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
where
generates an evolution family
,
X is a Banach space with the norm
, and
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
generates a compact
-semigroup. In [
5], Ding, Long, and N’Guérékata established new theorems on the existence of almost automorphic solutions to nonautonomous evolution equations
and
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
,
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
.
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
. 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 . Denote as the Banach space of all continuous functions from J to X with the norm . Using , we denote the space of all bounded linear operators from X to X, with the norm .
Assume the function
. Let
with the norm
. Then,
is a Banach space.
The following generalized Ascoli–Arzela theorem was obtained in [
11].
Lemma 1 ([
11]).
The set is relatively compact if and only if the following conditions hold:- (i)
for any , the set is equicontinuous on ;
- (ii)
uniformly for ;
- (iii)
for any , is relatively compact in X.
Remark 1. Lemma 1 was proven in [13] when and in [14] when , . Assume that
D is a nonempty subset of
X. Kuratowski’s measure of noncompactness
is said to be
where the diameter of
is given by diam
,
Proposition 1. Let and be two bounded sets of a Banach space X. Then, the following properties hold:
- (i)
if and only if is relatively compact in X;
- (ii)
if ;
- (iii)
for every and every nonempty subset ;
- (iv)
, where ;
- (v)
;
- (vi)
for any .
Lemma 2 ([
15]).
Let be a continuous function family. If there exists such thatthen is integrable on , and Lemma 3 (Mazur theorem). If is a compact subset of a Banach space X, then its convex closure is compact.
3. Some Lemmas
Let . We assume that is a linear operator acting from into X for each , and that generates a uniformly continuous evolution family , , satisfying the following properties:
- (i)
for each , where I is the identity operator in X;
- (ii)
for ;
- (iii)
for every and for each , the mapping is continuous.
An evolution operator U is said to be compact when is a compact operator for all , i.e., changes bounded sets into relatively compact sets.
Throughout of this paper, we utilize the following assumptions:
- (C1)
There exists a constant
, such that
- (C2)
is a mapping, such that is measurable for and is continuous for a.e. .
- (C3)
there exists a function
, such that
is continuous on
and
- (C4)
there exists a function
, such that
and
Definition 1. If satisfiesthen is a mild solution for the Cauchy problem (
1).
For any
, define an operator
as follows:
where
Clearly, the fixed points of the operator
are mild solutions for the problem (
1).
From (C4), there exists a constant
, such that
Let
Then,
is a nonempty, convex, and closed subset of
.
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 is equicontinuous.
For
, we obtain
For
, we have
Hence,
is equicontinuous.
Step II. We prove that is equicontinuous.
By (C4), for
, there exists
, such that
For
, in view of (
6), we obtain
For
,
, by (C4), we have
For
, we obtain
For
, if
, then
and
. Thus, for
Hence, for
,
Therefore,
is equicontinuous. Furthermore,
V is equicontinuous. □
Lemma 5. Assume that (C1)–(C4) hold. Then, uniformly for .
Proof. For any
,
which implies that
uniformly for
. This completes the proof. □
Lemma 6. Assume that (C1)–(C4) hold. Then, .
Proof. By Lemmas 4 and 5, one can find that
. For any
,
, from (
2), we obtain
For
,
. Therefore,
. □
Lemma 7. Assume that (C1)–(C4) hold. Then, is continuous.
Proof. Indeed, let
be a sequence in
, which is convergent to
. Consequently,
. From (C2), we obtain
. For any
, there exists
, such that (
6) holds. Thus, for
,
For each
,
. Using the Lebesgue dominated convergence theorem, we obtain
Thus, for
,
So,
as
. Hence,
is continuous. The proof is completed. □
4. Existence
Theorem 1. Assume that is compact for all . 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
if and only if the operator
has a fixed point
. From Lemmas 6 and 7, we know that
and
is continuous. In order to prove that
is a completely continuous operator, we need to prove that
is a relatively compact set. From Lemmas 4 and 5, the set
is equicontinuous and
uniformly for
. According to Lemma 1, we only need to prove that
is relatively compact in
X.
Clearly,
is relatively compact in
X. We only consider the case
. For
, we define the operator on
where
Since
is compact, it follows that
is compact and the set
is compact for all
. Then,
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
Therefore, the set
is relatively compact in
X for all
. Now, observe that
Thus,
is also a relatively compact set in
X for
. We include that
is a completely continuous operator. Hence, using the Schauder fixed-point theorem, we have that
has at least a fixed point
. Therefore
which implies that
is a mild solution for (
1). The proof is completed. □
By choosing different , we can provide some practical corollaries.
Corollary 1. Assume that is compact for all . Further, assume that (C1), (C2), and (C3) hold andThen, the Cauchy problem (1) has at least one mild solution. Proof. Choose , then condition (C4) is satisfied. □
Corollary 2. Assume that is compact for all . Further, assume that (C1), (C2), and (C3) hold andThen, the Cauchy problem (1) has at least one mild solution. Proof. Choose , then condition (C4) is satisfied. □
In the case that is noncompact for , we impose the following assumption:
- (C5)
there exist constants
, such that for any bounded
,
,
where 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 for all and , . From Lemma 6, , for . Let . We prove that the set V is relatively compact.
In view of Lemmas 4 and 5, the set V is equicontinuous and uniformly for . According to Lemma 1, we only need to prove that is relatively compact in X for .
Under the condition (C3), and using the properties of the measure of noncompactness along with Lemma 2, we have
On the other hand, using the properties of measure of noncompactness, for any
we have
Thus
where
. Therefore, using Gronwall’s inequality, we obtain that
, then
is relatively compact. Consequently, it follows from Lemma 1, that the set
V is relatively compact, i.e., there exists a convergent subsequence of
, such that
. Thus, through continuity of the operator
, we have
Therefore,
is a mild solution for (
1). The proof is completed. □
Corollary 3. Assume that (C1), (C2), (C3), and (C5) hold andThen, the Cauchy problem (1) has at least one mild solution. Corollary 4. Assume that (C1), (C2), (C3), and (C5) hold andThen, 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 . Their main idea was to first prove existence results on the bounded interval , and then extend these results to the unbounded interval . 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 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 equationwhere is continuous and is uniformly Hölder continuous in t, and are continuous functions. Consider and define by with domainThen, generates an evolution family satisfying assumption (C1). For , letand . Thus, under the above definitions of and , the system (15) can be represented using the abstract evolution problem (1). Furthermore, we assume that is compact and . Then, . By Corollary 2, problem (15) has at least one mild solution on .