1 Introduction

The aim of this paper is to investigate the existence and uniqueness of solutions for a nonlinear fractional differential equation (FDE) involving the Caputo fractional derivative of a function x with respect to another function \(\psi \):

$$\begin{aligned} \left\{ \begin{array}{l} {^CD_{a+}^{\alpha ,\psi }}x(t)=f(t,x(t)), \quad t\in [a,b]\\ x(a)=x_a\\ x'(a)=x^1_a\\ x(b)=K{I_{a+}^{\alpha ,\psi }}x(v), \end{array} \right. \end{aligned}$$
(1)

where \(\alpha \in (2,3)\), \(x_a,x^1_a,K,v\in \mathbb R\) with \(v\in (a,b]\), \(x\in C^2[a,b]\), \({^CD_{a+}^{\alpha ,\psi }}x\) is the \(\psi \)-Caputo fractional derivative of x (see (3)) and \(f:[a,b]\times \mathbb R\rightarrow \mathbb R\) is a continuous function. We assume that the restriction

$$\begin{aligned} N:=\frac{2K}{\Gamma (3+\alpha )}(\psi (v)-\psi (a))^{2+\alpha }-(\psi (b)-\psi (a))^2\not =0 \end{aligned}$$
(2)

is satisfied.

Fractional calculus can be seen as a generalization of ordinary calculus, in the sense that derivatives and integrals of arbitrary real or complex order are defined [13, 17, 19]. Given a function x with domain [a, b], the Riemann–Liouville fractional integral of x of order \(\alpha >0\) is given by the formula

$$\begin{aligned} {I_{a+}^{\alpha }} x(t):=\frac{1}{\Gamma (\alpha )}\int _a^t (t-\tau )^{\alpha -1}x(\tau )\,\hbox {d}\tau . \end{aligned}$$

When \(\alpha \) is an integer number, the fractional integral is simply an \(n-\)tuple integral. With respect to differentiation, two of the most common definitions are the Riemann–Liouville fractional derivative

$$\begin{aligned} {D_{a+}^{\alpha }} x(t):= \left( \frac{\hbox {d}}{\hbox {d}t}\right) ^n {I_{a+}^{n-\alpha }}x(t) \end{aligned}$$

and the Caputo fractional derivative

$$\begin{aligned} {^CD_{a+}^{\alpha }} x(t):={D_{a+}^{\alpha }} \left[ x(t)-\sum _{k=0}^{n-1}\frac{x^{(k)}(a)}{k!}(t-a)^k\right] , \end{aligned}$$

where \(n=[\alpha ]+1\). If \(\alpha \in \mathbb N\), then \({D_{a+}^{\alpha }} x(t)={^CD_{a+}^{\alpha }} x(t)= x^{(n)}(t)\).

However, these are not the only definitions, and we can find many more in the literature. To overcome this problem, one solution is to present general definitions and then study some of their properties. This can be done, e.g., considering fractional derivatives and fractional integrals with respect to another function \(\psi \) [13, Section 2.5], and for particular choices of \(\psi \), we recover, e.g., the Riemann–Liouville, the Hadamard, and the Erdélyi–Kober fractional derivatives and fractional integrals. In [5], a Caputo-type fractional derivative of a function with respect to another function is presented, and some of its main properties were studied. FDE is a branch of mathematics, where in the differential equations, ordinary derivatives are replaced by fractional operators (see the seminal papers [1, 8, 14]). Since there are several definitions for fractional derivatives, we find in the literature several papers dealings with similar subjects for different operators: the Riemann–Liouville [3, 7, 23], the Caputo [2, 18, 22], the Hadamard [15, 21], the Erdelyi–Kober [6, 20], etc. Fractional differential equations have been used to model more efficiently some real-life phenomena like viscoelasticity [12], electrical circuits [16], electromagnetism [10], in chemistry [9], and sound propagation [11] and has attracted the attention of numerous researchers.

