1 Introduction

In previous few decades, the area devoted to studying fractional calculus and derivatives and integrals of real or complex order has got proper attention. Fractional ordinary differential equations (FODEs) have numerous applications in the fields of physical science, biological sciences and engineering disciplines (for further details, see [1–7] and the references therein). The concerned area has the ability to model hereditary and memory process of many real world problems more comprehensively than classical derivatives and integral. Recently, some new differential operators of fractional order of nonsingular type have also attracted attention. Various researchers are working on it, it having also many applications in modeling real world problems. In this regard, some authors recently published very good work [8–10].

In real world problems, many phenomena face abrupt or sudden changes in their state of motion or rest. These abrupt changes are modeled by using impulsive differential equations. The area which addresses the aforementioned problems has been established very well in respect of ordinary derivatives and integrals. For the concerned investigations, researchers have used fixed point theory and nonlinear analysis tools. Further some results regarding numerical analysis corresponding to impulsive problems have also established [11, 12]. On the other hand the investigation of impulsive problems under the concept of fractional calculus has also developed properly. The problem related to the moments of sudden changes like heartbeat, earthquake, temperature, shock, and so forth occurs in our daily life problems. All of these sudden changes are concerned with the impulsive differential equations, for these types of sudden changes the investigation of impulsive differential equations is an important tool for analysis. Large numbers of articles can be found in this regard [13–17].

An important aspect to be investigated of the physical problems is whether the problem physically exists or not. For the existence theory, fixed point theory has been utilized in the last many years. But one thing which restricted the application of fixed point theory is the requirement of strong conditions. Because when we deal with problems of integral or differential equations by fixed point theory, we first convert it to operator equations for which we search fixed points. To achieve this goal often we impose strong compact conditions on the concerned operator, which restricts the applicability of the area to a limited class of problems. Therefore it is necessary to search some more sophisticated methods and theories to relax the criteria. Therefore Mawhin [18] used TDT for the first time to deal with integral equations. Later on Isaia [19] used the same theory to investigate some integral equations. During 2012 Wang [20] investigated some classes of initial value problems of FODEs by using the TDT including the following problem of impulsive FODEs:

$$ \textstyle\begin{cases} D^{\gamma } \varPsi (t)= h(t, \varPsi (t)),\quad t \neq t_{i}, 0< \gamma \le 1, \\ \delta \varPsi (t_{i})= I_{i}(\varPsi _{i}), \\ \varPsi (0)= \varPsi _{0}, \end{cases} $$

where \(i=1, 2, \ldots , l\),  h is a nonlinear function and \(I_{i}: \mathbb {R}\rightarrow \mathbb {R}\) is a nonlinear map which is the size of the jump at \(t_{i}\). As from present literature, the mentioned method has been used to investigate the usual initial and “boundary value problems” (BVPs) of FODEs (for details, see [21–23]).

On the other hand BVPs have many applications in engineering disciplines; see [24, 25]. The above-mentioned problems involve different kinds of boundary conditions like “Neumann boundary conditions and Dirichlet boundary conditions”. When the aforesaid boundary conditions are mixed then for the problem the mixed boundary conditions are called Robin boundary conditions (RBCs) or impedance BVPs. The concerned problems have many applications in heat transfer phenomena and electromagnetic theory. Further problems under RBCs are increasingly used in solving Sturm–Liouville equations which are largely used in engineering disciplines (see [26, 27]). Keeping in mind the mentioned literature, we investigate the following nonlinear problem of impulsive FODEs under RBCs for \(t\in [0, T]\):

$$ \textstyle\begin{cases} D^{\gamma } \varPsi (t)= h(t, \varPsi (t)), \quad t \neq t_{i}, 1< \gamma \le 2, \\ \delta \varPsi (t_{i})= I_{i}(\varPsi (t_{j})),\qquad \delta \varPsi ^{\prime }(t_{i})= J_{j}( \varPsi (t_{i})), \\ a \varPsi (0) + b \varPsi (T)= g_{1}(\varPsi ),\qquad c \varPsi ^{\prime }(0) + d \varPsi ^{\prime }(T)= g_{2}(\varPsi ), \end{cases} $$
(1.1)

where \(j=1, 2, \ldots , l \), \(0 < \gamma \leq 2 \) and the nonlinear function h is continuous and \(I_{j}\), \(J_{j}\) are nonlinear maps which determine the size of the jump at \(t_{i}\), where \(0< t_{1}< t_{0}< t_{2}< t_{3}\cdots <t_{l}\) and \(I_{i}(\varPsi (t_{i}))= \varPsi (t^{+}_{i})- \varPsi (t^{-}_{i}) \), \(J_{j}(\varPsi ^{\prime }(t_{i}))= \varPsi ^{\prime }(t^{+}_{i})- \varPsi ^{\prime }(t^{-}_{i}) \), the notations \(\varPsi (t^{+}_{i})\), \(\varPsi ^{-}(t^{+}_{i})\), and \(\varPsi (t^{-}_{i})\), \(\varPsi ^{-}(t^{-}_{i})\) are right and left limits, respectively, and \(D^{\gamma } \) represents the Caputo derivative of various orders where \(1< \gamma \leq 2\). We use TDT to establish some adequate results that ensure the existence of a solution to the considered problem. A pertinent example is given to demonstrate the main contribution.

2 Background materials

Some basic notions and results [1, 2] which we need for our analysis are given as follows.

Proposition 2.1

The following statements hold:

  1. (1)

    \(\psi (H)= 0\), iffHis relatively compact;

  2. (2)

    ψis of semi-norm, that is, \(\psi (\lambda H)= |\lambda |\psi (H)\)and\(\psi (H_{1}+ H_{2})\leq \psi (H_{1})+ \psi (H_{2})\);

  3. (3)

    \(H_{1}\subset H_{2}\)implies\(\psi (H_{1}) \leq \psi (H_{2}) \psi (H_{1} U H_{2})=\max \{{\psi (H_{1}), \psi (H_{2})}\} \);

  4. (4)

    \(\psi (\operatorname{conv} H)= \psi (H)\);

  5. (5)

    \(\psi (H^{-})= \psi (H)\).

Definition 2.1

Let \(\omega \subset Y\) and \(F: \omega \rightarrow Y\) a continuous bounded map, we say that F is ψ-Lipschitz if there exists \(k\geq 0\) such that \(\psi (F(H))\leq k\psi (H) \), for all \(H\subset \omega \). If \(k< 1\), then we say that F is a strict ψ− contraction.

Proposition 2.2

If\(F_{1}\), \(F_{2}\), \(F_{3}\), \(F_{4}: \omega \rightarrow Y \)areκ-Lipschitz maps with the constants\(m_{1}\), \(m_{2}\), \(m_{3}\), \(m_{4}\), respectively, then\(F_{1}, F_{2}, F_{3}, F_{4} : \omega \rightarrow Y\)areκ-Lipschitz with the constant\(m_{1}+ m_{2}+ m_{3}+ m_{4} \).

Proposition 2.3

If\(G : \omega \rightarrow Y\)is a compact mapping, then we callGψ-Lipschitz with zero constant.

Proposition 2.4

If\(G : \omega \rightarrow Y\)is Lipschitz with the constantk, thenGisψ-Lipschitz with the same constantk.

Theorem 2.1

Let\(F: Y \rightarrow Y\)beψ-condensing and

$$ T= \bigl\{ \varPsi \in Y: \textit{there exists } \lambda \in [0, T] \textit{ such that } \varPsi = \lambda F \varPsi \bigr\} . $$

IfTis a bounded set inY, so there exists\(r > 0\)such that\(T\subset B_{r}(0)\), then

$$ D\bigl(I- \lambda F, B_{r}(0), 0\bigr)= 1 \quad \textit{for all } \lambda \in [0, T]. $$

ThenFhas at least one fixed point and the set of fixed points ofFlies in\(B_{r}(0)\).

Lemma 2.1

The FODE wih\(\gamma >0\),

$$ D^{\gamma } \varPsi (t)= 0, $$

has a unique solution of the form\(\varPsi (t)= \sum_{n=0}^{m} c_{i}t^{m-1}\), and\(c_{i}\)are real constants and\(n= [\gamma ]+ 1\).

Lemma 2.2

The next consequence holds for FODE with\(\gamma >0\):

$$ I^{\gamma }D^{\gamma }x(t)= \varPsi (t)+ \sum _{i=0}^{m-1} \mathrm{c}_{i}t^{i}, $$

for arbitrary constant\(\mathrm{c}_{i}\in \mathbb {R}\), \(i=0\), \(1, 2, 3,\ldots , m-1\), \(m= [\gamma ]+ 1\).

