1 Introduction

Over the last few decades, fractional differential equations have been of interest to many researchers and have proven to be very powerful tools for describing real problems in physics, chemistry, biology, and other fields (see [6,7,8, 11, 12, 24] for example).

Recently, problems involving a non-linear p-Laplaican operator have gained its popularity and importance as a result of its distinguished applications in many diverse fields of science and engineering, such as viscoelasticity, non-Newtonian mechanics, electrochemistry, and so on. Some results have emerged for the existence and uniqueness of solutions to boundary value problems for fractional differential equations with p-Laplacian operator; we refer the reader to [10, 15, 18,19,20, 23, 27] and the references therein.

An important part of the qualitative theory of linear and nonlinear differential equations is the stability of Ulam–Hyers, originally formulated by Hyers and Ulam in 1940 [9, 25]. Also, the study of stability analysis of Hyers–Ulam and the Ulam–Hyers-Rassias for non-linear fractional differential equations is a hot topic of research and the study of this area has grown to be one of the most important subjects in the mathematical analysis. A general view of the development of the Ulam–Hyers and the Ulam–Hyers–Rassias stability theory for fractional differential equations can be found in [1, 3,4,5, 13, 16, 17, 26, 28].

Inspired by the above mentioned work, we are interested in studying the existence and uniqueness of solutions and the Ulam stability analysis for the following fractional p-Laplacian boundary value problem involving both the right Caputo fractional derivative and the left fractional derivative of Riemann–Liouville type:

$$\begin{aligned} (P)\left\{ \begin{array}{l} ^{C}D_{1^{-}}^{\beta }(\phi _{p}(D_{0^{+}}^{\alpha }u(t)))+f(t,u(t))=0,\text { }t\in \left[ 0,1\right] , \\ \phi _{p}(D_{0^{+}}^{\alpha }u(1))=0, \\ \left. t^{2-\alpha }u(t)\right| _{t=0}=0, \\ u(1)=\lambda \int _{0}^{\eta }u(s)ds, \end{array} \right. \end{aligned}$$

where \(1<\alpha <2,\) \(0<\beta <1,\) \(\lambda >0,\) \(0<\eta <1\), \(0<\lambda \eta ^{\alpha }<\alpha \) and \(\phi _{p}(u)=\left| u\right| ^{p-2}u,\) \( \phi _{p}^{-1}=\phi _{q},\frac{1}{p}+\frac{1}{q}=1.\) \(^{C}D_{1^{-}}^{\beta } \) and \(D_{0^{+}}^{\alpha }\) denote the right Caputo fractional derivative of order \(\beta \) and the left Riemann–Liouville fractional derivative of order \(\alpha \) and \(f:[0,1]\times \mathbb {R} \rightarrow \mathbb {R}\) is a continuous function.

The rest of this paper is organized as follows. We montion in the next section some definitions from fractional calculus theory and necessary lemmas. In Sect. 3, we will prove the existence and uniqueness of solution to the problem (P) by using the Banach’s contraction principle and Schauder’s Fixed Point. In Sect. 4, we present Ulam–Hyers and Ulam–Hyers–Rassias stability results for the nonlinear fractional differential equation. Examples are given to demonstrate the application of our main results.

2 Preliminaries

We recall some definitions and lemmas which are used further in this paper.

Definition 2.1

[14] The left and right Riemann–Liouville fractional integrals of order \(\mu >0\) of a function f are defined by

$$\begin{aligned} I_{0^{+}}^{\mu }f(t)= & {} \frac{1}{\Gamma (\mu )}\int _{0}^{t}(t-s)^{\mu -1}f(s)ds, \\ I_{1^{-}}^{\mu }f(t)= & {} \frac{1}{\Gamma (\mu )}\int _{t}^{1}(s-t)^{\mu -1}f(s)ds. \end{aligned}$$

Definition 2.2

[14] The left Riemann–Liouville fractional derivative and the right Caputo fractional derivative of ordre \(\mu >0\) of a function f are respectively:

$$\begin{aligned} D_{0^{+}}^{\mu }f(t)= & {} \frac{d^{n}}{dt^{n}}\left( I_{0^{+}}^{n-\mu }f\right) (t), \\ ^{C}D_{1^{-}}^{\mu }f(t)= & {} (-1)^{n}I_{1^{-}}^{n-\mu }f^{(n)}(t), \end{aligned}$$

where \(n=\left[ \mu \right] +1,\) \(\left[ \mu \right] \) denotes the integer part of the real number \(\mu \).

Lemma 2.3

[22] Let \(\phi _{p}\) be p-Laplacian operator. Then,

  1. (i)

    If \(1<p\le 2,\) \(uv>0\) and \(\left| u\right| ,\) \( \left| v\right| \ge m>0,\) then

    $$\begin{aligned} \left| \phi _{p}(u)-\phi _{p}(v)\right| \le (p-1)m^{p-2}\left| u-v\right| . \end{aligned}$$
  2. (ii)

    If \(p>2\) and \(\left| u\right| ,\) \(\left| v\right| \le M,\) then

    $$\begin{aligned} \left| \phi _{p}(u)-\phi _{p}(v)\right| \le (p-1)M^{p-2}\left| u-v\right| . \end{aligned}$$

