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., semigroup) in a Banach space X, and , are given Xvalued functions.
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 are compact. In [912] the semilinear evolution differential inclusions (1.1) were discussed when A generates a compact semigroup. Xue and Song [13] established the existence of mild solutions to the differential inclusions (1.1) when A generates an equicontinuous semigroup and is l.s.c. for a.e. . In [14] the author proved the existence of mild solutions of the differential inclusions (1.1) when A generates an equicontinuous semigroup and a Banach space X which is separable and uniformly smooth. In [15] Zhu and Li studied the differential inclusions (1.1) when F admits a strongly measurable selector. In [16] the differential inclusions (1.1) were discussed when is a family of linear (not necessarily bounded) operators. In [17] local and global existence results for differential inclusions with infinite delay in a Banach space were considered. Benchohra and Ntouyas [18] studied the secondorder initial value problems for delay integrodifferential inclusions. In [19,20] the impulsive multivalued semilinear neutral functional differential inclusions were discussed in the case that the linear semigroup is compact. The purpose of this paper is to continue the study of these authors. By using a new method, we prove the existence results of mild solutions for (1.1) under the following conditions of g and : g is either compact or Lipschitz continuous and is an equicontinuous semigroup. So, our work extends and improves many main results such as those in [812,14,15].
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 be a real Banach space. Let . A multivalued map is convex (closed) valued if is convex (closed) for all . We say that G is bounded on bounded sets if is bounded in X for each bounded set B of X. The map G is called upper semicontinuous (u.s.c.) on X if for each the set is a nonempty, closed subset of X, and if for each open set N of X containing , there exists an open neighborhood M of such that . Also, G is said to be completely continuous if is relatively compact for every bounded subset . If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph (i.e., , , imply ). Moreover, the following conclusions hold. Let and be closed for all , if G is u.s.c. and D is closed, then is closed. If is compact and D is closed, then G is u.s.c. if and only if is closed. Finally, we say that G has a fixed point if there exists such that .
We denote by the space of Xvalued continuous functions on with the norm , and by the space of Xvalued Bochner functions on with the norm .
A semigroup is said to be compact if is compact for any . If the semigroup is compact, then is equicontinuous at all with respect to x in all bounded subsets of X; i.e., the semigroup is equicontinuous. If A is the generator of an analytic semigroup or a differentiable semigroup , then is an equicontinuous semigroup (see [21]). In this paper, we suppose that A generates an equicontinuous semigroup on X. Since no confusion may occur, we denote by α the Hausdorff measure of noncompactness on both X and .
Definition 2.1 A function is a mild solution of (1.1) if
To prove the existence results in this paper, we need the following lemmas.
Lemma 2.2[22]
Ifis bounded, thenfor all, where. Furthermore, ifWis equicontinuous on, thenis continuous on, and.
Lemma 2.3[22]
Ifis a decreasing sequence of bounded closed nonempty subsets ofXand, thenis nonempty and compact inX.
Lemma 2.4[23]
Ifis uniformly integrable, thenis measurable and
Lemma 2.5[24]
If the semigroupis equicontinuous and, then the setis equicontinuous on.
Lemma 2.6[25]
IfWis bounded, then for each, there is a sequencesuch that
A countable set is said to be semicompact if
(a) it is integrably bounded: for a.e. and every , where ;
(b) the set is relatively compact for a.e. .
Lemma 2.7[26]
Every semicompact set is weakly compact in the space.
Ifis semicompact, thenis relatively compact inand, moreover, if, then
The map is said to be α contraction if there exists a positive constant such that
for any bounded closed subset .
Lemma 2.9[2730] (Fixed point theorem)
Ifis a nonempty, bounded, closed, convex and compact subset, the mapis upper semicontinuous witha nonempty, closed, convex subset ofWfor each, thenFhas at least one fixed point in W.
Lemma 2.10[26] (Fixed point theorem)
Ifis nonempty, bounded, closed and convex, the mapis a closedαcontraction map witha nonempty, convex and compact subset ofWfor each, thenFhas at least one fixed point inW.
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 semigroup generated by A is equicontinuous. We denote .
(2) is continuous and compact, there exist positive constants c and d such that , .
