This work concerns the study of the controllability for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces. We give sufficient conditions that ensure the controllability of the system by supposing that its undelayed part admits a resolvent operator in the sense of Grimmer, and by making use of the measure of noncompactness and the Mönch fixed-point Theorem. As a result, we obtain a generalization of the work of K. Balachandran and R. Sakthivel (Journal of Mathematical Analysis and Applications, 255, 447-457, (2001)) and a host of important results in the literature, without assuming the compactness of the resolvent operator. An example is given for illustration.
The dynamics of evolution processes is often subjected to abrupt changes such as shocks, harvesting, and natural disasters. Often these short-term perturbations are treated as having acted instantaneously or in the form of impulses [1]. The study of dynamical systems with impulsive effects is of great importance. Impulsive differential equations have become more important in recent years in some mathematical models of real processes and phenomena studied in control, physics, chemistry, population dynamics, aero- nautics and engineering. The concept of controllability plays an important role in many areas of applied mathematics. In recent years, significant progress has been made in the controllability of linear and nonlinear deterministic infinite dimensional systems, see for instance [2, 3, 4, 5, 6, 7, 8, 9, 10, 11] and the references therein. Many authors studied the controllability problem of nonlinear systems with delay in infinite dimensional Banach spaces; see for instance [2, 6, 9, 10, 11] etc and the references contained in them.
The controllability problem for nonlinear impulsive systems in infinite dimensional Banach spaces has been studied by several authors, see e.g., [6, 7, 11]. In [11], Selvi and Arjunan considered the following impulsive differential systems with finite delayRemark 1. Note that \((H_2)\) is satisfied in the modelling of Heat Conduction in materials with memory and viscosity. More details can be found in [17].
Let \(\mathcal{L}(X)\) be the Banach space of bounded linear operators on \(X\).Definition 1.[18] A resolvent operator \(\big(R(t)\big)_{t\geq0 }\) for Equation \((5)\) is a bounded operator valued function $$R\, :\ [0,+\infty)\ \longrightarrow \ \mathcal{L}(X)$$ such that
Theorem 1.[15] Assume that \((H_{1})\) and \((H_2)\) hold. Then, the linear Equation (5) has a unique resolvent operator \(\big(R(t)\big)_{t\geq0 }\).
Remark 2. In general, the resolvent operator \(\big(R(t)\big)_{t\geq0}\) for Equation (5) does not satisfy the semigroup law, namely, \(R(t+s) \ \neq \ R(t)R(s) \ \mbox{for some} \ \ t,\, s >0\, . \)
We have the following theorem that establishes the equivalence between the operator-norm continuity of the \(C_0\)-semigroup and the resolvent operator for integral equations.Theorem 2.[5] Let \(A\) be the infinitesimal generator of a \(C_0\)-semigroup \(\big(T(t)\big)_{t\geq0}\) and let \(\big(\gamma(t)\big)_{t\geq0}\) satisfy \((H_2)\). Then the resolvent operator \(\big(R(t)\big)_{t\geq0}\) for Equation \((5)\) is operator-norm continuous (or continuous in the uniform operator topology) for \(t>0\) if and only if \(\big(T(t)\big)_{t\geq0}\) is operator-norm continuous for \(t>0\). \label{normcontinuity}
Definition 2. Let \(u\in L^2(J,U)\) and \(\varphi\in\mathcal{P}\). A function \(x:\,]-\infty,b]\rightarrow X\) is called a mild solution of equation (4) if \(x(t)=\varphi(t)\ \ \text{for}\ \ t\in(-\infty,0],\ \Delta x(t_k)=I_k(x_{t_k}),\ \ k=1,2,\cdots,m\), the restriction of \(x\) to intervals \(J_k=(t_k,t_{k+1}]\ (k=0,\cdots,m)\) is continuous and the following integral equation is satisfied
Definition 3. Equation \((4)\) is said to be controllable on the interval \(J\) if for every \(\varphi\in\mathcal{P}\) and \( x_1\in X\), there exists a control \(u\in L^2(J,U)\) such that a mild solution \(x\) of Equation \((4)\) satisfies the condition \(x(b)=x_1\).
