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On the prescribed variable exponent mean curvature impulsive system boundary value problems
Boundary Value Problems volume 2014, Article number: 139 (2014)
Abstract
This paper investigates the existence of solutions for prescribed variable exponent mean curvature impulsive system, with periodic-like, Dirichlet, and Neumann boundary value conditions, respectively. The proof of our main result is based upon the Leray-Schauder degree. The sufficient conditions for the existence of solutions are given.
MSC: 34B37, 34B15.
1 Introduction
In this paper, we consider the existence of solutions for the prescribed variable exponent mean curvature system
where , with the following impulsive conditions:
and one of the following boundary value conditions:
where
are absolutely continuous; and p, q satisfy , . is called the variable exponent mean curvature operator; ; .
If and , then implies , and (4) is the periodic boundary value condition. Thus we call (4) the periodic-like boundary value condition.
The system (1)-(3) is called a prescribed variable exponent mean curvature impulsive system. It has three characteristics, i.e. impulsive, mean curvature and variable exponent. Let us simply introduce the three characteristics.
The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments, such as mechanical systems with impact, biological systems such as heart beats, blood flows, population dynamics, theoretical physics, radiophysics, pharmacokinetics, mathematical economy, chemical technology, electric technology, metallurgy, ecology, industrial robotics, biotechnology processes, chemistry, engineering, control theory, medicine, and so on. There are many results on the Laplacian impulsive differential equations boundary value problems (see [1]–[13]). The results as regards p-Laplacian impulsive differential equations boundary value problems are more difficult due to the nonlinearity of p-Laplacian (see [14]–[18]). Because of the impulsive conditions and the non-continuity of solutions, this paper has more difficulties than [19]. In many papers about the usual Laplacian impulsive problems, the authors use the impulsive condition , but we think that the condition (3) is better in this paper. Therefore, we should consider what kind of impulsive condition is suitable for prescribed variable exponent mean curvature problems, it is a difference between this paper with the usual Laplacian impulsive problems.
Simultaneously, system (1) is a kind of mean curvature system. This kind of problems has attracted more and more attentions recently (see [20]–[28]). In [19], the authors generalized the usual mean curvature systems to variable exponent mean curvature systems, and discuss the existence of solutions of (1) with periodic-like boundary value condition (without impulsive conditions). In [29], the authors dealt with the existence of solutions and nonnegative solutions of (1)-(3) with initial boundary value condition. This paper deals with the existence of solutions of (1)-(3) with periodic-like boundary value condition, Neumann boundary value condition, or Dirichlet boundary value condition, respectively. This paper was motivated by [19], [29]. Similar to [19], [29], the proof of our main result is based upon the Leray-Schauder degree, but this paper is more difficult than [19], [29].
System (1) is also a kind of variable exponent equations. The variable exponent equations arise from the study of nonlinear elasticity, electrorheological fluids and image restoration, etc. We refer readers to [30]–[33] for detailed application backgrounds. There are many results on this kind of problems [30]–[48]. Many results show that problems with variable exponent growth conditions are more complex than those with constant exponent growth conditions. For instance we have the following.
-
(a)
If is a bounded domain, the infimum of the eigenvalue of -Laplacian is zero in general, and the -Laplacian does not have the so-called first eigenfunction (see [39]); but the fact that is very important in the study of p-Laplacian problems, and the first eigenfunction of the p-Laplacian was used to construct subsolutions of a p-Laplacian problem successfully (see [49]).
-
(b)
For variable exponent function , the norm is not invariant for the translational coordinate transformation, i.e. under the usual Luxemburg norm , we have in general (see [50]).
-
(c)
In [51], the author generalized the Picone identities for half-linear elliptic operators with -Laplacians and applications to Sturmian comparison theory, but the formula is different from the constant exponent case.
In this paper, we investigate the existence of solutions for the prescribed variable exponent mean curvature impulsive differential system boundary value problems, the proof of our main result is based upon the Leray-Schauder degree.
