Fixed point problems for generalized contractions with applications

Abstract

In this paper, we investigate the conditions on the control mappings $\psi ,\varphi :(0,\infty )\rightarrow \mathbb{R}$ that guarantee the existence of the fixed points of the mapping $T:X\rightarrow P(X)$ satisfying the following inequalities:

$$ \psi \bigl(H(Tx,Ty)\bigr)\leq \varphi \bigl(d(x,y)\bigr) \quad \forall x,y\in X, \text{provided that } H(Tx,Ty)>0, $$

and

$$ \psi \bigl(H(Tx,Ty)\bigr)\leq \varphi \bigl(A(x,y)\bigr) \quad \forall x,y\in X, \text{provided that } H(Tx,Ty)>0, $$

where $A(x,y)=\max \{ d(x,y), d(x,Tx), d(y,Ty), (d(x,Ty) +d(Tx,y))/2 \}$, and $(X, d)$ is a metric space. The obtained fixed point results improve many earlier results on the set-valued contractions. As an application, we consider the existence of the solutions of an FDE.

1 Introduction and preliminaries

The metric fixed point theory was initiated by Banach (1922) [15]. This theory has been enriched with well-known structures and metric generalizations. Recently, Chifu [20] established common fixed point theorems endowed with directed graphs in extended b-metric spaces. Ozturk [31] proved a fixed point theorem involving a simulation function and F-contraction. Ozyurt [33] presented fixed point theorems covering a comparison function and large contractions. In [32] the author proved some results on $\alpha -\varphi$ contractions in Branciari b-metric spaces. The study of F-metric spaces attracted attention of many researchers, and in this direction, several papers were published (see [11, 12, 16]). The existence of solutions of FDEs, IEs, ODEs, and PDEs was investigated by applying various known fixed point results; see [4, 6, 9, 13, 17, 30, 38, 39] for details. A useful generalization of the Banach contraction principle is an F-contraction presented by Wordowski [40]; a survey on this contraction is given in [25]. The notion of F-contraction was generalized using various structures; see [1, 25, 30] for details. Boyd and Wong [18] introduced mapping $\psi :[0,\infty )\rightarrow [0,\infty )$ that satisfy the following conditions to generalize the Banach contraction principle:

• (1) $\psi (y)< y$ for all $y>0$,

• (2) $\lim_{x\rightarrow y+}\psi (x)< y$ for all $y>0$.

Boyd and Wong [18] used such mapping to present the following result.

Theorem 1.1

Let $(X,d)$ be a complete metric space, and let $T: X\rightarrow X$ be a mapping satisfying the following contractive condition:

$$ d(Tx,Ty)\leq \psi \bigl(d(x,y)\bigr) \quad \textit{for all } x,y\in X, $$

where $\psi :[0,\infty )\rightarrow [0,\infty )$ satisfies (1)–(2). Then T admits a unique fixed point u in X, and $(T^{n}x)\rightarrow u$ for all $x\in X$.

Note that Theorem 1.1 improves the fixed point theorems given by Rakotch [35] and Browder [19]. Recently, Proinov [34] presented a generalization of F-contraction, JS-contraction [24], and many recent results by introducing $(\phi ,\psi )$-generalized contractions. Using the mappings $\psi ,\varphi :(0,\infty )\rightarrow (-\infty , \infty )$, he introduced the following contractive-type conditions:

$$\begin{aligned}& \psi \bigl(d(Tx,Ty)\bigr)\leq \varphi \bigl(d(x,y)\bigr) \\& \quad \forall x,y\in X, \text{provided that } \min \bigl\{ d(x,y), d(Tx,Ty) \bigr\} >0, \end{aligned}$$

(1.1)

and

$$\begin{aligned}& \psi \bigl(d(Tx,Ty)\bigr)\leq \varphi \bigl(A(x,y)\bigr) \\& \quad \forall x,y\in X, \text{provided that } \min \bigl\{ A(x,y), d(Tx,Ty) \bigr\} >0, \end{aligned}$$

(1.2)

where $A(x,y)=\max \{ d(x,y), d(x,Tx), d(y,Ty), (d(x,Ty) +d(Tx,y))/2 \}$ and $T:X\rightarrow X$. In this paper, we call them $(\psi ,\varphi )$-contractions.

Proinov [34] established the following fixed point results.

Theorem 1.2

Let $(X,d)$ be a complete metric space, and let $T: X\rightarrow X$ be a mapping satisfying (1.1). Suppose the mappings $\psi ,\varphi :(0,\infty )\rightarrow (-\infty , \infty )$ satisfy the following conditions:

• (i) ψ is nondecreasing;

• (ii) $\varphi (y)< \psi (y)$ for all $y>0$;

• (iii) $\lim \sup_{y\rightarrow r+}\varphi (y)<\psi (r+)$ for all $r>0$.

Then T has a unique fixed point $p\in X$, and the iterative sequence $(T^{n}x)$ converges to p for all $x\in X$.

Theorem 1.3

([34])

Let $(X,d)$ be a complete metric space, and let $T: X\rightarrow X$ be a mapping satisfying (1.1). Suppose the mappings $\psi ,\varphi :(0,\infty )\rightarrow (-\infty , \infty )$ satisfy the following conditions:

• (i) $\varphi (y)<\psi (y)$ for all $y>0$;

• (ii) $\inf_{y>\epsilon }\psi (y)>-\infty$ for all $\epsilon >0$;

• (iii) if $(\psi (y_{n}))$ and $(\varphi (y_{n}))$ are convergent sequences such that $\lim_{n\rightarrow \infty }\psi (y_{n})=\lim_{n\rightarrow \infty } \varphi (y_{n})$ and $(\psi (y_{n}))$ is strictly decreasing, then $\lim_{n\rightarrow \infty }y_{n}=0$;

• (iv) $\lim \sup_{y\rightarrow \epsilon +}\varphi (y)<\lim \inf_{y \rightarrow \epsilon }\psi (y)$ or $\lim \sup_{y\rightarrow \epsilon }\varphi (y)<\lim \inf_{y \rightarrow \epsilon +}\psi (y)$ for all $\epsilon >0$;

• (v) T has a closed graph, or $\lim \sup_{y\rightarrow 0+}\varphi (y)<\lim \inf_{y\rightarrow \epsilon }\psi (y)$ for all $\epsilon >0$.

Then T has a unique fixed point $u\in X$, and the iterative sequence $(T^{n}x)$ converges to u for all $x\in X$.

Remark 1.4

Theorems 1.2 and 1.3 also hold if we replace contractive condition (1.1) with (1.2).

The following lemma is often seen in different papers (see [18, 34]) and provides a method to prove that a sequence to be Cauchy.

Lemma 1.5

Let $(X,d)$ be a metric space, and let $\{q_{n}\}\subset X$ be a sequence such that $\lim_{n\rightarrow \infty }d(q_{n}, q_{n+1})=0$. If the sequence $\{q_{n}\}$ is not Cauchy, then there exist two subsequences $\{q_{n_{k}}\}$ and $\{q_{m_{k}}\}$ and $\epsilon >0$ such that

$$ \lim_{k\rightarrow \infty }d(q_{n_{k}+1}, q_{m_{k}+1})=\epsilon + $$

(1.3)

and

$$ \lim_{k\rightarrow \infty }d(q_{n_{k}}, q_{m_{k}})=d(q_{n_{k}+1}, q_{m_{k}})=d(q_{n_{k}}, q_{m_{k}+1})= \epsilon . $$

(1.4)

For the control mappings $\psi ,\varphi :(0,\infty )\rightarrow \mathbb{R}$, the following conditions are needed for the upcoming results:

• (i) $\inf_{x>\epsilon } \psi (x)>-\infty$ for any $\epsilon >0$.

