Existence and uniqueness of solutions for multi-term nonlinear fractional integro-differential equations

Abstract

In this manuscript, by using the fixed point theorems, the existence and the uniqueness of solutions for multi-term nonlinear fractional integro-differential equations are reported. Two examples are presented to illustrate our results.

1 Introduction

The study of fractional differential equations ranges from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions. Fractional differential equations appear naturally in a number of fields such as physics, polymer rheology, regular variational in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electro-dynamics of complex medium, viscoelasticity, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, etc. An excellent account in the study of fractional differential equations can be found in [1, 2] and [3]. For more details and examples, one can study [4–13] and [14]. It is considerable that there are many works about fractional integro-differential equations (see, for example, [15–18] and [19]).

In 2007, Xinwei and Landong reviewed the existence of solutions for the nonlinear fractional differential equation

$${}^{c}D^{\alpha }u(t)=f(t,u(t){,}^c{D}^{\beta }u(t))(0<t<1)$$

with boundary values $u(0)=u^{\prime }(1)=0$ or $u^{\prime }(0)=u(1)=0$ or $u(0)=u(1)=0$, where $1<\alpha \le 2$, $0<\beta \le 1$, and f is continuous on $[0,1]\times \mathbb{R}\times \mathbb{R}$ [20]. In 2009, Su and Zhang studied the existence and uniqueness of solutions for the following nonlinear two-point fractional boundary value problem

$${}^{c}D^{\alpha }u(t)=f(t,u(t){,}^c{D}^{\beta }u(t))(0<t<1)$$

with boundary values $a_{1}u(0)-a_2{u}^{\prime }(0)=A$ and $b_{1}u(1)+b_2{u}^{\prime }(1)=B$, where α, β, $a_i$, $b_i$ ($i=1,2$) satisfy certain conditions [21]. In 2010, Ahmad and Sivasundaram studied the existence of solutions for the nonlinear fractional integro-differential equation

$${}^{c}D^{q}u(t)=f(t,u(t),(\varphi u)(t),(\psi u)(t))(0<t<1\text{ and }1<q\le 2)$$

with boundary values $u^{\prime }(0)+au({\eta }_1)=0$, $b{u}^{\prime }(1)+u({\eta }_2)=0$ and $0<{\eta }_1\le {\eta }_2<1$, where ${}^{c}D^q$ is the Caputo fractional derivative, $a,b\in (0,1)$, $f:[0,1]\times X\times X\times X\to X$ is continuous and for the mappings $\gamma ,\lambda :[0,1]\times [0,1]\to [0,\mathrm{\infty })$ with the property ${sup}_{t\in [0,1]}|{\int }_0^t\lambda (t,s)ds|<\mathrm{\infty }$ and ${sup}_{t\in [0,1]}|{\int }_0^t\gamma (t,s)ds|<\mathrm{\infty }$, the maps ϕ and ψ are defined by $(\varphi u)(t)={\int }_0^t\gamma (t,s)u(s)ds$ and $(\psi u)(t)={\int }_0^t\lambda (t,s)u(s)ds$. Here, X is a Banach space (see [22]).

2 Main results

2.1 The basic problem

In this paper, we study the existence and uniqueness of solutions for the multi-term nonlinear fractional integro-differential equation

$${}^{c}D^{\alpha }u(t)=f(t,u(t),(\varphi u)(t),(\psi u)(t){,}^c{D}^{{\beta }_1}u(t){,}^c{D}^{{\beta }_2}u(t),\dots {,}^c{D}^{{\beta }_n}u(t))(0<t<1)$$

(1)

with boundary values $u(0)+au(1)=0$ and $u^{\prime }(0)+b{u}^{\prime }(1)=0$, where $1<\alpha <2$, $0<{\beta }_i<1$, $\alpha -{\beta }_i\ge 1$, $a,b\ne -1$, $f:[0,1]\times {\mathbb{R}}^{n+3}\to \mathbb{R}$ is continuous, and for the mappings

$$\gamma ,\lambda :[0,1]\times [0,1]\to [0,\mathrm{\infty })$$

with the property ${sup}_{t\in [0,1]}|{\int }_0^t\lambda (t,s)ds|<\mathrm{\infty }$ and ${sup}_{t\in [0,1]}|{\int }_0^t\gamma (t,s)ds|<\mathrm{\infty }$, the maps ϕ and ψ are defined by $(\varphi u)(t)={\int }_0^t\gamma (t,s)u(s)ds$ and $(\psi u)(t)={\int }_0^t\lambda (t,s)u(s)ds$. In this way, we need the following result, which has been proved in [2].

Lemma 2.1 Let $\alpha >0$ and $n=[\alpha ]+1$. Then

$$I^{\alpha }{}^c{D}^{\alpha }u(t)=u(t)+c_0+c_{1}t+c_{2}t+\cdots +c_{n-1}{t}^{n-1},$$

where $c_0,c_1,\dots ,c_{n-1}$ are some real numbers.

The proof of the following result by using Lemma 2.1 is straightforward.

Lemma 2.2 Let $y\in C[0,1]$, $a,b\ne -1$ and $1<\alpha <2$. Then the problem ${}^{c}D^{\alpha }u(t)=y(t)$ with boundary values $u(0)+au(1)=0$ and $u^{\prime }(0)+b{u}^{\prime }(1)=0$ has the unique solution

$$\begin{array}{rcl}u(t)& =& \frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-s)}^{\alpha -1}y(s)ds-\frac{a}{(1+a)\mathrm{\Gamma }(\alpha )}{\int }_0^1{(1-s)}^{\alpha -1}y(s)ds\\ +\frac{ab-b(1+a)t}{(1+a)(1+b)\mathrm{\Gamma }(\alpha -1)}{\int }_0^1{(1-s)}^{\alpha -2}y(s)ds.\end{array}$$

2.2 Some results on solving the problem

Let $C(I)$ be the space of all continuous real-valued functions on $I=[0,1]$ and

$$X=\{u:u\in C(I){\text{ and }}^c{D}^{{\beta }_i}u\in C(I)(0<{\beta }_i<1)\text{ for }i=1,2,\dots ,n\}$$

endowed with the norm $\parallel u\parallel ={max}_{t\in I}|u(t){|+{\sum }_{i=1}^n{max}_{t\in I}|}^c{D}^{{\beta }_i}u(t)|$. It is known that $(X,\parallel \cdot \parallel )$ is a Banach space.

