Approximations of Solutions for an Impulsive Fractional Differential Equation with a Deviated Argument

Abstract

In the present work, we consider an impulsive fractional differential equation with a deviated argument in an arbitrary separable Hilbert space H. We obtain an associated integral equation and then consider a sequence of approximate integral equations. The existence and uniqueness of solutions to every approximate integral equation is obtained by using analytic semigroup and Banach fixed point theorem. Next we demonstrate the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. We study the Faedo–Galerkin approximation of the solution and establish some convergence results. Finally, we consider an example to show the effectiveness of obtained theory.

Introduction

In recent few decades, researcher has developed great interest in fractional calculus due to its wide applicability in science and engineering. Tools of fractional calculus have been available and applicable to deal with many physical and real world problems such as anomalous diffusion process, traffic flow, nonlinear oscillation of earthquake, real system characterized by power laws, critical phenomena, scale free process, describe viscoelastic materials and many others. The details on the theory and its applications can be found in [1–4] and papers [5–9] and references cited therein.

On the other hand, many real world processes and phenomena which are subjected during their development to short-term external influences can be modeled as impulsive differential equation with fractional order which have been used efficiently in modelling many practical problems. Their duration is negligible compared with the total duration of the entire process and phenomena. Such process is investigated in various fields such as biology, physics, control theory, population dynamics, economics, chemical technology, medicine and so on. For the study for impulsive differential equation, we refer to monograph [10, 11], and papers [12–23] and references given therein.

The purpose of this work is to establish the approximation of the solution to following differential equation with deviated argument in a separable Hilbert space $(H,\Vert \;\cdot \Vert ,(\cdot ,\cdot ))$

$$\begin{aligned} {}^{c} D^{q}_{0^+}x(t)= & {} -Ax(t)+f\left( t,x(t),x(a(x(t),t))\right) ,\nonumber \\ 0\le & {} t\le T_0<\infty ,\;\;t\ne t_i, \end{aligned}$$

(1)

$$\begin{aligned} \Delta x(t_i)= & {} I_i(x(t_i)),\;\;i=1,2,\cdots ,p,\;\; p\in {\mathbb {N}} \end{aligned}$$

(2)

$$\begin{aligned} x(0)= & {} u_0, \end{aligned}$$

(3)

where $0<q<1$, $^c D^q_{0^+}$ is the fractional derivative in Caputo sense with single base point 0, $0=t_0<t_1<\cdots <t_p<t_{p+1}=T_0$ are pre-fixed numbers, $\Delta x|_{t=t_i}=x(t_i^+)-x(t_i^-)$ and $x(t_i^+)=\lim _{h\rightarrow 0+}x(t_i+h)$ and $x(t_i^-)=\lim _{h\rightarrow 0-}x(t_i+h)$ denote the right and left limits of x(t) at $t=t_i$, respectively. In (1), $A:D(A)\subset H\rightarrow H$ is a closed, positive definite and self adjoint linear operator with dense domain D(A). We assume that $-A$ is the infinitesimal generator of an analytic semigroup of bounded linear operators on H. The functions $f:[0,T_0]\times H^2\rightarrow H$, $a:H\times [0,T_0]\rightarrow {\mathbb {R}}$, $I_i:H\rightarrow H$ are appropriate functions to be mentioned later. For more details of differential equation with deviating argument, we refer to papers [24–26] and references given therein.

In the present work, we investigate the Faedo–Galerkin approximations of the solutions for (1)–(3). The Faedo–Galerkin approximations of the solutions in a separable Hilbert space to the following system

$$\begin{aligned} x'(t)+Ax(t)= & {} M(x(t)),\;\;\;x(0)=u_0 \end{aligned}$$

(4)

has been studied first by Miletta [27] under the assumption that $-A$ is the infinitesimal generator of an analytic semigroup and the nonlinear function M is Lipschitz continuous on a ball in $D(A^\alpha )$, $0<\alpha <1$. Bahuguna and Srivastava [28] has discussed the more general cases. For a nice introduction on existence of an approximate solution and associated study of different problems are broadly talked about in the references [28–34].

The organization of the article is as follows: In Sect. 2, We provide some basic definitions, lemmas and theorems as preliminaries as these are useful for proving our results. In Sect. 3, we prove the existence and uniqueness of the approximate solutions by using analytic semigroup and Banach fixed point theorem. In Sect. 4, we show the convergence of the solution to each of the approximate integral equations with the limiting function which satisfies the associated integral equation and the convergence of the approximate Feado-Galerkin solutions will be shown in Sect. 5. In Sect. 6, we provide an example to illustrate the obtained theory.

Preliminaries and Assumptions

In this segment, some basic definitions, preliminaries, Theorems and Lemmas and assumptions which will be used to prove existence result, is stated.

Throughout the work, we assume that $(H,\Vert \;\cdot \Vert ,<\cdot ,\cdot >)$ is a separable Hilbert space. The symbol $C([0,T_0];H)$ stands for the Banach space of all the continuous functions from $[0,T_0]$ into H equipped with the norm $\Vert \;z(t)\Vert _C=\sup _{t\in [0,T_0]}\Vert \;z(t)\Vert _H$ and $L^p((0,T_0);H)$ stands for Banach space of all Bochner-measurable functions from $(0,T_0)$ to H with the norm

$$\begin{aligned} \Vert \;z\Vert _{L^p}=\left( \int _{(0,T_0)}\Vert \;z(s)\Vert ^p_H ds\right) ^{1/p}. \end{aligned}$$

Since $-A$ is the infinitesimal generator of an analytic semigroup of bounded linear operators $\{ {\mathcal {T}}(t);t\ge 0\}$. Therefore, there exist constants $C\ge 1$ and $\delta \ge 0$ such that $\Vert \;{\mathcal {T}}(t)\Vert \le Ce^{\delta t}$, $t\ge 0$. In addition, we note that

$$\begin{aligned} \Vert \;\frac{d^j}{dt^j} {\mathcal {T}}(t)\Vert \le M_j,\;\;t>t_0, \;t_0>0, \end{aligned}$$

(5)

where $M_j$ are some positive constants. Henceforth, without loss of generality, we might accept that ${\mathcal {T}}(t)$ is uniformly bounded by M i.e., $\Vert \;{\mathcal {T}}(t)\Vert \le M$ and $0\in \rho (-A)$ i.e., $-A$ is invertible. This permits us to define the positive fractional power $A^\alpha$ as closed linear operator with domain $D(A^\alpha )\subseteq H$ for $\alpha \in (0,1]$. Moreover, $D(A^\alpha )$ is dense in H with the norm

$$\begin{aligned} \Vert \;y\Vert _\alpha =\Vert \;A^\alpha y\Vert . \end{aligned}$$

(6)

Hence, we signify the space $D(A^\alpha )$ by $H_\alpha$ endowed with the $\alpha$-norm $(\Vert \;\cdot \Vert _\alpha )$. Also, we have that $H_\kappa \hookrightarrow H_\alpha$ for $0<\alpha <\kappa$ and therefore, the embedding is continuous. Then, we define $H_{-\alpha }=(H_\alpha )^*$, for each $\alpha >0$. The space $H_{-\alpha }$ stands for the dual space of $H_\alpha$, is a Banach space with the norm $\Vert \;z\Vert _{-\alpha }=\Vert \;A^{-\alpha }z\Vert$. For study on the fractional powers of closed linear operators, we refer to book by Pazy [35].

Lemma 2.1

Let $-A$ be the infinitesimal generator of an analytic semigroup $\{ {\mathcal {T}}(t):{t\ge 0}\}$ such that $\Vert \;{\mathcal {T}}(t)\Vert \le M$, for $t\ge 0$ and $0\in \rho (-A)$. Then,

• (i) For $0\le \alpha \le 1$, $H_\alpha$ is a Hilbert space.

• (ii) The operator $A^\alpha {\mathcal {T}}(t)$ is bounded for every $t>0$ and

  $$\begin{aligned} \Vert \;A{\mathcal {T}}(t)\Vert\le & {} M t^{-1}, \end{aligned}$$

  (7)
  $$\begin{aligned} \Vert \;A^\alpha {\mathcal {T}}(t)\Vert\le & {} M_\alpha t^{-\alpha }. \end{aligned}$$

  (8)

Now, we state some basic definitions and properties of fractional calculus.

Definition 2.1

The Riemann–Liouville fractional integral operator J is defined as

$$\begin{aligned} J^q_{0^+} F(t)=\frac{1}{\Gamma (q)}\int ^t_0(t-s)^{q-1}F(s)ds, \end{aligned}$$

(9)

where $F\in L^1((0,T_0);H)$ and $q>0$ is the order of the fractional integration.

Definition 2.2

The Riemann-Liouville fractional derivative is given as

$$\begin{aligned} ^{RL} D^q_{0+} F(t)=D^m_t J^{m-q}_{0+} F(t),\;m-1<q<m,\; m\in {\mathbb {N}}, \end{aligned}$$

(10)

where $D^m_t=\frac{d^m}{dt^m},\;F\in L^1((0,T_0);H),\;J^{m-q}_{0+}\in W^{m,1}((0,T_0);H)$.

Definition 2.3

The Caputo fractional derivative is given as

$$\begin{aligned} ^c D^q_{0^+} F(t)=\frac{1}{\Gamma (m-q)}\int ^t_0 (t-s)^{m-q-1}F^m(s)ds,\;\;m-1<q<m, \end{aligned}$$

(11)

where $F\in C^{m-1}((0,T_0);H)\cap L^1((0,T_0);H)$.

We denote by ${\mathcal {C}}^\alpha _t=PC([0,t];H_\alpha ),\;t\in (0,T_0]$ the space of all $H_\alpha$-valued functions on [0, t] such that x(t) is continuous on $t\ne t_i$, left continuous at $t=t_i$ and the right limit $x(t_i^{+})$ exists for $i=1,\cdots ,p$. It is clear that ${\mathcal {C}}^\alpha _t$ is a Banach space endowed with the norm

$$\begin{aligned} \Vert \;y\Vert _{t,\alpha }=\sup _{s\in [0,t]}\Vert \;y(s)\Vert _\alpha ,\;\;\;y\in {\mathcal {C}}^\alpha _t. \end{aligned}$$

For $0\le \alpha <1$, we define

$$\begin{aligned} {\mathcal {C}}^{\alpha -1}_t=\{x\in {\mathcal {C}}^\alpha _t:\Vert \;x(\tau )-x(s)\Vert \le {\mathcal {L}}|\tau -s|,\; \text {for all}\;\tau ,s\in [0,t] \}, \end{aligned}$$

(12)

where ${\mathcal {L}}>0$ is a appropriate constant to be defined later.

