Skip to main content

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.

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. (1)

    \(\psi (y)< y\) for all \(y>0\),

  2. (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:

  1. (i)

    ψ is nondecreasing;

  2. (ii)

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

  3. (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:

  1. (i)

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

  2. (ii)

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

  3. (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 \(\lim_{n\rightarrow \infty }y_{n}=0\);

  4. (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 \(\epsilon >0\);

  5. (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:

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

  4. (iv)

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

  5. (v)

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

  6. (vi)

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

  7. (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 \(\varphi :(0,\infty )\rightarrow (0,\infty )\).

  1. (viii)

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

  2. (ix)

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

  3. (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. (1)

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

  2. (2)

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

  3. (3)

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

  4. (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. (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 \(d ( a,b ) \leq q H(A,B)\).

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:

  1. (i)

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

  2. (ii)

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

  3. (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:

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

  4. (iv)

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

  5. (v)

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

  6. (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

  1. (i)

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

  2. (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].

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:

  1. (i)

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

  2. (ii)

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

  3. (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:

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

  4. (iv)

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

  5. (v)

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

  6. (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. (1)

    \(\max \{d(\sigma ,\varsigma ),d(\sigma ,T(\sigma )),d(\varsigma ,T( \varsigma ))\}\),

  2. (2)

    \(\max \{d(\sigma ,T(\sigma )),d(\varsigma ,T(\varsigma ))\}\),

  3. (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. (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. (5)

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

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:

  1. (I)

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

    $$\begin{aligned}& \bigl\vert K_{1} \bigl(\eta ,q(\eta ) \bigr)-K_{1} \bigl(\eta , \varsigma (\eta ) \bigr) \bigr\vert \\& \quad \leq \frac{e^{-\tau }\Gamma ( \beta +1 ) }{4M} \bigl\vert q (\eta )-\varsigma (\eta ) \bigr\vert \bigl(M= \min \bigl\{ f(q, \varsigma )| q,\varsigma \in \mathcal{C}_{0,1}\bigr\} \bigr); \end{aligned}$$
  2. (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. □

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.

Availability of data and materials

Not applicable.

References

  1. 1.

    Abbas, M., Berzig, M., Nazir, T., Karapinar, E.: Iterative approximation of fixed points for presic type F-contraction operators. UPB Sci. Bull., Ser. A, Appl. Math. Phys. 78(2), 147–160 (2016)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Acar, O., Altun, I.: Multivalued F-contractive mappings with a graph and some fixed point results. Publ. Math. (Debr.) 88, 305–317 (2016)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Acar, O., Durmaz, G., Minak, G.: Generalized multivalued F-contractions on complete metric space. Bull. Iran. Math. Soc. 40(6), 1469–1478 (2014)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Adigüzel, R.S., Aksoy, U., Karapinar, E., Erhan, I.M.: On the solution of a boundary value problem associated with a fractional differential equation. Math. Methods Appl. Sci. https://doi.org/10.1002/mma.665

  5. 5.

    Afshari, H., Aydi, H., Karapinar, E.: Some fixed point results for multivalued mappings in b-metric spaces. East Asian Math. J. 32, 319–332 (2016)

    Article  Google Scholar 

  6. 6.

    Afshari, H., Kalantari, S., Karapinar, E.: Solution of fractional differential equations via coupled fixed point. Electron. J. Differ. Equ. 2015, 286 (2015). http://ejde.math.txstate.edu

    MathSciNet  Article  Google Scholar 

  7. 7.

    Ahmad, J., Al-Mazrooei, A.E., Cho, Y.J., Yang, Y.O.: Fixed point results for generalized Θ-contractions. J. Nonlinear Sci. Appl. 10, 2350–2358 (2017)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Alizadeh, S., Baleanu, D., Rezapour, S.: Analyzing transient response of the parallel RCL circuit by using the Caputo–Fabrizio fractional derivative. Adv. Differ. Equ. 2020, 55 (2020)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Altun, I., Qasim, M.: Application of Perov type fixed point results to complex partial differential equations. Math. Methods Appl. Sci. 44(2), 2059–2070 (2021). https://doi.org/10.1002/mma.6915

    MathSciNet  Article  Google Scholar 

  10. 10.

    Amini-Harandi, A., Petrusel, A.: A fixed point theorem by altering distance technique in complete metric spaces. Miskolc Math. Notes 14, 11–17 (2013)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Asif, A., Nazam, M., Arshad, M., Kim, S.O.: F-metric, F-contraction and common fixed-point theorems with applications. Mathematics 7(7), 586 (2019). https://doi.org/10.3390/math7070586

    Article  Google Scholar 

  12. 12.

    Aydi, H., Karapinar, E., Mitrovic, Z.D., Rashid, T.: A remark on “Existence and uniqueness for a neutral differential problem with unbounded delay via fixed point results F-metric spaces”. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113, 3197–3206 (2019). https://doi.org/10.1007/s13398-019-00690-9

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Aydi, H., Karapinar, E., Yazidi, H.: Modified F-contractions via alpha-admissible mappings and application to integral equations. Filomat 31(5), 1141–1148 (2017) https://www.jstor.org/stable/10.2307/24902211

    MathSciNet  Article  Google Scholar 

  14. 14.

    Baleanu, D., Mohammadi, H., Rezapour, S.: Analysis of the model of HIV-1 infection of \(CD^{4+}\) T-cell with a new approach of fractional derivative. Adv. Differ. Equ. 2020, 71 (2020)

    Article  Google Scholar 

  15. 15.

    Banach, S.: Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales. Fundam. Math. 3, 133–181 (1922)

    Article  Google Scholar 

  16. 16.