(3) The multivalued operator satisfies the hypotheses: the set is nonempty.
(4) There exists such that for any bounded ,
(5) There exist a function and a nondecreasing continuous function such that
Remark 3.1 If , then for each (see Lasota and Opial [31]). If and , then the set is nonempty if and only if the function defined by belongs to (see Hu and Papageorgiou [32]).
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 sequencesand, where, such that, , then.
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
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 such that Γ maps W into itself.
In view of (3.1), we know there exists a constant such that
Then there exists a positive integer K such that
We denote , then is nonempty, bounded, closed and convex.
Therefore
and
which implies is a bounded operator.
Define , where means the closure of the convex hull in . Then is nonempty bounded closed convex on with . Let for all . Similarly to the above discussions, we know that for as and are both nonempty, closed, bounded and convex. Thus is a decreasing sequence consisting of subsets of . Moreover, set
then W is a convex, closed and bounded subset of and .
Now, we claim that W is nonempty and compact in . To do so, from Lemma 2.6, we know for arbitrary given , there exist sequences such that
Since this is true for arbitrary , we have
Because is decreasing for n, we can define
It implies that for all . By Lemma 2.2, we know that . Using Lemma 2.3, we obtain is nonempty and compact in .
(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 is closed. Let with in , and with in . Moreover, let be a sequence such that for any , and
As in , we know that is a bounded set of , we denote .
From hypothesis (5), we obtain
Then we have the set is integrably bounded for a.e. .
From hypothesis (4), we know
for a.e. . Then the set is relatively compact for a.e. .
So, the set is semicompact. By applying Lemma 2.7, it yields that is weakly compact in . We get that there exists such that . Therefore, we infer that
Further, we have
and hence
By Lemma 3.2, it implies that , i.e., . Therefore is closed. And hence Γ has closed values on W.
(4) Γ is u.s.c. on W.
Since is compact, W is closed and is closed, we can come to the conclusion that Γ is u.s.c. (see [30]).
Finally, due to fixed point Lemma 2.9, Γ has at least one point , and x is a mild solution to the semilinear evolution differential inclusions with the nonlocal conditions (1.1). Thus the proof is complete. □
Remark 3.4 In [812] the authors discuss the nonlocal initial value problem (1.1) when is compact. In [14] the existence of mild solutions of the differential inclusions (1.1) is proved when A generates an equicontinuous semigroup and Banach space X is separable and uniformly smooth. In this paper, by using a new method, we prove the operator Γ maps compact set W into itself. We do not impose any restriction on the coefficient , and we only require to be an equicontinuous semigroup. So, Theorem 3.3 generalizes and improves the related results in [812,14].
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 constantsuch that
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 , , , and . We cannot obtain a constant R such that
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 such that
So, Theorem 3.3 is better than Theorem 3.5.
Theorem 3.7[12]
If (1)(5) are satisfied and, then there is at least one mild solution for (1.1) provided that
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 such that for all . Therefore, , where .
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 is a bounded map with convex values and is closed on .
Now, we prove the values of Γ are compact in .
Let and . To prove that is compact, we have to show that has a subsequence converging to a point . We have such that
From hypothesis (5), we obtain
Then we have the set is integrably bounded for a.e. .
From hypothesis (4), we know
for a.e. . Then the set is relatively compact for a.e. .
So, the set is semicompact. By applying Lemma 2.7, it yields that is weakly compact in . We get that there exists such that . Therefore, we infer that
and
By Lemma 3.2, it implies that , i.e.,. Therefore Γ has compact values.
Next, we prove Γ is an α contraction map. For any , we have
From Lemma 2.6, we know for arbitrary given , there exist sequences such that
Since this is true for arbitrary , we have
Therefore, we obtain
Noting , therefore Γ is an α contraction map.
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 with a smooth boundary ∂Ω, is a smooth real function on .
We suppose that
(a) The differential operator is strongly elliptic [21].
(b) The function satisfies the following conditions:
(b_{1}) is a continuous function about r for a.e. .
(b_{2}) is measurable about for each fixed .
(b_{3}) For any , there is such that
(b_{4}) There exist and such that
Let and , then A generates an analytic semigroup on ([21]). We suppose
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 provided that hypotheses (3)(5) and (3.1) hold.
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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