1 Introduction

In most phenomena there appears usually a crisis. Our imagination as regards crises has effects on economy while there are distinct types of crisis-phenomena study in different fields of science such chemistry, social sciences, physics, mathematics, engineering and economy (see, for example, [1–7] and [8]). Considering the importance of modeling of crisis phenomena, some researchers are working and publishing in this area (see, for example, [9–13]). In 2016, Almeida, Bastos and Monteiro published a paper about modeling of some real phenomena by fractional differential equations [14]. As is well known, one of the best methods for mathematical describing this type phenomena is modeling of the problems as singular fractional integro-differential equations, which have been studied by researchers especially in recent decades (see, for example, [15–20] and [21]).

In 2010, Agarwal, O’Regan and Stanek investigated the existence of solutions for the problem \(D^{\alpha } u(t)+ f(t, u(t))=0\) with boundary conditions \(u'(0) = \cdots = u^{(n-1)} = 0\) and \(u(1)=\int _{0} ^{1} u(s)\,d\mu (s)\), where \(n \geq 2\), \(\alpha \in (n-1,n)\), \(\mu (s)\) is a functional of bounded variation with \(\int_{0} ^{1}d\mu (s) < 1\), and f may have a singularity at \(t=0\) [15]. They reviewed the existence of positive solutions for the system \(D^{\alpha } u_{i}(t)+ f_{i}(t,u_{1}(t),u_{2}(t))=0\) with boundary conditions \(u_{i}(0)=u'_{i}(0)=0\) and \(u_{i}(1) = \int_{0} ^{1} u_{i}(t)\,d\eta (t)\) for \(i=1,2\), where \(t \in (0,1)\), \(\alpha \in (2,3]\), \(\int_{0} ^{1} u_{i}(t)\,d\eta (t)\) denotes the Riemann–Stieltjes integral, \(f_{i} \in C([0,1] \times \mathbb{R}^{+} \times \mathbb{R} ^{+}, \mathbb{R})\) and \(D^{\alpha }\) is the Riemann–Liouville fractional derivative of order α [16]. In 2013, Bai and Qui studied the singular problem \(D^{\alpha } u+ f(t, u, D^{\gamma } u, D^{\mu } u)+ g(t, u, D^{\gamma } u, D^{\mu } u)=0\) with boundary conditions \(u(0)=u'(0)=u''(0)=u'''(0)=0\), where \(3< \alpha < 4\), \(0< \gamma <1\), \(1<\mu <2\), \(D^{\alpha }\) is the Caputo fractional derivative and f is a Caratheodory function on \([0,1] \times (0 , \infty)^{3}\) [17]. Recently, the multi-singular point-wise defined fractional integro-differential equation \(D^{\mu } x(t)+ f(t, x(t), x'(t), D^{\beta }x(t), I^{p}x(t)) =0\) with boundary conditions \(x'(0)=x(\xi)\) and \(x(1)=\int_{0}^{\eta }x(s)\,ds\) when \(\mu \in [2,3)\) and \(x'(0)=x(\xi)\), \(x(1)=\int_{0}^{\eta }x(s)\,ds\) and \(x^{(j)}(0)=0\) for \(j=2,\dots,[\mu ]-1\) when \(\mu \in [3,\infty)\) has been studied, where \(0\leq t\leq 1\), \(x \in C^{1}[0,1]\), \(\mu \in [2,\infty)\), \(\beta, \xi, \eta \in (0,1)\), \(p>1\), \(D^{\mu }\) is the Caputo fractional derivative of order μ and \(f:[0,1] \times \mathbb{R} ^{5} \to \mathbb{R}\) is a function such that \(f(t,\cdot ,\cdot ,\cdot ,\cdot )\) is singular at some points \(t\in [0,1]\) [19]. By using these ideas and providing a new method for modeling of crisis phenomena, we investigate the existence of solutions for the point-wise defined three steps crisis integro-differential equation

$$\begin{aligned} D^{\alpha } x(t)+ f\biggl(t, x(t), x'(t), D^{\beta }x(t), \int_{0}^{t} h( \xi) x(\xi)\,d\xi, \phi \bigl(x(t) \bigr)\biggr)=0 \end{aligned}$$
(1)

with boundary conditions \(x(1)=x(0)=x''(0)=x^{n}(0)=0\), where \(\alpha \geq 2\), \(\lambda, \mu, \beta \in (0,1)\), \(\phi: X \rightarrow X\) is a mapping such that \(\Vert \phi (x) - \phi (y)\Vert \leq \theta_{0} \Vert x-y\Vert + \theta_{1} \Vert x'-y' \Vert \) for some nonnegative real numbers \(\theta_{0}\) and \(\theta_{1} \in [0,\infty)\) and all \(x,y \in X\), \(D^{\alpha }\) is the Caputo fractional derivative of order α, \(f(t,x_{1}(t),\ldots, x_{5}(t))=f_{1}(t,x_{1}(t),\ldots, x_{5}(t))\) for all \(t\in [0,\lambda)\), \(f(t,x_{1}(t),\ldots, x_{5}(t))=f_{2}(t,x_{1}(t),\ldots, x_{5}(t))\) for all \(t\in [\lambda,\mu ]\) and \(f(t,x_{1}(t),\ldots, x _{5}(t))=f(t,x_{1}(t),\ldots, x_{5}(t))\) for all \(t\in (\mu,1]\), \(f_{1}(t,\cdot ,\cdot ,\cdot ,\cdot )\) and \(f_{3}(t,\cdot ,\cdot ,\cdot ,\cdot )\) are continuous on \([0,\lambda)\) and \((\mu,1]\) and \(f_{2}(t,\cdot ,\cdot ,\cdot ,\cdot )\) is multi-singular [19].

2 Preliminaries

Recall that \(D^{\alpha }x(t)+f(t)=0\) is a point-wise defined equation on \([0,1]\) if there exists a set \(E \subset [0,1]\) such that the measure of \(E^{c}\) is zero and the equation holds on E [19]. In this paper, we use \(\Vert \cdot \Vert _{1}\) for the norm of \(L ^{1} [0,1]\), \(\Vert \cdot \Vert \) for the sup norm of \(Y=C[0,1]\) and \(\Vert x \Vert _{*} = \max \{\Vert x\Vert , \Vert x'\Vert \} \) for the norm of \(X=C^{1}[0,1]\). As is well known, the Riemann–Liouville integral of order p with the lower limit \(a\geq 0\) for a function \(f:(a,\infty)\to \mathbb{R}\) is defined by \(I^{p}_{a^{+}}f(t)=\frac{1}{\Gamma (p)} \int_{a}^{t} (t-s)^{p-1} f(s)\,ds\), provided that the right-hand side is point-wise defined on \((a,\infty)\) [22]. We denote \(I^{p}_{0^{+}}f(t)\) by \(I^{p}f(t)\). Also, the Caputo fractional derivative of order \(\alpha >0\) is defined by \({}^{c}D^{\alpha }f(t)=\frac{1}{\Gamma (n- \alpha)} \int_{0}^{t} \frac{f^{n}(s)}{(t-s)^{\alpha +1-n}}\,ds\), where \(n=[\alpha ]+1\) and \(f:(a,\infty)\to \mathbb{R}\) is a function [22]. Let Ψ be the family of nondecreasing functions \(\psi:[0,\infty) \to [0,\infty)\) such that \(\sum_{n=1}^{\infty } \psi^{n}(t)<\infty \) for all \(t> 0\) (see [23]). One can check that \(\psi (t)< t\) for all \(t>0\). Let \((X,d)\) be a metric space and \(T:X \to X\) and \(\alpha:X \times X \to [0,\infty)\) two maps. Then T is called an α-admissible map whenever \(\alpha (x,y) \geq 1\) implies \(\alpha (Tx,Ty) \geq 1\) [23]. The map T is called an α-admissible map whenever \(\alpha (x,y) \geq 1\) implies \(\alpha (Tx,Ty) \geq 1\) [23]. Let \((X,d)\) be a metric space, \(\psi \in \Psi \) and \(\alpha:X \times X \to [0,\infty)\) a map. A self-map \(T:X \to X\) is called an α-ψ-contraction whenever \(\alpha (x,y) d(Tx,Ty) \leq \psi (d(x,y))\) for all \(x,y \in X\) [23]. To prove the existence of solutions, we need next results.

Lemma 2.1

([23])

Let \((X,d)\) be a complete metric space, \(\psi \in \Psi \), \(\alpha:X \times X \to [0,\infty)\) a map and \(T:X \to X\) an α-admissible α-ψ-contraction. If T is continuous and there exists \(x_{0} \in X\) such that \(\alpha (x_{0}, Tx_{0}) \geq 1\), then T has a fixed point.

Lemma 2.2

([24])

Let \(n-1\leq \alpha < n\) and \(x\in C(0,1) \cap L^{1}(0,1)\). Then we have \(I^{\alpha } D^{\alpha }x(t)=x(t)+ \sum_{i=0}^{n-1} c _{i}t^{i}\) for some real constants \(c_{0},\dots,c_{n-1}\).

Lemma 2.3

([21])

Let \(\beta > 0\) and \(\alpha >-1\). Then \(\int^{t}_{0} (t-s)^{ \alpha - 1} s^{\beta }\,ds = B(\beta + 1, \alpha) t^{\alpha + \beta }\), where \(B(\beta, \alpha) = \frac{\Gamma (\alpha) \Gamma (\beta)}{ \Gamma (\alpha +\beta)}\).

Lemma 2.4

([25])

Let E be a Banach space, \(P \subseteq E\) a cone and \(\Omega_{1}\), \(\Omega_{2}\) two bounded open balls of E centered at the origin with \(\overline{\Omega_{1}} \subset \Omega_{2}\). Suppose that \(F:P\cap (\bar{ \Omega }_{2} \backslash \Omega_{1}) \rightarrow P\) is a completely continuous operator such that either

(\(i_{1}\)):

\(\Vert F(x)\Vert \leq \Vert x\Vert \) for all \(x \in P \cap \partial \Omega _{1}\) and \(\Vert Fx\Vert \geq \Vert x\Vert \) for all \(x \in P \cap \partial \Omega _{2}\), or

(\(i_{2}\)):

\(\Vert Fx\Vert \geq \Vert x\Vert \) for all \(x \in P \cap \partial \Omega _{1}\) and \(\Vert Fx\Vert \leq \Vert x\Vert \) for all \(x \in P\cap \partial \Omega _{2}\)

holds. Then F has a fixed point in \(P\cap (\Omega_{2} \backslash \Omega_{1})\).

3 Main results

Now, we are ready for providing our results.

Lemma 3.1

Let \(\alpha \geq 2\), \(n=[\alpha ] +1\) and \(f \in L^{1}[0,1]\). A map u is a solution for the point-wise defined equation \(D^{\alpha }x(t) +f(t) = 0\) with boundary conditions \(x'(1)= x(0)=x''(0) = \cdots =x ^{n-1}(0)=0\) if and only if \(u(t)= \int^{1}_{0} G(t,s) f(s)\,ds\) for all \(t \in [0,1]\), where \(G(t,s)=\frac{t(1-s)^{\alpha -2}}{\Gamma (\alpha -1)}\) whenever \(0\leq t \leq s \leq 1\) and \(G(t,s)=\frac{t (1-s)^{ \alpha - 2}}{ \Gamma (\alpha -1)} - \frac{(t-s)^{\alpha -1}}{\Gamma ( \alpha)}\) whenever \(0\leq s \leq t \leq 1\).