In this paper, we continue the work started in [5], by considering a FDE of order \(\alpha \in (2,3)\), with mixed boundary conditions. The paper is organized as follows. Section 2 presents the notions that we will need, as well as some basic properties of the operators, and Sect. 3 presents the two results on the existence of solution for the FDE (1), using the Banach and the Leray–Schauder fixed-point theorems.

2 Review on Fractional Calculus

We deal with fractional derivatives and fractional integrals with respect to another function. To fix terminology, a real \(\alpha >0\) is the order of the fractional operators, \(x:[a,b]\rightarrow \mathbb R\) is an integrable function, and \(\psi :[a,b]\rightarrow \mathbb R\) is an increasing differentiable function such that \(\psi '(t)\not =0\), for all \(t\in [a,b]\).

The \(\psi \)-Riemann–Liouville fractional integral of x is given by

$$\begin{aligned} {I_{a+}^{\alpha ,\psi }}x(t):=\frac{1}{\Gamma (\alpha )}\int _a^t \psi '(\tau )(\psi (t)-\psi (\tau ))^{\alpha -1}x(\tau )\,\hbox {d}\tau , \end{aligned}$$

and the \(\psi \)-Riemann–Liouville fractional derivative of x by

$$\begin{aligned} {D_{a+}^{\alpha ,\psi }}x(t)&:= \left( \frac{1}{\psi '(t)}\frac{\hbox {d}}{\hbox {d}t}\right) ^n {I_{a+}^{n-\alpha ,\psi }}x(t)\\&=\frac{1}{\Gamma (n-\alpha )}\left( \frac{1}{\psi '(t)}\frac{\hbox {d}}{\hbox {d}t}\right) ^n\int _a^t \psi '(\tau )(\psi (t)-\psi (\tau ))^{n-\alpha -1}x(\tau )\,\hbox {d}\tau , \end{aligned}$$

where \(n=[\alpha ]+1\). The semigroup law is valid for fractional integrals, that is,

$$\begin{aligned} {I_{a+}^{\alpha ,\psi }}{I_{a+}^{\beta ,\psi }}x(t)={I_{a+}^{\alpha +\beta ,\psi }}x(t), \quad \forall \alpha ,\beta >0. \end{aligned}$$

The fractional derivative that we will deal in our work is a Caputo-type operator, and it is defined by the expression

$$\begin{aligned} {^CD_{a+}^{\alpha ,\psi }}x(t):={D_{a+}^{\alpha ,\psi }}\left[ x(t)-\sum _{k=0}^{n-1}\frac{x^{[k]}_\psi (a)}{k!}(\psi (t)-\psi (a))^k\right] , \end{aligned}$$
(3)

where \(x\in C^{n-1}[a,b]\), \(n=[\alpha ]+1\) for \(\alpha \notin \mathbb N\), \(n=\alpha \) for \(\alpha \in \mathbb N,\) and

$$\begin{aligned} x^{[k]}_\psi (t):=\left( \frac{1}{\psi '(t)}\frac{\hbox {d}}{\hbox {d}t}\right) ^kx(t). \end{aligned}$$

If x is of class \(C^n\), then the \(\psi \)-Caputo fractional derivative of x is given by (cf. [5, Theorem 3])

$$\begin{aligned} {^CD_{a+}^{\alpha ,\psi }}x(t)= {I_{a+}^{n-\alpha ,\psi }}\left( \frac{1}{\psi '(t)}\frac{\hbox {d}}{\hbox {d}t}\right) ^nx(t). \end{aligned}$$

Thus, if \(\alpha =m\in \mathbb N\), we have

$$\begin{aligned} {^CD_{a+}^{\alpha ,\psi }}x(t)=x^{[m]}_\psi (t), \end{aligned}$$

and for \(\alpha \notin \mathbb N\), we have

$$\begin{aligned} {^CD_{a+}^{\alpha ,\psi }}x(t)=\frac{1}{\Gamma (n-\alpha )}\int _a^t \psi '(\tau )(\psi (t)-\psi (\tau ))^{n-\alpha -1}x^{[n]}_\psi (\tau )\,\hbox {d}\tau . \end{aligned}$$

