**Theorem 3.1** *Let* *X* *be an ordered Banach space*, *whose positive cone* *P* *is normal with normal constant* *N*. *Assume that*$T(t)$ ($t\ge 0$) *is positive*, *the Cauchy problem* (1.1) *has a lower solution*${v}_{0}\in C(I,X)$*and an upper solution*${w}_{0}\in C(I,X)$*with*${v}_{0}\le {w}_{0}$, *and the following conditions are satisfied*:

(H

_{1})

*There exists a constant*$C\ge 0$*such that*$f(t,{x}_{2})-f(t,{x}_{1})\ge -C({x}_{2}-{x}_{1}),$

(3.1)

*for any*$t\in I$, *and*${v}_{0}(t)\le {x}_{1}\le {x}_{2}\le {w}_{0}(t)$. *That is*, $f(t,x)+Cx$*is increasing in* *x* *for*$x\in [{v}_{0}(t),{w}_{0}(t)]$.

(H_{2}) $g(u)$*is decreasing in* *u* *for*$u\in [{v}_{0},{w}_{0}]$.

(H_{3}) ${I}_{k}(x)$*is increasing in* *x* *for*$x\in [{v}_{0}(t),{w}_{0}(t)]$$(t\in I)$.

(H

_{4})

*There exists a constant*$L\ge 0$*such that*$\mu \left(\{f(t,{x}_{n})\}\right)\le L\mu (\{{x}_{n}\}),$

(3.2)

*for any*$t\in I$, *and increasing or decreasing monotonic sequence*$\{{x}_{n}\}\subset [{v}_{0}(t),{w}_{0}(t)]$.

(H_{5}) $\{g({u}_{n})\}$*is precompact in* *X*, *for any increasing or decreasing monotonic sequence*$\{{u}_{n}\}\subset [{v}_{0},{w}_{0}]$. *That is*, $\mu (\{g({u}_{n})\})=0$.

*Then the Cauchy problem* (1.1) *has the minimal and maximal mild solutions between*${v}_{0}$*and*${w}_{0}$, *which can be obtained by a monotone iterative procedure starting from*${v}_{0}$*and*${w}_{0}$, *respectively*.

*Proof* It is easy to see that

$-(A+CI)$ generates an positive analytic semigroup

$S(t)={e}^{-Ct}T(t)$. Let

$\mathrm{\Phi}(t)={\int}_{0}^{\mathrm{\infty}}{\zeta}_{\alpha}(\theta )S({t}^{\alpha}\theta )\phantom{\rule{0.2em}{0ex}}d\theta $,

$\mathrm{\Psi}(t)=\alpha {\int}_{0}^{\mathrm{\infty}}\theta {\zeta}_{\alpha}(\theta )S({t}^{\alpha}\theta )\phantom{\rule{0.2em}{0ex}}d\theta $. By Remark 2.14,

$\mathrm{\Phi}(t)$ (

$t\ge 0$) and

$\mathrm{\Psi}(t)$ (

$t\ge 0$) are positive. By (2.5) and Remark 2.8, we have that

$\parallel \mathrm{\Phi}(t)\parallel \le M,\phantom{\rule{2em}{0ex}}\parallel \mathrm{\Psi}(t)\parallel \le \frac{\alpha}{\mathrm{\Gamma}(\alpha +1)}M\triangleq {M}_{1},\phantom{\rule{1em}{0ex}}t\ge 0.$

(3.3)

