ \psi \bigl(d(x,Tx)+\varphi (x)+\varphi (Tx) \bigr)\leqslant \psi \bigl(d(x,x_{0})+ \varphi (x)+\varphi (x_{0}) \bigr)- \tau $$

for all $x,y\in X$ with $d(x,Tx)+\varphi (x)+\varphi (Tx)>0$, $\varphi (x_{0})=d(x_{0},Tx_{0})=0$, and $\varphi (x)\leqslant d(x,Tx)$ for all $x\in C_{x_{0},r}$ (resp., $D(x_{0},r)$), where $x_{0}\in X$, then the circle $C_{x_{0},r}$ (resp., the disc $D(x_{0},r)$) is a φ-fixed circle (resp., disc) of T.

Proof

Take $\phi (t)=\psi (t)-\tau$ in Corollary 4.2. □

Corollary 4.5

Let $(X,d)$ be a metric space, and let the number r be as in Definition 2.4. If there exist $\tau >0$ and nondecreasing function $\psi :{\mathbb{R}}_{+}\to \mathbb{R}$ such that $T:X\rightarrow X$ satisfies

$$ \psi \bigl(d(x,Tx) \bigr)\leqslant \psi \bigl(d(x,x_{0}) \bigr)- \tau $$

for all $x,y\in X$ with $d(x,Tx)>0$ and $d(x_{0},Tx_{0})=0$, where $x_{0}\in X$, then the circle $C_{x_{0},r}$ (resp., the disc $D(x_{0},r)$) is a fixed circle (resp., disc) of T.

Proof

Take $\varphi (t)=0$ in Corollary 4.4. □

Remark 4.3

Corollary 4.5, produced by only the nondecreasing property of ψ, is an improvement of Theorem 3.4 in [28].

Corollary 4.6

Let $(X,d)$ be a metric space, and let the number r be as in Definition 2.4. If there exist $k\in [0,1)$ and nondecreasing function $\psi :{\mathbb{R}}_{+}\to \mathbb{R}$ such that $T:X\rightarrow X$ satisfies

$$ \psi \bigl(d(x,Tx)+\varphi (x)+\varphi (Tx) \bigr)\leqslant \psi \bigl(d(x,x_{0})+ \varphi (x)+\varphi (x_{0}) \bigr)^{k} $$

for all $x,y\in X$ with $d(x,Tx)+\varphi (x)+\varphi (Tx)>0$, $\varphi (x_{0})=d(x_{0},Tx_{0})=0$, and $\varphi (x)\leqslant d(x,Tx)$ for all $x\in C_{x_{0},r}$ (resp., $D(x_{0},r)$), where $x_{0}\in X$, then the circle $C_{x_{0},r}$ (resp., the disc $D(x_{0},r)$) is a φ-fixed circle (resp., disc) of T.

Proof

Take $\phi (t)=\psi (t)^{k}$ in Theorem 4.1. □

Corollary 4.7

Let $(X,d)$ be a metric space, and let the number r be as in Definition 2.4. If there exist $k\in [0,1)$ and nondecreasing function $\psi :{\mathbb{R}}_{+}\to \mathbb{R}$ such that $T:X\rightarrow X$ satisfies

$$ \psi \bigl(d(x,Tx) \bigr)\leqslant \psi \bigl(d(x,x_{0}) \bigr)^{k} $$

for all $x,y\in X$ with $d(x,Tx)>0$ and $d(x_{0},Tx_{0})=0$, where $x_{0}\in X$, then the circle $C_{x_{0},r}$ (resp. the disc $D(x_{0},r)$) is a fixed circle (resp. disc) of T.

Proof

Take $\varphi (t)=0$ in Corollary 4.6. □

Remark 4.4

Corollary 4.7 shows that the condition $d(x_{0},Tx)=r$ for all $x\in C_{x_{0},r}$ of Corollary 3.8 in [26] can be weakened to $d(x_{0},Tx_{0})=0$.