One important function in fractional calculus is the Mittag–Leffler function, which is a generalization of the factorial function to real numbers, and it is given by

$$\begin{aligned} E_\alpha (z):=\sum _{k=0}^\infty \frac{z^k}{\Gamma (k\alpha +1)}. \end{aligned}$$

The derivative of a power function is given by

$$\begin{aligned} {^CD_{a+}^{\alpha ,\psi }}(\psi (t)-\psi (a))^{\beta -1}=\frac{\Gamma (\beta )}{\Gamma (\beta -\alpha )}(\psi (t)-\psi (a))^{\beta -\alpha -1}, \end{aligned}$$

where \(\beta \in \mathbb R\) with \(\beta >n\), and for \(\lambda \in \mathbb R\), the derivative of the Mittag–Leffler function is

$$\begin{aligned} {^CD_{a+}^{\alpha ,\psi }}E_\alpha (\lambda (\psi (t)-\psi (a))^\alpha )=\lambda E_\alpha (\lambda (\psi (t)-\psi (a))^\alpha ). \end{aligned}$$

The relation between fractional integrals and fractional derivatives is given next.

Theorem 1

[4] Let \(x:[a,b]\rightarrow \mathbb R\) be a function.

  1. 1.

    If x is continuous, then

    $$\begin{aligned} {^CD_{a+}^{\alpha ,\psi }}{I_{a+}^{\alpha ,\psi }}x(t)=x(t). \end{aligned}$$
  2. 2.

    If x is of class \(C^{n-1}\), then

    $$\begin{aligned} {I_{a+}^{\alpha ,\psi }}{^CD_{a+}^{\alpha ,\psi }}x(t)=x(t)-\sum _{k=0}^{n-1}\frac{x^{[k]}_\psi (a)}{k!}(\psi (t)-\psi (a))^k. \end{aligned}$$

We also refer to [5], where Theorem 1 is formulated for another class of functions. For our work, we need the following result.

Theorem 2

For \(\beta >0\) and \(\lambda \in \mathbb R\setminus \{0\}\), we have the following two formulas:

$$\begin{aligned} {I_{a+}^{\alpha ,\psi }}(\psi (t)-\psi (a))^{\beta -1}=\frac{\Gamma (\beta )}{\Gamma (\beta +\alpha )}(\psi (t)-\psi (a))^{\beta +\alpha -1} \end{aligned}$$

and

$$\begin{aligned} {I_{a+}^{\alpha ,\psi }}E_\alpha (\lambda (\psi (t)-\psi (a))^\alpha )=\frac{1}{\lambda }\left( E_\alpha (\lambda (\psi (t)-\psi (a))^\alpha )-1\right) . \end{aligned}$$

Proof

Direct computations lead to

$$\begin{aligned}&{I_{a+}^{\alpha ,\psi }}(\psi (t)-\psi (a))^{\beta -1}\\&\quad = \frac{1}{\Gamma (\alpha )}\int _a^t \psi '(\tau )(\psi (t)-\psi (\tau ))^{\alpha -1}(\psi (\tau )-\psi (a))^{\beta -1}\,\hbox {d}\tau \\&\quad = \frac{1}{\Gamma (\alpha )}(\psi (t)-\psi (a))^{\alpha -1}\int _a^t\psi '(\tau )\left( 1-\frac{\psi (\tau )-\psi (a)}{\psi (t)-\psi (a)}\right) ^{\alpha -1}(\psi (\tau )\\&\qquad -\psi (a))^{\beta -1}\,\hbox {d}\tau \\&\quad =\frac{1}{\Gamma (\alpha )}(\psi (t)-\psi (a))^{\beta +\alpha -1}\int _0^1(1-u)^{\alpha -1}u^{\beta -1}\,\hbox {d}u\\&\quad =\frac{1}{\Gamma (\alpha )}(\psi (t)-\psi (a))^{\beta +\alpha -1}\frac{\Gamma (\beta )\Gamma (\alpha )}{\Gamma (\beta +\alpha )} =\frac{\Gamma (\beta )}{\Gamma (\beta +\alpha )}(\psi (t)-\psi (a))^{\beta +\alpha -1}. \end{aligned}$$

