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Advances in Difference Equations

, 2018:135 | Cite as

On the existence of solutions of a three steps crisis integro-differential equation

  • Dumitru Baleanu
  • Khadijeh Ghafarnezhad
  • Shahram Rezapour
  • Mehdi Shabibi
Open Access
Research

Abstract

There are many natural phenomena including a crisis (such as a spate or contest) which could be described in three steps. We investigate the existence of solutions for a three step crisis integro-differential equation. We suppose that the second step is a point-wise defined singular fractional differential equation.

Keywords

Caputo derivative Point-wise defined singular equation Three steps crisis phenomena 

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, 2, 3, 4, 5, 6, 7] and [8]). Considering the importance of modeling of crisis phenomena, some researchers are working and publishing in this area (see, for example, [9, 10, 11, 12, 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, 16, 17, 18, 19, 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\(T:X \to X\)anα-admissibleα-ψ-contraction. IfTis continuous and there exists\(x_{0} \in X\)such that\(\alpha (x_{0}, Tx_{0}) \geq 1\), thenThas 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])

LetEbe a Banach space, \(P \subseteq E\)a cone and\(\Omega_{1}\), \(\Omega_{2}\)two bounded open balls ofEcentered 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. ThenFhas 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 mapuis 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 mapqdefined 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 alltand\(x_{1},\dots,x_{5}\in X\). 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.

Notes

Authors’ contributions

The main idea of this paper was proposed by the third author. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

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© The Author(s) 2018

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Dumitru Baleanu
    • 1
    • 2
  • Khadijeh Ghafarnezhad
    • 3
  • Shahram Rezapour
    • 3
    • 4
  • Mehdi Shabibi
    • 5
  1. 1.Department of MathematicsCankaya UniversityAnkaraTurkey
  2. 2.Institute of Space SciencesBucharestRomania
  3. 3.Department of MathematicsAzarbaijan Shahid Madani UniversityTabrizIran
  4. 4.Department of Medical Research, China Medical University HospitalChina Medical UniversityTaichungTaiwan
  5. 5.Department of MathematicsIslamic Azad University, Mehran BranchMehranIran

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