Proof

Let E be a subset of \([0,1]\) such that \(m(E^{c})=0\) and \(D^{\alpha }x(t) +f(t) = 0\) for all \(t \in E\). Here, m is the Lebesgue measure on \(\mathbb{R}\). Note that E is dense in \([0,1]\). Let \(f_{0} \in C[0,1]\) be a function such that \(f_{0}=f\) on E. Then we have

$$\begin{aligned} I^{\alpha }\bigl(f(t)\bigr) =& \frac{1}{\Gamma (\alpha)} \int_{0}^{t} (t-s)^{ \alpha -1}f(s)\,ds \\ =&\frac{1}{\Gamma (\alpha)}\biggl( \int_{[0,t]\cap E} (t-s)^{\alpha -1}f(s)\,ds + \int_{[0,t]\cap E^{c}} (t-s)^{\alpha }f(s)\,ds\biggr) \\ =&\frac{1}{\Gamma (\alpha)} \int_{[0,t]\cap E} (t-s)^{\alpha -1}f _{0}(s)\,ds \\ =&\frac{1}{\Gamma (\alpha)}\biggl( \int_{[0,t]\cap E} (t-s)^{\alpha -1}f _{0}(s)\,ds + \int_{[0,t]\cap E^{c}} (t-s)^{\alpha -1}f_{0}(s)\,ds\biggr) \\ =&\frac{1}{\Gamma (\alpha)} \int_{0}^{t} (t-s)^{\alpha -1}f_{0}(s) \,ds= I^{\alpha }\bigl(f_{0}(t)\bigr) \end{aligned}$$

for all \(t\in E\). Let \(t\in E^{c}\backslash \{0\}\). Choose a sequence \(\{ t_{n}\}_{n\geq 1}\) in E such that \(t_{n} \to t ^{-}\). Then

$$\begin{aligned} I^{\alpha }\bigl(f(t)\bigr) =& \frac{1}{\Gamma (\alpha)} \int_{0}^{t} (t-s)^{ \alpha -1}f(s)\,ds \\ =& \lim_{n\to \infty } \frac{1}{\Gamma (\alpha)} \int _{0}^{t_{n}} (t_{n} -s)^{\mu -1}f(s)\,ds = \lim_{n\to \infty } I^{ \alpha } \bigl(f(t_{n})\bigr) \\ =& \lim_{n\to \infty } I^{\alpha }\bigl(f_{0}(t_{n}) \bigr) =\lim_{n\to \infty } \frac{1}{\Gamma (\alpha)} \int_{0}^{t_{n}} (t_{n}-s)^{\alpha -1}f _{0}(s)\,ds \\ =& \frac{1}{\Gamma (\alpha)} \int_{0}^{t} (t-s)^{\alpha -1}f(s)\,ds \\ =&I^{\alpha }\bigl(f_{0}(t)\bigr). \end{aligned}$$

For \(t=0 \in E^{c} \), we get \(I^{\alpha }(f(t))=I^{\alpha }(f_{0}(t))=0\) and so \(I^{\alpha }(f(t))=I^{\alpha }(f_{0}(t))\) for all \(t\in [0,1]\). Thus, the equation \(D^{\alpha }x(t) +f(t) = 0\) equivalents to \(I^{\alpha }(D^{\alpha }x(t))= I^{\alpha }(-f_{0}(t))\) on \([0,1]\). By using Lemma 2.2 and the boundary condition, we get \(x(t)= - \frac{1}{ \Gamma (\alpha)} \int^{t}_{0} (t-s)^{\alpha - 1} y(s)\,ds + c_{1} t\) and so \(x'(t)= - \frac{1}{\Gamma (\alpha -1)} \int^{t}_{0} (t-s)^{\alpha - 2} y(s)\,ds + c_{1}\). Hence, \(x'(1)= - \frac{1}{\Gamma (\alpha -1)} \int^{1}_{0} (1-s)^{\alpha - 2} y(s)\,ds + c_{1}\). Since \(x'(1) = 0\), \(c_{1}= \frac{1}{\Gamma (\alpha -1)} \int^{1}_{0} (1-s)^{\alpha - 2} y(s)\,ds\) and so

$$x(t)= - \frac{1}{\Gamma (\alpha)} \int^{t}_{0} (t-s)^{\alpha - 1} y(s)\,ds + \frac{t}{\Gamma (\alpha -1)} \int^{1}_{0} (1-s)^{\alpha - 2} y(s)\,ds= \int^{1}_{0} G(t,s) y(s)\,ds, $$

where \(G(t,s)=\frac{t(1-s)^{\alpha -2}}{\Gamma (\alpha -1)}\) whenever \(0\leq t \leq s \leq 1\) and \(G(t,s)=\frac{t (1-s)^{\alpha - 2}}{ \Gamma (\alpha -1)} - \frac{(t-s)^{\alpha -1}}{\Gamma (\alpha)}\) whenever \(0\leq s \leq t \leq 1\). Also, an easy calculation shows that \(u(t)= \int^{1}_{0} G(t,s) f(s)\,ds\) is a solution for the equation with the boundary conditions. This completes the proof. □

Note that for the Green function \(G(t,s) \) in the last result we have \(G(t,s) \geq \frac{(\alpha -2)\vert t-s\vert ^{\alpha - 1}}{ \Gamma (\alpha)} \geq 0\), \(G(t,s) \leq \frac{t(1-s)^{\alpha -2}}{\Gamma ( \alpha -1)}\), \(\frac{ \partial }{ \partial t} G(t,s) \geq 0\) and \(\frac{\partial }{ \partial t}G(t,s) \leq \frac{(1-s)^{\alpha -2}}{\Gamma ( \alpha -1)}\) for all \(t,s \in [0,1]\). Also, G and \(\frac{ \partial }{ \partial t} G\) are continuous with respect to t. Consider the space \(X= C^{1}[0,1]\) with the norm \(\Vert \cdot \Vert _{*}\), where \(\Vert x\Vert _{*} = \max \{ \Vert x\Vert , \Vert x'\Vert \}\) and \(\Vert \cdot \Vert \) is the supremum norm on \(C[0,1]\). Let \(\lambda, \mu \in (0,1)\) with \(\lambda <\mu \). Suppose that \(f_{1}\) and \(f_{3}\) are continuous functions (with respect to the first variable) on \([0, \lambda ]\times X^{5}\) and \([\mu, 1] \times X^{5}\), respectively, and \(f_{2}\) is a function on \((\lambda, \mu)\times X^{5}\) which is singular at some points \(t\in (\lambda, \mu)\). Let f be a map on \([0,1]\times X^{5}\) such that \(f\vert _{[0, \lambda ]\times X^{5}}=f_{1}\), \(f\vert _{(\lambda,\mu)\times X^{5}} =f _{2}\) and \(f\vert _{[\mu,0]\times X^{5}}=f_{3}\). We denote this case briefly by \([\lambda, \mu, f=(f_{1},f_{2},f_{3})]\). Define the map \(F:X \to X\) by \(F_{x}(t)= \int_{0}^{1} G(t,s) f(s, x(s), x'(s), D^{ \beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi (x(s)))\,ds\) for all \(t\in [0,1]\). Note that the singular point-wise defined equation (1) has a solution \(u_{0}\in X\) if and only if \(u_{0}\) a fixed point of the map F.

Theorem 3.2

Let \([\lambda, \mu, f=(f_{1},f_{2},,f_{3})]\) with \(f_{1}(s,0,0,0,0,0)=f _{3}(t,0,0,0,0,0)=0\) for all \(s\in [0,\lambda ]\) and \(t\in [\mu, 1]\). Assume that there exist two maps \(H: X^{5} \to [0, \infty)\) and \(\Phi:(\lambda, \mu)\to [0, \infty)\) such that \(f_{2}(t,x_{1}, x _{2}, \ldots, x_{5}) \leq \Phi (t) H(x_{1}, x_{2}, \ldots, x_{5})\) for all \((x_{1},\ldots,x_{5}) \in X^{5}\) and almost all \(t \in (\lambda, \mu)\), where \(H: X^{5} \to [0,\infty) \) is nondecreasing with respect to all its components, \(\int_{\lambda }^{\mu } (1-s)^{\alpha -1} \Phi (s)\,ds <\infty \) and \(\lim_{z\to 0^{+}} \frac{H(z,z,z,z,z)}{z} =0\). Suppose that the map q defined by \(q(t)= \lim_{\max \{\Vert x_{1}\Vert ,\dots, \Vert x_{5}\Vert \} \to \infty } \frac{f_{2}(t, x_{1} x_{2}, \ldots, x_{5})}{\max \{\Vert x_{1}\Vert ,\dots, \Vert x_{5}\Vert \}} \) for almost all \(t \in (\lambda, \mu)\) has the property that \(\frac{ \alpha -2}{\Gamma (\alpha)} \int_{\lambda }^{\mu } (\mu -s)^{\alpha -2} q(s)\,ds >1\). Assume that there exist nonnegative real numbers \(l_{1},\dots,l_{5}\), \(l'_{1},\dots,l'_{5}\) and mappings \(a_{1},\dots,a_{5}:(\lambda,\mu)\to [0,\infty)\) and \(\Lambda_{1},\dots,\Lambda _{5}:X^{5}\to [0,\infty)\) such that \(\vert f_{1}(t, x_{1},\dots, x_{5})-f_{1}(t, y_{1},\dots, y_{5})\vert \leq \sum_{i=1}^{5} l_{i} \vert x_{i} - y_{i}\vert \),

$$\bigl\vert f_{2}(t, x_{1}, \ldots, x_{5}) - f_{2}(t, y_{1}, \ldots, y_{5})\bigr\vert \leq \sum_{i=1}^{5} a_{i}(t) \Lambda_{i}\bigl(\vert x_{1} - y_{1}\vert , \ldots, \vert x_{5} - y_{5}\vert \bigr) $$

and \(\vert f_{3}(t, x_{1},\dots, x_{5})-f_{3}(t, y_{1},\dots, y_{5})\vert \leq \sum_{i=1}^{5} l'_{i} \vert x_{i} - y_{i}\vert \) for all t and \(x_{1},\dots,x_{5}\in X\). If \(\lim_{z \to 0^{+}} \frac{\Lambda_{i}(z,z,z,z,z)}{z}= q_{i}<\infty \) and \([ \frac{L (1-(1- \lambda)^{\alpha -1})}{\Gamma (\alpha)} + \frac{L' }{\Gamma (\alpha)} (1-\mu)^{\alpha -1}] <1\) for \(i=1,\dots,5\), where \(m_{0}= \int _{0}^{1}\vert h(\xi)\vert \,d\xi \), \(L= l_{1}+l_{2}+ \frac{l_{3} }{\Gamma (2- \beta)}+ m_{0} l_{4}+ \theta_{0} l_{5}+\theta_{1} l_{5} \) and \(L'= l'_{1}+l'_{2}+ \frac{l'_{3} }{\Gamma (2-\beta)}+ m_{0} l'_{4}+ \theta_{0} l'_{5}+\theta_{1} l'_{5}\), then the problem (1) has a solution.

Proof

Consider the closed cone \(P=\{x \in X: x(t) \geq 0 \mbox{ and } x'(t) \geq 0 \mbox{ for all } t \in [0,1] \}\) in X. Let \(\epsilon >0\) be given, \(\{x_{n}\}_{n\geq 1}\) a sequence in X with \(x_{n} \to x\). Choose a natural number N such that \(\Vert x_{n} - x\Vert <\epsilon \) for all \(n\geq N\). Take \(\epsilon >0\) such that

$$\Biggl[ \frac{L (1-(1-\lambda)^{\alpha -1})}{\Gamma (\alpha)} + \frac{(q _{i}+\epsilon) \epsilon }{\Gamma (\alpha -1) } \sum _{i=1}^{5} M _{i}(\lambda, \mu) + \frac{L' }{\Gamma (\alpha)} (1-\mu)^{\alpha -1}\Biggr] < 1 $$

for \(i=1,\dots,5\), where \(M_{i}(\lambda, \mu)= \int_{\lambda }^{ \mu } (1-s)^{\alpha -2} a_{i}(s)\,ds\). Note that