For proving the main result of the paper we recall some properties of the measure of noncompactness and the Mönch fixed-point theorem.Definition 4. [19] Let \(D\) be a bounded subset of a normed space \(Y\). The Hausdorff measure of noncompactness ( shortly MNC) is defined by $$\beta(D)=\inf\Big\{\epsilon>0: D\ has\ a\ finite\ cover\ by\ balls\ of\ radius\ less\ than\ \epsilon\Big\}.$$
Theorem 3. [19] Let \(D,\ D_1,\ D_2\) be bounded subsets of a Banach space \(Y\). The Hausdorff MNC has the following properties:
Lemma 1. [19] Let \(M\subset \mathcal{PC}([a,b];X)\) be bounded and piecewise equicontinuous on \([a,b]\). Then \(\beta(M(t))\) is piecewise continuous for \(t\in[a,b]\) and \(\beta(M)=\sup\{\beta(M(t));\,t\in [a,b]\},\ \ \ \text{where}\ M(t)=\{x(t);\,x\in M\}.\)
Lemma 2. [19] Let \(M\subset\mathcal{C}([a,b];X)\) be bounded and equicontinuous. Then the set \(\overline{co}(M)\) is also bounded and equicontinuous.
To prove the controllability for Equation (4), we need the following results.Lemma 3.[4] If \((u_n)_{n\geq1}\) is a sequence of Bochner integrable functions from \(J\) into a Banach space \(Y\) with the estimation \(\|u_n(t)\|\leq\mu(t)\) for almost all \(t\in J\) and every \(n\geq1\), where \(\mu\in L^1(J,\mathbb{R})\), then the function \(\psi(t)\ =\ \beta(\{u_n(t):n\geq1\})\) belongs to \(L^1(J,\mathbb{R}^{+})\) and satisfies the following estimation \(\beta\left(\left\{\int_0^tu_n(s)ds\,:\ n\geq1\right\}\right)\ \leq\ 2\int_0^t\psi(s)ds.\)
We now state the following nonlinear alternative of Mönch’s type for selfmaps, which we shall use in the proof of the controllability of Equation \((4)\).Theorem 4. {[20](Mönch, 1980) Let \(\mathcal{K}\) be a closed and convex subset of a Banach space \(Z\) and \(0\in \mathcal{K}\). Assume that \(F:\mathcal{K}\rightarrow \mathcal{K}\) is a continuous map satisfying Mönch’s condition, namely, \(D\subseteq \mathcal{K}\) be countable and \( D\subseteq \overline{co}(\{0\}\cup F(D))\) implies \( \overline{D}\) is compact. Then \(F\) has a fixed point.
Theorem 5. Suppose that hypotheses \((H_3)-(H_5)\) hold and Equation (5) has a resolvent operator \(\big(R(t)\big)_{t\geq0}\) that is continuous in the operator-norm topology for \(t>0\). Then Equation \((4)\) is controllable on \(J\) provided that
Proof
Using \((H_3)\) and given an arbitrary function \(x\), we define the control as usual by the following formula;
$$u_x(t)\ =\ W^{-1}\left\{x_1-R(b)\varphi(0)-\int_0^bR(b-s)f(s,x_s)\,ds-\sum_{0< t_k< t}R(b-t_k)I_k(x_{t_k})\right\}(t)\qquad \text{for}\ t\in I.$$
For each \(x\in \mathcal{PC}\) such that \(x(0)=\varphi(0)\), we define its extension \(\widetilde{x}\) from \(]-\infty,b]\) to \(X\) as follows
\begin{equation*}
\widetilde{x}(t)=\left\{
\begin{array}{l}
x(t)\ \ \text{if}\ \ t\in[0,b],\\
\varphi(t) \ \ \text{if}\ \ t\in]-\infty,0].