Let

$D=[{v}_{0},{w}_{0}]$,

${J}_{1}^{\prime}=[{t}_{0},{t}_{1}]=[0,{t}_{1}]$,

${J}_{k}^{\prime}=({t}_{k-1},{t}_{k}]$,

$k=2,3,\dots ,m+1$. We define a mapping

$Q:D\to PC(I,X)$ by

$Qu(t)=\{\begin{array}{cc}\mathrm{\Phi}(t)[{x}_{0}-g(u)]+{\int}_{0}^{t}{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)[f(s,u(s))+Cu(s)]\phantom{\rule{0.2em}{0ex}}ds,\hfill & t\in {J}_{1}^{\prime},\hfill \\ \mathrm{\Phi}(t)[u({t}_{1})+{I}_{1}(u({t}_{1}))]\hfill \\ \phantom{\rule{1em}{0ex}}+{\int}_{{t}_{1}}^{t}{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)[f(s,u(s))+Cu(s)]\phantom{\rule{0.2em}{0ex}}ds,\hfill & t\in {J}_{2}^{\prime},\hfill \\ \vdots \hfill \\ \mathrm{\Phi}(t)[u({t}_{m})+{I}_{m}(u({t}_{m}))]\hfill \\ \phantom{\rule{1em}{0ex}}+{\int}_{{t}_{m}}^{t}{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)[f(s,u(s))+Cu(s)]\phantom{\rule{0.2em}{0ex}}ds,\hfill & t\in {J}_{m+1}^{\prime}.\hfill \end{array}$

(3.4)

Clearly,

$Q:D\to PC(I,X)$ is continuous. By Lemma 2.10,

$u\in D$ is a mild solution of problem (1.1) if and only if

For

${u}_{1},{u}_{2}\in D$ and

${u}_{1}\le {u}_{2}$, from the positivity of operators

$\mathrm{\Phi}(t)$ and

$\mathrm{\Psi}(t)$, (H

_{1}), (H

_{2}), and (H

_{3}), we have inequality

$Q{u}_{1}\le Q{u}_{2}.$

(3.6)

Now, we show that

${v}_{0}\le Q{v}_{0}$,

$Q{w}_{0}\le {w}_{0}$. Let

${D}^{\alpha}{v}_{0}(t)+A{v}_{0}(t)+C{v}_{0}(t)\triangleq \sigma (t)$. By Definition 2.5, Lemma 2.10, the positivity of operators

$\mathrm{\Phi}(t)$ and

$\mathrm{\Psi}(t)$, for

$t\in {J}_{1}^{\prime}$, we have that

$\begin{array}{rcl}{v}_{0}(t)& =& \mathrm{\Phi}(t){v}_{0}(0)+{\int}_{0}^{t}{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)\sigma (s)\phantom{\rule{0.2em}{0ex}}ds\\ \le & \mathrm{\Phi}(t)[{x}_{0}-g({v}_{0})]+{\int}_{0}^{t}{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)[f(s,{v}_{0}(s))+C{v}_{0}(s)]\phantom{\rule{0.2em}{0ex}}ds.\end{array}$

(3.7)

For

$t\in {J}_{2}^{\prime}$, we have that

$\begin{array}{rcl}{v}_{0}(t)& =& \mathrm{\Phi}(t)[{v}_{0}({t}_{1})+\mathrm{\Delta}{v}_{0}{|}_{t={t}_{1}}]+{\int}_{{t}_{1}}^{t}{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)\sigma (s)\phantom{\rule{0.2em}{0ex}}ds\\ \le & \mathrm{\Phi}(t)[{v}_{0}({t}_{1})+{I}_{1}({v}_{0}({t}_{1}))]+{\int}_{{t}_{1}}^{t}{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)\\ \times [f(s,{v}_{0}(s))+C{v}_{0}(s)]\phantom{\rule{0.2em}{0ex}}ds.\end{array}$

(3.8)

Continuing such a process interval by interval to

${J}_{m+1}^{\prime}$, by (3.4), we obtain that

${v}_{0}\le Q{v}_{0}$. Similarly, we can show that

$Q{w}_{0}\le {w}_{0}$. For

$u\in D$, in view of (3.6), then

${v}_{0}\le Q{v}_{0}\le Qu\le Q{w}_{0}\le {w}_{0}$. Thus,

$Q:D\to D$ is a continuous increasing monotonic operator. We can now define the sequences

${v}_{n}=Q{v}_{n-1},\phantom{\rule{2em}{0ex}}{w}_{n}=Q{w}_{n-1},\phantom{\rule{1em}{0ex}}n=1,2,\dots ,$

(3.9)

and it follows from (3.6) that

${v}_{0}\le {v}_{1}\le \cdots {v}_{n}\le \cdots \le {w}_{n}\le \cdots \le {w}_{1}\le {w}_{0}.$

(3.10)