Definition 2.4

(Ulam–Hyers stability [21]). The problem (P) is Ulam–Hyers (UH) stable if there exists a real number \(k>0\) such that for each \(\varepsilon >0 \) and for each \(v\in C^{1}([0,1], \mathbb {R})\) solution of the inequality

$$\begin{aligned} \left| ^{C}D_{1^{-}}^{\beta }(\phi _{p}(D_{0^{+}}^{\alpha }v(t)))+f(t,v(t))\right| \le \varepsilon ,\text { }t\in [0,1], \end{aligned}$$
(2.1)

there exists a solution \(u\in C^{1}([0,1],\mathbb {R})\) of the problem (P) such that

$$\begin{aligned} \left| v(t)-u(t)\right| \le k\varepsilon ,\text { }t\in [0,1]. \end{aligned}$$

Definition 2.5

(Generalized Ulam–Hyers stablility [21]). Assume that \(v\in C^{1}([0,1], \mathbb {R})\) satisfies the inequality (2.1) and \(u\in C([0,1], \mathbb {R})\) is a solution of problem (P). If there exists a function \(\theta _{f}\) \( \in \) \(C( \mathbb {R}^{+}, \mathbb {R}^{+})\) with \(\theta _{f}(0)=0\) satisfying

$$\begin{aligned} \left| v(t)-u(t)\right| \le \theta _{f}(\varepsilon ),\text { }t\in [0,1], \end{aligned}$$

then the problem (P) is said generalized Ulam-Hyres stable (GUH).

Definition 2.6

(Ulam–Hyers-Rassias stablility [21]). The problem (P) is Ulam–Hyers-Rassias stable (UHR) with respect to \(\theta _{f}\) \(\in \) \( C([0,1], \mathbb {R} ^{+})\) if there exists a real number \(k>0\) such that for each \(\varepsilon >0 \) and for each \(v\in C^{1}([0,1], \mathbb {R})\) solution of the inequality

$$\begin{aligned} \left| ^{C}D_{1^{-}}^{\beta }(\phi _{p}(D_{0^{+}}^{\alpha }v(t)))+f(t,v(t))\right| \le \varepsilon \theta _{f}(t),\text { }t\in [0,1], \end{aligned}$$
(2.2)

there exists a solution \(u\in C([0,1], \mathbb {R})\) of the problem (P) such that

$$\begin{aligned} \left| v(t)-u(t)\right| \le k\theta _{f}(t)\varepsilon ,\text { } t\in [0,1]. \end{aligned}$$

Definition 2.7

(Generalized Ulam–Hyers–Rassias stablility [21]). Assume that \( v\in C^{1}([0,1], \mathbb {R})\) satisfies the inequality in (2.2) and \(u\in C([0,1], \mathbb {R})\) is a solution of the problem (P). If there exists a real number \( k_{\theta _{f}}>0\ \)such that

$$\begin{aligned} \left| v(t)-u(t)\right| \le k_{\theta _{f}}\theta _{f}(t),\text { } t\in [0,1], \end{aligned}$$

then the problem (P) is said to be generalized Ulam-Hyres-Rassias stable (GUHR).

Remark 2.8

A function \(v\in C^{1}([0,1], \mathbb {R})\) is a solution of the inequality (2.1). If there is a function \(\Phi \in C([0,1], \mathbb {R}),\) which depends on v, such that

  1. 1.

    \(\left| \Phi (t)\right| \le \varepsilon ,\) for all \(t\in [0,1],\)

  2. 2.

    \(^{C}D_{1^{-}}^{\beta }(\phi _{p}(D_{0^{+}}^{\alpha }v(t)))=-f(t,v(t))+\Phi (t),\) \(t\in [0,1].\)

Remark 2.9

A function \(v\in C^{1}([0,1], \mathbb {R})\) is a solution of the inequality (2.2). If there is a function \(\Phi \in C([0,1], \mathbb {R}),\) which depends on v, such that

  1. 1.

    \(\left| \Phi (t)\right| \le \varepsilon \theta _{f}(t),\) for all \(t\in [0,1],\)

  2. 2.

    \(^{C}D_{1^{-}}^{\beta }(\phi _{p}(D_{0^{+}}^{\alpha }v(t)))=-f(t,v(t))+\Phi (t),\) \(t\in [0,1].\)

Theorem 2.10

(Arzela-Ascoli Theorem [2]). Let \(X\subset C([0,1], \mathbb {R})\). X is relatively compact if and only if X is uniformly bounded and equicontinuous.

Theorem 2.11

(Schauder fixed point theorem [2]). If \(\Omega \) is a nonempty closed bounded convex subset of a Banach space X and \(T: \Omega \rightarrow \Omega \) is completely continuous, then T has a fixed point in \(\Omega \).

Theorem 2.12

(Banach’s fixed point theorem [2]). Let \(\Omega \) be a nonempty closed convex subset of a Banach space X, then any contraction mapping \(T: \Omega \rightarrow \Omega \) has a unique fixed point.

3 Existence and uniqueness

Lemma 3.1

The boundary value problem

$$\begin{aligned} (P1)\left\{ \begin{array}{l} ^{C}D_{1^{-}}^{\beta }(\phi _{p}(D_{0^{+}}^{\alpha }u(t)))+h(t)=0,t\in [0,1], \\ \phi _{p}(D_{0^{+}}^{\alpha }u(1))=0, \\ \left. t^{2-\alpha }u(t)\right| _{t=0}=0, \\ u(1)=\lambda \int _{0}^{\eta }u(s)ds, \end{array} \right. \end{aligned}$$

has a unique solution, which is given by

$$\begin{aligned} u(t)=\int _{0}^{1}G(t,s)g(s)ds,\text { }t\in \left[ 0,1\right] , \end{aligned}$$
(3.1)