The notation\(Y= \{ \varPsi :[0, T] \rightarrow \mathbb {R}: \varPsi \in C(I^{\prime }) \}\)is used for the space. Further, \((Y, \| \cdot \|)\)is the Banach space under the norm\(\|\varPsi \|=\sup_{t\in [0, T]}|\varPsi (t)|\).

3 The existence results for the considered problem

Here we define some hypotheses for the existence and uniqueness theory for the problem under consideration and we name the space \(PC(J, R)= \varUpsilon \); is needed throughout this work.

\((A_{1})\):

Let there for \(\varPsi _{1}, \varPsi _{2} \in \varUpsilon \) exist constants \(K_{g_{1}}, K_{g_{2}} \in [0, 1)\) such that

$$ \bigl\vert g_{1}(\varPsi _{1})- g_{1}(\varPsi _{2}) \bigr\vert \leq K_{g_{1}} \vert \varPsi _{1}- \varPsi _{2} \vert ,\qquad \bigl\vert g_{2}(\varPsi _{1})- g_{2}(\varPsi _{2}) \bigr\vert \leq K_{g_{2}} \vert \varPsi _{1}- \varPsi _{2} \vert . $$
\((A_{2})\):

Let us for \(\varPsi \in \varUpsilon \) have some constants \(C_{g_{1}}, M_{g_{1}}, C_{g_{2}}, M_{g_{2}} \in [0, 1)\) such that

$$ \bigl\vert g_{1}(\varPsi ) \bigr\vert \leq C_{g_{1}} \vert \varPsi \vert + M_{g_{1}},\qquad \bigl\vert g_{2}(\varPsi ) \bigr\vert \leq C_{g_{2}} \vert \varPsi \vert + M_{g_{2}}. $$
\((A_{3})\):

Let us for \(\varPsi \in \varUpsilon \) have some constants \(C_{f}, M_{f} \in [0, 1)\) such that

$$ \bigl\vert h(t, \varPsi ) \bigr\vert \leq C_{f}+ \vert \varPsi \vert M_{f}. $$
\((A_{4})\):

Let us for \(\varPsi \in \mathbb {R}\) have some constants \(C_{1}, M_{1}, C_{2}, M_{2} \in [0, 1)\) such that

$$ \bigl\vert I_{j}(\varPsi ) \bigr\vert \leq C_{1} \vert \varPsi \vert + M_{1},\qquad \bigl\vert J_{j}(\varPsi ) \bigr\vert \leq C_{2} \vert \varPsi \vert + M_{2}. $$
\((A_{5})\):

Let \(I_{i}, J_{i}: \mathbb {R}\rightarrow \mathbb {R}\) and let us have some constants \(K^{i}_{I}, K^{i}_{J} \in [0, \frac{1}{m})\) such that

$$ \bigl\vert I_{i}(\varPsi _{1})- I_{i}(\varPsi _{2}) \bigr\vert \leq K^{i}_{I} \vert \varPsi _{1}- \varPsi _{2} \vert , \qquad \bigl\vert J_{i}(\varPsi _{1})- J_{i}(\varPsi _{2}) \bigr\vert \leq K^{i}_{J} \vert \varPsi _{1}- \varPsi _{2} \vert $$

and

$$ \bigl\vert h(t, \varPsi _{1})- h(t, \varPsi _{2}) \bigr\vert \leq K_{\varPsi } \vert \varPsi _{1}-\varPsi _{2} \vert , $$

for all \(\varPsi _{1}, \varPsi _{2} \in \mathbb {R}\) and \(i= 1,2,3,\ldots \) .

Definition 3.1

A function \(\varPsi \in \varUpsilon \) with its γ-derivative that exists on \([0, T]- \{ t_{1}, t_{2}, t_{3},\ldots , t_{m}\}\) is said to be the solution of the considered fractional impulsive problem under RBCs if it verifies the considered equation (1.1).

Lemma 3.1

Let \(\varPsi \in \varUpsilon \) be the solution of the fractional impulsive problem with \(\eta \in C([0,1], \mathbb {R})\) given by

$$ \textstyle\begin{cases} D^{\gamma } \varPsi (t)= \eta (t), \quad t \neq t_{i}, 1< \gamma \le 2, \\ \delta \varPsi (t_{j})= I_{j}(\varPsi (t_{j})), \qquad \delta \varPsi ^{\prime }(t_{j})= J_{j}( \varPsi (t_{j})),\quad j= 1, 2, 3, \ldots , l, \\ a \varPsi (0) + b \varPsi (T)= g_{1}(\varPsi ),\qquad c \varPsi ^{\prime }(0) + d \varPsi ^{\prime }(T)= g_{2}(\varPsi ), \end{cases} $$
(3.1)

if and only is Ψ is the solution of the fractional impulsive integral equation as

$$\begin{aligned} \varPsi (t)&= \frac{1}{\varGamma (\gamma )} \int _{t_{k}}^{t} (t- s)^{ \gamma -1} \eta (s)\,ds \\ &\quad {}+ \frac{1}{a+b} \Biggl( \frac{a}{\varGamma (\gamma )} \sum _{j=1}^{k} \int _{t_{j}-1}^{t_{j}}(t_{j}- s)^{\gamma -1} \eta (s)\,ds- \frac{b}{\varGamma (\gamma )}\sum _{j=1}^{k} \int _{t_{k}-1}^{T}(T- s)^{ \gamma -1} \eta (s)\,ds \Biggr) \\ &\quad {} + \Biggl( \sum_{j=1}^{k} (t- t_{j})+ \frac{bdT}{(a+b)(c+d)}- \frac{b}{a+b} \sum _{j=1}^{k}(T-t_{j})- \frac{td}{c+d} \Biggr) \\ &\quad {}\times\frac{1}{\varGamma (\gamma -1)} \sum _{j=1}^{k} \int _{t_{j}-1}^{t_{j}}(t_{j}- s)^{\gamma -2} \eta (s)\,ds \\ &\quad {}+ \biggl(\frac{bdT}{(a+b)(c+d)}- \frac{td}{c+d} \biggr) \frac{1}{\varGamma (\gamma -1)} \int _{t_{k}}^{T}(T- s)^{\gamma -2} \eta (s)\,ds+ \frac{a}{a+b} \sum_{j=1}^{k} I_{j}\bigl(\varPsi (t_{j})\bigr) \\ &\quad {}+ \Biggl( \sum_{j=1}^{k} (t- t_{j})+ \frac{bdT}{(a+b)(c+d)}- \frac{b}{a+b} \sum _{j=1}^{k}(T-t_{j})- \frac{td}{c+d} \Biggr) \sum_{j=1}^{k} J_{j}\bigl(\varPsi (t_{j})\bigr) \\ &\quad {}\times \biggl(\frac{bT}{(a+b)(c+d)}+ \frac{t}{c+d} \biggr) g_{2}(\varPsi )+ \frac{g_{1}(\varPsi )}{a+b}. \end{aligned}$$
(3.2)

Proof

Let Ψ is a solution of (3.1), then, for \(\eta \in C([0,1], \mathbb {R})\), \(t\in [0, T]\), and using Lemma 2.2 for the given problem, we have two constants \(f_{0}\), \(f_{1}\), that is,

$$ \begin{aligned} &\varPsi (t) = I^{\gamma }\eta (t)- f_{0}- f_{1}t,\quad t\in [0, t_{1}], \\ &\varPsi (t) = \frac{1}{\varGamma (\gamma )} \int _{0}^{t_{1}}(t_{1}-s)^{ \gamma -1} \eta (s)\,ds- f_{0}- f_{1}t, \quad t\in [0, t_{1}]. \end{aligned} $$
(3.3)

Differentiating we get

$$ \varPsi ^{\prime }(t) = \frac{1}{\varGamma (\gamma -1)} \int _{0}^{t_{1}}(t_{1}-s)^{ \gamma -2} \eta (s)\,ds- f_{1},\quad t\in [0, t_{1}]. $$
(3.4)

Likewise, for \(t\in (t_{1}, t_{2}]\), there are constants \(\mathbf{d}_{0}\), \(\mathbf{d}_{1}\) with

$$ \begin{aligned} &\varPsi (t) = I^{\gamma }\eta (t)- \mathbf{d}_{0}- \mathbf{d}_{1}(t- t_{1}), \quad t\in (t_{1}, t_{2}], \\ &\varPsi (t) = \frac{1}{\varGamma (\gamma )} \int _{t_{1}}^{t}(t_{1}-s)^{ \gamma -1} \eta (s) \,ds- \mathbf{d}_{0}- \mathbf{d}_{1}(t- t_{1}),\quad t \in [0, t_{1}]. \end{aligned} $$
(3.5)