Now, we introduce the following assumptions on A, f, a and $I_i\;(i=1,\cdots ,p)$:

• (A1) A is a closed, densely defined, positive definite and self-adjoint linear operator from $D(A)\subset H$ into H. We assume that operator A has the pure point spectrum

  $$\begin{aligned} 0<\lambda _0\le \lambda _1\le \lambda _2\le \cdots \le \lambda _m\le \cdots , \end{aligned}$$

  (13)

  with $\lambda _m\rightarrow \infty$ as $m\rightarrow \infty$ and a corresponding complete orthonormal system of eigenfunctions $\{\phi _j\}$, i.e.,

  $$\begin{aligned} A\phi _j=\lambda _j \phi _j,\;\;\text {and}\;\;<\phi _l,\phi _j>=\delta _{lj}, \end{aligned}$$

  (14)

  where

  $$\begin{aligned} \delta _{lj}={\left\{ \begin{array}{ll}1,\;\;j=l,\\ 0,\;\; \text {otherwise}. \end{array}\right. } \end{aligned}$$

• (A2) Let $W_1\subset Dom(f)$ be an open subset of ${\mathbb {R}}_+\times H_\alpha \times H_{\alpha -1}$, where $\alpha \in [0,1)$. For $(\tau ,x,y)\in W_1$, there is a neighborhood $U_1\subset W_1$ of $(\tau ,x,y)$ and positive constants ${\mathcal {L}}_f={\mathcal {L}}_f({\tau ,x,y,U_1})$ such that

  $$\begin{aligned} \Vert \;f(t,x_1,y_1)-f(s,x_2,y_2)\Vert \le {\mathcal {L}}_f[|t-s|^{\mu _1}+\Vert \;x_1-x_2\Vert _\alpha +\Vert \;y_1-y_2\Vert _{\alpha -1}],\nonumber \\ \end{aligned}$$

  (15)

  for all $(t, x_1, y_1),(s, x_2, y_2)\in U_1$ and $0<\mu _1\le 1$.

• (A3) For each $(x,\tau )\in W_2$, where $W_2\subset Dom(a)$ is an open subset of $H_\alpha \times {\mathbb {R}}_+$, there is a neighborhood $U_2\subset W_2$ of $(x,\tau )$ and positive constant ${\mathcal {L}}_a={\mathcal {L}}_a(x, \tau , U_2)$ such that $a(\cdot ,\cdot ):H_\alpha \times {\mathbb {R}}_+\rightarrow {\mathbb {R}}_+$, $a(\cdot , 0)=0$,

  $$\begin{aligned} |a(x_1,t_1)-(x_2,t_2)|\le {\mathcal {L}}_a\left[ \Vert \;x_1-x_2\Vert _{\alpha }+|t_1-t_2|^{\mu _2}\right] , \end{aligned}$$

  (16)

  for all $(x_1,t_1),(x_2,t_2)\in U_2$, $0<\mu _2\le 1$.

• (A4) All the function $I_i:H_\alpha \rightarrow H_\alpha , (i=1,\cdots ,p)$ are continuous function such that

  • (i) $\Vert \;I_i(u)\Vert _\alpha \le L_i$, for all $\alpha \in (0,1)$.

  • (ii) $\Vert \;I_i(u_1)-I_i(u_2)\Vert _\alpha \le N_i\Vert \;u_1-u_2\Vert _\alpha$, for all $u_1,u_2\in H_\alpha$.

where $L_{i}$ and $N_{i}, i$ = 1, $\ldots$, p are positive constants.

From [12], we adopt the following thought of solution.

Definition 2.4

A piecewise continuous function $x:[0,T_0]\rightarrow H$ is said to be a mild solution for the system (1)–(3) if $x\in {\mathcal {C}}^\alpha _{T_0}\cap {\mathcal {C}}^{\alpha -1}_{T_0}$ and satisfy the following impulsive integral equation

$$\begin{aligned} x(t)= & {} {\mathcal {S}}_q(t)u_0+\int ^t_0(t-s)^{q-1}{\mathcal {T}}_q(t-s)f(s,x(s),x(a(x(s),s)))ds\nonumber \\&+\sum _{0<t_i<t}{\mathcal {S}}_q(t-t_i)I_i(x(t_i)),\;\;\text {for all}\;\;t\in [0,T_0]. \end{aligned}$$

(17)

The operator ${\mathcal {S}}_q(t)$ and ${\mathcal {T}}_q(t)$ are defined as follows:

$$\begin{aligned} {\mathcal {S}}_q(t)= & {} \int ^\infty _0\zeta _q(\xi ){\mathcal {T}}(t^q\xi )d\xi , \end{aligned}$$

(18)

$$\begin{aligned} {\mathcal {T}}_q(t)= & {} q\int ^\infty _0\xi \zeta _q(\xi ){\mathcal {T}}(t^q\xi )d\xi , \end{aligned}$$

(19)

where $\zeta _q(\xi )=\frac{1}{q}\xi ^{1-1/q}\times \psi _q(\xi ^{-\frac{1}{q}})$ is a a probability density function defined on $(0,\infty )$ i.e., $\zeta _q(\xi )\ge 0$, $\int ^\infty _0\zeta _q(\xi )d\xi =1$ and

$$\begin{aligned} \psi _q(\xi )=\frac{1}{\pi }\sum _{n=1}^\infty (-1)^{n-1}\xi ^{-nq-1}\frac{\Gamma (nq+1)}{n!}\sin (n\pi q),\;\xi \in (0,\infty ). \end{aligned}$$

For more details of probability function and generalized functions, we refer to papers [36–39].

Lemma 2.2

The operator ${\mathcal {S}}_q(t),\;t\ge 0$ and ${\mathcal {T}}_q(t),\;t\ge 0$ are bounded linear operators and satisfy

• (i) $\Vert \;{\mathcal {S}}_q(t)y\Vert \le M\Vert \;y\Vert ,\;\Vert \;{\mathcal {T}}_q(t)y\Vert \le \frac{qM}{\Gamma (1+q)}\Vert y\Vert \;\text {and}\;\Vert \;A^\alpha {\mathcal {T}}_q(t)y\Vert \le \frac{qM_\alpha \Gamma (2-\alpha )t^{-q\alpha }}{\Gamma (1+q(1-\alpha ))}$ $\Vert \;y\Vert$, for any $y\in H$.

• (ii) The families $\{ {\mathcal {S}}_q(t):t\ge 0\}$ and $\{ {\mathcal {T}}_q(t):t\ge 0\}$ are strongly continuous.

• (iii) If ${\mathcal {T}}(t)$ is compact, then ${\mathcal {S}}_q(t)$ and ${\mathcal {T}}_q(t)$ are compact operators for any $t>0$.

Approximate Solutions and Convergence

The existence of approximate solutions for the problem (1) is established in this section.

Let ${\mathcal {H}}_n$ be the finite dimensional subspace of H which is spanned by $\{\phi _0,\phi _1,\cdots ,\phi _n\}$ and $P^n:H\rightarrow {\mathcal {H}}_n$ be the corresponding projection operator for $n=0,1,2,\cdots ,$. We define

$$\begin{aligned} f_n:{\mathbb {R}}_+\times H^2\rightarrow H,\;\; \text {and}\;\;I_{i,n}:H\rightarrow H, \end{aligned}$$

(20)

by

$$\begin{aligned} f_{n}\left( t,x(t),x(a(x(t),t))\right) =f\left( t, P^nx(t),P^nx(a(x(t),t))\right) , \end{aligned}$$

(21)

and

$$\begin{aligned} I_{i,n}(x)=I_i(P^n x),\;\forall \;x\in H,\;n=0,1,2,\ldots , \end{aligned}$$

(22)

for $i=1,2,\ldots ,p,$ respectively. We choose T, $0<T\le T_0$ sufficiently small such that

$$\begin{aligned}&T<\Big \{\frac{2R}{3}\left[ \frac{(1-\alpha )\Gamma (1+q(1-\alpha ))}{M_\alpha N_f\Gamma (2-\alpha )}\right] \Big \}^{\frac{1}{q(1-\alpha )}}, \end{aligned}$$

(23)

$$\begin{aligned}&\Vert \;[{\mathcal {S}}_q(t)-I]u_0\Vert _\alpha +M\sum _{i=1}^p L_i<\frac{R}{3}, \end{aligned}$$

(24)

$$\begin{aligned}&\frac{M_\alpha \Gamma (2-\alpha ) T^{q(1-\alpha )}}{(1-\alpha )\Gamma (1+q(1-\alpha ))}\left[ {\mathcal {L}}_f(1 +{\mathcal {L}}{\mathcal {L}}_a)\right] +M\sum _{i=1}^p N_i<1. \end{aligned}$$

(25)

Now, we consider

$$\begin{aligned} {\mathcal {B}}=\big \{y\in {\mathcal {C}}^\alpha _T\cap {\mathcal {C}}^{\alpha -1}_T: y(0)=u_0,\;\Vert \;y-u_0\Vert _{T,\alpha }\le R \big \}. \end{aligned}$$

(26)

By the assumptions $(A2)-(A3)$, we have that f is continuous on [0, T]. Therefore, there exist a constant $N_f>0$ such that

$$\begin{aligned} N_f={\mathcal {L}}_f\left[ T^{\mu _1} +R(1+{\mathcal {L}}{\mathcal {L}}_a)+{\mathcal {L}}{\mathcal {L}}_a T^{\mu _2}\right] +N,\;\;\text {where}\;\;N=\Vert \;f(0,u_0,u_0)\Vert , \end{aligned}$$

(27)

with

$$\begin{aligned} \Vert \;f(\tau ,x(\tau ),x(a(x(\tau ),\tau )))\Vert \le N_f,\;\;x\in H,\;\;\;\tau \in [0,T]. \end{aligned}$$

(28)

Now, we define the operator $Q_n$ on ${\mathcal {B}}$ as follows

$$\begin{aligned} Q_n x(t)= & {} {\mathcal {S}}_q(t)u_0 +\int ^t_0(t-s)^{q-1}{\mathcal {T}}_q(t-s)f_n(s,x(s),x(a(x(s),s)))ds \nonumber \\&+\sum _{i=1}^{p}{\mathcal {S}}_q(t-t_i)I_{i,n}(x(t_i)), \end{aligned}$$

(29)

for $\;t\in [0,T]$ and $x\in {\mathcal {B}}$.