$$\begin{aligned} \bigl\vert F_{x_{n}}(t)-F_{x}(t)\bigr\vert \leq& \int_{0}^{\lambda } G(t,s)\biggl\vert f_{1} \biggl(s, x _{n}(s), x'_{n}(s), D^{\beta }x_{n}(s), \int_{0}^{s} h(\xi) x_{n}( \xi)\,d\xi, \phi \bigl(x_{n}(s)\bigr)\biggr) \\ &{}- f_{1}\biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\biggr\vert \,ds \\ &{}+ \int_{\lambda }^{\mu } G(t,s) \biggl\vert f_{2} \biggl(s, x_{n}(s), x'_{n}(s), D^{ \beta }x_{n}(s), \int_{0}^{s} h(\xi) x_{n}(\xi)\,d\xi, \phi \bigl(x_{n}(s)\bigr)\biggr) \\ &{}- f_{2}\biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\biggr\vert \,ds \\ &{}+ \int_{\mu }^{1} G(t,s) \biggl\vert f_{3} \biggl(s, x_{n}(s), x'_{n}(s), D^{\beta }x _{n}(s), \int_{0}^{s} h(\xi) x_{n}(\xi)\,d\xi, \phi \bigl(x_{n}(s)\bigr)\biggr) \\ &{}- f_{3}\biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\biggr\vert \,ds \\ \leq& \int_{0}^{\lambda } G(t,s) \biggl( l_{1} \bigl\vert x_{n}(s) - x(s)\bigr\vert + l_{2} \bigl\vert x'_{n}(s) - x'(s)\bigr\vert + l_{3} \bigl\vert D^{\beta } (x_{n} - x) (s)\bigr\vert \\ &{}+ l_{4} \int_{0}^{s} \bigl\vert x_{n}(\xi) - x(\xi)\bigr\vert \,d\xi + l_{5} \bigl\vert \phi \bigl(x_{n}(s) - x(s)\bigr)\bigr\vert \biggr) \\ &{}+ \int_{\lambda }^{\mu } G(t,s) (a_{1}(s) \Lambda_{1}\biggl(\bigl\vert x_{n}(s) - x(s)\bigr\vert , \bigl\vert x'_{n}(s) - x'(s)\bigr\vert , \\ &{}\bigl\vert D^{\beta } (x_{n} - x) (s)\bigr\vert , \int_{0}^{s} \bigl\vert x_{n}(\xi) - x(\xi)\bigr\vert d \xi, \bigl\vert \phi \bigl(x_{n}(s) - x(s) \bigr)\bigr\vert \biggr) \\ &{}+\cdots+ a_{5}(s) \Lambda_{5}\biggl(\bigl\vert x_{n}(s) - x(s)\bigr\vert , \bigl\vert x'_{n}(s) - x'(s)\bigr\vert , \\ &{}\bigl\vert D^{\beta } (x_{n} - x) (s)\bigr\vert , \int_{0}^{s} \bigl\vert x_{n}(\xi) - x(\xi)\bigr\vert d \xi, \bigl\vert \phi \bigl(x_{n}(s) - x(s) \bigr)\bigr\vert \biggr) \\ &{}+ \int_{\mu }^{1} G(t,s) \biggl( l'_{1} \bigl\vert x_{n}(s) - x(s)\bigr\vert + l'_{2} \bigl\vert x'_{n}(s)- x'(s)\bigr\vert + l'_{3} \bigl\vert D^{\beta } (x_{n} - x) (s)\bigr\vert \\ &{}+ l'_{4} \int_{0}^{s} \bigl\vert x_{n}(\xi) - x(\xi)\bigr\vert \,d\xi + l'_{5} \bigl\vert \phi \bigl(x_{n}(s) - x(s)\bigr)\bigr\vert \biggr) \\ \leq& \int_{0}^{\lambda }G(t,s) \biggl( l_{1} \Vert x_{n} - x\Vert + l_{2} \bigl\Vert x'_{n} -x'\bigr\Vert + \frac{l_{3} }{\Gamma (2-\beta)} \bigl\Vert x'_{n} - x'\bigr\Vert \\ &{}+m_{0} l_{4} \Vert x_{n} - x\Vert + \theta_{0} l_{5} \Vert x_{n} - x\Vert +\theta _{1} l_{5} \bigl\Vert x'_{n} - x'\bigr\Vert \biggr) \\ &{}+ \int_{\lambda }^{\mu } G(t,s) (a_{1}(s) \Lambda_{1}\biggl( \Vert x_{n} - x\Vert , \bigl\Vert x'_{n} - x'\bigr\Vert , \frac{1 }{\Gamma (2-\beta)} \bigl\Vert x'_{n} - x'\bigr\Vert , \\ &{}m_{0} \Vert x_{n} - x\Vert , \theta_{0} l_{5} \Vert x_{n} - x\Vert +\theta_{1} l_{5} \bigl\Vert x'_{n} - x' \bigr\Vert \biggr) \\ &{}+\cdots+ a_{5}(s) \Lambda_{1}\biggl( \Vert x_{n} - x\Vert , \bigl\Vert x'_{n} - x'\bigr\Vert , \frac{1 }{ \Gamma (2-\beta)} \bigl\Vert x'_{n} - x'\bigr\Vert , \\ &{}m_{0} \Vert x_{n} - x\Vert , \theta_{0} l_{5} \Vert x_{n} - x\Vert +\theta_{1} l_{5} \bigl\Vert x'_{n} - x' \bigr\Vert \biggr)\,ds \\ &{}+ \int_{\mu }^{1} G(t,s) \biggl( l'_{1} \Vert x_{n} - x\Vert + l'_{2} \bigl\Vert x'_{n} - x'\bigr\Vert + \frac{l'_{3} }{\Gamma (2-\beta)} \bigl\Vert x'_{n} - x'\bigr\Vert \\ &{}+m_{0} l'_{4} \Vert x_{n} - x\Vert +\theta_{0} l'_{5} \Vert x_{n} - x\Vert +\theta _{1} l'_{5} \bigl\Vert x'_{n} - x'\bigr\Vert \biggr) \,ds \\ \leq& \biggl(l_{1} + l_{2}+ \frac{l_{3} }{\Gamma (2-\beta)}+ l_{4} m_{0} +l _{5} \theta_{0} + l_{5} \theta_{1}\biggr) \Vert x_{n} -x\Vert _{*} \int_{0}^{\lambda } G(t,s)\,ds \\ &{}+ \int_{\lambda }^{\mu } G(t,s) (\sum _{i=1}^{5} a_{i}(s) \Lambda_{i} \bigl( l \Vert x_{n} - x\Vert _{*}, l \Vert x_{n} - x\Vert _{*},l \Vert x_{n} - x\Vert _{*}, \\ &{}l \Vert x_{n} - x\Vert _{*}, l \Vert x_{n} - x\Vert _{*} \bigr)\,ds \\ &{}+\biggl(l'_{1} + l'_{2}+ \frac{l'_{3} }{\Gamma (2-\beta)}+ l'_{4} m_{0} +l'_{5} \theta_{0} + l'_{5} \theta_{1}\biggr) \Vert x_{n} -x\Vert _{*} \int_{\mu } ^{1} G(t,s)\,ds \end{aligned}$$

for all \(t \in [0,1]\), where \(l= \max \{1, \frac{1 }{\Gamma (2-\beta)}, m_{0}, \theta_{0}+\theta_{1} \}\). For each \(1\leq i \leq 5\) choose \(0<\delta_{i}(\epsilon)<\epsilon^{2} \) such that \(\frac{\Lambda_{i}(z,z,z,z,z)}{z}< q_{i}+\epsilon \) for all \(z\in (0, \delta_{i}(\epsilon)]\). Thus, \(\Lambda_{i}(z,z,z,z,z)<(q_{i}+\epsilon)z\) for all \(z\in (0,\delta_{i}(\epsilon)]\) and \(1\leq i \leq 5\). Put \(\delta:=\min_{1\leq i \leq 5}\delta_{i}(\epsilon)\). Then we have

$$\Lambda_{i}(\delta, \delta, \delta, \delta, \delta)< (q_{i}+ \epsilon)\delta < (q_{i}+\epsilon)\epsilon^{2}. $$

Let \(m_{1}\) be a natural number such that \(l\Vert x_{n} -x\Vert _{*}<\delta \) for all \(n\geq m_{1}\). This implies that \(\Lambda_{i}(l\Vert x_{n} -x\Vert _{*},\ldots , l\Vert x_{n} -x\Vert _{*})<\Lambda_{i}(\delta, \delta, \delta, \delta, \delta)<(q_{i}+\epsilon)\epsilon^{2}\) for all \(n\geq m_{1}\) and \(i=1,\dots,5\). Thus,

$$\begin{aligned} \bigl\vert F_{x_{n}}(t)-F_{x}(t)\bigr\vert \leq & L \Vert x_{n} -x\Vert _{*} \int_{0}^{\lambda } G(t,s)\,ds \\ &{}+ (q_{i}+\epsilon)\epsilon^{2} \int_{\lambda }^{\mu } G(t,s) \sum _{i=1}^{5} a_{i}(s)\,ds + L' \Vert x_{n} -x\Vert _{*} \int_{\mu }^{1} G(t,s)\,ds \end{aligned}$$

for all \(n\geq \max \{N, m_{1}\}\). This implies that

$$\begin{aligned}& \bigl\vert F_{x_{n}}(t)-F_{x}(t)\bigr\vert \\& \quad \leq \frac{L \epsilon t}{\Gamma (\alpha -1) } \int_{0}^{\lambda } (1-s)^{\alpha -2}\,ds \\& \qquad {}+ \frac{(q_{i}+\epsilon)\epsilon^{2} t}{\Gamma (\alpha -1) } \sum_{i=1}^{5} \int_{\lambda }^{\mu } (1-s)^{\alpha -2} a_{i}(s)\,ds + \frac{L' \epsilon t}{\Gamma (\alpha -1)} \int_{\mu }^{1} (1-s)^{ \alpha -2}\,ds \\& \quad = \frac{L \epsilon t (1-(1-\lambda)^{\alpha -1})}{\Gamma (\alpha)} + \frac{(q _{i}+\epsilon)\epsilon^{2} t}{\Gamma (\alpha -1) } \sum_{i=1}^{5} M _{i}(\lambda, \mu)+\frac{L' \epsilon t}{\Gamma (\alpha)} (1-\mu)^{ \alpha -1} \end{aligned}$$

for all \(n\geq \max \{N, m_{1}\}\) and \(t\in [0,1]\) and so

$$\begin{aligned} \Vert F_{x_{n}}-F_{x}\Vert \leq \Biggl[ \frac{L (1-(1-\lambda)^{\alpha -1})}{ \Gamma (\alpha)} + \frac{(q_{i}+\epsilon) \epsilon }{\Gamma (\alpha -1) } \sum_{i=1}^{5} M_{i}(\lambda, \mu) + \frac{L' }{\Gamma ( \alpha)} (1-\mu)^{\alpha -1}\Biggr] \epsilon < \epsilon. \end{aligned}$$

By using similar calculations, we get

$$\begin{aligned} \bigl\vert F'_{x_{n}}(t)-F'_{x}(t) \bigr\vert \leq& \int_{0}^{\lambda } \frac{\partial G}{ \partial t}(t,s)\biggl\vert f_{1}\biggl(s, x_{n}(s), x'_{n}(s), D^{\beta }x_{n}(s), \int_{0}^{s} h(\xi) x_{n}(\xi)\,d\xi, \phi \bigl(x_{n}(s)\bigr)\biggr) \\ &{}- f_{1}\biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\biggr\vert \,ds \\ &{}+ \int_{\lambda }^{\mu } \frac{\partial G}{\partial t}(t,s) \biggl\vert f_{2}\biggl(s, x_{n}(s), x'_{n}(s), D^{\beta }x_{n}(s), \int_{0}^{s} h(\xi) x_{n}( \xi)\,d\xi, \phi \bigl(x_{n}(s)\bigr)\biggr) \\ &{}- f_{2}\biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\biggr\vert \,ds \\ &{}+ \int_{\mu }^{1} \frac{\partial G}{\partial t}(t,s) \biggl\vert f_{3}\biggl(s, x_{n}(s), x'_{n}(s), D^{\beta }x_{n}(s), \int_{0}^{s} h(\xi) x_{n}(\xi)\,d\xi, \phi \bigl(x_{n}(s)\bigr)\biggr) \\ &{}-f_{3}\biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\biggr\vert \,ds \\ \leq& \int_{0}^{\lambda } \frac{\partial G}{\partial t}(t,s) \biggl( l_{1} \Vert x_{n} - x\Vert + l_{2} \bigl\Vert x'_{n} - x'\bigr\Vert + \frac{l_{3} }{\Gamma (2-\beta)} \bigl\Vert x'_{n} - x' \bigr\Vert \\ &{}+m_{0} l_{4} \Vert x_{n} - x\Vert + \theta_{0} l_{5} \Vert x_{n} - x\Vert +\theta _{1} l_{5} \bigl\Vert x'_{n} - x'\bigr\Vert \biggr)\,ds \\ &{}+ \int_{\lambda }^{\mu } \frac{\partial G}{\partial t}(t,s) (a_{1}(s) \Lambda_{1}\biggl( \Vert x_{n} - x \Vert , \bigl\Vert x'_{n} - x'\bigr\Vert , \frac{1 }{\Gamma (2- \beta)} \bigl\Vert x'_{n} - x'\bigr\Vert , \\ &{}m_{0} \Vert x_{n} - x\Vert , \theta_{0} l_{5} \Vert x_{n} - x\Vert +\theta_{1} l_{5} \bigl\Vert x'_{n} - x' \bigr\Vert \biggr) \\ &{}+\cdots+ a_{5}(s) \Lambda_{5}\biggl( \Vert x_{n} - x\Vert , \bigl\Vert x'_{n} - x'\bigr\Vert , \frac{1 }{ \Gamma (2-\beta)} \bigl\Vert x'_{n} - x'\bigr\Vert , \\ &{}m_{0} \Vert x_{n} - x\Vert , \theta_{0} l_{5} \Vert x_{n} - x\Vert +\theta_{1} l_{5} \bigl\Vert x'_{n} - x' \bigr\Vert \biggr)\,ds \\ &{}+ \int_{\mu }^{1} \frac{\partial G}{\partial t}(t,s) \biggl( l'_{1} \Vert x_{n}- x\Vert + l'_{2} \bigl\Vert x'_{n} - x'\bigr\Vert + \frac{l'_{3} }{\Gamma (2-\beta)} \bigl\Vert x'_{n} - x'\bigr\Vert \\ &{}+m_{0} l'_{4} \Vert x_{n} - x\Vert +\theta_{0} l'_{5} \Vert x_{n} - x\Vert +\theta _{1} l'_{5} \bigl\Vert x'_{n} - x'\bigr\Vert \biggr) \,ds \\ \leq& \Biggl[ \frac{L (1-(1-\lambda)^{\alpha -1})}{\Gamma (\alpha)} + \frac{(q _{i}+\epsilon) \epsilon }{\Gamma (\alpha -1) } \sum _{i=1}^{5} M _{i}(\lambda, \mu) + \frac{L' }{\Gamma (\alpha)} (1-\mu)^{\alpha -1}\Biggr] \epsilon \end{aligned}$$

for all \(n\geq \max \{N, m_{1}\}\) and \(t\in [0,1]\). Hence, \(\Vert F'_{x_{n}}-F'_{x}\Vert \leq \epsilon \) for sufficiently large n and so \(\Vert F_{x_{n}}-F_{x}\Vert _{*} = \max \{ \Vert F_{x_{n}}-F_{x}\Vert , \Vert F'_{x_{n}}-F'_{x}\Vert \} < \epsilon \) for sufficiently large n. This implies that \(F_{x_{n}} \to F_{x}\) in X. Now, we prove that F maps bounded sets into bounded sets of X. Let M be a bounded set of X. Choose \(r>0\) such that \(\Vert x\Vert _{*}< r\) for all \(x\in M\). Let \(x\in M\). Then

