Existence of Global Mild Solutions for Nonautonomous Abstract Evolution Equations

Abstract

In this paper, we investigate the Cauchy problem for nonautonomous abstract evolution equations of the form $y^{\prime }\left(t\right)=A\left(t\right)y\left(t\right)+f(t,y\left(t\right)),t\ge 0$, $y\left(0\right)=y_0$. We obtain new existence theorems for global mild solutions under both compact and noncompact evolution families $U(t,s)$. Our key method relies on the generalized Ascoli–Arzela theorem we previously obtained. Finally, an example is provided to illustrate the applicability of our results.

1. Introduction

Nonlinear evolution equations are an important area of research in nonlinear analysis and differential equations, with broad applicability across various fields. In a broad sense, partial differential equations containing the time variable t are collectively referred to as evolution equations. These equations are widely used in physics, mechanics, and other natural sciences to characterize the state or behavior of natural phenomena over time.In a narrower sense, the term refers to equations with practical application backgrounds, such as the wave equation, heat conduction equation, Schrödinger equation, fluid dynamics equation, KdV equation, reaction–diffusion equation, and other systems of coupled equations. Many of these equations can be transformed into abstract ordinary differential equations in Banach spaces using semigroup methods, all of which belong to the category of evolution equations. Thus, evolution equations have significant theoretical and applied value.

Generally speaking, it is difficult to obtain exact explicit solutions for nonlinear evolution equations, but we can indirectly study the properties of the solutions. In this way, although we cannot obtain exact solutions, we can characterize the behavior of the solutions through these properties.The search for the properties of solutions is a core problem in the study of nonlinear evolution equations, including the global existence, asymptotic behavior, and finite-time blow-up of solutions to initial or initial-boundary value problems.

In this paper, we consider the Cauchy problem for a nonautonomous abstract evolution equation on a semi-infinite interval

$$\left\{\begin{array}{c}{y}^{\prime }\left(t\right)=A\left(t\right)y\left(t\right)+f(t,y\left(t\right)),t\in [0,\infty ),\hfill \\ y\left(0\right)=y_0,\hfill \end{array}\right.$$

(1)

where $A\left(t\right):{\mathrm{\Omega }}_t\subset X\to X$ generates an evolution family $U(t,s)$, X is a Banach space with the norm $|·|$, and $f:[0,\infty )\times X\to X$ is a function that is defined later.

In recent years, many papers have studied Equation (1) on bounded intervals (see the monographs [1,2] and papers [3,4,5,6,7,8,9]). In [3], García-Falset studied the existence and asymptotic behavior of solutions for Cauchy problems with nonlocal initial datum generated by accretive operators in Banach spaces. Liu and Yuan [4] studied the existence of mild solutions for a class of semilinear evolution equations with non-local initial conditions in the case where $A\left(t\right)=A$ generates a compact $C_0$-semigroup. In [5], Ding, Long, and N’Guérékata established new theorems on the existence of almost automorphic solutions to nonautonomous evolution equations $u^{\prime }\left(t\right)=A\left(t\right)u\left(t\right)+f\left(t\right)$ and $u^{\prime }\left(t\right)=A\left(t\right)u\left(t\right)+f(t,u\left(t\right))$ in Banach spaces. Zhu and Li [6] established existence results for semilinear differential systems with nonlocal initial conditions in Banach spaces. They used fixed-point theorems combined with convex-power condensing operators. Ntouyas and Tsamotas [7] studied the global existence of solutions for semilinear evolution equations $x^{\prime }\left(t\right)=Ax\left(t\right)+f(t,x\left(t\right))$, $t\in [0,b]$ with nonlocal conditions, using a fixed-point analysis approach. However, only a few papers have addressed Equation (1) on unbounded intervals. In [8,9], Olszowy and Wȩdrychowicz provided existence theorems for mild solutions of semilinear evolution equations on both bounded and unbounded intervals. Their analysis is based on the concept of a measure of noncompactness in Fréchet spaces and the classical Tichonov fixed-point principle. Recently, Xu, Colao, and Muglia [10] studied the existence of mild solutions for a nonlocal semilinear evolution equation on an unbounded interval by means of an approximation solvability method, without assuming compactness on the evolution system or the nonlinearity. The method is based on a reduction to a finite-dimensional problem via projections, which is then solved using a fixed-point approach that includes a compactness criterion on $BC\left(\right[0,+\infty ),X)$.

On the other hand, it is well known that the classical Ascoli–Arzelà theorem is a powerful tool providing necessary and sufficient conditions for investigating the relative compactness of a family of abstract continuous functions; however, it is limited to finite compact intervals. In [11], the authors generalized the Ascoli–Arzelà theorem to infinite intervals and applied it to investigate an initial value problem for fractional evolution equations on an infinite interval.

