Abstract
Not considering the Green’s function, the present study starts to construct a cone formed by a nonlinear term in Banach spaces, and through the cone creates a convex closed set. We obtain the existence of solutions for the boundary values problems of nthorder impulsive singular nonlinear integrodifferential equations in Banach spaces by applying the Mönch fixed point theorem. An example is given to illustrate the main results.
MSC: 45J05, 34G20, 47H10.
Keywords:
impulsive singular integrodifferential equation; Banach spaces; boundary value problem; Mönch fixed point theorem; measure of noncompactness1 Introduction and preliminaries
By using the Schauder fixed point theorem, Guo [1] obtained the existence of solutions of initial value problems for nthorder nonlinear impulsive integrodifferential equations of mixed type on an infinite interval with infinite number of impulsive times in a Banach space. In [2], by using the fixed point theorem in a cone, Chen and Qin investigated the existence of multiple solutions for a class of boundary value problems of singular nonlinear integrodifferential equations of mixed type in Banach spaces. For singular differential equations in Banach spaces please see [39]. Generally based on Green’s function to construct a cone, but using the cone to study different nonlinear terms, we encountered difficulties, especially in infinite dimensional Banach spaces. In this paper, informed by the characteristics of the nonlinear term we construct a new cone, and through this cone create a convex closed set. On the new convex closed set, we apply the Mönch fixed point theorem to investigate the existence of solutions for the boundary value problems of nthorder impulsive singular nonlinear integrodifferential equations in Banach spaces. Finally, an example of scalar secondorder impulsive integrodifferential equations for an infinite system is offered. Because of difficulties of compactness arising from impulsiveness and the use of nthorder integrodifferential equations, a space is introduced. Let E be a real Banach space and . Let := { continuous at , left continuous at , and exists, }. Obviously is a Banach space with norm
Let := { exists and let it be continuous at , let and exist, }, where and represent the right and the left limits of at , respectively. For , we have
So observing the existence of and taking limits as in the above equality, we see that exists and
Similarly, we can show that exists. In the same way, we get the existence of . Define (, ). Then (), and, as is natural, in the following, is understood as . It is easy to see that is a Banach space with norm
Let P be a cone in E which defines a partial ordering in E by if and only if . P is said to be normal if there exists a positive constant N such that implies , where the smallest N is called the normal constant of P. For convenience, let . Let , in which and . For , we write . We consider the following singular boundary value problem (SBVP for short) for an nthorder impulsive nonlinear integrodifferential equation in E:
with (), . denotes the jump of at , i.e.,
and θ denotes the zero element of E.
Remark Obviously, , and is a normal cone of E if P is a normal cone of E. and P has the same normal constant N.
In the following, we assume that P is a normal cone. Let . A map is called a solution of SBVP (1) if it satisfies (1).
2 Several lemmas
To continue, let us formulate some lemmas.
Lemma 2.1Ifis bounded and the elements ofare equicontinuous on each (), then
in whichαdenotes the Kuratowski measure of noncompactness, ().
Proof For , it is easy to prove that
Next, we check that
In fact, for any , there is a division () such that
By hypothesis, the elements of are equicontinuous on each and there is a division:
such that
and
Let , (). By virtue of (5) and (6), we know that
Let . There is a division such that
Let F be the finite set of all maps into (). For , let . It is clear that . For any , , we have for some , and so
Consequently,
which implies . Since is arbitrary, we get
Finally, the conclusion follows from (3) and (10). For details of the Kuratowski measure of noncompactness, please see [10]. □
Lemma 2.2 (see [11])
Let us take a countable set (). For all, there issuch that, a.e, . Then, and
Lemma 2.3Supposeis bounded and equicontinuous on each (). Then, and
Proof By Theorem 1.2.2 of [10], the conclusion is obvious. □
Lemma 2.4Letbe two countable sets. Supposeand. Then
Proof The conclusion is obvious by Lemma 6 of [12]. □
Lemma 2.5 (see [13]) (The Mönch fixed point theorem)
LetEbe a Banach space. Assume thatis close and convex. Assume also thatis continuous with the further property that for some, we havecountable, implies thatCis relatively compact. ThenAhas a fixedpoint inD.
3 Main theorem and example
For convenience, we list the following conditions:
(H_{1}) There exist , (), () and () such that
where is nonincreasing, () and , are nondecreasing. And there exist , (, ) such that
(H_{2}) There exists a ( denotes the dual cone of ) such that . And for any , there exists a such that
(H_{3}) There exists a such that
where b, (), (), φ, (), (), (, ) and are defined as in conditions (H_{1}) and (H_{2}), and , .
(H_{4}) There exist (), such that
(), . There exist (, ) such that
Remark Obviously, condition (H_{4}) is satisfied automatically when E is finite dimensional.
Lemma 3.1Suppose conditions (H_{1}), (H_{2}) and (H_{3}) are satisfied. ThenQdefined by
is a nonempty, convex and closed subset of.
Proof Let
It is clear that . Since , for , by (11), one can see that
By conditions (H_{1}), (H_{2}), (H_{3}), and (11), we have
and . Therefore, and Q is a nonempty set.
