Analysis of Nonlinear Fractional Differential Equations Involving
Atangana–Baleanu–Caputo Derivative

Kishor D. Kucche 1

<EMAIL_ADDRESS>
Sagar T. Sutar 2

<EMAIL_ADDRESS>
1 Department of Mathematics, Shivaji University, Kolhapur-416 004,
Maharashtra, India.

2 Department of Mathematics, Vivekanand College (Autonomous), Kolhapur-416003,
Maharashtra, India.

###### Abstract

In the present paper, we determine the estimations on Atangana–Baleanu–Caputo
fractional derivative at extreme points. With the assistance of the
estimations obtained, we derive the comparison results. Peano’s type existence
results established for nonlinear fractional differential equations involving
Atangana–Baleanu–Caputo fractional derivative. The acquired comparison results
are then utilized to deal with the existence of local, extremal and global
solution.

Key words: Fractional differential equations, Atangana–Baleanu–Caputo
fractional derivative, Fractional differential inequalities, Comparison
results, Local and global existence.
2010 Mathematics Subject Classification: 26A33, 34A12, 34A40, 35B50, 34A99

##  1 Introduction

The fundamental theory of fractional calculus has been exhibited principally
with the two fractional derivative operators Riemann-Liouville and Caputo
fractional derivatives. The Riemann-Liouville (RL) fractional derivative plays
an important role mainly in the development of the theory of fractional
derivatives, integrals and for application to develop various theories in pure
mathematics. The RL fractional integral of the Caputo fractional derivative
generates the initial conditions in the form of classical derivatives. As a
result of which we have a clear physical interpretation in the modeling of
real-world phenomena in the form of fractional differential equations (FDEs).
Basic calculus and the theoretical development FDEs involving these two
fractional derivative operators have been excellently presented in the
monographs [1, 2, 3, 4, 5].

Kai et al. [6] researched the existence and uniqueness of solutions,
structural stability and the dependence of the solution on the order of the
differential equation and on the initial condition. Daftardar-Gejji et al. [7]
broadened these investigations for the system of fractional differential
equations. Lakshmikantham and Vatsala [8, 9] built up the theory of fractional
differential and integral inequalities and employed it to explore the
existence of extremal and global solutions of nonlinear FDEs. Basic
development relating to investigations on the theory of nonlinear FDEs can be
found in [10, 11, 12, 13] and ongoing advancements can be found in [14, 15]
and references therein.

Researchers working in the field of applied mathematics have been developing
the theory of fractional calculus in various directions by defining different
types of arbitrary order derivatives and integrals. It is very well known that
the traditional fractional derivative operators provide better mathematical
modeling of many real-world phenomena than the classical integer derivatives.
In spite of this reality, numerous researchers accept that that worthy
exactness may not be accomplished in the modeling of physical phenomena
involving memory effect in the whole of the time duration due to the presence
of a singular kernel in the definition of traditional fractional derivatives.

To eliminate the singular kernel, a non-singular fractional derivative
operator with exponential kernel is proposed in [16] , which is well known as
Caputo–Fabrizio (CF) fractional derivative. For the development of the theory
relating to FDEs involving CF derivative and it’s real-world application one
can refer[17, 18, 19, 20] and the references given therein. Motivated by the
investigations of [16], Atangana and Baleanu in [21] proposed new fractional
derivative having Mittag-Leffler (ML) function as its kernel, which is well
known as Atangana–Baleanu–Caputo (ABC) fractional derivative. The basic
calculus of ABC-fractional derivative can be found in [22, 23, 24, 25, 26].
Non-locality of ABC-fractional derivative with singular ML kernel effectively
permits taking care of the nonlocal dynamics, computational purposes and
capturing the various features of realistic systems more suitably.
Mathematical modeling via ABC-FDEs of the various outbreak, such as dengue
fever, the free motion of a coupled oscillator, a tumor-immune surveillance
mechanism etc. and efficient numerical method to tackle this has been explored
in [27, 28, 29, 30, 31, 32].

Jarad et al. [33] provided sufficient conditions for the existence and
uniqueness of the solution of nonlinear ABC-FDEs. Authors derived the Gronwall
inequality in the frame of Atangana-Baleanu fractional integral and through it
investigated Ulam-Hyers stability of nonlinear ABC-FDEs. Baleanu et al.[34]
considered the existence and uniqueness of solution nonlinear ABC-FDEs and
structured a numerical procedure dependent on the fractional Euler and
predictor-corrector technique. Syam et al.[35] determined existence and
uniqueness results for the linear and nonlinear ABC-FDEs and exhibited a
numerical technique dependent on the Cheby-shev collocation technique. Afshari
et al. [36] demonstrated the existence results for ABC-FDEs utilizing the
fixed point theorems for contractive mappings such as
$\alpha$-$\gamma$–Geraghty type, $\alpha$-type $\mathcal{F}$-contraction in
$\mathcal{F}$-complete metric space. Ravichandran et al. [38, 39, 40, 41]
examined the existence and uniqueness of solution for ABC-fractional
differential and integrodifferential equations. Shah et al. [42] analyzed the
qualitative theory of existence and stability theory of Ulam’s type for
evolution ABC-FDEs.

Even if, many researchers have investigated nonlinear ABC-FDEs, it ought to be
seriously analyzed for the qualitative properties. In this view, motivated by
the applications and the interesting literature on ABC-FDEs, on the line of
[8, 9, 25], we investigates estimation on ABC-fractional derivatives at
extreme points, comparison results, local and global existence of a solution
and extremal solution for nonlinear ABC-FDEs of the form

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega(\tau)$
$\displaystyle=f\left(\tau,\omega(\tau)\right),\tau\in J=[0,T],\;T>0,$ (1.1)
$\displaystyle\omega(0)$ $\displaystyle=\omega_{0},$ (1.2)

where $0<\alpha<1,~{}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}$ is the ABC-
fractional derivative operator,
$\omega,~{}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega\in C(J)$ and $f\in
C(J\times\mathbb{R},\mathbb{R})$ is continuous non-linear function. The local
existence of solution to ABC-FDEs (1.1)–(1.2) is based on Peano’s theorem and
the comparison results which we have derived in the present paper.

The novelty of the present paper is that we have obtained comparison results,
local and global existence of a solution, and extremal solution without
demanding the monotonicity and Holder continuity assumption on the nonlinear
function associated with ABC-FDEs (1.1)–(1.2). Further, we provided
alternative proofs to few results of [25] pertain to fractional integral
inequalities.

This paper is organized as follows. In section 2, we recall basic definitions
and results related with ABC-fractional derivative. Section 3 deals with
estimation on ABC-fractional derivative at extreme points. In section 4, we
derive comparison results for ABC-FDEs involving initial and boundary
conditions. Section 5 deals with local existence and extremal of solution of
ABC-FDEs (1.1)–(1.2). In section 5, the global existence of solution is proved
for ABC-FDEs (1.1)–(1.2).

##  2 Preliminaries

In this section, we recall some definitions and basic results about ABC-
fractional derivative operator and generalized Mittag–Leffler function.