In this paper, our aim is to present new results concerning the existence of global mild solutions for the problem (1) for both compact and noncompact evolution families $U(t,s)$. It is worth mentioning that when considering Equation (1) on an infinite interval, the method proposed in this paper seems to have more generalizability. We effectively apply the generalized Ascoli–Arzela theorem obtained in [11], along with fixed-point theorems and the measure of noncompactness, to the study of the existence of global mild solutions in this paper.

2. Preliminaries

We first introduce some notations and facts about the generalized of the Ascoli–Arzela theorem and measure of noncompactness. For more information, we refer to [11,12].

Let X be a Banach space with the norm $|·|$. Suppose that $J:=[0,\infty )={\mathbb{R}}^{+}$. Denote $C(J,X)$ as the Banach space of all continuous functions from J to X with the norm $\parallel u\parallel ={sup}_{t\in J}\left|u\left(t\right)\right|<\infty$. Using $\mathcal{B}\left(X\right)$, we denote the space of all bounded linear operators from X to X, with the norm ${\parallel ·\parallel }_{\mathcal{B}\left(X\right)}$.

Assume the function $h\in C\left(\right[0,\infty ),(0,\infty \left)\right)$. Let

$$\begin{array}{cc}\hfill C_h([0,\infty ),X)=& \left\{y\in C([0,\infty ),X):\underset{t\to \infty }{lim}h\left(t\right)\left|y\left(t\right)\right|=0\right\},\hfill \end{array}$$

with the norm ${\parallel y\parallel }_h={sup}_{t\in [0,\infty )}h\left(t\right)\left|y\left(t\right)\right|$. Then, $(C_h([0,\infty ),X),\parallel ·{\parallel }_h)$ is a Banach space.

The following generalized Ascoli–Arzela theorem was obtained in [11].

Lemma 1

([11]). The set $\mathrm{\Omega }\subset C_h([0,\infty ),X)$ is relatively compact if and only if the following conditions hold:

(i) 

for any $b>0$, the set $V=\{v:v(t)=h(t\left)y\right(t),y\in \mathrm{\Omega }\}$ is equicontinuous on $[0,b]$;

(ii) 

${lim}_{t\to \infty }h\left(t\right)\left|y\left(t\right)\right|=0$ uniformly for $y\in \mathrm{\Omega }$;

(iii) 

for any $t\in [0,\infty )$, $V\left(t\right)=\left\{v\right(t):v(t)=h(t\left)y\right(t),y\in \mathrm{\Omega }\}$ is relatively compact in X.

Remark 1.

Lemma 1 was proven in [13] when $h\left(t\right):=e^{-t}$ and in [14] when $h\left(t\right):=1/(1+t)$, $t\in (0,\infty )$.

Assume that D is a nonempty subset of X. Kuratowski’s measure of noncompactness $\chi$ is said to be

$$\chi \left(D\right)=inf\left\{d>0:D\subset \bigcup _{i=1}^n{M}_i\mathrm{and}\mathrm{diam}\left(M_i\right)\le d\right\},$$

where the diameter of $M_i$ is given by diam $\left(M_i\right)=sup\left\{\right|x-y|:x,y\in M_i\}$, $i=1,\dots ,n.$

Proposition 1.

Let $D_1$ and $D_2$ be two bounded sets of a Banach space X. Then, the following properties hold:

(i) 

$\chi \left(D_1\right)=0$ if and only if $D_1$ is relatively compact in X;

(ii) 

$\chi \left(D_1\right)\le \chi \left(D_2\right)$ if $D_1\subseteq D_2$;

(iii) 

$\chi \left\{\right\{x\}\cup D\}=\chi \left(D\right)$ for every $x\in X$ and every nonempty subset $B\subset X$;

(iv) 

$\chi \{D_1+D_2\}\le \chi \left(D_1\right)+\chi \left(D_2\right)$, where $D_1+D_2=\{x+y:x\in D_1,y\in D_2\}$;

(v) 

$\chi \{D_1\cup D_2\}\le max\{\chi \left(D_1\right),\chi \left(D_2\right)\}$;

(vi) 

$\chi \left(\kappa D\right)\le \left|\kappa \right|\chi \left(D\right)$ for any $\kappa \in \mathbb{R}$.

Lemma 2

([15]). Let ${\left\{u_n\left(t\right)\right\}}_{n=1}^{\infty }:[0,\infty )\to X$ be a continuous function family. If there exists $\xi \in L^1[0,\infty )$ such that

$$|u_n\left(t\right)|\le \xi \left(t\right),t\in [0,\infty ),n=1,2,\dots ,$$

then $\chi \left({\left\{u_n\left(t\right)\right\}}_{n=1}^{\infty }\right)$ is integrable on $[0,\infty )$, and

$$\chi \left(\left\{{\int }_0^t{u}_n\left(s\right)ds:n=1,2,\dots \right\}\right)\le 2{\int }_0^t\chi \left(\{u_n\left(s\right):n=1,2,\dots \}\right)ds.$$

Lemma 3

(Mazur theorem). If $\mathcal{K}$ is a compact subset of a Banach space X, then its convex closure $\overline{\mathrm{conv}\left(\mathcal{K}\right)}$ is compact.

