Existence of Solutions for a Non-Autonomous Evolution Equations with Nonlocal Conditions ()
1. Introduction
Recently, the evolution equation has been used to describe the state or process that changes with time in physics, mechanics, or other natural sciences. Byszewski [1] first investigated the nonlocal problems, they obtained the existence and uniqueness of mild solutions for nonlocal differential equations without impulsive conditions. Deng [2] pointed out that the nonlocal initial condition can be applied in physics with better effect than the classical initial condition
, and used the nonlocal conditions
to describe the diffusion phenomenon on a small amount of gas in a transparent tube. The aforementioned findings encourage more authors to focus on differential equations with nonlocal conditions. The integro-differential equations are usually applied to model processes which are subjected to abrupt changes at a certain time. They have wide applications in control, mechanics, electrical engineering fields, and so on. In 2010, Fan [3] discussed the existence and simulation of positive solutions for m-point fractional differential equations with derivative terms. In 2011, Tai [4] studied the exact controllability of fractional impulsive neutral functional integro-differential systems with nonlocal conditions by using the fractional power of operators and the Banach contraction mapping theorem. Consequently, to describe some physical phenomena, the nonlocal condition can be more useful than the standard initial condition
. The importance of nonlocal conditions has also been discussed in [5] [6] [7] [8] [9] .
All of researchers focus on the case that the differential operators in the main parts are independent of time t, which means that the problems are autonomous in previous researches. However, when treating some parabolic evolution equations, due to the frequent occurrence of such operators related to time t in applications, it is usually assumed that the partial differential operators depend on time t on account of this class of operators appears frequently in the applications, for the details please see Fu [10] ; Zhu [11] and Wang [12] . So it is meaningful to study the non-autonomous evolution equation, that is, the differential operators of the main part are related to time t.
In recent years, the existence and approximate controllability of the mild solution of the evolution equation are widely studied. Kalman [13] in 1963 introduced the concept of controllability firstly, and the concept has become an active area of investigation due to its great applications in the field of physics. There are various works on approximate controllability of systems represented by differential equations, integro-differential equations, differential inclusions, neutral functional differential equations, and impulsive differential equations of integer order in Banach spaces. Mahmudov [14] in 2008 studied the approximate controllability for the abstract evolution equations with nonlocal conditions in Hilbert spaces and obtained sufficient conditions for the approximate controllability of the semi-linear evolution equation. In 2017, Kumar [15] studied mild solution and fractional optimal control of semilinear system with fixed delay in Banach space. In 2018, Chen [16] discussed that approximate controllability of non-autonomous evolution system with nonlocal conditions and introduced a new Green’s function and constructed a control function involving Gramian controllability operator. He applied the Schauder’s fixed point theorem to proved the mild solutions of the evolution equations.
In 1995, Byszewski [17] obtained the existence and uniqueness of solutions to a class of abstract functional differential equations with nonlocal conditions of the form
where
is a constant,
,
and
are given functions, X is a Banach space,
,
,
,
. It pointed out that if
then we know that this can be applied to the motion of gas in physics.
In 2013, Fu and Zhang [10] obtained the exact null controllability for the following non-autonomous functional evolution system with nonlocal condition
where the state variable
takes values in a Hilbert space X and the control function
is given in Banach space
of admissible control functions, U is also a Hilbert space. The family
of linear operators generates a linear evolution system and B is a bounded linear operator from U into X.
,
,
and
, where
are given constants and
. A strong assumption is made for the nonlocal function g by [5] but no assumption is made for the nonlocal function in this paper.
In the above literature [18] - [25] , the authors make some assumptions about the nonlocal terms. Therefore, it seems natural to remove the strong constraints on the nonlocal function g. Motivated by all of the above-mentioned aspects, in this paper we consider the existence of mild solutions for the following non-autonomous evolution equation with nonlocal conditions
(1)
in Banach space X, where
is a constant.
is a family of (possibly unbounded) linear operators depending on time and having the domains
for every
, the control function
is given in Banach space
of admissible control functions, U is also a Banach space,
,
is a continuous nonlinear mapping, and B is a bounded linear operator from U to X,
,
,
are real numbers,
. The existence of mild solutions to the problem (1) is considered by using the Krasnoselskii’s fixed point theorem and the theory of evolution family. In additions, an example is given as an application of the results. We assumed that the existence of the mild solution to the evolution equation is related to its approximative controllability, which can lead to new ideas for the study of the existence and approximate controllability of the mild solution to the evolution equation. The interesting of this paper is that any assumptions are not imposed to the nonlocal terms and Green’s functions, and a new alternative method is applied to prove the existence of mild solutions.