Corollary 4.8

([34])

Let $(X,d)$ be a metric space, and let the number r be as in Definition 2.4. If there exists $0< k<1$ such that $T:X\rightarrow X$ satisfies

$$ d(x,Tx)>0\quad \Rightarrow\quad d(x,Tx)+\varphi (x)+\varphi (Tx)\leqslant k \bigl(d(x,x_{0})+ \varphi (x)+\varphi (x_{0}) \bigr) $$

(17)

and $\varphi (x)\leqslant d(x,Tx)$ for all $x\in C_{x_{0},r}$ (resp., $D(x_{0},r)$), where $x_{0}\in X$ with $\varphi (x_{0})=0$, then the circle $C_{x_{0},r}$ (resp., the disc $D(x_{0},r)$) is a φ-fixed circle (resp., disc) of T.

Proof

If $x_{0}\neq Tx_{0}$, then letting $x=x_{0}$ in (17), we obtain

$$ d(x_{0},Tx_{0})+\varphi (x_{0})+\varphi (Tx_{0})\leqslant k \bigl(d(x_{0},x_{0})+ \varphi (x_{0})+\varphi (x_{0}) \bigr)=0, $$

that is, $d(x_{0},Tx_{0})=0$. The result follows by letting $\psi (t)=t$ and $\phi (t)=kt$ in Theorem 4.1. □

Definition 4.2

Let $(X,d)$ be a metric space. A mapping $T:X\rightarrow X$ is called $(\psi ,\varphi ,\phi )_{x_{0}}$ type 2 contraction if there exist ψ and ϕ satisfying (i) and (ii) of Theorem 2.4 such that T satisfies

$$ \psi \bigl(\max \bigl\{ d(x,Tx),\varphi (x) \bigr\} +\varphi (Tx) \bigr)\leqslant \phi \bigl(\max \bigl\{ d(x,x_{0}),\varphi (x) \bigr\} + \varphi (x_{0}) \bigr) $$

(18)

for all $x\in X$ with $\max \{d(x,Tx),\varphi (x)\}+\varphi (Tx)>0$, where $x_{0}\in X$.

Theorem 4.2

Let $(X,d)$ be a metric space, and let $T:X\rightarrow X$ be a $(\psi ,\varphi ,\phi )_{x_{0}}$ type 2 contraction with the point $x_{0}\in X$ and the number r defined in Definition 2.4. If $\varphi (x_{0})=d(x_{0},Tx_{0})=0$ and $\varphi (x)\leqslant d(x,Tx)$ for all $x\in D_{x_{0},r}$, then $D_{x_{0},r}$ is a φ-fixed disc of T.

Proof

Firstly, we show that $x=Tx$ for all $x\in D_{x_{0},r}$.

Case 1: if $r=0$, then $D_{x_{0},r}=\{x_{0}\}$, and it is easy to see that $D_{x_{0},r}$ is a φ-fixed disc of T.

Case 2: if $r>0$, then assume that $x\neq Tx$ for all $x\in D_{x_{0},r}$. By (18), the definition of r, (ii), $\varphi (x_{0})=0$, and $x_{0}=Tx_{0}$, we have

$$\begin{aligned} \psi \bigl(\max \bigl\{ d(x,Tx),\varphi (x) \bigr\} +\varphi (Tx) \bigr)&\leqslant \phi \bigl( \max \bigl\{ d(x,x_{0}),\varphi (x) \bigr\} +\varphi (x_{0}) \bigr) \\ &=\phi \bigl(\max \bigl\{ d(x,x_{0}),\varphi (x) \bigr\} \bigr) \\ &< \psi \bigl(\max \bigl\{ d(x,x_{0}),\varphi (x) \bigr\} \bigr) \\ &\leqslant \psi \bigl(\max \bigl\{ r,\varphi (x) \bigr\} \bigr) \\ &\leqslant \psi \bigl(\max \bigl\{ d(x,Tx),\varphi (x) \bigr\} \bigr). \end{aligned}$$

By (i) and thus inequality we deduce that

$$ \max \bigl\{ d(x,Tx),\varphi (x) \bigr\} +\varphi (Tx)< \max \bigl\{ d(x,Tx),\varphi (x) \bigr\} . $$