Theorem 3.1

Suppose $(A1)-(A4)$ holds and $u_0\in D(A^\alpha )$, for $0\le \alpha <1$. Then, there exists a unique fixed point $x_n\in {\mathcal {C}}^\alpha _{T}\cap {\mathcal {C}}^{\alpha -1}_T$ of the map Q i.e., $Q_n x_n=x_n$ for each $n=0,1,2,\cdots ,$ and $x_n$ satisfies the following approximate integral equation

$$\begin{aligned} x_n(t)= & {} {\mathcal {S}}_q(t)u_0 +\int ^t_0(t-s)^{q-1}{\mathcal {T}}_q(t-s)f_n(s,x_n(s),x_n(a(x_n(s),s)))ds \nonumber \\&+\sum _{i=1}^{p}{\mathcal {S}}_q(t-t_i)I_{i,n}(x_n(t_i)), \end{aligned}$$

(30)

for $\;t\in [0,T]$.

Proof

To demonstrate the theorem, we first need to show that $Q_nx\in {\mathcal {C}}^\alpha _T\cap {\mathcal {C}}^{\alpha -1}_T$. It is clear that $Q_n:{\mathcal {C}}^\alpha _T\rightarrow {\mathcal {C}}^\alpha _T$. Now, it remains to show that $Q_n x\in {\mathcal {C}}^{\alpha -1}_T$. For $x\in {\mathcal {C}}^{\alpha -1}_T$, $0<\tau <t<T$, then we have

$$\begin{aligned}&\Vert \;Q_n x(t)-Q_n x(\tau )\Vert _{\alpha -1}\\&\quad \le \Vert \;\left[ {\mathcal {S}}_q(t) -{\mathcal {S}}_q(\tau )\right] u_0\Vert _{\alpha -1} +\int ^\tau _0\Vert \;(t-s)^{q-1}{\mathcal {T}}_q(t-s) -(\tau -s)^{q-1}{\mathcal {T}}_q(\tau -s)\Vert _{\alpha -1}\\&\qquad \times \Vert \;f_n(s,x(s),x(a(x(s),s)))\Vert ds\nonumber \\&\qquad +\int ^t_\tau \Vert (t-s)^{q-1}{\mathcal {T}}_q(t-s)\Vert _{\alpha -1}\Vert \; f_n(s,x(s),x(a(x(s),s)))\Vert ds,\\&\qquad +\sum _{i=1}^p \Vert \; \left[ {\mathcal {S}}_q(t-t_i)-{\mathcal {S}}_q(\tau -t_i)\right] I_{i,n}(x(t_i))\Vert _{\alpha -1}. \end{aligned}$$

From the first term of above inequality, we have

$$\begin{aligned} \left[ {\mathcal {S}}_q(t) -{\mathcal {S}}_q(\tau )\right] A^{\alpha -1}u_0 =\int ^\infty _0\zeta _q(\xi )\left[ {\mathcal {T}}(t^q\xi )-{\mathcal {T}}(\tau ^q\xi )\right] A^{\alpha -1}u_0d\xi , \end{aligned}$$

(31)

Also, we have that for each $z\in H$

$$\begin{aligned} \left[ {\mathcal {T}}(t^q\xi ) -{\mathcal {T}}(\tau ^q\xi )\right] z =\int ^t_\tau \frac{d}{ds}{\mathcal {T}}(s^q\xi )zds=\int ^t_\tau q\xi s^{q-1}A{\mathcal {T}}(s^q\xi )zds. \end{aligned}$$

(32)

Therefore, we estimate the first term as

$$\begin{aligned}&\int ^\infty _0\zeta _q(\xi )\Vert {\mathcal {T}}(t^q\xi )-{\mathcal {T}}(\tau ^q\xi )\Vert \Vert \;A^{\alpha -1}u_0\Vert d\xi \nonumber \\&\quad \le \int ^\infty _0\zeta _q(\xi )[\int ^t_\tau \Vert \frac{d}{ds}{\mathcal {T}}(s^q\xi )\Vert ]\Vert \;u_0\Vert _{\alpha -1}d\xi ,\nonumber \\&\quad \le \int ^\infty _0\zeta _q(\xi )[M_1(t-\tau )]\Vert \;u_0\Vert _{\alpha -1}d\xi ,\nonumber \\&\quad \le K_1(t-\tau )\int ^\infty _0\zeta _q(\xi )d\xi ,\nonumber \\&\quad =K_1 (t-\tau ), \end{aligned}$$

(33)

where $K_1=M_1\Vert \;u_0\Vert _{\alpha -1}$. The second integrals is estimated as

$$\begin{aligned}&\int ^\tau _0\Vert \;(t-s)^{q-1}{\mathcal {T}}_q(t-s) -(\tau -s)^{q-1}{\mathcal {T}}_q(\tau -s)\Vert _{\alpha -1}\Vert \; f_n(s,x(s),x(a(x(s),s)))\Vert ds\nonumber \\&\quad \le \int ^\tau _0\int ^\infty _0\zeta _q(\xi )\Vert \; \left[ \frac{d}{d\varsigma }{\mathcal {T}}((\varsigma -s)^q\xi )|_{\varsigma =t} -\frac{d}{d\varsigma }{\mathcal {T}}((\varsigma -s)^q\xi )|_{\varsigma =\tau }\right] A^{\alpha -2}\Vert \nonumber \\&\qquad \times \Vert \; f_n(s,x(s),x(a(x(s),s)))\Vert d\xi ds,\nonumber \\&\quad \le \int ^\tau _0\int ^\infty _0\zeta _q(\xi ) \left[ \int ^t_\tau \Vert \; A^{\alpha -2} \frac{d^2}{d\varsigma ^2} {\mathcal {T}}((\varsigma -s)^q\xi )\Vert d\varsigma \right] N_fd\xi ds,\nonumber \\&\quad \le \int ^\tau _0\int ^\infty _0\zeta _q(\xi )\left[ \Vert \;A^{\alpha -2}\Vert M_2(t-\tau )\right] N_fd\xi ds,\nonumber \\&\quad \le K_2(t-\tau ), \end{aligned}$$

(34)

where $K_2=\Vert \;A^{\alpha -2}\Vert M_2 N_f T$. The third integrals is estimated as

$$\begin{aligned}&\int ^t_\tau \Vert (t-s)^{q-1}{\mathcal {T}}_q(t-s)\Vert _{\alpha -1}\Vert \;f_n(s,x(s),x(a(x(s),s)))\Vert ds\nonumber \\&\quad \le \int ^t_\tau \int ^\infty _0\zeta _q(\xi )\Vert \;[q(t-s)^{q-1}\xi A {\mathcal {T}}((t-s)^q\xi )]A^{\alpha -2}\Vert \nonumber \\&\qquad \times \Vert \;f_n(s,x(s),x(a(x(s),s)))\Vert d\xi ds,\nonumber \\&\quad \le \int ^t_\tau \int ^\infty _0\zeta _q(\xi )\Vert \;\frac{d}{d\varsigma }{\mathcal {T}}((\varsigma -s)^q\theta )|_{\varsigma =t}A^{\alpha -2}\Vert N_fd\xi ds,\nonumber \\&\quad \le K_3(t-\tau ), \end{aligned}$$

(35)

where $K_3=M_1\Vert \;A^{\alpha -2}\Vert N_f$. Similarly, we estimate

$$\begin{aligned} \sum _{i=1}^p\Vert \; \left[ {\mathcal {S}}_q(t-t_i)-{\mathcal {S}}_q(\tau -t_i)\right] A^{\alpha -1}I_{i,n}(x(t_i))\Vert \le K_4(t-\tau ), \end{aligned}$$

(36)

where $K_4=M_1\Vert \;A^{-1}\Vert \sum _{i=1}^p L_i$.

Thus, from the inequality (33)–(36), we obtain that

$$\begin{aligned} \Vert \;Q_n x(t)-Q_n x(\tau )\Vert _{\alpha -1}\le {\mathcal {L}}(t-\tau ), \end{aligned}$$

(37)

for a positive suitable constant ${\mathcal {L}}$. Therefore, we conclude that $(Q_nx)\in {\mathcal {C}}^{\alpha -1}_T$. Hence, the $Q_n: {\mathcal {C}}^{\alpha -1}_T\rightarrow {\mathcal {C}}^{\alpha -1}_T$ is a well defined map.

Next, we prove that $Q_n:{\mathcal {B}}\rightarrow {\mathcal {B}}$. For $0\le t\le T$ and $x\in {\mathcal {B}}$, we get that $\Vert \;(Q_n x)(t) -u_0\Vert _\alpha$

$$\begin{aligned}\le & {} \Vert \;\left[ {\mathcal {S}}_q(t)-I\right] u_0\Vert _\alpha + \int ^t_0\Vert \;(t-s)^{q-1} {\mathcal {T}}_q(t-s)f_n(s,x(s),x(a(x(s),s)))\Vert _{\alpha }ds\nonumber \\&+\sum _{i=1}^p\Vert \; {\mathcal {S}}_q(t-t_i)I_{i,n}(x(t_i))\Vert _\alpha ,\nonumber \\\le & {} \Vert \;\left[ {\mathcal {S}}_q(t)-I\right] u_0\Vert _\alpha +\frac{q M_\alpha N_f\Gamma (2-\alpha )}{\Gamma (1+q(1-\alpha ))}\int ^t_0(t-s)^{q(1-\alpha )-1}ds +M\sum _{i=1}^p L_i,\nonumber \\\le & {} \Vert \;\left[ {\mathcal {S}}_q(t)-I\right] u_0\Vert _\alpha +\frac{M_\alpha N_f\Gamma (2-\alpha )T^{q(1-\alpha )}}{(1-\alpha )\Gamma (1+q(1-\alpha ))} +M\sum _{i=1}^p L_i, \end{aligned}$$

(38)