###### Definition 2.1

[35] Let $p\in[1,\infty)$ and $\Omega$ be an open subset of $\mathbb{R}$ the
Sobolev space $H^{p}(\Omega)$ is defined as

$H^{p}(\Omega)=\left\\{f\in L^{2}(\Omega):D^{\beta}f\in
L^{2}(\Omega),\text{for all }|\beta|\leq p\right\\}.$

###### Definition 2.2

[21] Let $\omega\in H^{1}(0,T)$ and $\alpha\in[0,1]$, the left
Atangana–Baleanu–Caputo fractional derivative of $\omega$ of order $\alpha$ is
defined by

${}^{ABC}{{}_{0}}\mathcal{D}_{\tau}^{\alpha}\omega(\tau)=\dfrac{B(\alpha)}{1-\alpha}\int_{0}^{\tau}\mathbb{E}_{\alpha}\left[-\dfrac{\alpha}{1-\alpha}(\tau-\sigma)^{\alpha}\right]\omega^{\prime}(\sigma)d\sigma,$

where $B(\alpha)>0$ is a normalization function satisfying $B(0)=B(1)=1$ and
$\mathbb{E}_{\alpha}$ is one parameter Mittag-Leffler function [3, 5] defined
by

$\mathbb{E}_{\alpha}(z)=\sum_{n=0}^{n=\infty}\frac{z^{n}}{\Gamma(n\alpha+1)}.$

The associated fractional integral is defined by

${}^{AB}{{}_{0}}I_{\tau}^{\alpha}\omega(\tau)=\dfrac{1-\alpha}{B(\alpha)}\omega(\tau)+\dfrac{\alpha}{B(\alpha)}~{}{{}_{0}}I_{\tau}^{\alpha}\omega(\tau).$

where

${{}_{0}}I_{\tau}^{\alpha}\omega(\tau)=\dfrac{1}{\Gamma(\alpha)}\int_{0}^{\tau}(\tau-\sigma)^{\alpha-1}\omega(\sigma)d\sigma,$

is the Riemann–Liouville fractional integral [3, 5] of $\omega$ of order
$\alpha$.

###### Lemma 2.1

[22] If $0<\alpha<1$, then
${}^{AB}{{}_{0}}I_{\tau}^{\alpha}\;\left({}^{ABC}{{}_{0}}\mathcal{D}_{\tau}^{\alpha}\omega(\tau)\right)=\omega(\tau)-\omega(0).$

###### Definition 2.3

[43, 44, 45] The generalized Mittag-Leffler function
$\mathbb{E}_{\alpha,\beta}^{\gamma}(z)$ for the complex numbers
$\alpha,\beta,\gamma$ with Re($\alpha)>0$ is defined as

$\mathbb{E}_{\alpha,\beta}^{\gamma}(z)=\sum_{k=0}^{\infty}\dfrac{(\gamma)_{k}}{\Gamma(\alpha
k+\beta)}\dfrac{z^{k}}{k!},$

where $(\gamma)_{k}$ is the Pochhammer symbol given by

$(\gamma)_{0}=1,\;(\gamma)_{k}=\gamma(\gamma+1)\cdots(\gamma+k-1),\;k=1,2,\cdots$

Note that,

$\mathbb{E}_{\alpha,\beta}^{1}(z)=\mathbb{E}_{\alpha,\beta}(z)~{}\mbox{and
}~{}\mathbb{E}_{\alpha,1}^{1}(z)=\mathbb{E}_{\alpha}(z).$

###### Lemma 2.2

[22] Let $0<\alpha<1$ and
$\beta,\sigma,\lambda\in\mathbb{C}\left(Re(\beta)>0\right)$. Then

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\left[\tau^{\beta-1}\;\mathbb{E}_{\alpha,\,\beta}^{\sigma}\,(\lambda\,\tau^{\alpha})\right]=\dfrac{B(\alpha)}{1-\alpha}\;\tau^{\beta-1}\;\mathbb{E}_{\alpha,\,\beta}^{1+\sigma}(\lambda\,\tau^{\alpha}).$

##  3 Estimates on ABC fractional derivatives at extreme points

###### Theorem 3.1

If $m$ is any differentiable function defined on $J$ such that
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m\in C(J)$ and there exists
$\tau_{0}\in(0,T]$ with $m(\tau_{0})=0,\;m(\tau)\leq 0,\;\tau\in[0,\tau_{0})$,
then ${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(\tau_{0})\geq 0$.

Proof: Using integration by parts we write

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(\tau)=\dfrac{B(\alpha)}{1-\alpha}\int_{0}^{\tau}\mathbb{E}_{\alpha}\left(-\dfrac{\alpha}{1-\alpha}(\tau-\sigma)^{\alpha}\right)m^{\prime}(\sigma)d\sigma$
$\displaystyle=\dfrac{B(\alpha)}{1-\alpha}\left\\{\left[\mathbb{E}_{\alpha}\left(-\dfrac{\alpha}{1-\alpha}(\tau-\sigma)^{\alpha}\right)m(\sigma)\right]_{\sigma=0}^{\sigma=\tau}-\int_{0}^{\tau}\left(\dfrac{d}{d\sigma}\mathbb{E}_{\alpha}\left(-\dfrac{\alpha}{1-\alpha}(\tau-\sigma)^{\alpha}\right)\right)m(\sigma)d\sigma\right\\}$
$\displaystyle=\dfrac{B(\alpha)}{1-\alpha}\left\\{m(\tau)-\mathbb{E}_{\alpha}\left(-\dfrac{\alpha}{1-\alpha}\tau^{\alpha}\right)m(0)-\int_{0}^{\tau}\left(\dfrac{d}{d\sigma}\mathbb{E}_{\alpha}\left(-\dfrac{\alpha}{1-\alpha}(\tau-\sigma)^{\alpha}\right)\right)m(\sigma)d\sigma\right\\}.$

Since $m(\tau_{0})=0$, we have

$\displaystyle^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(\tau_{0})$
$\displaystyle=-\dfrac{B(\alpha)}{1-\alpha}\left\\{\mathbb{E}_{\alpha}\left(-\dfrac{\alpha}{1-\alpha}\tau_{0}^{\alpha}\right)m(0)+\int_{0}^{\tau_{0}}\left(\dfrac{d}{d\sigma}\mathbb{E}_{\alpha}\left[-\dfrac{\alpha}{1-\alpha}(\tau_{0}-\sigma)^{\alpha}\right]\right)m(\sigma)d\sigma\right\\}.$
(3.1)

From [25], we have

$\mathbb{E}_{\alpha}(-\tau^{\alpha})=\int_{0}^{\infty}e^{-r\tau}K_{\alpha}(r)dr,\;0<\alpha<1,\;\mbox{for
all}~{}\tau>0,$

where

$K_{\alpha}(r)=\dfrac{1}{\pi}\dfrac{r^{\alpha-1}\sin(\alpha\pi)}{r^{2\alpha}+2r^{\alpha}\cos(\alpha\pi)+1}>0.$

Since
$K_{\alpha}(r),~{}e^{-r\,\left(\frac{\alpha}{1-\alpha}\right)^{\frac{1}{\alpha}}\tau_{0}}>0,\;\text{for}\;r>0~{}\mbox{and}~{}0<\alpha<1,$,
we have

$\mathbb{E}_{\alpha}\left(-\dfrac{\alpha}{1-\alpha}\tau_{0}^{\alpha}\right)=\mathbb{E}_{\alpha}\left(-\left(\left[\dfrac{\alpha}{1-\alpha}\right]^{\frac{1}{\alpha}}\tau_{0}\right)^{\alpha}\;\right)=\int_{0}^{\infty}e^{-r\,\left(\frac{\alpha}{1-\alpha}\right)^{\frac{1}{\alpha}}\tau_{0}}\,K_{\alpha}(r)\,dr>0.$
(3.2)

Since, $B(\alpha)>0$ and $m(0)\leq 0$, from (3.2) it follows that

$-\dfrac{B(\alpha)}{1-\alpha}\mathbb{E}_{\alpha}\left(-\dfrac{\alpha}{1-\alpha}\tau_{0}^{\alpha}\right)m(0)\geq
0.$ (3.3)