$$\begin{aligned} \bigl\vert F_{x}(t)\bigr\vert \leq & \biggl\vert \int_{0}^{\lambda } G(t,s) f_{1}\biggl(s, x(s), x'(s), D ^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\,ds \\ &{}+ \int_{\lambda }^{\mu } G(t,s) f_{2}\biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\,ds \\ &{}+ \int_{\mu }^{1} G(t,s) f_{3}\biggl(s, x(s), x'(s), D^{\beta }x(s), \int _{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\,ds\biggr\vert \\ \leq & \int_{0}^{\lambda } G(t,s) \biggl\vert f_{1} \biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr) \\ &{}-f _{1}(s,0,0,0,0,0)\biggr\vert \,ds \\ &{}+ \int_{0}^{\lambda } G(t,s) \bigl\vert f_{1}(s,0,0,0,0,0)\bigr\vert \,ds \\ &{}+ \int_{\lambda }^{\mu } G(t,s) \Phi (s) H\biggl(x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\,ds \\ &{}+ \int_{\mu }^{1} G(t,s) \biggl\vert f_{3} \biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr) \\ &{} -f_{3}(s,0,0,0,0,0)\biggr\vert \,ds \\ &{}+ \int_{\mu }^{1} G(t,s) \bigl\vert f_{3}(s,0,0,0,0,0) \bigr\vert \,ds \\ \leq & \int_{0}^{\lambda } G(t,s) \biggl(l_{1} \Vert x\Vert + l_{2} \bigl\Vert x'\bigr\Vert + l_{3} \bigl\Vert D^{\beta }x\bigr\Vert + l_{4} \Vert x\Vert \int_{0}^{s} \bigl\vert h(\xi)\bigr\vert \,d\xi +l_{5} \phi \bigl( \Vert x\Vert \bigr)\biggr)\,ds \\ &{}+ H\bigl(l \Vert x\Vert _{*},\ldots ,l \Vert x\Vert _{*}\bigr) \int_{\lambda }^{\mu } G(t,s) \Phi (s) \\ &{}+ \int_{\mu }^{1} G(t,s) \biggl(l'_{1} \Vert x\Vert + l'_{2} \bigl\Vert x' \bigr\Vert +l'_{3} \bigl\Vert D^{\beta }x\bigr\Vert \\ &{} + l'_{4} \Vert x\Vert \int_{0}^{s} \bigl\vert h(\xi)\bigr\vert \,d\xi + l'_{5} \phi \bigl( \Vert x\Vert \bigr)\biggr)\,ds \\ \leq & \frac{t}{\Gamma (\alpha -1)} \int_{0}^{\lambda } (1-s)^{ \alpha -2} \biggl(l_{1} \Vert x\Vert + l_{2} \bigl\Vert x'\bigr\Vert + \frac{l_{3}}{\Gamma (2-\beta)} \bigl\Vert x' \bigr\Vert \\ &{}+ l_{4} m_{0} \Vert x\Vert +l_{5} \theta_{0} \Vert x\Vert +l_{5} \theta_{1} \bigl\Vert x'\bigr\Vert \biggr)\,ds \\ &{}+ \frac{ H(l \Vert x\Vert _{*},\ldots ,l \Vert x\Vert _{*}) t}{\Gamma (\alpha -1)} \int_{\lambda }^{\mu } (1-s)^{\alpha -2} \Phi (s) \\ &{}+ \frac{t}{\Gamma (\alpha -1)} \int_{\mu }^{1} (1-s)^{\alpha -2} \biggl(l'_{1} \Vert x\Vert + l'_{2} \bigl\Vert x'\bigr\Vert , \frac{l'_{3}}{\Gamma (2-\beta)} \bigl\Vert x'\bigr\Vert + l'_{4} m_{0} \Vert x\Vert \\ &{}+l'_{5} \theta_{0} \Vert x\Vert +l'_{5} \theta_{1} \bigl\Vert x' \bigr\Vert \biggr)\,ds \\ \leq & \frac{t L}{\Gamma (\alpha -1)} \Vert x\Vert _{*} + \frac{H(l \Vert x\Vert _{*},\ldots ,l \Vert x\Vert _{*}) t}{\Gamma (\alpha -1)} \int_{\lambda }^{\mu } (1-s)^{ \alpha -2} \Phi (s)\,ds + \frac{t L}{\Gamma (\alpha -1)} \Vert x\Vert _{*} \end{aligned}$$

and so \(\Vert F_{x}\Vert \leq \frac{ L}{\Gamma (\alpha -1)} \Vert x\Vert _{*} + \frac{ H(l \Vert x\Vert _{*},\ldots ,l \Vert x\Vert _{*}) }{\Gamma (\alpha -1)} \int_{\lambda } ^{\mu } (1-s)^{\alpha -2} \Phi (s)\,ds + \frac{ L}{\Gamma (\alpha -1)} \Vert x\Vert _{*}\). By using similar calculations, we get \(\Vert F'_{x}\Vert \leq \frac{ L}{\Gamma (\alpha -1)} \Vert x\Vert _{*} + \frac{ H(l \Vert x\Vert _{*},\ldots ,l \Vert x\Vert _{*}) }{\Gamma (\alpha -1)} \int_{\lambda }^{\mu } (1-s)^{ \alpha -2} \Phi (s)\,ds + \frac{ L}{\Gamma (\alpha -1)} \Vert x\Vert _{*}\). This implies that

$$\begin{aligned} \Vert F_{x}\Vert _{*} =& \max \bigl\{ \Vert F_{x}\Vert , \bigl\Vert F'_{x}\bigr\Vert \bigr\} \\ \leq & \frac{ L}{\Gamma (\alpha -1)} \Vert x\Vert _{*} + \frac{ H(l \Vert x\Vert _{*},\ldots ,l \Vert x\Vert _{*}) }{\Gamma (\alpha -1)} \int_{\lambda }^{\mu } (1-s)^{ \alpha -2} \Phi (s)\,ds + \frac{ L}{\Gamma (\alpha -1)} \Vert x\Vert _{*} \\ < & \infty. \end{aligned}$$

This proves the claim. Since G and \(G'\) are continuous with respect to t, it is easy to check that \(F_{x}(t_{2}) \to F_{x}(t_{1})\) as \(t_{2} \to t_{1}\). By using the Arzela–Ascoli theorem, we get \(\overline{T(M)}\) is relatively compact and so \(F:P \to P\) is completely continuous. Since \(\lim_{z \to 0^{+}} \frac{H(z,z,z,z,z)}{z}=0\), one concludes that \(\lim_{\Vert x\Vert _{*} \to 0^{+}} \frac{H(l \Vert x\Vert _{*},\ldots ,l \Vert x\Vert _{*})}{l \Vert x\Vert _{*}}=0\). Let \(\epsilon >0\) be given. Choose \(\delta =\delta (\epsilon)>0\) such that \(\Vert x\Vert _{*}<\delta \) implies \(\frac{H(l \Vert x\Vert _{*},\ldots ,l \Vert x\Vert _{*})}{l \Vert x\Vert _{*}}< \epsilon \) and so \(H(l \Vert x\Vert _{*},\ldots ,l \Vert x\Vert _{*})< \epsilon l \Vert x\Vert _{*}\). Since \(\frac{L (1-(1-\lambda)^{\alpha -1})+L' (1-\mu)^{\alpha -1}}{ \Gamma (\alpha)} <1\), there exists \(\epsilon_{0}>0\) such that

$$\frac{L (1-(1-\lambda)^{\alpha -1})+L' (1-\mu)^{\alpha -1}}{\Gamma (\alpha)}+ \frac{\epsilon_{0} l \Vert \Phi \Vert ^{*}}{\Gamma (\alpha -1)} < 1, $$

where \(\Vert \Phi \Vert ^{*} = \int_{\lambda }^{\mu } (1-s)^{\alpha -2} \Phi (s)\,ds\). Let \(\delta_{0}= \delta (\epsilon_{0})\). Define \(\Omega_{1} = \{x \in X \text{ s.t. } \Vert x\Vert _{*}< \delta \}\). Then

$$\begin{aligned} \bigl\vert F_{x}(t)\bigr\vert \leq & \int_{0}^{\lambda } G(t,s) \biggl\vert f_{1} \biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\biggr\vert \,ds \\ &{}+ \int_{\lambda }^{\mu } G(t,s) \biggl\vert f_{2} \biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\biggr\vert \,ds \\ &{}+ \int_{\mu }^{1} G(t,s) \biggl\vert f_{3} \biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\biggr\vert \,ds \\ \leq & \int_{0}^{\lambda } G(t,s) \biggl(l_{1} \Vert x\Vert + l_{2} \bigl\Vert x'\bigr\Vert + \frac{l _{3}}{\Gamma (2-\beta)} \bigl\Vert x'\bigr\Vert + l_{4} m_{0} \Vert x\Vert \\ &{}+l_{5} \theta_{0} \Vert x\Vert +l_{5} \theta_{1} \bigl\Vert x'\bigr\Vert \biggr)\,ds \\ &{}+ \int_{\lambda }^{\mu } G(t,s) \Phi (s) H\biggl(x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\,ds \\ &{}+ \int_{\mu }^{1} G(t,s) \biggl(l'_{1} \Vert x\Vert + l'_{2} \bigl\Vert x' \bigr\Vert +\frac{l'_{3}}{ \Gamma (2-\beta)} \bigl\Vert x'\bigr\Vert + l'_{4} m_{0} \Vert x\Vert \\ &{} +l'_{5} \theta_{0} \Vert x\Vert +l'_{5} \theta_{1} \bigl\Vert x' \bigr\Vert \biggr)\,ds \\ \leq & \frac{t L}{\Gamma (\alpha - 1)} \Vert x\Vert _{*} \int_{0}^{\lambda } (1-s)^{\alpha -2}\,ds \\ &{}+ \frac{t H(l \Vert x\Vert _{*}, l \Vert x\Vert _{*}, l \Vert x\Vert _{*}, l \Vert x\Vert _{*}, l \Vert x\Vert _{*})}{\Gamma (\alpha -1)} \int_{\lambda }^{\mu } (1-s)^{\alpha -2} \Phi (s)\,ds \\ &{}+ \int_{\mu }^{1} \frac{t L'}{\Gamma (\alpha - 1)} \Vert x\Vert _{*} \int _{0}^{\lambda } (1-s)^{\alpha -2}\,ds \end{aligned}$$

for all \(x \in \Omega_{1}\) and \(t \in [0,1]\). Hence,

$$\begin{aligned} \Vert F_{x}\Vert \leq & \biggl[ \frac{L (1-(1-\lambda)^{\alpha -1})+L' (1- \mu)^{\alpha -1}}{\Gamma (\alpha)}+ \frac{\epsilon_{0} l \Vert \Phi \Vert ^{*}}{\Gamma (\alpha -1)}\biggr] \Vert x\Vert _{*}\leq \Vert x\Vert _{*}. \end{aligned}$$

Similarly, we get \(\Vert F'_{x}\Vert \leq \Vert x\Vert _{*} \) and so \(\Vert F_{x}\Vert _{*} \leq \Vert x\Vert _{*}\). Since \(\lim_{\max \Vert x_{i}\Vert \to \infty } \frac{f _{2}(t, x_{1} x_{2}, \ldots, x_{5})}{\max \Vert x_{i}\Vert } =q(t)\), there exists \(R= R(\epsilon)>0\) such that \(\max \Vert x_{i}\Vert >R(\epsilon)\) implies that \(\frac{f_{2}(t, x_{1} x_{2}, \ldots, x_{5})}{\max \Vert x_{i}\Vert } > q(t)- \epsilon \) and so \(f_{2}(t, x_{1} x_{2}, \ldots, x_{5})> (\max \Vert x_{i}\Vert ) (q(t) - \epsilon)\). Recall that

$$\frac{\alpha -2}{\Gamma (\alpha)} \int_{\lambda }^{\mu } (\mu -s)^{ \alpha -1}\,ds - \frac{\epsilon_{1} (\alpha -2) (\mu -\lambda)^{ \alpha }}{\Gamma (\alpha +1) } >1. $$

Choose \(R_{1} = R(\epsilon_{1})>0\). Put \(\Omega_{2} = \{ x \in X : \Vert x\Vert _{*}< R_{1} \}\). Then

$$\begin{aligned} \Vert F_{x}\Vert = & \sup_{t \in [0,1]} \bigl\vert F_{x}(t)\bigr\vert \\ \geq& \bigl\vert F_{x}(\mu)\bigr\vert \\ \geq& \int_{\lambda }^{\mu } G(t,s) f_{2}\biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\,ds \\ \geq & \int_{\lambda }^{\mu } \frac{ (\mu -s)^{\alpha -1}(\alpha -2)}{ \Gamma (\alpha)} \bigl(q(s) - \epsilon_{1}\bigr) \max \bigl\{ \Vert x\Vert , \bigl\Vert x'\bigr\Vert ,\ldots , \bigl\Vert \phi (x(s) \bigr\Vert \bigr\} \,ds \\ \geq & \Vert x\Vert _{*} \int_{\lambda }^{\mu } \frac{ (\mu -s)^{\alpha -1}( \alpha -2)}{\Gamma (\alpha)} \bigl(q(s) - \epsilon_{1}\bigr)\,ds \\ = & \Vert x\Vert _{*} \biggl[ \int_{\lambda }^{\mu } \frac{ (\mu -s)^{\alpha -1}( \alpha -2)}{\Gamma (\alpha)} q(s)\,ds - \epsilon_{1} \int_{\lambda } ^{\mu } \frac{ (\mu -s)^{\alpha -1}(\alpha -2)}{\Gamma (\alpha)}\,ds \biggr] \\ = & \Vert x\Vert _{*} \biggl[ \frac{ \alpha -2}{\Gamma (\alpha)} \int_{\lambda } ^{\mu } (\mu -s)^{\alpha -1} q(s)\,ds - \frac{ \epsilon_{1} (\alpha -2) (\mu -\lambda)^{\alpha }}{\Gamma (\alpha +1)} \biggr] > \Vert x\Vert _{*} \end{aligned}$$

for all \(x \in P \cap \partial \Omega_{2}\). Hence, \(\Vert F_{x}\Vert _{*} \geq \Vert x\Vert _{*}\) on \(P \cap \partial \Omega_{2}\). Now by using Lemma 2.4, \(F: X \to X\) has a fixed point on \(P\cap (\Omega_{2} \backslash \Omega_{1})\) which is a solution for the problem (1). □