This is a contradiction. So $x=Tx$. Since $\varphi (x)\leqslant d(x,Tx)$ for all $x\in D_{x_{0},r}$, we have

$$ \varphi (x)\leqslant d(x,Tx)=0. $$

So $\varphi (x)=0$, that is, $D_{x_{0},r}$ is a φ-fixed disc of T. □

Corollary 4.9

Let $(X,d)$ be a metric space, and let $T:X\rightarrow X$ be a $(\psi ,\varphi ,\phi )_{x_{0}}$ type 2 contraction with the point $x_{0}\in X$ and the number r as in Definition 2.4. If $\varphi (x_{0})=d(x_{0},Tx_{0})=0$ and $\varphi (x)\leqslant d(x,Tx)$ for all $x\in C_{x_{0},r}$, then the circle $C_{x_{0},r}$ is a φ-fixed circle of T.

Proof

Similar to that of Theorem 4.2. □

Remark 4.5

Replace $d(x,x_{0})$ in Theorem 4.2 and Corollary 4.9 with one of the following:

$$\begin{aligned}& Q_{1}(x,x_{0})=\max \biggl\{ d(x,x_{0}),d(x,Tx),d(x_{0},Tx_{0}), \frac{d(x,Tx_{0})+d(Tx,x_{0})}{2} \biggr\} ; \\& Q_{2}(x,x_{0})=a_{1}d(x,x_{0})+a_{2}d(x,Tx)+a_{3}d(x_{0},Tx_{0})+a_{4}d(x,Tx_{0})+a_{5}d(Tx,x_{0}), \\& \quad \text{where }a_{i}\in [0,1]\text{ with }\sum _{i=0}^{5}a_{i}=1; \\& \begin{aligned} Q_{3}(x,x_{0})={}&\max \biggl\{ d(x,x_{0}),vd(x,Tx)+(1-v)d(x_{0},Tx_{0}),(1-v)d(x,Tx)+vd(x_{0},Tx_{0}), \\ &{}\frac{d(x,Tx_{0})+d(Tx,x_{0})}{2} \biggr\} , \end{aligned} \end{aligned}$$

where $x\in X, v,\in [0,1]$. If we remove the condition $d(x_{0},Tx_{0})=0$ and add the condition $d(x_{0},Tx)\leqslant r$ for all $x\in C_{x_{0},r}$ (resp., $D(x_{0},r)$), then their results still hold.

Example 4.2

Let $X=\{-6,-3,-2,0,1,2,3,5\}$ be endowed with the usual metric $d(x,y)=|x-y|$. Consider the self-mapping $T:X\rightarrow X$ and the function $\varphi :X\rightarrow [0,\infty )$ defined by

$$ T(x)=\textstyle\begin{cases} x& \text{if } x\neq 5, \\ 2& \text{otherwise}. \end{cases} $$

and

$$ \varphi (x)=\textstyle\begin{cases} 0& \text{if } x\in \{-6,1,5\}, \\ x^{5}-13x^{3}+36x& \text{otherwise}. \end{cases} $$

We have $r=3$, $F(T)=X-\{5\}$, $Z_{\varphi}=\{-6,-3,-2,0,1,2,3,5\}$, and $F(T)\cap Z_{\varphi}=\{-6,-3,-2,0, 1,2,3\}$. Now we show that T is a type 1 $(\gamma ,\psi ,\varphi ,\phi )_{x_{0}}$ contraction with $x_{0}=0$, $\psi (t)=t$, and $\phi (t)=\frac{9}{10}t$. Indeed, when $x=5$, we obtain that

$$ \max \bigl\{ d \bigl(5,T(5) \bigr),\varphi (5) \bigr\} +\varphi \bigl(T(5) \bigr)=3 \leqslant \frac{9}{10} \bigl( \max \bigl\{ d(5,0),\varphi (5) \bigr\} + \varphi (0) \bigr). $$

Clearly, the conditions of Theorem 4.2 and Corollary 4.9 are satisfied by T, so we get that $C_{0,3}=\{-3,3\}$ is a φ-fixed circle of T and $D_{0,3}=\{-3,-2,0,1,2,3\}$ is a φ-fixed disc of T.