\end{array}
\right.%\eqno(2.2)
\end{equation*}
We define the space \(E_b=\Big\{x:]-\infty,b]\rightarrow X\ \text{such that}\ x|_{J}\in \mathcal{PC}\ \text{and}\ x_0\in\mathcal{P}\Big\},\) where where \(x|_{J}\) is the restriction of \(x\) to \(J\). We show, by using this control that the operator \(P:\,E_b\rightarrow E_b\) defined by
\begin{equation*}
(Px)(t)=
R(t)\varphi(0)+\displaystyle{\int_0^tR(t-s)\big[f(s,\widetilde{x}_s)+Cu_x(s)\big]\,ds}+\sum_{0< t_k< t}R(t-t_k)I_k(x_{t_k})\, \ \ \text{for}\ \ t\in I=[0,b]
\end{equation*}
has a fixed-point. This fixed point is then a mild solution of Equation \((4)\).
Observe that \((Px)(b)=x_1\).
This means that the control \(u_x\) steers the integrodifferential equation from \(\varphi\) to \(x_1\) in time \(b\) which implies that the Equation \((4)\) is controllable on \(J\).
For each \(\varphi\in\mathcal{P}\), we define the function \(y\in \mathcal{PC}\) by \(y(t)= R(t)\varphi(0)\) and its extension \(\widetilde{y}\) on \(]-\infty,0]\) by
\begin{equation*}
\widetilde{y}(t)=\left\{
\begin{array}{l}
y(t)\ \ \text{if}\ \ t\in[0,b], \\
\varphi(t)\ \ \text{if}\ \ t\in]-\infty,0].
\end{array}
\right.
\end{equation*}
For each \(z\in \mathcal{PC}\), set \(\widetilde{x}(t)=\widetilde{z}(t)+\widetilde{y}(t)\), where \(\widetilde{z}\) is the
extension by zero of the function \(z\) on \(]-\infty,0]\).
Observe that \(x\) satifies \((\ref{eqn3})\) if and only if \(z(0)=0\) and
$$z(t)= \int_0^tR(t-s)\big[f(s,\widetilde{z}_s+\widetilde{y}_s)+Cu_z(s)\big]\,ds+\sum_{0< t_k< t}R(t-t_k)I_k(z_{t_k}+\widetilde{y}_{t_k})\ \ \text{for}\ t\in[0,b],$$
where \(u_z(t)\ =\ W^{-1}\left\{x_1-R(b)\varphi(0)-\int_0^bR(b-s)f(s,\widetilde{z}_s+\widetilde{y}_s)\,ds-\sum_{0< t_k< t}R(b-t_k)I_k(z_{t_k}+\widetilde{y}_{t_k})\right\}(t).\)
Now let \(E_b^0=\Big\{z\in E_b\ \text{such that}\ z_0=0\Big\}.\)
Thus \(E_b^0\) is a Banach space provided with the supremum norm. Define the operator
\(\Gamma\, :\,E_b^0\rightarrow E_b^0\) by
\begin{equation*}
(\Gamma z)(t)=
\displaystyle{\int_0^tR(t-s)\big[f(s,\widetilde{z}_s+\widetilde{y}_s)+Cu_z(s)\big]\,ds}+\sum_{0< t_k< t}R(t-t_k)I_k(z_{t_k}+\widetilde{y}_{t_k})\ \ \text{for}\ \ t\in[0,b].
\end{equation*}
Note that the operator \(P\) has a fixed point if and only if \(\Gamma\) has one. So to prove that \(P\) has a fixed point, we only need to prove that \(\Gamma\) has one.
