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Solution of fractional differential equations in quasi-b-metric and b-metric-like spaces

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Abstract

In this article, using by α-admissible and \(\alpha _{qs^{p}}\)-admissible mappings, solutions of some fractional differential equations are investigated in quasi-b-metric and b-metric-like spaces.

Introduction and preliminaries

Throughout this paper we denote the set of continuous functions, b-metric space, b-metric-like space, and quasi-b-metric space by \(X=C(J)\), b-MS, b-MLS, and b-QMS, respectively, where \(J=[0,1]\).

In [24], the authors presented a new class of \(\alpha _{qs^{p}}\)-admissible mappings and proved some consequences in b-MLS. In 2016, Nawab Hussain et al. [10] stated some conclusions in ordered b-QMS.

The existence of a solution for problem

$$ D^{\kappa }w(\eta )=h\bigl(\eta ,w(\eta )\bigr)\quad \bigl(\eta \in [0,1], 1< \kappa \leq 2\bigr) $$
(1)

has been studied widely by many authors.

In [6], Baleanu, Rezapour and Mohammadi studied Eq. (1) by \(\alpha \mbox{-}\psi \)-contractions. Similar ideas have also been considered by some authors; see, for example, [2, 3, 8, 9, 14,15,16, 18,19,20], and the references therein.

In [1], the authors obtained some conclusions for \(\alpha \mbox{-}\psi \)-Geraghty type mappings in b-MS. Recently in [4], Afshari, Kalantari and Baleanu obtained solutions of equation (1) by \(\alpha \mbox{-}\psi \)-Geraghty type mappings in b-MS. In this paper, using α- and \(\alpha _{qs^{p}}\)-admissible mappings, we find solutions for some fractional differential equations in b-MLS and b-QMS.

Definition 1.1

([12, 17])

The Riemann–Liouville derivative for a continuous function h is defined by

$$ D^{\kappa }h(\eta )=\frac{1}{\varGamma (m-\kappa )}\biggl(\frac{d}{d\eta } \biggr)^{m} \int _{0}^{\eta }\frac{h(\zeta )}{(\eta -\zeta )^{\kappa -m+1}}\,d\zeta\quad \bigl(m=[\kappa ]+1\bigr), $$

where the right-hand side is defined on \((0,\infty )\).

Definition 1.2

([21])

Let \(g: X\rightarrow X\), where X is nonempty, and \(\alpha : X\times X\rightarrow [0,\infty )\) be given, then g is α-admissible if for \(s,t\in X\), \(\alpha (s,t)\geq 1\) implies \(\alpha (gs,gt)\geq 1\).

Definition 1.3

([5])

Let X be a nonempty set. The map \(b_{l}:X\times X\rightarrow \mathbb{R}^{+}\) is said to be metric-like on X if for any \(w,y,z\in X\), the following hold:

  1. (i)

    \(b_{l}(w,y)=0\) implies \(w=y\);

  2. (ii)

    \(b_{l}(w,y)=b_{l}(y,w)\);

  3. (iii)

    \(b_{l}(w,y)\leq s(b_{l}(w,z)+b_{l}(z,y))\).

The pair \((X,b_{l})\) called a b-MLS.

Let \(\alpha : X\times X\rightarrow [0,\infty )\) and \(p,q\geq 1\) be arbitrary constants, then \(g:X\rightarrow X\) is \(\alpha _{qs^{p}}\)-admissible if \(\alpha (w,y)\geq qs^{p}\) implies \(\alpha (gw,gy)\geq qs^{p}\) for all \(w,y\in X\). We further consider the following properties:

(\(H_{s^{p}}\)):

If \(\{w_{n}\}\subseteq X\) with \(w_{n}\rightarrow w \in X\) and \(\alpha (w_{n},w_{n+1})\geq s^{p}\), then there exists a subsequence \(\{w_{n_{k}}\}\) of \(\{w_{n}\}\) such that \(\alpha (w_{n _{k}},w)\geq s^{p}\) for all \(k\in N\).

Let Θ be the set of all mappings \(\gamma :[0,\infty )\rightarrow [0,1)\) such that \(\gamma (t_{n})\rightarrow 1\) implies that \(t_{n}\rightarrow 0\).

Proposition 1.4

([24])

Let \((X,b_{l})\) be a complete b-MLS with parameter \(s\geq 1\), let \(g:X\rightarrow X\) and \(\alpha : X\times X\rightarrow [0,\infty )\). Suppose

  1. (i)

    g is \(\alpha _{s^{p}}\)-admissible;

  2. (ii)

    There exists \(\gamma \in \varTheta \) such that

    $$ \alpha (w,y)b_{l}(gw,gy)\leq \gamma \bigl(b_{l}(w,y) \bigr)b_{l}(w,y); $$
    (2)
  3. (iii)

    There exists \(w_{0}\in X\) with \(\alpha (w_{0},gw_{0})\geq s ^{p}\);

  4. (iv)

    Either g is continuous or property (\(H_{s^{p}}\)) is satisfied.

Then g has a fixed point.

Main result

We endow X with

$$ b_{l}(w,y)=\max_{t\in J}\bigl( \bigl\vert w(t) \bigr\vert + \bigl\vert y(t) \bigr\vert \bigr)^{p}, $$
(3)

for \(w,y\in X\), where \(p>1\). Then \((X,b_{l})\) is a complete b-MLS with \(s=2^{p-1}\). Now we study the problem

$$ -D^{\kappa }w(\eta )=f\bigl(\eta , w(\eta )\bigr), \quad \eta \in (0,1), $$
(4)

with the boundary condition (BC)

$$ w(0)=w'(0)=w'(1)=0, \quad 2< \kappa < 3, $$
(5)

where \(f\in C(J\times [0,+\infty ),\mathbb{R})\) and \(D^{\kappa }\) is the Riemann–Liouville derivative.