3. Some Lemmas

Let $\mathrm{\Delta }:=\left\{\right(t,s)\in J\times J:0\le s\le t<+\infty \}$. We assume that $A\left(t\right)$ is a linear operator acting from ${\mathrm{\Omega }}_t\subset X$ into X for each $t\ge 0$, and that $A\left(t\right)$ generates a uniformly continuous evolution family $U(t,s)$, $(t,s)\in \mathrm{\Delta }$, satisfying the following properties:

(i)

$U(t,t)=I$ for each $t\ge 0$, where I is the identity operator in X;

(ii)

$U(t,\tau )U(\tau ,s)=U(t,s)$ for $0\le s\le \tau \le t<\infty$;

(iii)

$U(t,s)\in \mathcal{B}\left(X\right)$ for every $(t,s)\in \mathrm{\Delta }$ and for each $y\in X$, the mapping $(t,s)\to U(t,s)y$ is continuous.

An evolution operator U is said to be compact when $U(t,s)$ is a compact operator for all $t-s>0$, i.e., $U(t,s)$ changes bounded sets into relatively compact sets.

Throughout of this paper, we utilize the following assumptions:

(C1)

There exists a constant $M\ge 1$, such that

$${\parallel U(t,s)\parallel }_{\mathcal{B}\left(X\right)}\le M,\mathrm{for}\mathrm{any}(t,s)\in \mathrm{\Delta }.$$

(C2)

$f:[0,\infty )\times X\to X$ is a mapping, such that $f(·,x)$ is measurable for $x\in X$ and $f(t,·)$ is continuous for a.e. $t\in [0,\infty )$.

(C3)

there exists a function $g\in L^1({\mathbb{R}}^{+},{\mathbb{R}}^{+})$, such that ${\int }_0^{t}g\left(s\right)ds$ is continuous on $[0,\infty )$ and

$$\left|f\right(t,x\left)\right|\le g\left(t\right),\mathrm{for}\mathrm{a}.\mathrm{e}.t\in [0,\infty )\mathrm{and}\mathrm{any}x\in X.$$

(C4)

there exists a function $h\in C\left(\right[0,\infty ),(0,\infty \left)\right)$, such that ${lim}_{t\to \infty }h\left(t\right)=0$ and

$$\underset{t\to \infty }{lim}h\left(t\right){\int }_0^{t}g\left(s\right)ds=0.$$

Definition 1.

If $y\in C\left(\right[0,\infty ],X)$ satisfies

$$y\left(t\right)=U(t,0)y_0+{\int }_0^{t}U(t,s)f(s,y\left(s\right))ds,\mathrm{for}t\in [0,\infty ),$$

then $y\left(t\right)$ is a mild solution for the Cauchy problem (1).

For any $y\in C_h([0,\infty ),X)$, define an operator $\mathcal{S}$ as follows:

$$\left(\mathcal{S}y\right)\left(t\right)=\left({\mathcal{S}}_{1}y\right)\left(t\right)+\left({\mathcal{S}}_{2}y\right)\left(t\right),\mathrm{for}t\in [0,\infty ),$$

where

$$\left({\mathcal{S}}_{1}y\right)\left(t\right)=U(t,0)y_0,\left({\mathcal{S}}_{2}y\right)\left(t\right)={\int }_0^{t}U(t,s)f(s,y\left(s\right))ds,\mathrm{for}t\in [0,\infty ).$$

Clearly, the fixed points of the operator $\mathcal{S}$ are mild solutions for the problem (1).

From (C4), there exists a constant $r>0$, such that

$$\underset{t\in [0,\infty )}{sup}\left\{M|y_0|h\left(t\right)+Mh\left(t\right){\int }_0^{t}g\left(s\right)ds\right\}\le r.$$

(2)

Let

$$\mathrm{\Omega }=\{y\in C_h([0,\infty ),X):\parallel y{\parallel }_h\le r\}.$$

(3)

Then, $\mathrm{\Omega }$ is a nonempty, convex, and closed subset of $C_h([0,\infty ),X)$.

Let

$$V:=\left\{v:v\left(t\right)=h\left(t\right)\left(\mathcal{S}y\right)\left(t\right),y\in \mathrm{\Omega }\right\}.$$

Before presenting our main results, we first prove the following lemmas.

Lemma 4.

Assume that (C1)–(C4) hold. Then, the set V is equicontinuous.

Proof. 

Step I. We first prove that $\left\{v:v\left(t\right)=h\left(t\right)\left({\mathcal{S}}_{1}y\right)\left(t\right),u\in \mathrm{\Omega }\right\}$ is equicontinuous.

