1 Introduction

Fractional-order differential and integral operators have played a key role in modelling several real-world phenomena in a much better way than the classical operators. It has been mainly due to the capability of such operators to take into account hereditary and memory properties of the processes and materials involved in the phenomena. Although fractional derivatives for functions of one variable can be traced to the origin of calculus and were already introduced rigorously in the XIXth century by Liouville and Riemann, the development of linear and nonlinear equations involving fractional derivatives of functions of one or several variables, and in particular fractional Laplacians, is a more recent phenomenon. Rich mathematical concepts allow in general several approaches, and this is the case for the fractional Laplacian, which can be defined using Fourier analysis, functional calculus, singular integrals or Levy processes. In contrast to the Laplacian, which is a local operator, the fractional Laplacian is a paradigm of the vast family of nonlocal linear operators, and this has immediate consequences in the formulation of basic questions like the diffusion equation. Along this direction, the subject is of interest for the mathematical community. Examples include biomathematics, signal and image processing, biomedical and chemical processes, control theory, wave propagation. For more details and explanations, see the texts [1–4]. For the recent history of fractional calculus, we refer the reader to the paper [5].

The study of boundary value problems of linear and nonlinear classical and fractional-order differential equations involving different kinds of boundary conditions has been a topic of great interest due to the occurrence of such problems in pure and applied sciences. Besides classical two-point boundary conditions, nonlocal multi-point and integral boundary conditions have gained a significant attention. Nonlocal conditions [6] are used to describe some peculiarities of physical, chemical or other processes taking place at interior positions of the given domain, while integral boundary conditions provide a plausible and practical approach to model blood flow problems [7] and regularizing ill-posed parabolic backward problems in time partial differential equations [8]. For some recent works on fractional-order boundary value problems involving a variety of boundary conditions, for instance, see [9–22] and the references cited therein.

In this paper, we discuss the existence of solutions for a nonlinear boundary value problem of sequential fractional integro-differential equations with nonlocal strip conditions given by

$$\begin{aligned} \left\{ \begin{array} {cc}\displaystyle (^cD^q +k^cD^{q-1})x(t)=\nu f(t,x(t))+\omega I^j g(t,x(t)) ,\,\,t\in [0,1],\\ \displaystyle x(0)\!=\!\delta x(\sigma ), \, ax'(\zeta _1)\!+\!bx'(\zeta _2)\!=\!c\int _\eta ^\xi x'(s)\mathrm{d}s, 0<\sigma<\zeta _1<\eta<\xi<\zeta _2<1, \end{array} \right. \end{aligned}$$
(1.1)

where \(^cD^q \) denotes the Caputo fractional derivative of order q, \( 1<q\le 2\); \(I^j\) denotes Riemann–Liouville integral with \(0<j<1\); f, g are given continuous functions, \(0<\delta <1, k>0\) and \(a, b, c,\nu ,\omega \) are real constants.

In modelling processes of physics and engineering problems, substitution of one relationship involving derivatives (ordinary or fractional) into another one gives rise to sequential fractional derivatives. In this context, our present study is important. Moreover, problem (1.1) may be regarded as a fractional analogue of Cauchy-Euler boundary value problems. For instance, for \(q=2\), the differential operator in (1.1) takes the form \(D^2 +k D.\)

We organize the rest of the content of the paper as follows. In Sect. 2, we recall some preliminary concepts of fractional calculus and prove an auxiliary lemma which plays a pivotal role in the sequel. Section 3 contains main results, while a variant of the given problem is discussed in Sect. 4.

2 Background Material

This section is devoted to some preliminary concepts of fractional calculus that we need in the forthcoming analysis [1, 2].

Definition 2.1

The fractional integral of order r with the lower limit zero for a function f is defined as

$$\begin{aligned} I^{r} f(t)= \frac{1}{\Gamma (r)}\int _0^t\frac{f(s)}{(t-s)^{1-r}}\mathrm{d}s, \quad t>0,\quad r>0, \end{aligned}$$

provided the right-hand side is pointwise defined on \([0,\infty )\), where \(\Gamma (\cdot )\) is the gamma function, which is defined by \(\Gamma (r)=\int _{0}^{\infty }t^{r-1}\mathrm{e}^{-t}\mathrm{d}t\).

Definition 2.2

The Riemann–Liouville fractional derivative of order \( r>0,\ n-1<r<n,\ n\in N\), is defined as

where the function f(t) has absolutely continuous derivative up to order \((n-1)\).

Definition 2.3

The Caputo derivative of order r for a function \(f:[0,\infty )\rightarrow R\) can be written as

$$\begin{aligned} ^cD^r f(t)= D^r\left( f(t)-\sum _{k=0}^{n-1}\frac{t^k}{k!}f^{(k)}(0)\right) ,\quad t>0, \quad n-1<r<n. \end{aligned}$$

Remark 2.4

If \(f(t)\in C^{n}[0,\infty ),\) then

$$\begin{aligned} ^cD^{r}f(t)= \frac{1}{\Gamma (n-r)}\int _0^t \frac{f^{(n)}(s)}{(t-s)^{r+1-n}}\mathrm{d}s = I^{n-r}f^{(n)}(t),\ t>0,\ n-1<q<n. \end{aligned}$$

To define a solution of problem (1.1), we consider the following lemma.