• (ii) $\lim \inf_{x\rightarrow \epsilon +}\psi (x)>-\infty$ for any $\epsilon >0$.

• (iii) $\lim_{n\rightarrow \infty } \psi (x_{n})=-\infty$ implies $\lim_{n\rightarrow \infty } x_{n}=0$.

• (iv) $\lim_{n\rightarrow \infty } \varphi (x_{n})=0$ implies $\lim_{n \rightarrow \infty } x_{n}=0$, where $\{x_{n}\}$ is a bounded sequence.

• (v) $\lim \inf_{x\rightarrow \epsilon }\varphi (x)>0$ for all $\epsilon >0$.

• (vi) $\lim \sup_{x\rightarrow \epsilon }\varphi (x)<\lim \inf_{x \rightarrow \epsilon }\psi (x)$ for all $\epsilon >0$.

• (vii) If $\{x_{n}\}$ is a positive bounded sequence and if $\{\psi (x_{n})\}$ and $\{\varphi (x_{n})\}$ are two convergent sequences having the same limit, then $\lim_{n\rightarrow \infty } x_{n}=0$.

In the following conditions, we let $\varphi :(0,\infty )\rightarrow (0,\infty )$.

• (viii) If $\lim_{n\rightarrow \infty } x_{n}=\epsilon >0$, then $\lim \inf_{n\rightarrow \infty }\varphi (x_{n})>0$.

• (ix) $\lim \inf_{x\rightarrow \epsilon }\varphi (x)>0 \text{ for all } \epsilon >0$.

• (x) $\lim \sup_{x\rightarrow \epsilon }\varphi (x)>\lim \sup_{x \rightarrow \epsilon }\psi (x)-\lim \inf_{x\rightarrow \epsilon }\psi (x)$.

Using conditions (i)–(x), Proinov [34] obtained the following lemma.

Lemma 1.6

([34])

• (1) Let $\psi :(0,\infty )\rightarrow \mathbb{R}$. Then conditions (i), (ii), and (iii) are equivalent.

• (2) Let $\varphi :(0,\infty )\rightarrow \mathbb{R}$. Then condition (iv) implies (v).

• (3) Let $\varphi :(0,\infty )\rightarrow (0,\infty )$. Then conditions (viii), (vi), and (ix) are equivalent.

• (4) Let $\psi ,\varphi :(0,\infty )\rightarrow \mathbb{R}$ be two mappings satisfying conditions (vi) and (vii). Then $\lim_{n\rightarrow \infty } x_{n}=0$.

• (5) Let $\varphi :(0,\infty )\rightarrow (0,\infty )$ and $\psi :(0,\infty )\rightarrow \mathbb{R}$. Then condition (x) implies (iv).

Let $(\mathcal{A},d)$ be a metric space, let $P(\mathcal{A})$ denote the set of all nonempty subsets of $\mathcal{A}$, let $P_{cb}(\mathcal{A})$ denote the set of all nonempty closed bounded subsets of $\mathcal{A}$, and let $C(\mathcal{A})$ denote the compact subsets of $\mathcal{A}$.

Let $d(q,A)=\underset{a\in A}{\inf }d(q,a)$, and let the mapping

$$ H:P(\mathcal{A})\times P(\mathcal{A})\rightarrow [0,\infty ) $$

be defined by

$$ H(A,B)=\max \bigl\{ \underset{q\in A}{\sup }D(q,B), \underset{b\in B}{\sup }D(b,A) \bigr\} . $$

The mapping H satisfies all the axioms of metric and is known as the Hausdorff metric induced by the metric d.

Definition 1.7

Let $T:\mathcal{A}\rightarrow P(\mathcal{A})$ be a set-valued mapping. A point $\sigma \in \mathcal{A}$ is said to be a fixed point of T if $\sigma \in T(\sigma )$.

Definition 1.8

Let $T:\mathcal{A}\rightarrow P(\mathcal{A})$ and $f:\mathcal{A}\times \mathcal{A}\rightarrow [0,\infty )$. The mapping T is said to be strictly f-admissible if for all $q\in \mathcal{A}$ and $\varsigma \in T(q)$ with $f (q,\varsigma )> 1$, there exists $\omega \in T(\varsigma )$ such that $f(\varsigma ,\omega )> 1$.

Definition 1.9

Let $(\mathcal{A},d)$ be a metric space, and let $f:\mathcal{A}\times \mathcal{A}\rightarrow [0,\infty )$. The space $(\mathcal{A},d)$ is said to be strictly f-regular if for any sequence $\{q_{n}\}\subset \mathcal{A}$ such that $f(q_{n},q_{n+1})> 1$ for all $n\in \mathbb{N}$ and $q_{n}\rightarrow q$ as $n\rightarrow \infty$, we have $f(q_{n},q)>1$ for all $n\in \mathbb{N}$.

Definition 1.10

A mapping $T: (X,d)\rightarrow (X,d)$ is said to be asymptotically regular at a point x of X if

$$ \lim_{n\rightarrow \infty }\,d\bigl(T^{n}x,T^{n+1}x \bigr)=0. $$

If T is asymptotically regular at every point of X, then it is called an asymptotically regular mapping.

Lemma 1.11 plays a key role in the upcoming results.

Lemma 1.11

([29])

Let A and B be nonempty closed bounded subsets of a metric space $(\mathcal{A},d)$, and let $q>1$. Then for all $a\in A$, there exists $b\in B$ such that $d ( a,b ) \leq q H(A,B)$.

2 Set-valued $(\psi ,\varphi )_{f}$-contractions and related fixed point problems

In this section, we introduce set-valued $(\psi ,\varphi )_{f}$-contractions. We discuss their nature and generality. We investigate various conditions for the existence of fixed points of set-valued $(\psi ,\varphi )_{f}$-contractions.

Definition 2.1

Let $(\mathcal{A},d)$ be a metric space. A mapping $T:\mathcal{A}\rightarrow P_{cb}(\mathcal{A})$ is said to be a set-valued $(\psi ,\varphi )_{f}$-contraction if there exists $f:\mathcal{A}\times \mathcal{A}\rightarrow [0,\infty )$, such that

$$ \psi \bigl(f(q,\varsigma )H\bigl(T(q),T(\varsigma )\bigr)\bigr) \leq \varphi \bigl(d(q, \varsigma ) \bigr) $$

(2.1)

for all $q,\varsigma \in \mathcal{A}$ with $f(q,\varsigma )>1$ and $H(T(q),T(\varsigma ))>0$.

Remark 2.2

Inequality (2.1) reduces to multivalued F-contraction [2] if $\varphi (\sigma )=\psi (\sigma )-\tau$ for all $\sigma \in (0,\infty )$. Moreover, it turns into Nadler contraction [29] for $\psi (\sigma )=\ln (\sigma )$. Let $\psi :(0,\infty )\rightarrow (0,\infty )$ be a nondecreasing mapping, and let $\beta :(0,\infty )\rightarrow (0,1)$ be a mapping satisfying $\lim \sup_{y\rightarrow \epsilon +}\beta (y)<1$ for any $\epsilon >0$. Then substituting $\varphi (y)=\beta (y)\psi (y)$ and $\psi (y)=y$ for all $y>0$, we obtain a very famous multivalued Geraghty’s contraction discussed in [5].

The following theorem suggests a set of conditions for the existence of a fixed point of mapping T.