In view of inequality (3.3), Eq. (3.1) reduces to

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(\tau_{0})\geq-\dfrac{B(\alpha)}{1-\alpha}\int_{0}^{\tau_{0}}\left(\dfrac{d}{d\sigma}\mathbb{E}_{\alpha}\left(-\dfrac{\alpha}{1-\alpha}(\tau_{0}-\sigma)^{\alpha}\right)\right)m(\sigma)d\sigma.$

Again from [25], we have

$\dfrac{d}{d\tau}\mathbb{E}_{\alpha}\left(-\dfrac{\alpha}{1-\alpha}(\tau_{0}-\tau)^{\alpha}\right)\geq
0,$

Therefore, $m(\tau)\leq 0,\;\tau\in[0,\tau_{0})$ gives

$\dfrac{d}{d\tau}\mathbb{E}_{\alpha}\left[-\dfrac{\alpha}{1-\alpha}(\tau_{0}-\tau)^{\alpha}\right](-m(\tau))\geq
0,~{}\tau\in[0,\tau_{0}).$ (3.4)

Using the inequality (3.4) and the fact $B(\alpha)>0$, it follows that

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(\tau_{0})\geq\dfrac{B(\alpha)}{1-\alpha}\int_{0}^{\tau_{0}}\left(\dfrac{d}{d\sigma}\mathbb{E}_{\alpha}\left(-\dfrac{\alpha}{1-\alpha}(\tau_{0}-\sigma)^{\alpha}\right)\right)\left(-m(\sigma)\right)d\sigma\geq
0.$

This completes the proof. $\Box$

The dual of the Theorem 3.1 is also hold.

###### Theorem 3.2

If $m$ is any differentiable function defined on $J$ such that
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m\in C(J)$ and there exists
$\tau_{0}\in(0,T]$ with $m(\tau_{0})=0,\;m(\tau)\geq 0,\;\tau\in[0,\tau_{0})$,
then ${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(\tau_{0})\leq 0$.

Proof: One can observe that if $m(\tau)$ satisfies the assumptions of Theorem
3.2, then $(-m)(\tau)$ satisfies the conditions of Theorem 3.1. Hence by
applying Theorem 3.1 with $m$ replaced by $(-m)$ we obtain,
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}(-m)(\tau_{0})\geq 0$. This gives

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(\tau_{0})\leq 0.$

$\Box$

In the following Theorem, we give an alternative proof of Lemma 2.1 [25]
utilizing the outcome that we have gotten in Theorem 3.1.

###### Theorem 3.3

Let a differentiable function $f$ defined on $J$ attain its maximum at a point
$\tau_{0}\in J$ and ${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}f\in C(J)$.
Then the inequality
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}f(\tau_{0})\geq 0$ holds true.

Proof: Let $f_{\max}=f(\tau_{0})=\underset{\tau\in J}{\max}\;f(\tau)$. Define
$m(\tau)=f(\tau)-f_{\max},~{}\tau\in J$. Then $m$ is differentiable function
defined on $J$ such that, ${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m\in
C(J)$ and

$m(\tau_{0})=0,\;m(\tau)=f(\tau)-f_{\max}<0,\;\text{for all}\;\tau\in
J\,\backslash\,\\{\tau_{0}\\}$

Therefore, $m(\tau_{0})=0$ and $m(\tau)<0,\;\text{for
all}\;\tau\in[0,\tau_{0}).$ Since $m$ satisfies all the conditions of Theorem
3.1, by applying it we obtain
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(\tau_{0})\geq 0$. Since,

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(\tau)=\;^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\left(f(\tau)-f_{\max}\right)=\;^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}f(\tau),\;\tau\in
J,$

we have

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}f(\tau_{0})=\;^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(\tau_{0})\geq
0.$

$\Box$

In the next Theorem, we provide an alternating proof of Lemma 2.2 [25].

###### Theorem 3.4

Let a differentiable function $f$ defined on $J$ attain its minimum at a point
$\tau_{0}\in J$ and ${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}f\in C(J)$.
Then the inequality
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}f(\tau_{0})\leq 0$ holds true.

Proof: Let $f_{\min}=f(\tau_{0})=\underset{\tau\in J}{\min}f(\tau)$ and define
$m(\tau)=f(\tau)-f_{\min},~{}\tau\in J$. Then, one can complete the remaining
proof by applying Theorem 3.2 and following the similar types of steps as in
the proof of Theorem 3.3. $\Box$

##  4 Comparison Results

###### Theorem 4.1

Let $f\in C(J\times\mathbb{R},\mathbb{R})$. Let $v,w$ be any differentiable
functions on $J$ such that
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\;v,~{}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}w\in
C(J)$, satisfying

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}v(\tau)\leq
f\left(\tau,v(\tau)\right),\;\tau\in J,$

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}w(\tau)\geq
f\left(\tau,w(\tau)\right),\;\tau\in J,$

one of the above inequalities being strict.

Then $v(0)<w(0)$, implies

$v(\tau)<w(\tau),\;\tau\in J.$

Proof: Suppose that the conclusion of the theorem does not holds. Then by
continuity of $v,w$ there exits $\tau_{0}\in J$ such that

$v(\tau_{0})=w(\tau_{0})\;\text{and}\;v(\tau)<w(\tau)\;\text{for
all}\;\tau\in[0,\tau_{0}).$

Define $m(\tau)=v(\tau)-w(\tau),\;\tau\in J$. Then the function $m(\tau)$ is
differentiable on $J$ with ${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m\in
C(J)$ and $\tau_{0}\in J$ is such that

$m(\tau_{0})=0\;\text{and}\;m(\tau)<0\;\text{for all}\;\tau\in[0,\tau_{0}).$

Since $m$ satisfies all assumptions of Theorem 3.1, we get
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(\tau_{0})\geq 0$.

This gives

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}v(\tau_{0})\geq\;^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}w(\tau_{0}).$

Suppose that the inequality (i) is strict, then we get

$f\left(\tau_{0},v(\tau_{0})\right)>\;^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}v(\tau_{0})\geq\;^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}w(\tau_{0})\geq
f\left(\tau_{0},w(\tau_{0})\right).$

This is contradiction with $v(\tau_{0})=w(\tau_{0})$. Therefore, we must have

$v(\tau)<w(\tau),\;\text{for all}\;\tau\in J.$

This completes the proof of theorem. $\Box$

###### Theorem 4.2

Assume that the conditions of Theorem 4.1 holds (ii). Suppose that

$f(\tau,\omega)-f(\tau,\eta)\leq L(\omega-\eta),\;\text{for all
}\;\omega,\eta\in\mathbb{R}\;\text{with}\;\omega\geq\eta\;\text{and}\;0<L<\dfrac{B(\alpha)}{1-\alpha}.$

Then $v(0)\leq w(0)$ implies

$v(\tau)\leq w(\tau),\;\text{for all}\;\tau\in J.$

Proof: For $\epsilon>0$, we define

$w_{\epsilon}(\tau)=w(\tau)+\epsilon\mathbb{E}_{\alpha}(\tau^{\alpha}),\;\tau\in
J.$ (4.1)

By choice of $w$ and Lemma 2.2, the function $w_{\epsilon}$ is differentiable
on $J$ such that ${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}w_{\epsilon}\in
C(J)$ and

$w_{\epsilon}(0)=w_{0}+\epsilon>w(0).$

Since $w_{\epsilon}(\tau)\geq w(\tau),\;\tau\in J$, by using Lipschitz
condition on $f$, we have

$f(\tau,w_{\epsilon}(\tau))-f(\tau,w(\tau))\leq
L(w_{\epsilon}(\tau)-w(\tau))=L\epsilon\mathbb{E}_{\alpha}(\tau^{\alpha}).$