Differentiating, we get

$$\begin{aligned} \varPsi ^{\prime }(t) =& \frac{1}{\varGamma (\gamma -1)} \int _{t_{1}}^{t}(t_{1}-s)^{ \gamma -1} \eta (s)\,ds- \mathbf{d}_{1},\quad t\in (t_{1}, t_{2}], \end{aligned}$$
(3.6)

and

$$\begin{aligned}& \begin{aligned} &\varPsi \bigl(t^{-}_{1}\bigr) = \frac{1}{\varGamma (\gamma )} \int _{0}^{t_{1}}(t_{1}-s)^{ \gamma -1}- f_{0}- f_{1}t_{1},\quad t\in [0, t_{1}], \\ &\varPsi \bigl(t^{+}_{1}\bigr) = - f_{0}, \end{aligned} \end{aligned}$$
(3.7)
$$\begin{aligned}& \begin{aligned} &\varPsi ^{\prime }\bigl(t^{-}_{1}\bigr)= \frac{1}{\varGamma (\gamma -1)} \int _{t_{1}}^{t}(t_{1}-s)^{ \gamma -1} \eta (s)\,ds- \mathbf{d}_{1},\quad t\in (t_{1}, t_{2}], \\ &\varPsi ^{\prime }\bigl(t^{+}_{1}\bigr)= - \mathbf{d}_{1}.\end{aligned} \end{aligned}$$
(3.8)

Next applying the impulsive conditions after simplification we get the following:

$$ \begin{aligned} &{-}\mathbf{d}_{0} = \frac{1}{\varGamma (\gamma )} \int _{0}^{t_{1}}(t_{1}-s)^{ \gamma -1} \eta (s)\,ds- f_{0}- f_{1}t_{1}+ I_{1}\bigl(\varPsi (t_{1})\bigr), \\ &{-}\mathbf{d}_{1} = \frac{1}{\varGamma (\gamma - 1)} \int _{0}^{t_{1}}(t_{1}-s)^{ \gamma -2} \eta (s)\,ds- f_{0}+ J_{1}\bigl(\varPsi (t_{1}) \bigr). \end{aligned} $$
(3.9)

Putting these values in the given equation we get

$$\begin{aligned} \varPsi (t) =& \frac{1}{\varGamma (\gamma )} \int _{0}^{t_{1}}(t_{1}-s)^{ \gamma -1} \eta (s)\,ds+ \frac{1}{\varGamma (\gamma )} \int _{0}^{t_{1}}(t_{1}-s)^{ \gamma -1} \eta (s)\,ds \\ &{}+\frac{t- t_{1}}{\varGamma (\gamma - 1)} \int _{0}^{t_{1}}(t_{1}-s)^{ \gamma -1} \eta (s)\,ds \\ &{}+ (t- t_{1})J_{1}\bigl(\varPsi (t_{1})\bigr)+ I_{1}\bigl(\varPsi (t_{1}) \bigr)- f_{0}- f_{1}t . \end{aligned}$$
(3.10)

Reiterating the same procedure generally for \(t\in (t_{j-1}, t_{j}]\), we have

$$\begin{aligned} \varPsi (t) =& \frac{1}{\varGamma (\gamma )} \int _{0}^{t_{1}}(t_{1}-s)^{ \gamma -1} \eta (s)\,ds+ \frac{1}{\varGamma (\gamma )}\sum_{j=1}^{k} \int _{t_{j- 1}}^{t_{j}}(t_{1}-s)^{\gamma -1} \eta (s)\,ds \\ &{}+\sum_{j=1}^{k}(t- t_{j})\frac{1}{\varGamma (\gamma - 1)} \int _{t_{j-1}}^{t_{j}}(t_{j}-s)^{ \gamma -2} \eta (s)\,ds+ \sum_{j=1}^{k}(t- t_{j})J_{j}\bigl(\varPsi (t_{j})\bigr) \\ &{}+ \sum_{j=1}^{k}I_{j} \bigl(\varPsi (t_{j})\bigr)- f_{0}- f_{1}t. \end{aligned}$$
(3.11)

In view of the RBCs and after simplification, we have the subsequent values for the constant \(f_{0}\), \(f_{1}\),

$$\begin{aligned}& \begin{aligned} f_{0} &= \frac{b}{a+b} \Biggl(\frac{1}{\varGamma (\gamma )} \int _{t_{k}}^{T}(T- s)^{\gamma -1} \eta (s)\,ds+ \frac{1}{\varGamma (\gamma )} \sum_{j=1}^{k} \int _{t_{j-1}}^{t_{j}}(t_{j}- s)^{\gamma -1} \eta (s)\,ds \\ &\quad {} + \sum_{j=1}^{k}(T- t_{j}) \int _{t_{j-1}}^{t_{j}}(t_{j}- s)^{ \gamma -2}\frac{1}{\varGamma (\gamma )} \eta (s)\,ds+ \sum _{j=1}^{k}(T- t_{j})J_{j} \bigl( \varPsi (t_{j})\bigr) \\ &\quad {}+ \sum_{j=1}^{k}(T- t_{j})I_{j}\bigl(\varPsi (t_{j})\bigr) \Biggr) - \frac{bdT}{(a+b)(c+d)} \Biggl( \frac{1}{\varGamma (\gamma -1)} \int _{t_{k}}^{T}(T- s)^{\gamma -2} \eta (s)\,ds \\ &\quad {}+ \frac{1}{\varGamma (\gamma -1)} \sum_{j=1}^{k} \int _{t_{j}-1}^{t_{j}}(t_{j}- s)^{\gamma -2} \eta (s)\,ds \Biggr)- \frac{bdT}{(a+b)(c+d)}\sum_{j=1}^{k} J_{j}\bigl(\varPsi (t_{j})\bigr) \\ &\quad {}+ \biggl( \frac{bdT}{(a+b)(c+d)} \biggr)g_{2}\bigl(\varPsi (t_{j}) \bigr)- \biggl(\frac{1}{a+b} \biggr)g_{1}\bigl(\varPsi (t_{j})\bigr), \end{aligned} \\& \begin{aligned} f_{1} &= \frac{d}{d+c} \Biggl( \frac{1}{\varGamma (\gamma -1)} \int _{t_{k}}^{T}(T- s)^{\gamma -2} \eta (s)\,ds+ \frac{1}{\varGamma (\gamma -1)} \sum_{j=1}^{k} \int _{t_{j}-1}^{t_{j}}(t_{j}- s)^{\gamma -2} \eta (s)\,ds \\ &\quad {} + \sum_{j=1}^{k} J_{j} \bigl(\varPsi (t_{j})\bigr) \Biggr)- \biggl(\frac{1}{d+c} \biggr)g_{2}\bigl(\varPsi (t_{j})\bigr). \end{aligned} \end{aligned}$$

Putting these values in (3.11), we get the required solution given in (3.2) as

$$\begin{aligned} \varPsi (t) =& \frac{1}{\varGamma (\gamma )} \int _{t_{k}}^{t} (t- s)^{ \gamma -1} \eta (s)\,ds \\ &{}+ \frac{1}{\varGamma (\gamma )} \sum_{j=1}^{k} \int _{t_{j}-1}^{t_{j}}(t- s)^{\gamma -1}\eta (s)\,ds+ \frac{1}{\varGamma (\gamma -1)} \sum_{j=1}^{k} (t- t_{j}) \int _{t_{j}-1}^{t_{j}}(t- s)^{\gamma -2} \eta (s)\,ds \\ &{}+ \sum_{j=1}^{k} (t- t_{j}) J_{j}\bigl( \varPsi (t_{j})\bigr)+\sum _{j=1}^{k}I_{j} \bigl(\varPsi (t_{j})\bigr)- \frac{b}{a+b} \Biggl( \frac{1}{\varGamma (\gamma )} \int _{t_{k}}^{T}(T- s)^{\gamma -1} \eta (s)\,ds \\ &{}+ \frac{1}{\varGamma (\gamma )} \sum_{j=1}^{k} \int _{t_{j-1}}^{t_{j}}(t_{j}- s)^{\gamma -1} \eta (s)\,ds +\sum_{j=1}^{k}(T- t_{j}) \frac{1}{\varGamma (\gamma -1)} \int _{t_{j-1}}^{t_{j}}(t_{j}- s)^{ \gamma -2} \eta (s)\,ds \\ &{}+ \sum_{j=1}^{k}(T- t_{j}) J_{j}\bigl(\varPsi (t_{j})\bigr)+ \sum_{j=1}^{k}I_{j} \bigl(\varPsi (t_{j})\bigr) \Biggr) \\ &{}+ \frac{bdT}{(a+b)(c+d)} \Biggl( \frac{1}{\varGamma (\gamma -1)} \int _{t_{k}}^{T}(T- s)^{\gamma -2} \eta (s)\,ds \\ &{}+ \frac{1}{\varGamma (\gamma -1)} \sum_{j=1}^{k} \int _{t_{j}-1}^{t_{j}}(t_{j}- s)^{\gamma -2} \eta (s)\,ds+ \sum_{j=1}^{k}J_{j} \bigl(\varPsi (t_{j})\bigr)+ \frac{g_{2}(\varPsi )}{d} \Biggr) \\ &{}- \frac{td}{d+c} \Biggl( \frac{1}{\varGamma (\gamma -1)} \int _{t_{k}}^{T}(T- s)^{\gamma -2} \eta (s)\,ds \\ &{}+ \frac{1}{\varGamma (\gamma -1)} \sum_{j=1}^{k} \int _{t_{j}-1}^{t_{j}}(t_{j}- s)^{\gamma -2} \eta (s)\,ds+ \sum_{j=1}^{k}J_{j} \bigl(\varPsi (t_{j})\bigr) \Biggr) \\ &{}+ \biggl(\frac{t}{d+c} \biggr)g_{2}(\varPsi )+ \biggl(\frac{1}{a+b} \biggr)g_{1}( \varPsi ). \end{aligned}$$