Theorem 2.3

Let $(\mathcal{A},d)$ be an f-regular complete metric space. Let $T:\mathcal{A}\rightarrow P_{cb}(\mathcal{A})$ be an f-admissible mapping satisfying (2.1). Suppose the mappings $\psi ,\varphi :(0,\infty )\rightarrow (-\infty , \infty )$ satisfy the following conditions:

• (i) for any $q_{0}\in \mathcal{A}$, there exists $q_{1}\in T(q_{0})$ such that $f(q_{0},q_{1})\geq 1$;

• (ii) ψ is nondecreasing, and $\varphi (y)< \psi (y)$ for all $y>0$;

• (iii) $\lim \sup_{y\rightarrow r+}\varphi (y)<\psi (r+)$ for all $r>0$.

Then T admits a fixed point in $\mathcal{A}$.

Proof

Step 1. By assumption (i), for any $q_{0}\in \mathcal{A}$, there exists $q_{1}\in T(q_{0})$ such that $f(q_{0},q_{1})> 1$. Since T is an f-admissible mapping, there exists $q_{2}\in T(q_{1})$ such that $f(q_{1},q_{2})> 1$ and $q_{3}\in T(q_{2})$ such that $f(q_{2},q_{3})> 1$. In general, there exist $q_{n+1}\in T(q_{n})$ such that $f(q_{n},q_{n+1})> 1$ for all $n\geq 0$. Note that if $q_{n} \in T(q_{n})$, then $q_{n}$ is a fixed point of T for all $n\geq 0$. So we assume that $q_{n} \notin T(q_{n})$ for all $n\geq 0$. Thus $H(Tq_{n-1}, Tq_{n})>0$; otherwise, $q_{n}\in Tq_{n}$. Since $f(q_{n},q_{n+1})> 1$ and $T(q_{n})$, $T(q_{n+1})$ are closed and bounded sets for all $n\geq 0$, by Lemma 1.11 there exist $q_{n+1} \in T(q_{n})$ ($q_{n}\neq q_{n+1}$) such that $d(q_{n},q_{n+1})\leq f(q_{n-1},q_{n})H(T(q_{n-1}),T(q_{n}))$ for all $n\geq 1$. By first part of (ii) and (2.1) we have

$$ \psi \bigl(d(q_{n},q_{n+1})\bigr)\leq \psi \bigl(f(q_{n-1},q_{n})H\bigl(T(q_{n-1}),T(q_{n}) \bigr)\bigr) \leq \varphi \bigl(d(q_{n-1},q_{n}) \bigr). $$

By the second part of assumption (ii) we have

$$ \psi \bigl(d(q_{n},q_{n+1})\bigr)\leq \varphi \bigl(d(q_{n-1},q_{n}) \bigr)< \psi \bigl(d(q_{n-1},q_{n}) \bigr). $$

(2.2)

Since ψ is a nondecreasing mapping, $d(q_{n},q_{n+1})< d(q_{n-1},q_{n})$ for every $n\geq 1$. This shows that the sequence $\{d(q_{n-1},q_{n})\}$ is positively decreasing. Thus there exists $L\geq 0$ such that $\lim_{n\rightarrow \infty }d(q_{n-1},q_{n})=L+$. If $L>0$, then by (2.2) we obtain a contradiction to assumption (iii) as follows:

$$ \psi (L+)= \lim_{n\rightarrow \infty }\psi \bigl(d(q_{n},q_{n+1}) \bigr)\leq \lim_{n\rightarrow \infty }\sup \varphi \bigl(d(q_{n-1},q_{n}) \bigr)\leq \lim_{ \sigma \rightarrow L+}\sup \varphi (\sigma ). $$

Hence $L=0$, and, consequently, T is an asymptotically regular mapping.

Step 2. We show that $\{q_{n}\}$ is a Cauchy sequence. Assume on the contrary that $\{q_{n}\}$ is not a Cauchy sequence. In this case, by Lemma 1.5 there exist two subsequences $\{q_{n_{k}}\}$, $\{q_{m_{k}}\}$ of $\{q_{n}\}$ and $\epsilon >0$ such that (1.3) and (1.4) hold. By (1.3) we infer that $d(q_{n_{k}+1}, q_{m_{k}+1})>\epsilon$ and $f(q_{n_{k}}, q_{m_{k}})>1$ for all $k\geq 1$. Letting $q=q_{n_{k}}$ and $\varsigma =q_{m_{k}}$ in (2.1), we have

$$ \psi \bigl(d(q_{n_{k}+1}, q_{m_{k}+1})\bigr)\leq \psi \bigl(f(q_{n_{k}}, q_{m_{k}})H(Tq_{n_{k}}, Tq_{m_{k}})\bigr)\leq \varphi \bigl(d(q_{n_{k}}, q_{m_{k}})\bigr) \quad \text{for all } k \geq 1, $$

since if $a_{k}=d(q_{n_{k}+1}, q_{m_{k}+1})$ and $b_{k}=d(q_{n_{k}}, q_{m_{k}})$, then

$$ \psi (a_{k})\leq \varphi (b_{k})< \psi (b_{k}) \quad \text{for any } k\geq 1 \text{ implies that } a_{k}< b_{k}. $$

Since $\lim_{k\rightarrow \infty }a_{k}=\epsilon +$, we also have $\lim_{k\rightarrow \infty }b_{k}=\epsilon +$. Thus

$$ \psi (\epsilon +)=\lim_{k\rightarrow \infty }\psi (a_{k})\leq \lim_{k \rightarrow \infty }\sup \varphi (b_{k})\leq \lim _{\sigma \rightarrow \epsilon +}\varphi (\sigma ). $$

This is a contradiction to assumption (iii), and, consequently, $\{q_{n}\}$ is a Cauchy sequence in $(\mathcal{A},d)$. Since $(\mathcal{A},d)$ is a complete metric space, there exists $q^{*}\in \mathcal{A}$ such that $q_{n}\rightarrow q^{*}$ as $n\rightarrow \infty$, and the f-regularity of the space $(\mathcal{A},d)$ implies $f(q_{n}, q^{*})> 1$. We claim that $d(q^{*}, T(q^{*}))=0$. On the contrary, assume that $d(q^{*},T(q^{*}))>0$. Then there exists $n_{1}\in \mathbb{N}$ such that $d(q_{n},T(q^{*}))>0$ for each $n\geq n_{1}$. By (2.1)

$$ \psi \bigl(d\bigl(q_{n+1},T\bigl(q^{*}\bigr)\bigr)\bigr) \leq \psi \bigl(f\bigl(q_{n}, q^{*}\bigr)H \bigl(T(q_{n}),T\bigl(q^{*}\bigr)\bigr)\bigr) \leq \varphi \bigl(d\bigl(q_{n}, q^{*}\bigr) \bigr)< \psi \bigl(d \bigl(q_{n}, q^{*}\bigr) \bigr). $$

By the first part of assumption (ii) we have $d(q_{n+1},T(q^{*}))< d(q_{n}, q^{*})$. Taking the limit on both sides of the last inequality as $n\rightarrow \infty$, we have $d(q^{*},T(q^{*}))<0$. This implies $d(q^{*},T(q^{*}))=0$. Since $T(q^{*})$ is closed, $q^{*}\in T(q^{*})$. The uniqueness of $q^{*}$ is obvious from the contractive condition (2.1). □

The following theorem suggests another set of conditions for the existence of a fixed point of a self-mapping T satisfying (2.1).