Lemma 2.1

([23])

Given \(f\in C(J\times X,\mathbb{R})\) and \(2 <\kappa <3\), the unique solution of (4) with (BC) (5) is given by \(w(\eta )= \int _{0}^{1} G(\eta ,\zeta )f(\zeta ,w(\zeta ))\,d\zeta \), where

$$ G(\eta ,\zeta )=\textstyle\begin{cases} \frac{\eta ^{\kappa -1}(1-\zeta )^{\kappa -2}-(\eta -\zeta )^{\kappa -1}}{ \varGamma (\kappa )},& 0\leq \zeta \leq \eta \leq 1, \\ \frac{\eta ^{\kappa -1}(1-\zeta )^{\kappa -2}}{\varGamma (\kappa )},& 0 \leq \eta \leq \zeta \leq 1. \end{cases} $$
(6)

Lemma 2.2

([23])

The function \(G(\eta ,\zeta )\) defined by (6) satisfies the following condition:

$$ \frac{\eta ^{\kappa -1}\zeta (1-\zeta )^{\kappa -2}}{\varGamma (\kappa )} \leq G(\eta ,\zeta )\leq \frac{\zeta (1-\zeta )^{\kappa -2}}{\varGamma ( \kappa )}, \quad 0 \leq \eta ,\zeta \leq 1. $$

Theorem 2.3

Suppose there exists \(\varphi :\mathbb{R}^{2}\rightarrow \mathbb{R}\) such that

  1. (i)

    There exists \(p>1\) such that

    $$\begin{aligned}& \bigl\vert f\bigl(\eta ,w(\eta )\bigr) \bigr\vert + \bigl\vert f\bigl(\eta ,y(\eta )\bigr) \bigr\vert \\& \quad \leq \frac{1}{2^{p-1}}\varGamma ( \kappa +1) (\kappa -1) \bigl({\gamma \bigl( \bigl\vert w(\eta ) \bigr\vert + \bigl\vert y(\eta ) \bigr\vert \bigr)^{p}}\bigr)^{ \frac{1}{p}}\bigl( \bigl\vert w(\eta ) \bigr\vert + \bigl\vert y(\eta ) \bigr\vert \bigr), \end{aligned}$$

    for \(w\in C(J)\), \(\eta \in J\);

  2. (ii)

    Inequality \(\varphi (w(\eta ),y(\eta ))\geq 0\) implies

    $$ \varphi \biggl( \int _{0}^{1}G(\eta ,\zeta )f\bigl(\zeta ,w( \zeta )\bigr)\,d\zeta , \int _{0}^{1}G(\eta ,\zeta )f\bigl(\zeta ,y( \zeta )\bigr)\,d\zeta \biggr)\geq 0; $$
  3. (iii)

    If \(\{w_{n}\}\subseteq C(J)\), \(w_{n}\rightarrow w\) in \(C(J)\) and \(\varphi (w_{n},w_{n+1})\geq 0\), then there exists a subsequence \(\{w_{n_{k}}\}\) of \(\{w_{n}\}\) such that \(\varphi (w_{n _{k}},w)\geq 0\) for all \(k\in N\);

  4. (iv)

    There exists \(w_{0}\in C(J)\) with \(\varphi (w_{0}(\eta ),\int _{0}^{1}G(\eta ,\zeta )f(\zeta ,w_{0}(\zeta ))\,d\zeta )\geq 0\).

Then problem (4) has at least one solution in \((X,b_{l})\).

Proof

By Lemma 2.1, \(w\in C(J)\) is a solution of (4) if and only if it is a solution of \(w(\eta )=\int _{0}^{1} G(\eta ,\zeta )f(\zeta ,w(\zeta ))\,d\zeta \). Define \(T: C(J)\rightarrow C(J)\) by \(Tw(\eta )= \int _{0}^{1} G(\eta ,\zeta )f(\zeta ,w(\zeta ))\,d\zeta \), for all \(\eta \in J\). We find a fixed point of T. Observe that

$$\begin{aligned} &\bigl( \bigl\vert Tw(\eta ) \bigr\vert + \bigl\vert Ty(\eta ) \bigr\vert \bigr)^{p} \\ &\quad = \biggl( \biggl\vert \int _{0}^{1} G(\eta ,\zeta )f\bigl( \zeta ,w( \zeta )\bigr)\,d\zeta \biggr\vert + \biggl\vert \int _{0}^{1}G(\eta ,\zeta )f\bigl(\zeta ,y( \zeta )\bigr)\,d\zeta \biggr\vert \biggr)^{p} \\ &\quad \leq \biggl[ \int _{0}^{1} G(\eta ,\zeta ) \bigl\vert f \bigl(\zeta ,w(\zeta )\bigr) \bigr\vert + \int _{0}^{1}G(\eta ,\zeta ) \bigl\vert f \bigl(\zeta ,y(\zeta )\bigr) \bigr\vert \,d\zeta \biggr]^{p} \\ & \quad = \biggl[ \int _{0}^{1} G(\eta ,\zeta ) \bigl( \bigl\vert f\bigl(\zeta ,w(\zeta )\bigr) \bigr\vert + \bigl\vert f\bigl( \zeta ,y(\zeta )\bigr) \bigr\vert \bigr)\,d\zeta \biggr]^{p} \\ &\quad \leq \biggl[ \int _{0}^{1} G(\eta ,\zeta ) \frac{1}{2^{p-1}}\varGamma ( \kappa +1) (\kappa -1) \bigl({\gamma \bigl( \bigl\vert w(\eta ) \bigr\vert + \bigl\vert y(\eta ) \bigr\vert \bigr)^{p}}\bigr)^{ \frac{1}{p}}\bigl( \bigl\vert w(\eta ) \bigr\vert + \bigl\vert y(\eta ) \bigr\vert \bigr)d\eta \biggr]^{p} \\ &\quad \leq \frac{1}{2^{p(p-1)}}\gamma \bigl( \bigl\vert w(\eta ) \bigr\vert + \bigl\vert y(\eta ) \bigr\vert \bigr)^{p}\bigl( \bigl\vert w( \eta ) \bigr\vert + \bigl\vert y(\eta ) \bigr\vert \bigr)^{p}, \end{aligned}$$

with \(\varphi (w(\eta ),y(\eta ))\geq 0\). Define \(\alpha : C(J)\times C(J)\rightarrow [0,\infty )\) by

$$ \alpha (w,y)=\textstyle\begin{cases} 2^{p(p-1)},&\varphi (w(\eta ),y(\eta ))\geq 0, \eta \in J, \\ 0,&\text{else}. \end{cases} $$

So

$$ \alpha (w,y)b_{l}(Tw,Ty)\leq \gamma \bigl(b_{l}(w,y) \bigr)b_{l}(w,y),\quad \gamma \in S. $$