For each positive number \(q\), let \(B_q=\{z\in E_b^0:\|z\|\leq q\}\). Then, for any \(z\in B_q\), we have by axiom \((A_1)\) that
\begin{eqnarray}
\|z_s+y_s\|&\leq &\|z_s\|_{\mathcal{P}}+\|y_s\|_{\mathcal{P}}\\
&\leq&K(s)\|z(s)\|+M(s)\|z_0\|_{\mathcal{P}}+K(s)\|y(s)\|+M(s)\|y_0\|_{\mathcal{P}}\\
&\leq&K_b\|z(s)\|+K_b\|R(t)\|\|\varphi(0)\|+M_b\|\varphi\|_{\mathcal{P}}\\
&\leq&K_b\|z(s)\|+K_bR_bH\|\varphi\|_{\mathcal{P}}+M_b\|\varphi\|_{\mathcal{P}}\\
&\leq&K_b\|z(s)\|+\Big(K_bR_bH+M_b\Big)\|\varphi\|_{\mathcal{P}}\\
&\leq&K_b\,q+\Big(K_bR_bH+M_b\Big)\|\varphi\|_{\mathcal{P}}.
\end{eqnarray}
Thus, \(\|z_s+y_s\|\leq K_b\,q+\Big(K_bR_bH+M_b\Big)\|\varphi\|_{\mathcal{P}}=:q'.\) We shall prove the theorem in the following steps;
Step 1. We claim that there exists \(q>0\) such that \(\Gamma(B_q)\subset B_q\). We proceed by contradiction.
Assume that it is not true. Then for each positive number \(q\), there exists a function \(z^q\in B_q\), such that
\(\Gamma(z^q)\notin B_q,\ i.e.,\ \|(\Gamma z^q)(t)\|>q\) for some \(t\in [0,b]\).
Now we have that
\begin{eqnarray}
q &< & \Big\|(\Gamma z^q)(t)\Big\|\\
&\leq& R_b\int_0^b\Big\|f(s,\tilde{z}_s^q+\widetilde{y}_s)\Big\|\,ds+R_b\int_0^b\|Cu_{z^q}(s)\|\,ds+R_b\sum_{k=0}^mL_k(\|z_{t_k}+\widetilde{y}_{t_k}\|)\\
&\leq& R_b\int_0^b\Big\|f(s,\tilde{z}_s^q+\widetilde{y}_s)\Big\|\,ds+R_b\sum_{k=0}^mL_k(q')\\
&& +R_b\int_0^b\Big\|BW^{-1}\Big[x_1-R(b)\varphi(0)-\int_0^bR(b-s)f(s,\tilde{z}_s^q)\,ds-\sum_{0< t_k< t}R(b-t_k)I_k(z_{t_k}+\widetilde{y}_{t_k})\Big]\Big\|\,ds\\
&\leq& bR_bM_2M_3\left(\|x_1\|+R_b\|\varphi(0)\|+R_b\int_0^b\|f(s,\tilde{z}_s^q)\|\,ds+R_b\sum_{k=0}^mL_k(q')\right)\\
&& +R_b\int_0^b\Big\|f(s,\tilde{z}_s^q+\widetilde{y}_s)\Big\|\,ds+R_b\sum_{k=0}^mL_k(q')\\
&\leq& bR_bM_2M_3\left(\|x_1\|+R_bH\|\varphi\|_{\mathcal{B}}+R_b\int_0^bl_{q'}(s)\,ds+R_b\sum_{k=0}^mL_k(q')\right)+ R_b\int_0^bl_{q'}(s)\,ds+R_b\sum_{k=0}^mL_k(q'),
\end{eqnarray}
where \(q':=K_b\,q+q_0,\ \ \text{with}\ q_0:=\Big(K_bR_bH+M_b\Big)\|\varphi\|_{\mathcal{B}}.\) Hence
$$q\leq \Big(1+R_bM_2M_3b\Big)\left(R_b\int_0^bl_{q'}(s)\,ds+R_b\sum_{k=0}^mL_k(q')\right)+R_bM_2M_3b\Big(\|x_1\|+R_bH\|\varphi\|_{\mathcal{B}}\Big).$$
Dividing both sides by \(q\) and noting that \(q'=K_bq+q_0\rightarrow+\infty\) as \(q\rightarrow+\infty\), we obtain that
$$1\leq\Big(1+R_bM_2M_3b\Big)R_b\left(\frac{\displaystyle{\int_0^bl_{q'}(s)\,ds}+\sum_{k=0}^mL_k(q')}{q}\right)+\frac{R_bM_2M_3b\Big(\|x_1\|+R_bH\|\varphi\|_{\mathcal{B}}\Big)}{q}$$ and
$$\liminf_{q\rightarrow+\infty}\left(\frac{\displaystyle{\int_0^bl_{q'}(s)\,ds}+\sum_{k=0}^mL_k(q')}{q}\right)=\liminf_{q\rightarrow+\infty}\left(\frac{\displaystyle{\int_0^bl_{q'}(s)\,ds}}{q'}+\frac{\sum_{k=0}^mL_k(q')}{q'}\right)\frac{q'}{q}=\left(l+\sum_{k=0}^m\lambda_k\right)K_b.$$
Thus we have, \(1\leq\Big(1+R_bM_2M_3b\Big)R_b\left(l+\sum_{k=0}^m\lambda_k\right)K_b\), and this contradicts \((7)\). Hence for some positive number \(q\), we must have \(\Gamma(B_q)\subset B_q\).