For $0=t_1<t_2<\infty$, we obtain

$$\begin{array}{cc}\hfill |h\left(t_2\right)\left({\mathcal{S}}_{1}y\right)\left(t_2\right)-h_1\left(0\right)\left({\mathcal{S}}_{1}y\right)\left(0\right)|\le & |h\left(t_2\right)U(t_2,0)y_0-y_0|\to 0,\mathrm{as}{t}_2\to 0.\hfill \end{array}$$

(4)

For $0<t_1<t_2<\infty$, we have

$$\begin{array}{cc}\hfill & |h\left(t_2\right)\left({\mathcal{S}}_{1}y\right)\left(t_2\right)-h\left(t_1\right)\left({\mathcal{S}}_{1}y\right)\left(t_1\right)|\hfill \\ \hfill =& |h\left(t_2\right)U(t_2,0)y_0-h\left(t_1\right)U(t_1,0)y_0|\to 0,\mathrm{as}{t}_2\to t_1.\hfill \end{array}$$

(5)

Hence, $\left\{v:v\left(t\right)=h\left(t\right)\left({\mathcal{S}}_{1}y\right)\left(t\right),u\in \mathrm{\Omega }\right\}$ is equicontinuous.

Step II. We prove that $\left\{v:v\left(t\right)=h\left(t\right)\left({\mathcal{S}}_{2}y\right)\left(t\right),u\in \mathrm{\Omega }\right\}$ is equicontinuous.

By (C4), for $\epsilon >0$, there exists $T>0$, such that

$$Mh\left(t\right){\int }_0^{t}g\left(s\right)ds<{\displaystyle \frac{\epsilon }{2}},\mathrm{for}t>T.$$

(6)

For $t_1,t_2>T$, in view of (6), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & |h\left(t_2\right)\left({\mathcal{S}}_{2}y\right)\left(t_2\right)-h\left(t_1\right)\left({\mathcal{S}}_{2}y\right)\left(t_1\right)|\hfill \\ \hfill \le & \left|h\left(t_2\right){\int }_0^{t_2}U(t_2,s)f(s,y\left(s\right))ds\right|+\left|h\left(t_1\right){\int }_0^{t_1}U(t_1,s)f(s,y\left(s\right))ds\right|\hfill \\ \hfill \le & Mh\left(t_2\right){\int }_0^{t_2}g\left(s\right)ds+Mh\left(t_1\right){\int }_0^{t_1}g\left(s\right)ds<\epsilon .\hfill \end{array}\end{array}$$

(7)

For $t_1=0$, $0<t_2<T$, by (C4), we have

$$\begin{array}{cc}\hfill |h\left(t_2\right)\left({\mathcal{S}}_{2}y\right)\left(t_2\right)-h\left(0\right)\left({\mathcal{S}}_{2}y\right)\left(0\right)|=& \left|h\left(t_2\right){\int }_0^{t_2}U(t_2,s)f(s,y\left(s\right))ds\right|\hfill \\ \hfill \le & Mh\left(t_2\right){\int }_0^{t_2}g\left(s\right)ds\to 0,\mathrm{as}{t}_2\to 0.\hfill \end{array}$$

(8)

For $0<t_1<t_2\le T$, we obtain

$$\begin{array}{cc}\hfill & |h\left(t_2\right)\left({\mathcal{S}}_{2}y\right)\left(t_2\right)-h\left(t_1\right)\left({\mathcal{S}}_{2}y\right)\left(t_1\right)|\hfill \\ \hfill \le & \left|h\left(t_1\right){\int }_{t_1}^{t_2}U(t_2,s)f(s,y\left(s\right))ds\right|\hfill \\ \hfill & +\left|h\left(t_1\right){\int }_0^{t_1}\left(U(t_2,s)-U(t_1,s)\right)f(s,y\left(s\right))ds\right|\hfill \\ \hfill & +|h\left(t_2\right)-h\left(t_1\right)|\left|{\int }_0^{t_2}U(t_2,s)f(s,y\left(s\right))ds\right|\hfill \\ \hfill \le & \left|Mh\left(t_1\right){\int }_{t_1}^{t_2}g\left(s\right)ds\right|+\left|h\left(t_1\right)\underset{s\in [0,t_1]}{sup}\parallel U(t_2,s)-U(t_1,s)\parallel {\int }_0^{t_1}g\left(s\right)ds\right|\hfill \\ \hfill & +M|h\left(t_2\right)-h\left(t_1\right)|\left|{\int }_0^{t_2}g\left(s\right)ds\right|\to 0,\mathrm{as}{t}_2\to t_1.\hfill \end{array}$$

(9)