Using the condition on $L$, we have

$f(\tau,w(\tau))\geq
f(\tau,w_{\epsilon}(\tau))-L\epsilon\mathbb{E}_{\alpha}(\tau^{\alpha})>f(\tau,w_{\epsilon}(\tau))-\dfrac{B(\alpha)}{1-\alpha}\epsilon\mathbb{E}_{\alpha}(\tau^{\alpha}),\;\tau\in
J.$ (4.2)

By Lemma 2.2, we find

$^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\left(\mathbb{E}_{\alpha}(\tau^{\alpha})\right)=\;^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\left(\mathbb{E}^{1}_{\alpha,1}(\tau^{\alpha})\right)=\dfrac{B(\alpha)}{1-\alpha}\mathbb{E}^{2}_{\alpha,1}(\tau^{\alpha}),\;\tau\in
J.$ (4.3)

Since $(2)_{0}=1$ and $\dfrac{(2)_{k}}{k!}=k+1>1,\;k=1,2,\cdots$, we have

$\mathbb{E}^{2}_{\alpha,1}(\tau^{\alpha})=\sum_{k=0}^{\infty}\dfrac{z^{k}}{\Gamma(\alpha
k+1)}\dfrac{(2)_{k}}{k!}\geq\sum_{k=0}^{\infty}\dfrac{z^{k}}{\Gamma(\alpha
k+1)}=\mathbb{E}_{\alpha}(\tau^{\alpha}),\;\tau\in J.$ (4.4)

Using the inequality (4.4), Eq.(4.3) reduces to

$^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\left(\mathbb{E}_{\alpha}(\tau^{\alpha})\right)\geq\dfrac{B(\alpha)}{1-\alpha}\mathbb{E}_{\alpha}(\tau^{\alpha}),~{}\tau\in
J.$ (4.5)

Utilizing the inequalities (ii), (4.2) and (4.5), for any $\tau\in J$, we have

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\left(w_{\epsilon}(\tau)\right)$
$\displaystyle=\;^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\left[w(\tau)+\epsilon\mathbb{E}_{\alpha}(\tau^{\alpha})\right]$
$\displaystyle=\;^{ABC}{{}_{0}\mathcal{D}}_{t}^{\alpha}w(\tau)+\epsilon\;^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\mathbb{E}_{\alpha}(\tau^{\alpha})$
$\displaystyle\geq
f(\tau,w(\tau))+\epsilon\dfrac{B(\alpha)}{1-\alpha}\mathbb{E}_{\alpha}(\tau^{\alpha})$
$\displaystyle>f(\tau,w_{\epsilon}(\tau))-\dfrac{B(\alpha)}{1-\alpha}\epsilon\mathbb{E}_{\alpha}(\tau^{\alpha})+\epsilon\dfrac{B(\alpha)}{1-\alpha}\mathbb{E}_{\alpha}(\tau^{\alpha})$
$\displaystyle=f(\tau,w_{\epsilon}(\tau))$

Therefore,

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}w_{\epsilon}(\tau)>f\left(\tau,w_{\epsilon}(\tau)\right),\;\tau\in
J.$

Since $v(0)<w_{\epsilon}(0)$, by application of Theorem 3.1 with
$w(\tau)=w_{\epsilon}(\tau)$, for each $\epsilon>0$ we have

$v(\tau)<w_{\epsilon}(\tau),\;\tau\in J.$

Taking limit as $\epsilon\to 0$, in the above inequality and utilizing Eq.
(4.1), we obtain

$v(\tau)\leq w(\tau),\;\tau\in J.$

$\Box$

###### Corollary 4.3

If $m$ is any differentiable function defined on $J$ such that
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m\in C(J)$, $\tau\in J$ and

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(\tau)$
$\displaystyle\leq\dfrac{B(\alpha)}{1-\alpha}m(\tau),\;\tau\in J,$
$\displaystyle m(0)$ $\displaystyle=m_{0},$

then $m(\tau)\leq m_{0}\mathbb{E}_{\alpha}(\tau^{\alpha}),\;\tau\in J.$

Proof: Define

$\lambda(\tau)=m_{0}\mathbb{E}_{\alpha}(\tau^{\alpha}),\;\tau\in J.$

Then $\lambda(0)=m_{0}$. Further using the inequality (4.5), we have

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\lambda(\tau)\geq\dfrac{B(\alpha)}{1-\alpha}\lambda(\tau),\;\tau\in
J.$

Applying Theorem 4.2, with $v=m$ and $w=\lambda$, we get

$m(\tau)\leq\lambda(\tau)=m_{0}\mathbb{E}_{\alpha}(\tau^{\alpha}),\tau\in J.$

$\Box$

The following theorem is the comparison result for periodic boundary value
problems involving ABC-fractional derivative. The proof of the same one can
finish watching the comparable kind of steps of Theorem 2.6 [46].

###### Theorem 4.4

Let $v,w$ be any differentiable functions on $J$ such that
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\;v,^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}w\in
C(J),\;\tau\in J$, and $f\in C(J\times\mathbb{R},\mathbb{R})$ satisfying

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}v(\tau)\leq
f\left(\tau,v(\tau)\right),\;\tau\in J,\;v(0)\leq v(T)$

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}w(\tau)\geq
f\left(\tau,w(\tau)\right),\;\tau\in J,\;w(0)\geq w(T)$

If the function $f(\tau,\omega)$ is non increasing in $\omega$ for each $\tau$
then

$v(\tau)\leq w(\tau),\;\tau\in J.$

In the next Theorem, we will give an alternating proof of Lemma 2.3 [25]
without utilizing Cauchy-Schwartz inequality.

###### Theorem 4.5

If $m$ is any differentiable function defined on $J$ such that
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m\in C(J)$ then
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(0)=0$.

Proof: Let $m$ is any differentiable function defined on $J$ such that
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m\in C(J)$, $\tau\in J$. Since
$m^{\prime}$ exists and is continuous on $J$, we have $m,m^{\prime}\in C(J)$,
this implies $m,m^{\prime}\in L^{1}(J)$, therefore
$\underset{\tau\in[0,T]}{\sup}|m^{\prime}(\tau)|\leq M$. Using the definition
of ABC-fractional derivative operator,

$\displaystyle|^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(\tau)|$
$\displaystyle\leq\dfrac{B(\alpha)}{1-\alpha}\int_{0}^{\tau}\left|\mathbb{E}_{\alpha}\left(-\dfrac{\alpha}{1-\alpha}(\tau-\sigma)^{\alpha}\right)\right||m^{\prime}(\sigma)|d\sigma$
$\displaystyle\leq\dfrac{MB(\alpha)}{1-\alpha}\int_{0}^{\tau}\mathbb{E}_{\alpha}\left(\dfrac{\alpha}{1-\alpha}(\tau-\sigma)^{\alpha}\right)d\sigma=\dfrac{MB(\alpha)}{1-\alpha}\mathbb{E}_{\alpha,2}\left(\dfrac{\alpha}{1-\alpha}\tau^{\alpha}\right)\tau$

This gives, ${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(0)=0$. $\Box$

###### Corollary 4.6

If $\omega,~{}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega,~{}u\in C(J)$
and ${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega(\tau)=u(\tau)$, then
$u(0)=0$.

Proof: Proof follows from the Theorem 4.5. $\Box$

###### Remark 4.7

From the Corollary 4.6 it follows that the ABC-FDEs (1.1)–(1.2) and its
equivalent fractional integral equation is consistent only if
$f(0,\omega(0))=0$, where $\omega$ is differentiable function with
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega\in C(J)$.

##  5  Existence of Local and Extremal Solution

In this section, we determine the results about the existence of local and
extremal solutions of the ABC-FDEs (1.1)–(1.2) through the equivalent
fractional integral equation given in the following Lemma.

