In this section, we prove the existence results for FHDE (2.1) on the closed and bounded interval $J=[{t}_{0},{t}_{0}+a]$ under mixed Lipschitz and compactness conditions on the nonlinearities involved in it.

We place FHDE (2.1) in the space $C(J,\mathbb{R})$ of continuous real-valued functions defined on *J*. Define a supremum norm $\parallel \cdot \parallel $ in $C(J,\mathbb{R})$ by $\parallel x\parallel ={sup}_{t\in J}|x(t)|$. Clearly, $C(J,\mathbb{R})$ is a Banach algebra with respect to the above norm.

We prove the existence of a solution for FHDE (2.1) by a fixed point theorem in the Banach algebra due to Dhage [30].

**Definition 3.1** Let

*X* be a Banach space. A mapping

$T:X\to X$ is called

*φ*-Lipschitzian if there exists a continuous and nondecreasing function

$\phi :{R}^{+}\to {R}^{+}$ such that

$\parallel Tx-Ty\parallel \le \phi (\parallel x-y\parallel )$

for all $x,y\in X$, where $\phi (0)=0$.

Further, if *φ* satisfies the condition $\phi (r)<r$, $r>0$, then *T* is called a nonlinear contraction with a control function *φ*.

**Lemma 3.1** [30]

*Let* *S* *be a nonempty*,

*closed convex and bounded subset of the Banach algebra* *X* *and let* $A:X\to X$ *and* $B:S\to X$ *be two operators such that* - (a)
*A* *is nonlinear contraction*,

- (b)
*B* *is completely continuous*,

- (c)
$Ax+Bx\in S$ *for all* $x\in S$.

*Then the operator equation* $Ax+Bx=x$ *has a solution in* *S*.

We consider the following hypotheses in what follows.

(A_{0}) The function $x\mapsto x-f(t,x)$ is increasing in ℝ for all $t\in J$.

(A

_{1}) There exist constants

$M\ge L>0$ such that

$|f(t,x)-f(t,y)|\le \frac{L|x-y|}{M+|x-y|}$

for all $t\in J$ and $x,y\in \mathbb{R}$.

(A

_{3}) There exists a continuous function

$h\in C(J,\mathbb{R})$ such that

$|g(t,x)|\le h(t),\phantom{\rule{1em}{0ex}}t\in J,$

for all $x\in \mathbb{R}$.

**Lemma 3.2** [19]

*Let* $0<q<1$ *and* $u\in {L}^{1}(0,T)$.

(H_{1}) *The equality* ${D}^{q}{I}^{q}u(t)=u(t)$ *holds*.

(H

_{2})

*The equality* ${I}^{q}{D}^{q}u(t)=u(t)-\frac{{I}^{1-q}u({t}_{0})}{\mathrm{\Gamma}(q)}{(t-{t}_{0})}^{q-1}$

*holds almost everywhere on* *J*.

The following lemma is useful in what follows.

**Lemma 3.3** *Assume that hypothesis* (A

_{0})

*holds*.

*Then*,

*for any* $h\in C(J,\mathbb{R})$ *and* $0<q<1$,

*the function* $x\in C(J,\mathbb{R})$ *is a solution of the FHDE* ${D}^{q}[x(t)-f(t,x(t))]=h(t),\phantom{\rule{1em}{0ex}}t\in J,$

(3.1)

*and*
$x({t}_{0})={x}_{0},$

(3.2)

*if and only if* *x* *satisfies the hybrid integral equation* (

*HIE*)

$x(t)={x}_{0}-f({t}_{0},{x}_{0})+f(t,x(t))+\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{t}{(t-s)}^{q-1}h(s)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t\in J.$

(3.3)

*Proof* Let

*x* be a solution of the Cauchy problem (3.1) and (3.2). Since the Riemann-Liouville fractional integral

${I}^{q}$ is a monotone operator, thus we apply the fractional integral

${I}^{q}$ on both sides of (3.1). By Lemma 3.2, we have

${I}^{q}{D}^{q}[x(t)-f(t,x(t))]=x(t)-f(t,x(t))-\frac{{I}^{1-q}[x(t)-f(t,x(t))]{|}_{t={t}_{0}}}{\mathrm{\Gamma}(q)}{(t-{t}_{0})}^{q-1}={I}^{q}h(t),$

then by (3.2), we get

$x(t)-f(t,x(t))={I}^{q}h(t)+\frac{{I}^{1-q}[x(t)-f(t,x(t))]{|}_{t={t}_{0}}}{\mathrm{\Gamma}(q)}{(t-{t}_{0})}^{q-1}={x}_{0}-f({t}_{0},{x}_{0})+{I}^{q}h(t),$

*i.e.*,

$\begin{array}{rcl}x(t)& =& {x}_{0}-f({t}_{0},{x}_{0})+f(t,x(t))+{I}^{q}h(t)\\ =& {x}_{0}-f({t}_{0},{x}_{0})+f(t,x(t))+\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{t}{(t-s)}^{q-1}h(s)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t\in J.\end{array}$

Thus, (3.3) holds.