Rearranging terms so we can achieve the solution given in (3.2). On the other hand, we presume that Ψ is a solution of the given equation, it being straightforward we give the solution given by (3.2), by direct computation which satisfies (3.12). □

Corollary 3.1

In view of Lemma 3.1, the solution of the considered problem (1.1) is given by

$$\begin{aligned} \varPsi (t) =& \frac{1}{\varGamma (\gamma )} \int _{t_{k}}^{t} (t- s)^{ \gamma -1} h\bigl(s, \varPsi (s)\bigr)\,ds+ \frac{a}{(a+b)\varGamma (\gamma )} \sum_{j=1}^{k} \int _{t_{j}-1}^{t_{j}}(t_{j}- s)^{\gamma -1}h\bigl(s, \varPsi (s)\bigr)\,ds \\ &{}- \frac{b}{(a+b)\varGamma (\gamma )} \int _{t_{k}}^{T}(T- s)^{\gamma -1} h\bigl(s, \varPsi (s)\bigr)\,ds + \Biggl( \sum_{j=1}^{k} (t- t_{j})+ \frac{bdT}{(a+b)(c+d)} \\ &{}- \frac{b}{a+b} \sum _{j=1}^{k}(T-t_{j})- \frac{td}{c+d} \Biggr)\frac{1}{\varGamma (\gamma -1)} \sum _{j=1}^{k} \int _{t_{j}-1}^{t_{j}}(t_{j}- s)^{\gamma -2} h\bigl(s, \varPsi (s)\bigr)\,ds \\ &{}+ \biggl(\frac{bdT}{(a+b)(c+d)}- \frac{td}{c+d} \biggr) \frac{1}{\varGamma (\gamma -1)}\int _{t_{k}}^{T}(T- s)^{\gamma -2} h\bigl(s, \varPsi (s)\bigr)\,ds \\ &{} + \frac{a}{a+b} \sum_{j=1}^{k} I_{j}\bigl(\varPsi (t_{j})\bigr)+ \Biggl( \sum _{j=1}^{k} (t- t_{j})+ \frac{bdT}{(a+b)(c+d)}- \frac{b}{a+b} \sum_{j=1}^{k}(T-t_{j}) \\ &{}- \frac{td}{c+d} \Biggr) \sum_{j=1}^{k} J_{j}\bigl(\varPsi (t_{j})\bigr)+ \biggl( \frac{bT}{(a+b)(c+d)}+ \frac{t}{c+d} \biggr) g_{2}(\varPsi )+ \frac{g_{1}(\varPsi )}{a+b}. \end{aligned}$$
(3.12)

With the help of Corollary 3.1, the problem (3.12) is reduced to the fixed point problem as \(\varPsi = T(\varPsi )\). Keeping in mind this operator problem we develop results regarding the solution of the considered problem.

Next we define five operators to establish the main results as regards the existence and uniqueness as follows:

$$\begin{aligned}& F_{0} : \varUpsilon \rightarrow \varUpsilon , \\& \begin{aligned} F_{0}\varPsi (t) &= \frac{a}{a+b} \sum _{j=1}^{k} I_{j}\bigl(\varPsi (t_{j})\bigr) \\ &\quad {}+ \Biggl( \sum_{j=1}^{k} (t- t_{j})+ \frac{bdT}{(a+b)(c+d)}- \frac{b}{a+b} \sum _{j=1}^{k}(T-t_{j})- \frac{td}{c+d} \Biggr) \sum_{j=1}^{k} J_{j}\bigl(\varPsi (t_{j})\bigr) \\ &\quad {} + \biggl(\frac{bT}{(a+b)(c+d)}+ \frac{t}{c+d} \biggr) g_{2}(\varPsi )+ \frac{g_{1}(\varPsi )}{a+b}. \end{aligned} \end{aligned}$$

For the integral part we define the operators by

$$\begin{aligned}& F_{1} : \varUpsilon \rightarrow \varUpsilon , \\& F_{1}\varPsi (t) = \frac{1}{\varGamma (\gamma )} \int _{t_{k}}^{t} (t- s)^{ \gamma -1} h\bigl(s, \varPsi (s)\bigr)\,ds- \frac{b}{(a+b)\varGamma (\gamma )} \int _{t_{k}}^{T}(T- s)^{\gamma -1} h\bigl(s, \varPsi (s)\bigr)\,ds , \\& F_{2} : \varUpsilon \rightarrow \varUpsilon , \\& \begin{aligned} F_{2}\varPsi (t) &= \Biggl( \sum_{j=1}^{k} (t- t_{j})+ \frac{bdT}{(a+b)(c+d)}- \frac{b}{a+b} \sum _{j=1}^{k}(T-t_{j})- \frac{td}{c+d} \Biggr) \\ &\quad {} \times \frac{1}{\varGamma (\gamma -1)} \sum_{j=1}^{k} \int _{t_{j}-1}^{t_{j}}(t_{j}- s)^{\gamma -2} h\bigl(s, \varPsi (s)\bigr)\,ds, \end{aligned} \\& F_{3} : \varUpsilon \rightarrow \varUpsilon , \\& F_{3}\varPsi (t) = \biggl(\frac{bdT}{(a+b)(c+d)}- \frac{td}{c+d} \biggr) \frac{1}{\varGamma (\gamma -1)} \int _{t_{k}}^{T}(T- s)^{\gamma -2} h\bigl(s, \varPsi (s)\bigr)\,ds, \\& F_{4} : \varUpsilon \rightarrow \varUpsilon , \\& F_{4}\varPsi (t) = \frac{a}{(a+b)\varGamma (\gamma )} \sum _{j=1}^{k} \int _{t_{j}-1}^{t_{j}}(t_{j}- s)^{\gamma -1} h\bigl(s, \varPsi (s)\bigr)\,ds. \end{aligned}$$

Let \(T : \varUpsilon \rightarrow \varUpsilon \), then T is defined by

$$ T\varPsi (t)= F_{0}\varPsi (t)+ F_{1}\varPsi (t)+ F_{2}\varPsi (t)+ F_{3}\varPsi (t)+ F_{4}\varPsi (t). $$

Therefore, investigating the solution to the given (3.12) the problem is just like investigating the fixed point for the operator T.

Theorem 3.1

The operator\(F_{0}: \varUpsilon \rightarrow \varUpsilon \)is Lipschitz with constant\(K_{I}=\sum_{i=1}^{m}K^{i}_{I}\in [0, 1)\). Consequently\(F_{0}\)isψ-Lipschitz with the same constant\(K^{i}_{I}\in [0, 1)\). Further\(F_{0}\)verifies the following relation:

$$ \Vert F_{0}\varPsi \Vert \leq A+ B \Vert \varPsi \Vert , $$
(3.13)

where

$$\begin{aligned} A =& \biggl\vert \frac{a}{a+b} \biggr\vert M_{1}+ \biggl\vert lT+ \frac{bdt}{(a+b)(c+d)} \biggr\vert M_{2} \\ &{}+ \biggl\vert \frac{bT}{(a+b)(c+d)}+ \frac{T}{c+d} \biggr\vert M_{g_{2}}+ \biggl\vert \frac{a}{a+b} \biggr\vert M_{g_{1}} \end{aligned}$$
(3.14)

and

$$\begin{aligned} B =& \biggl\vert \frac{a}{a+b} \biggr\vert C_{1}+ \biggl\vert lT+ \frac{bdt}{(a+b)(c+d)} \biggr\vert C_{2} \\ &{}+ \biggl\vert \frac{bT}{(a+b)(c+d)}+ \frac{T}{c+d} \biggr\vert C_{g_{2}}+ \biggl\vert \frac{a}{a+b} \biggr\vert C_{g_{1}}. \end{aligned}$$
(3.15)