###### Lemma 5.1

[38, 35] The equivalent fractional integral equation to the the ABC-FDEs
(1.1)–(1.2) is given by

$\omega(\tau)=\omega_{0}+\dfrac{1-\alpha}{B(\alpha)}f(\tau,\omega(\tau))+\dfrac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{0}^{\tau}(\tau-\sigma)^{\alpha-1}f(\sigma,\omega(\sigma))d\sigma.$

Let $\epsilon>0$ be arbitrary. Consider the ABC-FDEs of the form,

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega_{\epsilon}(\tau)$
$\displaystyle=f\left(\tau,\omega_{\epsilon}(\tau)\right),\tau\in J,$ (5.1)
$\displaystyle\;\omega_{\epsilon}(\tau)|_{\tau=0}$
$\displaystyle=\omega_{\epsilon}(0),$ (5.2)

where
$\omega_{\epsilon},\;^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega_{\epsilon}\in
C(J)$ and $f\in C(J\times\mathbb{R},\mathbb{R})$. Then by Lemma 5.1, the
equivalent fractional integral equation of the ABC-FDEs (5.1)-(5.2) is given
by

$\omega_{\epsilon}(\tau)=\omega_{\epsilon}(0)+\dfrac{1-\alpha}{B(\alpha)}f(\tau,\omega_{\epsilon}(\tau))+\dfrac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{0}^{\tau}(\tau-\sigma)^{\alpha-1}f(\sigma,\omega_{\epsilon}(\sigma))d\sigma,\;\tau\in
J.$

###### Theorem 5.2

If the function $f\in
C(R_{0},\mathbb{R}),~{}R_{0}=\left\\{(\tau,\omega):\tau\in
J,|\omega-\omega_{0}|\leq b\right\\}$ is such that

$|f(\tau,\omega)|\leq M,\;\text{for all}\;(\tau,\omega)\in R_{0}$

and satisfies the Lipschitz type condition,

$|f(\tau_{1},\omega)-f(\tau_{2},\eta)|\leq
L_{1}|\tau_{1}-\tau_{2}|+L_{2}|\omega-\eta|,\;\tau_{1},\tau_{2}\in
J,\;\omega,\eta\in\mathbb{R},$

where $L_{1}>0$ and $0<L_{2}<\dfrac{B(\alpha)}{(1-\alpha)},$ then the family
$\left\\{\omega_{\epsilon}\right\\}$ of solution of the ABC-FDEs  (5.1)–(5.2)
is equicontinious on $J$.

Proof: Let $\epsilon>0$ be arbitrary. Let $\omega_{\epsilon}(\tau)$ be the
solution of the ABC-FDEs (5.1)–(5.2). Let $\tau_{1},\tau_{2}\in J$ with
$0<\tau_{1}\leq\tau_{2}<T$. Then by hypotheses, we have

$\displaystyle|\omega_{\epsilon}(\tau_{1})-\omega_{\epsilon}(\tau_{2})|$
$\displaystyle=\left|\left(\omega_{\epsilon}(0)+\dfrac{1-\alpha}{B(\alpha)}f(\tau_{1},\omega_{\epsilon}(\tau_{1}))+\dfrac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{0}^{\tau_{1}}(\tau_{1}-\sigma)^{\alpha-1}f(\sigma,\omega_{\epsilon}(\sigma))d\sigma\right)\right.$
$\displaystyle\qquad\left.-\left(\omega_{\epsilon}(0)+\dfrac{1-\alpha}{B(\alpha)}f(\tau_{2},\omega_{\epsilon}(\tau_{2}))+\dfrac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{0}^{\tau_{2}}(\tau_{2}-\sigma)^{\alpha-1}f(\sigma,\omega_{\epsilon}(\sigma))d\sigma\right)\right|$
$\displaystyle\leq\dfrac{1-\alpha}{B(\alpha)}|f(\tau_{1},\omega_{\epsilon}(\tau_{1}))-f(\tau_{2},\omega_{\epsilon}(\tau_{2}))|$
$\displaystyle\quad+\dfrac{\alpha}{B(\alpha)\Gamma(\alpha)}\left(\int_{0}^{\tau_{1}}((\tau_{1}-\sigma)^{\alpha-1}-(\tau_{2}-\sigma)^{\alpha-1})|f(\sigma,\omega_{\epsilon}(\sigma))|d\sigma+\int_{\tau_{1}}^{\tau_{2}}(\tau_{2}-\sigma)^{\alpha-1}|f(\sigma,\omega_{\epsilon}(\sigma))|d\sigma\right)$
$\displaystyle\leq\dfrac{1-\alpha}{B(\alpha)}\left(L_{1}|\tau_{1}-\tau_{2}|+L_{2}|\omega_{\epsilon}(\tau_{1})-\omega_{\epsilon}(\tau_{2})|\right)+\dfrac{M\alpha}{B(\alpha)\Gamma(\alpha)}\int_{0}^{\tau_{1}}((\tau_{1}-\sigma)^{\alpha-1}-(\tau_{2}-\sigma)^{\alpha-1})d\sigma$
$\displaystyle\quad+\dfrac{M\alpha}{B(\alpha)\Gamma(\alpha)}\int_{\tau_{1}}^{\tau_{2}}(\tau_{2}-\sigma)^{\alpha-1}d\sigma$

This gives,

$\displaystyle|\omega_{\epsilon}(\tau_{1})-\omega_{\epsilon}(\tau_{2})|\leq\dfrac{1}{\left(1-\dfrac{1-\alpha}{B(\alpha)}L_{2}\right)}\left[\dfrac{(1-\alpha)L_{1}}{B(\alpha)}(\tau_{2}-\tau_{1})+\dfrac{M}{B(\alpha)\Gamma(\alpha)}\left(2(\tau_{2}-\tau_{1})^{\alpha}+\tau_{1}^{\alpha}-\tau_{2}^{\alpha}\right)\right].$

Note that for any
$0<\alpha<1,\;(\tau_{2}-\tau_{1})^{\alpha}\leq(\tau_{2}-\tau_{1})$ and
$\tau_{1}^{\alpha}-\tau_{2}^{\alpha}\leq 0$. Therefore, we have

$|\omega_{\epsilon}(\tau_{1})-\omega_{\epsilon}(\tau_{2})|\leq\dfrac{\Gamma(\alpha)(1-\alpha)L_{1}+2M}{\Gamma(\alpha)\left\\{B(\alpha)-(1-\alpha)L_{2}\right\\}}(\tau_{2}-\tau_{1}).$

One can check that for any $\tilde{\epsilon}>0$ there exists
$\tilde{\delta}=\tilde{\epsilon}\dfrac{\Gamma(\alpha)\left\\{B(\alpha)-(1-\alpha)L_{2}\right\\}}{\Gamma(\alpha)(1-\alpha)L_{1}+2M}$
such that if $|\tau_{1}-\tau_{2}|<\tilde{\delta}$, then

$|\omega_{\epsilon}(\tau_{1})-\omega_{\epsilon}(\tau_{2})|<\tilde{\epsilon}.$

This proves that the family of solution $\left\\{\omega_{\epsilon}\right\\}$
of the ABC-FDEs (5.1)–(5.2) is equicontinious on $J$. $\Box$

###### Theorem 5.3

Assume that the conditions of Theorem 5.2 hold. If $M(1-\alpha)<bB(\alpha)$,
then the ABC-FDEs (1.1)–(1.2) has at least one solution on
$J^{\prime}=[0,\beta]$, where
$\beta=\min\left\\{T,\left[\dfrac{\Gamma(\alpha)(bB(\alpha)-M(1-\alpha))}{M}\right]^{\frac{1}{\alpha}}\right\\}$.