Let

$B=\{{v}_{n}\}$ and

${B}_{0}=\{{v}_{n-1}\}$,

$n=1,2,\dots $ . By (3.10) and the normality of the positive cone

*P*, then

*B* and

${B}_{0}$ are bounded. It follows from

${B}_{0}=B\cup \{{v}_{0}\}$ that

$\mu (B(t))=\mu ({B}_{0}(t))$ for

$t\in I$. Let

$\phi (t)=\mu (B(t))=\mu ({B}_{0}(t)),\phantom{\rule{1em}{0ex}}t\in I.$

(3.11)

From (H

_{4}), (H

_{5}), (3.3), (3.4), (3.9), (3.11), Lemma 2.16 and the positivity of operator

$\mathrm{\Psi}(t)$, for

$t\in {J}_{1}^{\prime}$, we have that

$\begin{array}{rcl}\phi (t)& =& \mu (B(t))=\mu (Q{B}_{0}(t))\\ =& \mu \left(\right\{\mathrm{\Phi}(t)[{x}_{0}-g({v}_{n-1})]\\ +{\int}_{0}^{t}{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)[f(s,{v}_{n-1}(s))+C{v}_{n-1}(s)]\phantom{\rule{0.2em}{0ex}}ds|n=1,2,\dots \})\\ \le & M\mu \left(\{g({v}_{n-1})\}\right)\\ +\mu \left(\{{\int}_{0}^{t}{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)[f(s,{v}_{n-1}(s))+C{v}_{n-1}(s)]\phantom{\rule{0.2em}{0ex}}ds|n=1,2,\dots \}\right)\\ \le & 2{\int}_{0}^{t}\mu \left(\{{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)[f(s,{v}_{n-1}(s))+C{v}_{n-1}(s)]|n=1,2,\dots \}\right)\phantom{\rule{0.2em}{0ex}}ds\\ \le & 2{M}_{1}{\int}_{0}^{t}{(t-s)}^{\alpha -1}(L+C)\mu ({B}_{0}(s))\phantom{\rule{0.2em}{0ex}}ds\\ =& 2{M}_{1}(L+C){\int}_{0}^{t}{(t-s)}^{\alpha -1}\phi (s)\phantom{\rule{0.2em}{0ex}}ds.\end{array}$

(3.12)

By (3.12) and Lemma 2.17, we obtain that

$\phi (t)\equiv 0$ on

${J}_{1}^{\prime}$. In particular,

$\mu (B({t}_{1}))=\mu ({B}_{0}({t}_{1}))=\phi ({t}_{1})=0$. This means that

$B({t}_{1})$ and

${B}_{0}({t}_{1})$ are precompact in

*X*. Thus,

${I}_{1}({B}_{0}({t}_{1}))$ is precompact in

*X* and

$\mu ({I}_{1}({B}_{0}({t}_{1})))=0$. For

$t\in {J}_{2}^{\prime}$, using the same argument as above for

$t\in {J}_{1}^{\prime}$, we have that

$\begin{array}{rcl}\phi (t)& =& \mu (B(t))=\mu (Q{B}_{0}(t))\\ =& \mu \left(\right\{\mathrm{\Phi}(t)[{v}_{n-1}({t}_{1})+{I}_{1}({v}_{n-1}({t}_{1}))]\\ +{\int}_{{t}_{1}}^{t}{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)[f(s,{v}_{n-1}(s))+C{v}_{n-1}(s)]\phantom{\rule{0.2em}{0ex}}ds|n=1,2,\dots \})\\ \le & M[\mu ({B}_{0}({t}_{1}))+\mu \left({I}_{1}({B}_{0}({t}_{1}))\right)]+2{M}_{1}(L+C){\int}_{{t}_{1}}^{t}{(t-s)}^{\alpha -1}\phi (s)\phantom{\rule{0.2em}{0ex}}ds\\ =& 2{M}_{1}(L+C){\int}_{{t}_{1}}^{t}{(t-s)}^{\alpha -1}\phi (s)\phantom{\rule{0.2em}{0ex}}ds.\end{array}$

(3.13)

By (3.13) and Lemma 2.17,

$\phi (t)\equiv 0$ on

${J}_{2}^{\prime}$. Then,

$\mu ({B}_{0}({t}_{2}))=\mu ({I}_{1}({B}_{0}({t}_{2})))=0$. Continuing such a process interval by interval to