Proof

Using \((A_{1})\) and \((A_{4})\), we have

$$\begin{aligned}& \sup_{t\in [0, T]} \vert F_{0}\varPsi _{1}- F_{0}\varPsi _{2} \vert \\& \quad = \sup_{t\in [0, T]} \Biggl| \frac{a}{a+b} \sum_{j=1}^{k} \bigl( I_{j}\bigl(\varPsi _{1}(t_{j})\bigr)- I_{j}\bigl(\varPsi _{2}(t_{j})\bigr) \bigr)+ \Biggl( \sum_{j=1}^{k} (t- t_{j})+ \frac{bdT}{(a+b)(c+d)} \\& \qquad {} - \frac{b}{a+b} \sum_{j=1}^{k}(T-t_{j})- \frac{td}{(c+d)} \Biggr) \bigl[J_{j}\bigl(\varPsi _{1}(t_{j})\bigr)- J_{j}\bigl(\varPsi _{2}(t_{j})\bigr) \bigr] + \biggl( \frac{bT}{(a+b)(c+d)} \\& \qquad {} + \frac{t}{c+d} \biggr) \bigl(g_{2}(\varPsi _{1})- g_{2}(\varPsi _{2}) \bigr)+ \frac{1}{a+b} \bigl(g_{1}(\varPsi _{1})- g_{1}(\varPsi _{2}) \bigr) \Biggr|. \end{aligned}$$

Further simplification yields

$$\begin{aligned} \Vert F_{0}\varPsi _{1}- F_{0}\varPsi _{2} \Vert \leq & \biggl\vert \frac{a}{a+b} \biggr\vert K^{i}_{1} \Vert \varPsi _{1}- \varPsi _{2} \Vert + \biggl\vert lT+ \frac{bdT}{(a+b)(c+d)} \biggr\vert K^{i}_{j} \Vert \varPsi _{1}- \varPsi _{2} \Vert \\ &{}+ \biggl\vert \frac{bT}{(a+b)(c+d)}+ \frac{T}{c+d} \biggr\vert k_{g_{2}} \Vert \varPsi _{1}- \varPsi _{2} \Vert + \biggl\vert \frac{1}{a+b} \biggr\vert K_{g_{1}} \Vert \varPsi _{1}- \varPsi _{2}\Vert . \end{aligned}$$

Hence one has

$$\begin{aligned} \Vert F_{0}\varPsi _{1}- F_{0}\varPsi _{2} \Vert \leq & \biggl( \biggl\vert \frac{a}{a+b} \biggr\vert K^{i}_{1}+ \biggl\vert lT+ \frac{bdT}{(a+b)(c+d)} \biggr\vert K^{i}_{j}+ \biggl\vert \frac{bT}{(a+b)(c+d)}+ \frac{T}{c+d} \biggr\vert k_{g_{2}} \\ &{}+ \biggl\vert \frac{1}{a+b} \biggr\vert K_{g_{1}} \biggr) \times \Vert \varPsi _{1}- \varPsi _{2} \Vert . \end{aligned}$$
(3.16)

Using

$$\begin{aligned} K_{1} =& \biggl( \biggl\vert \frac{a}{a+b} \biggr\vert K^{i}_{1}+ \biggl\vert lT+ \frac{bdT}{(a+b)(c+d)} \biggr\vert K^{i}_{j}+ \biggl\vert \frac{bT}{(a+b)(c+d)}+ \frac{T}{c+d} \biggr\vert k_{g_{2}} \\ &{}+ \biggl\vert \frac{1}{a+b} \biggr\vert K_{g_{1}} \biggr). \end{aligned}$$

Then (3.16) becomes

$$ \Vert F_{0}\varPsi _{1}- F_{0}\varPsi _{2} \Vert \leq K_{1} \Vert \varPsi _{1}- \varPsi _{2} \Vert .$$

Thus \(F_{0}\) is Lipschitz with constant \(K\in [0, 1)\).

For the growth relation using \((A_{2})\) and \((A_{5})\), we get the following:

$$\begin{aligned}& \begin{aligned} &\begin{aligned} \Vert F_{0}\varPsi \Vert &= \sup_{t\in [0, T]} \Biggl| \frac{a}{a+b} \sum_{j=1}^{k} I_{j}(\varPsi _{1}(t_{j}))+ \Biggl( \sum _{j=1}^{k} (t- t_{j})+ \frac{bdT}{(a+b)(c+d)} \\ &\quad {}- \biggl(\frac{b}{a+b}\biggr) \sum _{j=1}^{k}(T-t_{j})- \frac{td}{(c+d)} \Biggr) J_{j}(\varPsi (t_{j})) \\ &\quad {}+ \biggl(\frac{bT}{(a+b)(c+d)}+ \frac{t}{c+d} \biggr)g_{2}( \varPsi )+ \biggl(\frac{1}{a+b} \biggr) g_{1}(\varPsi ) \Biggr|, \end{aligned} \\ &\begin{aligned} \Vert F_{0}\varPsi \Vert &\leq \biggl\vert \frac{a}{a+b} \biggr\vert \bigl(C_{1} \Vert \varPsi \Vert + M_{1}\bigr)+ \biggl\vert lT+ \frac{bdT}{(a+b)(c+d)} \biggr\vert \bigl(C_{2} \Vert \varPsi \Vert + M_{2}\bigr) \\ &\quad {}+ \biggl\vert \frac{bT}{(a+b)(c+d)} + \frac{T}{c+d} \biggr\vert \bigl(C_{g_{2}} \Vert \varPsi \Vert + M_{g_{2}}\bigr) \\ &\quad {}+ \biggl\vert \frac{1}{a+b} \biggr\vert \bigl(C_{g_{1}} \Vert \varPsi \Vert + M_{g_{1}}\bigr). \end{aligned} \end{aligned} \end{aligned}$$
(3.17)

Then (3.17) becomes

$$\begin{aligned} \bigl\| \bigl|F_{0}\varPsi (t)\bigr|\bigr\| \leq A+ B \Vert \varPsi \Vert , \end{aligned}$$

where A and B are given in (3.14) and (3.15). □

Lemma 3.2

The operator

$$ F_{1}\varPsi (t): \varUpsilon \rightarrow \varUpsilon $$

is continuous and satisfies the following growth condition:

$$ \Vert F_{1}\varPsi \Vert \leq \biggl(1+ \biggl\vert \frac{b}{(a+b)} \biggr\vert \biggr) \frac{T^{\gamma }}{\varGamma (\gamma + 1)}\bigl( C_{f}+ M_{f} \Vert \varPsi \Vert \bigr). $$
(3.18)

Proof

Let \(\varPsi _{n}\) be a sequence in \(B_{k}= \{\varPsi \in Y: \|\varPsi \| \leq r\}\) such that \(\varPsi _{n}\rightarrow \varPsi \) as \(n\rightarrow \infty \).

This implies that

$$ \frac{(t- s)^{\gamma -1}}{\varGamma (\gamma )}\bigl(h(s, \varPsi _{n})- h(s, \varPsi )\bigr) \rightarrow 0,\quad n\rightarrow \infty , $$

and

$$ \frac{(T- s)^{\gamma -1}}{\varGamma (\gamma )}\bigl(h(s, \varPsi _{n})- h(s, \varPsi ) \bigr)\rightarrow 0,\quad n\rightarrow \infty . $$

Using the Lebesgue dominated convergence theorem, \(\|F_{1}(\varPsi _{n})- F_{1}(\varPsi )\| \rightarrow 0\) as \(n \rightarrow \infty \), this implies that \(F_{1}\) is continuous.

For the growth relation, using \((A_{3})\), we then find

$$\begin{aligned}& \begin{aligned} \bigl\vert F_{1}\bigl(\varPsi (t)\bigr) \bigr\vert &\leq \frac{1}{\varGamma (\gamma )} \int _{t_{k}}^{t} (t- s)^{\gamma -1} \bigl\vert h\bigl(s, \varPsi (s)\bigr) \bigr\vert \,ds \\ &\quad {}+ \biggl\vert \frac{b}{a+b} \biggr\vert \frac{1}{\varGamma (\gamma )} \int _{t_{k}}^{T}(T- s)^{\gamma -1} \bigl\vert h\bigl(s, \varPsi (s)\bigr) \bigr\vert \,ds , \end{aligned} \\& \sup_{t\in [0, T]} \bigl\vert F_{1}\bigl(\varPsi (t) \bigr) \bigr\vert \leq \sup_{t\in [0, T]} \biggl( \frac{(t-t_{k})^{\gamma }}{\varGamma (\gamma +1)}+ \biggl\vert \frac{b}{a+b} \biggr\vert \frac{(T-t_{k})^{\gamma }}{\varGamma (\gamma +1)} \biggr) \bigl( C_{f}+ M_{f} \Vert \varPsi \Vert \bigr) , \\& \Vert F_{1}\varPsi \Vert \leq \biggl(1+ \biggl\Vert \frac{b}{a+b} \biggr\Vert \biggr) \frac{T^{\gamma }}{\varGamma (\gamma + 1)}\bigl( C_{f}+ M_{f} \Vert \varPsi \Vert \bigr). \end{aligned}$$

 □

Theorem 3.2

The operator defined as\(F_{1}: Y\rightarrow Y\)is compact andψ-Lipschitz with constant zero.