Theorem 2.4

Let $(\mathcal{A},d)$ be an f-regular complete metric space. Let $T:\mathcal{A}\rightarrow P_{cb}(\mathcal{A})$ be an f-admissible mapping satisfying (2.1). Suppose mappings $\psi ,\varphi :(0,\infty )\rightarrow (-\infty , \infty )$ satisfy the following conditions:

• (i) for all $\sigma _{0}\in \mathcal{A}$, there exists $\sigma _{1}\in T(\sigma _{0})$ such that $f(\sigma _{0},\sigma _{1})\geq 1$;

• (ii) ψ is nondecreasing, and $\varphi (y)< \psi (y)$ for all $y>0$;

• (iii) $\inf_{\sigma >\epsilon } \psi (\sigma )>-\infty$;

• (iv) if the sequences $\{\psi (\sigma _{n})\}$ and $\{\varphi (\sigma _{n})\}$ converge to the same limit and $\{\psi (\sigma _{n})\}$ is strictly decreasing, then $\lim_{n\rightarrow \infty }\sigma _{n}=0$;

• (v) $\lim \sup_{\sigma \rightarrow \epsilon }\varphi (\sigma )< \lim \inf_{\sigma \rightarrow \epsilon +}\psi (\sigma )$ for all $\epsilon >0$;

• (vi) $\lim \sup_{\sigma \rightarrow \epsilon _{1}}\varphi (\sigma )< \lim \inf_{\sigma \rightarrow \epsilon }\varphi (\sigma )$ for all $\epsilon , \epsilon _{1}>0$.

Then T has a unique fixed point in $\mathcal{A}$.

Proof

For the proof, the first four conditions (i)–(iv) are needed to prove that T is asymptotically regular. Condition (v) is required to prove that $\{q_{n}\}$ is a Cauchy sequence, and condition (vi) is helpful to show the existence of a fixed point.

By assumption (i), for any $\sigma _{0}\in \mathcal{A}$, there exists $\sigma _{1}\in T(\sigma _{0})$ such that $f(\sigma _{0},\sigma _{1})> 1$. Since T is an f-admissible mapping, there exist $\sigma _{2}\in T(\sigma _{1})$ such that $f(\sigma _{1},\sigma _{2})> 1$ and $\sigma _{3}\in T(\sigma _{2})$ such that $f(\sigma _{2},\sigma _{3})> 1$. In general, there exist $\sigma _{n+1}\in T(\sigma _{n})$ such that $f(\sigma _{n},\sigma _{n+1})> 1$ for all $n\geq 0$. Note that if $\sigma _{n} \in T(\sigma _{n})$, then $\sigma _{n}$ is a fixed point of T for all $n\geq 0$. We assume that $\sigma _{n} \notin T(\sigma _{n})$ for all $n\geq 0$. Thus $H(T\sigma _{n-1}, T\sigma _{n})>0$; otherwise, $\sigma _{n}\in T\sigma _{n}$. Since $f(\sigma _{n},\sigma _{n+1})> 1$ and $T(\sigma _{n})$, $T(\sigma _{n+1})$ are closed bounded sets for all $n\geq 0$, by Lemma 1.11 there exists $\sigma _{n+1} \in T(\sigma _{n})$ $(\sigma _{n}\neq \sigma _{n+1})$ such that $d(\sigma _{n},\sigma _{n+1})\leq f(\sigma _{n-1},\sigma _{n})H(T( \sigma _{n-1}),T(\sigma _{n}))$ for all $n\geq 1$. By the first part of (ii) and (2.1) we have that for all $n\geq 1$,

$$\begin{aligned} \psi \bigl(d(\sigma _{n},\sigma _{n+1})\bigr) \leq& \psi \bigl(f(\sigma _{n-1},\sigma _{n})H\bigl(T( \sigma _{n-1}),T(\sigma _{n})\bigr)\bigr) \\ \leq& \varphi \bigl(d( \sigma _{n-1},\sigma _{n})\bigr)< \psi \bigl(d(\sigma _{n-1},\sigma _{n})\bigr). \end{aligned}$$

(2.3)

Inequality (2.3) shows that $\{\psi (d(\sigma _{n-1},\sigma _{n}))\}$ is a strictly decreasing sequence. Then it is either bounded below or not. If it is not bounded below, then by assumption (iii) and Lemma 1.6(1) we infer that $\lim_{n\rightarrow \infty }d(\sigma _{n-1},\sigma _{n})=0$. If it bounded below, then $\{\psi (d(\sigma _{n-1},\sigma _{n}))\}$ is a convergent sequence, and by (2.3) the sequence $\{\varphi (d(\sigma _{n-1},\sigma _{n}))\}$ also converges, and both have the same point of convergence. Thus by assumption (iv) we have $\lim_{n\rightarrow \infty }d(\sigma _{n-1},\sigma _{n})=0$. Hence T is asymptotically regular.

Following Step 2 of the proof of Theorem 2.3, we have

$$ \psi (a_{k})\leq \varphi (b_{k}),\quad \text{for any } k \geq 1. $$

(2.4)

By (1.3) and (1.4) we have $\lim_{k\rightarrow \infty }a_{k}=\epsilon +$ and $\lim_{k\rightarrow \infty }b_{k}=\epsilon$. By (2.4) we infer that

$$ \lim \inf_{\sigma \rightarrow \epsilon +}\psi (\sigma )\leq \lim \inf _{k\rightarrow \infty }\psi (a_{k})\leq \lim \sup _{k\rightarrow \infty }\varphi (b_{k})\leq \lim \sup _{\sigma \rightarrow \epsilon } \varphi (\sigma ). $$

This is a contradiction to (v), and hence $\{\sigma _{n}\}$ is a Cauchy sequence in $(\mathcal{A},d)$. Since $(\mathcal{A},d)$ is a complete metric space, there exists $\sigma ^{*}\in \mathcal{A}$ such that $\sigma _{n}\rightarrow \sigma ^{*}$ as $n\rightarrow \infty$.

Now we have to prove that the point of convergence $\sigma ^{*}$ is a fixed point of T. We consider two cases.

Case 1. If $d(\sigma _{n+1}, T\sigma ^{*})=0$ for some $n\geq 0$, then by the triangle property of d we obtain

$$ d\bigl(\sigma ^{*},T\sigma ^{*}\bigr)\leq d\bigl(\sigma ^{*},\sigma _{n+1}\bigr)+d\bigl( \sigma _{n+1}, T \sigma ^{*}\bigr)=d\bigl(\sigma ^{*},\sigma _{n+1}\bigr). $$

Taking the limit as $n\rightarrow \infty$ on both sides, we have $d(\sigma ^{*},T\sigma ^{*})\leq 0$. This implies $d(\sigma ^{*},T(\sigma ^{*}))=0$. Since $T(\sigma ^{*})$ is closed, $\sigma ^{*}\in T(\sigma ^{*})$.

Case 2. If $d(\sigma _{n+1}, T\sigma ^{*})>0$ for all $n\geq 0$, then by the f-regularity of the space $(\mathcal{A},d)$ we have $f(\sigma _{n}, \sigma ^{*})> 1$. By contractive condition (2.1) we have

$$ \psi \bigl(d\bigl(\sigma _{n+1},T\sigma ^{*}\bigr)\bigr) \leq \psi \bigl(f\bigl(\sigma _{n}, \sigma ^{*}\bigr)H \bigl(T \sigma _{n}, T\sigma ^{*}\bigr)\bigr)\leq \varphi \bigl(d\bigl(\sigma _{n},\sigma ^{*}\bigr)\bigr) \quad \text{for all }n\geq 0. $$

Let $a_{n}=d(\sigma _{n+1},T\sigma ^{*})$ and $b_{n}=d(\sigma _{n},\sigma ^{*})$. Then the last inequality reduces to

$$ \psi (a_{n})\leq \varphi (b_{n})\quad \text{for all }n\geq 0. $$

(2.5)

Let $\epsilon =d(\sigma ^{*},T\sigma ^{*})$. Then we observe that $a_{n}\rightarrow \epsilon$ and $b_{n}\rightarrow 0$ as $n\rightarrow \infty$. Applying the limits on (2.5), we have

$$ \lim \inf_{\sigma \rightarrow \epsilon }\psi (\sigma )\leq \lim \inf _{n \rightarrow \infty }\psi (a_{n}) \leq \lim \sup _{n\rightarrow \infty } \varphi (b_{n})\leq \lim \inf _{\sigma \rightarrow 0}\varphi (\sigma ). $$