Therefore, it gives that $Q_n({\mathcal {B}})\subset {\mathcal {B}}$. At long last, we will assert that $Q_n$ is a contraction map. For $x,\;y\in {\mathcal {B}}$ and $0\le t\le T$, we get that

$$\begin{aligned} \Vert \;(Q_n x)(t)-(Q_n y)(t)\Vert _\alpha\le & {} \int ^t_0\Vert \;(t-s)^{q-1}A^\alpha {\mathcal {T}}_q(t-s)\Vert \nonumber \\&\times \Vert f_n(s,x(s),x(a(x(s),s))) -f_n(s,y(s),y(a(y(s),s)))\Vert ds\nonumber \\&+\sum _{i=1}^p\Vert \; {\mathcal {S}}_q(t-t_i)\Vert \Vert \;I_{i,n}(x(t_i))-I_{i,n}(y(t_i))\Vert _\alpha . \end{aligned}$$

(39)

We have the following inequalities:

$$\begin{aligned} \Vert f_n(s,x(s),x(a(x(s),s)))-f_n(s,y(s),y(a(y(s),s)))\Vert \le {\mathcal {L}}_f[2+{\mathcal {L}}{\mathcal {L}}_a]\Vert \;x-y\Vert _{T,\alpha }. \end{aligned}$$

(40)

Similarly, we have

$$\begin{aligned} \Vert \;I_{i,n}(x(t_i))-I_{i,n}(y(t_i))\Vert \le N_i \Vert \;x-y\Vert _{T,\alpha }. \end{aligned}$$

(41)

Using (40), (41) in (39)and obtain that

$$\begin{aligned} \Vert \;(Q_n x)(t)-(Q_n y)(t)\Vert\le & {} \frac{qM_\alpha \Gamma (2-\alpha )}{\Gamma (1+q(1-\alpha ))}{\mathcal {L}}_f\left[ 2+ {\mathcal {L}}{\mathcal {L}}_a \right] \Vert \; x-y\Vert _{T,\alpha }\nonumber \\&\times \int ^t_0(t-s)^{q(1-\alpha )-1}ds +M\sum _{i=1}^p N_i \Vert \;x-y\Vert _{T,\alpha },\nonumber \\\le & {} \left[ \frac{M_\alpha \Gamma (2-\alpha )T^{q(1-\alpha )}}{(1-\alpha )\Gamma (1+q(1-\alpha ))}{\mathcal {L}}_f(2+{\mathcal {L}}{\mathcal {L}}_a)+M\sum _{i=1}^p N_i \right] \nonumber \\&\times \Vert \;x-y\Vert _{T,\alpha }. \end{aligned}$$

(42)

From the inequality (25), we get

$$\begin{aligned} \Vert \;(Q_n x)(t)-(Q_n y)(t)\Vert <\Vert \;x-y\Vert _{T,\alpha }. \end{aligned}$$

(43)

Therefore, it implies that the map $Q_n$ is a contraction map and has a unique fixed point $x_n\in {\mathcal {B}}$ i.e., $Q_n x_n=x_n$ and $x_n$ satisfies the approximate integral equation

$$\begin{aligned} x_n(t)= & {} {\mathcal {S}}_q(t)u_0 +\int ^t_0(t-s)^{q-1}{\mathcal {T}}_q(t-s)f_n(s,x_n(s),x_n(a(x_n(s),s)))ds\\&+\sum _{i=1}^{p}{\mathcal {S}}_q(t-t_i)I_{i,n}(x_n(t_i)), \end{aligned}$$

for $t\in [0,T]$. $\square$

Lemma 3.2

Assume that hypotheses $(A1)-(A4)$ hold. If $u_0\in D(A^\alpha )$, where $0< \alpha <1$, then $x_n(t)\in D(A^\upsilon )$ for all $t\in (0,T]$ with $0\le \upsilon <1$. Furthermore, if $u_0\in D(A)$ then $x_n(t)\in D(A^\upsilon )$ for all $t\in [0,T]$ with $0\le \upsilon <1$.

Proof

From Theorem (3.1), we have that there exists a unique $x_n\in {\mathcal {B}}\subset {\mathcal {C}}^{\alpha -1}_T$ such that $x_n$ satisfy the Eq. (30). Theorem 2.6.13 in Pazy [35] implies that $T(t):H\rightarrow D(A^\upsilon )$ for $t>0$ and $0\le \upsilon <1$ and for $0\le \upsilon \le \eta <1$, $D(A^\eta )\subseteq D(A^\upsilon )$. It is not difficult to see that H$\ddot{o}$lder continuity of $x_n$ might be made using the similar arguments from Eqs. (33)–(36). Additionally from Theorem 1.2.4 in Pazy [35], we have that $T(t)y\in D(A)$ if $y\in D(A)$. The result follows from these facts and the fact that $D(A)\subseteq D(A^\upsilon )$ for $0\le \upsilon \le 1$. This finishes the proof of Lemma. $\square$

Corollary 3.1

Suppose that the hypotheses $(A1)-(A4)$ hold. If $u_0\in D(A^\alpha )$ with $0<\alpha <1$, then for any $t_0\in (0,T]$, there exists a constant $U_{t_0}$ such that

$$\begin{aligned} \Vert A^\upsilon x_n(t)\Vert \le U_{t_0},\;\;n=1,2,3,\cdots , \end{aligned}$$

for all $t_0\le t\le T$ independent of n, where $0<\alpha <\upsilon <\beta$. Furthermore, if $u_0\in D(A)$, there exist a positive constant $U_0$ such that $\Vert A^\upsilon x_n(t)\Vert \le U_{0}$, $t\in [0,T]$, $n=1,2,\cdots ,$.

Proof

Let $u_0\in D(A^\alpha )$. Applying $A^\upsilon$ on the both the sides of (30) and $t_0\le t\le T$, we get

$$\begin{aligned}&\Vert \;A^\upsilon x_n(t)\Vert \nonumber \\&\quad \le \Vert \;A^\upsilon {\mathcal {S}}_q(t)u_0\Vert +\int ^t_0(t-s)^{q-1}\Vert A^\upsilon {\mathcal {T}}_q(t-s)\Vert \Vert \;f_n(s,x_n(s),x_n(a(x_n(s),s)))\Vert ds\nonumber \\&\qquad +\sum _{i=1}^p\Vert \;{\mathcal {S}}_q(t-t_i)A^\upsilon I_i(x(t_i))\Vert ,\nonumber \\&\quad \le M_{\upsilon }{t_0}^{-q\upsilon }\Vert \;u_0\Vert +\frac{qM_\upsilon N_f\Gamma (2-\upsilon )}{\Gamma (1+q(1-\upsilon ))}\int ^t_0(t-s)^{q(1-\upsilon )-1}ds+M\sum _{i=1}^p L_i,\nonumber \\&\quad \le M_{\upsilon }{t_0}^{-q\upsilon }\Vert \;u_0\Vert +\frac{M_\upsilon N_f\Gamma (2-\upsilon ) T^{q(1-\upsilon )}}{(1-\upsilon )\Gamma (1+q(1-\upsilon ))}+M\sum _{i=1}^p L_i, \end{aligned}$$

(44)

$$\begin{aligned}&\quad \le U_{t_0}. \end{aligned}$$

(45)

Again, for $0\le t\le T$ and $u_0\in D(A^\alpha )$, we have

$$\begin{aligned} \Vert \;A^\upsilon x_n(t)\Vert\le & {} M\Vert \;u_0\Vert _\upsilon +\frac{M_\upsilon N_f\Gamma (2-\upsilon ) T^{q(1-\upsilon )}}{(1-\upsilon )\Gamma (1+q(1-\upsilon ))}+M\sum _{i=1}^p L_i. \end{aligned}$$

(46)

Since, we might displace the first term in (44) by $M\Vert \;u_0\Vert _\upsilon$.

Moreover, if $u_0\in D(A)$, then $u_0\in D(A^\upsilon )$ for $0\le \upsilon <1$. Therefore, we can effortlessly get the required result. This finishes the proof of Lemma. $\square$

Convergence of Solutions

The convergence of the solution $x_n\in H_\alpha$ of the approximate integral Eq. (30) to a unique solution $x(\cdot )$ of the Eq. (17) on [0, T] is discussed in this section.

Theorem 4.1

Suppose that $(A1)-(A4)$ are satisfied. If $u_0\in D(A^\alpha )$, then

$$\begin{aligned} \lim _{m\rightarrow \infty }\sup _{\{n\ge m, t_0\le t\le T\}}\Vert \;x_n(t)-x_m(t)\Vert _\alpha =0, \end{aligned}$$

(47)

for every $t_0\in (0,T]$.

Proof

For $0<\alpha <\upsilon$, $n\ge m$ and $t\in (0,T]$, we have

$$\begin{aligned}&\Vert f_n(t,x_n(t),x_n(a(x_n(t),t))) -f_m(t,x_m(t),x_m(a(x_m(t),t)))\Vert \\&\quad \le \Vert f_n(t,x_n(t),x_n(a(x_n(t),t)))-f_n(t,x_m(t),x_m(a(x_m(t),t)))\Vert \\&\qquad +\Vert f_n(t,x_m(t),x_m(a(x_m(t),t)))-f_m(t,x_m(t),x_m(a(x_m(t),t)))\Vert ,\\&\quad \le {\mathcal {L}}_f\left[ 2+ {\mathcal {L}}{\mathcal {L}}_a\right] \Vert \;x_n(t) -x_m(t)\Vert _\alpha +{\mathcal {L}}_f \left[ \Vert \;(P^n-P^m)x_m(t)\Vert _\alpha \right. \\&\qquad \left. +\Vert A^{-1}\Vert \Vert \;(P^n-P^m)x_m(a(x_m(t),t))\Vert _\alpha \right] . \end{aligned}$$

Let $n>m$. Thus, ${\mathcal {H}}_m\subset {\mathcal {H}}_n$. Let ${\mathcal {H}}_m^{\top }$ be the orthogonal complement of ${\mathcal {H}}_m$ for each $m=0,1,\ldots ,$. Thus, we have ${\mathcal {H}}_n^{\top }\subset {\mathcal {H}}_m^{\top }$. Also, we have $H={\mathcal {H}}_m\oplus {\mathcal {H}}_m^{\top }={\mathcal {H}}_n\oplus {\mathcal {H}}_n^{\top }$. Let $y\in H$ be an arbitrary element. Then, $y=y_m+z_m$ with $y_m\in {\mathcal {H}}_m$ and $z_m\in {\mathcal {H}}_m{\top }$. Therefore, we have that $y_m\in {\mathcal {H}}_m=P^m y$. It is easy to see that $z_m\in {\mathcal {H}}_m^{\top }\rightarrow z_m=\sum _{i=m+1}^na_i\phi _i+z_m^{\prime }$, where $z_m^{\prime }\in {\mathcal {H}}_n^{\top }$. Let us take $y_m^{\prime }=\sum _{i=m+1}^na_i\phi _i$.