Theorem 2.3 Assume that there exist $\kappa \in (0,\alpha -1)$ and $\mu (t)\in L^{\frac{1}{\kappa }}([0,1],(0,\mathrm{\infty }))$ such that

$$\begin{array}{c}|f(t,x,y,w,u_1,u_2,\dots ,u_n)-f(t,x^{\prime },y^{\prime },w^{\prime },v_1,v_2,\dots ,v_n)|\hfill \\ \le \mu (t)(|x-x^{\prime }|+|y-y^{\prime }|+|w-w^{\prime }|+|u_1-v_1|+|u_2-v_2|+\cdots +|u_n-v_n|)\hfill \end{array}$$

for all $t\in [0,1]$ and $x,y,w,x^{\prime },y^{\prime },w^{\prime },u_1,u_2,\dots ,u_n,v_1,v_2,\dots ,v_n\in \mathbb{R}$. Then problem (1) has a unique solution whenever

$$\begin{array}{rcl}\mathrm{\Delta }& =& (1+{\gamma }_0+{\lambda }_0)[\frac{(1+2|a|){\mu }^{\ast }}{|1+a|\mathrm{\Gamma }(\alpha )}{\left(\frac{1-\kappa }{\alpha -\kappa }\right)}^{1-\kappa }+\frac{|b|(1+2|a|){\mu }^{\ast }}{|1+a||1+b|\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\\ +\sum _{i=1}^n(\frac{\mathrm{\Gamma }(\alpha -\kappa ){\mu }^{\ast }}{\mathrm{\Gamma }(\alpha -1)\mathrm{\Gamma }(\alpha -{\beta }_i-\kappa +1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\\ +\frac{|b|{\mu }^{\ast }}{|1+b|\mathrm{\Gamma }(2-{\beta }_i)\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\left)\right]<1,\end{array}$$

where ${\gamma }_0={sup}_{t\in I}|{\int }_0^t\gamma (t,s)ds|$, ${\lambda }_0={sup}_{t\in I}|{\int }_0^t\lambda (t,s)ds|$, ${\mu }^{\ast }={({\int }_0^1{(\mu (s))}^{\frac{1}{\kappa }}ds)}^{\kappa }$.

Proof Define the mapping $F:X\to X$ by

$$\begin{array}{rcl}(Fu)(t)& =& {\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))ds\\ -\frac{a}{(1+a)}{\int }_0^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}\\ \times f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))ds\\ +\frac{ab-b(1+a)t}{(1+a)(1+b)}{\int }_0^1\frac{{(1-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\\ \times f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))ds.\end{array}$$

For each $u,v\in X$ and $t\in [0,1]$, by using the Hölder inequality, we have

$$\begin{array}{c}|(Fu)(t)-(Fv)(t)|\hfill \\ =\left|{\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}\right(f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))\hfill \\ -f(s,v(s),(\varphi v)(s),(\psi v)(s){,}^c{D}^{{\beta }_1}v(s){,}^c{D}^{{\beta }_2}v(s),\dots {,}^c{D}^{{\beta }_n}v(s)))ds\hfill \\ -\frac{a}{(1+a)}{\int }_0^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}(f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))\hfill \\ -f(s,v(s),(\varphi v)(s),(\psi v)(s){,}^c{D}^{{\beta }_1}v(s){,}^c{D}^{{\beta }_2}v(s),\dots {,}^c{D}^{{\beta }_n}v(s)))ds\hfill \\ +\frac{ab-b(1+a)t}{(1+a)(1+b)}{\int }_0^1\frac{{(1-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\left(f\right(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\hfill \\ \dots {,}^c{D}^{{\beta }_n}u(s))-f(s,v(s),(\varphi v)(s),(\psi v)(s){,}^c{D}^{{\beta }_1}v(s){,}^c{D}^{{\beta }_2}v(s),\dots {,}^c{D}^{{\beta }_n}v(s)))ds|\hfill \\ \le {\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}|f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))\hfill \\ -f(s,v(s),(\varphi v)(s),(\psi v)(s){,}^c{D}^{{\beta }_1}v(s){,}^c{D}^{{\beta }_2}v(s),\dots {,}^c{D}^{{\beta }_n}v(s))|ds\hfill \\ +\frac{|a|}{|1+a|}{\int }_0^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}|f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))\hfill \\ -f(s,v(s),(\varphi v)(s),(\psi v)(s){,}^c{D}^{{\beta }_1}v(s){,}^c{D}^{{\beta }_2}v(s),\dots {,}^c{D}^{{\beta }_n}v(s))|ds\hfill \\ +\frac{|ab-b(1+a)t|}{|1+a||1+b|}{\int }_0^1\frac{{(1-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\left|f\right(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\hfill \\ \dots {,}^c{D}^{{\beta }_n}u(s))-f(s,v(s),(\varphi v)(s),(\psi v)(s){,}^c{D}^{{\beta }_1}v(s){,}^c{D}^{{\beta }_2}v(s),\dots {,}^c{D}^{{\beta }_n}v(s))|ds\hfill \\ \le {\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}\mu (s)(|u(s)-v(s)|+|(\varphi u)(s)-(\varphi v)(s)|+|(\psi u)(s)-(\psi v)(s)|\hfill \\ +{|}^c{D}^{{\beta }_1}u(s){-}^c{D}^{{\beta }_1}v(s){|+|}^c{D}^{{\beta }_2}u(s){-}^c{D}^{{\beta }_2}v(s){|+\cdots +|}^c{D}^{{\beta }_n}u(s){-}^c{D}^{{\beta }_n}v(s)\left|\right)ds\hfill \\ +\frac{|a|}{|1+a|}{\int }_0^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}\mu (s)(|u(s)-v(s)|+|(\varphi u)(s)-(\varphi v)(s)|+|(\psi u)(s)-(\psi v)(s)|\hfill \\ +{|}^c{D}^{{\beta }_1}u(s){-}^c{D}^{{\beta }_1}v(s){|+|}^c{D}^{{\beta }_2}u(s){-}^c{D}^{{\beta }_2}v(s){|+\cdots +|}^c{D}^{{\beta }_n}u(s){-}^c{D}^{{\beta }_n}v(s)\left|\right)ds\hfill \\ +\frac{|b|(1+2|a|)}{|1+a||1+b|}{\int }_0^1\frac{{(1-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\mu (s)(|u(s)-v(s)|+|(\varphi u)(s)-(\varphi v)(s)|\hfill \\ +|(\psi u)(s)-(\psi v)(s)|+{|}^c{D}^{{\beta }_1}u(s){-}^c{D}^{{\beta }_1}v(s)|\hfill \\ +{|}^c{D}^{{\beta }_2}u(s){-}^c{D}^{{\beta }_2}v(s){|+\cdots +|}^c{D}^{{\beta }_n}u(s){-}^c{D}^{{\beta }_n}v(s)\left|\right)ds\hfill \\ \le \frac{(1+{\gamma }_0+{\lambda }_0)\parallel u-v\parallel }{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-s)}^{\alpha -1}\mu (s)ds\hfill \\ +\frac{|a|(1+{\gamma }_0+{\lambda }_0)\parallel u-v\parallel }{|1+a|\mathrm{\Gamma }(\alpha )}{\int }_0^1{(1-s)}^{\alpha -1}\mu (s)ds\hfill \\ +\frac{|b|(1+2|a|)(1+{\gamma }_0+{\lambda }_0)\parallel u-v\parallel }{|1+a||1+b|\mathrm{\Gamma }(\alpha -1)}{\int }_0^1{(1-s)}^{\alpha -2}\mu (s)ds\hfill \\ \le \frac{(1+{\gamma }_0+{\lambda }_0)\parallel u-v\parallel }{\mathrm{\Gamma }(\alpha )}{\left({\int }_0^t{\left({(t-s)}^{\alpha -1}\right)}^{\frac{1}{1-\kappa }}ds\right)}^{1-\kappa }{\left({\int }_0^t{(\mu (s))}^{\frac{1}{\kappa }}ds\right)}^{\kappa }\hfill \\ +\frac{|a|(1+{\gamma }_0+{\lambda }_0)\parallel u-v\parallel }{|1+a|\mathrm{\Gamma }(\alpha )}{\left({\int }_0^1{\left({(1-s)}^{\alpha -1}\right)}^{\frac{1}{1-\kappa }}ds\right)}^{1-\kappa }{\left({\int }_0^1{(\mu (s))}^{\frac{1}{\kappa }}ds\right)}^{\kappa }\hfill \\ +\frac{|b|(1+2|a|)(1+{\gamma }_0+{\lambda }_0)\parallel u-v\parallel }{|1+a||1+b|\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times {\left({\int }_0^1{\left({(1-s)}^{\alpha -2}\right)}^{\frac{1}{1-\kappa }}ds\right)}^{1-\kappa }{\left({\int }_0^1{(\mu (s))}^{\frac{1}{\kappa }}ds\right)}^{\kappa }\hfill \\ \le \frac{{\mu }^{\ast }(1+{\gamma }_0+{\lambda }_0)\parallel u-v\parallel }{\mathrm{\Gamma }(\alpha )}{\left(\frac{1-\kappa }{\alpha -\kappa }\right)}^{1-\kappa }+\frac{|a|{\mu }^{\ast }(1+{\gamma }_0+{\lambda }_0)\parallel u-v\parallel }{|1+a|\mathrm{\Gamma }(\alpha )}{\left(\frac{1-\kappa }{\alpha -\kappa }\right)}^{1-\kappa }\hfill \\ +\frac{|b|(1+2|a|){\mu }^{\ast }(1+{\gamma }_0+{\lambda }_0)\parallel u-v\parallel }{|1+a||1+b|\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\hfill \\ \le (1+{\gamma }_0+{\lambda }_0)[\frac{(1+2|a|){\mu }^{\ast }}{|1+a|\mathrm{\Gamma }(\alpha )}{\left(\frac{1-\kappa }{\alpha -\kappa }\right)}^{1-\kappa }\hfill \\ +\frac{|b|(1+2|a|){\mu }^{\ast }}{|1+a||1+b|\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }]\parallel u-v\parallel .\hfill \end{array}$$