Considering (ii), \(\alpha (w,y)\geq 2^{p(p-1)}=s^{p}\) implies \(\varphi (w(\eta ),y(\eta ))\geq 0\) and \(\varphi (T(w),T(y))\geq 0\) implies \(\alpha (T(w),T(y))\geq 2^{p(p-1)}=s^{p}\), \(w\in C(J)\). Thus, T is α-admissible. From (iv), there exists \(w_{0}\) \(\in C(J)\) with \(\alpha (w_{0},Tw_{0})\geq 1\). By (iii) and Proposition 1.4, we notice that \(w^{*}\in C(J)\) with \(w^{*}=Tw ^{*}\). □

Corollary 2.4

Suppose that for \(\eta \in J\) and \(w,y\in C(J)\) there exists \(p>1\) such that

$$ \bigl\vert f\bigl(\eta ,w(\eta )\bigr) \bigr\vert + \bigl\vert f\bigl(\eta ,y(\eta )\bigr) \bigr\vert \leq \frac{45\sqrt{\pi }}{2^{p+3}} \bigl({\gamma \bigl( \bigl\vert w(\eta ) \bigr\vert + \bigl\vert y(\eta ) \bigr\vert \bigr)^{p}\bigr)} ^{\frac{1}{p}}\bigl( \bigl\vert w(\eta ) \bigr\vert + \bigl\vert y(\eta ) \bigr\vert \bigr), $$

also conditions (ii)(v) from Theorem 2.3 hold for f, where \(G(\eta ,\zeta )\) is given in (6). Then the problem

$$ -\frac{D^{\frac{5}{2}}}{D\eta }w(\eta )=f\bigl(\eta ,w(\eta )\bigr), \quad \eta \in J, $$
(7)

where

$$ w(0)=w^{\prime }(0)=w^{\prime }(1)=0, $$

has at least one solution in \((X,b_{l})\).

Lemma 2.5

([13])

If \(f\in C(J\times [0,\infty ),\mathbb{R})\), then the problem

$$ \begin{aligned} &D_{0+}^{\kappa }z(\eta )+f \bigl(\eta ,z(\eta )\bigr)=0\quad (0< \eta < 1, 1< \kappa < 2), \\ & z(0)=z(1)=0. \end{aligned} $$
(8)

has a unique positive solution

$$ z(\eta )= \int _{0}^{1} G(\eta ,\zeta )f\bigl(\zeta ,z( \zeta )\bigr)\,d\zeta , $$

where \(G(\eta ,\zeta )\) is as follows:

$$ G(\eta ,\zeta )=\frac{1}{\varGamma (\kappa )}\textstyle\begin{cases} (\eta (1-\zeta ))^{\kappa -1}-(\eta -\zeta )^{\kappa -1},&\zeta \leq \eta , \\ (\eta (1-\zeta ))^{\kappa -1},&\eta \leq \zeta . \end{cases} $$
(9)

Lemma 2.6

([22])

Function \(G(\eta ,\zeta )\) in Lemma 2.5 has the following feature:

$$ \frac{\kappa -1}{\varGamma (\kappa )}\eta ^{\kappa -1}(1-\eta ) (1-\zeta )^{ \kappa -1} \zeta \leq G(\eta ,\zeta )\leq \frac{1}{\varGamma (\kappa )} \eta ^{\kappa -1}(1-\eta )^{\kappa -1}(1-\zeta )^{\kappa -2}, $$

where \(\eta ,\zeta \in J\), \(1<\kappa <2\).

From Theorem 2.11, we get the following result.

Corollary 2.7

Suppose for \(\eta \in J\) and \(w,y\in C(J)\) there exists \(p>1\) such that

$$ \bigl\vert f\bigl(\eta ,w(\eta )\bigr) \bigr\vert + \bigl\vert f\bigl(\eta ,y(\eta )\bigr) \bigr\vert \leq \frac{1}{M2^{p-1}} {\gamma \bigl(\bigl( \bigl\vert w( \eta ) \bigr\vert + \bigl\vert y(\eta ) \bigr\vert \bigr)^{p}\bigr)}^{\frac{1}{p}}\bigl( \bigl\vert w(\eta ) \bigr\vert + \bigl\vert y(\eta ) \bigr\vert \bigr), $$

where \(M=\sup_{\eta \in J} \int _{0}^{1}G(\eta ,\zeta )\,d\zeta \), also conditions (ii)(iv) from Theorem 2.3 are satisfied, where \(G(\eta ,\zeta )\) is given in (9). Then problem (8) has at least one solution.

Example 2.8

Endow \(X=C(J)\) with

$$ b_{l}(w,y)=\max_{\eta \in J}\bigl( \bigl\vert w(\eta ) \bigr\vert + \bigl\vert y(\eta ) \bigr\vert \bigr)^{2}, $$
(10)

then \((X,d)\) is a complete b-MLS with \(s=2\).

Let \(\varphi (w,y)=wy\) and \(w_{n}(\eta )= \frac{\eta {n^{2}}}{n^{2}+1}\). We consider \(f:J\times X\to \mathcal{R ^{+}}\) and the following periodic boundary value problem for \(w,y\in X\):

$$ -D^{\frac{5}{2}}w(\eta )=f\bigl(\eta , w(\eta )\bigr), \quad \eta \in (0,1), $$
(11)

with the boundary condition (BC)

$$ w(0)=w'(0)=w'(1)=0, $$

where f satisfies in the following condition:

$$ \bigl\vert f\bigl(\eta ,w(\eta )\bigr) \bigr\vert + \bigl\vert f\bigl(\eta ,y(\eta )\bigr) \bigr\vert \leq \frac{45\sqrt{\pi }}{64} \bigl({\gamma \bigl( \bigl\vert w(\eta ) \bigr\vert + \bigl\vert y(\eta ) \bigr\vert \bigr)^{2}}\bigr)^{ \frac{1}{2}}\bigl( \bigl\vert w(\eta ) \bigr\vert + \bigl\vert y(\eta ) \bigr\vert \bigr). $$

If \(w_{0}(\eta )=\eta \) then

$$ \varphi \biggl(w_{0}(\eta ), \int _{0}^{1}G(\eta ,\zeta )h\bigl(\zeta ,w_{0}( \zeta )\bigr)\,d\zeta \biggr)\geq 0, $$

for all \(\eta \in J\), also \(\varphi (w(\eta ),y(\eta ))=w(\eta )y( \eta )\geq 0\) implies that

$$ \varphi \biggl( \int _{0}^{1}G(\eta ,\zeta )f\bigl(\zeta ,w(\zeta )\bigr)\,d\zeta , \int _{0}^{1}G(\eta ,\zeta )f\bigl(\zeta ,y( \zeta )\bigr)\,d\zeta \biggr)\geq 0. $$

It is obvious that condition (iii) in Theorem 2.4 holds. Hence, from Theorem 2.4 problem (7) has at least one solution.