Step 2. \(\Gamma:\,E_b^0\rightarrow E_b^0\) is continuous. In fact let \(\Gamma:=\Gamma_1+\Gamma_2\), where
$$(\Gamma_1z)(t)=\int_0^tR(t-s)f(s,\widetilde{z}_s+\widetilde{y}_s)\,ds+\sum_{k=0}^mR(t-t_k)I_k(z_{t_k}+\widetilde{y}_{t_k})\ \ \ \text{and}\ \ \ \ \ (\Gamma_2z)(t)=\int_0^tR(t-s)Cu_z(s)\,ds.$$
Let \(\{z^n\}_{n\geq1}\subset E_b^0\) with \(z^n\rightarrow z\) in \(E_b^0\). Then there exists a number \(q>1\) such that \(\|z^n(t)\|\leq q\) for all \(n\)
and \(a.e.\ \,t\in J\). So \(z^n,\,z\in B_q\).
By \((H_4)-(i),\,\, f(t,\tilde{z}_t^n+\widetilde{y}_t)\rightarrow f(t,\widetilde{z}_t+\widetilde{y}_t)\) for each \(t\in [0,b]\).
Also, by \((H_5)-(i),\,\,I_k(z_{t_k}^n+\widetilde{y}_{t_k})\rightarrow I_k(z_{t_k}+\widetilde{y}_{t_k})\) for each \(t\in [0,b]\).
And by \((H_4)-(ii)\),
\(\|f(t,\tilde{z}_t^n+\widetilde{y}_t)-f(t,\widetilde{z}_t+\widetilde{y}_t)\|\leq 2l_{q’}(t).\)
Then we have
$$\|\Gamma_1z^n-\Gamma_1z\|_{\mathcal{P}}\leq R_b\int_0^b\|f(s,\tilde{z}_s^n+\widetilde{y}_s)-f(s,\widetilde{z}_s+\widetilde{y}_s)\|\,ds+R_b\sum_{k=0}^m\|I_k(z_{t_k}^n+\widetilde{y}_{t_k})-I_k(z_{t_k}+\widetilde{y}_{t_k})\|\longrightarrow0,\ as\ n\rightarrow+\infty$$
by dominated convergence Theorem. Also we have that
$$\|\Gamma_2z^n-\Gamma_2z\|_{\mathcal{P}}\leq R_b^2M_2M_3b\left(\int_0^b\|f(s,\tilde{z}_s^n+\widetilde{y}_s)-f(s,\widetilde{z}_s+\widetilde{y}_s)\|\,ds+\sum_{k=0}^m\|I_k(z_{t_k}^n+\widetilde{y}_{t_k})-I_k(z_{t_k}+\widetilde{y}_{t_k})\right)\longrightarrow0,$$
by dominated convergence Theorem.