Corollary 4.10

Let $(X,d)$ be a metric space, and let the number r be as in Definition 2.4. If there exist $\tau >0$ and nondecreasing function $\psi :{\mathbb{R}}_{+}\to \mathbb{R}$ such that $T:X\rightarrow X$ satisfies

$$ \psi \bigl(\max \bigl\{ d(x,Tx),\varphi (x) \bigr\} +\varphi (Tx) \bigr)\leqslant \psi \bigl(\max \bigl\{ d(x,x_{0}),\varphi (x) \bigr\} +\varphi (x_{0}) \bigr)-\tau $$

for all $x,y\in X$ with $\max \{d(x,Tx),\varphi (x)\}+\varphi (Tx)>0$ and $\varphi (x)\leqslant d(x,Tx)$ for all $x\in C_{x_{0},r}$ (resp., $D(x_{0},r)$), where $x_{0}\in X$ with $\varphi (x_{0})=d(x_{0},Tx_{0})=0$, then the circle $C_{x_{0},r}$ (resp., the disc $D(x_{0},r)$) is a φ-fixed circle (resp., disc) of T.

Proof

Take $\phi (t)=\psi (t)-\tau$ in Theorem 4.2 and Corollary 4.9. □

Corollary 4.11

Let $(X,d)$ be a metric space, and let the number r be as in Definition 2.4. If there exist $k\in [0,1)$ and nondecreasing function $\psi :{\mathbb{R}}_{+}\to \mathbb{R}$ such that $T:X\rightarrow X$ satisfies

$$ \psi \bigl(\max \bigl\{ d(x,Tx),\varphi (x) \bigr\} +\varphi (Tx) \bigr)\leqslant \psi \bigl(\max \bigl\{ d(x,x_{0}),\varphi (x) \bigr\} +\varphi (x_{0}) \bigr)^{k} $$

for all $x,y\in X$ with $d(x,Tx)+\varphi (x)+\varphi (Tx)>0$ and $\varphi (x)\leqslant d(x,Tx)$ for all $x\in C_{x_{0},r}$ (resp., $D(x_{0},r)$), where $x_{0}\in X$ with $\varphi (x_{0})=d(x_{0},Tx_{0})=0$, then the circle $C_{x_{0},r}$ (resp., the disc $D(x_{0},r)$) is a φ-fixed circle (resp., disc) of T.

Proof

Take $\phi (t)=\psi (t)^{k}$ in Theorem 4.2 and Corollary 4.9. □

Corollary 4.12

([34])

Let $(X,d)$ be a metric space, and let the number r be as in Definition 2.4. If there exist $0< k<1$ such that $T:X\rightarrow X$ satisfies

$$ \max \bigl\{ d(x,Tx),\varphi (x) \bigr\} +\varphi (Tx)\leqslant k \bigl(\max \bigl\{ d(x,x_{0}), \varphi (x) \bigr\} +\varphi (x_{0}) \bigr) $$

and $\varphi (x)\leqslant d(x,Tx)$ for all $x\in C_{x_{0},r}$ (resp., $D(x_{0},r)$), where $x_{0}\in X$ with $\varphi (x_{0})=0$, then the circle $C_{x_{0},r}$ (resp., the disc $D(x_{0},r)$) is a φ-fixed circle (resp., disc of T).

Proof

The result follows by Theorem 4.2 and Corollary 4.9 similarly to the proof of Corollary 4.8. □

Definition 4.3

Let $(X,d)$ be a metric space. A mapping $T:X\rightarrow X$ is called a $(\psi ,\varphi ,\phi )_{x_{0}}$ type 3 contraction if there exist ψ and ϕ satisfying (i) and (ii) of Theorem 2.4 such that

$$ \psi \bigl(\max \bigl\{ d(x,Tx),\varphi (x),\varphi (Tx) \bigr\} \bigr)\leqslant \phi \bigl(\max \bigl\{ d(x,x_{0}),\varphi (x), \varphi (x_{0}) \bigr\} \bigr), $$

(19)

for all $x\in X$ with $\max \{d(x,Tx),\varphi (x),\varphi (Tx)\}>0$, where $x_{0}\in X$.