Also, we have

$$\begin{array}{c}{|}^c{D}^{{\beta }_i}(Fu)(t){-}^c{D}^{{\beta }_i}(Fv)(t)|\hfill \\ =|{\int }_0^t\frac{{(t-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}{(Fu)}^{\prime }(s)ds-{\int }_0^t\frac{{(t-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}{(Fv)}^{\prime }(s)ds|\hfill \\ =\left|{\int }_0^t\frac{{(t-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}\right({\int }_0^s\frac{{(s-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau \hfill \\ -\frac{b}{1+b}{\int }_0^1\frac{{(1-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau )ds\hfill \\ -{\int }_0^t\frac{{(t-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}({\int }_0^s\frac{{(s-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,v(\tau ),(\varphi v)(\tau ),(\psi v)(\tau ){,}^c{D}^{{\beta }_1}v(\tau ){,}^c{D}^{{\beta }_2}v(\tau ),\dots {,}^c{D}^{{\beta }_n}v(\tau ))d\tau \hfill \\ -\frac{b}{1+b}{\int }_0^1\frac{{(1-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,v(\tau ),(\varphi v)(\tau ),(\psi v)(\tau ){,}^c{D}^{{\beta }_1}v(\tau ){,}^c{D}^{{\beta }_2}v(\tau ),\dots {,}^c{D}^{{\beta }_n}v(\tau ))d\tau \left)ds\right|\hfill \\ \le {\int }_0^t\frac{{(t-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}({\int }_0^s\frac{{(s-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times |f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))\hfill \\ -f(\tau ,v(\tau ),(\varphi v)(\tau ),(\psi v)(\tau ){,}^c{D}^{{\beta }_1}v(\tau ){,}^c{D}^{{\beta }_2}v(\tau ),\dots {,}^c{D}^{{\beta }_n}v(\tau ))\left|d\tau \right)ds\hfill \\ +\frac{|b|}{|1+b|}{\int }_0^t\frac{{(t-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}\hfill \\ \times \left({\int }_0^1\frac{{(1-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\right|f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))\hfill \\ -f(\tau ,v(\tau ),(\varphi v)(\tau ),(\psi v)(\tau ){,}^c{D}^{{\beta }_1}v(\tau ){,}^c{D}^{{\beta }_2}v(\tau ),\dots {,}^c{D}^{{\beta }_n}v(\tau ))\left|d\tau \right)ds\hfill \\ \le \frac{(1+{\gamma }_0+{\lambda }_0)\parallel u-v\parallel }{\mathrm{\Gamma }(1-{\beta }_i)\mathrm{\Gamma }(\alpha -1)}{\int }_0^t{(t-s)}^{-{\beta }_i}({\int }_0^s{(s-\tau )}^{\alpha -2}\mu (\tau )d\tau )ds\hfill \\ +\frac{|b|(1+{\gamma }_0+{\lambda }_0)\parallel u-v\parallel }{|1+b|\mathrm{\Gamma }(1-{\beta }_i)\mathrm{\Gamma }(\alpha -1)}{\int }_0^t{(t-s)}^{-{\beta }_i}({\int }_0^1{(1-\tau )}^{\alpha -2}\mu (\tau )d\tau )ds\hfill \\ \le \frac{{\mu }^{\ast }(1+{\gamma }_0+{\lambda }_0)\parallel u-v\parallel }{\mathrm{\Gamma }(1-{\beta }_i)\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }{\int }_0^t{(t-s)}^{-{\beta }_i}{s}^{\alpha -\kappa -1}ds\hfill \\ +\frac{|b|{\mu }^{\ast }(1+{\gamma }_0+{\lambda }_0)\parallel u-v\parallel }{|1+b|\mathrm{\Gamma }(1-{\beta }_i)\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }{\int }_0^t{(t-s)}^{-{\beta }_i}ds\hfill \\ \le \frac{{\mu }^{\ast }(1+{\gamma }_0+{\lambda }_0)\parallel u-v\parallel }{\mathrm{\Gamma }(1-{\beta }_i)\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }{\int }_0^1{(1-\xi )}^{-{\beta }_i}{\xi }^{\alpha -\kappa -1}d\xi \hfill \\ +\frac{|b|{\mu }^{\ast }(1+{\gamma }_0+{\lambda }_0)\parallel u-v\parallel }{|1+b|\mathrm{\Gamma }(2-{\beta }_i)\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }.\hfill \end{array}$$

Since $B(\alpha -\kappa ,1-{\beta }_i)={\int }_0^1{(1-\xi )}^{-{\beta }_i}{\xi }^{\alpha -\kappa -1}d\xi =\frac{\mathrm{\Gamma }(\alpha -\kappa )\mathrm{\Gamma }(1-{\beta }_i)}{\mathrm{\Gamma }(\alpha -{\beta }_i-\kappa +1)}$, we obtain

$$\begin{array}{rcl}{|}^c{D}^{{\beta }_i}(Fu)(t){-}^c{D}^{{\beta }_i}(Fv)(t)|& \le & (1+{\gamma }_0+{\lambda }_0)[\frac{\mathrm{\Gamma }(\alpha -\kappa ){\mu }^{\ast }}{\mathrm{\Gamma }(\alpha -1)\mathrm{\Gamma }(\alpha -{\beta }_i-\kappa +1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\\ +\frac{|b|{\mu }^{\ast }}{|1+b|\mathrm{\Gamma }(2-{\beta }_i)\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }]\parallel u-v\parallel \end{array}$$

for all $i=1,2,\dots ,n$. Hence, we get

$$\begin{array}{c}\parallel Fu-Fv\parallel \hfill \\ \le (1+{\gamma }_0+{\lambda }_0)[\frac{(1+2|a|){\mu }^{\ast }}{|1+a|\mathrm{\Gamma }(\alpha )}{\left(\frac{1-\kappa }{\alpha -\kappa }\right)}^{1-\kappa }+\frac{|b|(1+2|a|){\mu }^{\ast }}{|1+a||1+b|\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\hfill \\ +\sum _{i=1}^n(\frac{\mathrm{\Gamma }(\alpha -\kappa ){\mu }^{\ast }}{\mathrm{\Gamma }(\alpha -1)\mathrm{\Gamma }(\alpha -{\beta }_i-\kappa +1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\hfill \\ +\frac{|b|{\mu }^{\ast }}{|1+b|\mathrm{\Gamma }(2-{\beta }_i)\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\left)\right]\parallel u-v\parallel =\mathrm{\Delta }\parallel u-v\parallel .\hfill \end{array}$$