For the second formula,

$$\begin{aligned} {I_{a+}^{\alpha ,\psi }}E_\alpha (\lambda (\psi (t)-\psi (a))^\alpha )&= \sum _{k=0}^\infty \frac{\lambda ^k}{\Gamma (k\alpha +1)}{I_{a+}^{\alpha ,\psi }}(\psi (t)-\psi (a))^{k\alpha }\\&= \sum _{k=0}^\infty \frac{\lambda ^k}{\Gamma (k\alpha +\alpha +1)}(\psi (t)-\psi (a))^{k\alpha +\alpha }\\&=\frac{1}{\lambda }\sum _{k=1}^\infty \frac{\lambda ^k}{\Gamma (k\alpha +1)} (\psi (t)-\psi (a))^{k\alpha }\\&=\frac{1}{\lambda }\left( E_\alpha (\lambda (\psi (t)-\psi (a))^\alpha )-1\right) . \end{aligned}$$

\(\square \)

3 Main Results

Our next result establishes an equivalence between problem (1) and a Volterra integral equation.

Theorem 3

Problem (1) is equivalent to the integral equation

$$\begin{aligned} x(t)={I_{a+}^{\alpha ,\psi }}f(t,x(t))+x_a+\frac{x^1_a}{\psi '(a)}(\psi (t)-\psi (a))+\gamma _x(\psi (t)-\psi (a))^2, \quad t\in [a,b], \end{aligned}$$
(4)

where

$$\begin{aligned} \gamma _x:= & {} \Big [{I_{a+}^{\alpha ,\psi }}f(b,x(b))-K {I_{a+}^{2\alpha ,\psi }}f(v,x(v))+x_a\Big (1-\frac{K}{\Gamma (1+\alpha )}(\psi (v)-\psi (a))^\alpha \Big )\nonumber \\&\quad +\,\frac{x^1_a}{\psi '(a)}\Big (\psi (b)-\psi (a)-\frac{K}{\Gamma (2+\alpha )}(\psi (v)-\psi (a))^{1+\alpha }\Big )\Big ] / N.\end{aligned}$$
(5)

Proof

Since \({^CD_{a+}^{\alpha ,\psi }}x(t)=f(t,x(t))\), applying the fractional integral to both sides of the equation and using Theorem 1, we arrive at the relation

$$\begin{aligned} x(t)={I_{a+}^{\alpha ,\psi }}f(t,x(t))+\gamma _0+\gamma _1(\psi (t)-\psi (a))+\gamma _2(\psi (t)-\psi (a))^2, \end{aligned}$$

for some constants \(\gamma _0,\gamma _1,\gamma _2\). Since \(x(a)=x_a\) and \(x'(a)=x^1_a\), we deduce that \(\gamma _0=x_a\) and \(\gamma _1=x^1_a/\psi '(a)\). To determine \(\gamma _2\), first observe that

$$\begin{aligned} {I_{a+}^{\alpha ,\psi }}x(t)&= {I_{a+}^{\alpha ,\psi }}\Big ( {I_{a+}^{\alpha ,\psi }}f(t,x(t))+x_a+\frac{x^1_a}{\psi '(a)}(\psi (t)-\psi (a)){+}\gamma _2(\psi (t)-\psi (a))^2 \Big )\\&={I_{a+}^{2\alpha ,\psi }}f(t,x(t))+\frac{x_a}{\Gamma (1+\alpha )}(\psi (t)-\psi (a))^\alpha +\frac{x^1_a}{\psi '(a)\Gamma (2+\alpha )}(\psi (t)\\&\quad -\,\psi (a))^{1+\alpha }+\frac{2\gamma _2}{\Gamma (3+\alpha )}(\psi (t)-\psi (a))^{2+\alpha }.\\ \end{aligned}$$

