Abstract
In this paper, we use a new method to study semilinear evolution differential inclusions with nonlocal conditions in Banach spaces. We derive conditions for F and g for the existence of mild solutions. The results obtained here improve and generalize many known results.
MSC: 34A60, 34G20.
Keywords:
semilinear evolution differential inclusions; mild solutions; measure of noncompactness; upper semicontinuous1 Introduction
In this paper, we discuss the nonlocal initial value problem
where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear
operators (i.e.,
The study of nonlocal evolution equations was initiated by Byszewski [1]. Since these represent mathematical models of various phenomena in physics, Byszewski’s
work was followed by many others [27]. Subsequently, many authors have contributed to the study of the differential inclusions
(1.1). Differential inclusions (1.1) were first considered by Aizicovici and Gao [8] when g and
The organization of this work is as follows. In Section 2, we recall some definitions and facts about setvalued analysis and the measure of noncompactness. In Section 3, we give the existence of mild solutions of the nonlocal initial value problem (1.1). In Section 4, an example is given to show the applications of our results.
2 Preliminaries
Let
We denote by
A
Definition 2.1 A function
(1)
(2)
To prove the existence results in this paper, we need the following lemmas.
Lemma 2.2[22]
If
Lemma 2.3[22]
If
Lemma 2.4[23]
If
Lemma 2.5[24]
If the semigroup
Lemma 2.6[25]
IfWis bounded, then for each
A countable set
(a) it is integrably bounded:
(b) the set
Lemma 2.7[26]
Every semicompact set is weakly compact in the space
If
as
The map
for any bounded closed subset
Lemma 2.9[2730] (Fixed point theorem)
If
Lemma 2.10[26] (Fixed point theorem)
If
3 Main results
In this section, by using the measure of noncompactness and fixed point theorems, we give the existence results of the nonlocal initial value problem (1.1). Here we list the following hypotheses.
(1) The
(2)
(3) The multivalued operator
(4) There exists
for a.e.
(5) There exist a function
for all
Remark 3.1 If
The following lemma plays a crucial role in the proof of the main theorem.
Lemma 3.2[26]
Under assumptions (3)(5), if we consider sequences
Now we give the existence results under the above hypotheses.
Theorem 3.3If (1)(5) are satisfied, then there is at least one mild solution for (1.1) provided that there exists a constantRwith
Proof Define the operator
We shall show that the multivalued Γ has at least one fixed point. The fixed point is then a mild solution of the problem (1.1).
(1) We contract a bounded, convex, closed and compact set
In view of (3.1), we know there exists a constant
where
Then there exists a positive integer K such that
Hence, we get
We denote
For any
Therefore
and
which implies
Define
then W is a convex, closed and bounded subset of
Now, we claim that W is nonempty and compact in
Since this is true for arbitrary
Because
Let
It implies that
(2) We shall show that Γ is closed on W with closed convex values. It is very easy to see that Γ has convex values.
Let us now verity that
As
From hypothesis (5), we obtain
Then we have the set
From hypothesis (4), we know
for a.e.
So, the set
Further, we have
and hence
By Lemma 3.2, it implies that
(4) Γ is u.s.c. on W.
Since
Finally, due to fixed point Lemma 2.9, Γ has at least one point
Remark 3.4 In [812] the authors discuss the nonlocal initial value problem (1.1) when
Theorem 3.5[15]
If (1)(5) are satisfied, then there is at least one mild solution for (1.1) provided that there exists a constant
Proof In view of (3.4), we get
From Theorem 3.3, the nonlocal initial value problem (1.1) has at least one mild solution. □
Remark 3.6 If
By using Theorem 3.5, we do not know whether or not equation (1.1) has a mild solution.
But we know there exists a constant
So, Theorem 3.3 is better than Theorem 3.5.
Theorem 3.7[12]
If (1)(5) are satisfied and
Proof In view of (3.5), we get there exists a constant R such that
By Theorem 3.3, we complete the proof of this theorem. □
Next, we give the existence result for (1.1) when g is Lipschitz continuous.
We suppose that:
(6) There exists a constant
Theorem 3.8If (1) and (3)(6) are satisfied and
then there is at least one mild solution for (1.1) provided that there exists a constantRsatisfying
Proof With the same arguments as given in the first portion of the proof of Theorem 3.3,
we know
Now, we prove the values of Γ are compact in
Let
From hypothesis (5), we obtain
Then we have the set
From hypothesis (4), we know
for a.e.
So, the set
and
By Lemma 3.2, it implies that
Next, we prove Γ is an α contraction map. For any
From Lemma 2.6, we know for arbitrary given
Since this is true for arbitrary
Therefore, we obtain
Noting
Finally, due to Lemma 2.10, Γ has at least one fixed point. This completes the proof. □
4 An example
In this section, as an application of our main results, an example is presented. We consider the following partial differential equation:
where Ω is a bounded domain in
We suppose that
(a) The differential operator
(b) The function
(b_{1})
(b_{2})
(b_{3}) For any
for all
uniformly for
(b_{4}) There exist
for all
Let
From [33], we obtain g satisfies hypothesis (2).
Then equation (4.1) can be regarded as the following nonlocal semilinear evolution equation:
By using Theorem 3.3, the problem (4.1) has at least one mild solution
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. The research was supported by the Scientific Research Foundation of Nanjing Institute of Technology (No: QKJA2011009).
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