Since $\mathrm{\Delta }<1$, F is a contraction mapping, therefore, by using the Banach contraction principle, F has a unique fixed point, which is the unique solution of problem (1) by using Lemma 2.2. □

Corollary 2.4 Assume that there exists $L>0$ such that

$$\begin{array}{c}|f(t,x,y,w,u_1,u_2,\dots ,u_n)-f(t,x^{\prime },y^{\prime },w^{\prime },v_1,v_2,\dots ,v_n)|\hfill \\ \le L(|x-x^{\prime }|+|y-y^{\prime }|+|w-w^{\prime }|+|u_1-v_1|+|u_2-v_2|+\cdots +|u_n-v_n|)\hfill \end{array}$$

for all $t\in [0,1]$ and $x,y,w,x^{\prime },y^{\prime },w^{\prime },u_1,u_2,\dots ,u_n,v_1,v_2,\dots ,v_n\in \mathbb{R}$. Then problem (1) has a unique solution whenever

$$\begin{array}{r}(1+{\gamma }_0+{\lambda }_0)[\frac{(1+2|a|)(1+(\alpha +1)|b|)L}{|1+a||1+b|\mathrm{\Gamma }(\alpha +1)}\\ +\sum _{i=1}^n(\frac{L}{\mathrm{\Gamma }(\alpha -{\beta }_i+1)}+\frac{|b|L}{|1+b|\mathrm{\Gamma }(2-{\beta }_i)\mathrm{\Gamma }(\alpha )})]<1,\end{array}$$

where ${\gamma }_0={sup}_{t\in I}|{\int }_0^t\gamma (t,s)ds|$, ${\lambda }_0={sup}_{t\in I}|{\int }_0^t\lambda (t,s)ds|$.

Now, we restate the Schauder’s fixed point theorem, which is needed to prove next result (see Theorem 1.10.16 in [23]).

Theorem 2.5 Let E be a closed, convex and bounded subset of a Banach space X, and let $F:E\to E$ be a continuous mapping such that $F(E)$ is a relatively compact subset of X. Then F has a fixed point in E.

Theorem 2.6 Let $f:[0,1]\times {\mathbb{R}}^{n+3}\to \mathbb{R}$ be a continuous function such that there exists a constant $l\in (0,\alpha -1)$ and a real-valued function $m(t)\in L^{\frac{1}{l}}([0,1],(0,\mathrm{\infty }))$ such that

$$\left|f(t,x,y,w,u_1,u_2,\dots ,u_n)\right|\le m\left(t\right)+d{\left|x\right|}^{\rho }+d^{\prime }{\left|y\right|}^{\rho \prime }+d^{\prime \prime }{\left|w\right|}^{\rho \prime \prime }+d_1{\left|u_1\right|}^{{\rho }_1}+d_2{\left|u_2\right|}^{{\rho }_2}+\dots d_n{\left|u_n\right|}^{\rho n},$$

(*)

where $d,d^{\prime },d^{″},d_i\ge 0$ and $0<\rho ,{\rho }^{\prime },{\rho }^{″},{\rho }_i<1$ for $i=1,2,\dots ,n$, or

$$\begin{array}{c}|f(t,x,y,w,u_1,u_2,\dots ,u_n)|\hfill \\ \le d{|x|}^{\rho }+d^{\prime }{|y|}^{{\rho }^{\prime }}+d^{″}{|w|}^{{\rho }^{″}}+d_1{|u_1|}^{{\rho }_1}+d_2{|u_2|}^{{\rho }_2}+\cdots +d_n{|u_n|}^{{\rho }_n},\hfill \end{array}$$

where $d,d^{\prime },d^{″},d_i>0$ and $\rho ,{\rho }^{\prime },{\rho }^{″},{\rho }_i>1$ for $i=1,2,\dots ,n$. Then problem (1) has a solution.

Proof First, suppose that f satisfy condition (∗). Define $B_r=\{u\in X,\parallel u\parallel \le r\}$, where

$$\begin{array}{r}\begin{array}{rl}r\ge & max\{{((n+4)Ad)}^{\frac{1}{1-\rho }},{((n+4)A{d}^{\prime }{\gamma }_0^{p^{\prime }})}^{\frac{1}{1-{\rho }^{\prime }}},{((n+4)A{d}^{″}{\lambda }_0^{p^{″}})}^{\frac{1}{1-{\rho }^{″}}},{((n+4)A{d}_1)}^{\frac{1}{1-{\rho }_1}},\\ {((n+4)A{d}_2)}^{\frac{1}{1-{\rho }_2}},\dots ,{((n+4)A{d}_n)}^{\frac{1}{1-{\rho }_n}},(n+4)K\},\end{array}\\ \begin{array}{rl}K=& \frac{(1+2|a|)M}{|1+a|\mathrm{\Gamma }(\alpha )}{\left(\frac{1-l}{\alpha -l}\right)}^{1-l}+\frac{|b|(1+2|a|)M}{|1+a||1+b|\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\sum _{i=1}^n(\frac{\mathrm{\Gamma }(\alpha -l)M}{\mathrm{\Gamma }(\alpha -1)\mathrm{\Gamma }(\alpha -{\beta }_i-l+1)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\frac{|b|M}{|1+b|\mathrm{\Gamma }(2-{\beta }_i)\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}),\end{array}\\ A=\frac{(1+2|a|)(1+(1+\alpha )|b|)}{|1+a||1+b|\mathrm{\Gamma }(\alpha +1)}+\sum _{i=1}^n(\frac{1}{\mathrm{\Gamma }(\alpha -{\beta }_i+1)}+\frac{|b|}{|1+b|\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(2-{\beta }_i)})\end{array}$$

and $M={({\int }_0^1{(m(t))}^{\frac{1}{l}}ds)}^l$. Note that $B_r$ is a closed, bounded and convex subset of the Banach space X. For each $u\in B_r$, we have