Therefore, $y=y_m+y_m^{\prime }+z_m^{\prime }$ and $P^n y=y_m +y_m^{\prime }$. Thus,

$$\begin{aligned} P^{n} y-P^{m} y=y_m^{\prime }=\sum _{i=m+1}^n a_i\phi _i. \end{aligned}$$

If, $y=\sum _{i=1}^\infty a_i\phi _i$. Then, we get $\Vert y\Vert ^2=\sum _{i=1}^\infty |a_i|^2$.

Since, $A^{\alpha -\upsilon }\phi _i=\lambda _i^{\alpha -\upsilon }\phi$. Hence, we get

$$\begin{aligned} \Vert A^{\alpha -\upsilon }(P^n-P^m)y\Vert ^2= & {} <A^{\alpha -\upsilon }(P^n-P^m)y,A^{\alpha -\upsilon }(P^n-P^m)y>,\nonumber \\= & {} <\sum _{i=m+1}^n a_i A^{\alpha -\upsilon }\phi _i, \sum _{j=m+1}^n a_j A^{\alpha -\upsilon }\phi _j>,\nonumber \\= & {} <\sum _{i=m+1}^n a_i \lambda _i^{\alpha -\upsilon }\phi _i, \sum _{j=m+1}^n a_j \lambda _j^{\alpha -\upsilon }\phi _j>,\nonumber \\= & {} \sum _{i,j=m+1}^n a_i a_j\lambda _{i}^{\alpha -\upsilon }\lambda _j^{\alpha -\upsilon }<\phi _i,\phi _j>,\nonumber \\\le & {} \lambda ^{2(\alpha -\upsilon )}_{m+1}\left( \sum _{i=m+1}^n |a_i|^2 \right) ,\nonumber \\\le & {} \frac{1}{\lambda _m^{2(\upsilon -\alpha )}}\Vert y\Vert ^2. \end{aligned}$$

(48)

Thus, we have the following estimation

$$\begin{aligned} \Vert \;(P^n-P^m)x_m(t)\Vert _\alpha \le \Vert \;A^{\alpha -\upsilon }(P^n-P^m)A^{\upsilon }x_m(t)\Vert \le \frac{1}{\lambda _m^{\upsilon -\alpha }}\Vert \;A^{\upsilon }x_m(t)\Vert . \end{aligned}$$

Thus, we obtain

$$\begin{aligned}&\Vert \;f_n(t,x_n(t),x_n(a(x_n(t),t)))-f_m(t,x_m(t),x_m(a(x_m(t),t)))\Vert \nonumber \\&\quad \le {\mathcal {L}}_f[2+ {\mathcal {L}}{\mathcal {L}}_a]\Vert \; x_n(t)-x_m(t)\Vert _\alpha +{\mathcal {L}}_f \left[ \frac{1}{\lambda _m^{\upsilon -\alpha }}\Vert \; A^{\upsilon }x_m(t)\Vert \right. \nonumber \\&\qquad \left. +\frac{\Vert \;A^{-1}\Vert }{\lambda _m^{\upsilon -\alpha }}\Vert \; A^{\upsilon }x_m(a(x_m(t),t))\Vert \right] . \end{aligned}$$

(49)

Similarly, we estimate

$$\begin{aligned} \Vert \;I_{i,n}(x_n(t_i))-I_{i,m}(x_m(t_i))\Vert \le N_i \left[ \Vert \;x_n(t_i)-x_m(t_i)\Vert _\alpha +\frac{1}{\lambda _m^{\upsilon -\alpha }}\Vert \; A^{\upsilon }x_m(t_i)\Vert \right] .\quad \end{aligned}$$

(50)

We choose $t_{0}^{\prime }$ such that $0<t_0^{\prime }<t<T$, we have

$$\begin{aligned}&\Vert \;x_n(t)-x_m(t)\Vert _\alpha \nonumber \\&\quad \le \left( \int ^{t_0^{\prime }}_0 +\int ^t_{t_0^{\prime }}\right) (t-s)^{q-1}\Vert \; A^\alpha {\mathcal {T}}_q(t-s)\Vert \nonumber \\&\qquad \times \Vert \; f_n(t,x_n(t),x_n(a(x_n(t),t))) -f_m \left( t,x_m(t),x_m(a(x_m(t),t))\right) \Vert ds\nonumber \\&\qquad +\sum _{i=0}^p \Vert \;{\mathcal {S}}_q(t-t_i)\Vert \Vert \; I_{i,n}(x_n(t_i)) -I_{i,m}(x_m(t_i))\Vert _\alpha , \end{aligned}$$

(51)

we estimate the first integral as

$$\begin{aligned}&\int ^{t_0^{\prime }}_0(t-s)^{q-1}\Vert \; A^\alpha {\mathcal {T}}_q(t-s)\Vert \Vert \; f_n(t,x_n(t),x_n(a(x_n(t),t)))\nonumber \\&\qquad -f_m(t,x_m(t),x_m(a(x_m(t),t)))\Vert ds\nonumber \\&\quad \le \int ^{t_0^{\prime }}_0(t-s)^{q-1}\Vert \; A^\alpha {\mathcal {T}}_q(t-s)\Vert 2N_fds,\nonumber \\&\quad \le \frac{2N_fM_\alpha \Gamma (2-\alpha )}{(1-\alpha )\Gamma (1 +q(1-\alpha ))}[t^{q(1-\alpha )} -(t-t_0^{\prime })^{q(1-\alpha )}],\nonumber \\&\quad \le \frac{2N_fM_\alpha \Gamma (2-\alpha )}{(1-\alpha )\Gamma (1 +q(1-\alpha ))}(t-b_1t_0^{\prime })^{q(1-\alpha )-1}t_0^{\prime },\; 0<b_1<1,\nonumber \\&\quad \le \frac{2N_fM_\alpha \Gamma (2-\alpha )}{(1 -\alpha )\Gamma (1+q(1-\alpha ))}(t_0-t_0^{\prime })^{q(1 -\alpha )-1}t_0^{\prime }. \end{aligned}$$

(52)

The second integral is estimated as

$$\begin{aligned}&\int ^t_{t_0^{\prime }}(t-s)^{q-1}\Vert \; A^\alpha {\mathcal {T}}_q(t-s)\Vert \Vert \; f_n(t,x_n(t),x_n(a(x_n(t),t)))\nonumber \\&\qquad -f_m(t,x_m(t),x_m(a(x_m(t),t)))\Vert ds\nonumber \\&\quad \le \frac{qM_\alpha \Gamma (2-\alpha )}{\Gamma (1+q(1 -\alpha ))}\int ^t_0(t-s)^{q-1}\left\{ {\mathcal {L}}_f\left[ 2+ {\mathcal {L}}{\mathcal {L}}_a\right] \Vert \; x_n(s)-x_m(s)\Vert _\alpha \right. \nonumber \\&\qquad \left. +{\mathcal {L}}_f \left[ \frac{1}{\lambda _m^{\upsilon -\alpha }}\Vert \; A^{\upsilon }x_m(s)\Vert +\frac{\Vert \;A^{-1}\Vert }{\lambda _m^{\upsilon -\alpha }}\Vert \; A^{\upsilon }x_m(a(x_m(s),s))\Vert \right] \right\} ds,\nonumber \\&\quad \le \frac{qM_\alpha {\mathcal {L}}_f \Gamma (2-\alpha )}{\Gamma (1+q(1-\alpha ))}\left[ \left( 1+\Vert \; A^{-1}\Vert \right) \frac{U_{t_0^{\prime }}T^{q(1-\alpha )}}{q(1- \alpha )\lambda _m^{\upsilon -\alpha }} +(2+{\mathcal {L}}{\mathcal {L}}_a)\int ^t_{t_0^{\prime }}(t-s)^{q(1-\alpha )-1}\right. \nonumber \\&\qquad \left. \times \Vert \;x_n(s)-x_m(s)\Vert _\alpha ds\right] . \end{aligned}$$

(53)

Thus, we have

$$\begin{aligned}&\Vert \;x_n(t)-x_m(t)\Vert _\alpha \nonumber \\&\quad \le D_1 t_0^{\prime }+\frac{D_2}{\lambda _m^{\upsilon -\alpha }}+ D_3\Vert \;x_n(t)-x_m(t)\Vert _{\alpha }+ D_4\int ^t_0(t-s)^{q(1-\alpha )-1}\nonumber \\&\qquad \times \Vert \;x_n(s)-x_m(s)\Vert _{\alpha }ds, \end{aligned}$$

(54)

where

$$\begin{aligned} D_1= & {} \frac{2N_fM_\alpha \Gamma (2-\alpha )}{(1-\alpha )\Gamma (1+ q(1-\alpha ))}(T-t_0^{\prime })^{q(1-\alpha )-1}, \end{aligned}$$

(55)

$$\begin{aligned} D_2= & {} \frac{qM_\alpha {\mathcal {L}}_f \Gamma (2-\alpha )}{\Gamma (1+q(1-\alpha ))}\times (1 +\Vert \;A^{-1}\Vert )\frac{U_{t_0^{\prime }}T^{q(1-\alpha )}}{q(1-\alpha )}+M\sum _{i=1}^pN_i, \end{aligned}$$

(56)

$$\begin{aligned} D_3= & {} M\sum _{i=1}^pN_i, \end{aligned}$$

(57)

$$\begin{aligned} D_4= & {} \frac{qM_\alpha {\mathcal {L}}_f \Gamma (2-\alpha )}{\Gamma (1+q(1-\alpha ))}(2+{\mathcal {L}}{\mathcal {L}}_a), \end{aligned}$$

(58)

Since $1-M\sum _{i=1}^pN_i>0$, we have

$$\begin{aligned}&\Vert \;x_n(t)-x_m(t)\Vert _\alpha \\&\quad \le \frac{1}{1-D_3}\left[ {D_1}t_0^{\prime }+\frac{D_2}{\lambda _m^{\upsilon -\alpha }} +{D_4}\int ^t_0(t-s)^{q(1-\alpha )-1}\Vert \;x_n(s)-x_m(s)\Vert _{\alpha }ds\right] . \end{aligned}$$