    Bera, A., Garai, H., Damjanovic, B., Chanda, A.: Some interesting results on F-metric spaces. Filomat 33(10), 3257–3268 (2019). https://doi.org/10.2298/FIL1910257B

    MathSciNet  Article  Google Scholar 

  17. 17.

    Binh, T.T., Luc, N.H., O’Regan, D., Can, N.H.: On an initial inverse problem for a diffusion equation with a conformable derivative. Adv. Differ. Equ. 2019, 481 (2019). https://doi.org/10.1186/s13662-019-2410-z

    MathSciNet  Article  Google Scholar 

  18. 18.

    Boyd, D.W., Wong, J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Browder, F.E.: On the convergence of successive approximations for nonlinear functional equations. Indag. Math. 30, 27–35 (1968)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Chifu, C.: Common fixed point results in extended b-metric spaces endowed with a directed graph. Results Nonlinear Anal. 2(1), 18–24 (2019)

    Google Scholar 

  21. 21.

    Ćirić, L.B.: Generalized contractions and fixed-point theorems. Publ. Inst. Math. 26, 19–26 (1971)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Durmaz, G., Altun, I.: Fixed point results for α-admissible multivalued F-contractions. Miskolc Math. Notes 17(1), 187–199 (2016)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Geraghty, M.A.: On contractive mappings. Proc. Am. Math. Soc. 40, 604–608 (1973)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Jleli, M., Samet, B.: A new generalization of the Banach contraction principle. J. Inequal. Appl. 2014, 38 (2014)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Karapinar, E., Fulga, A., Agarwal, R.P.: A survey: F-contractions with related fixed point results. J. Fixed Point Theory Appl. 22, 69 (2020). https://doi.org/10.1007/s11784-020-00803-7

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Li, Z., Jiang, S.: Fixed point theorems of JS-quasi-contractions. Fixed Point Theory Appl. 2016, 40 (2016)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Lukács, A., Kajánto, S.: Fixed point theorems for various types of F-contractions in complete b-metric spaces. Fixed Point Theory 19, 321–334 (2018)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Moradi, S.: Fixed point of single-valued cyclic weak φF-contraction mappings. Filomat 28, 1747–1752 (2014)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Nadler, S.B.: Multivalued contraction mappings. Pac. J. Math. 30, 475–488 (1969)

    Article  Google Scholar 

  30. 30.

    Nazam, M.: On \(J_{c}\)-contraction and related fixed point problem with applications. Math. Methods Appl. Sci. 43(17), 10221–10236 (2020). https://doi.org/10.1002/mma.6689

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Ozturk, A.: A fixed point theorem for mappings with an F-contractive iterate. Adv. Theory Nonlinear Anal. Appl. 3(4), 231–236 (2019). https://doi.org/10.31197/atnaa.644325

    Article  Google Scholar 

  32. 32.

    Ozyurt, S.G.: On some alpha-admissible contraction mappings on Branciari b-metric spaces. Adv. Theory Nonlinear Anal. Appl. 1(1), 1–13 (2017). https://doi.org/10.31197/atnaa.318445

    Article  MATH  Google Scholar 

  33. 33.

    Ozyurt, S.G.: A fixed point theorem for extended large contraction mappings. Results Nonlinear Anal. 1(1), 46–48 (2018)

    Google Scholar 

  34. 34.

    Proinov, P.D.: Fixed point theorems for generalized contractive mappings in metric spaces. J. Fixed Point Theory Appl. 22, 21 (2020). https://doi.org/10.1007/s11784-020-0756-1

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Rakotch, E.: A note on contractive mappings. Proc. Am. Math. Soc. 13, 459–465 (1962)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Secelean, N.A.: Weak F-contractions and some fixed point results. Bull. Iranian Math. Soc. 42, 779–798 (2016)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Skof, F.: Teoremi di punto fisso per applicazioni negli spazi metrici. Atti Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat. 111, 323–329 (1977)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Tuan, N.H., Thach, T.N., Can, N.H., O’Regan, D.: Regularization of a multidimensional diffusion equation with conformable time derivative and discrete data. Math. Methods Appl. Sci. 44(4), 2879–2891 (2021). https://doi.org/10.1002/mma.6133

    MathSciNet  Article  Google Scholar 

  39. 39.

    Tuan, N.H., Zhou, Y., Long, L.D., Can, N.H.: Identifying inverse source for fractional diffusion equation with Riemann–Liouville derivative. Comput. Appl. Math. 39, 75 (2020). https://doi.org/10.1007/s40314-020-1103-2

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Wardowski, D.: Fixed points of new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012, 94 (2012)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Wardowski, D., Van Dung, N.: Fixed points of F-weak contractions on complete metric spaces. Demonstr. Math. 1, 146–155 (2014)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to express our sincere gratitude to the anonymous referee for his/her helpful comments, which helped to improve the quality of the manuscript.

Funding

Funding is not available.

Author information

Affiliations

Authors

Contributions

The authors equally conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Corresponding authors

Correspondence to Muhammad Nazam or Choonkil Park.

Ethics declarations

Ethics approval and consent to participate

We would like to mention that this papere does not contain any studies with animals and does not involve any studies over human being.

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nazam, M., Park, C. & Arshad, M. Fixed point problems for generalized contractions with applications. Adv Differ Equ 2021, 247 (2021). https://doi.org/10.1186/s13662-021-03405-w

Download citation

MSC

  • 47H10
  • 26E05
  • 26E25

Keywords

  • Fixed point
  • \((\psi ,\varphi )\)-contraction
  • Complete metric space
  • Applications