Definition 2.9

([11])

Let X be a nonempty set, \(s\geq 1\), and suppose \(q_{b}: X\times X \rightarrow [0,\infty )\), for all \(w,y\in X\), satisfies the following:

(\(q_{b_{1}}\)):

\(q_{b}(w,y)=0\) if and only if \(w=y\);

(\(q_{b_{2}}\)):

\(q_{b}(w,y)\leq s(q_{b}(w,z)+q_{b}(z,y))\) for all \(w,y,z\in X\).

The pair \((X,q_{b})\) is called a b-QMS.

Theorem 2.10

([10])

Let \((X,q_{b})\) be a complete b-QMS, \(g:X\rightarrow X\), and suppose there exists \(\alpha :X\times X\rightarrow [0,\infty )\) with

$$ \alpha (w,y)q_{b}(gw,gy)\leq kq_{b}(w,y), $$
(12)

for all \(w,y\in X\), \(k\in [0,s^{-1})\). Also assume

  1. (i)

    g is α-admissible;

  2. (ii)

    There exists \(w_{0}\in X\) such that \(\alpha (w_{0},gw_{0}) \geq 1\);

  3. (iii)

    If \(w_{n}\rightarrow w\), then \(\lim \sup_{n\rightarrow \infty } q_{b}(w_{n},y)\geq q_{b}(w,y)\), for all \(y \in X\);

  4. (iv)

    If \(\{w_{n}\}\subseteq X\), \(\alpha (w_{n},w_{n+1})\geq 1\), for all \(n\in N\), and \(w_{n}\rightarrow w\in X\), then there exists \(\{w_{n(k)}\}\) of \(\{w_{n}\}\) with \(\alpha (w_{n(k)},w)\geq 1\), for \(k\in N\).

Then there exists \(w\in X\) with \(g(w)=w\).

Let \(q_{b}:X\times X\to [0,\infty )\) be given by

$$ q_{b}(w,y)=\textstyle\begin{cases} \Vert (w-y)^{2} \Vert _{\infty }+ \Vert w \Vert _{\infty},& w,y\in X, w\neq y, \\ 0& \text{otherwise}, \end{cases} $$
(13)

where

$$ \Vert w \Vert _{\infty }=\sup_{\eta \in J} \bigl\vert w(\eta ) \bigr\vert . $$

Then \((X,q_{b})\) is a complete b-QMS with \(s=2\), but \((X,q_{b})\) is not b-MS.

Theorem 2.11

Suppose

  1. (i)

    There exists \(k\in [0,\frac{1}{2})\) such that \(|f(\eta ,w( \eta ))|\leq k\varGamma (\kappa +1)(\kappa -1)\| w \| _{ \infty }\), and

    $$ \bigl\vert f\bigl(\eta ,w(\eta )\bigr)-f\bigl(\eta ,y(\eta )\bigr) \bigr\vert \leq k\varGamma (\kappa +1) (1- \kappa ) \bigl\Vert (w-y)^{2} \bigr\Vert _{\infty } $$

    for \(w,y\in C(J)\), \(\eta \in J\).

  2. (ii)

    Inequality \(\varphi (w(\eta ),y(\eta ))\geq 0\) implies

    $$ \varphi \biggl( \int _{0}^{1}G(\eta ,\zeta )f\bigl(\zeta ,w( \zeta )\bigr)\,d\zeta , \int _{0}^{1}G(\eta ,\zeta )f\bigl(\zeta ,y( \zeta )\bigr)\,d\zeta \biggr)\geq 0; $$
  3. (iii)

    If \(w_{n}\rightarrow w\), \(w_{n},w\in C(J)\), then

    $$ \limsup_{n\rightarrow \infty } \bigl( \bigl\Vert (w_{n}-y)^{2} \bigr\Vert _{\infty }+ \Vert w_{n} \Vert _{\infty }\bigr)\geq \bigl\Vert (w-y)^{2} \bigr\Vert _{\infty }+ \Vert w \Vert _{\infty } $$
  4. (iv)

    If \(\{w_{n}\}\subseteq C(J)\), \(w_{n}\rightarrow w\) in \(C(J)\) and \(\varphi (w_{n},w_{n+1})\geq 0\) then there exists \(\{w_{n(i)}\}\) of \(\{w_{n}\}\), with \(\varphi (w_{n(i)},w)\geq 0\) for \(i\in N\).

  5. (v)

    There exists \(w_{0}\in C(J)\) with \(\varphi (w_{0}(\eta ),\int _{0}^{1}G(\eta ,\zeta )f(\zeta ,w_{0}(\zeta ))\,d\zeta )\geq 0\).

Then problem (4) has at least one solution.

Proof

By Lemma 2.1, \(w\in C(J)\) is a solution of (4) if and only if it is a solution of \(w(\eta )=\int _{0}^{1} G(\eta ,\zeta )f(\zeta ,w(\zeta ))\,d\zeta \). We define \(T : C(J)\rightarrow C(J)\) by \(Tw(\eta )=\int _{0}^{1} G(\eta ,\zeta )f(\zeta ,w(\zeta ))\,d\zeta \) for all \(\eta \in J\). For \(w\in C(J)\) with \(\varphi (w(\eta ),y(\eta )) \geq 0\) and \(\eta \in J\), using (i), we have

$$\begin{aligned} & \bigl\vert Tw(\eta )-Ty(\eta ) \bigr\vert ^{2}+ \bigl\vert Tw(\eta ) \bigr\vert \\ &\quad = \biggl\vert \int _{0}^{1} G(\eta , \zeta ) \bigl(f\bigl( \zeta ,w(\zeta )\bigr)-f\bigl(\zeta ,y(\zeta )\bigr)\bigr)\,d\zeta \biggr\vert ^{2} \\ &\qquad {}+ \biggl\vert \int _{0}^{1} G(\eta ,\zeta )f\bigl(\zeta ,w( \zeta )\bigr)\,d\zeta \biggr\vert \\ &\quad \leq \biggl( \int _{0}^{1} G(\eta ,\zeta ) \bigl\vert f \bigl(\zeta ,w(\zeta )\bigr)-f\bigl(\zeta ,y( \zeta )\bigr) \bigr\vert \,d\zeta \biggr)^{2} + \int _{0}^{1} G(\eta ,\zeta ) \bigl\vert f \bigl(\zeta ,w(\zeta )\bigr) \bigr\vert \,d\zeta \\ &\quad \leq \biggl( \int _{0}^{1} G(\eta ,\zeta )k\varGamma (\kappa +1) (1-\kappa ) \bigl\Vert (w-y)^{2} \bigr\Vert _{\infty }\,d \zeta \biggr)^{2} \\ &\qquad {}+ \int _{0}^{1} G(\eta ,\zeta )k\varGamma (\kappa +1) (1-\kappa ) \Vert w \Vert _{\infty }\,d\zeta \\ &\quad \leq k\bigl(\bigl( \bigl\Vert (w-y)^{2} \bigr\Vert _{\infty }\bigr)^{2}+ \Vert w \Vert _{\infty } \bigr)=kq_{b}(w,y). \end{aligned}$$