By Lemma 5.6.7 in [35], we have that there exists a constant ${\mathcal {K}}$ such that

$$\begin{aligned} \Vert \;x_n(t)-x_m(t)\Vert _\alpha\le & {} \frac{1}{1-D_3}[{D_1}t_0^{\prime }+ \frac{D_2}{\lambda _m^{\upsilon -\alpha }}]{\mathcal {K}}, \end{aligned}$$

(59)

taking supremum over $[t_0,T]$ and letting $m\rightarrow \infty$, we obtain

$$\begin{aligned} \lim _{m\rightarrow \infty }\sup _{\{n\ge m, t_0\le t\le T\}}\Vert \;x_n(t)-x_m(t)\Vert _\alpha \le \frac{D_1}{(1-D_3)}t_0^{\prime } {\mathcal {K}}. \end{aligned}$$

(60)

As $t_0^{\prime }$ is arbitrary, therefore the right hand side may be made as small as desired by taking $t_0^{\prime }$ sufficiently small. This completes the proof of the Theorem. $\square$

Proposition 4.2

If $u_0\in D(A)$, then there exist a Cauchy sequence $x_n\in {\mathcal {B}}$ on [0, T] i.e.,

$$\begin{aligned} \Vert \;x_n-x_m\Vert _{T,\alpha }\rightarrow 0, \end{aligned}$$

(61)

as $m,n\rightarrow \infty$.

Proof

Taking $t_0=0$ in the proof of Theorem 4.1, we replace the term $(t_0-t_0^{\prime })^{q(1-\alpha )-1}t_0^{\prime }$ by $(1-b_1)^{q(1-\alpha )-1}{t_0^{\prime }}^{1-\alpha }$ in Eq. 52 and the constant $U_{t_0^{\prime }}$ by the constant $U_0$ from the Lemma 3.1 and Corollary 3.1. $\square$

Theorem 4.3

Suppose that (A1)–(A4) are satisfied and $u_0\in D(A^\alpha )$. Then, there exists a unique $x_n\in {\mathcal {B}}$, satisfying

$$\begin{aligned} x_n(t)= & {} {\mathcal {S}}_q(t) u_0+\int ^t_0(t-s)^{q-1}{\mathcal {T}}_q(t-s)f_n(s,x_n(s), x_n(a(x_n(s),s)))ds\\&+\sum _{i=1}^p{\mathcal {S}}_q(t-t_i)I_{i,n}(x_n(t_i)),\;t\in [0,T], \end{aligned}$$

and $x\in {\mathcal {B}}$, satisfying

$$\begin{aligned} x(t)= & {} {\mathcal {S}}_q(t) u_0+\int ^t_0(t-s)^{q-1}{\mathcal {T}}_q(t-s)f(s,x(s), x(a(x(s),s)))ds\\&+\sum _{i=1}^p{\mathcal {S}}_q(t-t_i)I_i(x(t_i)),\;t\in [0,T], \end{aligned}$$

such that $x_n$ converges to x in ${\mathcal {B}}$ i.e., $x_n\rightarrow x$ as $n\rightarrow \infty$.

Proof

Let $u_0\in D(A^\alpha )$. For $0<t\le T$, $A^\alpha x_n(t)\rightarrow A^\alpha x(t)$ as $n\rightarrow \infty$ and $x(0)=x_n(0)=u_0$ for all n. Also, for $t\in [0,T]$, we have $A^\alpha x_n(t)\rightarrow A^\alpha x(t)$ as $n\rightarrow \infty$ in H. Since $x_n\in {\mathcal {B}}$, therefore it follows that $x\in {\mathcal {B}}$ and

$$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{t_0\le t\le T}\Vert \;x_n(t)-x(t)\Vert _\alpha =0,\;\;\text {for any}\;t_0\in (0,T]. \end{aligned}$$

(62)

Also, we have

$$\begin{aligned}&\Vert \;f_n(t,x_n(t),x_n(a(x_n(t),t)))-f(t,x(t),x(a(x(t),t)))\Vert \nonumber \\&\quad \le {\mathcal {L}}_f\left[ 2+{\mathcal {L}}{\mathcal {L}}_a \right] \Vert \; x_n(t)-x(t)\Vert _\alpha +{\mathcal {L}}_f\left[ \Vert \;(P^n-I)x(t)\Vert _\alpha \right. \nonumber \\&\qquad \left. +\Vert \;A^{-1}\Vert \;\Vert \;(P^n-I)x(a(x(t),t))\Vert _\alpha \right] \rightarrow 0, \end{aligned}$$

(63)

as $n\rightarrow \infty$. For $0<t_0<t$, we rewrite 30 as

$$\begin{aligned} x_n(t)= & {} {\mathcal {S}}_q(t) u_0+(\int ^{t_0}_0 +\int ^t_{t_0})(t-s)^{q-1}{\mathcal {T}}_q(t-s)f_n(s,x_n(s), x_n(a(x_n(s),s)))ds \\&+\sum _{i=1}^p{\mathcal {S}}_q(t-t_i)I_{i,n}(x_n(t_i)). \end{aligned}$$

We may estimate the first integral as

$$\begin{aligned} \Vert \int ^{t_0}_0(t-s)^{q-1}{\mathcal {T}}_q(t-s)f_n(s,x_n(s), x_n(a(x_n(s),s)))ds\Vert\le & {} \frac{q M N_f}{\Gamma (1+q)}{T}^{q-1}t_0,\qquad \end{aligned}$$

(64)

Thus, we deduce that

$$\begin{aligned}&\Vert \;x_n(t)-{\mathcal {S}}_q(t) u_0-\sum _{i=1}^p{\mathcal {S}}_q(t-t_i)I_{i,n}(x_n(t_i)) -\int ^t_{t_0}(t-s)^{q-1}{\mathcal {T}}_q(t-s)\nonumber \\&\quad \times f_n(s,x_n(s), x_n(a(x_n(s),s)))ds\Vert \le \left[ \frac{q M N_f}{\Gamma (1+q)}{T}^{q-1}\right] t_0. \end{aligned}$$

(65)

Letting $n\rightarrow \infty$ in the above inequality, we obtain

$$\begin{aligned}&\Vert \;x(t)-{\mathcal {S}}_q(t) u_0-\sum _{i=1}^p{\mathcal {S}}_q(t-t_i)I_{i}(x(t_i)) -\int ^t_{t_0}(t-s)^{q-1}{\mathcal {T}}_q(t-s)\nonumber \\&\qquad \times f(s,x(s), x(a(x(s),s)))ds\Vert \nonumber \\&\quad \le \left[ \frac{q M N_f}{\Gamma (1+q)}{T}^{q-1}\right] t_0. \end{aligned}$$

(66)

Since $t_0$ is arbitrary, we deduce that x satisfies the integral Eq. (17).

Now, we shall show the uniqueness of the solution to Eq. (17). Let $x_1$ and $x_2$ be the two solutions of the (17). We have

$$\begin{aligned} \Vert \;x_1(t)-x_2(t)\Vert _\alpha\le & {} \int ^t_0(t-s)^{q-1}\Vert \;A^\alpha {\mathcal {T}}_q(t-s)\Vert \Vert \;f(s,x_1(s), x_1(a(x_1(s,)s)))\nonumber \\&-f(s,x_2(s), x_2(a(x_2(s,)s)))\Vert ds\nonumber \\&+\sum _{i=1}^p \Vert \;{\mathcal {S}}_q(t-t_i)\Vert \Vert \;I_i(x_1(t_i))-I_i(x_2(t_i))\Vert ,\nonumber \\\le & {} \frac{q M_\alpha \Gamma (2-\alpha )}{\Gamma (1+q(1-\alpha ))}{\mathcal {L}}_f(2+ {\mathcal {L}}{\mathcal {L}}_a)\Vert \;x_1-x_2\Vert _{T,\alpha }\nonumber \\&\,\,\int ^t_0(t-s)^{q(1-\alpha )-1}ds+M\sum _{i=1}^p N_i \Vert \;x_1-x_2\Vert _{T,\alpha },\nonumber \\\le & {} \left[ \frac{M_\alpha \Gamma (2-\alpha )T^{q(1-\alpha )}}{(1-\alpha )\Gamma (1+q(1-\alpha ))}{\mathcal {L}}_f(2+{\mathcal {L}}{\mathcal {L}}_a)+M\sum _{i=1}^p N_i \right] \nonumber \\&\times \Vert \;x_1-x_2\Vert _{T,\alpha }, \end{aligned}$$

(67)

By Lemma 5.6.7 in Pazy [35], we obtain that

$$\begin{aligned} \Vert \;x_1(t)-x_2(t)\Vert =0. \end{aligned}$$

(68)

Also, we have that

$$\begin{aligned} \Vert \;x_1(t)-x_2(t)\Vert \le \frac{1}{\lambda _0^\alpha }\Vert \;x_1(t)-x_2(t)\Vert _\alpha , \end{aligned}$$

(69)

From (68) and (69), we deduce that $u_1=u_2$ on [0, T]. Hence, the theorem is proved. $\square$

Faedo–Galerkin Approximations

In this section, we consider the Faedo–Galerkin Approximation of a solution and show the convergence results for such an approximation.

We know that for any $0< T < T_0$, we have a unique $x\in {\mathcal {C}}^\alpha _T$ satisfying the following integral equation

$$\begin{aligned} x(t)= & {} {\mathcal {S}}_q(t) u_0+\int ^t_0(t-s)^{q-1}{\mathcal {T}}_q(t-s)f(s,x(s), x(a(x(s),s)))ds\nonumber \\&+\sum _{i=1}^p{\mathcal {S}}_q(t-t_i)I_{i}(x(t_i)),\;\; \end{aligned}$$

(70)

for $0<t<T_0$.