Proof

Clearly \(F_{1}\) satisfies the growth condition so \(F_{1}\) is bounded on \(B_{k}= \{\varPsi \in Y: \|\varPsi \| \leq r\}\).

Let \(\varPsi \in B_{k}\), we have

$$ \Vert F_{1}\varPsi \Vert \leq \biggl(1+ \biggl\vert \frac{b}{a+b} \biggr\vert \biggr) \frac{T^{\gamma }}{\varGamma (\gamma + 1)}\bigl( C_{f}+ M_{f} \Vert \varPsi \Vert \bigr)\leq Q_{1}. $$

Hence \(F_{1}\) is bounded.

Further for \(0\leq t_{1}\leq t_{2}\leq T\), we show \(F_{1}\) is equi-continuous.

$$\begin{aligned}& \bigl\vert F_{1}\bigl(\varPsi (t_{1})\bigr)- F_{1}\bigl(\varPsi (t_{2})\bigr) \bigr\vert \\& \quad \leq \frac{1}{\varGamma (\gamma )} \int _{t_{k}}^{t_{1}} (t_{1}- s)^{\gamma -1} \bigl\vert h\bigl(s, \varPsi (s)\bigr) \bigr\vert \,ds- \frac{1}{\varGamma (\gamma )} \int _{t_{k}}^{t_{2}} (t_{2}- s)^{\gamma -1} \bigl\vert h\bigl(s, \varPsi (s)\bigr) \bigr\vert \,ds \\& \qquad {} + \frac{b}{a+b}\frac{1}{\varGamma (\gamma )} \int _{t_{k}}^{T}(T- s)^{ \gamma -1} \bigl\vert h\bigl(s, \varPsi (s)\bigr) \bigr\vert \,ds \\& \qquad {}- \frac{b}{a+b} \frac{1}{\varGamma (\gamma )} \int _{t_{k}}^{T}(T- s)^{\gamma -1} \bigl\vert h\bigl(s, \varPsi (s)\bigr) \bigr\vert \,ds. \end{aligned}$$

One has

$$ \bigl\vert F_{1}\bigl(\varPsi (t_{1})\bigr)- F_{1}\bigl(\varPsi (t_{2})\bigr) \bigr\vert \leq \frac{( C_{f}+ M_{f} \Vert \varPsi \Vert )}{\varGamma (\gamma +1)} \bigl( (t_{1}- t_{k})^{ \gamma }- (t_{2}- t_{k})^{\gamma } \bigr).$$

As \(t_{1}\rightarrow t_{2}\), then \(|F_{1}(\varPsi (t_{1}))- F_{1}(\varPsi (t_{2}))|\rightarrow 0\). Hence \(F_{1}\) is equi-continuous so \(F_{1}\) is compact.

Then, by Proposition 2.3, \(F_{1}\) is ψ-Lipschitz with constant zero. □

Lemma 3.3

The operator

$$ F_{2}: \varUpsilon \rightarrow \varUpsilon $$

is continuous and satisfies the following growth condition:

$$ \Vert F_{2}\varPsi \Vert \leq \biggl(l+ \biggl\vert \frac{bd}{(a+b)(c+d)} \biggr\vert \biggr) \frac{T^{\gamma +1}}{\varGamma (\gamma )}\bigl( C_{f}+ M_{f} \Vert \varPsi \Vert \bigr). $$
(3.19)

Proof

Let \(\varPsi _{n}\) be a sequence in \(B_{k}= \{\varPsi \in Y: \|\varPsi \| \leq r\}\) such that \(\varPsi _{n}\rightarrow \varPsi \) as \(n\rightarrow \infty \).

This implies that

$$ \frac{(t_{i}- s)^{\gamma -2}}{\varGamma (\gamma -1)}\bigl(h(s, \varPsi _{n})- h(s, \varPsi )\bigr) \rightarrow 0, \quad n\rightarrow \infty , $$

Using the Lebesgue dominated convergence theorem, we see that

$$ \bigl\Vert F_{2}(\varPsi _{n})- F_{2}(\varPsi ) \bigr\Vert \rightarrow 0 \quad \textit{as } n \rightarrow \infty . $$

This implies that \(F_{2}\) is continuous.

Further for the growth relation using \((A_{3})\), then one has

$$\begin{aligned} \bigl\vert F_{2}\bigl(\varPsi (t)\bigr) \bigr\vert \leq & \Biggl\vert \sum_{j=1}^{k} (t- t_{j})+ \frac{bdT}{(a+b)(c+d)}- \frac{b}{a+b} \sum _{j=1}^{k}(T-t_{j})- \frac{td}{c+d} \Biggr\vert \\ &{}\times \sum_{j=1}^{k} \int _{t_{j}-1}^{t_{j}} \frac{(t_{j}- s)^{\gamma -2}}{\varGamma (\gamma )} \bigl\vert h\bigl(s, \varPsi (s)\bigr) \bigr\vert \,ds , \end{aligned}$$

which yields

$$ \sup_{t\in [0,T]} \bigl\vert F_{2}\bigl(\varPsi (t) \bigr) \bigr\vert \leq \sup_{t\in [0,T]} \Biggl\vert \sum _{j=1}^{k} (t- t_{j})+ \frac{bdT}{(a+b)(c+d)} \Biggr\vert \times \int _{t_{j}-1}^{t_{j}} \frac{(t_{j}- s)^{\gamma -2}}{\varGamma (\gamma )} \bigl\vert h\bigl(s, \varPsi (s)\bigr) \bigr\vert \,ds.$$

Thus one has

$$ \bigl\Vert F_{2}(\varPsi ) \bigr\Vert \leq \biggl\vert l+ \frac{bd}{(a+b)(c+d)} \biggr\vert \times \frac{T^{\gamma +1}}{\varGamma (\gamma )} \bigl( C_{f}+ M_{f} \Vert \varPsi \Vert \bigr). $$

This is the required relation. □

Lemma 3.4

The operator defined as\(F_{2}: Y\rightarrow Y\)is compact and\(F_{2}\)isψ-Lipschitz with constant zero.

Proof

It can be easily derived similarly to Lemma 3.3. □

Lemma 3.5

The operator

$$\begin{aligned} F_{3}: \varUpsilon \rightarrow \varUpsilon \end{aligned}$$

is continuous and satisfies the following growth condition:

$$ \Vert F_{3}\varPsi \Vert \leq \biggl\vert \frac{bd}{(a+b)(c+d)} \biggr\vert \times \frac{T^{\gamma +1}}{\varGamma (\gamma )}\bigl( C_{f}+ M_{f} \Vert \varPsi \Vert \bigr).$$
(3.20)

Proof

Let \(\varPsi _{n}\) be a sequence in \(B_{k}= \{\varPsi \in Y: \|\varPsi \| \leq r\}\) such that \(\varPsi _{n}\rightarrow \varPsi \) as \(n\rightarrow \infty \).

This implies that

$$ \frac{(T- s)^{\gamma -2}}{\varGamma (\gamma -1)}\bigl(h(s, \varPsi _{n})- h(s, \varPsi )\bigr) \rightarrow 0,\quad n\rightarrow \infty . $$

Using the Lebesgue dominated convergence theorem

$$ \bigl\Vert F_{3}(\varPsi _{n})- F_{3}(\varPsi ) \bigr\Vert \rightarrow 0 \quad \text{as } n \rightarrow \infty . $$

This implies that \(F_{3}\) is continuous.

For the growth relation using \((A_{3})\), then

$$\begin{aligned}& \bigl\vert F_{3}\bigl(\varPsi (t)\bigr) \bigr\vert \leq \biggl\vert \frac{bdT}{(a+b)(c+d)}- \frac{td}{c+d} \biggr\vert \times \frac{1}{\varGamma (\gamma -1)} \int _{t_{k}}^{T}(T- s)^{\gamma -2} \bigl\vert h\bigl(s, \varPsi (s)\bigr) \bigr\vert \,ds , \\& \sup_{t\in [0,T]} \bigl\vert F_{3}\bigl(\varPsi (t) \bigr) \bigr\vert \leq \sup_{t\in [0,T]} \biggl\vert \frac{bdT}{(a+b)(c+d)} \biggr\vert \times \int _{t_{k}}^{T}(T- s)^{\gamma -2} \bigl\vert h\bigl(s, \varPsi (s)\bigr) \bigr\vert \,ds. \end{aligned}$$

Hence one has

$$ \bigl\Vert F_{3}(\varPsi ) \bigr\Vert \leq \biggl\vert \frac{bd}{(a+b)(c+d)} \biggr\vert \times \frac{T^{\gamma +1}}{\varGamma (\gamma )}\bigl( C_{f}+ M_{f} \Vert \varPsi \Vert \bigr). $$

This is the required relation. □

Lemma 3.6

The operator defined by\(F_{3}: Y\rightarrow Y\)is compact and\(F_{3}\)isψ-Lipschitz with constant zero.