Proof: Fix $\delta>0$. Let $\omega_{0}\in C[-\delta,0]$ be any real valued
function satisfying the conditions

$\omega_{0}(0)=\omega_{0},\;|\omega_{0}(\tau)-\omega_{0}|\leq b.$

For any $\epsilon$, $0<\epsilon<\delta$, define
$\beta_{1}=\min\left\\{\beta,\epsilon\right\\}$. Consider the ABC-fractional
delay differential equations

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega_{\epsilon}(\tau)$
$\displaystyle=f\left(\tau,\omega_{\epsilon}(\tau-\epsilon)\right),\tau\in[0,\beta_{1}],$
(5.3) $\displaystyle\omega_{\epsilon}(\tau)$
$\displaystyle=\omega_{0}(\tau),\tau\in[-\delta,0].$ (5.4)

Then by Lemma 5.1, equivalent fractional integral equ. of the ABC-FDEs
(5.3)–(5.4) is

$\omega_{\epsilon}(\tau)=\begin{cases}\omega_{0}(\tau),\;\tau\in[-\delta,0],\\\
\omega_{0}+\dfrac{1-\alpha}{B(\alpha)}f(\tau,\omega_{\epsilon}(\tau-\epsilon))\\\
\;\;+\dfrac{\alpha}{B(\alpha)\Gamma(\alpha)}\large\int_{0}^{\tau}(\tau-\sigma)^{\alpha-1}f(\sigma,\omega_{\epsilon}(\sigma-\epsilon))d\sigma,\;\tau\in[0,\beta_{1}].\end{cases}$
(5.5)

In the view of Corollary 4.6, the ABC-FDEs (5.3)–(5.4) is consistent only if

$f(0,\omega_{\epsilon}(-\epsilon))=f(0,\omega_{0}(-\epsilon))=0$ (5.6)

One can observe that $\omega_{\epsilon}$ is continuous on
$[-\delta,\beta_{1}]$ expect possibly at $\tau=0$. Using continuity of $f$ and
Eq.(5.6), we have

$\displaystyle\lim\limits_{\tau\to 0^{+}}\omega_{\epsilon}(\tau)$
$\displaystyle=\lim\limits_{\tau\to
0^{+}}\left(\omega_{0}+\dfrac{1-\alpha}{B(\alpha)}f(\tau,\omega_{\epsilon}(\tau-\epsilon))\;+\dfrac{\alpha}{B(\alpha)\Gamma(\alpha)}\large\int_{0}^{\tau}(\tau-\sigma)^{\alpha-1}f(\sigma,\omega_{\epsilon}(\sigma-\epsilon))d\sigma\right)$
$\displaystyle=\omega_{0}+f(0,\omega_{0}(-\epsilon))=\omega_{0}$

Hence the function $\omega_{\epsilon}(\tau):[-\delta,\beta_{1}]\to\mathbb{R}$
is continuous. Note that,

$|\omega_{\epsilon}(\tau)-\omega_{0}|=|\omega_{0}(\tau)-\omega_{0}|\leq
b,\;\tau\in[-\delta,0].$ (5.7)

Also for any $\tau\in[0,\beta_{1}]$,

$\displaystyle|\omega_{\epsilon}(\tau)-\omega_{0}|$
$\displaystyle\leq\dfrac{1-\alpha}{B(\alpha)}|f(\tau,\omega_{\epsilon}(\tau-\epsilon))|+\dfrac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{0}^{\tau}(\tau-\sigma)^{\alpha-1}|f(\sigma,\omega_{\epsilon}(\sigma-\epsilon))|d\sigma$
$\displaystyle\leq\dfrac{(1-\alpha)M}{B(\alpha)}+\dfrac{M\alpha}{B(\alpha)\Gamma(\alpha)}\int_{0}^{\tau}(t-\sigma)^{\alpha-1}d\sigma$
$\displaystyle=\dfrac{M}{B(\alpha)}\left(1-\alpha+\dfrac{\tau^{\alpha}}{\Gamma(\alpha)}\right)\leq\dfrac{M}{B(\alpha)}\left(1-\alpha+\dfrac{\beta_{1}^{\alpha}}{\Gamma(\alpha)}\right).$
(5.8)

Since
$\beta_{1}\leq\beta\leq\left[\dfrac{\Gamma(\alpha)(bB(\alpha)-M(1-\alpha))}{M}\right]^{\frac{1}{\alpha}}$,
we have

$\dfrac{\beta_{1}^{\alpha}}{\Gamma(\alpha)}\leq\dfrac{M}{B(\alpha)}\left(\dfrac{bB(\alpha)}{M}-1+\alpha\right)$
(5.9)

From (5) and (5.9), we have

$|\omega_{\epsilon}(\tau)-\omega_{0}|\leq b,\tau\in[0,\beta_{1}]$ (5.10)

From (5.7) and (5.10), we have

$|\omega_{\epsilon}(\tau)-\omega_{0}|\leq b,\tau\in[-\delta,\beta_{1}].$
(5.11)

If $\beta_{1}<\beta$, we consider fractional integral Eq.(5.5) on the interval
$[-\delta,\beta_{2}]$, where $\beta_{2}=\min\\{\beta,2\epsilon\\}$, such that

$|\omega_{\epsilon}(\tau)-\omega_{0}|\leq b,\;\tau\in[-\delta,\beta_{2}].$

Continuing in this way, $\omega_{\epsilon}(\tau)$ can be extended to
$[-\delta,\beta]$, such that

$|\omega_{\epsilon}(\tau)-\omega_{0}|\leq b,\;\tau\in[-\delta,\beta].$

This gives

$|\omega_{\epsilon}(\tau)|\leq|\omega_{\epsilon}(\tau)-\omega_{0}|+|\omega_{0}|\leq
b+|\omega_{0}|,\;\tau\in[-\delta,\beta].$

Therefore

$\|\omega_{\epsilon}\|\leq b+|\omega_{0}|,\;\tau\in[-\delta,\beta],$

and hence $\left\\{\omega_{\epsilon}\right\\}$ is uniformly bounded family of
function defined on $[-\delta,\beta]$. Since, the hypothesis of Lemma 5.1 are
satisfied, the family $\left\\{\omega_{\epsilon}\right\\}$ is equicontinious.
Applying Ascoli-Arzela’s theorem there exists a decreasing sequence
$\left\\{\epsilon_{n}\right\\}$ with $\epsilon_{n}>0$ for all $n$ and
$\epsilon_{n}\to 0$ such that $\omega_{\epsilon_{n}}\to\omega$, $\omega\in
C([0,\beta],\mathbb{R})$ uniformly on $[0,\beta]$. Since $f\in
C(R_{0},\mathbb{R})$, we have
$f(\tau,\omega_{\epsilon_{n}}(\tau-\epsilon_{n}))\to f(\tau,\omega)$. By
replacing $\epsilon$ with $\epsilon_{n}$ in equation (5.5) and then taking
limit as $n\to\infty$, we obtain

$\omega(\tau)=\omega_{0}+\dfrac{1-\alpha}{B(\alpha)}f(\tau,\omega(\tau))+\dfrac{\alpha}{B(\alpha)\Gamma(\alpha)}\int_{0}^{\tau}(\tau-\sigma)^{\alpha-1}f(s,\omega(s))d\sigma,\;\tau\in[0,\beta],$

which gives the required solution of the ABC-FDEs (1.1)–(1.2). $\Box$

###### Theorem 5.4

Assume that the conditions of Theorem 5.2 hold. If
$(2M+b)(1-\alpha)<bB(\alpha)$, then the ABC-FDEs (1.1)–(1.2) has extremal
solution on $J^{\prime\prime}=[0,\beta_{0}]$, where
$\beta_{0}=\min\left\\{T,\left[\dfrac{(bB(\alpha)-(2M+b)(1-\alpha))\Gamma(\alpha)}{(2M+b)}\right]^{\frac{1}{\alpha}}\right\\}$.