Conversely, assume that

*x* satisfies HIE (3.3). Then applying

${D}^{q}$ on both sides of (3.3), (3.1) is satisfied. Again, substituting

$t={t}_{0}$ in (3.3) yields

$x({t}_{0})-f({t}_{0},x({t}_{0}))={x}_{0}-f({t}_{0},{x}_{0}).$

The map $x\mapsto x-f(t,x)$ is increasing in ℝ for all $t\in J$, the map $x\mapsto x-f({t}_{0},x)$ is injective in ℝ, hence $x({t}_{0})={x}_{0}$. The proof is completed. □

Now, we are in a position to prove the following existence theorem for FHDE (2.1).

**Theorem 3.1** *Assume that hypotheses* (A_{0})-(A_{2}) *hold*. *Then FHDE* (2.1) *has a solution defined on* *J*.

*Proof* Set

$X=C(J,\mathbb{R})$ and define a subset

*S* of

*X* defined by

$S=\{x\in X\mid \parallel x\parallel \le N\},$

(3.4)

where $N=|{x}_{0}-f({t}_{0},{x}_{0})|+L+{F}_{0}+\frac{{a}^{q}}{\mathrm{\Gamma}(q+1)}{\parallel h\parallel}_{{L}^{1}}$ and ${F}_{0}={sup}_{t\in J}|f(t,0)|$.

Clearly,

*S* is a closed, convex and bounded subset of the Banach algebra

*X*. Now, using the hypotheses (A

_{0})-(A

_{2}), it can be shown by an application of Lemma 3.3, FHDE (2.1) is equivalent to the nonlinear HIE

$x(t)={x}_{0}-f({t}_{0},{x}_{0})+f(t,x(t))+\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{t}{(t-s)}^{q-1}g(s,x(s))\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t\in J.$

(3.5)

Define two operators

$A:X\to X$ and

$B:S\to X$ by

$Ax(t)=f(t,x(t)),\phantom{\rule{1em}{0ex}}t\in J,$

(3.6)

and

$Bx(t)={x}_{0}-f({t}_{0},{x}_{0})+\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{t}{(t-s)}^{q-1}g(s,x(s))\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t\in J.$

(3.7)

Then HIE (3.5) is transformed into the operator equation as

$Ax(t)+Bx(t)=x(t),\phantom{\rule{1em}{0ex}}t\in J.$

(3.8)

We will show that the operators *A* and *B* satisfy all the conditions of Lemma 3.1.

First, we show that

*A* is a Lipschitz operator on

*X* with the Lipschitz constant

*L*. Let

$x,y\in X$. Then by hypothesis (A

_{1}),

$|Ax(t)-Ay(t)|=|f(t,x(t))-f(t,y(t))|\le \frac{L|x(t)-y(t)|}{M+|x(t)-y(t)|}\le \frac{L\parallel x-y\parallel}{M+\parallel x-y\parallel}$

for all

$t\in J$. Taking supremum over

*t*, we obtain

$\parallel Ax-Ay\parallel \le \frac{L\parallel x-y\parallel}{M+\parallel x-y\parallel}$

for all $x,y\in X$. This shows that *A* is a nonlinear contraction on *X* with a control function *φ* defined by $\phi =\frac{Lr}{M+r}$.

Next, we show that

*B* is a compact and continuous operator on

*S* into

*X*. First, we show that

*B* is continuous on

*S*. Let

$\{{x}_{n}\}$ be a sequence in

*S* converging to a point

$x\in S$. Then, by the Lebesgue dominated convergence theorem,

$\begin{array}{rl}\underset{n\to \mathrm{\infty}}{lim}B{x}_{n}(t)& ={x}_{0}-f({t}_{0},{x}_{0})+\underset{n\to \mathrm{\infty}}{lim}\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{t}{(t-s)}^{q-1}g(s,{x}_{n}(s))\phantom{\rule{0.2em}{0ex}}ds\\ ={x}_{0}-f({t}_{0},{x}_{0})+\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{t}{(t-s)}^{q-1}\underset{n\to \mathrm{\infty}}{lim}g(s,{x}_{n}(s))\phantom{\rule{0.2em}{0ex}}ds\\ ={x}_{0}-f({t}_{0},{x}_{0})+\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{t}{(t-s)}^{q-1}g(s,x(s))\phantom{\rule{0.2em}{0ex}}ds\\ =Bx(t)\end{array}$

for all $t\in J$. This shows that *B* is a continuous operator on *S*.