where

$$\begin{aligned} G(t,s)= & {} G_{1}(t,s)+G_{2}(t,s),\text { }t,s\in \left[ 0,1\right] , \end{aligned}$$
(3.2)
$$\begin{aligned} G_{1}(t,s)= & {} \frac{1}{\Gamma (\alpha )}\left\{ \begin{array}{ll} t^{\alpha -1}(1-s)^{\alpha -1}-(t-s)^{\alpha -1}, &{} 0\le s\le t\le 1,\\ t^{\alpha -1}(1-s)^{\alpha -1}, &{} 0\le t\le s\le 1, \end{array} \right. \end{aligned}$$
(3.3)
$$\begin{aligned} G_{2}(t,s)= & {} \frac{t^{\alpha -1}\alpha \lambda }{(\alpha -\lambda \eta ^{\alpha })}\int _{0}^{\eta }G_{1}(t,s)dt,\text { }t,s\in \left[ 0,1\right] , \end{aligned}$$
(3.4)
$$\begin{aligned} g(s)= & {} \phi _{q}(I_{1^{-}}^{\beta }h)(s), s\in \left[ 0,1\right] . \end{aligned}$$
(3.5)

Proof

Applying the fractional integral operator \(I_{1^{-}}^{\beta }\) to the first equation of (P1), we get

$$\begin{aligned} \phi _{p}(D_{0^{+}}^{\alpha }u(s))=\frac{-1}{\Gamma (\beta )} \int _{s}^{1}(\tau -s)^{\beta -1}h(\tau )d\tau +c. \end{aligned}$$
(3.6)

By the boundary value condition \(\phi _{p}(D_{0^{+}}^{\alpha }u(1))=0\), we have \(c=0,\) consequently,

$$\begin{aligned} \phi _{p}(D_{0^{+}}^{\alpha }u(t))=-I_{1^{-}}^{\beta }h(t), \end{aligned}$$

and then,

$$\begin{aligned} D_{0^{+}}^{\alpha }u(s)=-\phi _{q}\left( \frac{1}{\Gamma (\beta )} \int _{s}^{1}(\tau -s)^{\beta -1}h(\tau )d\tau \right) . \end{aligned}$$

Letting

$$\begin{aligned} g(s)=\phi _{q}\left( \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}h(\tau )d\tau \right) , \end{aligned}$$

so

$$\begin{aligned} D_{0^{+}}^{\alpha }u(t)=-g(t), \end{aligned}$$

we arrive at

$$\begin{aligned} u(t)=-\frac{1}{\Gamma (\alpha )}\int _{0}^{t}(t-s)^{\alpha -1}g(s)ds+c_{1}t^{\alpha -1}+c_{2}t^{\alpha -2}. \end{aligned}$$
(3.7)

Condition \(\left. t^{2-\alpha }u(t)\right| _{t=0}=0\), imply that \(c_{2}=0\); i.e.,

$$\begin{aligned} u(t)=-I_{0^{+}}^{\alpha }g(t)+c_{1}t^{\alpha -1}. \end{aligned}$$
(3.8)

By using the condition \(u(1)=\lambda \int _{0}^{\eta }u(s)ds,\) we obtain

$$\begin{aligned} u(1)=-I_{0^{+}}^{\alpha }g(1)+c_{1}=\lambda \int _{0}^{\eta }u(s)ds, \end{aligned}$$
(3.9)

which implies

$$\begin{aligned} c_{1}=\frac{1}{\Gamma (\alpha )}\int _{0}^{1}(1-s)^{\alpha -1}g(s)ds+\lambda \int _{0}^{\eta }u(s)ds. \end{aligned}$$

Hence

$$\begin{aligned} u(t) =&\,-\frac{1}{\Gamma (\alpha )}\int _{0}^{t}(t-s)^{\alpha -1}g(s)ds+\frac{ t^{\alpha -1}}{\Gamma (\alpha )}\int _{0}^{1}(1-s)^{\alpha -1}g(s)ds \nonumber \\&+t^{\alpha -1}\lambda \int _{0}^{\eta }u(s)ds. \end{aligned}$$
(3.10)

As a result,

$$\begin{aligned} u(t)=\int _{0}^{1}G_{1}(t,s)g(s)ds+t^{\alpha -1}\lambda \int _{0}^{\eta }u(s)ds. \end{aligned}$$
(3.11)

where \(G_{1}\) is defined by (3.3). From (3.11), we have

$$\begin{aligned} \int _{0}^{\eta }u(t)dt=\frac{\alpha }{\alpha -\lambda \eta ^{\alpha }} \int _{0}^{\eta }\int _{0}^{1}G_{1}(t,s)g(s)dsdt. \end{aligned}$$
(3.12)

Substituting (3.12) into (3.11), we obtain

$$\begin{aligned} u(t)&=\int _{0}^{1}G_{1}(t,s)g(s)ds+\frac{\alpha \lambda t^{\alpha -1}}{ \alpha -\lambda \eta ^{\alpha }}\int _{0}^{\eta }\int _{0}^{1}G_{1}(t,s)g(s)dsdt \\&=\int _{0}^{1}G_{1}(t,s)g(s)ds+\int _{0}^{1}G_{2}(t,s)g(s)ds \\&=\int _{0}^{1}G(t,s)g(s)ds, \end{aligned}$$

where G and \(G_{2}\) are defined by (3.2) and (3.4) respectively. Inversement, we prove that the function \(u\in C([0,1], \mathbb {R})\) defined by (3.1) is solution of problem (P1). It is easy to verify that the function u satisfies the first equation of the problem (P1) , and it is easy to prove the first and second condition of the problem (P1). For the last condition we have,