Thus \(\|\Gamma z^n-\Gamma z\|\leq\|\Gamma_1z^n-\Gamma_1z\|+\|\Gamma_2z^n-\Gamma_2z\|\longrightarrow0,\ as\ n\rightarrow+\infty.\) Hence \(\Gamma\) is continuous on \(E_b^0\).
Step 3. \(\Gamma(B_q)\) is equicontinuous on \([0,b]\). In fact let \(t_1,\,t_2\in J_k,\ \ t_1< t_2\) and \(z\in B_q\), we have
\begin{eqnarray}
&&\|(\Gamma z)(t_2)-(\Gamma z)(t_1)\|\leq\int_0^{t_1}\|R(t_2-s)-R(t_1-s)\|\|f(s,\widetilde{z}_s+\widetilde{y}_s)+Cu_z(s)\|\,ds\\
&& +\sum_{0< t_k< t_1}\|R(t_2-t_k)-R(t_1-t_k)\|\|I_k(z_{t_k}+\widetilde{y}_{t_k})\|+\sum_{t_1\leq t_k< t_2}\|R(t_1-t_k)\|\|I_k(z_{t_k}+\widetilde{y}_{t_k})\|\\
&& +\int_{t_1}^{t_2}\|R(t_2-s)\|\|f(s,\widetilde{z}_s+\widetilde{y}_s)+Cu_z(s)\|\,ds\\
&&\leq\int_0^{t_1}\|R(t_2-s)-R(t_1-s)\|l_{q'}(s)\,ds\\
&& +\int_0^{t_1}\|R(t_2-s)-R(t_1-s)\|M_2M_3\left(\|x_1\|+R_bH\|\varphi\|_{\mathcal{B}}+R_b\int_0^bl_{q'}(\tau)\,d\tau+\sum_{k=0}^mL_k(q')\right)\,ds\\
&& +\sum_{0< t_k< t_1}\|R(t_2-t_k)-R(t_1-t_k)\|L_k(q')+R_b\sum_{t_1\leq t_k< t_2}L_k(q')+\int_{t_1}^{t_2}\|R(t_2-s)\|l_{q'}(s)\,ds\\
&& +\int_{t_1}^{t_2}\|R(t_2-s)\|M_2M_3\left(\|x_1\|+R_bH\|\varphi\|_{\mathcal{B}}+R_b\int_0^bl_{q'}(\tau)\,d\tau+\sum_{k=0}^mL_k(q')\right)\,ds.
\end{eqnarray}
By the continuity of \(\big(R(t)\big)_{t\geq0}\) in the operator-norm toplogy, the dominated convergence Theorem, we conclude that the right hand side of the above inequality tends to zero and independent of \(z\) as \(t_2\rightarrow t_1\). Hence
\(\Gamma(B_q)\) is equicontinuous on \(J\).
Step 4. We show that the Mönch’s condition holds.
Suppose that \(D\subseteq B_q\) is countable and \(D\subseteq\overline{co}(\{0\}\cup \Gamma(D))\). We shall show that \(\beta(D)=0\), where \(\beta\) is the Hausdorff
MNC. Without loss of generality, we may assume that \(D=\{z^n\}_{n\geq1}\).