$$\begin{array}{rcl}|(Fu)(t)|& =& |{\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))ds\\ -\frac{a}{(1+a)}{\int }_0^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}\\ \times f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))ds\\ +\frac{ab-b(1+a)t}{(1+a)(1+b)}{\int }_0^1\frac{{(1-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\\ \times f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))ds|\\ \le & {\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}\left|f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))\right|ds\\ +\frac{|a|}{|1+a|}{\int }_0^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}\\ \times \left|f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))\right|ds\\ +\frac{|b|(1+2|a|)}{|1+a||1+b|}{\int }_0^1\frac{{(1-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\\ \times \left|f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))\right|ds\\ \le & {\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}m(s)ds\\ +(d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n}){\int }_0^t\frac{{(t-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}ds\\ +\frac{|a|}{|1+a|}{\int }_0^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}m(s)ds\\ +\frac{|a|}{|1+a|}(d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n})\\ \times {\int }_0^1\frac{{(1-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}ds+\frac{|b|(1+2|a|)}{|1+a||1+b|}{\int }_0^1\frac{{(1-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}m(s)ds\\ +\frac{|b|(1+2|a|)}{|1+a||1+b|}(d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n})\\ \times {\int }_0^1\frac{{(1-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}ds\\ \le & \frac{1}{\mathrm{\Gamma }(\alpha )}{\left({\int }_0^t{\left({(t-s)}^{\alpha -1}\right)}^{\frac{1}{1-l}}ds\right)}^{1-l}{\left({\int }_0^t{(m(s))}^{\frac{1}{l}}ds\right)}^l\\ +\frac{|a|}{|1+a|\mathrm{\Gamma }(\alpha )}{\left({\int }_0^1{\left({(1-s)}^{\alpha -1}\right)}^{\frac{1}{1-l}}ds\right)}^{1-l}{\left({\int }_0^1{(m(s))}^{\frac{1}{l}}ds\right)}^l\\ +\frac{|b|(1+2|a|)}{|1+a||1+b|\mathrm{\Gamma }(\alpha -1)}{\left({\int }_0^1{\left({(1-s)}^{\alpha -2}\right)}^{\frac{1}{1-l}}ds\right)}^{1-l}{\left({\int }_0^1{(m(s))}^{\frac{1}{l}}ds\right)}^l\\ +\frac{(1+2|a|)(1+(1+\alpha )|b|)}{|1+a||1+b|\mathrm{\Gamma }(\alpha +1)}\\ \times (d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n})\\ \le & \frac{(1+2|a|)M}{|1+a|\mathrm{\Gamma }(\alpha )}{\left(\frac{1-l}{\alpha -l}\right)}^{1-l}+\frac{|b|(1+2|a|)M}{|1+a||1+b|\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\frac{(1+2|a|)(1+(1+\alpha )|b|)}{|1+a||1+b|\mathrm{\Gamma }(\alpha +1)}\\ \times (d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n}).\end{array}$$

Also, we have

$$\begin{array}{rcl}{|}^c{D}^{{\beta }_i}(Fu)(t)|& =& |{\int }_0^t\frac{{(t-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}{(Fu)}^{\prime }(s)ds|\\ =& \left|{\int }_0^t\frac{{(t-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}\right({\int }_0^s\frac{{(s-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau \\ -\frac{b}{1+b}{\int }_0^1\frac{{(1-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ),\\ {}^{c}D^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau )\left)d\tau \right)ds|\\ \le & {\int }_0^t\frac{{(t-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}\left({\int }_0^s\frac{{(s-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\right|f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ),\\ {}^{c}D^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau )\left)\right|d\tau )ds\\ +\frac{|b|}{|1+b|}{\int }_0^t\frac{{(t-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}\left({\int }_0^1\frac{{(1-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\right|f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ),\\ {}^{c}D^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau )\left)\right|d\tau )ds\\ \le & {\int }_0^t\frac{{(t-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}({\int }_0^s\frac{{(s-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}m(\tau )d\tau )ds\\ +(d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n})\\ \times {\int }_0^t\frac{{(t-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}\left({\int }_0^s\frac{{(s-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}d\tau \right)ds\\ +\frac{|b|}{|1+b|}{\int }_0^t\frac{{(t-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}({\int }_0^1\frac{{(1-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}m(\tau )d\tau )ds\\ +\frac{|b|}{|1+b|}(d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n})\\ \times {\int }_0^t\frac{{(t-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}\left({\int }_0^1\frac{{(1-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}d\tau \right)ds\\ \le & \frac{1}{\mathrm{\Gamma }(\alpha -1)\mathrm{\Gamma }(1-{\beta }_i)}{\int }_0^t{(t-s)}^{-{\beta }_i}\\ \times \left[{\left({\int }_0^s{\left({(s-\tau )}^{\alpha -2}\right)}^{\frac{1}{1-l}}d\tau \right)}^{1-l}{\left({\int }_0^s{(m(\tau ))}^{\frac{1}{l}}d\tau \right)}^l\right]ds\\ +\frac{(d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n})}{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(1-{\beta }_i)}\\ \times {\int }_0^t{(t-s)}^{-{\beta }_i}{s}^{\alpha -1}ds+\frac{|b|}{|1+b|\mathrm{\Gamma }(\alpha -1)\mathrm{\Gamma }(1-{\beta }_i)}\\ \times {\int }_0^t{(t-s)}^{-{\beta }_i}\left[{\left({\int }_0^1{\left({(1-\tau )}^{\alpha -2}\right)}^{\frac{1}{1-l}}d\tau \right)}^{1-l}{\left({\int }_0^1{(m(\tau ))}^{\frac{1}{l}}d\tau \right)}^l\right]ds\\ +\frac{|b|(d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n})}{|1+b|\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(2-{\beta }_i)}\\ \le & \frac{M}{\mathrm{\Gamma }(\alpha -1)\mathrm{\Gamma }(1-{\beta }_i)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}{\int }_0^t{(t-s)}^{-{\beta }_i}{s}^{\alpha -l-1}ds\\ +\frac{(d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n})}{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(1-{\beta }_i)}\\ \times {\int }_0^t{(t-s)}^{-{\beta }_i}{s}^{\alpha -1}ds\\ +\frac{|b|M}{|1+b|\mathrm{\Gamma }(\alpha -1)\mathrm{\Gamma }(1-{\beta }_i)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}{\int }_0^t{(t-s)}^{-{\beta }_i}ds\\ +\frac{|b|(d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n})}{|1+b|\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(2-{\beta }_i)}\\ \le & \frac{M}{\mathrm{\Gamma }(\alpha -1)\mathrm{\Gamma }(1-{\beta }_i)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}{\int }_0^1{(1-\xi )}^{-{\beta }_i}{\xi }^{\alpha -l-1}d\xi \\ +\frac{(d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n})}{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(1-{\beta }_i)}\\ \times {\int }_0^1{(1-\xi )}^{-{\beta }_i}{\xi }^{\alpha -1}d\xi \\ +\frac{|b|M}{|1+b|\mathrm{\Gamma }(\alpha -1)\mathrm{\Gamma }(2-{\beta }_i)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\frac{|b|(d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n})}{|1+b|\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(2-{\beta }_i)}.\end{array}$$

