In this section we give the existence results for the problem (1.1) under different conditions on g and {I}_{i} when the semigroup is not compact and f is not compact or Lipschitz continuous, by using Lemma 2.7 and the generalized βcondensing operator. More precisely, Theorem 3.1 is concerned with the case that compactness conditions are satisfied. Theorem 3.4 deals with the case that Lipschitz conditions are satisfied. And mixedtype conditions are considered in Theorem 3.5 and Theorem 3.6.
Let r be a finite positive constant, and set {B}_{r}=\{x\in X:\parallel x\parallel \le r\}, {W}_{r}=\{u\in PC([0,b];X):u(t)\in {B}_{r},t\in [0,b]\}. We define the solution map G:PC([0,b];X)\to PC([0,b];X) by
(Gu)(t)=T(t)g(u)+{\int}_{0}^{t}T(ts)f(s,u(s))\phantom{\rule{0.2em}{0ex}}\mathrm{d}s+\sum _{0<{t}_{i}<t}T(t{t}_{i}){I}_{i}(u({t}_{i}))
(3.1)
with
for all t\in [0,b]. It is easy to see that u is the mild solution of the problem (1.1) if and only if u is a fixed point of the map G.
We list the following hypotheses:
(Hf) f:[0,b]\times X\to X satisfies the following conditions:

(i)
f(t,\cdot ):X\to X is continuous for a.e. t\in [0,b] and f(\cdot ,x):[0,b]\to X is measurable for all x\in X. Moreover, for any r>0, there exists a function {\rho}_{r}\in {L}^{1}([0,b],R) such that
\parallel f(t,x)\parallel \le {\rho}_{r}(t)
for a.e. t\in [0,b] and x\in {B}_{r}.

(ii)
there exists a constant L>0 such that for any bounded set D\subset X,
\beta (f(t,D))\le L\beta (D)
(3.2)
for a.e. t\in [0,b].
(Hg1) g:PC([0,b];X)\to X is continuous and compact.
(HI1) {I}_{i}:X\to X is continuous and compact for i=1,\dots ,s.
Theorem 3.1 Assume that the hypotheses (HA), (Hf), (Hg1), (HI1) are satisfied, then the nonlocal impulsive problem (1.1) has at least one mild solution on[0,b]provided that there exists a constantr>0such that
M[\underset{u\in {W}_{r}}{sup}\parallel g(u)\parallel +{\parallel {\rho}_{r}\parallel}_{{L}^{1}}+\underset{u\in {W}_{r}}{sup}\sum _{i=1}^{s}\parallel {I}_{i}(u({t}_{i}))\parallel ]\le r.
(3.3)
Proof We will prove that the solution map G has a fixed point by using the fixed point theorem about the βconvexpower condensing operator.
Firstly, we prove that the map G is continuous on PC([0,b];X). For this purpose, let {\{{u}_{n}\}}_{n=1}^{\mathrm{\infty}} be a sequence in PC([0,b];X) with {lim}_{n\to \mathrm{\infty}}{u}_{n}=u in PC([0,b];X). By the continuity of f with respect to the second argument, we deduce that for each s\in [0,b], f(s,{u}_{n}(s)) converges to f(s,u(s)) in X. And we have
\begin{array}{rcl}{\parallel G{u}_{n}Gu\parallel}_{PC}& \le & M[\parallel g({u}_{n})g(u)\parallel +\sum _{i=1}^{s}\parallel {I}_{i}({u}_{n}({t}_{i})){I}_{i}(u({t}_{i}))\parallel ]\\ +M{\int}_{0}^{b}\parallel f(s,{u}_{n}(s))f(s,u(s))\parallel \phantom{\rule{0.2em}{0ex}}\mathrm{d}s.\end{array}
Then by the continuity of g, {I}_{i} and using the dominated convergence theorem, we get {lim}_{n\to \mathrm{\infty}}G{u}_{n}=Gu in PC([0,b];X).