The last inequality is a contradiction to assumption (vi) if $\epsilon >0$. Thus we have $d(\sigma ^{*},T\sigma ^{*})= 0$. Hence $\sigma ^{*}\in T\sigma ^{*}$, that is, $\sigma ^{*}$ is a fixed point of T. The uniqueness of $\sigma ^{*}$ is obvious from the contractive condition (2.1). □

Note that Theorems 2.3 and 2.4 reduce to the Nadler fixed point theorem [15] if $\psi (y)=y$ and $\varphi (y)=\lambda y$ for all $y>0$ and $0\leq \lambda <1$. If $\psi (y)=y$ for all $y>0$, then they reduce to the multivalued version of the Boyd–Wong fixed point theorem (Theorem 1.1). By substituting $\varphi (y)=\psi (y)-\tau$ into Theorems 2.3 and 2.4 we obtain an improvement of fixed point theorems established in [2, 22] and of the results presented by Secelean [36] and Lukacs and Kajanto [27] as follows.

Corollary 2.5

Let $(\mathcal{A},d)$ be an f-regular complete metric space, and let $T: \mathcal{A}\rightarrow P_{cb}(\mathcal{A})$ be a set-valued strictly f-admissible mapping satisfying the following inequality:

$$ \psi \bigl(f(x,y)H(Tx,Ty)\bigr)\leq \psi \bigl(d(x,y)\bigr))-\tau \quad \forall x,y\in \mathcal{A}, \textit{provided that } H(Tx,Ty)>0, $$

where $\psi :(0,\infty )\rightarrow \mathbb{R}$ is a nondecreasing mapping, and $\tau >0$. If for any initial guess $\sigma _{0}\in \mathcal{A}$, there exists $\sigma _{1}\in T(\sigma _{0})$ such that $f(\sigma _{0},\sigma _{1})\geq 1$, then T has a unique fixed point in $\mathcal{A}$.

If ψ is lower semicontinuous and φ is upper semicontinuous, then Theorem 2.4 is an improvement of the Amini–Harandi–Petrusel fixed point theorem [10]. If we take $\varphi (y)=h(\psi (y))$ in Theorem 2.3, we obtain the following improvement of Moradi’s theorem [28].

Corollary 2.6

Let $(\mathcal{A},d)$ be a f-regular complete metric space, and let $T: \mathcal{A}\rightarrow P_{cb}(\mathcal{A})$ be a set-valued strictly f-admissible mapping satisfying the following inequality:

$$ \psi \bigl(f(x,y)H(Tx,Ty)\bigr)\leq h\bigl(\psi \bigl(d(x,y)\bigr)\bigr)\quad \forall x,y\in \mathcal{A}, \textit{provided } H(Tx,Ty)>0, $$

where

• (i) $h:I\rightarrow [0,\infty )$ is an upper semicontinuous mapping such that $h(y)< y$ for all $y\in I\subset \mathbb{R}$;

• (ii) $\psi :(0,\infty )\rightarrow I$ is nondecreasing.

If for any initial guess $\sigma _{0}\in \mathcal{A}$, there exists $\sigma _{1}\in T(\sigma _{0})$ such that $f(\sigma _{0},\sigma _{1})\geq 1$, then T has a unique fixed point in $\mathcal{A}$.

Taking $h(y)=y^{r}$ with $r\in (0,1)$ in Corollary 2.6, we obtain the following result.

Corollary 2.7

Let $(\mathcal{A},d)$ be an f-regular complete metric space, and let $T: \mathcal{A}\rightarrow P_{cb}(\mathcal{A})$ be a set-valued strictly f-admissible mapping satisfying the following inequality:

$$ \psi \bigl(f(x,y)H(Tx,Ty)\bigr)\leq \bigl(\psi \bigl(d(x,y)\bigr) \bigr)^{r}\quad \forall x,y\in \mathcal{A}, \textit{provided that } H(Tx,Ty)>0, $$

where, $\psi :(0,\infty )\rightarrow (0,1)$ is a nondecreasing mapping. If for any initial guess $\sigma _{0}\in \mathcal{A}$, there exists $\sigma _{1}\in T(\sigma _{0})$ such that $f(\sigma _{0},\sigma _{1})\geq 1$. Then T has a unique fixed point in $\mathcal{A}$.

It is obvious that Corollary 2.7 improves the Jleli–Samet fixed point theorem [24] and the results presented by Ahmad et al. [7] and Li and Jiang [26].

We also note that an improvement of particular case of the Skof fixed point theorem [37] can be obtained by taking $\varphi (y)=\lambda \psi (y)$ in Theorems 2.3 and 2.4 as follows.

Corollary 2.8

Let $(\mathcal{A},d)$ be an f-regular complete metric space, and let $T: \mathcal{A}\rightarrow P_{cb}(\mathcal{A})$ be a set-valued strictly f-admissible mapping satisfying the following inequality:

$$ \psi \bigl(f(x,y)H(Tx,Ty)\bigr)\leq \lambda \psi \bigl(d(x,y)\bigr)\quad \forall x,y \in \mathcal{A}, \textit{provided that } H(Tx,Ty)>0, $$

where $\psi :(0,\infty )\rightarrow (0,\infty )$ is a nondecreasing mapping, and $\lambda \in (0,1)$. If for any initial guess $\sigma _{0}\in \mathcal{A}$, there exists $\sigma _{1}\in T(\sigma _{0})$ such that $f(\sigma _{0},\sigma _{1})\geq 1$, then T has a unique fixed point in $\mathcal{A}$.

Let us consider a nondecreasing mapping $\psi :(0,\infty )\rightarrow (0,\infty )$ and a mapping $\beta :(0,\infty )\rightarrow (0,1)$ satisfying $\lim \sup_{y\rightarrow \epsilon +}\beta (y)<1$ for any $\epsilon >0$. Then taking $\varphi (y)=\beta (y)\psi (y)$ and $\psi (y)=y$ for all $y>0$ in Theorem 2.3, we obtain an improvement of the well-known Geraghty fixed point theorem [23].

3 Theorems on generalized $(\psi ,\varphi )_{f}$-contractions

Since the generalized $(\psi ,\varphi )_{f}$-contractions are not $(\psi ,\varphi )_{f}$-contractions in general, in this section, we give some fixed-point results for the class of generalized $(\psi ,\varphi )_{f}$-contractions defined below.

Definition 3.1

Let $(\mathcal{A},d)$ be a metric space. A mapping $T:\mathcal{A}\rightarrow P(\mathcal{A})$ is said to be a set-valued generalized $(\psi ,\varphi )_{f}$-contraction if there exists $f:\mathcal{A}\times \mathcal{A}\rightarrow [0,\infty )$ such that

$$ \psi \bigl(f(q,\varsigma )H\bigl(T(q),T(\varsigma )\bigr)\bigr) \leq \varphi \bigl(A(q, \varsigma ) \bigr) $$

(3.1)

for all $q,\varsigma \in \mathcal{A}$ with $f(q,\varsigma )>1$ and $H(T(q),T(\varsigma ))>0$, where

$$ A(q,\varsigma )=\max \bigl\{ d(q,\varsigma ), d(q,Tq), d(\varsigma ,T \varsigma ), \bigl(d(q,T\varsigma ) +d(Tq,\varsigma )\bigr)/2 \bigr\} . $$

The following theorems generalize many fixed point theorems involving Ciric type contractions. For Ćirić contraction and related fixed-point results, see ([3, 21, 41]).

