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

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.

1 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:

• (i) $b_{l}(w,y)=0$ implies $w=y$;

• (ii) $b_{l}(w,y)=b_{l}(y,w)$;

• (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

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

• (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)

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

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

Then g has a fixed point.

2 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

• (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 $w\in C(J)$, $\eta \in J$;

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

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

• (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 $k\in N$;

• (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

• (i) g is α-admissible;

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

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

• (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

• (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 $w,y\in C(J)$, $\eta \in J$.

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

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

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

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

• (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 $i\in N$.

• (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

• (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 $w\in C(J)$, $\eta \in J$.

• (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

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

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

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

• (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 $i\in N$;

• (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

• (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.