$$\begin{aligned} \lambda \int _{0}^{\eta }u\left( t\right) dt&=\lambda \int _{0}^{\eta } \int _{0}^{1}G_{1}(t,s)g(s)dsdr +\frac{\alpha \lambda ^{2}}{ \alpha -\lambda \eta ^{\alpha }}\int _{0}^{\eta }t^{\alpha -1}\left( \int _{0}^{\eta }\int _{0}^{1}G_{1}(t,s)g(s)dsdr \right) dt \\&=\lambda \int _{0}^{\eta } \int _{0}^{1}G_{1}(t,s)g(s)dsdr +\frac{ \lambda ^{2}\eta ^{\alpha }}{ \alpha -\lambda \eta ^{\alpha }}\int _{0}^{\eta }\int _{0}^{1}G_{1}(t,s)g(s)dsdr \\&=\frac{\alpha \lambda }{\alpha -\lambda \eta ^{\alpha }} \int _{0}^{\eta }\int _{0}^{1}G_{1}\left( t,s\right) g\left( s\right) dsdr. \end{aligned}$$

Other hand, we have

$$\begin{aligned} G_{1}\left( 1,s\right) =0,\text { for all }s\in \left[ 0,1\right] , \end{aligned}$$

or

$$\begin{aligned} u\left( t\right) =\int _{0}^{1}G_{1}\left( t,s\right) g\left( s\right) ds+ \frac{\alpha \lambda t^{\alpha -1}}{\alpha -\lambda \eta ^{\alpha }} \int _{0}^{\eta }\int _{0}^{1}G_{1}\left( t,s\right) g\left( s\right) dsdr, \end{aligned}$$

then

$$\begin{aligned} u\left( 1\right) =\frac{\alpha \lambda }{\alpha -\lambda \eta ^{\alpha }} \int _{0}^{\eta }\int _{0}^{1}G_{1}\left( t,s\right) g\left( s\right) dsdr, \end{aligned}$$

we conclude that

$$\begin{aligned} u\left( 1\right) =\lambda \int _{0}^{\eta }u\left( t\right) dt. \end{aligned}$$

\(\square \)

Lemma 3.2

The function G defined by (3.2) is continuous on \([0,1]\times [0,1]\) and satisfies

  1. 1.

    \(G(t,s)\ge 0\) for \(t,s\in [0,1],\)

  2. 2.

    \(G(t,s)\le \delta \), where \(\delta =\frac{\alpha -\lambda \eta ^{\alpha }+\alpha \lambda \eta }{\Gamma (\alpha )(\alpha -\lambda \eta ^{\alpha })}\) for \(t,s\in [0,1].\)

Proof

Firstly, observing the expression of function G, it is easy to see that \(G(t,s)\ge 0\) for \(t,s\in [0,1].\)

Secondly, by definition of function \( G_{1}\), it is clear that \(G_{1}(t,s)\le \frac{1}{\Gamma (\alpha )}\) for all \(t,s\in [0,1],\) on the other hand,

$$\begin{aligned} G_{2}(t,s)&=\frac{t^{\alpha -1}\alpha \lambda }{(\alpha -\lambda \eta ^{\alpha })}\int _{0}^{\eta }G_{1}(t,s)dt, \\&\le \frac{\alpha \lambda \eta }{\Gamma (\alpha )(\alpha -\lambda \eta ^{\alpha })}, \end{aligned}$$

then

$$\begin{aligned} G(t,s)=G_{1}(t,s)+G_{2}(t,s)\le \frac{\alpha -\lambda \eta ^{\alpha }+\alpha \lambda \eta }{\Gamma (\alpha )(\alpha -\lambda \eta ^{\alpha })}, \text { } \end{aligned}$$

\(\square \)

for all \(t,s\in [0,1].\)

Let the Banach space \(E=C([0,1], \mathbb {R} )\) be endowed with the norm

$$\begin{aligned} \left\| u\right\| _{\infty }=\underset{t\in [0,1]}{max}|u(t)|. \end{aligned}$$

We define the operator \(T:E\rightarrow E\) by

$$\begin{aligned} Tu(t)=\int _{0}^{1}G(t,s)\phi _{q}\left( \frac{1}{\Gamma (\beta )} \int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,u(\tau ))d\tau \right) ds. \end{aligned}$$

To prove the uniqueness of the solution of the problem (P),  we used the Banach contraction mapping principle. For this, we make the following assumptions:

(H1):

There exists a positive continuous function w(t) such that

$$\begin{aligned} \left| f(t,u(t))\right| \le w(t),\text { }\left( t,u\right) \in [0,1]\times \mathbb {R}. \end{aligned}$$
(H2):

There exists a positive constant L such that for all \( u,v\in \mathbb {R}\)

$$\begin{aligned} \left| f(t,u)-f(t,v)\right| \le L\left| u-v\right| ,\text { for all }t\in [0,1]. \end{aligned}$$

Theorem 3.3

Suppose that \(1<p\) \(\le 2\). Assume (H1) and (H2) holds. If

$$\begin{aligned} \frac{LM^{q-2}\delta (q-1)}{\Gamma (\beta +2)}<1, \end{aligned}$$

where

$$\begin{aligned} M=\frac{1}{\Gamma (\beta +1)}\max _{t\in [0,1]}w(t), \end{aligned}$$

then the problem (P) has a unique solution.