Lemma 2.5

For \(h \in C[0,1],\) the unique solution of linear fractional differential equation

$$\begin{aligned} \displaystyle (^{c}D^q +k ^cD^{q-1})x(t)=h(t), \, \, \, 1<q\le 2,t\in [0,1] , \end{aligned}$$
(2.1)

subject to the boundary conditions given by (1.1) is

$$\begin{aligned} \nonumber x(t)= & {} \int _0^t\mathrm{e}^{-k(t-s)} I^{q-1} h(s)\mathrm{d}s+ \frac{\delta }{1-\delta }\int _0^\sigma \mathrm{e}^{-k(\sigma -s)} I^{q-1} h(s)\mathrm{d}s\\\nonumber&+\,\frac{1}{G}\left[ e^{-kt}+\frac{\delta \mathrm{e}^{-k\sigma }-1}{1-\delta }\right] \Bigg [-k\Bigg ( a\int _0^{\zeta _1}\mathrm{e}^{-k(\zeta _1-s)} I^{q-1} h(s)\mathrm{d}s\\\nonumber&+\, b\int _0^{\zeta _2}\mathrm{e}^{-k(\zeta _2-s)} I^{q-1} h(s)\mathrm{d}s-c\int _\eta ^\xi \int _0^s \mathrm{e}^{-k(s-u)} I^{q-1} h(u)\mathrm{d}u\mathrm{d}s\Bigg )\\&+\, aI^{q-1}h(\zeta _1)+bI^{q-1}h(\zeta _2)-c\int _\eta ^\xi I^{q-1}h(s)\mathrm{d}s\Bigg ], \end{aligned}$$
(2.2)

where

$$\begin{aligned} \displaystyle G=c (\mathrm{e}^{-k\xi }-\mathrm{e}^{-k\eta })+k(a\mathrm{e}^{-k\zeta _1}+b\mathrm{e}^{-k\zeta _2}) \ne 0, \, \, I^{q-1}h(t) =\int _0^t\frac{(t-s)^{q-2}}{\Gamma (q-1)}h(s)\mathrm{d}s. \end{aligned}$$
(2.3)

Proof

The general solution of (2.1), as argued in [23], can be written as

$$\begin{aligned} \displaystyle x(t)=A \mathrm{e}^{-kt} +\int _0^t \mathrm{e}^{-k(t-s)} I^{q-1} h(s)\mathrm{d}s+B. \end{aligned}$$
(2.4)

Using the boundary condition given by (1.1) into (2.4) , we find that

$$\begin{aligned} A= & {} \frac{1}{G}\Bigg [ -k\Bigg (a\int _0^{\zeta _1}\mathrm{e}^{-k(\zeta _1-s)} I^{q-1} h(s)\mathrm{d}s+b\int _0^{\zeta _2}\mathrm{e}^{-k(\zeta _2-s)} I^{q-1} h(s)\mathrm{d}s\\&-\,\, c\int _\eta ^\xi \int _0^s \mathrm{e}^{-k(s-u)} I^{q-1} h(u)\mathrm{du}\mathrm{d}s\Bigg ) +aI^{q-1}h(\zeta _1)+bI^{q-1}h(\zeta _2)\\&-\,\, c\int _\eta ^\xi I^{q-1}h(s)\mathrm{d}s\Bigg ] \end{aligned}$$

and

$$\begin{aligned} B= & {} \frac{1}{1-\delta }\Bigg \{\frac{(\delta \mathrm{e}^{-k\sigma }-1)}{G}\Bigg [ -k\Bigg (a\int _0^{\zeta _1}\mathrm{e}^{-k(\zeta _1-s)} I^{q-1} h(s)\mathrm{d}s\\&+\, b\int _0^{\zeta _2}\mathrm{e}^{-k(\zeta _2-s)} I^{q-1} h(s)\mathrm{d}s -c\int _\eta ^\xi \int _0^s \mathrm{e}^{-k(s-u)} I^{q-1} h(u)\mathrm{d}u\mathrm{d}s\Bigg )\\&+\, aI^{q-1}h(\zeta _1)+bI^{q-1}h(\zeta _2)-c\int _\eta ^\xi I^{q-1}h(s)\mathrm{d}s\Bigg ]\\&+\, \delta \int _0^\sigma \mathrm{e}^{-k(\sigma -s)} I^{q-1} h(s)\mathrm{d}s\Bigg \}. \end{aligned}$$

Inserting the values of A and B in (2.4) yields the solution (2.2). The converse of the theorem follows by direct computation. This completes the proof. \(\square \)

3 Main Results

In view of Lemma 2.5, we define an operator \({\mathcal {H}}\): \({\mathcal {P}} \longrightarrow {\mathcal {P}} \) as

$$\begin{aligned} \nonumber ({\mathcal {H}}x)(t)= & {} \int _0^t\mathrm{e}^{-k(t-s)}\psi _x(s) \mathrm{d}s+ \frac{\delta }{1-\delta }\int _0^\sigma \mathrm{e}^{-k(\sigma -s)} \psi _x(s)\mathrm{d}s\\\nonumber&+\, \frac{1}{G}\Bigg [\mathrm{e}^{-kt}+\frac{\delta \mathrm{e}^{-k\sigma }-1}{1-\delta }\Bigg ]\Bigg [-k\Bigg ( a\int _0^{\zeta _1}\mathrm{e}^{-k(\zeta _1-s)}\psi _x(s)\mathrm{d}s\\\nonumber&+\, b\int _0^{\zeta _2}\mathrm{e}^{-k(\zeta _2-s)} \psi _x(s)\mathrm{d}s-c\int _\eta ^\xi \int _0^s \mathrm{e}^{-k(s-u)} \psi _x(u)\mathrm{d}u\mathrm{d}s\Bigg )\\&+\, a\psi _x(\zeta _1)+b\psi _x(\zeta _2)-c\int _\eta ^\xi \psi _x(s)\mathrm{d}s\Bigg ], \end{aligned}$$
(3.1)

where

$$\begin{aligned} \psi _x(s)=\nu \int _0^s\frac{(s-u)^{q-2}}{\Gamma (q-1)}f(u,x(u))du+\omega \int _0^s\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}g(u,x(u))du, \end{aligned}$$
(3.2)

G is given by (2.3), \({\mathcal {P}}= C([0,1],{\mathbb {R}}) \) denote the Banach space of all continuous functions from [0,1] into \( {\mathbb {R}}\) endowed with the usual norm \(\Vert x\Vert =sup\{|x(t)|,t\in [0,1]\} \).