Proof: We give the proof only for the existence of maximal solution of the
ABC-FDEs (1.1)–(1.2), as the proof of existence of minimal solution one can
complete on similar lines.

For $0<\epsilon\leq\frac{b}{2},$ consider the ABC-FDEs

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega(\tau)$
$\displaystyle=f\left(\tau,\omega(\tau)\right)+\epsilon:=f_{\epsilon}\left(\tau,\omega(\tau)\right),\tau\in
J,$ (5.12) $\displaystyle\omega(0)$
$\displaystyle=\omega_{0}+\epsilon=\omega(0,\epsilon).$ (5.13)

Define $R_{\epsilon}=\left\\{(\tau,\omega):\tau\in
J,~{}|\omega-\omega(0,\epsilon)|\leq\dfrac{b}{2}\right\\}$. Clearly
$R_{\epsilon}\subset R_{0}.$ Consider the function
$f_{\epsilon}:R_{\epsilon}\to\mathbb{R}$ defined by

$f_{\epsilon}\left(\tau,\omega(\tau)\right)=f\left(\tau,\omega(\tau)\right)+\epsilon.$

Then, $f_{\epsilon}$ satisfies the Lipschitz type condition with same
Lipschitz constants $L_{1},L_{2}$ as defined in Theorem 5.2. Further,

$|f_{\epsilon}\left(\tau,\omega(\tau)\right)|\leq
M+\frac{b}{2},\;(\tau,\omega)\in R_{\epsilon}.$

Since $f_{\epsilon}$ satisfies all assumptions of Theorem 5.3, by applying it,
the ABC-FDEs (5.12)–(5.13) has at least one solution $\omega(\tau,\epsilon)$
on $J^{\prime\prime}$.

Let $0<\epsilon_{2}<\epsilon_{1}\leq\epsilon$, then we have

$\displaystyle\omega(0,\epsilon_{2})=\omega_{0}+\epsilon_{2}$
$\displaystyle<\omega_{0}+\epsilon_{1}=\omega(0,\epsilon_{1});$
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega(\tau,\epsilon_{2})$
$\displaystyle=f\left(\tau,\omega(\tau,\epsilon_{2})\right)+\epsilon_{2},\;\tau\in
J^{\prime\prime};$
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega(\tau,\epsilon_{1})$
$\displaystyle>f\left(\tau,\omega(\tau,\epsilon_{1})\right)+\epsilon_{2},\;\tau\in
J^{\prime\prime}.$

Note that $\omega(\tau,\epsilon_{1})$ and $\omega(\tau,\epsilon_{2})$
respectively are lower and upper solutions of ABC-FDEs, with
$\omega(0,\epsilon_{2})<\omega(0,\epsilon_{1})$. Therefore by applying Theorem
4.1 we have,

$\omega(\tau,\epsilon_{2})<\omega(\tau,\epsilon_{1}),\;\tau\in
J^{\prime\prime}.$

Next we show that the family $\left\\{\omega(\tau,\epsilon)\right\\}$ of
solutions of the ABC-FDEs (5.12)–(5.13) is uniformally bounded on
$J^{\prime\prime}$. Proceeding as in the proof of Theorem 5.3, for any
$\tau\in J^{\prime\prime}$, we have

$\displaystyle|\omega(\tau,\epsilon)-\omega(0,\epsilon)|\leq\dfrac{2M+b}{2B(\alpha)}\left(1-\alpha+\dfrac{\beta_{0}^{\alpha}}{\Gamma(\alpha)}\right),\tau\in
J^{\prime\prime}.$

Since
$\beta_{0}\leq\left[\dfrac{(bB(\alpha)-(2M+b)(1-\alpha))\Gamma(\alpha)}{(2M+b)}\right]^{\frac{1}{\alpha}},$
above inequality reduces to

$|\omega(\tau,\epsilon)-\omega(0,\epsilon)|\leq\dfrac{2M+b}{2B(\alpha)}\left(1-\alpha+\dfrac{bB(\alpha)}{2M+b}-1+\alpha\right)\\\
\leq\dfrac{b}{2}<b,\;\tau\in J^{\prime\prime}.$

Since $f_{\epsilon}$ satisfies assumptions of Theorem 5.2, the family
$\left\\{\omega(\tau,\epsilon)\right\\}$ is equicontinious on $J$. Applying
Ascoli-Arzela’s theorem there exists a decreasing sequence
$\left\\{\epsilon_{n}\right\\}$ with $\epsilon_{n}>0$ for all $n$ and
$\epsilon_{n}\to 0$ such that $\omega(\tau,\epsilon_{n})\to\eta(\tau)$
uniformly on $J^{\prime\prime}$, where $\eta\in C([0,\beta_{0}],\mathbb{R})$
uniformly on $[0,\beta]$. By uniform continuity of $f_{\epsilon}$, we have

$f_{\epsilon_{n}}(\tau,\omega(\tau,\epsilon_{n}))\to
f(\tau,\eta(\tau)),\;\tau\in J^{\prime\prime}.$

By replacing $\epsilon$ with $\epsilon_{n}$ in equation (5.12) and then taking
limit as $n\to\infty$, we obtain

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\eta(\tau)$
$\displaystyle=f\left(\tau,\eta(\tau)\right),\tau\in J^{\prime\prime},$
$\displaystyle\eta(0)$ $\displaystyle=\omega_{0}.$

This proves that $\eta(\tau)$ is a solution of ABC-FDEs(1.1)–(1.2).

It remains to prove that $\eta(\tau)$ is the maximal solution of the ABC-FDEs
(1.1)–(1.2). Let $\omega(\tau)$ be any solution of (1.1)–(1.2) on
$J^{\prime\prime}$. Then for any $\epsilon>0$,

$\displaystyle\omega_{0}=\omega(0)$
$\displaystyle<\omega_{0}+\epsilon=\omega(0,\epsilon);$
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega(\tau)$
$\displaystyle>f\left(\tau,\omega(\tau,\epsilon_{2})\right)+\epsilon,\;\tau\in
J^{\prime\prime};$
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega(\tau,\epsilon)$
$\displaystyle=f\left(\tau,\omega(\tau,\epsilon)\right)+\epsilon,\;\tau\in
J^{\prime\prime}.$

Note that $\omega(\tau)$ and $\omega(\tau,\epsilon)$ respectively are lower
and upper solutions of ABC-FDEs, with $\omega_{0}<\omega_{0}+\epsilon$.
Therefore by applying Theorem 4.1 we have,

$\omega(\tau)<\omega(\tau,\epsilon),\;\tau\in J^{\prime\prime}.$

Taking limit as $\epsilon\to 0$, we obtain

$\omega(\tau)\leq\eta(\tau),\;\tau\in J^{\prime\prime}.$

This proves that $\eta(\tau)$ is the maximal solution of the ABC-FDEs
(1.1)–(1.2) on $J"$. $\Box$

##  6 Existence of Global Solution

###### Theorem 6.1

Assume that $m$ is any differentiable function defined on $J$ such that
${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m\in C(J)$ and $g\in
C(J\times\mathbb{R}_{+},\mathbb{R}_{+})$ such that

$^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(\tau)\leq
g(\tau,m(\tau)),\;\tau\in J$ (6.1)

and let $\eta(\tau)$ is the maximal solution of the ABC-FDEs

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}u(\tau)$
$\displaystyle=g\left(\tau,u(\tau)\right),\tau\in J,\;$ $\displaystyle u(0)$
$\displaystyle=u_{0}.$