Proof

The proof is so easy that we omit it. □

Lemma 3.7

The operator

$$ F_{4}: \varUpsilon \rightarrow \varUpsilon $$

is continuous and satisfies the following growth condition:

$$ \Vert F_{4}\varPsi \Vert \leq \biggl\vert \frac{a}{a+b} \biggr\vert \times \frac{lT^{\gamma }}{\varGamma (\gamma +1)}\bigl( C_{f}+ M_{f} \Vert \varPsi \Vert \bigr).$$
(3.21)

Proof

Let \(\varPsi _{n}\) be a sequence in \(B_{k}= \{\varPsi \in Y: \|\varPsi \| \leq r\}\) such that \(\varPsi _{n}\rightarrow \varPsi \) as \(n\rightarrow \infty \).

This implies that

$$ \frac{(t_{j}- s)^{\gamma -1}}{\varGamma (\gamma )}\bigl(h(s, \varPsi _{n})- h(s, \varPsi )\bigr) \rightarrow 0,\quad n\rightarrow \infty , $$

Using the Lebesgue dominated convergence theorem, \(\|F_{4}(\varPsi _{n})- F_{4}(\varPsi )\| \rightarrow 0\) as \(n \rightarrow \infty \). This implies that \(F_{4}\) is continuous.

For the growth relation using \((A_{3})\), then

$$\begin{aligned}& \bigl\vert F_{4}\bigl(\varPsi (t)\bigr) \bigr\vert \leq \biggl\vert \frac{a}{a+b} \biggr\vert \frac{1}{\varGamma (\gamma )} \sum _{j=1}^{k} \int _{t_{j}-1}^{t_{j}}(t_{j}- s)^{\gamma -1} \bigl\vert h\bigl(s, \varPsi (s)\bigr) \bigr\vert \,ds , \\& \sup_{t\in [0, T]} \bigl\vert F_{4}\bigl(\varPsi (t) \bigr) \bigr\vert \leq \sup_{t\in [0, T]} \biggl\vert \frac{a}{a+b} \biggr\vert \times \frac{(t_{j}- t_{j-1})^{\gamma }}{\varGamma (\gamma +1)} \times \bigl( C_{f}+ M_{f} \Vert \varPsi \Vert \bigr) , \\& \Vert F_{4}\varPsi \Vert \leq \biggl\vert \frac{a}{a+b} \biggr\vert \times \frac{lT^{\gamma }}{\varGamma (\gamma +1)}\bigl( C_{f}+ M_{f} \Vert \varPsi \Vert \bigr). \end{aligned}$$

This is the required relation. □

Lemma 3.8

The operator defined by\(F_{4}: Y\rightarrow Y\)is compact and\(F_{4}\)isψ-Lipschitz with constant zero.

Proof

The proof is so easy that we omit it. □

Next we show in combined form thta the four operators \(F_{1}\), \(F_{2}\), \(F_{3}\) and \(F_{4}\), satisfy the growth condition and are continuous, then we show the operators \(F_{1}\), \(F_{2}\), \(F_{3}\) and \(F_{4}\) are ψ-Lipschitz with constants zero and co-mpact.

Theorem 3.3

The operators\(F_{1}\), \(F_{2}\), \(F_{3}\)and\(F_{4}: \varUpsilon \rightarrow \varUpsilon \), are continuous and satisfy the following relation:

$$\begin{aligned}& \Vert F_{1}\varPsi \Vert + \Vert F_{2}\varPsi \Vert + \Vert F_{3}\varPsi \Vert + \Vert F_{4}\varPsi \Vert \\& \quad \leq \biggl(1+ \biggl\vert \frac{b}{(a+b)} \biggr\vert \frac{T^{\gamma }}{\varGamma (\gamma + 1)}+ \biggl\vert l+ \frac{bd}{(a+b)(c+d)} \biggr\vert \frac{T^{\gamma +1}}{\varGamma (\gamma )} \\& \qquad {} + \biggl\vert \frac{bd}{(a+b)(c+d)} \biggr\vert \times \frac{T^{\gamma +1}}{\varGamma (\gamma )}+ \biggl\vert \frac{a}{(a+b)} \biggr\vert \times \frac{lT^{\gamma }}{\varGamma (\gamma +1)} \biggr) \\& \qquad {} \times \bigl( C_{f}+ M_{f} \Vert \varPsi \Vert \bigr). \end{aligned}$$
(3.22)

Since\(F_{1}\), \(F_{2}\), \(F_{3}\)and\(F_{4}\), are continuous, \(F_{1}+ F_{2}+ F_{3}+ F_{4}\), is continuous.

Lemma 3.9

The operators\(F_{1}\), \(F_{2}\), \(F_{3}\)and\(F_{4}: \varUpsilon \rightarrow \varUpsilon \), are compact andψ-Lipschitz with constant zero.

Proof

Since \(F_{1}\), \(F_{2}\), \(F_{3}\) and \(F_{4}\), are compact, \(F_{1}+ F_{2}+ F_{3}+ F_{4}\), is compact, then, by Proposition 2.3, it is also ψ-Lipschitz with zero constant. □

Theorem 3.4

Presume that\((A_{1})\), \((A_{2})\), \((A_{3})\)and\((A_{4})\), hold, then the problem3.2has at least one solution\(\varPsi \in \varUpsilon \)and the set of the solutions of the problem3.2is bounded inϒ.

Proof

Let \(F_{0}\), \(F_{1}\), \(F_{2}\), \(F_{3}\) and \(F_{4}, T: \varUpsilon \rightarrow \varUpsilon \) be the operators defined in the section above. They are bounded and continuous. Furthermore, \(F_{0}\) is ψ-Lipschitz with constant \(K\in [0, 1)\) and \(F_{1}\), \(F_{2}\), \(F_{3}\) and \(F_{4}\) are ψ-Lipschitz with zero constants. Let

$$ H = \bigl\{ \varPsi \in \varUpsilon : \text{there exist } \lambda \in [0, 1] \text{ such that } \varPsi = \lambda T\varPsi \bigr\} . $$

We show that H is bounded in ϒ. Let \(\varPsi \in H\), \(\lambda \in [0, 1]\) such that \(\|\varPsi \| = \lambda \|T\varPsi \|\). It follows from (3.13) and (3.22) that

$$\begin{aligned}& \Vert \varPsi \Vert \leq \vert \lambda \vert \bigl( \Vert F_{0}\varPsi \Vert + \Vert F_{1}\varPsi \Vert + \Vert F_{2} \varPsi \Vert + \Vert F_{3}\varPsi \Vert + \Vert F_{4}\varPsi \Vert \bigr), \\& \begin{aligned} \Vert \varPsi \Vert &\leq \vert \lambda \vert \biggl[A+ B \Vert \varPsi \Vert + \biggl(1+ \biggl\vert \frac{b}{a+b} \biggr\vert \biggr) \frac{T^{\gamma }}{\varGamma (\gamma + 1)}+ \biggl(l+ \biggl\vert \frac{bd}{(a+b)(c+d)} \biggr\vert \biggr) \frac{T^{\gamma +1}}{\varGamma (\gamma )} \\ &\quad {}+ \biggl\vert \frac{bd}{(a+b)(c+d)} \biggr\vert \frac{T^{\gamma +1}}{\varGamma (\gamma )}+ \biggl\vert \frac{a}{a+b} \biggr\vert \times \frac{\operatorname{lT}^{\gamma }}{\varGamma (\gamma +1)} \biggr] \bigl( M_{f} \Vert \varPsi \Vert + C_{f}\bigr). \end{aligned} \end{aligned}$$

This inequality shows that H is bounded; if not, then we assume that \(\zeta = \|\varPsi \| \rightarrow \infty \). Dividing both sides of the above inequality by \(\|\varPsi \|\), we have

$$\begin{aligned} 1 \leq & \vert \lambda \vert \biggl[ \frac{ A+ B \Vert \varPsi \Vert + (1+ \vert \frac{b}{a+b} \vert )\frac{T^{\gamma }}{\varGamma (\gamma + 1)}+ (l+ \vert \frac{bd}{(a+b)(c+d)} \vert )\frac{T^{\gamma +1}}{\varGamma (\gamma )} }{\zeta } \\ &{}+ \frac{ \vert \frac{bd}{(a+b)(c+d)} \vert \frac{T^{\gamma +1}}{\varGamma (\gamma )}+ \vert \frac{a}{a+b} \vert \times \frac{lT^{\gamma }}{\varGamma (\gamma +1)} }{\zeta } \biggr] \times \bigl( M_{f} \Vert \varPsi \Vert + C_{f}\bigr). \end{aligned}$$