For $0<t_1<T<t_2$, if $t_2\to t_1$, then $t_2\to T$ and $t_1\to T$. Thus, for $y\in \mathrm{\Omega }$

$$\begin{array}{cc}\hfill & |h\left(t_2\right)\left({\mathcal{S}}_{2}y\right)\left(t_2\right)-h\left(t_1\right)\left({\mathcal{S}}_{2}y\right)\left(t_1\right)|\hfill \\ \hfill \le & |h\left(t_2\right)\left({\mathcal{S}}_{2}y\right)\left(t_2\right)-h\left(T\right)\left({\mathcal{S}}_{2}y\right)\left(T\right)|\hfill \\ \hfill & +|h\left(T\right)\left({\mathcal{S}}_{2}y\right)\left(T\right)-h\left(t_1\right)\left({\mathcal{S}}_{2}y\right)\left(t_1\right)|\to 0,\mathrm{as}{t}_2\to t_1.\hfill \end{array}$$

(10)

Hence, for $0\le t_1<t_2<\infty$,

$$|h\left(t_2\right)\left({\mathcal{S}}_{2}y\right)\left(t_2\right)-h\left(t_1\right)\left({\mathcal{S}}_{2}y\right)\left(t_1\right)|\to 0,\mathrm{as}{t}_2\to t_1.$$

Therefore, $\left\{v:v\left(t\right)=h\left(t\right)\left({\mathcal{S}}_{2}y\right)\left(t\right),u\in \mathrm{\Omega }\right\}$ is equicontinuous. Furthermore, V is equicontinuous. □

Lemma 5.

Assume that (C1)–(C4) hold. Then, ${lim}_{t\to \infty }h\left(t\right)\left|\left(\mathcal{S}y\right)\left(t\right)\right|=0$ uniformly for $y\in \mathrm{\Omega }$.

Proof. 

For any $y\in \mathrm{\Omega }$,

$$\begin{array}{cc}\hfill h\left(t\right)\left|\right(\mathcal{S}y\left)\right(t\left)\right|\le & \left|U(t,0)y_0\right|+\left|{\int }_0^{t}U(t,s)f(s,y\left(s\right))ds\right|\hfill \\ \hfill \le & M|y_0|h\left(t\right)+Mh\left(t\right){\int }_0^{t}g\left(s\right)ds,\hfill \end{array}$$

(11)

which implies that ${lim}_{t\to \infty }h\left(t\right)\left|\left(\mathcal{S}y\right)\left(t\right)\right|=0$ uniformly for $y\in \mathrm{\Omega }$. This completes the proof. □

Lemma 6.

Assume that (C1)–(C4) hold. Then, $\mathcal{S}\mathrm{\Omega }\subset \mathrm{\Omega }$.

Proof. 

By Lemmas 4 and 5, one can find that $\mathcal{S}\mathrm{\Omega }\subset C_h([0,\infty ),X)$. For any $t>0$, $y\in \mathrm{\Omega }$, from (2), we obtain

$$\begin{array}{cc}\hfill h\left(t\right)\left|\right(\mathcal{S}y\left)\right(t\left)\right|\le & M|y_0|h\left(t\right)+Mh\left(t\right){\int }_0^{t}g\left(s\right)ds\le r.\hfill \end{array}$$

(12)

For $t=0$, $\left|\right(\mathcal{S}y\left)\right(0\left)\right|=0<r$. Therefore, $\mathcal{S}\mathrm{\Omega }\subset \mathrm{\Omega }$. □

Lemma 7.

Assume that (C1)–(C4) hold. Then, $\mathcal{S}$ is continuous.

Proof. 

Indeed, let ${\left\{y_n\right\}}_{n=1}^{\infty }$ be a sequence in $\mathrm{\Omega }$, which is convergent to $y\in \mathrm{\Omega }$. Consequently, ${lim}_{n\to \infty }{y}_n\left(t\right)=y\left(t\right),\mathrm{for}t\in (0,\infty )$. From (C2), we obtain ${lim}_{n\to \infty }f(t,y_n\left(t\right))=f(t,y\left(t\right))$. For any $\epsilon >0$, there exists $T>0$, such that (6) holds. Thus, for $t>T$,

$$|h\left(t\right)\left(\mathcal{S}{y}_n\right)\left(t\right)-h\left(t\right)\left(\mathcal{S}y\right)\left(t\right)|\le 2Mh\left(t\right){\int }_0^{t}g\left(s\right)ds<\epsilon .$$

(13)

For each $t\in [0,T]$, $|f(s,y_n\left(s\right))-f(s,y\left(s\right))|\le 2g\left(s\right)$. Using the Lebesgue dominated convergence theorem, we obtain

$${\int }_0^t|f(s,y_n\left(s\right))-f(s,y\left(s\right))|ds\to 0,\mathrm{as}n\to \infty .$$

Thus, for $t\in [0,T]$,

$$\begin{array}{cc}\hfill & |h\left(t\right)\left(\mathcal{S}{u}_n\right)\left(t\right)-h\left(t\right)\left(\mathcal{S}y\right)\left(t\right)|\hfill \\ \hfill \le & h\left(t\right){\int }_0^t\left|U(t-s)(f(s,y_n\left(s\right))-f(s,y\left(s\right)))\right|ds\hfill \\ \hfill \le & Mh\left(t\right){\int }_0^t|f(s,y_n\left(s\right))-f(s,y\left(s\right))|ds\to 0,\mathrm{as}n\to \infty .\hfill \end{array}$$

So, $\parallel \mathcal{S}{y}_n-\mathcal{S}y\parallel \to 0$ as $n\to \infty$. Hence, $\mathcal{S}$ is continuous. The proof is completed. □

4. Existence

Theorem 1.