We transform problem (1.1) to an equivalent fixed point problem as \( x = {\mathcal {H}} x\) and note that the existence of a fixed point for the operator \({\mathcal {H}}\) implies the existence of a solution for problem (1.1).

For the sake of computational convenience, we set

$$\begin{aligned} \nonumber \beta= & {} \frac{(1-\mathrm{e}^{-k})}{k}\Bigg (\frac{|\nu |}{\Gamma (q)}+\frac{|w|}{\Gamma (q+j)}\Bigg )+\frac{\delta (1-\mathrm{e}^{-k\sigma })}{k(1-\delta )} \Bigg (\frac{|\nu |\sigma ^{q-1}}{\Gamma (q)}+\frac{|w|\sigma ^{q+j-1}}{\Gamma (q+j)}\Bigg )\\\nonumber&+\, \frac{1}{|G|}\Bigg (1+\frac{|\delta \mathrm{e}^{-k\sigma }-1|}{(1-\delta )}\Bigg )\Bigg [|a|(1-\mathrm{e}^{-k\zeta _1})\Bigg (\frac{|\nu |\zeta _1^{q-1}}{\Gamma (q)}+\frac{|w|\zeta _1^{q+j-1}}{\Gamma (q+j)}\Bigg )\\\nonumber&+\, |b|(1-\mathrm{e}^{-k\zeta _2})\Bigg (\frac{|\nu |\zeta _2^{q-1}}{\Gamma (q)}+\frac{|w|\zeta _2^{q+j-1}}{\Gamma (q+j)}\Bigg )+|a|\Bigg (\frac{|\nu |\zeta _1^{q-1}}{\Gamma (q)} +\frac{|w|\zeta _1^{q+j-1}}{\Gamma (q+j)}\Bigg )\\ \nonumber&+\, |b|\Bigg (\frac{|\nu |\zeta _2^{q-1}}{\Gamma (q)}+\frac{|w|\zeta _2^{q+j-1}}{\Gamma (q+j)}\Bigg )+|c|\Bigg (\frac{|\nu (\xi ^{q}-\eta ^{q})|}{\Gamma (q+1)}+\frac{|w(\xi ^{q+j}-\eta ^{q+j})|}{\Gamma (q+j+1)}\Bigg )\\&+\, k|c|\Bigg (\frac{q|\nu (\xi ^{q+1}-\eta ^{q+1})|}{\Gamma (q+2)}+\frac{(q+j)|w(\xi ^{q+j+1}-\eta ^{q+j+1})|}{\Gamma (q+j+2)}\Bigg )\Bigg ]. \end{aligned}$$
(3.3)

Theorem 3.1

Let \(f,g:[0,1]\times {\mathbb {R}}\longrightarrow {\mathbb {R}}\) be continuous functions such that the following conditions hold:

\((A_1)\) :

\(|f(t,x) -f(t,y)|\le \ell _1|x-y|, \, \, |g(t,x) -g(t,y)|\le \ell _2|x-y|, \forall t\in [0,1],x,y\in {\mathbb {R}},\ell _1,\ell _2 >0.\)

Then the problem (1.1) has a unique solution if \(\ell \beta <1 \), where \( \ell =\max \{\ell _1,\ell _2\}\) and \(\beta \) is given by (3.3).

Proof

Let us define \(\alpha =\max \{\alpha _1,\alpha _2\},\) where \( \alpha _1,\alpha _2\) are positive real numbers given by \( \sup _{t\in [0,1]} |f(t,0)|=\alpha _1 \), \( \sup _{t\in [0,1]} |g(t,0)|=\alpha _2. \) To satisfy the hypothesis of Banach’s contraction mapping principle, as a first step, we show that \( {{\mathcal {H}}} B_r \subset B_r \), where \( B_r=\{x\in {\mathcal {P}}:\Vert x\Vert \le r\},\) \( r\ge \beta \alpha /(1-\beta \ell ),\) and the operator \({{\mathcal {H}}}\) is given by (3.1). For \( x\in B_r, t\in [0,1]\), we have the estimates: \(|f(t,x(t))|=|f(t,x(t))-f(t, 0)+f(t,0)| \le \ell _1 r+\alpha _1, \, |g(t,x(t))|=|g(t,x(t))-g(t, 0)+g(t,0)| \le \ell _2 r+\alpha _2. \) Using these estimates together with \( \ell =\max \{\ell _1,\ell _2\}\) and \(\alpha =\max \{\alpha _1,\alpha _2\},\) we get