Proof

By condition (H1), we have that

$$\begin{aligned} \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,u(\tau ))d\tau&\le \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}w(\tau )d\tau \\&\le \frac{\max _{t\in [0,1]}\left( w(t)\right) }{\Gamma (\beta )} \int _{s}^{1}(\tau -s)^{\beta -1}d\tau \nonumber \\&\le M \nonumber . \end{aligned}$$
(3.13)

By Lemma (2.3)(i) and by (3.13), for any \(u,v\in E\), we obtain

$$\begin{aligned} \left| Tu(t)-Tv(t)\right|&\le \left| \int _{0}^{1}G(t,s)\phi _{q}\left( \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,u(\tau ))d\tau \right) ds\right. \\&\left. -\int _{0}^{1}G(t,s)\phi _{q}\left( \frac{1}{\Gamma (\beta )} \int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,v(\tau ))d\tau \right) ds\right| \\&\le \frac{M^{q-2}\delta (q-1)}{\Gamma (\beta )}\int _{0}^{1}\left( \int _{s}^{1}(\tau -s)^{\beta -1}\left| f(\tau ,u(\tau ))-f(\tau ,v(\tau ))\right| d\tau \right) ds \\&\le \frac{M^{q-2}\delta (q-1)}{\Gamma (\beta )}\int _{0}^{1}\left( \int _{s}^{1}(\tau -s)^{\beta -1}\left| f(\tau ,u(\tau ))-f(\tau ,v(\tau ))\right| d\tau \right) ds \\&\le \frac{M^{q-2}\delta (q-1)}{\Gamma (\beta )}\int _{0}^{1}\left( \int _{s}^{1}(\tau -s)^{\beta -1}L\left\| u-v\right\| _{\infty }d\tau \right) ds \\&\le \frac{LM^{q-2}\delta (q-1)}{\Gamma (\beta +2)}\left\| u-v\right\| _{\infty }. \end{aligned}$$

This implies that \(T:E\rightarrow E\) is a contraction mapping. By means of the Banach contraction mapping principle, we get that T has a unique fixed point which is a solution of problem (P). \(\square \)

Now, we use the Schauder’s fixed point theorem to investigate the existence results for the problem (P). To prove the main result, we make the following assumption:

(H3):

There exist \(a_{1},a_{2}\in C([0,1], \mathbb {R}_{+})\) ,  \(0\le \nu <p-1\) such that

$$\begin{aligned} \left| f(t,u(t))\right| \le a_{1}(t)+a_{2}(t)\left| u\right| ^{\nu },\text { }\left( t,u\right) \in [0,1]\times \mathbb {R}. \end{aligned}$$

Let

$$\begin{aligned} \Omega =\left\{ u\in E,\text { }\left\| u\right\| _{\infty }<R\right\} , \end{aligned}$$

where R is chosen such that

$$\begin{aligned} R\ge \max \left\{ \left( \frac{2A_{1}\delta }{\Gamma (\beta +1)^{q-1}(\beta (q-1)+1)}\right) ^{q-1},\left( \frac{2A_{2}\delta }{\Gamma (\beta +1)^{q-1}(\beta (q-1)+1)}\right) ^{\frac{q-1}{1-\nu (q-1)}}\right\} , \end{aligned}$$

where \(A_{1}=\underset{t\in [0,1]}{\max }a_{1}(t),\) \(A_{2}=\underset{ t\in [0,1]}{\max }a_{2}(t).\)

Theorem 3.4

Assume (H3) hold. Then the problem (P) has at least one solution in \( \Omega \).

Proof

For convenience, the proof will be done in several steps.

Claim 1. We shall show that T is continuous.

Consider the sequence \((u_{n})_{n}\) converges to u in E, then for every \( t\in [0,1]\) we have

$$\begin{aligned} \left| Tu_{n}(t)-Tu(t)\right|&= \left| \int _{0}^{1}G(t,s)\phi _{q}\left( \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,u_{n}(\tau ))d\tau \right) ds\right. \\&\qquad -\left. \int _{0}^{1}G(t,s)\phi _{q}\left( \frac{1}{\Gamma (\beta )} \int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,u(\tau ))d\tau \right) ds\right| \\ \le&\int _{0}^{1}G(t,s) \left| \phi _{q}\left( \frac{1}{\Gamma (\beta ) }\int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,u_{n}(\tau ))d\tau \right) \right. \\&\qquad -\left. \phi _{q}\left( \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,u(\tau ))d\tau \right) \right| ds\\ \le&\delta \int _{0}^{1} \left| \phi _{q}\left( \frac{1}{\Gamma (\beta ) }\int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,u_{n}(\tau ))d\tau \right) \right. \\&\qquad -\left. \phi _{q}\left( \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,u(\tau ))d\tau \right) \right| ds. \end{aligned}$$

From the continuity of f,  \(\phi _{q}\) and by the dominated convergence theorem, we get \(\left\| Tu_{n}-Tu\right\| _{\infty }\rightarrow 0\) as \(n\rightarrow \infty .\) So, the operator T is continuous.

Claim 2. We show that \(T(\Omega )\subset \Omega \).

By condition (H3),  for all \(t\in [0,1]\) and \(u\in \Omega ,\) we have

$$\begin{aligned} \left| Tu(t)\right|\le & {} \int _{0}^{1}G(t,s)\left( \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,u(\tau ))d\tau \right) ^{q-1}ds \\\le & {} \int _{0}^{1}G(t,s)\left( \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}(a_{1}(t)+a_{2}(t)\left| u\right| ^{\nu })d\tau \right) ^{q-1}ds \\\le & {} \frac{\delta }{\Gamma (\beta +1)^{q-1}\left[ \beta (q-1)+1\right] } \left[ A_{1}+A_{2}R^{\nu }\right] ^{q-1} \\\le & {} R, \end{aligned}$$

which implies that \(T(\Omega )\subset \Omega \) and the set \(T(\Omega )\) is uniformly bounded.