Now, we show that

*B* is a compact operator on

*S*. It is enough to show that

$B(S)$ is a uniformly bounded and equicontinuous set in

*X*. On the one hand, let

$x\in S$ be arbitrary. Then by hypothesis (A

_{2}),

$\begin{array}{rl}|Bx(t)|& =|{x}_{0}-f({t}_{0},{x}_{0})|+\left|\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{t}{(t-s)}^{q-1}g(s,x(s))\phantom{\rule{0.2em}{0ex}}ds\right|\\ \le |{x}_{0}-f({t}_{0},{x}_{0})|+\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{t}{(t-s)}^{q-1}\left|g(s,x(s))\right|\phantom{\rule{0.2em}{0ex}}ds\\ \le |{x}_{0}-f({t}_{0},{x}_{0})|+\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{t}{(t-s)}^{q-1}h(s)\phantom{\rule{0.2em}{0ex}}ds\\ \le |{x}_{0}-f({t}_{0},{x}_{0})|+\frac{{a}^{q}}{\mathrm{\Gamma}(q+1)}{\parallel h\parallel}_{{L}^{1}},\end{array}$

for all

$t\in J$. Taking supremum over

*t*,

$\parallel Bx\parallel \le |{x}_{0}-f({t}_{0},{x}_{0})|+\frac{{a}^{q}}{\mathrm{\Gamma}(q+1)}{\parallel h\parallel}_{{L}^{1}}$

for all $x\in S$. This shows that *B* is uniformly bounded on *S*.

On the other hand, let

${t}_{1},{t}_{2}\in J$ with

${t}_{1}<{t}_{2}$. Then, for any

$x\in S$, one has

$\begin{array}{rcl}|Bx({t}_{1})-Bx({t}_{2})|& =& |\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{{t}_{1}}{({t}_{1}-s)}^{q-1}g(s,x(s))\phantom{\rule{0.2em}{0ex}}ds-\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{{t}_{2}}{({t}_{2}-s)}^{q-1}g(s,x(s))\phantom{\rule{0.2em}{0ex}}ds|\\ \le & |\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{{t}_{1}}{({t}_{1}-s)}^{q-1}g(s,x(s))\phantom{\rule{0.2em}{0ex}}ds-\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{{t}_{1}}{({t}_{2}-s)}^{q-1}g(s,x(s))\phantom{\rule{0.2em}{0ex}}ds|\\ +|\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{{t}_{1}}{({t}_{2}-s)}^{q-1}g(s,x(s))\phantom{\rule{0.2em}{0ex}}ds-\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{{t}_{2}}{({t}_{2}-s)}^{q-1}g(s,x(s))\phantom{\rule{0.2em}{0ex}}ds|\\ \le & \frac{{\parallel h\parallel}_{{L}^{1}}}{\mathrm{\Gamma}(q+1)}[|{({t}_{2}-{t}_{0})}^{q}-{({t}_{1}-{t}_{0})}^{q}-{({t}_{2}-{t}_{1})}^{q}|+{({t}_{2}-{t}_{1})}^{q}].\end{array}$

Hence, for

$\epsilon >0$, there exists a

$\delta >0$ such that

$|{t}_{1}-{t}_{2}|<\delta \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}|Bx({t}_{1})-Bx({t}_{2})|<\epsilon ,$

for all ${t}_{1},{t}_{2}\in J$ and for all $x\in S$. This shows that $B(S)$ is an equicontinuous set in *X*. Now, the set $B(S)$ is a uniformly bounded and equicontinuous set in *X*, so it is compact by the Arzela-Ascoli theorem. As a result, *B* is a complete continuous operator on *S*.

Next, we show that hypothesis (c) of Lemma 3.1 is satisfied. Let

$x\in S$. Then, by assumption (A

_{1}), we have

$\begin{array}{rcl}|Ax(t)+Bx(t)|& \le & |Ax(t)|+|Bx(t)|\\ \le & |{x}_{0}-f({t}_{0},{x}_{0})|+\left|f(t,x(t))\right|+\left|\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{t}{(t-s)}^{q-1}g(s,y(s))\phantom{\rule{0.2em}{0ex}}ds\right|\\ \le & |{x}_{0}-f({t}_{0},{x}_{0})|+[|f(t,x(t))-f(t,0)|+|f(t,0)|]\\ +\left(\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{t}{(t-s)}^{q-1}\left|g(s,x(s))\right|\phantom{\rule{0.2em}{0ex}}ds\right)\\ \le & |{x}_{0}-f({t}_{0},{x}_{0})|+L+{F}_{0}+(\frac{1}{\mathrm{\Gamma}(q)}{\int}_{{t}_{0}}^{t}{(t-s)}^{q-1}h(s)\phantom{\rule{0.2em}{0ex}}ds)\\ \le & |{x}_{0}-f({t}_{0},{x}_{0})|+L+{F}_{0}+\frac{{T}^{q}}{\mathrm{\Gamma}(q+1)}{\parallel h\parallel}_{{L}^{1}}.\end{array}$

Taking supremum over

*t*,

$\parallel x\parallel \le |{x}_{0}-f({t}_{0},{x}_{0})|+L+{F}_{0}+\frac{{T}^{q}}{\mathrm{\Gamma}(q+1)}{\parallel h\parallel}_{{L}^{1}}=N.$

Thus, all the conditions of Lemma 3.1 are satisfied and hence the operator equation $Ax+Bx=x$ has a solution in *S*. As a result, FHDE (2.1) has a solution defined on *J*. This completes the proof. □