$$\begin{aligned} \Vert ({\mathcal {H}}x)\Vert\le & {} |\nu |(\ell _1 r+\alpha _1)\sup _{t\in [0,1]}\Bigg \{\int _0^t\mathrm{e}^{-k(t-s)} \int _0^s\frac{(s-u)^{q-2}}{\Gamma (q-1)}\mathrm{d}u\mathrm{d}s\\&+\, \frac{\delta }{(1-\delta )}\int _0^\sigma \mathrm{e}^{-k(\sigma -s)} \int _0^s\frac{(s-u)^{q-2}}{\Gamma (q-1)}\mathrm{d}u\mathrm{d}s\\&+ \frac{1}{|G|}\Bigg [\mathrm{e}^{-kt}+\frac{|\delta \mathrm{e}^{-k\sigma }-1|}{(1-\delta )}\Bigg ]\Bigg [k\Bigg ( |a|\int _0^{\zeta _1}\mathrm{e}^{-k(\zeta _1-s)} \int _0^s\frac{(s-u)^{q-2}}{\Gamma (q-1)}\mathrm{d}u\mathrm{d}s\\&+\, |b|\int _0^{\zeta _2}\mathrm{e}^{-k(\zeta _2-s)} \int _0^s\frac{(s-u)^{q-2}}{\Gamma (q-1)}\mathrm{d}u\mathrm{d}s\\&+\, |c|\int _\eta ^\xi \int _0^s \mathrm{e}^{-k(s-u)} \int _0^u\frac{(u-\rho )^{q-2}}{\Gamma (q-1)}d\rho \mathrm{d}u\mathrm{d}s\Bigg )\\&+\, |a|\int _0^{\zeta _1} \frac{(\zeta _1-s)^{q-2}}{\Gamma (q-1)}\mathrm{d}s+|b| \int _0^{\zeta _2}\frac{(\zeta _2-s)^{q-2}}{\Gamma (q-1)}\mathrm{d}s\\&+\, |c| \int _\eta ^\xi \int _0^{s}\frac{(s-u)^{q-2}}{\Gamma (q-1)}\mathrm{d}u\mathrm{d}s\Bigg ]\Bigg \}\\&+\, |\omega |(\ell _2 r+\alpha _2)\sup _{t\in [0,1]}\Bigg \{\int _0^t\mathrm{e}^{-k(t-s)} \int _0^s\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}\mathrm{d}u\mathrm{d}s\\&+\, \frac{\delta }{(1-\delta )}\int _0^\sigma \mathrm{e}^{-k(\sigma -s)} \int _0^s\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}\mathrm{d}u\mathrm{d}s\\&+\, \frac{1}{|G|}\Bigg [\mathrm{e}^{-kt}+\frac{|\delta \mathrm{e}^{-k\sigma }-1|}{(1-\delta )}\Bigg ]\Bigg [k\Bigg ( |a| \int _0^{\zeta _1}\mathrm{e}^{-k(\zeta _1-s)} \int _0^s\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}\mathrm{d}u\mathrm{d}s\\&+\, |b|\int _0^{\zeta _2}\mathrm{e}^{-k(\zeta _2-s)} \int _0^s\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}\mathrm{d}u\mathrm{d}s\\&+\, |c|\int _\eta ^\xi \int _0^s \mathrm{e}^{-k(s-u)} \int _0^u\frac{(u-\rho )^{q+j-2}}{\Gamma (q+j-1)}d\rho \mathrm{d}u\mathrm{d}s\Bigg )\\&+\, |a|\int _0^{\zeta _1} \frac{(\zeta _1-s)^{q+j-2}}{\Gamma (q+j-1)}\mathrm{d}s+|b| \int _0^{\zeta _2}\frac{(\zeta _2-s)^{q+j-2}}{\Gamma (q+j-1)}\mathrm{d}s\\&+\, |c| \int _\eta ^\xi \int _0^{s}\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}\mathrm{d}u\mathrm{d}s\Bigg ]\Bigg \}\\\le & {} (\ell r+\alpha )\beta \le r. \end{aligned}$$

This shows that \({{\mathcal {H}}} B_r \subset B_r.\) Next, using the condition \((A_1)\), we obtain