For \(w\in C(J)\), \(\eta \in J\) with \(\varphi (w(\eta ),y(\eta ))\geq 0\), we have

$$ \bigl\Vert (Tw-Ty)^{2} \bigr\Vert _{\infty }+ \Vert Tw \Vert _{ \infty }\leq kq_{b}(w,y). $$

Define \(\alpha : C(J)\times C(J)\rightarrow [0,\infty )\) by

$$ \alpha (w,y)=\textstyle\begin{cases} 1,&\varphi (w(\eta ),y(\eta ))\geq 0, \eta \in J, \\ 0,&\text{else}. \end{cases} $$

Then we have

$$ \alpha (w,y)q_{b}a(Tw,Ty) \leq q_{b}a(Tw,Ty)\leq kq_{b}(w,y), $$

from (ii); \(\alpha (w,y)\geq 1\) implies \(\varphi (w(\eta ),y(\eta )) \geq 0\), and \(\varphi (T(w),T(y))\geq 0\) implies \(\alpha (T(w), T(y)) \geq 1\), \(w\in C(J)\).

Thus, T is α-admissible. From \((v)\), there exists \(w_{0}\) \(\in C(J)\) with \(\alpha (w_{0},Tw_{0})\geq 1\). By (iii), (iv) and Theorem 2.10, we find that \(w^{*}\in C(J)\) with \(w^{*}=Tw^{*}\). □

Corollary 2.12

Suppose for \(\eta \in J\) and \(w\in C(J)\) there exists \(k\in [0, \frac{1}{2})\), \(\varphi :\mathbb{R}^{2}\rightarrow \mathbb{R}\) such that

$$ \begin{aligned} &\bigl\vert f\bigl(\eta ,w(\eta )\bigr) \bigr\vert \leq k \frac{45\sqrt{\pi }}{16} \Vert w \Vert _{\infty }, \\ &\bigl\vert f \bigl(\eta ,w(\eta )\bigr)-f\bigl(\eta ,y(\eta )\bigr) \bigr\vert \leq k \frac{45\sqrt{ \pi }}{16} \bigl\Vert (w-y)^{2} \bigr\Vert _{\infty }. \end{aligned} $$
(14)

Also assume that conditions (ii)(v) from Theorem 2.11 hold for f, where \(G(\eta ,\zeta )\) is given in (6). Then the problem

$$ -\frac{D^{\frac{5}{2}}}{D\eta }w(\eta )=f\bigl(\eta ,w(\eta )\bigr), \quad \eta \in J, \qquad w(0)=w^{\prime }(0)=w^{\prime }(1)=0, $$

has at least one solution.

Proof

By using Lemma 2.2,

$$ 0\leq \int _{0}^{1}G(\eta ,\zeta )\,d\zeta \leq \frac{16}{45\sqrt{ \pi }}, \quad \eta \in J. $$
(15)

By employing (14), (15) and in accordance with 2.11, we obtain

$$ \bigl\Vert (Tw-Ty)^{2} \bigr\Vert _{\infty }+ \Vert Tw \Vert _{ \infty }\leq k\bigl(\bigl( \bigl\Vert (w-y)^{2} \bigr\Vert _{\infty }\bigr)^{2}+ \Vert w \Vert _{\infty }\bigr)=kq_{b}(w,y). $$

The rest of proof is similar to that of Theorem 2.11. □

Corollary 2.13

Suppose for \(\eta \in J\) and \(w,y\in C(J)\) there exist \(k\in [0, \frac{1}{2})\) such that

$$ \bigl\vert f\bigl(\eta ,w(\eta )\bigr)-f\bigl(\eta ,y(\eta )\bigr) \bigr\vert \leq \frac{k}{M} \bigl\Vert (w-y)^{2} \bigr\Vert _{\infty }, \qquad \bigl\vert f\bigl(\eta ,w(\eta )\bigr) \bigr\vert \leq \frac{k}{M} \Vert w \Vert _{\infty }, $$

\(M=\sup_{\eta \in J} \int _{0}^{1}G(\eta ,\zeta )\,d\zeta \), also conditions (ii)(iv) from Theorem 2.11 are satisfied, where \(G(\eta ,\zeta )\) is given in (9). Then problem (8) has at least one solution.

Definition 2.14

([12, 17])

For a continuous function \(h:[0,\infty )\rightarrow \mathbb{R}\), the Caputo derivative of fractional order κ is defined by

$$ {}^{c}D^{\kappa }h(\eta )=\frac{1}{\varGamma (m-\kappa )} \int _{0}^{\eta }( \eta -\zeta )^{m-\kappa -1}h^{(m)}( \zeta )\,d\zeta , $$

where \(m-1<\kappa <m\), \(m=[\kappa ]+1\), and \([\kappa ]\) denotes the integer part of κ.

We consider

$$ {}^{c}D^{\kappa }w(\eta )+f\bigl(\eta ,w( \eta )\bigr)=0, \quad 0< \eta < 1, 2< \kappa < 3, $$
(16)

with boundary conditions (BC)

$$ w(0)=w^{\prime \prime }(0)=0, \qquad w(1)=\lambda \int _{0}^{1}w(\zeta )\,d\zeta . $$
(17)

Lemma 2.15

([7])

Let \(2<\kappa <3\), \(\lambda \neq 0\) and \(f\in C([0,T]\times X, \mathbb{R})\) be given. Then Eq. (16) with (BC) (17) has a unique solution given by

$$ w(\eta )= \int _{0}^{1}G(\eta ,\zeta )f\bigl(\zeta ,w( \zeta )\bigr)\,d\zeta , $$

where

$$ G(\eta ,\zeta )=\textstyle\begin{cases} \frac{2\eta (1-\zeta )^{\kappa -1})(\kappa -\lambda +\lambda \zeta )-(2- \lambda )\kappa (\eta -\zeta )^{\kappa -1})}{(2-\lambda )\varGamma ( \kappa +1)},&0\leq \zeta \leq \eta \leq 1, \\ \frac{2\eta (1-\zeta )^{\kappa -1})(\kappa -\lambda +\lambda \zeta )}{(2- \lambda )\varGamma (\kappa +1)},&0\leq \eta \leq \zeta \leq 1. \end{cases} $$
(18)

From Lemma 2.15 and Theorem 2.11, we get the following conclusion.