${J}_{m+1}^{\prime}$, we can prove that

$\phi (t)\equiv 0$ on every

${J}_{k}^{\prime}$,

$k=1,2,\dots ,m+1$. This means

$\{{v}_{n}(t)\}$ (

$n=1,2,\dots $) is precompact in

*X* for every

$t\in I$. So,

$\{{v}_{n}(t)\}$ has a convergent subsequence in

*X*. In view of (3.10), we can easily prove that

$\{{v}_{n}(t)\}$ itself is convergent in

*X*. That is, there exist

$\underline{u}(t)\in X$ such that

${v}_{n}(t)\to \underline{u}(t)$ as

$n\to \mathrm{\infty}$ for every

$t\in I$. By (3.4) and (3.9), we have that

${v}_{n}(t)=\{\begin{array}{c}\mathrm{\Phi}(t)[{x}_{0}-g({v}_{n-1})]\hfill \\ \phantom{\rule{1em}{0ex}}+{\int}_{0}^{t}{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)[f(s,{v}_{n-1}(s))+C{v}_{n-1}(s)]\phantom{\rule{0.2em}{0ex}}ds,\hfill & t\in {J}_{1}^{\prime},\hfill \\ \mathrm{\Phi}(t)[{v}_{n-1}({t}_{1})+{I}_{1}({v}_{n-1}({t}_{1}))]\hfill \\ \phantom{\rule{1em}{0ex}}+{\int}_{{t}_{1}}^{t}{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)[f(s,{v}_{n-1}(s))+C{v}_{n-1}(s)]\phantom{\rule{0.2em}{0ex}}ds,\hfill & t\in {J}_{2}^{\prime},\hfill \\ \vdots \hfill \\ \mathrm{\Phi}(t)[{v}_{n-1}({t}_{m})+{I}_{m}({v}_{n-1}({t}_{m}))]\hfill \\ \phantom{\rule{1em}{0ex}}+{\int}_{{t}_{m}}^{t}{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)[f(s,{v}_{n-1}(s))+C{v}_{n-1}(s)]\phantom{\rule{0.2em}{0ex}}ds,\hfill & t\in {J}_{m+1}^{\prime}.\hfill \end{array}$

(3.14)

Let

$n\to \mathrm{\infty}$, then by Lebesgue-dominated convergence theorem, we have that

$\underline{u}(t)=\{\begin{array}{cc}\mathrm{\Phi}(t)[{x}_{0}-g(\underline{u})]+{\int}_{0}^{t}{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)[f(s,\underline{u}(s))+C\underline{u}(s)]\phantom{\rule{0.2em}{0ex}}ds,\hfill & t\in {J}_{1}^{\prime},\hfill \\ \mathrm{\Phi}(t)[\underline{u}({t}_{1})+{I}_{1}(\underline{u}({t}_{1}))]\hfill \\ \phantom{\rule{1em}{0ex}}+{\int}_{{t}_{1}}^{t}{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)[f(s,\underline{u}(s))+C\underline{u}(s)]\phantom{\rule{0.2em}{0ex}}ds,\hfill & t\in {J}_{2}^{\prime},\hfill \\ \vdots \hfill \\ \mathrm{\Phi}(t)[\underline{u}({t}_{m})+{I}_{m}(\underline{u}({t}_{m}))]\hfill \\ \phantom{\rule{1em}{0ex}}+{\int}_{{t}_{m}}^{t}{(t-s)}^{\alpha -1}\mathrm{\Psi}(t-s)[f(s,\underline{u}(s))+C\underline{u}(s)]\phantom{\rule{0.2em}{0ex}}ds,\hfill & t\in {J}_{m+1}^{\prime}.\hfill \end{array}$

(3.15)

Then $\underline{u}\in C(I,X)$ and $\underline{u}=Q\underline{u}$. Similarly, we can prove that there exists $\overline{u}\in C(I,X)$ such that $\overline{u}=Q\overline{u}$. By (3.6), if $u\in D$, and *u* is a fixed point of *Q*, then ${v}_{1}=Q{v}_{0}\le Qu=u\le Q{w}_{0}={w}_{1}$. By induction, ${v}_{n}\le u\le {w}_{n}$. By (3.10) and taking the limit as $n\to \mathrm{\infty}$, we conclude that ${v}_{0}\le \underline{u}\le u\le \overline{u}\le {w}_{0}$. That means that $\underline{u}$, $\overline{u}$ are the minimal and maximal fixed points of *Q* on $[{v}_{0},{w}_{0}]$, respectively. By (3.5), they are the minimal and maximal mild solutions of the Cauchy problem (1.1) on $[{v}_{0},{w}_{0}]$, respectively. □