Theorem 4.3

Let $(X,d)$ be a metric space, and let $T:X\rightarrow X$ be a $(\psi ,\varphi ,\phi )_{x_{0}}$ type 3 contraction with the point $x_{0}\in X$ and the number r as in Definition 2.4. If $\varphi (x_{0})=d(x_{0},Tx_{0})=0$ and $\varphi (x)\leqslant d(x,Tx)$ for all $x\in D_{x_{0},r}$, then the circle $D_{x_{0},r}$ is a φ-fixed disc of T.

Proof

Firstly, we show that $x=Tx$ for all $x\in D_{x_{0},r}$.

Case 1: if $r=0$, then $D_{x_{0},r}=\{x_{0}\}$, and it is easy to see that $D_{x_{0},r}$ is a φ-fixed disc of T.

Case 2: if $r>0$, the assume that $x\neq Tx$ for all $x\in D_{x_{0},r}$. By (19), the definition of r, $(\phi 1)$, $\varphi (x_{0})=0$, and $x_{0}=Tx_{0}$, we have

$$\begin{aligned} \psi \bigl(\max \bigl\{ d(x,Tx),\varphi (x),\varphi (Tx) \bigr\} \bigr)&\leqslant \phi \bigl( \max \bigl\{ d(x,x_{0}),\varphi (x),\varphi (x_{0}) \bigr\} \bigr) \\ &=\phi \bigl(\max \bigl\{ d(x,x_{0}),\varphi (x) \bigr\} \bigr) \\ &< \psi \bigl(\max \bigl\{ d(x,x_{0}),\varphi (x) \bigr\} \bigr) \\ &\leqslant \psi \bigl(\max \bigl\{ r,\varphi (x) \bigr\} \bigr) \\ &\leqslant \psi \bigl(\max \bigl\{ d(x,Tx),\varphi (x) \bigr\} \bigr) \\ &\leqslant \psi \bigl(d(x,Tx) \bigr). \end{aligned}$$

By $(\psi 1)$ and this inequality we deduce that

$$ \max \bigl\{ d(x,Tx),\varphi (x) \bigr\} \leqslant \max \bigl\{ d(x,Tx),\varphi (x), \varphi (Tx) \bigr\} < d(x,Tx). $$

This is a contradiction. So $x=Tx$. Since $\varphi (x)\leqslant d(x,Tx)$ for all $x\in D_{x_{0},r}$, we have

$$ \varphi (x)\leqslant d(x,Tx)=0. $$

So $\varphi (x)=0$, that is, $D_{x_{0},r}$ is a φ-fixed disc of T. □

Corollary 4.13

Let $(X,d)$ be a metric space, and let $T:X\rightarrow X$ be a $(\psi ,\varphi ,\phi )_{x_{0}}$ type 3 contraction with the point $x_{0}\in X$ and the number r as in Definition 2.4. If $\varphi (x_{0})=d(x_{0},Tx_{0})=0$ and $\varphi (x)\leqslant d(x,Tx)$ for all $x\in C_{x_{0},r}$, then the circle $C_{x_{0},r}$ is a φ-fixed circle of T.

Proof

Similar to that of Theorem 4.3. □

Remark 4.6

Replace $d(x,x_{0})$ in Theorem 4.3 and Corollary 4.13 with one of the following:

$$\begin{aligned}& Q_{1}(x,x_{0})=\max \biggl\{ d(x,x_{0}),d(x,Tx),d(x_{0},Tx_{0}), \frac{d(x,Tx_{0})+d(Tx,x_{0})}{2}\biggr\} ; \\& Q_{2}(x,x_{0})=a_{1}d(x,x_{0})+a_{2}d(x,Tx)+a_{3}d(x_{0},Tx_{0})+a_{4}d(x,Tx_{0})+a_{5}d(Tx,x_{0}), \\& \quad \text{where }a_{i}\in [0,1]\text{ with }\sum _{i=0}^{5}a_{i}=1; \\& \begin{aligned} Q_{3}(x,x_{0})={}&\max \biggl\{ d(x,x_{0}),vd(x,Tx)+(1-v)d(x_{0},Tx_{0}),(1-v)d(x,Tx)+vd(x_{0},Tx_{0}), \\ &{}\frac{d(x,Tx_{0})+d(Tx,x_{0})}{2}\biggr\} , \end{aligned} \end{aligned}$$

where $x\in X, v,\in [0,1]$. If we remove the condition $d(x_{0},Tx_{0})=0$ and add the condition $d(x_{0},Tx)\leqslant r$, for all $x\in C_{x_{0},r}$ (resp., $D(x_{0},r)$), then their results still hold.