The rest of this paper is organized as follows. Some basic definitions, lemmas and properties are introduced in Section 2. In Section 3, the existence of mild solutions to the non-autonomous evolution equations (1) is proved. An example is given to illustrate the main results in Section 4. The main contents of this article are summarized in Section 5.
2. Preliminaries
In this section, we introduce some notations, definitions, and preliminary facts which are used throughout this paper.
Let X and U be two real Banach spaces with norms
and
. We denote by
the Banach space of all continuous functions from interval J into X equipped with the supremum norm
,
.
And by
the Banach space of all linear and bounded operators in E endowed with the topology defined by the operator norm. Let
be the Banach space of all U value Bochner square integrable functions defined on J with the norm
,
,
Suppose that a family of linear operators
satisfies the following assumptions:
(A1) The family
is a closed linear operator;
(A2) For each
, the resolvent
of linear operator
exists for all
such that
, and there also exists
such that
;
(A3) There exist
and
such that
for all
and
;
(A4) For each
and some
, the resolvent set
of linear operator
is compact.
Because of these conditions, the family
generates a unique linear evolution system, or called linear evolution family
, and there exists a family of bounded linear operators
with norm
such that
can be represented as
(2)
where
denotes the analytic semigroup with infinitesimal generator
.
Lemma 2.1 [16] The family of linear operators
satisfies the following conditions:
1) the mapping
is continuous, for each
,
and
;
2)
for
, and
;
3)
is a compact operator whenever
;
4) There holds, if
,
, and
,
Condition (A4) ensures the generated evolution operator satisfies 3) (see [12] , Proposition 2.1). Hence, there exists a constant
, such that
for all
. (3)
Definition 2.1 [16] The evolution family
is continuous and maps bounded subsets of X into pre-compact subsets of X.
Lemma 2.2 [18] Let
be a compact evolution system in X. Then for each
,
is continuous by operator norm for
.
Let Y be another separable reflexive Banach space, whose norm is also denoted by
, in which the control function
takes its values, E a bounded subset of Y. Denoted by
is a class of nonempty closed and convex subsets of Y. We suppose that the multi-valued map
is graph measurable,
. The admissible control set
is defined by
Obviously,
(see [26] ) and
is bounded, closed and convex.
In order to discuss the existence of solutions to (1), we consider the following linear non-autonomous evolution equations:
(4)
exists a unique solution
expressed by
(5)
If
, the function u given by (5) belongs to
, which is known as a mild solution of the (4).
Assume that the condition
(H0)
holds. From (3) and the assumption (H0) one gets that
(6)
From the (6), we obtain
By operator spectrum theorem, the operator
exists and is bounded. Moreover, by Neumann expression, we have
To prove our main results, for any
, we consider the following nonlocal problem of linear non-autonomous evolution equation
(7)
Lemma 2.3 If the condition (H0) is satisfied, then a function
is said to be a mild solution of the system (7), which is
(8)
Proof. It is well-known that (7) has a unique mild solution
expressed by
(9)
Particularly,
(10)
From (7) and (10), we have
(11)
From (9) and (11), we know that
satisfies (3).
Inversely, we can verify directly that the function
given by (8) is a mild solution of (7).
Definition 2.2 A function
is said to be a mild solution of nonlocal problem (1), if for any
,
satisfies the integral equation
Lemma 2.4 [18] (Krasnoselskii’s fixed point theorem). Let W be a closed, convex and nonempty subset of Banach space X. Let operators
satisfy
1) if
, then
;
2)
is a contraction;
3)
is compact and continuous.
Then the operator
has at least one fixed point in W.
Lemma 2.5 [18] If Ω is a compact subset of a Banach space X, its convex closure is compact.