Since $B(\alpha -l,1-{\beta }_i)={\int }_0^1{(1-\xi )}^{-{\beta }_i}{\xi }^{\alpha -l-1}d\xi =\frac{\mathrm{\Gamma }(\alpha -l)\mathrm{\Gamma }(1-{\beta }_i)}{\mathrm{\Gamma }(\alpha -{\beta }_i-l+1)}$ and, on the other hand, $B(\alpha ,1-{\beta }_i)={\int }_0^1{(1-\xi )}^{-{\beta }_i}{\xi }^{\alpha -1}d\xi =\frac{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(1-{\beta }_i)}{\mathrm{\Gamma }(\alpha -{\beta }_i+1)}$, we conclude that

$$\begin{array}{rcl}{|}^c{D}^{{\beta }_i}(Fu)(t)|& \le & \frac{\mathrm{\Gamma }(\alpha -l)M}{\mathrm{\Gamma }(\alpha -1)\mathrm{\Gamma }(\alpha -{\beta }_i-l+1)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\frac{|b|M}{|1+b|\mathrm{\Gamma }(\alpha -1)\mathrm{\Gamma }(2-{\beta }_i)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\frac{d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n}}{\mathrm{\Gamma }(\alpha -{\beta }_i+1)}\\ +\frac{|b|(d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n})}{|1+b|\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(2-{\beta }_i)}\end{array}$$

for all $i=1,2,\dots ,n$. Thus,

$$\begin{array}{rcl}\parallel Fu\parallel & \le & \frac{(1+2|a|)M}{|1+a|\mathrm{\Gamma }(\alpha )}{\left(\frac{1-l}{\alpha -l}\right)}^{1-l}+\frac{|b|(1+2|a|)M}{|1+a||1+b|\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\sum _{i=1}^n[\frac{\mathrm{\Gamma }(\alpha -l)M}{\mathrm{\Gamma }(\alpha -1)\mathrm{\Gamma }(\alpha -{\beta }_i-l+1)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}\\ +\frac{|b|M}{|1+b|\mathrm{\Gamma }(\alpha -1)\mathrm{\Gamma }(2-{\beta }_i)}{\left(\frac{1-l}{\alpha -l-1}\right)}^{1-l}]\\ +(d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n})\\ \times (\frac{(1+2|a|)(1+(1+\alpha )|b|)}{|1+a||1+b|\mathrm{\Gamma }(\alpha +1)}+\sum _{i=1}^n[\frac{1}{\mathrm{\Gamma }(\alpha -{\beta }_i+1)}+\frac{|b|}{|1+b|\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(2-{\beta }_i)}])\\ =& K+(d{r}^{\rho }+d^{\prime }{\gamma }_0^{p^{\prime }}{r}^{{\rho }^{\prime }}+d^{″}{\lambda }_0^{p^{″}}{r}^{{\rho }^{″}}+d_1{r}^{{\rho }_1}+d_2{r}^{{\rho }_2}+\cdots +d_n{r}^{{\rho }_n})A\\ \le & \frac{r}{n+4}\times (n+4)=r.\end{array}$$

Hence, F maps $B_r$ into $B_r$. Now, suppose that f satisfy the second condition. In this case, choose

$$\begin{array}{rcl}0& <& r\\ \le & min\{{\left(\frac{1}{(n+3)Ad}\right)}^{\frac{1}{\rho -1}},{\left(\frac{1}{(n+3)A{d}^{\prime }{\gamma }_0^{p^{\prime }}}\right)}^{\frac{1}{{\rho }^{\prime }-1}},{\left(\frac{1}{(n+3)A{d}^{″}{\lambda }_0^{p^{″}}}\right)}^{\frac{1}{{\rho }^{″}-1}},\\ {\left(\frac{1}{(n+3)A{d}_1}\right)}^{\frac{1}{{\rho }_1-1}},{\left(\frac{1}{(n+3)A{d}_2}\right)}^{\frac{1}{{\rho }_2-1}},\dots ,{\left(\frac{1}{(n+3)A{d}_n}\right)}^{\frac{1}{{\rho }_n-1}}\}.\end{array}$$

By using similar arguments, one can show that $\parallel Fu\parallel \le \frac{r}{n+3}\times (n+3)=r$, and so F maps $B_r$ into $B_r$. Since f is continuous, it is easy to get that F is also continuous. Now, we show that F is completely continuous operator on $B_r$. For each $u\in B_r$, put

$$N=\underset{t\in I}{max}\left\{f(t,u(t),(\varphi u)(t),(\psi u)(t){,}^c{D}^{{\beta }_1}u(t){,}^c{D}^{{\beta }_2}u(t),\dots {,}^c{D}^{{\beta }_n}u(t))\right\}+1.$$

For each $t_1,t_2\in I$ with $t_1<t_2$, we have

$$\begin{array}{c}|(Fu)(t_2)-(Fu)(t_1)|\hfill \\ =|{\int }_0^{t_2}\frac{{(t_2-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))ds\hfill \\ -{\int }_0^{t_1}\frac{{(t_1-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))ds\hfill \\ +\frac{b(t_1-t_2)}{1+b}{\int }_0^1\frac{{(1-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))ds|\hfill \\ =|{\int }_0^{t_1}\frac{{(t_2-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))ds\hfill \\ +{\int }_{t_1}^{t_2}\frac{{(t_2-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))ds\hfill \\ -{\int }_0^{t_1}\frac{{(t_1-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))ds\hfill \\ +\frac{b(t_1-t_2)}{1+b}{\int }_0^1\frac{{(1-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))ds|\hfill \\ \le {\int }_0^{t_1}\frac{{(t_2-s)}^{\alpha -1}-{(t_1-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}\hfill \\ \times \left|f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))\right|ds\hfill \\ +{\int }_{t_1}^{t_2}\frac{{(t_2-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}\left|f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))\right|ds\hfill \\ +\frac{|b|(t_2-t_1)}{|1+b|}{\int }_0^1\frac{{(1-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times \left|f(s,u(s),(\varphi u)(s),(\psi u)(s){,}^c{D}^{{\beta }_1}u(s){,}^c{D}^{{\beta }_2}u(s),\dots {,}^c{D}^{{\beta }_n}u(s))\right|ds\hfill \\ \le N{\int }_0^{t_1}\frac{{(t_2-s)}^{\alpha -1}-{(t_1-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}ds+N{\int }_{t_1}^{t_2}\frac{{(t_2-s)}^{\alpha -1}}{\mathrm{\Gamma }(\alpha )}ds\hfill \\ +\frac{N|b|(t_2-t_1)}{|1+b|}{\int }_0^1\frac{{(1-s)}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}ds\hfill \\ =\frac{N}{\mathrm{\Gamma }(\alpha +1)}(t_2^{\alpha }-t_1^{\alpha })+\frac{N|b|}{|1+b|\mathrm{\Gamma }(\alpha )}(t_2-t_1).\hfill \end{array}$$