Corollary 2.16

Suppose for \(\eta \in J\) and \(w,y\in C(J)\) there exists \(k\in [0, \frac{1}{2})\), such that

$$\begin{aligned}& \bigl\vert f\bigl(\eta ,w(\eta )\bigr) \bigr\vert \leq \frac{k(2-\lambda )\varGamma (\kappa )}{2} \Vert w \Vert _{\infty }, \\& \bigl\vert f\bigl(\eta ,w(\eta ) \bigr)-f\bigl(\eta ,y(\eta )\bigr) \bigr\vert \leq \frac{k(2-\lambda )\varGamma (\kappa )}{2} \bigl\Vert (w-y)^{2} \bigr\Vert _{\infty }, \end{aligned}$$

where \(0<\lambda <2\); also suppose that conditions (ii)(iv) from Theorem 2.11 are satisfied, where \(G(\eta ,\zeta )\) is given in (18). Then (16) with (BC) (17) has at least one solution.

Let \((X,q_{b})\) be given in (13). For

$$ {}^{c}D^{\kappa }w(\eta )=f\bigl(\eta ,w( \eta )\bigr) \quad (\eta \in J, 1< \kappa \leq 2), $$
(19)

with

$$ w(0)=0,\qquad w(1)= \int _{0}^{\xi }w(\zeta )\,d\zeta \quad (0< \xi < 1), $$

where \(f:J\times X\rightarrow \mathbb{R}\) is continuous, we have the following result.

Theorem 2.17

Assume

  1. (i)

    There exists \(k\in [0,\frac{1}{2})\) such that \(|f(\eta ,w( \eta ))|\leq \frac{k}{2}\frac{\varGamma (\kappa +1)}{5}\| w \| _{\infty }\), and

    $$ \bigl\vert f\bigl(\eta ,w(\eta )\bigr)-f\bigl(\eta ,y(\eta )\bigr) \bigr\vert \leq \sqrt{\frac{k}{2}}\frac{ \varGamma (\kappa +1)}{5} \bigl\Vert (w-y)^{2} \bigr\Vert _{\infty } $$

    for \(w\in C(J)\), \(\eta \in J\).

  2. (ii)

    Inequality \(\varphi (w(\eta ),y(\eta ))\geq 0\) implies \(\varphi (T(w(\eta )),T(y(\eta )))\geq 0\), where \(T:C(J)\rightarrow C(J)\) is defined by

    $$\begin{aligned} Tw(\eta ) :=&\frac{1}{\varGamma (\kappa )} \int _{0}^{1}(\eta -\zeta )^{ \kappa -1}f \bigl(\zeta ,w(\zeta )\bigr) \,d\zeta \\ &{}-\frac{2\eta }{(2-\xi ^{2})\varGamma (\kappa )} \int _{0}^{1}(1-\zeta )^{\kappa -1}f\bigl( \zeta ,w(\zeta )\bigr)\, d \zeta \\ &{}+\frac{2\eta }{(2-\xi ^{2})\varGamma (\kappa )} \int _{0}^{\xi }\biggl( \int _{0} ^{\zeta }(\zeta -n)^{\kappa -1}f \bigl(n,w(n)\bigr)\,dn\biggr)\,d\zeta \quad (\eta \in J); \end{aligned}$$
  3. (iii)

    If \(w_{n}\rightarrow w\), \(w_{n},w\in C(J)\), then

    $$ \limsup_{n\rightarrow \infty } \bigl( \bigl\Vert (w_{n}-y)^{2} \bigr\Vert _{\infty }+ \Vert w_{n} \Vert _{\infty }\bigr)\geq \bigl\Vert (w-y)^{2} \bigr\Vert _{\infty }+ \Vert w \Vert _{\infty }; $$
  4. (iv)

    If \(\{w_{n}\}\subseteq C(J)\), \(w_{n}\rightarrow w\) in \(C(J)\) and \(\varphi (w_{n},w_{n+1})\geq 0\) then there exists \(\{w_{n(i)}\}\) of \(\{w_{n}\}\), with \(\varphi (w_{n(i)},w)\geq 0\) for \(i\in N\);

  5. (v)

    There exists \(w_{0}\in C(J)\) with \(\varphi (w_{0}(\eta ),T(w _{0}(\eta )))\geq 0\).

Then (19) has at least one solution.

Proof

Function \(w\in C(J)\) is a solution of (19) if and only if it is a solution of

$$\begin{aligned} w(\eta ) =&\frac{1}{\varGamma ({\kappa })} \int _{0}^{1}(\eta -\zeta )^{ \kappa -1}f \bigl(\zeta ,w(\zeta )\bigr) \,d\zeta -\frac{2\eta }{(2-{\xi }^{2}) \varGamma (\kappa )} \int _{0}^{1}(1-\zeta )^{\kappa -1}f\bigl( \zeta ,w(\zeta )\bigr)\, d \zeta \\ &{}+\frac{2\eta }{(2-{\xi }^{2})\varGamma (\kappa )} \int _{0}^{{\xi }}\biggl( \int _{0}^{\zeta }(\zeta -n)^{\kappa -1}f \bigl(n,w(n)\bigr)\,dn\biggr)\,d\zeta \quad ( \eta \in J). \end{aligned}$$