Example 3.1

Define the map d on \([0.1,0.9]\) by \(d(t)=\frac{1}{c(t)}\) whenever \(t\in [0.1,0.9]\cap \mathbb{Q}\) where \(c(t)=0\) on \([0.1,0.9]\cap \mathbb{Q}\) and \(d(t)=10\) whenever \(t\in [0.1,0.9]\cap \mathbb{Q}^{c}\). Now, consider the point-wise defined fractional integro-differential equation \(D^{\frac{7}{2}} x(t) +f(t, x(t), x'(t), D^{\frac{1}{2}} x(t), \int_{0}^{t} x(s)\,ds, D^{\frac{1}{3}} x(t))=0\), where

$$f(t, x_{1}, x_{2}, x_{3}, x_{4}, x_{5})=\textstyle\begin{cases} t \sum_{i=1}^{5} x_{i} & 0 \leq t < 0.1, \\ d(t) H(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) & 0.1 \leq t \leq 0.9, \\ (1-t) \sum_{i=1}^{5} x_{i} & 0.9 < t \leq 1, \end{cases} $$

and \(H(x_{1}, x_{2}, x_{3}, x_{4}, x_{5})= \sum_{i=1}^{5} \frac{ \Vert x_{i} \Vert ^{2}}{ 1+ \Vert x_{i} \Vert }\). Put \(f_{1}(t, x_{1}, x_{2}, x_{3}, x _{4}, x_{5})= t \sum_{i=1}^{5} x_{i}\),

$$f_{2}(t, x_{1}, x_{2}, x_{3}, x_{4}, x_{5})= d(t) H(x_{1}, x_{2}, x _{3}, x_{4}, x_{5}), $$

and \(f_{3}(t, x_{1}, x_{2}, x_{3}, x_{4}, x_{5})= (1-t) \sum_{i=1} ^{5} x_{i}\). Note that

$$\begin{aligned}& f_{1}(t, 0, 0, 0, 0, 0)= f_{3}(t, 0, 0, 0, 0, 0) =0, \\& f_{1}(t, x_{1}, x_{2}, x_{3}, x_{4}, x_{5})- f_{1}(t, y_{1}, y_{2}, y _{3}, y_{4}, y_{5}) \leq t \sum_{i=1}^{5} \Vert x_{i} - y_{i} \Vert \leq 0.1 \sum_{i=1}^{5} \Vert x_{i} - y_{i} \Vert , \\& \bigl\vert f_{2}(t, x_{1}, x_{2}, x_{3}, x_{4}, x_{5})- f_{2}(t, y_{1}, y_{2}, y_{3}, y_{4}, y_{5})\bigr\vert \\& \quad = d(t) \sum_{i=1}^{5} \biggl\vert \frac{ \Vert x_{i}\Vert ^{2}}{1+ \Vert x_{i} \Vert } - \frac{ \Vert y_{i} \Vert ^{2}}{1+ \Vert y_{i} \Vert }\biggr\vert \\& \quad = d(t) \sum_{i=1}^{5} \biggl\vert \frac{ \Vert x_{i} \Vert ^{2} - \Vert x_{i} \Vert ^{2} \Vert y_{i} \Vert - \Vert y_{i} \Vert ^{2} - \Vert y_{i} \Vert ^{2} \Vert x_{i}\Vert }{(1+ \Vert x_{i}\Vert ) (1+ \Vert y_{i} \Vert )}\biggr\vert \\& \quad = d(t) \sum_{i=1}^{5} \biggl\vert \frac{ \Vert x_{i} \Vert ^{2} - \Vert y_{i} \Vert ^{2} + \Vert x_{i} \Vert ( \Vert x_{i} \Vert \Vert y_{i} \Vert - \Vert y_{i} \Vert ^{2}) }{(1+ \Vert x_{i}\Vert ) (1+ \Vert y_{i} \Vert )}\biggr\vert \\& \quad = d(t) \sum_{i=1}^{5} \biggl\vert \frac{ (\Vert x_{i} \Vert - \Vert y_{i} \Vert ) (\Vert x_{i} \Vert + \Vert y_{i} \Vert ) + \Vert x_{i} \Vert ( \Vert x_{i} \Vert - \Vert y_{i} \Vert ) \Vert y_{i} \Vert }{(1+ \Vert x_{i}\Vert ) (1+ \Vert y_{i} \Vert )}\biggr\vert \\& \quad = d(t) \sum_{i=1}^{5} \biggl\vert \frac{ (\Vert x_{i} \Vert - \Vert y_{i} \Vert ) (\Vert x_{i} \Vert + \Vert y_{i} \Vert + \Vert x_{i} \Vert \Vert y_{i} \Vert )}{\Vert x_{i}\Vert + \Vert y_{i} \Vert + \Vert x_{i}\Vert \Vert y_{i} \Vert +1 }\biggr\vert \\& \quad \leq d(t) \sum_{i=1}^{5} \biggl\vert \frac{ (\Vert x_{i} \Vert - \Vert y_{i} \Vert ) ( \Vert x_{i} \Vert + \Vert y_{i} \Vert + \Vert x_{i} \Vert \Vert y_{i} \Vert )}{\Vert x_{i}\Vert + \Vert y_{i} \Vert + \Vert x_{i}\Vert \Vert y_{i} \Vert }\biggr\vert \\& \quad = d(t) \sum_{i=1}^{5} \bigl\vert \Vert x_{i} \Vert - \Vert y_{i} \Vert \bigr\vert \leq d(t) \sum_{i=1}^{5} \Vert x_{i} - y_{i} \Vert \\& \quad := d(t) \sum_{i=1}^{5} \Lambda_{i} (x_{1} - y_{1},\ldots, x_{5} - y _{5}) \end{aligned}$$

and \(f_{3}(t, x_{1}, x_{2}, x_{3}, x_{4}, x_{5})- f_{3}(t, y_{1}, y _{2}, y_{3}, y_{4}, y_{5}) \leq 0.1\sum_{i=1}^{5} \Vert x_{i} - y_{i}\Vert \), where \(\Lambda_{i} (x_{1} ,\ldots, x_{5} ) = \Vert x_{i}\Vert \) for \(i=1,\dots,5\). Note that

$$\begin{aligned}& L=\biggl(l_{1}+l_{2}+ \frac{l_{3} }{\Gamma (2-\beta)}+ m_{0} l_{4}+ \theta _{0} l_{5}+\theta_{1} l_{5}\biggr) \\& \hphantom{L} = (0.1 + 0.1+ \frac{0.1}{\Gamma (2- \frac{1}{2})} + 0.1+ \frac{0.1}{\Gamma (2-\frac{1}{3})}< 0.4, \\& L'=\biggl(l'_{1}+l'_{2}+ \frac{l'_{3} }{\Gamma (2-\beta)}+ m_{0} l'_{4}+ \theta_{0} l'_{5}+\theta_{1} l'_{5}\biggr) \\& \hphantom{L}=(0.1 + 0.1+ \frac{0.1}{\Gamma (2- \frac{1}{2})} + 0.1+ \frac{0.1}{\Gamma (2-\frac{1}{3})}< 0.4, \end{aligned}$$

and \(\lim_{z \to 0^{+}} \frac{\Lambda_{i} (z,z,z,z,z)}{z} = 1:= q_{i}\) for \(i=1,\dots,5\). Then we have

$$\frac{L (1-(1-\lambda)^{\alpha -1})}{\Gamma (\alpha)} + \frac{L' }{ \Gamma (\alpha)} (1-\mu)^{\alpha -1} < \frac{0.4 (1-(1-0.1)^{ \frac{5}{2}})}{\Gamma (\frac{7}{2})} +\frac{0.4(1-0.9)^{\frac{5}{2}}}{ \Gamma (\frac{7}{2})} < 1 $$

and for almost all \(t \in [0,1]\)

$$\begin{aligned} q(t) :=& \lim_{\max \Vert x_{i}\Vert \to \infty } \frac{f_{2}(t, x_{1} x_{2}, \ldots, x_{5})}{\max \Vert x_{i}\Vert } = d(t) \lim _{\max \Vert x_{i}\Vert \to \infty } \frac{ \sum_{i=1}^{5} \frac{ \Vert x_{i} \Vert ^{2}}{1+ \Vert x_{i} \Vert }}{ \max \Vert x_{i}\Vert } \\ \geq& d(t) \lim_{ \Vert x_{r}\Vert \to \infty } \frac{ \Vert x_{i} \Vert ^{2}}{\Vert x_{r}\Vert (1+ \Vert x_{r} \Vert )} =d(t) \lim _{ \Vert x_{r}\Vert \to \infty } \frac{ \Vert x_{r} \Vert }{1+ \Vert x_{r} \Vert } = d(t), \end{aligned}$$

where \(\Vert x_{r}\Vert = \max_{1 \leq i \leq 5} \Vert x_{i}\Vert \). Thus, we obtain

$$\frac{\alpha -2}{\Gamma (\alpha)} \int_{\lambda }^{\mu } (\mu -s)^{ \alpha -2} q(s)\,ds \geq \frac{\frac{3}{2}}{\Gamma (\frac{7}{2})} \int_{0.1}^{0.9} 10 (0.9-s)^{\frac{3}{2}}\,ds > 1. $$

Now, by using Theorem 3.2, the problem has a solution.

Theorem 3.3

Let \([\lambda, \mu, f=(f_{1},f_{2},,f_{3})]\) with \(f_{1}(s,0,0,0,0,0)=f _{3}(t,0,0,0,0,0)=0\) for all \(s\in [0,\lambda ]\) and \(t\in [\mu, 1]\). Assume that there exist nonnegative functions \(a \in L^{1} [0,\lambda ]\), \(c \in L^{1} [\mu, 1]\) and \(b_{1},\dots, b_{5}: [\lambda, \mu ] \to \mathbb{R}\) with \(\hat{b_{i}}:= (1-t)^{\alpha -2} b_{i}(t) \in L^{1}[\lambda, \mu ]\) (\(i=1,\dots,5\)) such that \(\vert f_{1}(t, x_{1}, \ldots, x_{5}) - f_{1}(t, y_{1}, \ldots, y_{5})\vert \leq a(t) \sum_{i=1} ^{5} \Vert x_{i} - y_{i} \Vert \),

$$\bigl\vert f_{2}(t, x_{1}, \ldots, x_{5}) - f_{2}(t, y_{1}, \ldots, y_{5})\bigr\vert \leq \sum_{i=1}^{5} b_{i}(t) \Vert x_{i} - y_{i} \Vert , $$

and \(\vert f_{3}(t, x_{1}, \ldots, x_{5}) - f_{3}(t, y_{1}, \ldots, y_{5})\vert \leq c(t) \sum_{i=1}^{5} \Vert x_{i} - y_{i} \Vert \) for all \(x_{1},\dots, x_{5}, y_{1},\dots, y_{5} \in X\) and almost all \(t \in [0,1]\). Suppose that there exist a natural number \(n_{0}\) and nonnegative functions \(\phi_{1},\dots, \phi_{n_{0}}\) with \(\hat{\phi _{i}}:=(1-t)^{ \alpha -2} \phi_{i}(t) \in L^{1}[\lambda, \mu ]\) and nonnegative and nondecreasing with respect to all components maps \(\Lambda_{1},\dots, \Lambda_{n_{0}}: X^{5} \to [0, \infty)\) with \(\lim_{z \to 0^{+}} \frac{ \Lambda_{i}(z,z,z,z,z)}{z} =0\) such that \(\vert f_{2}(t, x_{1},\dots, x_{5})\vert \leq \sum_{i=1}^{n_{0}} \phi_{i} \Lambda_{i} (x_{1},\dots, x _{5})\) for all \((x_{1},\dots, x_{5}) \in X\) and almost all \(t \in [\lambda, \mu ]\). If \((2+ \frac{1}{\Gamma (2- \beta)}+ m _{0} + \theta_{0} + \theta_{1})( \Vert a\Vert _{[0, \lambda ]} + \sum_{i=1} ^{5} \Vert \hat{b_{i}}\Vert + (1-\mu)^{\alpha - 2} \Vert c\Vert _{[1, \mu ]} ) < \Gamma (\alpha -1)\), then the problem (1) has a solution.