Claim 3. We show that the \(T\left( \Omega \right) \) is an equicontinuous set.

Let \(u\in \Omega ,\) \(0\le t_{1}\le t_{2}\le 1.\) We have

$$\begin{aligned} \left| Tu(t_{2})-Tu(t_{1})\right|&=\left| \int _{0}^{1}G(t_{2},s)\phi _{q}\left( \frac{1}{\Gamma (\beta )} \int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,u(\tau ))d\tau \right) ds\right. \\&\qquad \left. -\int _{0}^{1}G(t_{1},s)\phi _{q}\left( \frac{1}{\Gamma (\beta )} \int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,u(\tau ))d\tau \right) ds\right| \\&\le \int _{0}^{1}\left| G(t_{2},s)-G(t_{1},s)\right| \left( \frac{1 }{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}\left| f(\tau ,u(\tau ))\right| d\tau \right) ^{q-1}ds \\&\le \int _{0}^{t_{1}}\left| G(t_{2},s)-G(t_{1},s)\right| \left( \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}\left| f(\tau ,u(\tau ))\right| d\tau \right) ^{q-1}ds \\&\qquad +\int _{t_{1}}^{t_{2}}\left| G(t_{2},s)-G(t_{1},s)\right| \left( \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}\left| f(\tau ,u(\tau ))\right| d\tau \right) ^{q-1}ds \\&\qquad +\int _{t_{2}}^{1}\left| G(t_{2},s)-G(t_{1},s)\right| \left( \frac{1 }{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}\left| f(\tau ,u(\tau ))\right| d\tau \right) ^{q-1}ds \\&\le \frac{\left( A_{1}+A_{2}R^{\nu }\right) ^{q-1}}{\Gamma \left( \alpha +1\right) \left( \Gamma (\beta +1)\right) ^{q-1}}\left[ \left( 1+\lambda ^{*}\right) \left( t_{2}^{\alpha -1}-t_{1}^{\alpha -1}\right) -\left( t_{2}^{\alpha }-t_{1}^{\alpha }\right) \right] , \end{aligned}$$

where \(\lambda ^{*}=\frac{\lambda \eta }{\left( \alpha -\lambda \eta ^{\alpha }\right) \Gamma \left( \alpha -1\right) }.\) As \(t_{2}\rightarrow t_{1}\), the right-hand side of the above inequality tends to zero, therefore \(T\left( \Omega \right) \) is an equicontinuous set. Therefore by the Arzela-Ascoli implies that T is completely continuous. According to the Schauder’s fixed point theorem, the operator T has at least one fixed point \(u\in \Omega \) which is a solution of the problem (P). \(\square \)

4 Ulam stability results

Theorem 4.1

Assume \(1<p\) \(\le 2.\) Suppose that (H1) and (H2) holds. If

$$\begin{aligned} \frac{LM^{^{q-2}}\delta (q-1)}{\Gamma (\beta +2)}<1, \end{aligned}$$

then the problem (P) is UH stable.

Proof

Assume \(1<p\) \(\le 2.\) Suppose v is a solution of the following inequality for \(\varepsilon \in (0,1]\)

$$\begin{aligned} \left| ^{C}D_{1^{-}}^{\beta }(\phi _{p}(D_{0^{+}}^{\alpha }v(t)))+f(t,v(t))\right| \le \varepsilon , \text { for all } t\in [0,1]. \end{aligned}$$
(4.1)

By Remark 2.8, we have

$$\begin{aligned} v(t)=\int _{0}^{1}G(t,s)\phi _{q}\left( \frac{1}{\Gamma (\beta )} \int _{s}^{1}(\tau -s)^{\beta -1}\left( f(\tau ,v(\tau ))-\Phi (\tau )\right) d\tau \right) ds. \end{aligned}$$

Other hand, from (H1) we obtain

$$\begin{aligned} \left| \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}\left( f(\tau ,v(\tau ))-\Phi (\tau )\right) d\tau \right|\le & {} \frac{1}{ \Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}\left| f(\tau ,v(\tau ))-\Phi (\tau )\right| d\tau \\\le & {} \frac{1}{\Gamma (\beta +1)}\max _{t\in [0,1]}w(t)+\frac{ \varepsilon }{\Gamma (\beta +1)} \\\le & {} M+\frac{1}{\Gamma (\beta +1)}=M_{0}, \end{aligned}$$

then, by Remark 2.8 and by Lemma (2.3)(i), we have

$$\begin{aligned} \left| v(t)-Tv(t)\right|\le & {} \int _{0}^{1}G(t,s)\left( \frac{1}{ \Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}\left| \Phi (\tau )\right| d\tau \right) ds \\\le & {} \frac{M_{0}^{^{q-2}}\delta (q-1)}{\Gamma (\beta +2)}\varepsilon . \end{aligned}$$