Secondly, we claim that G{W}_{r}\subseteq {W}_{r}. In fact, for any u\in {W}_{r}\subset PC([0,b];X), from (3.1) and (3.3), we have
\begin{array}{rcl}\parallel (Gu)(t)\parallel & \le & \parallel T(t)g(u)\parallel +\parallel {\int}_{0}^{t}T(ts)f(s,u(s))\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\parallel +\sum _{0<{t}_{i}<t}\parallel T(t{t}_{i}){I}_{i}(u({t}_{i}))\parallel \\ \le & M[\parallel g(u)\parallel +{\parallel {\rho}_{r}\parallel}_{{L}^{1}}+\sum _{i=1}^{s}\parallel {I}_{i}(u({t}_{i}))\parallel ]\\ \le & r,\end{array}
for each t\in [0,b]. It implies that G{W}_{r}\subseteq {W}_{r}.
Now, we show that G{W}_{r} is equicontinuous on {J}_{0}=[0,{t}_{1}], {J}_{i}=({t}_{i},{t}_{i+1}] and is also equicontinuous at t={t}_{i}^{+}, i=1,\dots ,s. Indeed, we only need to prove that G{W}_{r} is equicontinuous on [{t}_{1},{t}_{2}], as the cases for other subintervals are the same. For u\in {W}_{r}, {t}_{1}\le s<t\le {t}_{2}, we have, using the semigroup property,
\begin{array}{rcl}\parallel T(t)g(u)T(s)g(u)\parallel & =& \parallel T(s)[T(ts)T(0)]g(u)\parallel \\ \le & M\parallel [T(ts)T(0)]g(u)\parallel .\end{array}
Thus, {G}_{1}{W}_{r} is equicontinuous on [{t}_{1},{t}_{2}] due to the compactness of g and the strong continuity of T(\cdot ). The same idea can be used to prove the equicontinuity of {G}_{3}{W}_{r} on [{t}_{1},{t}_{2}]. That is, for u\in {W}_{r}, {t}_{1}\le s<t\le {t}_{2}, we have
\parallel T(t{t}_{1}){I}_{1}(u({t}_{1}))T(s{t}_{1}){I}_{1}(u({t}_{1}))\parallel \le M\parallel [T(ts)T(0)]{I}_{1}(u({t}_{1}))\parallel ,
which implies the equicontinuity of {G}_{3}{W}_{r} on [{t}_{1},{t}_{2}] due to the compactness of {I}_{1} and the strong continuity of T(\cdot ). Moreover, from Lemma 2.9, we have that {G}_{2}{W}_{r} is equicontinuous on [0,b]. Therefore, we have that the functions in G{W}_{r}=({G}_{1}+{G}_{2}+{G}_{3}){W}_{r} are equicontinuous on each [{t}_{i},{t}_{i+1}], i=0,1,\dots ,s.
Set W=\overline{conv}G({W}_{r}), where \overline{conv} means the closure of convex hull. It is easy to verify that G maps W into itself and W is equicontiuous on each \overline{{J}_{i}}=[{t}_{i},{t}_{i+1}], i=0,1,\dots ,s. Now, we show that G:W\to W is a convexpower condensing operator. Take {x}_{0}\in W, we shall prove that there exists a positive integral {n}_{0} such that
\beta ({G}^{({n}_{0},{x}_{0})}(B))<\beta (B),
for every nonprecompact bounded subset B\subset W.