Proof

First we show that F is a continuous map on X. Let \(x_{1}, x_{2} \in X\) and \(t \in [0,1]\). Then

$$\begin{aligned} \bigl\vert F_{x_{1}}(t)-F_{x_{2}}(t)\bigr\vert \leq& \int_{0}^{\lambda } G(t,s)\biggl\vert f_{1} \biggl(s, x _{1}(s), x'_{1}(s), D^{\beta }x_{1}(s), \int_{0}^{s} h(\xi) x_{1}( \xi)\,d\xi, \phi \bigl(x_{1}(s)\bigr)\biggr) \\ &{}- f_{1}\biggl(s, x_{2}(s), x'_{2}(s), D^{\beta }x_{2}(s), \int_{0}^{s} h( \xi) x_{2}(\xi)\,d\xi, \phi \bigl(x_{2}(s)\bigr)\biggr)\biggr\vert \,ds \\ &{}+ \int_{\lambda }^{\mu } G(t,s) \biggl\vert f_{2} \biggl(s, x_{1}(s), x'_{1}(s), D^{ \beta }x_{1}(s), \int_{0}^{s} h(\xi) x_{1}(\xi)\,d\xi, \phi \bigl(x_{1}(s)\bigr)\biggr)\,ds \\ &{}-f_{2}\biggl(s, x_{2}(s), x'_{2}(s), D^{\beta }x_{2}(s), \int_{0}^{s} h( \xi) x_{2}(\xi)\,d\xi, \phi \bigl(x_{2}(s)\bigr)\biggr)\biggr\vert \,ds \\ &{}+ \int_{\mu }^{1} G(t,s) \biggl\vert f_{3} \biggl(s, x_{1}(s), x'_{1}(s), D^{\beta }x _{1}(s), \int_{0}^{s} h(\xi) x_{1}(\xi)\,d\xi, \phi \bigl(x_{1}(s)\bigr)\biggr)\,ds \\ &{}-f_{3}\biggl(s, x_{2}(s), x'_{2}(s), D^{\beta }x_{2}(s), \int_{0}^{s} h( \xi) x_{2}(\xi)\,d\xi, \phi \bigl(x_{2}(s)\bigr)\biggr)\biggr\vert \,ds \\ \leq& \int_{0}^{\lambda } \frac{t(1-s)^{\alpha -2}}{\Gamma (\alpha -1)} a(s) \biggl( \bigl\vert x_{1}(s) - x _{2}(s)\bigr\vert + \bigl\vert x'_{1}(s) - x_{2}'(s)\bigr\vert + \bigl\vert D^{\beta } (x_{1} - x_{2}) (s)\bigr\vert \\ &{}+ \int_{0}^{s} \bigl\vert x_{1}(\xi) - x_{2}(\xi)\bigr\vert \,d\xi + \bigl\vert \phi \bigl(x_{1}(s) - x_{2}(s)\bigr)\bigr\vert \biggr)\,ds \\ &{}+ \int_{\lambda }^{\mu } \frac{t(1-s)^{\alpha -2}}{\Gamma (\alpha -1)} \biggl(b_{1}(s) \bigl\vert x_{1}(s) - x_{2}(s) \bigr\vert +b_{2}(s) \bigl\vert x'_{1}(s) - x_{2}'(s)\bigr\vert \\ &{}+b_{3}(s)\bigl\vert D^{\beta } (x_{1} - x_{2}) (s)\bigr\vert \\ &{} + b_{4}(s) \int_{0}^{s} \bigl\vert x_{1}(\xi) - x_{2}(\xi)\bigr\vert \,d\xi +b_{5}(s) \bigl\vert \phi \bigl(x_{1}(s) - x_{2}(s)\bigr)\bigr\vert \biggr)\,ds \\ &{}+ \int_{\mu }^{1} \frac{t(1-s)^{\alpha -2}}{\Gamma (\alpha -1)} c(s) \biggl( \bigl\vert x_{1}(s) - x_{2}(s)\bigr\vert \\ &{}+ \bigl\vert x'_{1}(s) - x_{2}'(s) \bigr\vert + \bigl\vert D^{\beta } (x_{1} -x_{2}) (s)\bigr\vert \\ &{}+ \int_{0}^{s} \bigl\vert x_{1}(\xi) - x_{2}(\xi)\bigr\vert \,d\xi + \bigl\vert \phi \bigl(x_{1}(s) - x_{2}(s)\bigr)\bigr\vert \biggr)\,ds \\ \leq& \frac{t}{\Gamma (\alpha -1)} \int_{0}^{\lambda } (1-s)^{\alpha -2} a(s) \biggl( \Vert x_{1} - x_{2}\Vert + \bigl\Vert x'_{1} - x_{2}'\bigr\Vert + \frac{\Vert x'_{1} - x'_{2}\Vert }{\Gamma (2 - \beta)} \\ &{}+ m_{0} \Vert x_{1} - x_{2}\Vert + \theta_{0} \Vert x_{1} - x_{2}\Vert + \theta_{1} \bigl\Vert x'_{1} - x'_{2}\bigr\Vert \biggr)\,ds \\ &{}+\frac{t}{\Gamma (\alpha -1)} \int_{\lambda }^{\mu } (1-s)^{\alpha -2} \biggl(b_{1}(s) \Vert x_{1} - x_{2}\Vert +b_{2}(s) \bigl\Vert x'_{1} - x_{2}'\bigr\Vert \\ &{}+b_{3}(s) \frac{\Vert x'_{1} - x'_{2}\Vert }{\Gamma (2 - \beta)}+ b_{4}(s) m _{0} \Vert x_{1} - x_{2}\Vert \\ &{} +b_{5}(s) \bigl(\theta_{0} \Vert x_{1} - x_{2}\Vert + \theta_{1} \bigl\Vert x'_{1} - x'_{2}\bigr\Vert \bigr)\biggr)\,ds \\ &{}+\frac{t}{\Gamma (\alpha -1)} \int_{\mu }^{1} (1-s)^{\alpha -2} c(s) \biggl( \Vert x_{1} - x_{2}\Vert + \bigl\Vert x'_{1} - x_{2}'\bigr\Vert + \frac{\Vert x'_{1} - x'_{2}\Vert }{ \Gamma (2 - \beta)} \\ &{}+ m_{0} \Vert x_{1} - x_{2}\Vert + \theta_{0} \Vert x_{1} - x_{2}\Vert + \theta_{1} \bigl\Vert x'_{1} - x'_{2}\bigr\Vert \biggr)\,ds \\ \leq& \frac{t(2+ \frac{1}{\Gamma (2- \beta)} +m_{0} + \theta_{0}+ \theta_{1})}{\Gamma (\alpha -1)} \Vert x_{1} - x_{2}\Vert _{*} \int_{0}^{ \lambda } (1-s)^{\alpha -2} a(s)\,ds \\ &{}+ \frac{t(2+ \frac{1}{\Gamma (2- \beta)} +m_{0} + \theta_{0}+ \theta _{1})}{\Gamma (\alpha -1)} \Vert x_{1} - x_{2}\Vert _{*} \sum_{i=1}^{5} \int_{\lambda }^{\mu } (1-s)^{\alpha -2} b_{i}(s)\,ds \\ &{}+ \frac{t(2+ \frac{1}{\Gamma (2- \beta)} +m_{0} + \theta_{0}+ \theta _{1})}{\Gamma (\alpha -1)} \Vert x_{1} - x_{2}\Vert _{*} \int_{\mu }^{1} (1-s)^{ \alpha -2} c(s)\,ds \\ \leq& \frac{t(2+ \frac{1}{\Gamma (2- \beta)} +m_{0} + \theta_{0}+ \theta_{1})}{\Gamma (\alpha -1)} \Vert x_{1} - x_{2}\Vert _{*} \Biggl[ \int_{0}^{ \lambda } a(s)\,ds \\ &{}+ \sum_{i=1}^{5} \int_{\lambda }^{\mu } (1- s)^{\alpha -2} b_{i}(s)\,ds + \int_{\mu }^{1} c(s)\,ds \Biggr] \end{aligned}$$

and so

$$\begin{aligned} \Vert F_{x_{1}}-F_{x_{2}}\Vert \leq & \frac{(2+ \frac{1}{\Gamma (2- \beta)} +m_{0} + \theta_{0}+ \theta_{1})}{\Gamma (\alpha -1)} \Biggl[ \Vert a\Vert _{[0,\lambda ]} \\ &{}+ \sum_{i=1}^{5} \Vert \hat{ b_{i}}\Vert _{[\lambda, \mu ]} + \Vert c\Vert _{[ \mu, 1]} \Biggr] \Vert x_{1} - x_{2}\Vert _{*}. \end{aligned}$$

By using similar calculations, we get

$$\begin{aligned}& \bigl\vert F'_{x_{1}}(t)-F'_{x_{2}}(t) \bigr\vert \\& \quad \leq \int_{0}^{\lambda } \frac{\partial G}{\partial t}(t,s) \biggl\vert f_{1}\biggl(s, x_{1}(s), x'_{1}(s), D^{\beta }x_{1}(s), \int_{0}^{s} h(\xi) x_{1}(\xi)\,d\xi, \phi \bigl(x_{1}(s)\bigr)\biggr) \\& \qquad {}- f_{1}\biggl(s, x_{2}(s), x'_{2}(s), D^{\beta }x_{2}(s), \int_{0}^{s} h( \xi) x_{2}(\xi)\,d\xi, \phi \bigl(x_{2}(s)\bigr)\biggr)\biggr\vert \,ds \\& \qquad {}+ \int_{\lambda }^{\mu } \frac{\partial G}{\partial t}(t,s) \biggl\vert f_{2}\biggl(s, x _{1}(s), x'_{1}(s), D^{\beta }x_{1}(s), \int_{0}^{s} h(\xi) x_{1}( \xi)\,d\xi, \phi \bigl(x_{1}(s)\bigr)\biggr)\,ds \\& \qquad {}-f_{2}\biggl(s, x_{2}(s), x'_{2}(s), D^{\beta }x_{2}(s), \int_{0}^{s} h( \xi) x_{2}(\xi)\,d\xi, \phi \bigl(x_{2}(s)\bigr)\biggr)\biggr\vert \,ds \\& \qquad {}+ \int_{\mu }^{1} \frac{\partial G}{\partial t}(t,s) \biggl\vert f_{3}\biggl(s, x_{1}(s), x'_{1}(s), D^{\beta }x_{1}(s), \int_{0}^{s} h(\xi) x_{1}(\xi)\,d\xi, \phi \bigl(x_{1}(s)\bigr)\biggr)\,ds \\& \qquad {}- f_{3}\biggl(s, x_{2}(s), x'_{2}(s), D^{\beta }x_{2}(s), \int_{0}^{s} h( \xi) x_{2}(\xi)\,d\xi, \phi \bigl(x_{2}(s)\bigr)\biggr)\biggr\vert \,ds \\& \quad \leq \int_{0}^{\lambda } \frac{t(1-s)^{\alpha -2}}{\Gamma (\alpha -1)} a(s) \biggl( \bigl\vert x_{1}(s) - x_{2}(s)\bigr\vert + \bigl\vert x'_{1}(s) - x_{2}'(s)\bigr\vert + \bigl\vert D^{\beta } (x_{1} - x_{2}) (s)\bigr\vert \\& \qquad {}+ \int_{0}^{s} \bigl\vert x_{1}(\xi) - x_{2}(\xi)\bigr\vert \,d\xi + \bigl\vert \phi \bigl(x_{1}(s) - x_{2}(s)\bigr)\bigr\vert \biggr)\,ds \\& \qquad {}+ \int_{\lambda }^{\mu } \frac{t(1-s)^{\alpha -2}}{\Gamma (\alpha -1)} \biggl(b_{1}(s) \bigl\vert x_{1}(s) - x_{2}(s) \bigr\vert +b_{2}(s) \bigl\vert x'_{1}(s) - x_{2}'(s)\bigr\vert \\& \qquad {}+b_{3}(s)\bigl\vert D^{\beta } (x_{1} - x_{2}) (s)\bigr\vert \\& \qquad {} + b_{4}(s) \int_{0}^{s} \bigl\vert x_{1}(\xi) - x_{2}(\xi)\bigr\vert \,d\xi +b_{5}(s) \bigl\vert \phi \bigl(x_{1}(s) - x_{2}(s)\bigr)\bigr\vert \biggr)\,ds \\& \qquad {}+ \int_{\mu }^{1} \frac{t(1-s)^{\alpha -2}}{\Gamma (\alpha -1)} c(s) \biggl( \bigl\vert x_{1}(s) - x_{2}(s)\bigr\vert + \bigl\vert x'_{1}(s) - x_{2}'(s)\bigr\vert + \bigl\vert D^{\beta } (x_{1} -x_{2}) (s)\bigr\vert \\& \qquad {}+ \int_{0}^{s} \bigl\vert x_{1}(\xi) - x_{2}(\xi)\bigr\vert \,d\xi + \bigl\vert \phi \bigl(x_{1}(s) - x_{2}(s)\bigr)\bigr\vert \biggr)\,ds \\& \quad \leq \frac{t(2+ \frac{1}{\Gamma (2- \beta)} +m_{0} + \theta_{0}+ \theta_{1})}{\Gamma (\alpha -1)} \Vert x_{1} - x_{2}\Vert _{*} \int_{0}^{ \lambda } (1-s)^{\alpha -2} a(s)\,ds \\& \qquad {}+ \frac{t(2+ \frac{1}{\Gamma (2- \beta)} +m_{0} + \theta_{0}+ \theta _{1})}{\Gamma (\alpha -1)} \Vert x_{1} - x_{2}\Vert _{*} \sum_{i=1}^{5} \int_{\lambda }^{\mu } (1-s)^{\alpha -2} b_{i}(s)\,ds \\& \qquad {}+ \frac{t(2+ \frac{1}{\Gamma (2- \beta)} +m_{0} + \theta_{0}+ \theta _{1})}{\Gamma (\alpha -1)} \Vert x_{1} - x_{2}\Vert _{*} \int_{\mu }^{1} (1-s)^{ \alpha -2} c(s)\,ds \\& \quad \leq \frac{t(2+ \frac{1}{\Gamma (2- \beta)} +m_{0} + \theta_{0}+ \theta_{1})}{\Gamma (\alpha -1)} \Vert x_{1} - x_{2}\Vert _{*} \Biggl[ \int_{0}^{ \lambda } a(s)\,ds \\& \qquad {}+ \sum_{i=1}^{5} \int_{\lambda }^{\mu } (1- s)^{\alpha -2} b_{i}(s)\,ds + \int_{\mu }^{1} c(s)\,ds \Biggr] \end{aligned}$$

and so

$$\begin{aligned} \bigl\Vert F'_{x_{1}}-F'_{x_{2}}\bigr\Vert \leq & \frac{(2+ \frac{1}{\Gamma (2- \beta)} +m_{0} + \theta_{0}+ \theta_{1})}{\Gamma (\alpha -1)} \\ &{}\times \Biggl[ \Vert a\Vert _{[0,\lambda ]}+ \sum_{i=1}^{5} \Vert \hat{b_{i}} \Vert _{[\lambda, \mu ]} + \Vert c\Vert _{[ \mu, 1]} \Biggr] \Vert x_{1} - x_{2}\Vert _{*}. \end{aligned}$$