On the other hand, for each $i\in \{1,2,\dots ,n\}$, we have

$$\begin{array}{c}{|}^c{D}^{{\beta }_i}(Fu)(t_2){-}^c{D}^{{\beta }_i}(Fu)(t_1)|\hfill \\ =|{\int }_0^{t_2}\frac{{(t_2-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}{(Fu)}^{\prime }(s)ds-{\int }_0^{t_1}\frac{{(t_1-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}{(Fu)}^{\prime }(s)ds|\hfill \\ =\left|{\int }_0^{t_2}\frac{{(t_2-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}\right({\int }_0^s\frac{{(s-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau \hfill \\ -\frac{b}{1+b}{\int }_0^1\frac{{(1-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau )ds\hfill \\ -{\int }_0^{t_1}\frac{{(t_1-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}({\int }_0^s\frac{{(s-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau \hfill \\ -\frac{b}{1+b}{\int }_0^1\frac{{(1-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau \left)ds\right|\hfill \\ =\left|{\int }_0^{t_1}\frac{{(t_2-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}\right({\int }_0^s\frac{{(s-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau \hfill \\ -\frac{b}{1+b}{\int }_0^1\frac{{(1-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau )ds\hfill \\ +{\int }_{t_1}^{t_2}\frac{{(t_2-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}({\int }_0^s\frac{{(s-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau \hfill \\ -\frac{b}{1+b}{\int }_0^1\frac{{(1-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau )ds\hfill \\ -{\int }_0^{t_1}\frac{{(t_1-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}({\int }_0^s\frac{{(s-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau \hfill \\ -\frac{b}{1+b}{\int }_0^1\frac{{(1-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau \left)ds\right|\hfill \\ =|{\int }_0^{t_1}\frac{{(t_2-s)}^{-{\beta }_i}-{(t_1-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}\hfill \\ \times \left({\int }_0^s\frac{{(s-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}f\right(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ),\hfill \\ {}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau \hfill \\ -\frac{b}{1+b}{\int }_0^1\frac{{(1-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau )ds\hfill \\ +{\int }_{t_1}^{t_2}\frac{{(t_2-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}({\int }_0^s\frac{{(s-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau \hfill \\ -\frac{b}{1+b}{\int }_0^1\frac{{(1-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))d\tau \left)ds\right|\hfill \\ \le {\int }_0^{t_1}\frac{{(t_1-s)}^{-{\beta }_i}-{(t_2-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}\hfill \\ \times \left({\int }_0^s\frac{{(s-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\right|f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ),\hfill \\ {}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau )\left)\right|d\tau )ds\hfill \\ +\frac{|b|}{|1+b|}{\int }_0^{t_1}\frac{{(t_1-s)}^{-{\beta }_i}-{(t_2-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}\hfill \\ \times \left({\int }_0^1\frac{{(1-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\right|f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ),\hfill \\ {}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau )\left)\right|d\tau )ds\hfill \\ +{\int }_{t_1}^{t_2}\frac{{(t_2-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}({\int }_0^s\frac{{(s-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times \left|f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))\right|d\tau )ds\hfill \\ +\frac{|b|}{|1+b|}{\int }_{t_1}^{t_2}\frac{{(t_2-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}({\int }_0^1\frac{{(1-\tau )}^{\alpha -2}}{\mathrm{\Gamma }(\alpha -1)}\hfill \\ \times \left|f(\tau ,u(\tau ),(\varphi u)(\tau ),(\psi u)(\tau ){,}^c{D}^{{\beta }_1}u(\tau ){,}^c{D}^{{\beta }_2}u(\tau ),\dots {,}^c{D}^{{\beta }_n}u(\tau ))\right|d\tau )ds\hfill \\ \le \frac{N}{\mathrm{\Gamma }(\alpha )}{\int }_0^{t_1}\frac{{(t_1-s)}^{-{\beta }_i}-{(t_2-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}{s}^{\alpha -1}ds\hfill \\ +\frac{N|b|}{|1+b|\mathrm{\Gamma }(\alpha )}{\int }_0^{t_1}\frac{{(t_1-s)}^{-{\beta }_i}-{(t_2-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}ds\hfill \\ +\frac{N}{\mathrm{\Gamma }(\alpha )}{\int }_{t_1}^{t_2}\frac{{(t_2-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}{s}^{\alpha -1}ds+\frac{N|b|}{|1+b|\mathrm{\Gamma }(\alpha )}{\int }_{t_1}^{t_2}\frac{{(t_2-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}ds\hfill \\ \le \frac{(1+2|b|)N}{|1+b|\mathrm{\Gamma }(\alpha )}{\int }_0^{t_1}\frac{{(t_1-s)}^{-{\beta }_i}-{(t_2-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}ds\hfill \\ +\frac{(1+2|b|)N}{|1+b|\mathrm{\Gamma }(\alpha )}{\int }_{t_1}^{t_2}\frac{{(t_2-s)}^{-{\beta }_i}}{\mathrm{\Gamma }(1-{\beta }_i)}ds\hfill \\ \le \frac{(1+2|b|)N}{|1+b|\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(2-{\beta }_i)}[(t_2^{1-{\beta }_i}-t_1^{1-{\beta }_i})+2{(t_2-t_1)}^{1-{\beta }_i}].\hfill \end{array}$$

Hence,

$$\begin{array}{c}\parallel Fu(t_2)-Fu(t_1)\parallel \hfill \\ \le \frac{N}{\mathrm{\Gamma }(\alpha +1)}(t_2^{\alpha }-t_1^{\alpha })+\frac{N|b|}{|1+b|\mathrm{\Gamma }(\alpha )}(t_2-t_1)\hfill \\ +\sum _{i=1}^n\frac{(1+2|b|)N}{|1+b|\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(2-{\beta }_i)}[(t_2^{1-{\beta }_i}-t_1^{1-{\beta }_i})+2{(t_2-t_1)}^{1-{\beta }_i}],\hfill \end{array}$$

which implies that $\parallel Fu(t_2)-Fu(t_1)\parallel \to 0$ as $t_1\to t_2$. Thus, F is equi-continuous and uniformly bounded. By using the Arzela-Ascoli theorem, one can get that F is completely continuous. Now, by using Theorem 2.5, F has a fixed point in $B_r$. Therefore, problem (1) has a solution. □

Corollary 2.7 Let $f:[0,1]\times {\mathbb{R}}^{n+3}\to \mathbb{R}$ be a continuous function such that there exists a constant $l\in (0,\alpha -1)$ and a real-valued function $m(t)\in L^{\frac{1}{l}}([0,1],(0,\mathrm{\infty }))$ such that $|f(t,x,y,w,u_1,u_2,\dots ,u_n)|\le m(t)$ for all $t\in [0,1]$ and $x,y,w,u_1,u_2,\dots ,u_n\in \mathbb{R}$. Then problem (1) has a solution.

Finally, we should emphasize the importance of anti-periodic conditions, which appear in the case that $a=1$ and $b=1$ in the problem. For more details about this note, one can review the papers [24] and [25].

2.3 Examples

Example 2.1 Consider the boundary value problem

$$\{\begin{array}{c}{}^{c}D^{\frac{7}{4}}u(t)=\frac{e^{-\pi t}}{24\sqrt{\pi }+e^{-\pi t}}[\frac{sint+e^t}{1+t^3}+\frac{|u(t)|}{1+|u(t)|}+\frac{e^{-\pi t}cos\pi t}{1+t^2}(1+\frac{|(\varphi u)(t){+}^c{D}^{\frac{1}{2}}u(t)|}{1+|(\varphi u)(t){+}^c{D}^{\frac{1}{2}}u(t)|})\hfill \\ \phantom{{}^{c}D^{\frac{7}{4}}u(t)=}+\frac{1+{sin}^2\pi t}{2(t^{\frac{3}{2}}+4)}((\psi u)(t)+\frac{{|}^c{D}^{\frac{3}{4}}u(t)|}{1+{|}^c{D}^{\frac{3}{4}}u(t)|})],\hfill \\ u(0)+u(1)=0,u^{\prime }(0)+u^{\prime }(1)=0,\hfill \end{array}$$

(2)

where $(\varphi u)(t)={\int }_0^t\frac{e^{-(s-t)}}{8}u(s)ds$ and $(\psi u)(t)={\int }_0^t\frac{e^{-(s-t)/2}}{8}u(s)ds$ with ${\gamma }_0=\frac{e-1}{8}$ and ${\lambda }_0=\frac{\sqrt{e}-1}{4}$. Then, we have

$$\begin{array}{c}|f(t,u(t),(\varphi u)(t),(\psi u)(t){,}^c{D}^{\frac{1}{2}}u(t){,}^c{D}^{\frac{3}{4}}u(t))\hfill \\ -f(t,v(t),(\varphi v)(t),(\psi v)(t){,}^c{D}^{\frac{1}{2}}v(t){,}^c{D}^{\frac{3}{4}}v(t))|\hfill \\ \le \frac{1}{24\sqrt{\pi }}(|u(t)-v(t)|+|(\varphi u)(t)-(\varphi v)(t)|\hfill \\ +|(\psi u)(t)-(\psi v)(t)|+{|}^c{D}^{\frac{1}{2}}u(t){-}^c{D}^{\frac{1}{2}}v(t)|\hfill \\ +{|}^c{D}^{\frac{3}{4}}u(t){-}^c{D}^{\frac{3}{4}}v(t)\left|\right).\hfill \end{array}$$