Since \(x(b)=K{I_{a+}^{\alpha ,\psi }}x(v)\), we determine the value of \(\gamma _2\), thus ending the first part of the proof. For the converse, if x satisfies equation (4), applying the fractional derivative to both sides of the equation and attending that

$$\begin{aligned} {^CD_{a+}^{\alpha ,\psi }}x_a={^CD_{a+}^{\alpha ,\psi }}(\psi (t)-\psi (a))={^CD_{a+}^{\alpha ,\psi }}(\psi (t)-\psi (a))^2=0, \end{aligned}$$

we obtain that \({^CD_{a+}^{\alpha ,\psi }}x(t)=f(t,x(t))\). \(\square \)

Motivated by Theorem 3, we define the functional F by

$$\begin{aligned} F(x)(t):={I_{a+}^{\alpha ,\psi }}f(t,x(t))+x_a+\frac{x^1_a}{\psi '(a)}(\psi (t)-\psi (a))+\gamma _x(\psi (t)-\psi (a))^2, \end{aligned}$$
(6)

where \(\gamma _x\) is given by (5). In the following, given a function \(y:[c,d]\rightarrow \mathbb R\), we consider the usual norm \(\Vert y\Vert :=\sup _{t\in [c,d]}|y(t)|.\)

Theorem 4

If there exists an integrable function \(L:[a,b]\rightarrow \mathbb R^+_0\) such that

$$\begin{aligned} |f(t,x_1)-f(t,x_2)|\le L(t)|x_1-x_2|, \quad \forall t\in [a,b], \, \forall x_1,x_2\in \mathbb R, \end{aligned}$$

there exists then a real \(h>0\) (with \(h\le b-a\)) for which the FDE (1) has a unique solution in \([a,a+h]\).

Proof

Let \(h>0\) be such that

$$\begin{aligned} \sup _{t\in [a,a+h]}\left\{ {I_{a+}^{\alpha ,\psi }}L(t)+\Big [{I_{a+}^{\alpha ,\psi }}L(b)+K{I_{a+}^{2\alpha ,\psi }}L(v)\Big ]\frac{(\psi (t)-\psi (a))^2}{N} \right\} <1, \end{aligned}$$
(7)

and U the set

$$\begin{aligned} U:=\{x\in C^2[a,a+h]: \, {^CD_{a+}^{\alpha ,\psi }}x(t) \, \text{ exists } \text{ and } \text{ is } \text{ continous } \text{ in } [a,a+h]\}. \end{aligned}$$

It is enough to prove that \(F:U\rightarrow U\) is a contraction. Let us see that F is well defined, that is, \(F(U)\subseteq U\). Given a function \(x\in U\), we have that the map \(t\mapsto F(x)(t)\) is of class \(C^2\) and

$$\begin{aligned} {^CD_{a+}^{\alpha ,\psi }}F(x)(t)=f(t,x(t)) \end{aligned}$$

is continuous. Now, let \(x_1,x_2\in U\) be arbitrary. Then,

$$\begin{aligned} \Vert F(x_1)-F(x_2)\Vert&\le \sup _{t\in [a,a+h]}\Big \{{I_{a+}^{\alpha ,\psi }}|f(t,x_1(t))-f(t,x_2(t))| + \Big [{I_{a+}^{\alpha ,\psi }}|f(b,x_1(b))\\&\quad -\,f(b,x_2(b))|\\&\quad +\,K {I_{a+}^{2\alpha ,\psi }}|f(v,x_1(v))-f(v,x_2(v))|\Big ]\frac{(\psi (t)-\psi (a))^2}{N}\Big \}\\&\le \sup _{t\in [a,a+h]}\Big \{ {I_{a+}^{\alpha ,\psi }}L(t)+\Big [{I_{a+}^{\alpha ,\psi }}L(b)\\&\quad +\,K{I_{a+}^{2\alpha ,\psi }}L(v)\Big ]\frac{(\psi (t)-\psi (a))^2}{N}\Big \}\Vert x_1-x_2\Vert , \end{aligned}$$

which proves that F is a contraction. Using the Banach fixed-point theorem, problem (1) has a unique solution. \(\square \)

Recall that a family of functions \(\Upsilon \) is equicontinuous if for every \(\epsilon >0\), there exists some \(\delta >0\) such that \(d(x(t_1),x(t_2))<\epsilon \) for all \(x \in \Upsilon \) and all \(t_1,t_2\) such that \(d(t_1,t_2)<\delta \).