Suppose v is a solution of the inequality (4.1), we obtain

$$\begin{aligned} \left| v(t)-u(t)\right|&=\left| v(t)-\int _{0}^{1}G(t,s)\phi _{q}\left( \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,u(\tau ))d\tau \right) ds\right| \\&=\left| v(t)-\int _{0}^{1}G(t,s)\phi _{q}\left( \frac{1}{\Gamma (\beta ) }\int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,v(\tau ))d\tau \right) ds\right. \\&\qquad \left. +\int _{0}^{1}G(t,s)\phi _{q}\left( \frac{1}{\Gamma (\beta )} \int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,v(\tau ))d\tau \right) ds\right. \\&\qquad \left. -\int _{0}^{1}G(t,s)\phi _{q}\left( \frac{1}{\Gamma (\beta )} \int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,u(\tau ))d\tau \right) ds\right| \\&\le \left| v(t)-Tv(t)\right| +\left| Tv(t)-Tu(t)\right| \\&\le \frac{M_{0}^{^{q-2}}\delta (q-1)}{\Gamma (\beta +2)}\varepsilon + \frac{LM^{^{q-2}}\delta (q-1)}{\Gamma (\beta +2)}\left\| v-u\right\| _{\infty }, \end{aligned}$$

then

$$\begin{aligned} \left| v(t)-u(t)\right| \le \frac{M_{0}^{^{q-2}}\delta (q-1)}{ \Gamma (\beta +2)-LM^{^{q-2}}\delta (q-1)}\varepsilon ,\text { }t\in [0,1], \end{aligned}$$

where \( M_{0}=M+\frac{1}{\Gamma (\beta +1)} \). Setting

$$\begin{aligned} k=\frac{M_{0}^{^{q-2}}\delta (q-1)}{\Gamma (\beta +2)-LM^{^{q-2}}\delta (q-1) }, \end{aligned}$$

we obtain

$$\begin{aligned} \left| v(t)-u(t)\right| \le k\varepsilon . \end{aligned}$$
(4.2)

Therefore, the problem (P) is UH stable. \(\square \)

Remark 4.2

By setting \(\theta _{f}(\varepsilon )=k\varepsilon \) in (4.2), we obtain \(\theta _{f}(0)=0\) and then Problem (P) is GUH stable.

Theorem 4.3

Assume that \(1<p\) \(\le 2\). Suppose that the conditions (H1) and (H2) holds. If

$$\begin{aligned} \frac{LM^{^{q-2}}\delta (q-1)}{\Gamma (\beta +2)}<1, \end{aligned}$$

and if there exists a constant \(k_{\theta _{f}}>0\) such that

$$\begin{aligned} \frac{M_{0}^{^{q-2}}(q-1)}{\Gamma (\beta )}\int _{0}^{1}G(t,s)\left[ \int _{s}^{1}(\tau -s)^{\beta -1}\left| \Phi (\tau )\right| d\tau \right] ds\le \varepsilon k_{\theta _{f}}\theta _{f}(t),\text { }t\in \left[ 0,1\right] , \end{aligned}$$

where \(\theta _{f}\in C([0,1], \mathbb {R}^{+}).\) Then the problem (P) is UHR stable.

Proof

Suppose v is a solution of the following inequality

$$\begin{aligned} \left| ^{C}D_{1^{-}}^{\beta }(\phi _{p}(D_{0^{+}}^{\alpha }v(t)))+f(t,v(t))\right| \le \varepsilon \theta _{f}(t),\text {for all }\ t\in [0,1]. \end{aligned}$$
(4.3)

From Remark 2.9, we have

$$\begin{aligned} v(t)=\int _{0}^{1}G(t,s)\phi _{q}\left( \frac{1}{\Gamma (\beta )} \int _{s}^{1}(\tau -s)^{\beta -1}\left( f(\tau ,v(\tau ))-\Phi (\tau )\right) d\tau \right) ds. \end{aligned}$$

Other hand, for \(\varepsilon \in (0,1].\) We have

$$\begin{aligned} \left| \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}\left( f(\tau ,v(\tau ))-\Phi (\tau )\right) d\tau \right| \le M_{0}, \end{aligned}$$

then

$$\begin{aligned} \left| v(t)-Tv(t)\right|&\le \int _{0}^{1}G(t,s)\left| \phi _{q}\left( \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}\left( f(\tau ,v(\tau ))-\Phi (\tau )\right) d\tau \right) \right. \\&\qquad \left. -\phi _{q}\left( \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,v(\tau )d\tau \right) \right| ds \\&\le \frac{M_{0}^{^{q-2}}(q-1)}{\Gamma (\beta )}\int _{0}^{1}G(t,s) \int _{s}^{1}(\tau -s)^{\beta -1}\left| \Phi (\tau )\right| d\tau ds\\&\le k_{\theta _{f}}\varepsilon \theta _{f}(t). \end{aligned}$$

Suppose v is a solution of the inequality (4.1), we have

$$\begin{aligned} \left| v(t)-u(t)\right|&=\left| v(t)-\int _{0}^{1}G(t,s)\phi _{q}\left( \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,u(\tau ))d\tau \right) ds\right| \\&=\left| v(t)-\int _{0}^{1}G(t,s)\phi _{q}\left( \frac{1}{\Gamma (\beta )}\int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,v(\tau ))d\tau \right) ds\right. \\&\qquad \left. +\int _{0}^{1}G(t,s)\phi _{q}\left( \frac{1}{\Gamma (\beta )} \int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,v(\tau ))d\tau \right) ds\right. \\&\left. -\int _{0}^{1}G(t,s)\phi _{q}\left( \frac{1}{\Gamma (\beta )} \int _{s}^{1}(\tau -s)^{\beta -1}f(\tau ,u(\tau ))d\tau \right) ds\right| \\&\le \left| v(t)-Tv(t)\right| +\left| Tv(t)-Tu(t)\right| \\&\le k_{\theta _{f}}\varepsilon \theta _{f}(t)+\frac{LM^{q-2}\delta (q-1)}{ \Gamma (\beta +2)}\left\| v-u\right\| _{\infty }, \end{aligned}$$

then

$$\begin{aligned} \left| v(t)-u(t)\right| \le \frac{\Gamma (\beta +2)k_{\theta _{f}}}{ \Gamma (\beta +2)-LM^{q-2}\delta (q-1)}\varepsilon \theta _{f}(t),t\in \left[ 0,1\right] , \end{aligned}$$

setting

$$\begin{aligned} d=\frac{\Gamma (\beta +2)k_{\theta _{f}}}{\Gamma (\beta +2)-LM^{q-2}\delta (q-1)}, \end{aligned}$$

we obtain

$$\begin{aligned} \left| v(t)-u(t)\right| \le \varepsilon d\theta _{f}(t),\text { } t\in \left[ 0,1\right] . \end{aligned}$$
(4.4)

Therefore, the problem (P) is UHR stable. \(\square \)

Remark 4.4

By putting \(\varepsilon =1\) in (4.4), we deduce that problem (P) is GUHR stable.