Then (19) is replaceable to get \(w^{*}\in C(J)\), with \(Tw^{*}=w^{*}\). Let \(w\in C(J)\) with \(\varphi (w(\eta ),y(\eta )) \geq 0\), \(\eta \in J\). By (i), we have

$$\begin{aligned}& \bigl\vert Tw(\eta )-Ty(\eta ) \bigr\vert ^{2}+ \bigl\vert Tw(\eta ) \bigr\vert \\& \quad = \biggl\vert \frac{1}{\varGamma (\kappa )} \int _{0}^{1}(\eta -\zeta )^{\kappa -1}f \bigl(\zeta ,w(\zeta )\bigr)\,d\zeta \\& \qquad {} -\frac{2\eta }{(2-\xi ^{2})\varGamma (\kappa )} \int _{0}^{1}(1-\zeta )^{ \kappa -1}f\bigl( \zeta ,w(\zeta )\bigr)\,d\zeta \\& \qquad {} +\frac{2\eta }{(2-\xi ^{2})\varGamma (\kappa )} \int _{0}^{\xi }\biggl( \int _{0} ^{\zeta }(\zeta -n)^{\kappa -1}f \bigl(n,w(n)\bigr)\,dn\biggr)\,d\zeta \\& \qquad {} -\frac{1}{\varGamma (\alpha )} \int _{0}^{1}(\eta -\zeta )^{\kappa -1}f \bigl( \zeta ,y(\zeta )\bigr)\,d\zeta \\& \qquad {} +\frac{2\eta }{(2-\xi ^{2})\varGamma (\kappa )} \int _{0}^{1}(1-\zeta )^{ \kappa -1}f\bigl( \zeta ,y(\zeta )\bigr)\,d\zeta \\& \qquad {} -\frac{2\eta }{(2-\xi ^{2})\varGamma (\kappa )} \int _{0}^{\xi }\biggl( \int _{0} ^{\zeta }(\zeta -n)^{\kappa -1}f \bigl(n,y(n)\bigr)\,dn\biggr)\,d\zeta \biggr\vert ^{2} \\& \qquad {} + \biggl\vert \frac{1}{\varGamma (\kappa )} \int _{0}^{1}(\eta -\zeta )^{\kappa -1}f \bigl( \zeta ,w(\zeta )\bigr) \,d\zeta -\frac{2\eta }{(2-\xi ^{2})\varGamma (\kappa )} \int _{0}^{1}(1-\zeta )^{\kappa -1}f\bigl( \zeta ,w(\zeta )\bigr)\,d\zeta \\& \qquad {} +\frac{2\eta }{(2-\xi ^{2})\varGamma (\kappa )} \int _{0}^{\xi }\biggl( \int _{0} ^{\zeta }(\zeta -n)^{\kappa -1}f \bigl(n,w(n)\bigr)\,dn\biggr)\,d\zeta \biggr\vert \\& \quad \leq \biggl[\frac{1}{\varGamma (\kappa )} \int _{0}^{1} \vert \eta -\zeta \vert ^{\kappa -1} \bigl\vert f\bigl( \zeta ,w(\zeta )\bigr)-f\bigl(\zeta ,y( \zeta )\bigr) \bigr\vert \,d\zeta \\& \qquad {} +\frac{2\eta }{(2-\xi ^{2})\varGamma (\kappa )} \int _{0}^{1} \vert 1-\zeta \vert ^{ \kappa -1} \bigl\vert f\bigl(\zeta ,w(\zeta )\bigr)-f\bigl(\zeta ,y( \zeta )\bigr) \bigr\vert \,d\zeta \\& \qquad {} +\frac{2\eta }{(2-\xi ^{2})\varGamma (\kappa )} \int _{0}^{\xi } \biggl\vert \int _{0} ^{\zeta } \vert \zeta -n \vert ^{\kappa -1} \bigl\vert f\bigl(n,w(n)\bigr)-f\bigl(n,y(n)\bigr) \bigr\vert \,dn \biggr\vert \,d\zeta \biggr]^{2} \\& \qquad {} +\frac{1}{\varGamma (\kappa )} \int _{0}^{1} \bigl\vert (\eta -\zeta ) \bigr\vert ^{\kappa -1} \bigl\vert f\bigl( \zeta ,w(\zeta )\bigr) \bigr\vert \,d\zeta \\& \qquad {}+\frac{2\eta }{(2-\xi ^{2})\varGamma (\kappa )} \int _{0}^{1} \bigl\vert (1-\zeta ) \bigr\vert ^{\kappa -1} \bigl\vert f\bigl(\zeta ,w(\zeta )\bigr) \bigr\vert \,d\zeta \\& \qquad {} +\frac{2\eta }{(2-\xi ^{2})\varGamma (\kappa )} \int _{0}^{\xi }\biggl( \int _{0} ^{\zeta } \bigl\vert (\zeta -n) \bigr\vert ^{\kappa -1} \bigl\vert f\bigl(n,w(n)\bigr) \bigr\vert \,dn \biggr)\,d\zeta \\& \quad \leq \biggl(\frac{\varGamma (\kappa +1)}{5}\biggr)^{2} \frac{k}{2} \Vert w-y \Vert _{\infty }^{2} \biggl[\sup \biggl( \int _{0}^{1} \vert \eta -\zeta \vert ^{\kappa -1}\, d \zeta +\frac{2\eta }{(2-\xi ^{2})\varGamma (\kappa )} \int _{0}^{1} \vert 1- \zeta \vert ^{\kappa -1} \,d\zeta \\& \qquad {} +\frac{2\eta }{(2-\xi ^{2})\varGamma (\kappa )} \int _{0}^{\xi }\biggl( \int _{0} ^{\zeta } \vert \zeta -n \vert ^{\kappa -1}\,dn\biggr)\,d\zeta \biggr)\biggr]^{2} \\& \qquad {} +\frac{\varGamma (\kappa +1)}{5}\frac{k}{2} \Vert w-y \Vert _{ \infty }\biggl[\sup \biggl( \int _{0}^{1} \vert \eta -\zeta \vert ^{\kappa -1}\,d\zeta +\frac{2 \eta }{(2-\xi ^{2})\varGamma (\kappa )} \int _{0}^{1} \vert 1-\zeta \vert ^{\kappa -1}\, d \zeta \\& \qquad {} +\frac{2\eta }{(2-\xi ^{2})\varGamma (\kappa )} \int _{0}^{\xi }\biggl( \int _{0} ^{\zeta } \vert \zeta -n \vert ^{\kappa -1}\,dn\biggr)\,d\zeta \biggr)\biggr] \leq k\bigl( \Vert w-y \Vert _{\infty }^{2}+ \Vert w-y \Vert _{\infty } \bigr) \end{aligned}$$

for each \(w,y\in C(J)\) with \(\varphi (w(\eta ),y(\eta ))\geq 0\), \(\eta \in J\), and

$$ \bigl\Vert (Tw-Ty)^{2} \bigr\Vert _{\infty }+ \Vert Tw \Vert _{ \infty }\leq kq_{b}(w,y). $$