Then $m(0)\leq u(0)$, implies $m(\tau)\leq\eta(\tau),\;\tau\in J.$

Proof: Let $\epsilon>0$ be arbitrary. Let $u(\tau,\epsilon)$ be a solution of
the ABC-FDEs of the form

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}u(\tau)$
$\displaystyle=g\left(\tau,u(\tau)\right)+\epsilon,\tau\in J,$ $\displaystyle
u(0)$ $\displaystyle=u_{0}+\epsilon.$

Therefore,

$\displaystyle^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}u(\tau,\epsilon)$
$\displaystyle>g\left(\tau,u(\tau,\epsilon)\right),\tau\in J,$ $\displaystyle
u(0)$ $\displaystyle=u_{0}+\epsilon$ (6.2)

Therefore $m$ is lower solution and $u(\tau,\epsilon)$ is upper solution of
the ABC-FDE

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega(\tau)=g\left(\tau,\omega(\tau)\right),\tau\in
J.$

Further,

$m(0)\leq u(0)<u(0)+\epsilon=u(0,\epsilon).$

By applying Theorem 4.1, we obtain

$m(\tau)\leq u(\tau,\epsilon),\;\tau\in J,\;\epsilon>0.$

Taking $\epsilon\to 0$, and following similar approach as in the proof of
Theorem 5.4, we obtain

$m(\tau)\leq\eta(\tau),\;\tau\in J.$

$\Box$

###### Theorem 6.2

Assume that $f\in
C([0,\infty)\times\mathbb{R},\mathbb{R})~{}\mbox{and}~{}\;g\in
C([0,\infty)\times\mathbb{R}_{+},\mathbb{R}_{+})$ are such that
$|f(\tau,\omega)|\leq g(\tau,|\omega|)$. Further, suppose that there exists
local solution $\omega(\tau,\omega_{0})$ of the ABC-FDEs

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega(\tau)$
$\displaystyle=f\left(\tau,\omega(\tau)\right),\tau\in[0,\infty),$ (6.3)
$\displaystyle\omega(0)$ $\displaystyle=\omega_{0},$ (6.4)

and maximal solution $\eta(\tau)$ of

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}u(\tau)$
$\displaystyle=g\left(\tau,u(\tau)\right),\tau\in[0,\infty),$ $\displaystyle
u(0)$ $\displaystyle=u_{0}\geq 0.$

Then the largest interval of existence of solution $\omega(\tau,\omega_{0})$
of the (6.3)–(6.4) such that $|\omega_{0}|<u_{0}$ is $[0,\infty)$.

Proof: By assumption there exists a local solution $\omega(\tau,\omega_{0})$
of the (6.3)–(6.4) on the interval $[0,\beta)$, where $\beta<\infty$ with
$|\omega_{0}|<u_{0}$. Suppose on contrary $\beta$ can not be increased
further.

Define $m(\tau)=|\omega(\tau,\omega_{0})|,\;\tau\in[0,\beta)$. As
$|f|^{\prime}\leq|f^{\prime}|$, for any $\tau\in[0,\beta)$, we find

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(\tau)$
$\displaystyle=\dfrac{B(\alpha)}{1-\alpha}\int_{0}^{\tau}\mathbb{E}_{\alpha}\left(-\dfrac{\alpha}{1-\alpha}(\tau-\sigma)^{\alpha}\right)m^{\prime}(\sigma)d\sigma$
$\displaystyle=\dfrac{B(\alpha)}{1-\alpha}\int_{0}^{\tau}\mathbb{E}_{\alpha}\left(-\dfrac{\alpha}{1-\alpha}(\tau-\sigma)^{\alpha}\right)|\omega(\sigma,\omega_{0}|^{\prime}d\sigma$
$\displaystyle\leq\dfrac{B(\alpha)}{1-\alpha}\int_{0}^{\tau}\mathbb{E}_{\alpha}\left(-\dfrac{\alpha}{1-\alpha}(\tau-\sigma)^{\alpha}\right)|\omega^{\prime}(\sigma,\omega_{0}|d\sigma$
$\displaystyle=\left|\dfrac{B(\alpha)}{1-\alpha}\int_{0}^{\tau}\mathbb{E}_{\alpha}\left(-\dfrac{\alpha}{1-\alpha}(\tau-\sigma)^{\alpha}\right)\omega^{\prime}(\sigma,\omega_{0})d\sigma\right|$
$\displaystyle=|^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega(\tau,\omega_{0})|$

Therefore,

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}m(\tau)\leq|^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega(\tau,\omega_{0})|=|f(\tau,\omega(\tau,\omega_{0}))|\leq
g(\tau,|\omega(\tau,\omega_{0})|)=g(\tau,m(\tau)),\;\tau\in[0,\beta).$

This implies $m(\tau)$ is lower solution of

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}u(\tau)$
$\displaystyle=g\left(\tau,u(\tau)\right),\tau\in[0,\beta).$ (6.5)

Further by assumption $\eta(\tau)$ is the maximal solution of the problem
(6.5). Since $m(0)=|\omega_{0}|\leq u_{0}$, by applying Theorem 4.1, we obtain

$m(\tau)=|\omega(\tau,\omega_{0})|\leq\eta(\tau),\;\tau\in[0,\beta).$

By assumption $\eta(\tau)$ exists on $[0,\infty)$. Therefore by continuity of
$g$ on $[0,\beta]\times\mathbb{R}_{+}$, there exists $M>0$ such that,

$|g\left(\tau,\eta(\tau)\right)|\leq M,\;\tau\in[0,\beta).$

Let $0\leq\tau_{1}\leq\tau_{2}<\beta$. Then following similar steps as in the
proof of Theorem 4.1, we have

$|\omega(\tau_{1},\omega_{0})-\omega(\tau_{2},\omega_{0})|\leq\dfrac{\Gamma(\alpha)(1-\alpha)L_{1}+2M}{\Gamma(\alpha)\left[B(\alpha)-(1-\alpha)L_{2}\right]}(\tau_{2}-\tau_{1}).$

From above inequality it follows that,

$\lim\limits_{\tau_{1},\;\tau_{2}\to\beta^{-}}\omega(\tau_{1},\omega_{0})=\lim\limits_{\tau_{2},\;\tau_{1}\to\beta^{-}}\omega(\tau_{2},\omega_{0}),$

for any $\tau_{1},\tau_{2}$ with $0\leq\tau_{1}\leq\tau_{2}<\beta$. This
implies that $\lim\limits_{\tau\to\beta^{-}}\omega(\tau,\omega_{0})$ exists.
Let

$\omega(\beta,\omega_{0})=\lim\limits_{\tau\to\beta^{-}}\omega(\tau,\omega_{0}).$

Then by assumption the ABC-FDEs

${}^{ABC}{{}_{0}\mathcal{D}}_{\tau}^{\alpha}\omega(\tau)$
$\displaystyle=f\left(\tau,\omega(\tau)\right),\tau\geq\beta,$
$\displaystyle\omega(\beta)$ $\displaystyle=\omega_{\beta},$

has a local solution. This implies that $\omega(\tau,\omega_{0})$ can be
continued beyond $\beta$ which is a contradiction our assumption. Hence every
solution $\omega(\tau,\omega_{0})$ of the ABC-FDEs (6.3)–(6.4) exists on
$[0,\infty)$. $\Box$

## Conclusion

The comparison results, local, extremal, and global existence of a solution,
derived without demanding the monotonicity and Holder continuity assumption on
the nonlinear function involving ABC-FDEs. The estimations on ABC-fractional
derivative and the comparison results obtained to ABC-FDEs one can use to
research the existence, uniqueness and qualitative properties of solutions for
various classes of ABC-FDEs.

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