Assume that $U(t,s)$ is compact for all $t-s>0$. Further, suppose that (C1)–(C4) hold. Then, the Cauchy problem (1) has at least one mild solution.

Proof. 

Clearly, the problem (1) admits a mild solution $y\in \mathrm{\Omega }$ if and only if the operator $\mathcal{S}$ has a fixed point $y\in \mathrm{\Omega }$. From Lemmas 6 and 7, we know that $\mathcal{S}\mathrm{\Omega }\subset \mathrm{\Omega }$ and $\mathcal{S}$ is continuous. In order to prove that $\mathcal{S}$ is a completely continuous operator, we need to prove that $\mathcal{S}\mathrm{\Omega }$ is a relatively compact set. From Lemmas 4 and 5, the set $V=\left\{v:v\left(t\right)=h\left(t\right)\left(\mathcal{S}y\right)\left(t\right),y\in \mathrm{\Omega }\right\}$ is equicontinuous and ${lim}_{t\to \infty }\left|h\left(t\right)\left(\mathcal{S}y\right)\left(t\right)\right|=0$ uniformly for $u\in \mathrm{\Omega }$. According to Lemma 1, we only need to prove that $V\left(t\right)=\left\{v\left(t\right):v\left(t\right)=h\left(t\right)\left(\mathcal{S}y\right)\left(t\right),u\in \mathrm{\Omega }\right\}$ is relatively compact in X.

Clearly, $V\left(0\right)$ is relatively compact in X. We only consider the case $t>0$. For $0<\epsilon <t$, we define the operator on $\mathrm{\Omega }$

$$\begin{array}{c}\hfill \left({\mathcal{S}}_{\epsilon }y\right)\left(t\right):=U(t,0)y_0+\left({\mathcal{S}}_2^{\epsilon }y\right)\left(t\right),\end{array}$$

where

$$\left({\mathcal{S}}_2^{\epsilon }y\right)\left(t\right)={\int }_0^{t-\epsilon }U(t,s)f(s,y\left(s\right))ds.$$

Since $U(t,s)$ is compact, it follows that $U(t,0)$ is compact and the set ${\mathcal{K}}_{\epsilon }:=\{U(t,s)f(s,y\left(s\right)):y\in \mathrm{\Omega },0\le s\le t-\epsilon \}$ is compact for all $\epsilon >0$. Then, $\overline{\mathrm{conv}\left({\mathcal{K}}_{\epsilon }\right)}$ is also a compact set by Lemma 3. By using the mean-value theorem for the Bochner integrals (see [16], Corollary 8, page 48), we obtain that

$$\left({\mathcal{S}}_2^{\epsilon }y\right)\left(t\right)\in t\overline{\mathrm{conv}\left({\mathcal{K}}_{\epsilon }\right)},\mathrm{for}\mathrm{any}t\ge 0.$$

Therefore, the set $\{\left({\mathcal{S}}_2^{\epsilon }y\right)\left(t\right),y\in \mathrm{\Omega }\}$ is relatively compact in X for all $\epsilon >0$. Now, observe that

$$\begin{array}{c}\hfill |h\left(t\right)\left(\mathcal{S}y\right)\left(t\right)-h\left(t\right)\left({\mathcal{S}}_{\epsilon }y\right)\left(t\right)|\le Mh\left(t\right){\int }_{t-\epsilon }^{t}g\left(s\right)ds\to 0,\epsilon \to 0.\end{array}$$

Thus, $V\left(t\right)$ is also a relatively compact set in X for $t\in [0,\infty )$. We include that $\mathcal{S}$ is a completely continuous operator. Hence, using the Schauder fixed-point theorem, we have that $\mathcal{S}$ has at least a fixed point $y^{*}\in \mathrm{\Omega }$. Therefore

$$y^{*}\left(t\right)=U(t,0)y_0+{\int }_0^{t}U(t,s)f(s,y^{*}\left(s\right))ds,t\in [0,\infty ),$$

which implies that $y^{*}$ is a mild solution for (1). The proof is completed. □

By choosing different $h\left(t\right)$, we can provide some practical corollaries.

Corollary 1.

Assume that $U(t,s)$ is compact for all $t-s>0$. Further, assume that (C1), (C2), and (C3) hold and

$$\underset{t\to \infty }{lim}{\displaystyle \frac{1}{1+t}}{\int }_0^{t}g\left(s\right)ds=0.$$

Then, the Cauchy problem (1) has at least one mild solution.