From Lemma 2.2 and Lemma 2.8, noticing the compactness of g and {I}_{i}, we have
\begin{array}{rcl}\beta (\left({G}^{(1,{x}_{0})}B\right)(t))& =& \beta ((GB)(t))\\ \le & \beta (T(t)g(B))+\beta ({\int}_{0}^{t}T(ts)f(s,B(s))\phantom{\rule{0.2em}{0ex}}\mathrm{d}s)\\ +\beta (\sum _{0<{t}_{i}<t}T(t{t}_{i}){I}_{i}(B({t}_{i})))\\ \le & {\int}_{0}^{t}\beta (T(ts)f(s,B(s)))\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \le & M{\int}_{0}^{t}\beta \left(f(s,B(s))\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \le & M{\int}_{0}^{t}L\beta (B(s))\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \le & MLt\beta (B)\end{array}
for t\in [0,b]. Further,
\begin{array}{rcl}\beta (\left({G}^{(2,{x}_{0})}B\right)(t))& =& \beta (\left(G\overline{conv}\{{G}^{(1,{x}_{0})}B,{x}_{0}\}\right)(t))\\ \le & \beta (T(t)g\left(\overline{conv}\{{G}^{(1,{x}_{0})}B,{x}_{0}\}\right))\\ +\beta ({\int}_{0}^{t}T(ts)f(s,\overline{conv}\{{G}^{(1,{x}_{0})}B(s),{x}_{0}(s)\})\phantom{\rule{0.2em}{0ex}}\mathrm{d}s)\\ +\beta (\sum _{0<{t}_{i}<t}T(t{t}_{i}){I}_{i}\left(\overline{conv}\{{G}^{(1,{x}_{0})}B({t}_{i}),{x}_{0}({t}_{i})\}\right))\\ \le & \beta ({\int}_{0}^{t}T(ts)f(s,\overline{conv}\{{G}^{(1,{x}_{0})}B(s),{x}_{0}(s)\})\phantom{\rule{0.2em}{0ex}}\mathrm{d}s)\\ \le & M{\int}_{0}^{t}\beta \left(f(s,\overline{conv}\{{G}^{(1,{x}_{0})}B(s),{x}_{0}(s)\})\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \le & M{\int}_{0}^{t}L\beta \left(\overline{conv}\{{G}^{(1,{x}_{0})}B(s),{x}_{0}(s)\}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \le & ML{\int}_{0}^{t}\beta (\left({G}^{(1,{x}_{0})}B\right)(s))\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \le & ML{\int}_{0}^{t}MLs\beta (B)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \le & {M}^{2}{L}^{2}\frac{{t}^{2}}{2!}\beta (B)\end{array}
for t\in [0,b]. We can continue this iterative procedure and get that
\beta (\left({G}^{(n,{x}_{0})}B\right)(t))\le \frac{{M}^{n}{L}^{n}{b}^{n}}{n!}\beta (B)
for t\in [0,b]. As {G}^{(n,{x}_{0})}(B) is equicontinuous on each [{t}_{i},{t}_{i+1}], by Lemma 2.7, we have that
\beta \left({G}^{(n,{x}_{0})}B\right)=\underset{t\in [0,b]}{sup}\beta (\left({G}^{(n,{x}_{0})}B\right)(t))\le \frac{{M}^{n}{L}^{n}{b}^{n}}{n!}\beta (B).
By the fact that \frac{{M}^{n}{L}^{n}{b}^{n}}{n!}\to 0 as n\to \mathrm{\infty}, we know that there exists a large enough positive integral {n}_{0} such that
\frac{{M}^{{n}_{0}}{L}^{{n}_{0}}{b}^{{n}_{0}}}{{n}_{0}!}<1,
which implies that G:W\to W is a convexpower condensing operator. From Lemma 2.5, G has at least one fixed point in W, which is just a mild solution of the nonlocal impulsive problem (1.1). This completes the proof of Theorem 3.1. □
Remark 3.2 By using the method of the measure of noncompactness, we require f to satisfy some proper conditions of MNC, but do not require the compactness of a semigroup T(t). Note that if f is compact or Lipschitz continuous, then the condition (Hf)(ii) is satisfied. And our work improves many previous results, where they need the compactness of T(t) or f, or the Lipschitz continuity of f. In the proof, Lemma 2.7 plays an important role for the impulsive differential equations, which provides us with the way to calculate the measure of noncompactness in PC([0,b];X). The use of noncompact measures in functional differential and integral equations can also be seen in [18–20, 22].
Remark 3.3 When we apply DarboSadovskii’s fixed point theorem to get the fixed point of a map, a strong inequality is needed to guarantee its condensing property. By using the βconvexpower condensing operator developed by Sun et al.[29], we do not impose any restrictions on the coefficient L. This generalized condensing operator also can be seen in Liu et al.[30], where nonlinear Volterra integral equations are discussed.