$$\begin{aligned}&\Vert ({\mathcal {H}}x)-({\mathcal {H}}y)\Vert \\&\quad \le \sup _{t\in [0,1]}\Bigg \{\int _0^t\mathrm{e}^{-k(t-s)}\Bigg (|\nu | \int _0^s\frac{(s-u)^{q-2}}{\Gamma (q-1)}|f(u,x(u))-f(u,y(u))|du\\&\qquad + |\omega |\int _0^s\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}|g(u,x(u))-g(u,y(u))|du\Bigg )\mathrm{d}s\\&\qquad + \frac{\delta }{(1-\delta )}\int _0^\sigma \mathrm{e}^{-k(\sigma -s)} \Bigg (|\nu |\int _0^s\frac{(s-u)^{q-2}}{\Gamma (q-1)}|f(u,x(u))-f(u,y(u))|du\\&\qquad + |\omega |\int _0^s\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}|g(u,x(u))-g(u,y(u))|du\Bigg )\mathrm{d}s\\&\qquad + \frac{1}{|G|}\Bigg [\mathrm{e}^{-kt}+\frac{|\delta \mathrm{e}^{-k\sigma }-1|}{(1-\delta )}\Bigg ]\Bigg [k\Bigg (|a|\int _0^{\zeta _1}\mathrm{e}^{-k(\zeta _1-s)}\Bigg (|\nu | \int _0^s\frac{(s-u)^{q-2}}{\Gamma (q-1)}\\&\qquad \times |f(u,x(u))-f(u,y(u))|du\\&\qquad + |\omega | \int _0^s\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}|g(u,x(u))-g(u,y(u))|du\Bigg )\mathrm{d}s\\&\qquad + |b|\int _0^{\zeta _2}\mathrm{e}^{-k(\zeta _2-s)} \Bigg (|\nu | \int _0^s\frac{(s-u)^{q-2}}{\Gamma (q-1)}|f(u,x(u))-f(u,y(u))|du\\&\qquad + |\omega | \int _0^s\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}|g(u,x(u))-g(u,y(u))|du\Bigg )\mathrm{d}s\\&\qquad + |c|\int _\eta ^\xi \int _0^s \mathrm{e}^{-k(s-u)} \Bigg (|\nu | \int _0^u\frac{(u-\rho )^{q-2}}{\Gamma (q-1)}|f(\rho ,x(\rho ))-f(\rho ,y(\rho ))|d\rho \\&\qquad + |\omega | \int _0^u\frac{(u-\rho )^{q+j-2}}{\Gamma (q+j-1)}|g(\rho ,x(\rho ))-|g(\rho ,y(\rho ))|d\rho \Bigg )\mathrm{d}u\mathrm{d}s\Bigg )\\&\qquad + |a\nu |\int _0^{\zeta _1} \frac{(\zeta _1-s)^{q-2}}{\Gamma (q-1)}|f(s,x(s))-f(s,y(s))|\mathrm{d}s\\&\qquad + |a\omega | \int _0^{\zeta _1}\frac{(\zeta _1-s)^{q+j-2}}{\Gamma (q+j-1)}|g(s,x(s))-g(s,y(s))|\mathrm{d}s\\&\qquad + |b\nu | \int _0^{\zeta _2}\frac{(\zeta _2-s)^{q-2}}{\Gamma (q-1)}|f(s,x(s))-f(s,y(s))|\mathrm{d}s\\&\qquad + |b\omega | \int _0^{\zeta _2}\frac{(\zeta _2-s)^{q+j-2}}{\Gamma (q+j-1)}|g(s,x(s))-g(s,y(s))|\mathrm{d}s\\&\qquad + |c\nu | \int _\eta ^\xi \int _0^{s}\frac{(s-u)^{q-2}}{\Gamma (q-1)}|f(u,x(u))-f(u,y(u))|\mathrm{d}u\mathrm{d}s\\&\qquad + |c\omega | \int _\eta ^\xi \int _0^{s}\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}|g(u,x(u))-g(u,y(u))|\mathrm{d}u\mathrm{d}s\Bigg ]\Bigg \}\\&\quad \le \ell \beta ||x-y||. \end{aligned}$$

Since \(\ell \beta <1,\) the operator \({\mathcal {H}}\) is a contraction. Thus, it follows by the contraction mapping principle (Banach fixed point theorem) that the problem (1.1) has a unique solution on [0, 1]. \(\square \)

Now we establish the existence of solutions for problem (1.1) by means of Krasnoselskii’s fixed point theorem.

Lemma 3.2

(Krasnoselskii [24]) Let M be a closed, convex, bounded and nonempty subset of a Banach space X. Let A, B be the operators such that \(Ax+By\in M\) whenever \(x,y\in M\), A is compact and continuous, and B is a contraction mapping. Then, there exists \(z\in M\) such that \(z= Az+Bz\).

Theorem 3.3

Let \( f,g:[0,1] \times {\mathbb {R}} \rightarrow {\mathbb {R}} \) be continuous functions satisfying \( (A_1)\), and

\((A_2)\) :

\(|f(t,x)|\le \mu _1(t), \, \,|g(t,x)|\le \mu _2(t), \forall (t,x)\in [0,1] \times {\mathbb {R}}\) with \( \mu _1 ,\mu _2\in C([0,1], {\mathbb {R^+}})\).

Then the problem(1.1) has at least one solution on [0, 1] if \( \ell \gamma < 1 \), where

$$\begin{aligned} \gamma =\beta -\frac{(1-\mathrm{e}^{-k})}{k}\Bigg (\frac{|\nu |}{\Gamma (q)}+\frac{|w|}{\Gamma (q+j)}\Bigg ), \end{aligned}$$

\(\beta \) is given by (3.3) and \(\sup _{t\in [0,1]}|\mu _i(t)|=\Vert \mu _i\Vert ,i=1,2. \)

Proof

Let us define \(B_a=\{x\in {\mathcal {P}}:\Vert x\Vert \le a \}\) with \( a \ge \Vert \mu \Vert \beta ,\) and consider the operators \( \varphi _1 \) and \( \varphi _2 \) on \( B_a \) as