Also, we have a unique solution $x_n\in {\mathcal {C}}^\alpha _T$ of the approximate integral equation

$$\begin{aligned} x_n(t)= & {} {\mathcal {S}}_q(t) u_0+\int ^t_0(t-s)^{q-1}{\mathcal {T}}_q(t-s)f_n(s,x_n(s), x_n(a(x_n(s),s)))ds\nonumber \\&+\sum _{i=1}^p{\mathcal {S}}_q(t-t_i)I_{i,n}(x_n(t_i)),\;\;\nonumber \\ \end{aligned}$$

(71)

Applying the projection on above equation, then Faedo–Galerkin approximation is given by $v_n(t)=P^n x_n(t)$ satisfying

$$\begin{aligned} P^n x_n(t)= & {} v_n(t)\nonumber \\= & {} {\mathcal {S}}_q(t)P^n u_0+\int ^t_0(t-s)^{q-1}{\mathcal {T}}_q(t-s)P^nf_n(s,x_n(s), x_n(a(x_n(s),s)))ds\nonumber \\&+\sum _{i=1}^p{\mathcal {S}}_q(t-t_i)P^nI_{i,n}(x_n(t_i)),\nonumber \\= & {} {\mathcal {S}}_q(t)P^n u_0+\int ^t_0(t-s)^{q-1}{\mathcal {T}}_q(t-s)P^nf(s,v_n(s), v_n(a(v_n(s),s)))ds\nonumber \\&+\sum _{i=1}^p{\mathcal {S}}_q(t-t_i)P^nI_{i}(v_n(t_i)). \end{aligned}$$

(72)

Let solution $x(\cdot )$ of (70) and $v_n(\cdot )$ of (72), have the representation

$$\begin{aligned} x(t)= & {} \displaystyle \sum _{i=0}^{\infty }\alpha _i(t)]\phi _i,\;\;\; \alpha _i(t)=(x(t),\phi _i),\ \ i=0,1,2\ldots , \end{aligned}$$

(73)

$$\begin{aligned} v_n(t)= & {} \displaystyle \sum _{i=0}^{n}\alpha _i^{n}(t)\phi _i,\;\;\; \alpha _i^{n}(t)=(v_n(t),\phi _i),\ \ i=0,1,2\ldots , \end{aligned}$$

(74)

Using (74) in (72) and taking inner product with $\phi _i$, we obtain a system of fractional order integro-differential equation of the form

$$\begin{aligned} \frac{d^q}{dt^q}\alpha _i^{n}(t)+\lambda _i\alpha _i^{n}(t)= & {} F_i^{n}(t, \alpha _0^n(t),\alpha _1^n(t)...,\alpha _n^n(t)), \end{aligned}$$

(75)

$$\begin{aligned} \Delta \alpha _i^n(t_k)= & {} I_i^n(\alpha _i^n(t_k)),\;k=1,\ldots ,p, \end{aligned}$$

(76)

$$\begin{aligned} \alpha ^n_i(0)= & {} \varphi _i, \end{aligned}$$

(77)

where

$$\begin{aligned} F_i^{n}= & {} \left( f\left( t,\sum _{i=0}^n\alpha _i^n \phi _i, \sum _{i=0}^n\tau _i^n\phi _i \right) ,\phi _i \right) , \end{aligned}$$

(78)

$$\begin{aligned} \tau _i^n= & {} \alpha _i^n\left( a \left( \alpha _0^n, \alpha _1^n,\ldots , \alpha _n^n(t)\right) \right) , \end{aligned}$$

(79)

$$\begin{aligned} I_i^n= & {} \left( I_k\left( \sum _{k=1}^p \sum _{i=1}^n \alpha _i^n(t_k)\phi _i\right) ,\phi _i \right) , \end{aligned}$$

(80)

$$\begin{aligned} \varphi _i= & {} (u_0,\phi _i),\;\;\text {for}\;\;i=1,2,\ldots ,n. \end{aligned}$$

(81)

For the convergence of $\alpha _i^n$ to $\alpha _i$, we have the following convergence theorem.

Theorem 5.1

Let us assume that $(A1)--(A4)$ are satisfied and $u_0\in D(A^\alpha )$. Then there exist a unique function $v_n\in {\mathcal {B}}$ given as

$$\begin{aligned} v_n(t)= & {} {\mathcal {S}}_q(t)P^n u_0+\int ^t_0(t-s)^{q-1}{\mathcal {T}}_q(t-s)f_n(s,v_n(s), v_n(a(v_n(s),s)))ds\nonumber \\&+\sum _{i=1}^p{\mathcal {S}}_q(t-t_i)P^nI_{i}(v_n(t_i)), \end{aligned}$$

(82)

and $x\in {\mathcal {B}}$ satisfying

$$\begin{aligned} x(t)= & {} {\mathcal {S}}_q(t) u_0+\int ^t_0(t-s)^{q-1}{\mathcal {T}}_q(t-s)f(s,x(s), x(a(x(s),s)))ds\nonumber \\&+\sum _{i=1}^p{\mathcal {S}}_q(t-t_i)I_{i}(x(t_i)),\;\; \end{aligned}$$

(83)

for $t\in [0,T_0]$, such that $v_n\rightarrow x$ as $n\rightarrow \infty$ in ${\mathcal {B}}$ and x satisfies the Eq. (17) on $[0,T_0]$.

The system (75)–(77) determines the $\alpha _i^n$’s. It can easily be investigated that

$$\begin{aligned} A^\alpha [x(t)-v(t)]= & {} A^\alpha \left[ \sum _{i=0}^\infty (\alpha _i(t)-\alpha _i^n(t))\phi _i \right] = \sum _{i=0}^{\infty }\lambda _i^\alpha (\alpha _i(t) -\alpha _i^n)\phi _i, \end{aligned}$$

(84)

Thus, we conclude that

$$\begin{aligned} \Vert \;A^\alpha [x(t)-v(t)]\Vert ^2\ge \sum _{i=0}^n \lambda _i^{2\alpha }(\alpha _i(t)-\alpha _i^n(t))^2. \end{aligned}$$

(85)

Theorem 5.2

Let us assume that $(A1)--(A4)$ are satisfied. Then, we have the following results

• (a) If $u_0\in D(A^\alpha )$, then

  $$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{t\in [t_0,T_0]} \left[ \sum _{i=0}^n\lambda _i^{2\alpha }(\alpha _i(t)-\alpha _i^n(t))^2\right] =0, \end{aligned}$$

  (86)

  for any $0<t_0\le T_0$.

• (b) If $u_0\in D(A)$, then

  $$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{t\in [0,T_0]} \left[ \sum _{i=0}^n\lambda _i^{2\alpha }(\alpha _i(t)-\alpha _i^n(t))^2 \right] =0, \end{aligned}$$

  (87)

for any $0\le t\le T_0$.

The statement of this hypothesis takes after from the facts specified above and the following results.

Corollary 5.1

Assume that $(A1)--(A4)$ are satisfied. Then

• (a) If $u_0\in D(A^\alpha )$, then

  $$\begin{aligned} \sup _{t\in [t_0,T_0]}\Vert \;v_n(t)-v_m(t)\Vert _{\alpha }\rightarrow 0,\;\;\text {as}\;\;m,n\rightarrow \infty , \end{aligned}$$

  (88)

  for any $0< t_0 \le {T_0}<T_{\max }$.

• (b) If $u_0\in D(A)$, then

  $$\begin{aligned} \sup _{t\in [0,T_0]}\Vert \;v_n(t)-v_m(t)\Vert _\alpha \rightarrow 0,\;\;\text {as}\;\;m,n\rightarrow \infty . \end{aligned}$$

  (89)

Proof

For $n\ge m$ and $0\le \alpha <\upsilon$, we get

$$\begin{aligned} \Vert \;v_n(t)-v_m(t)\Vert _\alpha= & {} \Vert \;P^n x_n(t)-P^m x_n(t)\Vert _\alpha ,\nonumber \\\le & {} \Vert \;P^n[x_n(t)-x_m(t)]\Vert _\alpha +\Vert \; (P^n-P^m)x_m(t)\Vert _\alpha ,\nonumber \\\le & {} \Vert \;x_n(t)-x_m(t)\Vert _\alpha +\frac{1}{\lambda _{m}^{\upsilon -\alpha }}\Vert \; A^\upsilon x_m(t)\Vert . \end{aligned}$$

(90)

If $u_0\in D(A^\alpha )$ then the result in (a) follows from Theorem 4.1, If $u_0\in D(A)$, (b) follows from Proposition 4.2. $\square$

Application

In this section, we present an example to show the feasibility of our abstract result.

Let us consider following fractional differential equation with impulsive conditions in the separable Hilbert space H

$$\begin{aligned} \frac{\partial ^q w}{\partial t^q}= & {} \frac{\partial ^2 w}{\partial u^2}+\widetilde{P}(u, w(u, t))+\widetilde{H}(t, u, w(u, t)), \;(u,t)\in (0,1)\nonumber \\&\times \left( 0,\frac{1}{2}\right) \cup \left( \frac{1}{2}, 1\right) , \end{aligned}$$

(91)

$$\begin{aligned} \delta w|_{t=\frac{1}{2}}= & {} \frac{2w(\frac{1}{2})^{-}}{2+w(\frac{1}{2})^{-}}, \end{aligned}$$

(92)

$$\begin{aligned} w(0,t)= & {} w(1, t)=0, \end{aligned}$$

(93)

$$\begin{aligned} w(x,0)= & {} w_0(u),\;\;\;u\in (0,1), \end{aligned}$$

(94)

where $0<q<1$, $\widetilde{H}:{\mathbb {R}}^+\times [0,1]\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is a nonlinear function which is measurable in u, locally H$\ddot{o}$lder continuous in first argument t, locally Lipschitz continuous in w and uniformly in u. The function $\widetilde{P}$ is given as

$$\begin{aligned} \widetilde{P}(u, w(u, t))=\int _0^u \mathcal (G)(u,y)w(y, h(t)|w(y,t)|)dy, \end{aligned}$$

(95)

here, $h:{\mathbb {R}}^+\rightarrow {\mathbb {R}}^+$ is assumed to be locally H$\ddot{o}$lder continuous in t with $h(0)=0$ and ${\mathcal {G}}\in C^1([0,1]\times [0,1], {\mathbb {R}})$.

Now, we take $H=L^2((0,1),{\mathbb {R}})$ and operator A as $Aw=d^2 w/dx^2$ with domain $D(A)=H^2(0,1)\cap H^1_0(0,1)$. Let $\alpha =1/2$, then $H_{1/2}=D(A^{1/2})=H^1_0(0,1)$ is a Banach space with norm $\Vert w\Vert _{1/2}:=\Vert A^{1/2} w\Vert$, for $w\in D(A^{1/2})$ and $H_{-1/2}=(H^1_0(0,1))^*=H^{-1}(0,1)\equiv H^1(0,1)$ is dual space of the space $H_{1/2}$.