**Corollary 3.2** *Let* *X* *be an ordered Banach space*, *whose positive cone* *P* *is regular*. *Assume that*$T(t)$ ($t\ge 0$) *is positive*, *the Cauchy problem* (1.1) *has a lower solution*${v}_{0}\in C(I,X)$*and an upper solution*${w}_{0}\in C(I,X)$*with*${v}_{0}\le {w}_{0}$, (*H*_{1}), (*H*_{2}), *and* (*H*_{3}) *hold*. *Then the Cauchy problem* (1.1) *has the minimal and maximal mild solutions between*${v}_{0}$*and*${w}_{0}$, *which can be obtained by a monotone iterative procedure starting from*${v}_{0}$*and*${w}_{0}$, *respectively*.

*Proof* Since

*P* is regular, any ordered-monotonic and ordered-bounded sequence in

*X* is convergent. For

$t\in I$, let

$\{{x}_{n}\}$ be an increasing or decreasing sequence in

$[{v}_{0}(t),{w}_{0}(t)]$. By (H

_{1}),

$\{f(t,{x}_{n})+C{x}_{n}\}$ is an ordered-monotonic and ordered-bounded sequence in

*X*. Then,

$\mu (\{f(t,{x}_{n})+C{x}_{n}\})=\mu (\{{x}_{n}\})=0$. By the properties of the measure of noncompactness, we have

$\mu \left(\{f(t,{x}_{n})\}\right)\le \mu \left(\{f(t,{x}_{n})+C{x}_{n}\}\right)+C\mu (\{{x}_{n}\})=0.$

(3.16)

So, (H_{4}) holds. Let $\{{u}_{n}\}$ be an increasing or decreasing sequence in $[{v}_{0},{w}_{0}]$. By (H_{2}), $\{g({u}_{n})\}$ is an ordered-monotonic and ordered-bounded sequence in *X*. Then $\{g({u}_{n})\}$ is precompact in *X*. Thus, (H_{5}) holds. By Theorem 3.1, the proof is then complete. □

**Corollary 3.3** *Let* *X* *be an ordered and weakly sequentially complete Banach space*, *whose positive cone* *P* *is normal with normal constant* *N*. *Assume that*$T(t)$ ($t\ge 0$) *is positive*, *the Cauchy problem* (1.1) *has a lower solution*${v}_{0}\in C(I,X)$*and an upper solution*${w}_{0}\in C(I,X)$*with*${v}_{0}\le {w}_{0}$, (*H*_{1}), (*H*_{2}), *and* (*H*_{3}) *hold*. *Then the Cauchy problem* (1.1) *has the minimal and maximal mild solutions between*${v}_{0}$*and*${w}_{0}$, *which can be obtained by a monotone iterative procedure starting from*${v}_{0}$*and*${w}_{0}$, *respectively*.

*Proof* In an ordered and weakly sequentially complete Banach space, the normal cone *P* is regular. Then the proof is complete. □

**Corollary 3.4** *Let* *X* *be an ordered and reflective Banach space*, *whose positive cone* *P* *is normal with normal constant* *N*. *Assume that*$T(t)$ ($t\ge 0$) *is positive*, *the Cauchy problem* (1.1) *has a lower solution*${v}_{0}\in C(I,X)$*and an upper solution*${w}_{0}\in C(I,X)$*with*${v}_{0}\le {w}_{0}$, (*H*_{1}), (*H*_{2}), *and* (*H*_{3}) *hold*. *Then the Cauchy problem* (1.1) *has the minimal and maximal mild solutions between*${v}_{0}$*and*${w}_{0}$, *which can be obtained by a monotone iterative procedure starting from*${v}_{0}$*and*${w}_{0}$, *respectively*.

*Proof* In an ordered and reflective Banach space, the normal cone *P* is regular. Then the proof is complete. □