Theorem 3.2

Let $(\mathcal{A},d)$ be an f-regular complete metric space. Let $T:\mathcal{A}\rightarrow C(\mathcal{A})$ be an f-admissible mapping satisfying (3.1). Suppose mappings $\psi ,\varphi :(0,\infty )\rightarrow (-\infty , \infty )$ satisfy the following conditions:

• (i) for all $q_{0}\in \mathcal{A}$, there exists $q_{1}\in T(q_{0})$ such that $f(q_{0},q_{1})\geq 1$;

• (ii) ψ is nondecreasing, and $\varphi (y)< \psi (y)$ for all $y>0$;

• (iii) $\lim \sup_{y\rightarrow r+}\varphi (y)<\psi (r+)$ for all $r>0$.

Then T admits a fixed point in $\mathcal{A}$.

Proof

Let $q_{0}\in \mathcal{A}$ be an arbitrary initial guess. Following the arguments in Step 1 of the proof of Theorem 2.3, we have $d(q_{n},q_{n+1})\leq f(q_{n-1},q_{n})H(T(q_{n-1}),T(q_{n}))$ for all $n\geq 1$. By the first part of (ii) and (3.1) we have

$$ \psi \bigl(d(q_{n},q_{n+1})\bigr)\leq \psi \bigl(f(q_{n-1},q_{n})H\bigl(T(q_{n-1}),T(q_{n}) \bigr)\bigr) \leq \varphi \bigl(A(q_{n-1},q_{n}) \bigr). $$

Since $T(x)$ is compact for all $x\in \mathcal{A}$, there exists $q_{n}\in T(q_{n-1})$ such that $d(q_{n-1},q_{n})=d(q_{n-1}, T(q_{n-1}))$ for all $n\geq 1$ and

$$\begin{aligned} &\psi \bigl(d(q_{n},q_{n+1})\bigr) \\ &\quad \leq \varphi \bigl(A(q_{n-1},q_{n}) \bigr) \\ &\quad = \varphi \bigl(\max \bigl\{ d(q_{n-1},q_{n}), d \bigl(q_{n-1}, T(q_{n-1})\bigr),d\bigl(q_{n}, T(q_{n})\bigr), d\bigl(q_{n-1}, T(q_{n}) \bigr) \\ &\qquad {}+d\bigl(q_{n}, T(q_{n-1})\bigr)/2 \bigr\} \bigr) \\ &\quad = \varphi \bigl(\max \bigl\{ d(q_{n-1},q_{n}),d(q_{n}, q_{n+1}) \bigr\} \bigr). \end{aligned}$$

If $d(q_{n-1},q_{n})< d(q_{n}, q_{n+1})$, then $\psi (d(q_{n},q_{n+1}))\leq \varphi (d(q_{n}, q_{n+1}) )$, which is a contradiction to the second part of assumption (ii). Thus we have $d(q_{n-1},q_{n})>d(q_{n}, q_{n+1})$ and

$$ \psi \bigl(d(q_{n},q_{n+1})\bigr)\leq \varphi \bigl(d(q_{n-1}, q_{n}) \bigr). $$

By the second part of assumption (ii) we have

$$ \psi \bigl(d(q_{n},q_{n+1})\bigr)\leq \varphi \bigl(d(q_{n-1},q_{n}) \bigr)< \psi \bigl(d(q_{n-1},q_{n}) \bigr). $$

(3.2)

Since ψ is a nondecreasing mapping, $d(q_{n},q_{n+1})< d(q_{n-1},q_{n})$ for every $n\geq 1$. This shows that the sequence $\{d(q_{n-1},q_{n})\}$ is positively decreasing. Thus there exists $L\geq 0$ such that $\lim_{n\rightarrow \infty }d(q_{n-1},q_{n})=L+$. If $L>0$, then by (3.2) we obtain a contradiction to assumption (iii) as follows:

$$ \psi (L+)= \lim_{n\rightarrow \infty }\psi \bigl(d(q_{n},q_{n+1}) \bigr)\leq \lim_{n\rightarrow \infty }\sup \varphi \bigl(d(q_{n-1},q_{n}) \bigr)\leq \lim_{ \sigma \rightarrow L+}\sup \varphi (\sigma ). $$

Hence $L=0$, and, consequently, T is an asymptotically regular mapping.

Now we show that $\{q_{n}\}$ is a Cauchy sequence. Assume on the contrary that the sequence $\{q_{n}\}$ is not Cauchy. In this case, by Lemma 1.5 there exist two subsequences $\{q_{n_{k}}\}$, $\{q_{m_{k}}\}$ of $\{q_{n}\}$ and $\epsilon >0$ such that (1.3) and (1.4) hold. By (1.3) we infer that $d(q_{n_{k}+1}, q_{m_{k}+1})>\epsilon$ and $f(q_{n_{k}}, q_{m_{k}})>1$ for all $k\geq 1$. Letting $q=q_{n_{k}}$ and $\varsigma =q_{m_{k}}$ in (3.1), we have

$$ \psi \bigl(d(q_{n_{k}+1}, q_{m_{k}+1})\bigr)\leq \psi \bigl(f(q_{n_{k}}, q_{m_{k}})H(Tq_{n_{k}}, Tq_{m_{k}})\bigr)\leq \varphi \bigl(A(q_{n_{k}}, q_{m_{k}})\bigr) \quad \text{for all } k \geq 1. $$

If $a_{k}=d(q_{n_{k}+1}, q_{m_{k}+1})$ and $b_{k}=A(q_{n_{k}}, q_{m_{k}})$, then

$$ \psi (a_{k})\leq \varphi (b_{k})< \psi (b_{k})\quad \text{for any } k\geq 1 \text{ implies that } a_{k}< b_{k}. $$

Since $\lim_{k\rightarrow \infty }a_{k}=\epsilon +$, $\lim_{k\rightarrow \infty }b_{k}=\epsilon +$. Thus

$$ \psi (\epsilon +)=\lim_{k\rightarrow \infty }\psi (a_{k})\leq \lim_{k \rightarrow \infty }\sup \varphi (b_{k})\leq \lim _{\sigma \rightarrow \epsilon +}\varphi (\sigma ). $$

This is a contradiction to assumption (iii), and, consequently, $\{q_{n}\}$ is a Cauchy sequence in $(\mathcal{A},d)$. Since $(\mathcal{A},d)$ is a complete metric space, there exists $q^{*}\in \mathcal{A}$ such that $q_{n}\rightarrow q^{*}$ as $n\rightarrow \infty$, and the f-regularity of the space $(\mathcal{A},d)$ implies $f(q_{n}, q^{*})> 1$. We claim that $d(q^{*}, T(q^{*}))=0$. On the contrary, assume that $d(q^{*},T(q^{*}))>0$. Then there exists $n_{1}\in \mathbb{N}$ such that $d(q_{n},T(q^{*}))>0$ for each $n\geq n_{1}$. By (3.1)

$$ \psi \bigl(d\bigl(q_{n+1},T\bigl(q^{*}\bigr)\bigr)\bigr) \leq \psi \bigl(f\bigl(q_{n}, q^{*}\bigr)H \bigl(T(q_{n}),T\bigl(q^{*}\bigr)\bigr)\bigr) \leq \varphi \bigl(A\bigl(q_{n}, q^{*}\bigr) \bigr)< \psi \bigl(A \bigl(q_{n}, q^{*}\bigr) \bigr). $$

By the first part of assumption (ii) we have $d(q_{n+1},T(q^{*}))< A(q_{n}, q^{*})$. Applying the limit as $n\rightarrow \infty$ on both sides of the last inequality, we have $d(q^{*},T(q^{*}))< d(q^{*},T(q^{*}))$, a contradiction, and thus $d(q^{*},T(q^{*}))=0$. Since $T(q^{*})$ is compact, $q^{*}\in T(q^{*})$. □