Since \(\Gamma\) maps \(B_q\) into an equicontinuous family, \(\Gamma(D)\) is also
equicontinuous on \(J\). By \((H_3)-(ii)\), \((H_4)-(iii)\) and Lemma 3, we have that
\begin{align*}
& \beta\Big(\{u_{z^n}(t)\}_{n\geq1}\Big) = \beta\left(W^{-1}\left\{x_1-R(b)\varphi(0)-\int_0^bR(t-b)f\Big(s,\{\tilde{z}_s^n+\widetilde{y}_s\}_{n\geq1}\Big)\,ds-\sum_{0< t_k< t}R(b-t_k)I_k(z_{t_k}^n+\widetilde{y}_{t_k})\right\}\right)\\
&\leq L_W(t)\beta\left(\left\{x_1-R(b)\varphi(0)\right\}\right)+L_W(t)\beta\left(\left\{\int_0^bR(t-b)f\Big(s,\{\tilde{z}_s^n+\widetilde{y}_s\}_{n\geq1}\Big)\,ds\right\}_{n\geq1}\right) \end{align*}
\begin{align*}
&\;\;\;+L_W(t)\beta\left(\left\{\sum_{0< t_k< t}R(b-t_k)I_k(z_{t_k}^n+\widetilde{y}_{t_k})\right\}_{n\geq1}\right)\\
&\leq 2R_bL_W(t)\left(\int_0^bh(s)\beta\left(\left\{\tilde{z}_s^n\right\}_{n\geq1}+\{\widetilde{y}_s\}\right)ds\right)+R_bL_W(t)\sum_{k=0}^m\beta\left(\left\{I_k(z_{t_k}^n+\widetilde{y}_{t_k})\right\}_{n\geq1}\right)\\
&\leq 2R_bL_W(t)\left(\int_0^bh(s)\Big[\beta\left(\left\{\tilde{z}_s^n\right\}_{n\geq1}\right)+\beta\Big(\{\widetilde{y}_s\}\Big)\Big]\,ds\right)+R_bL_W(t)\sum_{k=0}^m\alpha_k\sup_{-\infty< \theta\leq0}\beta\left(\left\{z_{t_k}^n+\widetilde{y}_{t_k}\right\}_{n\geq1}\right)\\
&\leq 2R_bL_W(t)\left(\int_0^bh(s)\beta\left(\left\{\tilde{z}_s^n\right\}_{n\geq1}\right)\,ds\right)+R_bL_W(t)\sum_{k=0}^m\alpha_k\sup_{-\infty< \theta\leq0}\beta\left(\left\{z_{t_k}^n\right\}_{n\geq1}\right),
\end{align*}
since \(\Big\{\widetilde{y}_s:\ s\in[0,b]\Big\}\)is compact, so
\begin{align*} &\leq 2R_bL_W(t)\left(\int_0^bh(s)\displaystyle\sup_{-\infty< \theta\leq0}\beta\left(\left\{\tilde{z}_s^n(\theta)\right\}_{n\geq1}\right)\,ds\right)+R_bL_W(t)\sum_{k=0}^m\alpha_k\sup_{-\infty< \theta\leq0}\beta\left(\left\{z_{t_k}^n\right\}_{n\geq1}\right),
\end{align*}
by Lemma 1, since \(D=\{z^n\}_{n\geq1}\) is equicontinuous, we obtain
\begin{align*} &\leq 2R_bL_W(t)\left(\int_0^bh(s)\,ds\right)\displaystyle\sup_{0\leq t\leq b}\beta\left(\left\{z^n(t)\right\}_{n\geq1}\right)+R_bL_W(t)\sum_{k=0}^m\alpha_k\sup_{0\leq \tau_k\leq t_k}\beta\left(\left\{z^n(\tau_k)\right\}_{n\geq1}\right).