In the following, by using Lemma 2.7 and DarboSadovskii’s fixed point theorem, we give the existence results of the problem (1.1) under Lipschitz conditions and mixedtype conditions, respectively.
We give the following hypotheses:
(Hg2) g:PC([0,b];X)\to X is Lipschitz continuous with the Lipschitz constant k.
(HI2) {I}_{i}:X\to X is Lipschitz continuous with the Lipschitz constant {k}_{i}; that is,
\parallel {I}_{i}(x){I}_{i}(y)\parallel \le {k}_{i}\parallel xy\parallel ,
for x,y\in X, i=1,\dots ,s.
Theorem 3.4 Assume that the hypotheses (HA), (Hf), (Hg2), (HI2) are satisfied, then the nonlocal impulsive problem (1.1) has at least one mild solution on[0,b]provided that
M(k+Lb+\sum _{i=1}^{s}{k}_{i})<1
(3.4)
and (3.3) are satisfied.
Proof From the proof of Theorem 3.1, we have that the solution operator G is continuous and maps {W}_{r} into itself. It remains to show that G is βcondensing in {W}_{r}.
By the conditions (Hg2) and (HI2), we get that {G}_{1}+{G}_{3}:{W}_{r}\to PC([0,b];X) is Lipschitz continuous with the Lipschitz constant M(k+{\sum}_{i=1}^{s}{k}_{i}). In fact, for u,v\in {W}_{r}, we have
Thus, from Lemma 2.2(7), we obtain that
\beta (({G}_{1}+{G}_{3}){W}_{r})\le M(k+\sum _{i=1}^{s}{k}_{i})\beta ({W}_{r}).
(3.5)
For the operator ({G}_{2}u)(t)={\int}_{0}^{t}T(ts)f(s,u(s))\phantom{\rule{0.2em}{0ex}}\mathrm{d}s, from Lemma 2.6, Lemma 2.8 and Lemma 2.9, we have
\begin{array}{rcl}\beta ({G}_{2}{W}_{r})& =& \underset{t\in [0,b]}{sup}\beta (({G}_{2}{W}_{r})(t))\\ \le & \underset{t\in [0,b]}{sup}{\int}_{0}^{t}\beta (T(ts)f(s,{W}_{r}(s)))\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \le & \underset{t\in [0,b]}{sup}M{\int}_{0}^{t}L\beta ({W}_{r}(s))\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \le & MLb\beta ({W}_{r}).\end{array}
(3.6)
Combining (3.5) and (3.6), we have
\begin{array}{rcl}\beta (G{W}_{r})& \le & \beta (({G}_{1}+{G}_{3}){W}_{r})+\beta ({G}_{2}{W}_{r})\\ \le & M(k+Lb+\sum _{i=1}^{s}{k}_{i})\beta ({W}_{r}).\end{array}
From the condition (3.4), M(k+Lb+{\sum}_{i=1}^{s}{k}_{i})<1, the solution map G is βcondensing in {W}_{r}. By DarboSadovskii’s fixed point theorem, G has a fixed point in {W}_{r}, which is just a mild solution of the nonlocal impulsive problem (1.1). This completes the proof of Theorem 3.4. □
Among the previous works on nonlocal impulsive differential equations, few are concerned with the mixedtype conditions. Here, by using Lemma 2.7, we can also deal with the mixedtype conditions in a similar way.
Theorem 3.5 Assume that the hypotheses (HA), (Hf), (Hg1), (HI2) are satisfied, then the nonlocal impulsive problem (1.1) has at least one mild solution on[0,b]provided that
M(Lb+\sum _{i=1}^{s}{k}_{i})<1
(3.7)
and (3.3) are satisfied.
Proof We will also use DarboSadovskii’s fixed point theorem to obtain a fixed point of the solution operator G. From the proof of Theorem 3.1, we have that G is continuous and maps {W}_{r} into itself.