Theorem 5

Suppose that there exist

  1. 1.

    Two continuous functions \(F_1,F_2:[a,b]\rightarrow \mathbb R\) and a nondecreasing function \(\Lambda :\mathbb R_0^+\rightarrow \mathbb R_0^+\) such that, for all \(t\in [a,b]\) and \(x\in \mathbb R\), we have

    $$\begin{aligned} |f(t,x)|\le F_1(t)\Lambda (|x|)+F_2(t), \end{aligned}$$
  2. 2.

    A real \(R>0\) such that

    $$\begin{aligned}&\omega _1\Lambda (R)+\omega _2+|x_a|+\frac{|x^1_a|}{\psi '(a)}(\psi (b)-\psi (a))\\&\quad +\,\Big |x_a\left[ 1-\frac{K}{\Gamma (1+\alpha )}(\psi (v)-\psi (a))^\alpha \right] \\&\quad +\,\frac{x^1_a}{\psi '(a)}\Big [\psi (b)-\psi (a)-\frac{K}{\Gamma (2+\alpha )}(\psi (v)\\&\quad -\,\psi (a))^{1+\alpha }\Big ] \Big | \frac{(\psi (b)-\psi (a))^2}{N}\le R, \end{aligned}$$

    where

    $$\begin{aligned} w_i:= & {} \sup _{t\in [a,b]} {I_{a+}^{\alpha ,\psi }}|F_i(t)|+({I_{a+}^{\alpha ,\psi }}|F_i(b)| +K {I_{a+}^{2\alpha ,\psi }}|F_i(v)| )\frac{(\psi (b)-\psi (a))^2}{N},\\&\quad \quad i=1,2. \end{aligned}$$

Then, the Cauchy problem (1) has at least one solution in [a, b].

Proof

Let F be the functional given by the expression (6) and consider the closed ball

$$\begin{aligned} B_R:=\left\{ x\in C^2[a,b]\,:\, \Vert x\Vert \le R \right\} . \end{aligned}$$

Step 1:\(F(B_R)\subseteq B_R\).

Given \(x\in B_R\), we have the following

$$\begin{aligned} \Vert F(x)\Vert&\le \sup _{t\in [a,b]}\Big \{ {I_{a+}^{\alpha ,\psi }}|f(t,x(t))|+|x_a|+\frac{|x^1_a|}{\psi '(a)}(\psi (t)-\psi (a))+|\gamma _x|(\psi (t)\\&\quad -\psi (a))^2\Big \}\\&\le \sup _{t\in [a,b]} {I_{a+}^{\alpha ,\psi }}|F_1(t)\Lambda (|x(t)|)+F_2(t)|+|x_a|+\frac{|x^1_a|}{\psi '(a)}(\psi (b)-\psi (a))\\&\quad +\Big [{I_{a+}^{\alpha ,\psi }}|F_1(b)\Lambda (|x(b)|)+F_2(b)|\\&\quad +K {I_{a+}^{2\alpha ,\psi }}|F_1(v)\Lambda (|x(v)|){+}F_2(v)|{+}\Big |x_a\Big [1-\frac{K}{\Gamma (1+\alpha )}(\psi (v)-\psi (a))^\alpha \Big ]\\&\quad +\frac{x^1_a}{\psi '(a)}\big [\psi (b)-\psi (a)- \frac{K}{\Gamma (2{+}\alpha )}(\psi (v)-\psi (a))^{1+\alpha }\big ]\Big |\Big ]\frac{(\psi (b)-\psi (a))^2}{N}\\&\le \omega _1\Lambda (R)+\omega _2+|x_a|+\frac{|x^1_a|}{\psi '(a)}(\psi (b)-\psi (a))\\&\quad +\Big |x_a\Big [1-\frac{K}{\Gamma (1+\alpha )}(\psi (v)-\psi (a))^\alpha \Big ]\\&\quad +\frac{x^1_a}{\psi '(a)}\Big [\psi (b)-\psi (a)-\frac{K}{\Gamma (2+\alpha )}(\psi (v)\\&\quad -\psi (a))^{1+\alpha }\Big ]\Big | \frac{(\psi (b)-\psi (a))^2}{N}\le R. \end{aligned}$$

Step 2:\(F(B_R)\) is equicontinuous.