Now, for each $u\in (0,1)$, we may consider the function $f:{\mathbb {R}}^+\times H_{1/2}\times H_{-1/2}\rightarrow H$ defined as

$$\begin{aligned} f(t, w, z)(u)=\widetilde{P}(u, z)+\widetilde{H}(t, u, w), \end{aligned}$$

(96)

with $\widetilde{P}:[0,1]\times H_{-1/2}\rightarrow H$ which is defined by

$$\begin{aligned} \widetilde{P}(u, z)=\int ^u_0{\mathcal {G}}(u,y)z(y)dy, \end{aligned}$$

(97)

and ${\mathcal {H}}:{\mathbb {R}}^+\times [0,1]\times H_{1/2}\rightarrow H$ fulfills following conditions

$$\begin{aligned} \Vert {\mathcal {H}}(t,u, w)\Vert \le {\mathcal {Q}}(u, t)(1+\Vert w\Vert _{1/2}), \end{aligned}$$

(98)

where ${\mathcal {Q}}(\cdot , t)\in H$ and ${\mathcal {Q}}$ continuous in its second arguments.

For $w\in D(A)$ and $\lambda \in {\mathbb {R}}$ with $-Aw=\lambda w$, we obtain

$$\begin{aligned} <-A w, w>=<\lambda w, w>, \end{aligned}$$

and $\Vert w'\Vert _{L^2}=\lambda \Vert w\Vert _{L^2}$. This gives that $\lambda >0$. Let $w(u)=C_1\sin (\sqrt{\lambda }u)+C_2\cos (\sqrt{\lambda }u)$ be the solution of the equation $-Aw=w''=\lambda w$. We use the boundary condition and get $C_2=0$ and $\lambda =\lambda _n=n^2\pi ^2$ for each $n\in {\mathbb {N}}$. Therefore, we get

$$\begin{aligned} w_n(u)=C_1\sin (\sqrt{\lambda _n}u), \;\;\;{\textit{text}}~\,{\textit{for each}}\;\;n\in {\mathbb {N}}, \end{aligned}$$

(99)

and $<w_n, w_m>=0,\;\;m\ne n$, $<w_n, w_n>=1$.

For $w\in D(A)$, there exists a sequence of real numbers $\{\beta _n\}$ such that

$$\begin{aligned} w(u)=\sum _{n\in {\mathbb {N}}}\beta _n w_n(u),\;\;\sum _{n\in {\mathbb {N}}}(\beta _n)^2<+\infty ,\;\;\sum _{n\in {\mathbb {N}}}(\lambda _n)^2(\beta _n)^2<+\infty . \end{aligned}$$

We also have

$$\begin{aligned} A^{1/2} w(u)=\sum _{n\in {\mathbb {N}}}\sqrt{\lambda _n}\beta _n w_n(u),\;\;\;w\in D(A^{1/2}) \end{aligned}$$

(100)

with $\sum _{n\in {\mathbb {N}}}\lambda _n(\beta _n)^2<+\infty$. The semigroup ${\mathcal {S}}(t)$ have the following expression as

$$\begin{aligned} {\mathcal {S}}(t)w=\sum _{n=1}^\infty \exp (n^2 t)<w, w_n>w_n, \end{aligned}$$

(101)

here, $\{w_n\}$, $n=1, 2, \cdots$ denotes the orthogonal set of eigenfunctions of A defined by the (99). Now, we will show that (A2)-(A3) are verified. For (A2), we have that $\widetilde{P}:[0,1]\times H_{-1/2}\rightarrow H$ defined by

$$\begin{aligned} \widetilde{P}(u, z(u,t))=\int ^u_0 {\mathcal {G}}(u,y)z(y,t)dy, \end{aligned}$$

and $z(u,t)=z(y, h(t)|z(y,t)|)$. Thus, for each $u\in [0,1]$, we obtain

$$\begin{aligned} |\widetilde{P}(u, z_1(u,))-\widetilde{P}(u, z_2(u,))|\le & {} \int ^u_0|{\mathcal {G}}(u,y)|\cdot |(z_1-z_2)(y,\cdot )|dy,\nonumber \\\le & {} \Vert {\mathcal {G}}\Vert _{\infty }\int ^u_0|(z_1-z_2)(y,\cdot )|dy. \end{aligned}$$

(102)

Since $z_1, z_2\in H^1(0,1)$. Therefore, applying the Minkowski’s integral inequality and getting

$$\begin{aligned} |\widetilde{P}(u, z_1(u,))-\widetilde{P}(u, z_2(u,))|^2_{L^2(0,1)}\le & {} \Vert {\mathcal {G}}\Vert ^2_\infty \int ^1_0\int ^y_0|(z_1-z_2)(y,\cdot )|^2 dxdy,\nonumber \\\le & {} \Vert {\mathcal {G}}\Vert ^2_\infty \int ^1_0 y|(z_1-z_2)(y,\cdot )|^2 dy,\nonumber \\\le & {} \Vert {\mathcal {G}}\Vert ^2_\infty \Vert z_1-z_2\Vert ^2_{L^2(0,1)}. \end{aligned}$$

(103)

Since we have

$$\begin{aligned} \frac{\partial }{\partial u}\widetilde{P}(u, z(u,\cdot ))={\mathcal {G}}(u,u)z(u,\cdot )+\int ^u_0\frac{\partial {\mathcal {G}}}{\partial u}(u,u)z(y,\cdot )dy. \end{aligned}$$

(104)

Thus, we estimate

$$\begin{aligned} \Vert \frac{\partial }{\partial u}\widetilde{P}(u, z_1(u,\cdot ))-\frac{\partial }{\partial u}\widetilde{P}(u, z_2(u,\cdot ))\Vert _{L^2(0,1)}\le \left( \Vert {\mathcal {G}}\Vert _\infty +\Vert \frac{\partial {\mathcal {G}}}{\partial u}\Vert _\infty \right) \Vert z_1-z_2\Vert _{L^2(0,1)}. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert \widetilde{P}(u, z_1(u,\cdot ))-\widetilde{P}(u, z_2(u,\cdot ))\Vert _{H^1(0,1)}\le & {} \left( 2\Vert {\mathcal {G}}\Vert _\infty +\Vert \frac{\partial {\mathcal {G}}}{\partial u}\Vert _\infty \right) \Vert z_1-z_2\Vert _{L^2(0,1)},\\\le & {} \left( 2\Vert {\mathcal {G}}\Vert _\infty +\Vert \frac{\partial {\mathcal {G}}}{\partial u}\Vert _\infty \right) \Vert z_1-z_2\Vert _{H^1(0,1)}, \end{aligned}$$

The assumption on $\widetilde{H}$ gives that there exist constants $B_2>0$ and $\mu \in (0, 1]$ such that

$$\begin{aligned} \Vert \widetilde{H}(t, u, w_1)-\widetilde{H}(s, u, w_2)\Vert _{H^1_0(0,1)}\le B_2(|t-s|^\mu +\Vert w_1-w_2\Vert _{H^1_0(0,1)}), \end{aligned}$$

(105)

for all $t,s\in [0,1]$, $u\in (0,1)$ and $w_1,\;w_2\in H^1_0(0,1)$. Therefore, $f:[0,1]\times H^1_0(0,1)\times H^1(0,1)\rightarrow L^2(0,1)$ defined by $f=\widetilde{P}+\widetilde{H}$ fulfills the assumption (A2).

Next, we will show that $a: H^1_0(0,1)\times {\mathbb {R}}^+\rightarrow {\mathbb {R}}^+$ which is defined as $a(w(u,t), t)=h(t)|w(u,t)|$, fulfill the assumption (A3). For $t\in [0,1]$

$$\begin{aligned} |a(w,t)|= & {} |h(t)|w(u,t)||,\nonumber \\\le & {} \Vert h\Vert _\infty \times \Vert w\Vert _\infty \le \Vert h\Vert _\infty \Vert w\Vert _{H^1_0(0,1)}, \end{aligned}$$

(106)

In the above inequality, we have used the following embedding $H^1_0(0,1)\subset C[0,1]$. By the H$\ddot{\mathrm{o}}$lder continuity of h, we have that there exist $L_h>0$ and $\theta _1\in (0,1]$ such that

$$\begin{aligned} |h(t)-h(s)|\le L_h|t-s|,\;\;t,s\in [0,1]. \end{aligned}$$

(107)

Furthermore, for $w_1,w_2\in H^1_0(0,1)$, we have

$$\begin{aligned} |a(w_1, t)-a(w_2, s)|= & {} |h(t)\left[ |w_1(u,t)|-|w_2(u,s)| \right] +(h(t)-h(s))w_2(u,s)|,\\\le & {} \Vert h\Vert _\infty \Vert w_1-w_2\Vert _\infty +L_h|t-s|^\theta \Vert w_2\Vert _\infty ,\\\le & {} \Vert h\Vert _\infty \Vert w_1-w_2\Vert _{H^1_0(0,1)}+L_h|t-s|^\theta \Vert w_2\Vert _\infty ,\\\le & {} \max \{\Vert h\Vert _\infty , L_h \Vert w_2\Vert _\infty \}\left( \Vert w_1-w_2\Vert _{H^1_0(0,1)}+|t-s|^\theta \right) . \end{aligned}$$

For $w_1, w_2\in D(A^{1/2})$, we have

$$\begin{aligned} \Vert I_i(w_1)-I_i(w_2)\Vert _{1/2}\le \frac{2\Vert w_1-w_2\Vert _{1/2}}{\Vert (2+w_1)(2+w_2)\Vert _{1/2}}\le \frac{1}{2}\Vert w_1-w_2\Vert _{1/2}. \end{aligned}$$

Thus, all the results of this section to obtain the main results can be applied.

For the particular case, we can take following example

$$\begin{aligned} f(t,w(t),w(a(w(t),t)))= & {} \frac{3}{\sin (w(\frac{1}{2}w(t)))+4},\;\;t\in [0,1],\\ I_i(w(t_i))= & {} \frac{|w(t_i^{-})|}{9+|w(t_i^{-})|} \end{aligned}$$

where $L_a=\frac{1}{2}$ $L_F=3/16$ and $L_I=1/9$.