Taking the limit as \(\zeta \rightarrow \infty \), we get the relation \(1\leq 0\), which is not possible. Then we have the possibility that T is bounded in ϒ and has at least one fixed point. □

Theorem 3.5

Under the hypothesis\((A_{5})\)and if\(\mathbf{K}<1\)holds, where

$$\begin{aligned} \mathbf{K} =& \biggl[ \biggl(\frac{T^{\gamma }}{\varGamma (\gamma +1)}+ \biggl\vert \frac{a}{a+b} \biggr\vert \times \frac{lT^{\gamma }}{\varGamma (\gamma +1)}+ \biggl\vert \frac{b}{a+b} \biggr\vert \times \frac{T^{\gamma }}{\varGamma (\gamma +1)}+ \biggl\vert l \\ &{}+ \frac{bd}{(a+b)(c+d)} \biggr\vert \times \frac{lT^{\gamma +1}}{\varGamma (\gamma )} + \biggl\vert \frac{bd}{(a+b)(c+d)} \biggr\vert \times \frac{T^{\gamma }}{\varGamma (\gamma )} \biggr)K_{\varPsi }+ \biggl\vert \frac{a}{a+b} \biggr\vert K_{I}^{i} \\ &{}+ \biggl\vert l+ \frac{bd}{(a+b)(c+d)} \biggr\vert K_{J}^{i}+ \biggl\vert \frac{bT}{(a+b)(c+d)}+ \frac{1}{c+d} \biggr\vert K_{g_{2}}+ \biggl\vert \frac{1}{a+b} \biggr\vert K_{g_{1}} \biggr], \end{aligned}$$

the considered problem (1.1) has a unique solution.

Proof

Consider \(\varPsi _{1}, \varPsi _{2} \in \varUpsilon \) to be two solutions of the given problem (1.1), then

$$\begin{aligned}& \Vert T\varPsi _{1}- T\varPsi _{2} \Vert \leq \biggl[ \frac{K_{\varPsi }T^{\gamma }}{\varGamma (\gamma +1)}+ \biggl\vert \frac{a}{a+b} \biggr\vert \times \frac{K_{\varPsi } l T^{\gamma }}{\varGamma (\gamma +1)}+ \biggl\vert \frac{b}{a+b} \biggr\vert \times \frac{K_{\varPsi }T^{\gamma }}{\varGamma (\gamma +1)} \\& \hphantom{\Vert T\varPsi _{1}- T\varPsi _{2} \Vert \leq{}}{}+ \biggl\vert l+ \frac{bd}{(a+b)(c+d)} \biggr\vert \times \frac{K_{\varPsi }lT^{\gamma +1}}{\varGamma (\gamma )} + \biggl\vert \frac{bd}{(a+b)(c+d)} \biggr\vert \times \frac{K_{\varPsi }T^{\gamma }}{\varGamma (\gamma )} \\& \hphantom{\Vert T\varPsi _{1}- T\varPsi _{2} \Vert \leq{}}{}+ \biggl\vert \frac{a}{a+b} \biggr\vert K_{I}^{i}+ \biggl\vert l+ \frac{bd}{(a+b)(c+d)} \biggr\vert K_{J}^{i} \\& \hphantom{\Vert T\varPsi _{1}- T\varPsi _{2} \Vert \leq{}}{} + \biggl\vert \frac{bT}{(a+b)(c+d)}+ \frac{1}{c+d} \biggr\vert K_{g_{2}}+ \biggl\vert \frac{1}{a+b} \biggr\vert K_{g_{1}} \biggr] \Vert \varPsi _{1}-\varPsi _{2} \Vert , \\& \Vert T\varPsi _{1}- T\varPsi _{2} \Vert \leq \biggl[ \biggl( \frac{T^{\gamma }}{\varGamma (\gamma +1)}+ \biggl\vert \frac{a}{a+b} \biggr\vert \times \frac{lT^{\gamma }}{\varGamma (\gamma +1)}+ \biggl\vert \frac{b}{a+b} \biggr\vert \times \frac{T^{\gamma }}{\varGamma (\gamma +1)} \\& \hphantom{\Vert T\varPsi _{1}- T\varPsi _{2} \Vert \leq{}}{}+ \biggl\vert l+ \frac{bd}{(a+b)(c+d)} \biggr\vert \times \frac{lT^{\gamma +1}}{\varGamma (\gamma )}+ \biggl\vert \frac{bd}{(a+b)(c+d)} \biggr\vert \times \frac{T^{\gamma }}{\varGamma (\gamma )} \biggr)K_{\varPsi } \\& \hphantom{\Vert T\varPsi _{1}- T\varPsi _{2} \Vert \leq{}}{}+ \biggl\vert \frac{a}{a+b} \biggr\vert K_{I}^{i}+ \biggl\vert l+ \frac{bd}{(a+b)(c+d)} \biggr\vert K_{J}^{i}+ \biggl\vert \frac{bT}{(a+b)(c+d)}+ \frac{1}{c+d} \biggr\vert K_{g_{2}} \\& \hphantom{\Vert T\varPsi _{1}- T\varPsi _{2} \Vert \leq{}}{} + \biggl\vert \frac{1}{a+b} \biggr\vert K_{g_{1}} \biggr] \Vert \varPsi _{1}-\varPsi _{2} \Vert . \end{aligned}$$

Thus T is a contraction mapping and by the Banach contraction theorem T has a unique fixed point. Thus the considered problem has a unique solution. □

4 Test problem

Example 4.1

Let \(\gamma = \frac{3}{2}\), \(T= 1\), \(a= b= 1\), \(c= d= 0.4\), and the subsequent problem be

$$ \textstyle\begin{cases} D^{\frac{3}{2}} \varPsi (t)= \frac{\sin (t)}{16+ \vert \varPsi (t) \vert }, \quad t\in [0, 1], t \neq \frac{1}{5}, \\ \delta \varPsi (\frac{1}{5} )= I_{j}(\varPsi (\frac{1}{5} ))= \frac{4+ \vert \varPsi \vert ^{2}}{30+ \vert \varPsi \vert ^{2}}, \\ \delta \varPsi ^{\prime } (\frac{1}{5} )= J_{j}(\varPsi ^{\prime } (\frac{1}{5} ))= \frac{20+ \vert \varPsi \vert ^{2}}{30+ \vert \varPsi \vert ^{2}}, \\ \varPsi (0)+ \varPsi (1)= \frac{\cos (\varPsi )}{10}, \qquad 0.4 \varPsi ^{\prime }(0) + 0.4 \varPsi ^{\prime }(1)= \frac{\sin (\varPsi )}{20}. \end{cases} $$

As regards Theorem 3.4, it is not difficult to see that all the hypotheses are satisfied. Thus, the said problem has a solution in ϒ. For the uniqueness we presume \(K_{\varPsi }= \frac{1}{16}\), \(K_{g_{1}}= \frac{1}{10}\), \(K_{g_{2}}= \frac{1}{20}\), \(K^{1}_{I}=\frac{1}{30}\), \(K^{1}_{J} =\frac{1}{30}\), \(l=1\), and by simple computation we have

$$\begin{aligned} K =& \biggl(\frac{1}{\frac{3}{4}\times \sqrt{(\pi )}}+ \frac{0.5\times 1}{\frac{3}{4}\times \sqrt{(\pi )}}+ \frac{0.5}{\frac{3}{4}\times \sqrt{(\pi )}}+ \frac{1.25}{\frac{1}{2}\times \sqrt{(\pi )}}+ \frac{0.25}{\frac{1}{2}\times \sqrt{(\pi )}} \biggr)\times \frac{1}{16} \\ &{}+ \biggl(0.5 \times \frac{1}{30} \biggr)+ \biggl(0.25 \times \frac{1}{30} \biggr)+ \biggl(1.875\times \frac{1}{20} \biggr)+ \biggl(0.5 \times \frac{1}{10} \biggr) \\ =&\frac{357}{1000}. \end{aligned}$$
(4.1)

From (4.1), we see \(K<1\). Hence all the conditions of Theorem 3.5 hold. Consequently the given problem (1.1) under RBCs has a unique solution in ϒ.

5 Conclusion

With the support of a degree method of topology, we found sufficient conditions for the impulsive problem (1.1) and by a well-known Banach contraction theorem the uniqueness conditions have been obtained. A test example is presented and it is observed that our results are effective under some conditions.