This implies that

$$\begin{aligned} \Vert F_{x_{1}}-F_{x_{2}}\Vert _{*} \leq & \frac{(2+ \frac{1}{\Gamma (2- \beta)} +m_{0} + \theta_{0}+ \theta_{1})}{\Gamma (\alpha -1)} \\ &{}\times \Biggl[ \Vert a\Vert _{[0,\lambda ]} + \sum_{i=1}^{5} \Vert \hat{b_{i}} \Vert _{[\lambda, \mu ]} + \Vert c\Vert _{[ \mu, 1]} \Biggr] \Vert x_{1} - x_{2}\Vert _{*} \end{aligned}$$

and so \(F_{x_{1}} \to F_{x_{2}}\) in X as \(x_{2} \to x_{1}\). Thus, F is continuous on X. We have \(\lim_{z \to 0^{+}} \frac{\Lambda _{i}(z,z,z,z,z)}{z} =0\), \(\lim_{z \to 0^{+}} \frac{\Lambda_{i}(lz,lz,lz,lz,lz)}{z} =0\), where \(l = \max \{ 1, \frac{1}{ \Gamma (2-\beta)}, m_{0}, \theta_{0} + \theta_{1} \}\). Let \(\epsilon >0\) be given. Choose \(\delta_{i} := \delta_{i}(\epsilon) >0\) such that \(0< z \leq \delta_{i}\) implies that \(\lim_{z \to 0^{+}} \frac{ \Lambda_{i}(lz,lz,lz,lz,lz)}{z} < \epsilon \) for \(1 \leq i \leq n_{0}\). Hence, \(\Lambda_{i}(lz,lz,lz,lz,lz)< \epsilon z\) for \(0< z \leq \delta _{i}\) and so \(\Lambda_{i}(lz,lz,lz,lz,lz)< \epsilon z\) for all \(1 \leq i \leq n_{0}\) and \(z \in (0. \delta ]\), where \(\delta:= \delta (\epsilon) = \min_{1 \leq i \leq n_{0}} \{ \delta_{i} \}\). Since

$$\biggl(2+ \frac{1}{\Gamma (2- \beta)}+ m_{0} + \theta_{0} + \theta_{1}\biggr) \bigl( \Vert a\Vert _{[0, \lambda ]} + (1- \mu)^{\alpha - 2} \Vert c\Vert _{[1, \mu ]} \bigr) < \Gamma (\alpha -1), $$

there exists \(\epsilon_{0}>0\) such that

$$\biggl(2+ \frac{1}{\Gamma (2- \beta)}+ m_{0} + \theta_{0} + \theta_{1}\biggr) \bigl( \Vert a\Vert _{[0, \lambda ]} + (1- \mu)^{\alpha - 2} \Vert c\Vert _{[1, \mu ]} \bigr) + \epsilon_{0} \sum_{i=1}^{n_{0}} \Vert \hat{\phi _{i}} \Vert _{[\lambda, \mu ]} < \Gamma (\alpha -1). $$

Let \(r= \delta (\epsilon_{0})\). Then \(\Lambda_{i}(lz,lz,lz,lz,lz)< \epsilon_{0} z\) for all \(1 \leq i \leq n_{0}\) and for \(z \in (0, r]\). Put \(C= \{ x \in X : \Vert x\Vert _{*}< r \}\). Define the map \(\alpha: X^{2} \to [0, \infty)\) by \(\alpha (x,y)=1\) whenever \(x,y\in C\) and \(\alpha (x,y)=0\) otherwise. We show that F is α-admissible. Let \(x, y \in X\) be such that \(\alpha (x,y) \geq 1\). Then \(x,y \in C\), \(\Vert x\Vert _{*}< r\) and \(\Vert y\Vert _{*}< r\). Let \(t \in [0,1]\). Then we have

$$\begin{aligned} \bigl\vert F_{x}(t)\bigr\vert \leq& \int_{0}^{\lambda } G(t,s)\biggl\vert f_{1} \biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\biggr\vert \,ds \\ &{}+ \int_{\lambda }^{\mu } G(t,s) \biggl\vert f_{2} \biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\biggr\vert \,ds \\ &{}+ \int_{\mu }^{1} G(t,s) \biggl\vert f_{3} \biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\biggr\vert \,ds \\ \leq& \frac{t}{\Gamma (\alpha -1)} \int_{0}^{\lambda } (1-s)^{\alpha -2} \biggl\vert f_{1}\biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr) \\ &{}-f_{1}(s,0, 0, 0, 0, 0)\biggr\vert \,ds + \frac{t}{\Gamma (\alpha -1)} \int_{0} ^{\lambda } (1-s)^{\alpha -2} \bigl\vert f_{1}(s,0, 0, 0, 0, 0)\bigr\vert \,ds \\ &{}+\frac{t}{\Gamma (\alpha -1)} \int_{\lambda }^{\mu } (1-s)^{\alpha -2} \sum _{i=1}^{n_{0}} \phi_{i}(s) \Lambda_{i} \biggl(x(s), x'(s), D^{\beta }x(s), \\ &{} \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\,ds \\ &{}+\frac{t}{\Gamma (\alpha -1)} \int_{\mu }^{1} (1-s)^{\alpha -2} \biggl\vert f _{3}\biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr) \\ &{}- f_{3}(s,0, 0, 0, 0, 0)\biggr\vert \,ds \\ &{}+ \frac{t}{\Gamma (\alpha -1)} \int_{0}^{\lambda } (1-s)^{\alpha -2} \bigl\vert f_{3}(s,0, 0, 0, 0, 0)\bigr\vert \,ds \\ \leq& \frac{t}{\Gamma (\alpha -1)} \int_{0}^{\lambda } (1-s)^{\alpha -2} a(s) \biggl( \Vert x\Vert + \bigl\Vert x'\bigr\Vert + \frac{\Vert x'\Vert }{\Gamma (2-\beta)} \\ &{}+ m_{0} \Vert x\Vert + \theta_{0} \Vert x\Vert + \theta_{1} \Vert x\Vert \biggr)\,ds \\ &{}+ \frac{t}{\Gamma (\alpha -1)} \int_{\lambda }^{\mu } (1-s)^{\alpha -2} \sum _{i=1}^{n_{0}} \phi_{i}(s) \Lambda_{i} \biggl(\Vert x\Vert + \bigl\Vert x'\bigr\Vert + \frac{ \Vert x'\Vert }{\Gamma (2-\beta)} \\ &{} + m_{0} \Vert x\Vert + \theta_{0} \Vert x \Vert + \theta _{1} \Vert x\Vert \biggr)\,ds \\ &{}+\frac{t}{\Gamma (\alpha -1)} \int_{\mu }^{1} (1-s)^{\alpha -2} c(s) \biggl( \Vert x\Vert + \bigl\Vert x'\bigr\Vert + \frac{\Vert x'\Vert }{\Gamma (2-\beta)} \\ &{} + m_{0} \Vert x\Vert + \theta_{0} \Vert x \Vert + \theta_{1} \Vert x\Vert \biggr)\,ds \\ \leq& \frac{t}{\Gamma (\alpha -1)}\Biggl( \biggl[2+ \frac{1}{\Gamma (2-\beta)}+ m _{0} + \theta_{0}+\theta_{1}\biggr] \Vert x \Vert _{*} \int_{0}^{\lambda } \sup (1-s)^{ \alpha -2} a(s)\,ds \\ &{}+\sum_{i=1}^{n_{0}} \int_{\lambda }^{\mu } (1-s)^{\alpha -2} \phi _{i}(s) \Lambda_{i} \bigl(l \Vert x\Vert _{*}, l \Vert x\Vert _{*}, l \Vert x\Vert _{*}, l \Vert x\Vert _{*}, l \Vert x\Vert _{*} \bigr)\,ds \\ &{}+ \biggl[2+ \frac{1}{\Gamma (2-\beta)}+ m_{0} + \theta_{0}+ \theta_{1}\biggr] \Vert x\Vert _{*} \int_{\mu }^{1} \sup (1-s)^{\alpha -2} c(s)\,ds \Biggr) \\ \leq& \frac{1}{\Gamma (\alpha -1)}\Biggl( \biggl[2+ \frac{1}{\Gamma (2-\beta)}+ m _{0} + \theta_{0}+\theta_{1}\biggr] \bigl(\Vert a\Vert _{[0, \lambda ]}+ (1-\mu)^{ \alpha -2}\Vert c\Vert _{[\mu, 1]} \bigr) r \\ &{}+\sum_{i=1}^{n_{0}} \Lambda_{i} (lr, lr, lr, lr, lr) \int_{\lambda }^{\mu } \hat{\phi _{i}} (s)\,ds \Biggr) \\ =&\frac{1}{\Gamma (\alpha -1)}\Biggl( \biggl[2+ \frac{1}{\Gamma (2-\beta)}+ m_{0} + \theta_{0}+\theta_{1}\biggr] \bigl(\Vert a\Vert _{[0, \lambda ]}+ (1-\mu)^{\alpha -2} \Vert c\Vert _{[\mu, 1]}\bigr) r \\ &{} + \sum_{i=1}^{n_{0}} \Vert \hat{\phi _{i}}\Vert \Lambda_{i} (lr, lr, lr, lr, lr)\Biggr) \\ \leq& \frac{1}{\Gamma (\alpha -1)}\Biggl( \biggl[2+ \frac{1}{\Gamma (2-\beta)}+ m _{0} + \theta_{0}+\theta_{1}\biggr] \bigl(\Vert a\Vert _{[0, \lambda ]}+ (1-\mu)^{ \alpha -2}\Vert c\Vert _{[\mu, 1]} \bigr) \\ &{}+\epsilon_{0} \sum_{i=1}^{n_{0}} \Vert \hat{\phi _{i}}\Vert \Biggr) r \\ < & \frac{1}{\Gamma (\alpha -1)} \Gamma (\alpha -1)r=r, \end{aligned}$$

and so \(\Vert F_{x}\Vert < r\). Similarly one can prove that \(\Vert F'_{x}\Vert < r\) and so \(\Vert F_{x}\Vert _{*} =\max \{ \Vert F_{x}\Vert , \Vert F'_{x}\Vert \} < r\). Hence, \(F_{x} \in C\) and by same reason \(F_{y} \in C\). This implies that \(\alpha (F_{x},F_{y}) \geq 1\) and so F is α-admissible. Also, \(\alpha (x_{0}, F_{x_{0}}) \geq 1\) for all \(x_{0} \in C\) (note that C is nonempty). Let \(x, y \in X\) and \(t \in [0,1]\). Then we have