Example 4.3

In Examples 4.1 and 4.2, clearly, the conditions of Theorem 4.3 and Corollary 4.13 are satisfied by T, the circles $C_{0,3}$ and $C_{-2,4}$ are φ-fixed circles of T, and the discs $D_{0,3}$ and $D_{-2,4}$ are φ-fixed discs of T.

Remark 4.7

By Example 4.3 note that the φ-fixed circle (disc) of T is not unique.

Corollary 4.14

Let $(X,d)$ be a metric space, and let the number r be as in Definition 2.4. If there exist $\tau >0$ and a nondecreasing function $\psi :{\mathbb{R}}_{+}\to \mathbb{R}$ such that $T:X\rightarrow X$ satisfies

$$ \psi \bigl(\max \bigl\{ d(x,Tx),\varphi (x),\varphi (Tx) \bigr\} \bigr)\leqslant \psi \bigl(\max \bigl\{ d(x,x_{0}),\varphi (x),\varphi (x_{0}) \bigr\} \bigr)-\tau $$

for all $\max \{d(x,Tx),\varphi (x),\varphi (Tx)\}>0$ and $\varphi (x)\leqslant d(x,Tx)$ for all $x\in C_{x_{0},r}$ (resp., $D(x_{0},r)$), where $x_{0}\in X$ with $\varphi (x_{0})=d(x_{0},Tx_{0})=0$, then the circle $C_{x_{0},r}$ (resp., the disc $D(x_{0},r)$) is a φ-fixed circle (resp., disc) of T.

Proof

Similar to that of Corollary 4.10. □

Corollary 4.15

Let $(X,d)$ be a metric space, and let the number r be as in Definition 2.4. If there exist $k\in [0,1)$ and a nondecreasing function $\psi :{\mathbb{R}}_{+}\to \mathbb{R}$ such that $T:X\rightarrow X$ satisfies

$$ \psi \bigl(\max \bigl\{ d(x,Tx),\varphi (x),\varphi (Tx) \bigr\} \bigr)\leqslant \psi \bigl(\max \bigl\{ d(x,x_{0}),\varphi (x),\varphi (x_{0}) \bigr\} \bigr)^{k} $$

for all $x,y\in X$ with $\max \{d(x,Tx),\varphi (x),\varphi (Tx)\}>0$ and $\varphi (x)\leqslant d(x,Tx)$ for all $x\in C_{x_{0},r}$ (resp., $D(x_{0},r)$), where $x_{0}\in X$ with $\varphi (x_{0})=d(x_{0},Tx_{0})=0$, then the circle $C_{x_{0},r}$ (resp., the disc $D(x_{0},r)$) is a φ-fixed circle (resp., disc) of T.

Proof

Similar to that of Corollary 4.11. □

Corollary 4.16

([34])

Let $(X,d)$ be a metric space, and let the number r be as in Definition 2.4. If there exists $0< k<1$ such that $T:X\rightarrow X$ satisfies

$$ d(x,Tx)>0\quad \Rightarrow\quad \max \bigl\{ d(x,Tx),\varphi (x),\varphi (Tx) \bigr\} \leqslant k\max \bigl\{ d(x,x_{0}),\varphi (x),\varphi (x_{0}) \bigr\} $$

and $\varphi (x)\leqslant d(x,Tx)$ for all $x\in C_{x_{0},r}$ (resp., $D(x_{0},r)$), where $x_{0}\in X$ with $\varphi (x_{0})=0$, then the circle $C_{x_{0},r}$ (resp., the disc $D(x_{0},r)$) is a φ-fixed circle (resp., disc) of T.