Now, we check that Q is a convex subset of . In fact, for any , , we write , which means . It is clear that
By virtue of the characters of elements of Q and the characters of φ, we have
In the same way,
and
Therefore, . Thus, Q is a convex subset of . It is clear that Q is a closed subset of . So the conclusion holds. □
Lemma 3.2Assume that conditions (H_{1}), (H_{2}) and (H_{3}) are satisfied. Then, where the operatorAis defined by
and
and
Analogously, for , it is easy to see
Hence,
Differentiating (15) times, we get
where is understood as θ for . Similarly, () exist. Hence,
Let
Since
and (; ), it follows from (15), (16), (17) and (18) that
It is clear that
Since , by (22), (26) and condition (H_{2}), we have
Now, we show that
By (15), (19), (21), conditions (H_{1}) and (H_{3}) imply
which implies that (29) is true. By (24), (26) to (29), the conclusion holds. □
Lemma 3.3Suppose conditions (H_{1}) to (H_{4}) are satisfied. Let
and
and
Proof In order to avoid the singularity, given , let
By conditions (H_{1}), (H_{2}) and (H_{3}), for any , , we have
By virtue of absolute continuity of the Lebesgue integrable function, we have
in which, denotes the Hausdorff distance between and . Therefore,
Now, we show
In fact, by Lemma 2.2, we have
On the other hand, for , it follows from (19) and (21) that
and
Taking , , by (16), (17) and (18), one can see that
Therefore, by condition (H_{4}) and (36), for , it is easy to get
Since B is a bounded set of and is a bounded set, is equicontinuous on each (). By Lemma 2.3, it is easy to get
Substituting (41) into (40), we get (31).
Similarly, we obtain (32) and our conclusion holds. □
Lemma 3.4Let conditions (H_{1}) to (H_{3}) be satisfied. is a solution of SBVP (1), if and only ifis a fixed point of the operatorAdefined by (15).
Proof First of all, by mathematical induction, for , Taylor’s formula with the integral remainder term holds,
In fact, as , for , let , it is easy to see that
Adding these together, we get
that is,
This proves that (42) is true for .
Suppose (42) is true for , i.e., for , the next formula holds:
Now we check that (42) is also true for n. In fact, suppose . Then , by (43), one can see
Substituting the above equation into (44), we get
So, (42) is also true for n. By mathematical induction, (42) holds.
Suppose is a solution of SBVP (1). By (42), we can see that
Substituting
into (47), by (15), we get . So u is a fixed point of the operator A defined by (15) in Q.
Conversely, if is a fixed point of the operator A, i.e., u is a solution of the following impulsive integrodifferential equation:
Then, by (15), similar to (26), from the derivative of both sides of the above equation one can draw the following conclusions:
So, we have
and
Hence
It follows from (48) and (50) that . By (48), (49) and (50), we have
and
It is easy to see by (48) and (50)
By (51) to (54), u is a solution of SBVP (1). □
Theorem 3.1Let conditions (H_{1}) to (H_{4}) be satisfied. Assume that
Then SBVP (1) has at least a solution.
Proof We will use Lemma 2.5 to prove our conclusion. By (H_{1})(H_{3}), from Lemma 3.2, we know .
We affirm that is continuous. In fact, let , , (as ). From the continuity of f and (, ) and the definition of A, by virtue of the Lebesgue dominated convergence theorem, we see that
For (fixed), we have (). We also see that is bounded and is equicontinuous on each . By Lemma 2.1, it is easy to get
i.e., is a relatively compact set in . The reduction to absurdity is used to prove that A is continuous. Suppose . Then , such that
On the other hand, since is a relatively compact set in , there exists a subsequence of which converges to . Without loss of generality, we may assume itself converges to y, that is,
By virtue of (56), we see that . Obviously, this is in contradiction to (58). Hence,
By Lemma 2.4, for any countable , which satisfies , one can see
By virtue of the character of noncompactness, it is easy to get ,
For any fixed , , by condition (H_{4}) and Lemma 3.3, we have
Similarly, by Lemma 3.3, for , we have
Let . It is clear that . By (61) and (62), for , it is easy to see that
Since , by (64) and (65), we know that . It is easy to see that is bounded and the elements of are equicontinuous on each (). It follows from (61) and Lemma 2.1 that
Hence, B is a relatively compact set. By Lemma 2.5 (the Mönch fixed point theorem), A has at least a fixed point , and by Lemma 3.4, is the solution of SBVP (1) which means conclusion holds. □
An application of Theorem 3.1 is as follows.
Example Consider the following infinite system of scalar nonlinear second order impulsive integrodifferential equations:
Conclusion. The infinite system (67) has at least a () solution, , , , .
Proof Let , with norm . We have the cone . Obviously P is a normal cone in E. Taking (), it is easy to see , and . The infinite system (67) can be regarded as a SBVP of the form (1) in E. In this situation,
in which
which implies
Since
and
That is, . Obviously, . By (71), we can see
On the other hand, from (68) and (70), we have
It is easy to get
It follows from (72), (73) and (74) that
Taking
and
by (69) and (72), condition (H_{1}) holds.
For any , define φ by . It is easy to see , and . For any , let
Therefore, condition (H_{2}) is satisfied. It follows from (68) that
Now, we check that condition (H_{3}) is true. In fact, it is easy to get
there exists a sufficient large such that
which implies that condition (H_{3}) is satisfied.
Let
in which
where
For any
by (71) and (78), it is easy to get
Hence, the relative compactness of in follows directly from a known result (see [14]): a bounded set X of is relatively compact if and only if
That is,
For any
by (78), one can get
in which . Since and , by (81), it is easy to see
which implies
Similarly, by (78),
and
By (80), (82), (83) and (84), it is easy to get
In the same way,
Taking
the condition (H_{4}) follows from (86) and (87). We can calculate and get
Therefore, by Theorem 3.1, the conclusion holds. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors typed, read and approved the final manuscript.
Acknowledgements
The authors would like to thank the reviewers for carefully reading this article and making valuable comments and suggestions. The project is supported by the National Natural Science Foundation of P.R. China (71272119), Social Science Foundation of Shandong Province (13CJRJ07) and Teaching and Research Projects of Qi Lu Normal University (201306).
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