Suppose \(\alpha : C(J)\times C(J)\rightarrow [0,\infty )\) is defined by

$$ \alpha (w,y)=\textstyle\begin{cases} 1,&\varphi (w(\eta ),y(\eta ))\geq 0, \eta \in J, \\ 0,&\text{else}, \end{cases} $$

then

$$ \alpha (w,y)q_{b}(Tw,Ty) \leq q_{b}(Tw,Ty)\leq kq_{b}(w,y), $$

for \(w,y\in C(J)\). By Theorem 2.10, the result is obtained by the process of the proof of Theorem 2.11. □

Here, we find a positive solution for

$$ \frac{{}^{c}D^{\kappa }}{D\eta }w(\eta )=f\bigl(\eta ,w(\eta )\bigr), \quad 0< \kappa \leq 1, \eta \in J, $$
(20)

where

$$ w(0)+ \int _{0}^{1}w(\zeta )\,d\zeta =w(1). $$

We note that \({}^{c}D^{\nu }\) is the Caputo derivative of order ν. We consider the Banach space of continuous functions on J endowed with the sup norm. We have the following lemma.

Lemma 2.18

([7])

Let \(0<\kappa \leq 1\) and \(h\in C([0,T]\times X,\mathbb{R})\) be given. Then the equation

$$ {}^{c}D^{\kappa }w(\eta )=f\bigl(\eta ,w(\eta )\bigr) \quad \bigl(\eta \in [0,T], T\geq 1\bigr), $$

with

$$ w(0)+ \int _{0}^{T}w(\zeta )\,d\zeta =w(T), $$

has a unique solution given by

$$ w(\eta )= \int _{0}^{T}G(\eta ,\zeta )f\bigl(\zeta ,w( \zeta )\bigr)\,d\zeta , $$

where \(G(\eta ,\zeta )\) is defined by

$$\begin{aligned} G(\eta ,\zeta )=\textstyle\begin{cases} \frac{-(T-\zeta )^{\kappa }+\kappa T(\eta -\zeta )^{\kappa -1}}{T \varGamma (\kappa +1)}+\frac{(T-\zeta )^{\kappa -1}}{T\varGamma (\kappa )},&0 \leq \zeta < \eta , \\ \frac{-(T-\zeta )^{\kappa }}{T\varGamma (\kappa +1)}+\frac{(T-\zeta )^{ \kappa -1}}{T\varGamma (\kappa )},&\eta \leq \zeta < T. \end{cases}\displaystyle \end{aligned}$$
(21)

From Lemma 2.18 and Theorem 2.11, we get the following conclusion.

Corollary 2.19

Assume

  1. (i)

    There exists \(k\in [0,\frac{1}{2})\) such that \(|f(\eta ,w( \eta ))|\leq \frac{51k}{80}\| w \| _{\infty }\), and

    $$ \bigl\vert f\bigl(\eta ,w(\eta )\bigr)-f\bigl(\eta ,y(\eta )\bigr) \bigr\vert \leq \frac{51k}{80} \bigl\Vert (w-y)^{2} \bigr\Vert _{\infty } $$

    for \(w,y\in C(J)\), \(\eta \in J\).

Suppose that conditions (ii)(iv) from Theorem 2.11 are met, where \(G(\eta ,\zeta )\) is given in (21), then the following problem has at least one solution:

$$ {}^{c}D^{\frac{1}{2}}w(\eta )=f\bigl(\eta ,w(\eta )\bigr) \quad \bigl(\eta \in [0,1]\bigr), \qquad w(0)+ \int _{0}^{1}w(\zeta )\,d\zeta =w(1). $$

Example 2.20

Let \(X=C(J)\) and \(q_{b}:X\times X\to [0,\infty )\) be given by

$$ q_{b}(w,y)=\textstyle\begin{cases} \Vert (w-y)^{2} \Vert _{\infty }+ \Vert w \Vert _{\infty },& w,y\in X, w\neq y, \\ 0,& \text{otherwise}. \end{cases} $$
(22)

Then \((X,d)\) is a complete b-QMS with \(s=2\), but is not a b-metric space.

Let \(\theta (w,y)=w^{3}y^{3}\), \(w_{n}(\eta )=\frac{\eta }{n^{2}+1}\). We consider \(f:J\times [0,5]\to [0,5]\) and the periodic boundary value problem

$$ {}^{c}D^{\frac{1}{2}}w(\eta )=f\bigl(\eta ,w( \eta )\bigr) \quad (\eta \in J), $$
(23)

with

$$ w(0)=0,\qquad w(1)= \int _{0}^{\xi }w(\zeta )\,d\zeta \quad (0< \xi < 1), $$

and suppose there exists \(k\in [0,\frac{1}{2})\) such that f satisfies in the following condition:

$$ \bigl\vert f\bigl(\eta ,w(\eta )\bigr) \bigr\vert \leq \frac{51k}{80} \Vert w \Vert _{\infty },\qquad \bigl\vert f\bigl(\eta ,w(\eta ) \bigr)-f\bigl(\eta ,y(\eta )\bigr) \bigr\vert \leq \frac{51k}{80} \bigl\Vert (w-y)^{2} \bigr\Vert _{\infty } $$

when \(\eta \in J\) and \(w(\eta ),y(\eta )\in [0,5]\). If \(w_{0}(\eta )= \eta \), then

$$ \theta \biggl(w_{0}(\eta ), \int _{0}^{1}G(\eta ,\zeta )f\bigl(\zeta ,y_{0}(\zeta )\bigr)\,d\zeta \biggr)\geq 0, $$

for all \(\eta \in J\), also \(\theta (w(\eta ),y(\eta ))={w(\eta )}^{3} {y(\eta )}^{3}\geq 0\) implies that

$$ \theta \biggl( \int _{0}^{1}G(\eta ,\zeta )f(\zeta ,w_{(}\zeta ))d\zeta_{,} \int _{0}^{1}G(\eta ,\zeta )f\bigl(\zeta ,y( \zeta )\bigr)\,d\zeta \biggr)\geq 0. $$

It is obvious that conditions (iii) and (iv) in Corollary 2.19 hold. Hence, from Corollary 2.19 problem (23) has at least one solution.

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Keywords

  • Fractional differential equation
  • \(\alpha _{qs^{p}}\)-admissible mappings
  • Quasi-b-metric and b-metric-like spaces