Proof

Similar to that of Corollary 4.12. □

5 Applications

In this section, we consider the following nonlinear integral equation:

$$ \eta (u)=\zeta (u)+ \int ^{\beta}_{\alpha}\pi \bigl(u,v,\eta (v) \bigr)\,dv, $$

(20)

where $\alpha ,\beta \in \mathbb{R}$ with $\alpha \leqslant \beta$, $\eta \in C[\alpha ,\beta ]$ (the set of all continuous functions from $[\alpha ,\beta ]$ to $\mathbb{R}$), and $\zeta :[\alpha ,\beta ]\to \mathbb{R}$ and $\pi :{[\alpha ,\beta ]}^2\times \mathbb{R}\to \mathbb{R}$ are given continuous functions. Let $X=C[\alpha ,\beta ]$ be endowed with the standard metric $d(\eta ,\theta )=\sup_{u\in{\alpha ,\beta}} |\eta (u)- \theta (u)|$. It is obvious that $(X,d)$ is a complete metric space. Define $T:X\rightarrow X$ by

$$ T\eta (u)=\zeta (u)+ \int ^{\beta}_{\alpha}\pi \bigl(u,v,\eta (v) \bigr)\,dv\quad \text{for } u\in [\alpha ,\beta ]. $$

(21)

Clearly, the solution of (20) is equivalent to the fixed point of T in (21). Next, we will prove our result as follows.

Theorem 5.1

If there exists $\tau >0$ such that

$$ \psi \biggl( \int ^{\beta}_{\alpha} \bigl\vert \pi (v,u,\eta )-\pi (v,u,\theta ) \bigr\vert \,du \biggr) \leqslant \psi \biggl( \int ^{\beta}_{\alpha}\frac{1}{\beta -\alpha} \bigl( \vert \eta - \theta \vert \bigr)\,du \biggr)-\tau $$

for all $\eta ,\theta \in X$ and $u\in [\alpha ,\beta ]$, where $\psi :{\mathbb{R}}_{+}\to \mathbb{R}$ is nondecreasing, then the integral equation (20) has a unique solution.

Proof

Define the control functions $\gamma =\gamma _{1}$, $\varphi (t)=0$, and $\phi (t)=\psi (t)-\tau$ for $t\in X$. Then

$$\begin{aligned} \psi ( \vert T\eta -T\theta \vert &=\psi \biggl( \biggl\vert \int ^{\beta}_{\alpha}\pi \bigl(v,u,\eta (u) \bigr)\,du- \int ^{\beta}_{\alpha}\pi \bigl(v,u,\theta (u) \bigr)\,du \biggr\vert \biggr) \\ &\leqslant \psi \biggl( \int ^{\beta}_{\alpha} \bigl\vert \pi \bigl(v,u,\eta (u) \bigr)-\pi \bigl(v,u, \theta (u) \bigr) \bigr\vert \,du \biggr) \\ &\leqslant \psi \biggl( \int ^{\beta}_{\alpha}\frac{1}{\beta -\alpha} \bigl( \vert \eta - \theta \vert \bigr)\,du \biggr)-\tau \\ &\leqslant \sup_{u\in [\alpha ,\beta ]} \psi \biggl( \int ^{\beta}_{ \alpha}\frac{1}{\beta -\alpha} \bigl( \vert \eta -\theta \vert \bigr)\,du \biggr)-\tau \\ &\leqslant \psi \bigl(d(\eta -\theta ) \bigr)-\tau . \end{aligned}$$

This implies that

$$ \psi \bigl(\gamma \bigl(d(T\eta ,T\theta ) \bigr),\varphi (T\eta ),\varphi (T \theta ) \bigr) \leqslant \phi \bigl(\gamma \bigl(d(\eta ,\theta ),\varphi (\eta ), \varphi ( \theta ) \bigr) \bigr). $$

It is obvious that T satisfies the conditions Theorem 3.1. So T has a unique fixed point, that is, the integral equation (20) has a unique solution. □

6 Conclusions

In this paper, we present six novel contractions and obtain the corresponding results in a metric space. On the other hand, we improve and expand some recent results in a metric space. Next, we use some simple instances and an application to show the validity of our main results. Finally, regarding the main results of this paper, we draw some inferences. Due to the importance of the fixed point theory, we consider possible future research directions.

There are some works in the future:

(i) replace or change some conditions in our main theorems;

(ii) extend our results to another metric space, like fuzzy metric space [35], b-metric space [36], and so on;

(iii) apply our main results and techniques to solve fractional differential equations [37, 38].