$$\begin{aligned} (\varphi _1)(t)= & {} \int _0^t\mathrm{e}^{-k(t-s)}\Bigg (\nu \int _0^s\frac{(s-u)^{q-2}}{\Gamma (q-1)}f(u,x(u))du+\omega \int _0^s\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}g(u,x(u))du\Bigg )\mathrm{d}s,\\ (\varphi _2)(t)= & {} \frac{\delta }{1-\delta }\int _0^\sigma \mathrm{e}^{-k(\sigma -s)} \Bigg (\nu \int _0^s\frac{(s-u)^{q-2}}{\Gamma (q\!-\!1)}f(u,x(u))du\!+\!\omega \int _0^s\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}g(u,x(u))du\Bigg )\mathrm{d}s\\&+\, \frac{1}{G}\Bigg [\mathrm{e}^{-kt}+\frac{\delta \mathrm{e}^{-k\sigma }-1}{1-\delta }\Bigg ]\Bigg [-k\Bigg ( a\int _0^{\zeta _1}\mathrm{e}^{-k(\zeta _1-s)}\Bigg (\nu \int _0^s\frac{(s-u)^{q-2}}{\Gamma (q-1)}f(u,x(u))du\\&+\, \omega \int _0^s\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}g(u,x(u))du\Bigg )\mathrm{d}s+b\int _0^{\zeta _2}\mathrm{e}^{-k(\zeta _2-s)}\Bigg (\nu \int _0^s\frac{(s-u)^{q-2}}{\Gamma (q-1)}f(u,x(u))du\\&+\, \omega \int _0^s\frac{(s\!-\!u)^{q+j-2}}{\Gamma (q\!+\!j\!-\!1)}g(u,x(u))du\Bigg )\mathrm{d}s \!-\!c\int _\eta ^\xi \int _0^s \mathrm{e}^{-k(s-u)} \Bigg (\nu \int _0^u\frac{(u-\rho )^{q-2}}{\Gamma (q-1)}f(\rho ,x(\rho ))d\rho \\&+\, \omega \int _0^u\frac{(u-\rho )^{q+j-2}}{\Gamma (q+j-1)}g(\rho ,x(\rho ))d\rho \Bigg )\mathrm{d}u\mathrm{d}s\Bigg )\\&+\, a\nu \int _0^{\zeta _1} \frac{(\zeta _1-s)^{q-2}}{\Gamma (q-1)}f(s,x(s))\mathrm{d}s+a\omega \int _0^{\zeta _1}\frac{(\zeta _1-s)^{q+j-2}}{\Gamma (q+j-1)}g(s,x(s))\mathrm{d}s\\&+\, b\nu \int _0^{\zeta _2}\frac{(\zeta _2-s)^{q-2}}{\Gamma (q-1)}f(s,x(s))\mathrm{d}s+b\omega \int _0^{\zeta _2}\frac{(\zeta _2-s)^{q+j-2}}{\Gamma (q+j-1)}g(s,x(s))\mathrm{d}s\\- & {} c\nu \int _\eta ^\xi \int _0^{s}\frac{(s-u)^{q-2}}{\Gamma (q-1)}f(u,x(u))\mathrm{d}u\mathrm{d}s-c\omega \int _\eta ^\xi \int _0^{s}\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}g(u,x(u))\mathrm{d}u\mathrm{d}s\Bigg ]. \end{aligned}$$

For \( x,y \in B_a \), we find that

$$\begin{aligned} \Vert (\varphi _1x) +(\varphi _2y) \Vert\le & {} \frac{(1-\mathrm{e}^{-k})}{k}\Bigg (\frac{|\nu |\Vert \mu _1\Vert }{\Gamma (q)}+\frac{|w|\Vert \mu _2\Vert }{\Gamma (q+j)}\Bigg )\\&+\, \frac{\delta (1-\mathrm{e}^{-k\sigma })}{k(1-\delta )}\Bigg (\frac{|\nu |\sigma ^{q-1}\Vert \mu _1\Vert }{\Gamma (q)}+\frac{|w|\alpha ^{q+j-1}\Vert \mu _2\Vert }{\Gamma (q+j)}\Bigg )\\&+\, \frac{1}{|G|}\Bigg (\mathrm{e}^{-k}+\frac{|\delta \mathrm{e}^{-k\alpha }-1|}{1-\delta }\Bigg )\Bigg [\\&|a|(1-\mathrm{e}^{-k\zeta _1})\Bigg (\frac{|\nu |\zeta _1^{q-1}\Vert \mu _1\Vert }{\Gamma (q)}+\frac{|w|\zeta _1^{q+j-1}\Vert \mu _2\Vert }{\Gamma (q+j)}\Bigg )\\&+\, |b|(1-\mathrm{e}^{-k\zeta _2})\Bigg (\frac{|\nu |\zeta _2^{q-1}\Vert \mu _1\Vert }{\Gamma (q)}+\frac{|w|\zeta _2^{q+j-1}\Vert \mu _2\Vert }{\Gamma (q+j)}\Bigg )\\&+\, |a|\Bigg (\frac{|\nu |\zeta _1^{q-1}\Vert \mu _1\Vert }{\Gamma (q)}+\frac{|w|\zeta _1^{q+j-1}\Vert \mu _2\Vert }{\Gamma (q+j)}\Bigg )\\&+\, |b|\Bigg (\frac{|\nu |\zeta _2^{q-1}\Vert \mu _1\Vert }{\Gamma (q)}+\frac{|w|\zeta _2^{q+j-1}\Vert \mu _2\Vert }{\Gamma (q+j)}\Bigg )\\&+\, |c|\Bigg (\frac{|\nu (\xi ^{q+1}-\eta ^{q})|\Vert \mu _1\Vert }{\Gamma (q+1)}+\frac{|w(\xi ^{q+j}-\eta ^{q+j})|\Vert \mu _2\Vert }{\Gamma (q+j+1)}\Bigg )\\&+\, k|c|\Bigg (\frac{q|\nu (\xi ^{q+1}-\eta ^{q+1})|\Vert \mu _1\Vert }{\Gamma (q+2)}+\frac{(q+j)|w(\xi ^{q+j+1}-\eta ^{q+j+1})|\Vert \mu _2\Vert }{\Gamma (q+j+2)}\Bigg )\Bigg ]\\\le & {} a . \end{aligned}$$

Thus, \( \varphi _1 x +\varphi _2 y \in B_a \). It follows from assumption \((A_1)\) together with (3.4) that \(\varphi _2\) is a contraction mapping. \(\square \)

Continuities of f and g imply that the operator \( \varphi _1\) is continuous. Also, \(\varphi _1\) is uniformly bounded on \(B_a\) as

$$\begin{aligned} \Vert (\varphi _1x)\Vert \le \frac{(1-\mathrm{e}^{-k})\Bigg [|\nu | \Vert \mu _1\Vert \Gamma (q+j)+|w|\Vert \mu _2\Vert \Gamma (q)\Bigg ]}{k\Gamma (q+j)\Gamma (q)} \end{aligned}$$

Now, we prove the compactness of the operator \(\varphi _1\). In view of \((A_1)\), we define \(\sup _{(t,x)\in [0,1]\times B_a } |f(t,x)|=\overline{f}\), \(\sup _{(t,x)\in [0,1]\times B_a } |g(t,x)|=\overline{g}\).