Proof. 

Choose $h\left(t\right)=1/(1+t)$, then condition (C4) is satisfied. □

Corollary 2.

Assume that $U(t,s)$ is compact for all $t-s>0$. Further, assume that (C1), (C2), and (C3) hold and

$$\underset{t\to \infty }{lim}{e}^{-t}{\int }_0^{t}g\left(s\right)ds=0.$$

Then, the Cauchy problem (1) has at least one mild solution.

Proof. 

Choose $h\left(t\right)=e^{-t}$, then condition (C4) is satisfied. □

In the case that $U(t,s)$ is noncompact for $t>0$, we impose the following assumption:

(C5)

there exist constants $K>0$, such that for any bounded $D\subseteq X$, $t\in [0,\infty )$,

$$\chi \left(f\right(t,D\left)\right)\le K\chi \left(D\right),\mathrm{for}\mathrm{a}.\mathrm{e}.t\in [0,\infty ),$$

where $\chi$ is the Kuratowski’s measure of noncompactness.

Theorem 2.

Assume that (C1)–(C5) hold. Then, the Cauchy problem (1) has at least one mild solution.

Proof. 

Let $y_0\left(t\right)=U(t,0)y_0$ for all $t\in [0,\infty )$ and $y_{n+1}=\mathcal{S}{y}_n$, $n=0,1,2,\cdots$. From Lemma 6, $\mathcal{S}{y}_n\in \mathrm{\Omega }$, for $y_n\in \mathrm{\Omega }$. Let $V={\{v_n:v_n\left(t\right)=h\left(t\right)\left(\mathcal{S}{y}_n\right)\left(t\right),y_n\in \mathrm{\Omega }\}}_{n=0}^{\infty }$. We prove that the set V is relatively compact.

In view of Lemmas 4 and 5, the set V is equicontinuous and ${lim}_{t\to \infty }h\left(t\right)\left|\left(F{y}_n\right)\left(t\right)\right|=0$ uniformly for $y_n\in \mathrm{\Omega }$. According to Lemma 1, we only need to prove that $V\left(t\right)=\{v_n\left(t\right):v_n\left(t\right)=h\left(t\right)\left(F{y}_n\right)\left(t\right),$$y_n\in \mathrm{\Omega }{\}}_{n=0}^{\infty }$ is relatively compact in X for $t\in [0,\infty )$.

Under the condition (C3), and using the properties of the measure of noncompactness along with Lemma 2, we have

$$\begin{array}{cc}\hfill & \chi \left({\left\{h\left(t\right)\left(F{y}_n\right)\left(t\right)\right\}}_{n=0}^{\infty }\right)\hfill \\ \hfill =& \chi \left({\left\{h\left(t\right)U(t,0)y_0+h\left(t\right){\int }_0^{t}U(t,s)f(s,y_n\left(s\right))ds\right\}}_{n=0}^{\infty }\right)\hfill \\ \hfill =& \chi \left({\left\{h\left(t\right){\int }_0^{t}U(t,s)f(s,y_n\left(s\right))ds\right\}}_{n=0}^{\infty }\right)\hfill \\ \hfill \le & 2Mh\left(t\right){\int }_0^t\chi \left(f(s,{\left\{y_n\left(s\right)\right\}}_{n=0}^{\infty })\right)ds\hfill \\ \hfill \le & 2MKh\left(t\right){\int }_0^t\chi \left({\left\{y_n\left(s\right)\right\}}_{n=0}^{\infty }\right)ds\hfill \\ \hfill \le & 2MKh\left(t\right){\int }_0^t{\displaystyle \frac{1}{h\left(s\right)}}\chi \left({\left\{h\left(s\right)y_n\left(s\right)\right\}}_{n=0}^{\infty }\right)ds.\hfill \end{array}$$

On the other hand, using the properties of measure of noncompactness, for any $t\in [0,\infty )$ we have

$$\begin{array}{cc}\hfill \chi \left({\left\{h\left(t\right)y_n\left(t\right)\right\}}_{n=0}^{\infty }\right)=& \chi \left(\left\{h\left(t\right)y_0\left(t\right)\right\}\cup {\left\{h\left(t\right)y_n\left(t\right)\right\}}_{n=1}^{\infty }\right)\hfill \\ \hfill =& \chi \left({\left\{h\left(t\right)y_n\left(t\right)\right\}}_{n=1}^{\infty }\right)\hfill \\ \hfill =& \chi \left(V\right(t\left)\right).\hfill \end{array}$$

Thus

$$\chi \left(V\left(t\right)\right)\le 2MK{H}^{*}{\int }_0^t{\displaystyle \frac{1}{h\left(s\right)}}\chi \left(V\left(s\right)\right)ds,$$