3. Existence of Mild Solutions
In this section, the existence of mild solutions of the non-autonomous evolution Equation (1) is we considered. The proof is based on the Krasnoselskii’s fixed point theorem and the theory of evolution system. We further assume the following conditions:
(H1) The function
satisfies that for every
, the function
is continuous and for each
, the function
is strongly measurable, there exists a function
such that
for all
and
,
(H2) There exists a function
such that
for all
and
,
(H3)
, and
,
(H4) The function
and there exists a constant
, such that
,
.
Theorem 3.1 Let the evolution family
generated by
is compact. Suppose also that the assumption (H0)-(H4) are satisfied. Then the nonlocal problem (1) has at least one mild solution on J provided that
Proof. Defined the operator
, where
(12)
(13)
By Definition 2.2, we can know that the mild solution of nonlocal problem (1) is equivalent to the fixed point of operator Q. In the following, we will prove that the operator Q admits a fixed point by applying the Krasnoselskii’s fixed point theorem. The proof is divided into five steps.
Step 1.
for some
. For any
, let
.
If this is not true, for each
, there exists
,
for all
. From the definition of Q and hypotheses (H0)-(H4), we have
Dividing on both sides by r and taking the lower limit as
, we obtain
, which is a contradiction to the condition in Theorem 3.1. Thus,
for some
.
Step 2.
is a contraction operator.
For any
,
, (12), and (H0)-(H4) imply
which yields that
. Hence
is a contraction operator in
.
Step 3.
is continuous. Let
with
in
. Then by the continuity of f, we have
(14)
In addition, since
(15)
and the Lebesgue’s dominated convergence theorem follows that
which implies that
is continuous.
Step 4.
is equi-continuous in
. For any
and
, by (13) and (H0)-(H4), we have
where
;
.
If
,
, the conclusion is obvious. If
, we choose
small enough, by the conditions (H0), (H1) and (H2), we have
, as
.
Therefore,
, as
, which means that the operator
is equi-continuous.
Step 5. The set
s relatively compact in X for each
. Obviously, the set
is relatively compact in X. Let
, for any
and
, we define an operator
by
It follows from the boundedness of
and (H1) that the set
is relatively compact and depend on the compactness of
. Then,
is a compact set depend on Lemma 2.5. By the mean value theorem of Bochner integrals, we can get
for all
. Thus, the set
is relatively compact in X for every
. Moreover, by (13) and (H0) and (H1), we have
which means that
. So we have proved that there is a family of relatively compact sets
arbitrarily close to the set
. Thus, the set
is relatively compact in X for every
.
By Steps 3-5, thanks to the Ascoli-Arzela theorem, we deduce that the operator
is compact and continuous in
. By the Krasnoselskii’s fixed point theorem we can get the operator Q has at least one fixed point in
, which is the mild solution of the evolution Equation (1) on J. This completes the proof of Theorem 3.1.
4. Example
In this section, we provide a correct example to illustrate our abstract results.
Example 4.1 Consider the following non-autonomous partial differential equation with nonlocal problem:
(16)
where
is a continuously differentiable function and satisfies
(17)
and
is a constant,
,
. Let
with the norm
and inner product
. Consider the operator A on X defined by
where
The A generates a compact and analytic C0-semigroup in C, A has a discrete spectrum, and its eigenvalues are
with the corresponding normalized eigenvectors
. Define the operator
on A by
with domain
The family
generates an strongly continuous evolution family
defined by
(18)
A direct calculation gives
(17) and (18) means that
(see [9] , [22] )
For any
, we define
In conclusion, from
, we know that the assumption (H0) hold. From the definition of nonlinear term f and bounded linear
operator A combined with the above discussion, we can easily verify that the assumptions (H1)-(H4) are satisfied with
,
.
Therefore, the non-autonomous partial differential Equation (16) is equivalent to the problem (1). According to theorem 3.1, we know that (16) has at least one mild solution
.
5. Conclusion
In this paper, the existence of solutions to non-autonomous evolution equations with nonlocal conditions is studied. By using the development family theory, solution operators and the application of Krasnoselskiis fixed point theorem, the existence of solutions of non-autonomous evolution equations is obtained.
Funding
This work is supported by the Key Project of Yili Normal University (YSPY 2022014), the Research and Innovation Team of Yili Normal University (CXZK 2021016), the Natural Science Foundation of Xinjiang Uygur Autonomous Region (2023D01C51) and the Xuzhou Science and Technology Plan Project (No. KC23058).