Let \(t_1,t_2\in [a,b]\) with \(t_1>t_2\) and

$$\begin{aligned} M:=\max _{(t,x)\in [a,b]\times B_R}|f(t,x)|. \end{aligned}$$

Then,

$$\begin{aligned} |F(x)(t_1)-F(x)(t_2)|&\le \frac{M}{\Gamma (\alpha +1)}\left[ (\psi (t_1)-\psi (a))^\alpha -(\psi (t_2)-\psi (a))^\alpha \right] \\&\quad +\frac{|x^1_a|}{\psi '(a)}(\psi (t_1)-\psi (t_2))\\&\quad +|\gamma _x|\left[ (\psi (t_1)-\psi (a))^2-(\psi (t_2)-\psi (a))^2 \right] \end{aligned}$$

which converges to zero as \(t_1\rightarrow t_2\). By the Ascoli–Arzela Theorem, \(F(B_R)\) is contained in a compact set. Finally, we now prove that there are no \(x\in \partial B_R\) and \(\lambda \in (0,1)\) such that \(x=\lambda F(x)\). For that purpose, suppose that such x and \(\lambda \) exist. In this case,

$$\begin{aligned} R=\Vert x\Vert < \Vert F(x)\Vert \le R, \end{aligned}$$

as we have seen before, which proves our claim. In conclusion, by the Leray–Schauder alternative, we prove the existence of a fixed point to F.

4 Examples

Example 1

Fix a kernel \(\psi :[0,1]\rightarrow \mathbb R\) and the fractional order \(\alpha =2.5\). Consider the FDE

$$\begin{aligned} \left\{ \begin{array}{l} {^CD_{0+}^{2.5,\psi }} x(t)=\Gamma (4.5)(\psi (t)-\psi (0)), \quad t\in [0,1]\\ x(0)=0\\ x'(0)=0\\ x(1)=K{I_{0+}^{2.5,\psi }} x(0.5), \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} K=\frac{6!(\psi (1)-\psi (0))^{3.5}}{\Gamma (4.5)(\psi (0.5)-\psi (0))^6}. \end{aligned}$$

It is easy to check that the solution to this problem is the function \(x(t)=(\psi (t)-\psi (0))^{3.5}\). We can apply Theorem 4, by considering \(L(t)=0\), for all t, and conclude about the existence and uniqueness of the solution in the interval [0, 1].

Example 2

For our second example, consider the problem

$$\begin{aligned} \left\{ \begin{array}{l} {^CD_{0+}^{2.5,\psi }} x(t)=x(t), \quad t\in [0,1]\\ x(0)=1\\ x'(0)=0\\ x(1)=K{I_{0+}^{2.5,\psi }} x(0.5), \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} K=\frac{E_{2.5}((\psi (1)-\psi (0))^{2.5})}{E_{2.5}((\psi (0.5)-\psi (0))^{2.5})-1}. \end{aligned}$$

We assume that condition (2) is satisfied. The solution is the function \(x(t)=E_{2.5}((\psi (t)-\psi (0))^{2.5})\). We can apply Theorem 4, by considering \(L(t)=1\). Alternatively, applying Theorem 5 with \(F_1(t)=1\), \(F_2(t)=0\), and \(\Lambda (x)=x\), we conclude about the existence of the solution on the interval [0,1].

5 Conclusion

In this paper, we deal with fractional differential equations of order \(\alpha \in (2,3)\), with initial and end-point conditions. First, we prove an equivalent problem by rewritten the FDE as a Volterra integral equation. Then, using some known fixed-point theorems, we prove the existence and uniqueness of solution. Using the techniques exemplified in this paper, other fixed-point theorems could be applied.