(14)

where $H^{*}={max}_{t\in [0,\infty )}h\left(t\right)\}$. Therefore, using Gronwall’s inequality, we obtain that $\chi \left(V\right(t\left)\right)=0$, then $V\left(t\right)$ is relatively compact. Consequently, it follows from Lemma 1, that the set V is relatively compact, i.e., there exists a convergent subsequence of ${\left\{y_{n_k}\right\}}_{k=0}^{\infty }$, such that ${lim}_{k\to \infty }{y}_{n_k}=y^{*}\in \mathrm{\Omega }$. Thus, through continuity of the operator $\mathcal{S}$, we have

$$y^{*}=\underset{k\to \infty }{lim}{y}_{n_k}=\underset{k\to \infty }{lim}\mathcal{S}{y}_{n_k-1}=\mathcal{S}\left(\underset{k\to \infty }{lim}{y}_{n_k-1}\right)=\mathcal{S}{y}^{*}.$$

Therefore, $y^{*}$ is a mild solution for (1). The proof is completed. □

Corollary 3.

Assume that (C1), (C2), (C3), and (C5) hold and

$$\underset{t\to \infty }{lim}{\displaystyle \frac{1}{1+t}}{\int }_0^{t}g\left(s\right)ds=0.$$

Then, the Cauchy problem (1) has at least one mild solution.

Corollary 4.

Assume that (C1), (C2), (C3), and (C5) hold and

$$\underset{t\to \infty }{lim}{e}^{-t}{\int }_0^{t}g\left(s\right)ds=0.$$

Then, the Cauchy problem (1) has at least one mild solution.

Remark 2.

In [8,9], Olszowy and Wȩdrychowicz established some existence theorems for mild solutions under the condition $\chi \left(f(t,D)\right)\le L\left(t\right)\chi \left(D\right),L\in L^1$. Their main idea was to first prove existence results on the bounded interval $[0,T]$, and then extend these results to the unbounded interval $[0,\infty )$. In this paper, new sufficient conditions for the existence of global mild solutions are directly provided by using the generalized Ascoli–Arzela theorem, especially in cases where the evolution family $U(t,s)$ is compact and noncompact. Theorems 1 and 2 can be easily extended to the nonlocal problems studied in [8,9].

Example 1.

Consider the following partial differential equation

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial z}{\partial t}(t,x)=a(t,x)\frac{{\partial }^{2}z}{{\partial }^{2}x}(t,x)+F(t,z(t,x)),t\in [0,\infty ),x\in [0,\pi ]}\hfill \\ z(t,0)=z(t,\pi )=0,t\in [0,\infty ),\hfill \\ z(0,x)=\varphi \left(x\right),x\in [0,\pi ],\hfill \end{array}\right.$$

(15)

where $a:[0,\infty )\times [0,\pi ]\to \mathbb{R}$ is continuous and is uniformly Hölder continuous in t, $F:[0,\infty )\times \mathbb{R}\to \mathbb{R}$ and $\varphi :[0,\pi ]\to \mathbb{R}$ are continuous functions.

Consider $E=L^2(0,\pi ],\mathbb{R})$ and define $A\left(t\right)$ by $A\left(t\right)w=a(t,x)w^{\prime \prime }$ with domain

$$D\left(A\right)=\{w\in E:w,w^{\prime }\mathrm{are}\mathrm{absolutely}\mathrm{continuous},w^{\prime \prime }\in E,w\left(0\right)=w\left(\pi \right)=0\}.$$

Then, $A\left(t\right)$ generates an evolution family $U(t,s)$ satisfying assumption (C1).

For $x\in [0,\pi ]$, let

$$y\left(t\right)\left(x\right)=z(t,x),t\in [0,\infty ),$$

$$f(t,y)\left(x\right)=F(t,z(t,x\left)\right),t\in [0,\infty ),$$

and $y_0=\varphi \left(x\right)$. Thus, under the above definitions of $f,y_0$ and $A(·)$, the system (15) can be represented using the abstract evolution problem (1).

Furthermore, we assume that $U(t,s)$ is compact and $\left|f(t,·)\right|\le m{e}^{wt},m>0,0<w<1,t\in [0,\infty )$. Then, ${lim}_{t\to \infty }{e}^{-t}{\int }_0^{t}m{e}^{ws}ds=0$. By Corollary 2, problem (15) has at least one mild solution on $[0,\infty )$.

5. Conclusions

In this paper, using the generalized Ascoli–Arzela theorem and some analytical techniques, we investigated the existence of global mild solutions for nonautonomous evolution equations on infinite intervals. Some sufficient conditions for the existence of global mild solutions are provided for two cases—where the evolution operator is compact and noncompact. In particular, we did not assume that $f(t,·)$ is continuous and/or satisfies the Lipschitz condition. It is worth mentioning that the generalized Ascoli–Arzela theorem can be used to study impulsive differential equations, stochastic evolution equations, and fractional evolution equations.