Consequently, we have

$$\begin{aligned}&\Vert (\varphi _1x)(t_2)-(\varphi _1x)(t_1)\Vert \\&\quad \le \Bigg |\mathrm{e}^{-kt_2}-\mathrm{e}^{-kt_1}\Bigg |\int _0^{t_1}\mathrm{e}^{ks}\Bigg (|\nu |\overline{f} \int _0^s\frac{(s-u)^{q-2}}{\Gamma (q-1)}du+|\omega |\overline{g} \int _0^s\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}du\Bigg )\mathrm{d}s\\&\qquad +\int _{t_1}^{t_2}\mathrm{e}^{-k(t_2-s)}\Bigg (|\nu |\overline{f} \int _0^s\frac{(s-u)^{q-2}}{\Gamma (q-1)}du+|\omega |\overline{g} \int _0^s\frac{(s-u)^{q+j-2}}{\Gamma (q+j-1)}du\Bigg )\mathrm{d}s, \end{aligned}$$

which is independent of x and tends to zero as \(t_2\longrightarrow t_1\). Thus, \(\varphi _1\) is relatively compact on \(B_a\). Hence, by the \( Arzel\acute{a}-Ascoli\) theorem, \(\varphi _1\) is compact on \(B_a\). Thus, all the assumptions of lemma 3.2 are satisfied, and hence problem(1.1) has at least one solution on [0, 1].

Example 3.4

Consider a boundary value problem of integro-differential equations of fractional order given by

$$\begin{aligned} \left\{ \begin{array} {cc} \displaystyle (^cD^q +k ^{c}D^{q-1})x(t)=\nu f(t,x(t))+\omega I^j g(t,x(t)) ,\,\,t\in [0,1],\\ \displaystyle x(0)=\delta x(\sigma ), \, ax'(\zeta _1)\!+\!bx'(\zeta _2)\!=\!c\int _\eta ^\xi x'(s)\mathrm{d}s, 0\!<\!\sigma<\zeta _1<\eta<\xi<\zeta _2<1, \end{array} \right. \end{aligned}$$
(3.4)

where

$$\begin{aligned} f(t,x)= & {} 3\times 10^{-5}\frac{ \cos x}{\sqrt{t^2+16}}+2\times 10^{-4}\frac{ |x|}{1+|x|}+\frac{t^2}{\sqrt{1+t^2}}, \,\\ g(t,x)= & {} 2\times 10^{-3} \cos x +t^2. \end{aligned}$$

With the given data \(q=6/5, j=1/4,\delta =1/2, \sigma =1/9, a=1/2, b=1/3, c=1, \zeta _1=1/5, \zeta _2=4/5, \xi =2/3, \eta =1/2,k=1/2, \nu =\omega =1,\) it is found that \(\ell _1=2.075\times 10^{-4},\ell _2=2\times 10^{-3}\) as \(|f(t,x)-f(t,y)|\le (2.075\times 10^{-4})|x-y|, |g(t,x)-g(t,y)|\le (2\times 10^{-3})|x-y|,\)and \(\beta \approx 18.7078.\) Clearly \(\ell =\max \{\ell _1,\ell _2\}=2\times 10^{-3}\) and \(\ell \beta <1\). Thus, all the assumptions of Theorem 3.1 are satisfied. Hence, by the conclusion of Theorem 3.1, problem (3.4) has a unique solution on [0, 1].

Example 3.5

Consider the problem (3.4) with

$$\begin{aligned} | f(t,x)|\le & {} \mu _1(t),\,\,\, | g(t,x)|\le \mu _2(t),\,\,\ \forall \in [0,1]\times {\mathbb {R}} \,\,\, with\,\,\, \mu _1(t),\mu _2(t)\\&\in C([0,1], {\mathbb {R^+}}). \end{aligned}$$

Clearly \( | f(t,x)|\le \frac{ 3\times 10^{-5}}{\sqrt{t^2+16}}+2\times 10^{-4}+\frac{t^2}{\sqrt{1+t^2}}\) and \(| g(t,x)|\le 2\times 10^{-3} +t^2\).

We find that \( \gamma \approx 16.9622 ,\,\,\ \ell \gamma <1\). Thus, by Theorem 3.3, there exists at least one solution for the problem (3.4).

4 A Variant Problem

In this Section, we consider a variant of problem (1.1) given by

$$\begin{aligned} \left\{ \begin{array} {cc} \displaystyle (^cD^q +k ^{c}D^{q-1})x(t)=\nu f(t,x(t))+\omega I^j g(t,x(t)) ,\,\,t\in [0,1]\\ \displaystyle x(0)=0 , \, ax(\zeta _1)+bx(\zeta _2)=c\int _\eta ^\xi x(s)\mathrm{d}s, 0<\zeta _1<\eta<\xi<\zeta _2<1, \end{array} \right. \end{aligned}$$
(4.1)

where the functions and parameters are the same as defined in problem (1.1).