Theorem 3.3

Let $(\mathcal{A},d)$ be an f-regular complete metric space. Let $T:\mathcal{A}\rightarrow C(\mathcal{A})$ be an f-admissible mapping satisfying (3.1). Suppose the mappings $\psi ,\varphi :(0,\infty )\rightarrow (-\infty , \infty )$ satisfy the following conditions:

• (i) for all $\sigma _{0}\in \mathcal{A}$, there exists $\sigma _{1}\in T(\sigma _{0})$ such that $f(\sigma _{0},\sigma _{1})\geq 1$;

• (ii) ψ is nondecreasing, and $\varphi (y)< \psi (y)$ for all $y>0$;

• (iii) $\inf_{\sigma >\epsilon } \psi (\sigma )>-\infty$;

• (iv) if the sequences $\{\psi (\sigma _{n})\}$ and $\{\varphi (\sigma _{n})\}$ converge to the same limit and $\{\psi (\sigma _{n})\}$ is strictly decreasing, then $\lim_{n\rightarrow \infty }\sigma _{n}=0$;

• (v) $\lim \sup_{\sigma \rightarrow \epsilon }\varphi (\sigma )< \lim \inf_{\sigma \rightarrow \epsilon +}\psi (\sigma )$ for all $\epsilon >0$;

• (vi) $\lim \sup_{\sigma \rightarrow \epsilon _{1}}\varphi (\sigma )< \lim \inf_{\sigma \rightarrow \epsilon }\varphi (\sigma )$ for all $\epsilon , \epsilon _{1}>0$.

Then T has a fixed point in $\mathcal{A}$.

Proof

This proof can be obtained by following the proofs of Theorems 2.4 and 3.2. We omit the details. □

For single-valued mappings, we have the following result.

Theorem 3.4

Let $(\mathcal{A}, d )$ be an f-regular complete metric space, and let $T:X \rightarrow X$ be a strictly f-admissible mapping satisfying following inequality:

$$ \tau +\psi \bigl(f(\sigma ,\varsigma )\,d\bigl(T(\sigma ),T(\varsigma )\bigr) \bigr)\leq \psi \bigl( A(\sigma ,\varsigma )\bigr) $$

(3.3)

for all $\sigma ,\varsigma \in \mathcal{A}$ with $d(T(\sigma ),T(\varsigma ))>0$, where $\psi :(0,\infty )\rightarrow \mathbb{R}$ is a nondecreasing mapping, and $\tau >0$. If for any initial guess $\sigma _{0}\in \mathcal{A}$, there exists $\sigma _{1}= T(\sigma _{0})$ such that $f(\sigma _{0},\sigma _{1})\geq 1$, then T admits a unique fixed point.

Proof

Setting $\varphi (y)=\psi (y)-\tau$ for all $y>0$ and letting $T(x)$ to be a singleton set for all $x\in \mathcal{A}$ in Theorem 3.2, we have required result. □

Remark 3.5

It is noted in [27] that the Riech and Hardy–Roger contractions are reducible to the Ćirić contraction (also called generalized contraction). Thus Theorems 3.2, 3.3, and 3.4 remains true if we replace $A(\sigma ,\varsigma )$ by anyone of the following:

• (1) $\max \{d(\sigma ,\varsigma ),d(\sigma ,T(\sigma )),d(\varsigma ,T( \varsigma ))\}$,

• (2) $\max \{d(\sigma ,T(\sigma )),d(\varsigma ,T(\varsigma ))\}$,

• (3) $\max \{ d(\sigma ,\varsigma ), \frac{d(\sigma ,T(\sigma ))+d(\varsigma ,T(\varsigma ))}{2}, \frac{d(\varsigma ,T(\sigma ))+d (\sigma ,T(\varsigma ))}{2} \}$,

• (4) $ad(\sigma ,\varsigma )+b(d(\sigma ,T(\sigma ))+d(\varsigma ,T( \varsigma )))+c(d(\varsigma ,T(\sigma ))+d(\sigma ,T(\varsigma )))$ with $a+b+c<1$,

• (5) $ad(\sigma ,\varsigma )+bd(\sigma ,T(\sigma ))+cd(\varsigma ,T( \varsigma ))$ with $a+b+c<1$.

4 Applications to fractional differential equations

Lacroix (1819) introduced and investigated several applicable properties of fractional differentials. Recently, various new models involving the Caputo–Fabrizio derivative (CFD) were discovered and analyzed [8, 14, 38, 39]. In the following, we investigate one of these models in metric spaces. We introduce some notations for this purpose.

Let $\mathcal{C}_{0,1}$ be the space of continuous functions $w: [ 0,1 ]\rightarrow \mathbb{R}$. Define the metric $d:\mathcal{C}_{0,1}\times \mathcal{C}_{0,1}\rightarrow [0,\infty )$ by

$$ d(w,g)= \Vert w-g \Vert _{\infty }=\max_{\nu \in [ 0,1 ] } \bigl\vert w(\nu )-g(\nu ) \bigr\vert \quad \text{for }w,g\in \mathcal{C}_{0,1}. $$

Then the space $(\mathcal{C}_{0,1}, d)$ is a complete metric space. Let $f:\mathcal{C}_{0,1}\times \mathcal{C}_{0,1}\rightarrow (1,\infty )$ be defined by

$$ f(r,t)=e^{ \Vert r+t \Vert _{\infty }} \quad \text{for } r,t\in \mathcal{C}_{0,1}. $$

Let $K_{1}: [ 0,1 ] \times \mathbb{R}\rightarrow \mathbb{R}$ be a continuous mapping. We will investigate the CFDE

$$ ^{C}D^{\beta }q ( \nu ) =K_{1}\bigl(\nu ,q(\nu ) \bigr) $$

(4.1)

with boundary conditions

$$ q ( 0 ) =0,\qquad Iq ( 1 ) =q^{\prime } ( 0 ). $$

Here $^{C}D^{\beta }$ denotes the CFD of order β defined by

$$ ^{C}D^{\beta }K_{1} ( \nu ) = \frac{1}{\Gamma ( n-\beta ) } \int _{0}^{\nu } ( \nu - \eta ) ^{n-\beta -1}K_{1}^{n}(\eta ))\,d\eta , $$

where

$$ n-1< \beta < n\quad \text{and}\quad n= [ \beta ] +1, $$

and $I^{\beta }K_{1}$ is given by

$$ I^{\beta }K_{1} (\nu ) = \frac{1}{\Gamma ( \beta ) } \int _{0}^{\nu } ( \nu - \eta ) ^{\beta -1}K_{1}(\eta )\,d\eta \quad \text{with }\beta >0. $$

Then equation (4.1) can be modified to

$$ q (\nu ) =\frac{1}{\Gamma ( \beta ) } \int _{0}^{ \nu } (\nu -\eta ) ^{\beta -1}K_{1} \bigl(\eta ,q(\eta )\bigr)\,d\eta +\frac{2\nu }{\Gamma ( \beta ) } \int _{0}^{1} \int _{0}^{ \eta } (\eta -u ) ^{\beta -1}K_{1} \bigl(u,q(u)\bigr)\,du\,d\eta . $$

Theorem 4.1

Equation (4.1) admits a solution in $\mathcal{C}_{0,1}$ provided that:

• (I) there exists $\tau >0$ such that for all $q,\varsigma \in \mathcal{C}_{0,1}$, we have

  $$\begin{aligned}& \bigl\vert K_{1} \bigl(\eta ,q(\eta ) \bigr)-K_{1} \bigl(\eta , \varsigma (\eta ) \bigr) \bigr\vert \\& \quad \leq \frac{e^{-\tau }\Gamma ( \beta +1 ) }{4M} \bigl\vert q (\eta )-\varsigma (\eta ) \bigr\vert \bigl(M= \min \bigl\{ f(q, \varsigma )| q,\varsigma \in \mathcal{C}_{0,1}\bigr\} \bigr); \end{aligned}$$

• (II) there exists $q_{0}\in \mathcal{C}_{0,1}$ such that for all $\nu \in [ 0,1 ]$, we have

  $$\begin{aligned} q_{0} ( \nu ) \leq& \frac{1}{\Gamma ( \beta ) } \int _{0}^{\nu } (\nu - \eta ) ^{\beta -1}K_{1}\bigl(\eta ,q_{0}(\eta ) \bigr)\,d\eta \\ &{} + \frac{2\nu }{\Gamma ( \beta ) } \int _{0}^{1} \int _{0}^{ \eta } (\eta -u ) ^{\beta -1}K_{1} \bigl(u,q_{0}(u) \bigr)\,du\,d\eta . \end{aligned}$$

Proof

Consistently with the notations introduced, define the mapping $R:\mathcal{C}_{0,1} \rightarrow \mathcal{C}_{0,1}$ by

$$\begin{aligned} R \bigl(q (\nu ) \bigr) =& \frac{1}{\Gamma ( \beta ) } \int _{0}^{\nu } (\nu - \eta ) ^{\beta -1}K_{1}\bigl(\eta ,q(\eta )\bigr)\,d\eta \\ &{}+ \frac{2\nu }{\Gamma ( \beta ) } \int _{0}^{1} \int _{0}^{ \eta } (\eta -u ) ^{\beta -1}K_{1} \bigl(u,q(u)\bigr)\,du\,d\eta . \end{aligned}$$

By (II) there exists $q_{0} \in \mathcal{C}_{0,1}$ such that $q_{n}=R^{n}(q_{0}))$. The continuity of the mapping $K_{1}$ leads to the continuity of the mapping R on $\mathcal{C}_{0,1}$. It is easy to verify the assumptions of Theorem 3.4. Let us verify the contractive condition (3.3) of Theorem 3.4.

$$\begin{aligned}& \bigl\vert R \bigl(q (\nu ) \bigr)-R \bigl(\varsigma (\nu ) \bigr) \bigr\vert = \left \vert \textstyle\begin{array}{ll} \frac{1}{\Gamma ( \beta ) }\int _{0}^{\nu } (\nu - \eta )^{\beta -1}K_{1}(\eta ,q(\eta ) )\,d\eta &\\ \quad {}-\frac{1}{\Gamma ( \beta ) }\int _{0}^{\nu } (\nu - \eta ) ^{\beta -1}K_{1}(\eta ,\varsigma (\eta ))\,d\eta &\\ \quad {}+\frac{2\nu }{\Gamma ( \beta ) }\int _{0}^{1}\int _{0}^{ \eta } (\eta -u ) ^{\beta -1}K_{1}(u,q(u) )\,du\,d\eta &\\ \quad {}-\frac{2\nu }{\Gamma ( \beta ) }\int _{0}^{1}\int _{0}^{ \eta } (\eta -u ) ^{\beta -1}K_{1}(u,\varsigma (u))\,du\,d \eta \end{array}\displaystyle \right \vert \quad \text{implies } \\& \bigl\vert R \bigl(q (\nu ) \bigr)-R \bigl( \varsigma (\nu ) \bigr) \bigr\vert \\& \quad \leq \biggl\vert \int _{0}^{\nu } \biggl( \frac{1}{\Gamma ( \beta ) } ( \nu -\eta ) ^{ \beta -1}K_{1}\bigl(\eta ,q(\eta )\bigr)- \frac{1}{\Gamma ( \beta ) } (\nu -\eta ) ^{ \beta -1}K_{1}\bigl(\eta , \varsigma (\eta )\bigr) \biggr)\,d\eta \biggr\vert \\& \qquad {}+ \biggl\vert \int _{0}^{1} \int _{0}^{\eta } \biggl( \frac{2}{\Gamma ( \beta ) } ( \eta -u ) ^{ \beta -1}K_{1}\bigl(\eta ,q(\eta )\bigr)- \frac{2}{\Gamma ( \beta ) } (\eta -u )^{ \beta -1}K_{1}\bigl(u, \varsigma (u)\bigr) \biggr)\,du\,d\eta \biggr\vert \\& \quad \leq \frac{1}{\Gamma (\beta ) } \frac{e^{-\tau }\Gamma (\beta +1 ) }{4M}\cdot \int _{0}^{ \nu } (\nu -\eta ) ^{\beta -1} \bigl(q(\eta )-\varsigma (\eta )\bigr)\,d\eta \\& \qquad {}+\frac{2}{\Gamma ( \beta ) } \frac{e^{-\tau }\Gamma (\beta +1 ) }{4M}\cdot \int _{0}^{1} \int _{0}^{\eta } (\eta -u ) ^{\beta -1} \bigl(\varsigma (u)-q(u) \bigr)\,du\,d\eta \\& \quad \leq \frac{1}{\Gamma ( \beta ) } \frac{e^{-\tau }\Gamma (\beta +1 ) }{4M}\cdot d(q, \varsigma )\cdot \int _{0}^{\nu } (\nu -\eta ) ^{\beta -1}\,d\eta \\& \qquad {}+\frac{2}{\Gamma ( \beta ) } \frac{e^{-\tau }\Gamma (\beta )\cdot \Gamma ( \beta +1 ) }{4M\Gamma ( s )\cdot \Gamma ( \beta +1 ) } \cdot d(q,\varsigma )\cdot \int _{0}^{1} \int _{0}^{\eta } (\eta -u ) ^{\beta -1}\,du\,d \eta \\& \quad \leq \biggl( \frac{e^{-\tau }\Gamma (\beta )\cdot \Gamma (\beta +1 ) }{4M\Gamma (\beta )\cdot \Gamma ( \beta +1 ) } \biggr)\cdot d(q,\varsigma )+2e^{-\tau }B ( \beta +1,1 ) \frac{\Gamma ( \beta )\cdot \Gamma ( \beta +1 ) }{4M\Gamma ( \beta )\cdot \Gamma ( \beta +1 ) } \cdot d(q,\varsigma ) \\& \quad \leq \frac{e^{-\tau }}{4M}d(q,\varsigma )+\frac{e^{-\tau }}{2M}d(q, \varsigma )< \frac{e^{-\tau }}{M}d(q,\varsigma ) \end{aligned}$$

where B is the beta mapping. The last inequality can be written as

$$ Md \bigl( R ( q ), R ( \varsigma ) \bigr) \leq f(q,\varsigma )\,d\bigl( R ( q ), R ( \varsigma ) \bigr) \leq e^{-\tau }\psi (q,\varsigma ). $$

(4.2)

Define the mapping $\psi (q(\nu ))=\ln (q(\nu ))$ for $q, \varsigma \in \mathcal{C}_{0,1}$. Then inequality (4.2) can be written as

$$ \tau +\psi \bigl(f(q,\varsigma )\,d\bigl(R ( q ), R ( \varsigma )\bigr) \bigr)\leq \psi \bigl( \psi (q,\varsigma ) \bigr). $$

By Theorem 3.4 the self-mapping R admits a fixed point, and hence equation (4.1) has a solution. □

5 Conclusion

The $(\psi ,\varphi )_{f}$-contractions are general enough to contain famous contractions. The theorems give a general criterion for the existence of unique fixed points of the self-mappings satisfying $(\psi ,\varphi )_{f}$-contractions. We investigated the existence of a solution to a fractional differential equation through fixed point methodology.