5 Illustrative examples

5.1 Example 1

Consider the fractional boundary value problem

$$\begin{aligned} (P2)\left\{ \begin{array}{l} ^{C}D_{1^{-}}^{3/5}(\phi _{2}(D_{0^{+}}^{4/3}u(t)))+\frac{1}{100} (1+t)({1+|u(t)|}^{1/5}) =0,t\in [0,1], \\ \phi _{2}(D_{0^{+}}^{4/3}u(1))=0, \\ \left. t^{2/3}u(t)\right| _{t=0}=0, \\ u(1)=\lambda \int _{0}^{1/2}u(s)ds. \end{array} \right. \end{aligned}$$

We have \(\alpha =\frac{4}{3},\) \(\beta =\frac{3}{5},\) \(p=3,\) \(q=\frac{3}{2},\) \( \eta =\frac{1}{2},\) \(\lambda =2,\) \(\delta =3.\,8868\) and

$$\begin{aligned} f(t,u)=\frac{1}{100} (1+t)({1+|u(t)|}^{1/5}),\text { }t\in \left[ 0,1\right] ,u\in \mathbb {R,} \end{aligned}$$

satisfy

$$\begin{aligned} \vert f(t,u)\vert \le \frac{1}{100}(1+t)+\frac{1}{100}(1+t)\vert u \vert ^{\frac{1}{5}}. \end{aligned}$$

Then, we get \(\nu =\frac{1}{5}\), \( A_{1}=A_{2}=\frac{1}{50} \) and \( R=\frac{1}{2} \ge \max \lbrace 1. 0683,1. 1163\rbrace \) such that condition (H3) hold. By Theorem (3.4), we get that the problem (P2) has at least one solution.

5.2 Example 2

Consider the following boundary value problem

$$\begin{aligned} (P3)\left\{ \begin{array}{l} ^{C}D_{1^{-}}^{1/2}(\phi _{3/2}(D_{0^{+}}^{3/2}u(t)))+\frac{1}{10} (1-t)^{2}\left( \frac{1+|u(t)|}{2+|u(t)|}\right) =0,t\in [0,1], \\ \phi _{3/2}(D_{0^{+}}^{3/2}u(1))=0, \\ \left. t^{1/2}u(t)\right| _{t=0}=0, \\ u(1)=\lambda \int _{0}^{1/2}u(s)ds, \end{array} \right. \end{aligned}$$

Here, we have \(\alpha =\frac{3}{2},\) \(\beta =\frac{1}{2},\) \(p=\frac{3}{2},\) \(q=3,\) \( \eta =\frac{1}{2},\) \(\lambda =4,\) \(\delta =40.\,588\) and

$$\begin{aligned} f(t,u)=\frac{1}{10}(1-t)^{2}\left( \frac{1+|u|}{2+|u|}\right) ,\text { }t\in \left[ 0,1\right] ,u\in \mathbb {R.} \end{aligned}$$

There exists a function \(w(t)=\frac{1}{10}(1-t)^{2}\ge 0\), for all \(t\in \left[ 0,1\right] \). Such that

$$\begin{aligned} \left| f(t,u(t))\right| \le w(t),\text { }t\in \left[ 0,1\right] , \end{aligned}$$

and

$$\begin{aligned} \left| f(t,u)-f(t,v)\right|= & {} \left| \frac{1}{10}(1-t)^{2}(\vert u\vert -\vert v\vert )\right| \\\le & {} \frac{1}{10}\left| u-v\right| ,\text { }t\in \left[ 0,1\right] ,\text { }u,v\in \mathbb {R.} \end{aligned}$$

In view of Theorem (3.3)

$$\begin{aligned} \frac{LM^{q-2}\delta (q-1)}{\Gamma (\beta +2)}=0.459\,36<1. \end{aligned}$$

Therefore, we conclude that the problem (P3) has a unique solution. Similarly, this implies that the solution of problem (P3) is UH stable with \(k=474.\,35\) and hence GUH stable. By setting

$$\begin{aligned} \theta _{f}(t)=1.\,128\,4+39.\,46t^{1/2}, \text { }t\in \left[ 0,1\right] , \end{aligned}$$

we ensure that the conditions of Theorem (4.3) are satisfied. So, the problem (P3) is UHR stable by respect to \(\theta _{f}\) with \(d=474.\,35\) and also, GUHR stable.

6 Conclusion

In this article, we have discussed the existence and uniqueness of solution for the nonlinear mixed-type fractional differential equations by applying some fixed point theorems (Banach’s Contraction Principle, Schauder’s Fixed Point Theorem). We have also show the Ulam stability of the solutions. The presence of the p-Laplacian operator, with left Riemann–Liouville and right Caputo fractional derivatives in the posed problem makes it more complicated and interesting. Similar problems can be generalized to fractional differential equations with other type of derivative and with higher order in future works.