Put $\mu (t)=\frac{1}{24\sqrt{\pi }}\in L^4([0,1],(0,\mathrm{\infty }))$, $\kappa =\frac{1}{4}$ and ${\mu }^{\ast }={({\int }_0^1{(\frac{1}{24\sqrt{\pi }})}^{4}ds)}^{\frac{1}{4}}=\frac{1}{24\sqrt{\pi }}$. Since $\mathrm{\Gamma }(\frac{3}{2})=\frac{\sqrt{\pi }}{2}$, $\mathrm{\Gamma }(\frac{3}{4})\approx 1/2\text{,}254$, $\mathrm{\Gamma }(\frac{5}{4})\approx 0/9\text{,}064$ and $\mathrm{\Gamma }(\frac{7}{4})\approx 0/9\text{,}191$, we have

$$\begin{array}{c}(1+{\gamma }_0+{\lambda }_0)[\frac{3{\mu }^{\ast }}{2\mathrm{\Gamma }(\alpha )}{\left(\frac{1-\kappa }{\alpha -\kappa }\right)}^{1-\kappa }+\frac{{\mu }^{\ast }}{4\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }\hfill \\ +\frac{\mathrm{\Gamma }(\alpha -\kappa ){\mu }^{\ast }}{\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }(\frac{1}{\mathrm{\Gamma }(\alpha -{\beta }_1-\kappa +1)}+\frac{1}{\mathrm{\Gamma }(\alpha -{\beta }_2-\kappa +1)})\hfill \\ +\frac{{\mu }^{\ast }}{2\mathrm{\Gamma }(\alpha -1)}{\left(\frac{1-\kappa }{\alpha -\kappa -1}\right)}^{1-\kappa }(\frac{1}{\mathrm{\Gamma }(2-{\beta }_1)}+\frac{1}{\mathrm{\Gamma }(2-{\beta }_2)})]\hfill \\ =(1+\frac{e-1}{8}+\frac{\sqrt{e}-1}{4})[\frac{1}{16\sqrt{\pi }\mathrm{\Gamma }(\frac{7}{4})}{\left(\frac{1}{2}\right)}^{\frac{3}{4}}+\frac{1}{96\sqrt{\pi }\mathrm{\Gamma }(\frac{3}{4})}{\left(\frac{3}{2}\right)}^{\frac{3}{4}}\hfill \\ +\frac{\mathrm{\Gamma }(\frac{3}{2})}{24\sqrt{\pi }\mathrm{\Gamma }(\frac{3}{4})}{\left(\frac{3}{2}\right)}^{\frac{3}{4}}(\frac{1}{\mathrm{\Gamma }(2)}+\frac{1}{\mathrm{\Gamma }(\frac{7}{4})})+\frac{1}{48\sqrt{\pi }\mathrm{\Gamma }(\frac{3}{4})}{\left(\frac{3}{2}\right)}^{\frac{3}{4}}(\frac{1}{\mathrm{\Gamma }(\frac{3}{2})}+\frac{1}{\mathrm{\Gamma }(\frac{5}{4})})]\hfill \\ \approx 0/1\text{,}466<1.\hfill \end{array}$$

Thus, by using Theorem 2.3, the boundary value problem (2) has a unique solution.

Example 2.2 Consider the boundary value problem

$$\{\begin{array}{c}{}^{c}D^{\alpha }u(t)=\frac{\lambda e^{-\pi t}}{\sqrt{1+t^2}}+\frac{cos\pi t}{\sqrt{\pi +|u(t){|+|}^c{D}^{\frac{1}{2}}u(t)|}}{(u(t))}^{{\sigma }_1}+\frac{e^{-\pi t}(1+{sin}^{2}u(t))}{{(t+6)}^3}{((\varphi u)(t))}^{{\sigma }_2}\hfill \\ \phantom{{}^{c}D^{\alpha }u(t)=}+\frac{tu(t)}{(4+t^2)(1+|u(t)|)}{((\psi u)(t))}^{{\sigma }_3}\hfill \\ \phantom{{}^{c}D^{\alpha }u(t)=}+\frac{(1+\alpha ){(t-\frac{1}{2})}^2}{\mathrm{\Gamma }(\alpha )(1+|u(t){+}^c{D}^{\frac{3}{2}}u(t)|)}{\sum }_{k=1}^4(\frac{sink\pi t}{2^k}){({}^{c}D^{{\beta }_k}u(t))}^{{\delta }_k},\hfill \\ u(0)+\frac{1}{2}u(1)=0,u^{\prime }(0)+\frac{2}{3}{u}^{\prime }(1)=0,\hfill \end{array}$$

(3)

where $\alpha =\frac{9}{5}$, ${\beta }_1=\frac{1}{2}$, ${\beta }_2=\frac{4}{5}$, ${\beta }_3=\frac{2}{3}$, ${\beta }_4=\frac{1}{7}$, $\lambda \in [0,\mathrm{\infty })$, $(\varphi u)(t)={\int }_0^t\frac{s^2{e}^{-(s-t)/2}}{s^3+4}u(s)ds$ and $(\psi u)(t)={\int }_0^t\frac{{(t-s)}^5}{\sqrt{1+s^2}}u(s)ds$. Then, we have

$$\begin{array}{c}\left|f(t,u(t),(\varphi u)(t),(\psi u)(t){,}^c{D}^{{\beta }_1}u(t){,}^c{D}^{{\beta }_2}u(t){,}^c{D}^{{\beta }_3}u(t){,}^c{D}^{{\beta }_4}u(t))\right|\hfill \\ \le m(t)+\frac{1}{\sqrt{\pi }}{|u(t)|}^{{\sigma }_1}+\frac{1}{108}{|(\varphi u)(t)|}^{{\sigma }_2}+\frac{1}{4}{|(\psi u)(t)|}^{{\sigma }_3}+\sum _{k=1}^4\frac{(1+\alpha )}{\mathrm{\Gamma }(\alpha )2^{k+2}}{|}^c{D}^{{\beta }_k}u(t){|}^{{\delta }_k},\hfill \end{array}$$

where $m(t)=\frac{\lambda e^{-\pi t}}{\sqrt{1+t^2}}$ for all $t\in [0,1]$. If $0<{\sigma }_j<1$, $0<{\delta }_i<1$ for $j=1,2,3$ and $i=1,2,3,4$, then assumption (∗) holds. If $\lambda =0$ and ${\sigma }_j,{\delta }_i>1$ for $j=1,2,3$ and $i=1,2,3,4$, then the second condition of Theorem 2.6 holds. Thus, by using Theorem 2.6, the boundary value problem (3) has a solution.

3 Conclusions

Fractional nonlinear differential equations, fractional integro-differential equations and their applications represent a topic of high interest in the area of fractional calculus and its applications in various fields of science and engineering [7]. In this article, based on the Schauder’s fixed point theorem, we have proved some existence results for the multi-term nonlinear fractional integro-differential equation (1).