Subsequently, we show that G is βcondensing in {W}_{r}. From the compactness of g and the strong continuity of T(\cdot ), we get that \{T(\cdot )g(u):u\in {W}_{r}\} is equicontinuous on [0,b]. Then by Lemma 2.6, we have that
\beta ({G}_{1}{W}_{r})=\underset{t\in [0,b]}{sup}\beta (({G}_{1}{W}_{r})(t))=\underset{t\in [0,b]}{sup}\beta (T(t)g({W}_{r}))=0.
(3.8)
On the other hand, for u,v\in {W}_{r}, we have
\begin{array}{rcl}{\parallel {G}_{3}u{G}_{3}v\parallel}_{PC}& =& \underset{t\in [0,b]}{sup}\parallel \sum _{0<{t}_{i}<t}T(t{t}_{i})({I}_{i}(u({t}_{i})){I}_{i}(v({t}_{i})))\parallel \\ \le & M\sum _{i=1}^{s}\parallel {I}_{i}(u({t}_{i})){I}_{i}(v({t}_{i}))\parallel \\ \le & M\sum _{i=1}^{s}{k}_{i}{\parallel uv\parallel}_{PC}.\end{array}
Then by Lemma 2.2(7), we obtain that
\beta ({G}_{3}{W}_{r})\le M\sum _{i=1}^{s}{k}_{i}\beta ({W}_{r}).
(3.9)
Combining (3.6), (3.8) and (3.9), we get that
\beta (G{W}_{r})\le \beta ({G}_{1}{W}_{r})+\beta ({G}_{2}{W}_{r})+\beta ({G}_{3}{W}_{r})\le M(Lb+\sum _{i=1}^{s}{k}_{i})\beta ({W}_{r}).
From the condition (3.7), the map G is βcondensing in {W}_{r}. So, G has a fixed point in {W}_{r} due to DarboSadovskii’s fixed point theorem, which is just a mild solution of the nonlocal impulsive problem (1.1). This completes the proof of Theorem 3.5. □
Theorem 3.6 Assume that the hypotheses (HA), (Hf), (Hg2), (HI1) are satisfied, then the nonlocal impulsive problem (1.1) has at least one mild solution on[0,b]provided that
and (3.3) are satisfied.
Proof From the proof of Theorem 3.1, we have that the solution operator G is continuous and maps {W}_{r} into itself. In the following, we shall show that G is βcondensing in {W}_{r}.
By the Lipschitz continuity of g, we have that for u,v\in {W}_{r},
{\parallel {G}_{1}u{G}_{1}v\parallel}_{PC}=\underset{t\in [0,b]}{sup}\parallel T(t)[g(u)g(v)]\parallel \le Mk{\parallel uv\parallel}_{PC},
which implies that
\beta ({G}_{1}{W}_{r})\le Mk\beta ({W}_{r}).
(3.11)
Similar to the discussion in Theorem 3.1, from the compactness of {I}_{i} and the strong continuity T(\cdot ), we get that {G}_{3}{W}_{r} is equicontinuous on each \overline{{J}_{i}}=[{t}_{i},{t}_{i+1}], i=0,1,\dots ,s. Then by Lemma 2.7, we have that
\beta ({G}_{3}{W}_{r})=\underset{t\in [0,b]}{sup}\beta (({G}_{3}{W}_{r})(t))\le \sum _{i=1}^{s}\beta (T(t{t}_{i}){I}_{i}({W}_{r}({t}_{i})))=0.
(3.12)
Combining (3.6), (3.11) and (3.12), we have that
\beta (G{W}_{r})\le \beta ({G}_{1}{W}_{r})+\beta ({G}_{2}{W}_{r})+\beta ({G}_{3}{W}_{r})\le M(k+Lb)\beta ({W}_{r}).
From condition (3.10), the map G is βcondensing in {W}_{r}. So, G has a fixed point in {W}_{r} due to DarboSadovskii’s fixed point theorem, which is just a mild solution of the nonlocal impulsive problem (1.1). This completes the proof of Theorem 3.6. □
Remark 3.7 With the assumption of compactness on the associated semigroup, the existence of mild solutions to functional differential equations has been discussed in [6, 23–25]. By using the method of the measure of noncompactness, we deal with the four cases of impulsive differential equations in a unified way and get the existence results when the semigroup in not compact.