$$\begin{aligned} \bigl\vert F_{x}(t)-F_{y}(t)\bigr\vert \leq & \int_{0}^{\lambda } G(t,s)\biggl\vert f_{1} \biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr) \\ &{} - f_{1}\biggl(s, y(s), y'(s), D^{\beta }y(s), \int_{0}^{s} h(\xi) y( \xi)\,d\xi, \phi \bigl(y(s) \bigr)\biggr)\biggr\vert \,ds \\ &{}+ \int_{\lambda }^{\mu } G(t,s) \biggl\vert f_{2} \biggl(s, x(s), x'(s), D^{\beta }x(s), \int_{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\,ds \\ &{} - f_{2}\biggl(s, y(s), y'(s), D^{\beta }y(s), \int_{0}^{s} h(\xi) y( \xi)\,d\xi, \phi \bigl(y(s) \bigr)\biggr)\biggr\vert \,ds \\ &{}+ \int_{\mu }^{1} G(t,s) \biggl\vert f_{3} \biggl(s, x(s), x'(s), D^{\beta }x(s), \int _{0}^{s} h(\xi) x(\xi)\,d\xi, \phi \bigl(x(s) \bigr)\biggr)\,ds \\ &{} - f_{3}(s, y), y'(s), D^{\beta }y(s), \int_{0}^{s} h(\xi) y( \xi)\,d\xi, \phi \bigl(y(s) \bigr))\biggr\vert \,ds \\ \leq & \frac{t}{\Gamma (\alpha -1)} \int_{0}^{\lambda } (1-s)^{ \alpha -2} a(s) \biggl( \Vert x - y \Vert + \bigl\Vert x' - y' \bigr\Vert + \bigl\Vert D^{\beta } (x - y)\bigr\Vert \\ &{}+ \int_{0}^{s} \bigl\vert h(\xi)\bigr\vert \Vert x - y \Vert \,d\xi + \phi \bigl(\Vert x - y\Vert \bigr)\biggr)\,ds \\ &{}+ \frac{t}{\Gamma (\alpha -1)} \int_{\lambda }^{\mu } (1-s)^{\alpha -2} \biggl(b_{1}(s) \Vert x - y \Vert +b_{2}(s) \bigl\Vert x' - y' \bigr\Vert \\ &{}+ b_{3}(s) \bigl(D^{\beta } \Vert x - y \Vert \bigr)+ b_{4}(s) \int_{0}^{s} \bigl\vert h(\xi)\bigr\vert \Vert x - y \Vert \,d\xi +b_{5}(s) \phi \bigl(\Vert x - y\Vert \bigr)\biggr)\,ds \\ &{}+ \frac{t}{\Gamma (\alpha -1)} \int_{\mu }^{1} (1-s)^{\alpha -2} c(s) \biggl(\Vert x - y \Vert + \bigl\Vert x' - y' \bigr\Vert + \bigl\Vert D^{\beta } (x - y)\bigr\Vert \\ &{}+ \int_{0}^{s} \bigl\vert h(\xi)\bigr\vert \Vert x - y \Vert \,d\xi + \phi \bigl(\Vert x - y\Vert \bigr)\biggr)\,ds \\ \leq & \frac{t}{\Gamma (\alpha -1)} \int_{0}^{\lambda } (1-s)^{ \alpha -2} a(s) \biggl( \Vert x - y\Vert + \bigl\Vert x' - y'\bigr\Vert + \frac{\Vert x' - y'\Vert }{\Gamma (2 - \beta)} \\ &{}+ m_{0} \Vert x - y\Vert + \theta_{0} \Vert x - y\Vert + \theta_{1} \bigl\Vert x' - y' \bigr\Vert \biggr)\,ds \\ &{}+ \frac{t}{\Gamma (\alpha -1)} \int_{\lambda }^{\mu } (1-s)^{\alpha -2} \biggl(b_{1}(s) \Vert x - y\Vert +b_{2}(s) \bigl\Vert x' - y'\bigr\Vert \\ &{}+ b_{3}(s) \frac{\Vert x' - y'\Vert }{\Gamma (2 - \beta)}+ b_{4}(s) m_{0} \Vert x - y\Vert \\ &{}+b_{5}(s) \bigl( \theta_{0} \Vert x - y\Vert + \theta_{1} \bigl\Vert x' - y'\bigr\Vert \bigr)\biggr)\,ds \\ &{}+ \frac{t}{\Gamma (\alpha -1)} \int_{\mu }^{1} (1-s)^{\alpha -2} c(s) \biggl( \Vert x - y\Vert + \bigl\Vert x' - y'\bigr\Vert + \frac{\Vert x' - y'\Vert }{\Gamma (2 - \beta)} \\ &{}+ m_{0} \Vert x - y\Vert + \theta_{0} \Vert x - y\Vert + \theta_{1} \bigl\Vert x' - y' \bigr\Vert \biggr)\,ds \\ \leq & \frac{t(2+ \frac{1}{\Gamma (2- \beta)} +m_{0} + \theta_{0}+ \theta_{1})}{\Gamma (\alpha -1)} \Vert x - y\Vert _{*} \Biggl[ \int_{0}^{\lambda } \sup (1-s)^{\alpha -2} a(s)\,ds \\ &{}+ \sum_{i=1}^{5} \int_{\lambda }^{\mu } (1-s)^{\alpha -2} b_{i}(s)\,ds + \int_{\mu }^{1} \sup (1-s)^{\alpha -2} c(s)\,ds \Biggr] \\ \leq & \frac{1}{\Gamma (\alpha -1)} \biggl(2+ \frac{1}{\Gamma (2- \beta)}+ m_{0} + \theta_{0} + \theta_{1}\biggr) \\ &{}\times \Biggl( \Vert a\Vert _{[0, \lambda ]} + \sum_{i=1} ^{5} \Vert \hat{b_{i}}\Vert + (1-\mu)^{\alpha - 2} \Vert c\Vert _{[1, \mu ]} \Biggr) \Vert x - y\Vert _{*} \\ :=&\psi \bigl( \Vert x - y\Vert _{*} \bigr). \end{aligned}$$

Similarly, one can show that \(\Vert F'_{x} - F'_{y}\Vert \leq \psi ( \Vert x - y\Vert _{*} )\) and so \(\alpha (x,y) \Vert F_{x} - F_{y}\Vert _{*} \leq \psi ( d(x,y))\) for all \(x,y\in X\). We have

$$\frac{1}{\Gamma (\alpha -1)} \biggl(2+ \frac{1}{\Gamma (2- \beta)}+ m_{0} + \theta_{0} + \theta_{1}\biggr) \Biggl( \Vert a\Vert _{[0, \lambda ]} + \sum_{i=1}^{5} \Vert \hat{b_{i}}\Vert + (1-\mu)^{\alpha - 2} \Vert c\Vert _{[1, \mu ]} \Biggr) < 1, $$

\(\psi \in \Psi \). By using Lemma 2.2, F has a fixed point which is a solution for the problem (1). □

Example 3.2

Consider the problem \(D^{\frac{9}{2}} x(t) + f(t, x(t), x'(t), D^{\frac{1}{2}} x(t), \int _{0}^{t} x(\xi)\,d\xi, I^{\frac{1}{3}} x(t))=0\), where

$$f(t, x_{1}, \ldots, x_{5})=\textstyle\begin{cases} f_{1}(t, x_{1}, \ldots, x_{5}):= \sin t(\sum_{i=1}^{5} \Vert x_{i}\Vert ) & t \in [0, 0.2), \\ f_{2}(t, x_{1}, \ldots, x_{5}):= \frac{0.2}{p(t)} \sum_{i=1}^{5} \frac{ \Vert x_{i}\Vert ^{2}}{1+ \Vert x_{i}\Vert } & t \in [0.2, 0.7], \\ f_{3}(t, x_{1}, \ldots, x_{5}):= t(\sum_{i=1}^{5} \Vert x_{i}\Vert ) & t \in [0.7, 1], \end{cases} $$

and \(p(t)=0\) whenever \(t\in [0.2,0.07]\cap \mathcal{Q}\) and \(p(t)= \sqrt{t}\) whenever \(t\in [0.2,0.07]\cap \mathcal{Q}^{c}\). Put \(a(t) = \sin t\), \(b_{1}(t)=\cdots =b_{5}(t)= \frac{1}{p(t)}\) and \(c(t) = t\) for all t. Note that

$$\begin{aligned}& \bigl\vert f_{1}(t, x_{1}, \ldots, x_{5}) - f_{1}(t, y_{1}, \ldots, y_{5})\bigr\vert = \sin t \Biggl\vert \sum_{i=1}^{5} \Vert x_{i}\Vert - \Vert y_{i}\Vert \Biggr\vert \leq \sin t \sum_{i=1}^{5} \Vert x_{i} - y_{i}\Vert , \\& \bigl\vert f_{2}(t, x_{1}, \ldots, x_{5}) - f_{2}(t, y_{1}, \ldots, y_{5})\bigr\vert \\& \quad = \frac{0.2}{p(t)} \Biggl\vert \sum_{i=1}^{5} \frac{ \Vert x_{i}\Vert ^{2}}{1+ \Vert x_{i}\Vert } - \frac{ \Vert y_{i}\Vert ^{2}}{1+ \Vert y_{i}\Vert }\Biggr\vert \\& \quad = \frac{0.2}{p(t)}\Biggl\vert \sum_{i=1}^{5} \frac{ \Vert x_{i}\Vert ^{2}+ \Vert x_{i}\Vert ^{2} \Vert y_{i}\Vert - \Vert x_{i}\Vert \Vert y_{i}\Vert ^{2} - \Vert y_{i}\Vert ^{2}}{(1+ \Vert x_{i}\Vert ) (1+ \Vert y_{i}\Vert )}\Biggr\vert \\& \quad = \frac{0.2}{p(t)}\Biggl\vert \sum_{i=1}^{5} \frac{ (\Vert x_{i}\Vert +\Vert y_{i}\Vert )( \Vert x_{i}\Vert -\Vert y_{i}\Vert ) + \Vert x_{i}\Vert (\Vert x_{i}\Vert -\Vert y_{i}\Vert ) \Vert y_{i}\Vert }{(1+ \Vert x_{i}\Vert ) (1+ \Vert y_{i}\Vert )}\Biggr\vert \\& \quad = \frac{0.2}{p(t)}\Biggl\vert \sum_{i=1}^{5} \frac{ (\Vert x_{i}\Vert -\Vert y_{i}\Vert ) ( \Vert x_{i}\Vert + \Vert y_{i}\Vert + \Vert x_{i}\Vert \Vert y_{i}\Vert ) }{(1+\Vert x_{i}\Vert + \Vert y_{i}\Vert + \Vert x_{i}\Vert \Vert y_{i}\Vert }\Biggr\vert \\& \quad \leq \frac{0.2}{p(t)}\Biggl\vert \sum_{i=1}^{5} \Vert x_{i}\Vert -\Vert y_{i}\Vert \Biggr\vert \leq \frac{0.2}{p(t)} \sum_{i=1}^{5} \Vert x_{i} - y_{i}\Vert . \end{aligned}$$

Define \(\Lambda_{i} (x_{1},\ldots, x_{5}) = \frac{ \Vert x_{i}\Vert ^{2}}{1+ \Vert x_{i}\Vert }\) for \(i=1,\dots,5\). Then \(\lim_{z \to 0^{+}} \frac{\Lambda _{i} (z, z, z, z, z)}{z} =0\) for all i. Put \(b_{i}(t) = \phi_{i}(t) = \frac{0.2}{p(t)}\) for all i, \(n_{0} = 5\) and \(\beta = \frac{1}{2}\). Since \(\vert \int_{0}^{t} x(\xi)\,d\xi \vert \leq t \Vert x\Vert \leq \Vert x\Vert \), put \(m_{0} = 1\). Since \(\vert I^{\frac{1}{3}}x(t)\vert = \vert \frac{1}{\Gamma (\frac{1}{3})} \int_{0}^{t} (t-s)^{ \frac{1}{3}-1 } x(s)\,ds\vert \leq \frac{1}{ \Gamma (\frac{1}{3})} \int_{0}^{t} \vert (t-s)^{ \frac{1}{3}-1 } x(s) \vert \,ds \leq \frac{\Vert x\Vert }{\Gamma (\frac{1}{3})} \int_{0}^{t} \frac{ds}{(t-s)^{ \frac{2}{3} }} \leq \frac{\Vert x\Vert }{\Gamma (\frac{1}{2})}\), we put \(\theta_{0} = \frac{1}{\Gamma (\frac{1}{3})}\) and \(\theta_{1}=0\). Note that \(\Vert a\Vert _{[0, \lambda ]} = \int_{0}^{0.2} \sin t \,dt \leq 0.02\), \(\Vert \hat{b_{i}}\Vert _{[ \lambda, \mu ]} = \int_{0.2}^{0.7} \frac{0.2}{ \sqrt{t}}\,dt \leq 0.08\), \(\Vert c\Vert _{[\mu, 1]} = \int_{0.7}^{1} t \,dt = 0.045\) and

$$\begin{aligned}& \biggl(2 + \frac{1}{\Gamma (2-\beta)}+ m_{0} + \theta_{0} + \theta_{0} + \theta_{1}\biggr) \Biggl(\Vert a\Vert _{[0, \lambda ]}+ \sum_{i=1}^{5} \Vert \hat{b_{i}}\Vert + (1-\mu)^{\alpha -2}\Vert c\Vert _{[\mu, 1]}\Biggr) \\& \quad \leq \biggl(2+ \frac{1}{\Gamma (\frac{3}{2})}+ 1 + \frac{1}{\Gamma ( \frac{1}{2})}\biggr) \Biggl(0.02+ \sum_{i=1}^{5} 0.8 + (1- 0.7)^{\frac{7}{2}} 0.045\Biggr) \\& \quad \leq \biggl(3+ \frac{2}{\sqrt{\pi }}+ \frac{1}{\sqrt{\pi }}\biggr) (0.421) < \Gamma \biggl(\frac{7}{2}\biggr) = \Gamma (\alpha -1). \end{aligned}$$

Now by using Theorem 3.3, the problem has a solution.

4 Conclusions

Most natural phenomena include crisis and it is important we could model this type phenomena. Researchers are going to use fractional integro-differential equations for modeling of crisis phenomena. In this work, we investigate the existence of solutions for a three steps crisis integro-differential equation by considering this assumption that the second step is a point-wise defined singular fractional differential equation, while the first and third parts have natural treatments.