As argued for problem (1.1), the fixed point problem associated with (4.1) is \( x = {\mathcal {H}}_v x,\) where \({\mathcal {H}}_v\): \({\mathcal {P}} \longrightarrow {\mathcal {P}} \) is defined by

$$\begin{aligned} \nonumber ({\mathcal {H}}_v x)(t)= & {} \int _0^t\mathrm{e}^{-k(t-s)}\psi _x(s)\mathrm{d}s+\frac{\mathrm{e}^{-kt}-1}{G_v}\Bigg [c\int _\eta ^\xi \int _0^s \mathrm{e}^{-k(s-u)} \psi _x(u)\mathrm{d}u\mathrm{d}s\\&-\, a\int _0^{\zeta _1}\mathrm{e}^{-k(\zeta _1-s)} \psi _x(s)\mathrm{d}s-b\int _0^{\zeta _2}\mathrm{e}^{-k(\zeta _2-s)}\psi _x(s)\mathrm{d}s\Bigg ], \end{aligned}$$
(4.2)

where

$$\begin{aligned} \displaystyle G_v=a(\mathrm{e}^{-k\zeta _1}-1)+b(\mathrm{e}^{-k\zeta _2}-1)+c \Bigg (\frac{\mathrm{e}^{-k\xi }-\mathrm{e}^{-k\eta }}{k}+\xi -\eta \Bigg ) \ne 0, \end{aligned}$$
(4.3)

and \(\psi _x\) is defined by (3.2).

Now we state the existence results for problem (4.1) without proofs. One can obtain the proofs using the methods of proofs employed in Sect. 3.

Theorem 4.1

Let \(f,g:[0,1]\times {\mathbb {R}}\longrightarrow {\mathbb {R}}\) be a continuous function satisfying the assumption \((A_1).\) Then problem (4.1) has a unique solution if \(\ell \beta _v<1 \), where \( \ell =\max \{\ell _1,\ell _2\}\) and

$$\begin{aligned} \nonumber \beta _v= & {} \frac{(1-\mathrm{e}^{-k})}{k}\Bigg (\frac{|\nu |}{\Gamma (q)}+\frac{|\omega |}{\Gamma (q+j)}\Bigg )+\frac{(1-\mathrm{e}^{-k})}{k |G_v|}\Bigg [k|c|\Bigg (\frac{q|\nu (\xi ^{q+1}-\eta ^{q+1})|}{\Gamma (q+2)}\\\nonumber&+\, (q+j)\frac{|\omega (\xi ^{q+j+1}-\eta ^{q+j+1})|}{\Gamma (q+j+2)}\Bigg )+|a|(1-\mathrm{e}^{-k\zeta _1})\Bigg (\frac{|\nu |\zeta _1^{q-1}}{\Gamma (q)} +\frac{|\omega |\zeta _1^{q+j-1}}{\Gamma (q+j)}\Bigg )\\&+\, |b|(1-\mathrm{e}^{-k\zeta _2})\Bigg (\frac{|\nu |\zeta _2^{q-1}}{\Gamma (q)}+\frac{|\omega |\zeta _2^{q+j-1}}{\Gamma (q+j)}\Bigg )\Bigg ]. \end{aligned}$$
(4.4)

Theorem 4.2

Let \( f,g:[0,1] \times {\mathbb {R}} \rightarrow {\mathbb {R}} \) be continuous function satisfying \( (A_1)\) and \((A_2). \) Then problem(4.1) has at least one solution on [0, 1] if \( \ell \gamma _v < 1 \), where

$$\begin{aligned} \gamma _v=\beta _v-\frac{(1-\mathrm{e}^{-k})}{k}\Bigg (\frac{|\nu |}{\Gamma (q)}+\frac{|\omega |}{\Gamma (q+j)}\Bigg )>0,\end{aligned}$$
(4.5)

and \(\beta _v\) is given by (4.4).

Example 4.3

Consider a boundary value problem of integro-differential equations of fractional order given by

$$\begin{aligned} \left\{ \begin{array} {cc} \displaystyle (^cD^q +k ^{c}D^{q-1})x(t)=\nu f(t,x(t))+\omega I^j g(t,x(t)) ,\,\,t\in [0,1]\\ \displaystyle x(0)=0 , \, ax(\zeta _1)+bx(\zeta _2)=c\int _\eta ^\xi x(s)\mathrm{d}s, 0<\zeta _1<\eta<\xi<\zeta _2<1, \end{array} \right. \end{aligned}$$
(4.6)

where

$$\begin{aligned} f(t,x)= & {} \frac{x}{\sqrt{t+256}} + \frac{t}{16} \tan ^{-1} x +\mathrm{e}^{-t}\cos (t^2+1), \, \\&\quad g(t,x)=\frac{t}{14}\sin x+\frac{\mathrm{e}^{-t}}{14} \Bigg (\frac{|x|}{1+|x|}\Bigg ) + \cos t. \end{aligned}$$

With the given data \(q=3/2, j=1/2, a=1/2, b=1/3, c=1, \zeta _1=1/5, \zeta _2=2/3, \xi =1/2, \eta =1/3,k=1, \nu =2,\omega =1,\) it is found that \(\ell _1=1/8,\ell _2=1/7\) as \(|f(t,x)-f(t,y)|\le (1/8)|x-y|, |g(t,x)-g(t,y)|\le (1/7)|x-y|,\)and \(\beta _v\approx 4.5523.\) Clearly \(\ell =\max \{\ell _1,\ell _2\}=1/7\) and \(\ell \beta <1\). Thus, all the assumptions of Theorem 4.1 are satisfied. Hence, by the conclusion of Theorem 4.1, problem (4.6) has a unique solution on [0, 1].