\end{align*}
This implies that
\begin{align*}
&\beta\Big(\{(\Gamma z^n)(t)\}_{n\geq1}\Big)\leq \beta\left(\left\{\int_0^tR(t-s)f(s,\{\tilde{z}_s^n+\widetilde{y}_s\}_{n\geq1})\,ds\right\}_{n\geq1}\right)+\beta\left(\left\{\int_0^tR(t-s)u_{z^n}(s)\,ds\right\}_{n\geq1}\right)\\
&\;\;\;+\beta\left(\left\{\sum_{0< t_k< t}R(b-t_k)I_k(z_{t_k}^n+\widetilde{y}_{t_k})\right\}_{n\geq1}\right)\\
&\leq2R_b\left(\int_0^bh(s)\,ds\right)\displaystyle\sup_{0\leq t\leq b}\beta\left(\left\{z^n(t)\right\}_{n\geq1}\right)+R_b\sum_{k=0}^m\alpha_k\sup_{0\leq \tau_k\leq t_k}\beta\left(\left\{z^n(\tau_k)\right\}_{n\geq1}\right)\\
& \;\;\;+2R_bM_2\left(\int_0^bL_W(s)\,ds\right)2R_b\left(\int_0^bh(s)\,ds\right)\displaystyle\sup_{0\leq t\leq b}\beta\left(\left\{z^n(t)\right\}_{n\geq1}\right)\\&\;\;\;+2R_b^2M_2\left(\int_0^bL_W(s)\,ds\right)\sum_{k=0}^m\alpha_k\sup_{0\leq \tau_k\leq t_k}\beta\left(\left\{z^n(\tau_k)\right\}_{n\geq1}\right)\\
&\leq 2R_b\|h\|_{L^1}\displaystyle\sup_{0\leq t\leq b}\beta\left(\left\{z^n(t)\right\}_{n\geq1}\right)+R_b\sum_{k=0}^m\alpha_k\sup_{0\leq \tau_k\leq t_k}\beta\left(\left\{z^n(\tau_k)\right\}_{n\geq1}\right)\\&\;\;\; +2R_bM_2\|L_W\|_{L^1}2R_b\|h\|_{L^1}\displaystyle\sup_{0\leq t\leq b}\beta\left(\left\{z^n(t)\right\}_{n\geq1}\right)+2R_b^2M_2\|L_W\|_{L^1}\sum_{k=0}^m\alpha_k\sup_{0\leq \tau_k\leq t_k}\beta\left(\left\{z^n(\tau_k)\right\}_{n\geq1}\right).
\end{align*}
It follows that
\begin{align*}
& \beta\Big(\Gamma (D)(t)\Big) \leq 2R_b\|h\|_{L^1}\displaystyle\sup_{0\leq t\leq b}\beta\left(D(t)\right)+R_b\sum_{k=0}^m\alpha_k\sup_{0\leq t\leq b}\beta\left(D(t)\right) +2R_bM_2\|L_W\|_{L^1}2R_b\|h\|_{L^1}\displaystyle\sup_{0\leq t\leq b}\beta\left(D(t)\right)\\
& \;\;\;+2R_b^2M_2\|L_W\|_{L^1}\sum_{k=0}^m\alpha_k\sup_{0\leq t\leq b}\beta\left(D(t)\right)\\
&\leq \left(2R_b\|h\|_{L^1}+R_b\sum_{k=0}^m\alpha_k+2R_bM_2\|L_W\|_{L^1}2R_b\|h\|_{L^1}+2R_b^2M_2\|L_W\|_{L^1}\sum_{k=0}^m\alpha_k\right)\sup_{0\leq t\leq b}\beta\left(D(t)\right)\\
&\leq \Big(1+2R_bM_2\|L_W\|_{L^1}\Big)\left(2R_b\|h\|_{L^1}+R_b\sum_{k=0}^m\alpha_k\right)\sup_{0\leq t\leq b}\beta\left(D(t)\right).
\end{align*}
Since \(D\) and \(\Gamma(D)\) are equicontinuous on \([0,b]\) and \(D\) is bounded, it follows by Lemma 1 that
\(\beta\Big(\Gamma(D)\Big)\leq \tau\beta\Big(D\Big)\), where \(\tau\) is as defined in \((H_5)\). Thus from the Mönch condition, we get that
\(\beta\Big(D\Big)\leq\beta\Big(\overline{co}(\{0\}\cup \Gamma(D)\Big)=\beta\Big(\Gamma(D)\Big)\leq \tau\beta\Big(D\Big),\) and since \(\tau< 1\), this implies \(\beta\Big(D\Big)=0\), which implies that \(D\) is relatively compact as desired in \(B_q\) and the Mönch condition is satisfied. We conclude by Theorem 4, that \(\Gamma\) has a fixed point \(z\) in \(B_q\). Then \(x=z+y\) is a fixed point of \(P\) in \(E_b\)
and thus equation \((4)\) is controllable on \([0,b]\).
Example 1. Consider the partial functional integrodifferential system of the form