This paper is divided into four sections; in the second section, we will discuss the existence of solutions of variable exponent mean curvature impulsive system periodic-like boundary value problems. In the third section, we will discuss the existence of solutions of variable exponent mean curvature impulsive system Dirichlet boundary value problems. Finally, in the fourth section, we will discuss the existence of solutions of variable exponent mean curvature impulsive system Neumann boundary value problems.
2 Periodic-like boundary value problems
In this section, we will discuss the existence of solutions of variable exponent mean curvature impulsive system periodic-like boundary value problems, i.e. the existence of solutions of (1)-(4). In order to do that, we give the following notations and basic assumptions:
For any , will denote the j th component of v; the inner product in will be denoted by ; will denote the absolute value and the Euclidean norm on . Denote , , , , , where , . Denote the interior of , . Let ; . For any , we denote . Obviously, is a Banach space with the norm , is a Banach space with the norm . In the following, and will be simply denoted by PC and , respectively. Denote , and the norm in is .
Let , the function is assumed to be Caratheodory, and by this we mean:
-
(i)
for almost every the function is continuous;
-
(ii)
for each the function is measurable on J;
-
(iii)
for each there is a such that, for almost every and every with , , one has .
We say a function is a solution of (1) if with absolutely continuous on , , which satisfies (1) a.e. on J.
In fact, the above notations and basic assumptions will be used in throughout the paper.
2.1 Preliminary
In this subsection, we will make some preparations.
Lemma 2.1
(see [19])
φ is a continuous function and satisfies the following.
-
(i)
For any , is strictly monotone, i.e.
-
(ii)
For any fixed , is a homeomorphism from to
For any , we denote by the inverse operator of , then
Let us now consider the following simple problem:
with the following impulsive boundary value conditions:
where , ; and satisfies .
Denote
Let , then . If u is a solution of (7) with (8), by integrating (7) from 0 to t, we find that
Denote . Define operator F: as
From (9) and (ii) in Lemma 2.1, we can see that
Denote
By (9), we have
the boundary value conditions imply that
Denote , . It is easy to see that ρ is dependent on a, b and h. For fixed , , we define
Denote
If (7) with (8) has a solution in , we must have .
Denote with the norm
then is a Banach space.
Denote
Lemma 2.2
The functionhas the following properties:
(10):For any fixed, the equation
has a unique solution.
(20):For any fixed, , contains the open ball, and then one defines a mapping.
(30):The function, defined in (20), is continuous and bounded.
(40):For any fixed, , the setis open and unbounded in.
Proof
(10) From the definition of , for any fixed , , the equation has at least one solution. Since is strictly monotone, we can see that is strictly monotone. Thus, the equation has a unique solution .
(20) Let , then . Denote
Obviously, for any , for any , is well defined, i.e.,
Observe that and for any .
Since for any , we have
This means that for any .
For any fixed , let us consider
It is easy to see that has no solution on ∂A for any . According to the homotopy invariance property of the Brouwer degree, we can see that possesses a solution in A. Thus , and one then defines a mapping .
(30) For any , from the definition of , we have
Since , we have
Thus the mapping ρ is bounded.
Now, let us prove the continuity of ρ. Let is a convergent sequence in , and as , where . Since is a bounded sequence, it contains a convergent subsequence . We may assume that as .
If is well defined, i.e., . Since , letting , we have . Combining with (10), we get . This means that ρ is continuous.
It only remains to prove that is well defined.
Denote by an open ball which centered at such that is well defined on . According to Lemma 2.1, we have
It is easy to see that there exists a neighborhood of in , is a neighborhood of , is a neighborhood of , such that, for each , is well defined on . In addition, the mapping
is easily seen to be continuous in . Thus, there exists a neighborhood of such that
For any fixed , let us consider
Obviously, has no solution on for any .
For any , according to the homotopy invariance property of the Brouwer degree, we conclude that the equation has its (unique) solution on . Since as , we have when j is large enough, and . Since as , . This means that is well defined.
(40) Let , then . Denote . For any , , and , we can see the following function is well defined:
If , then there exists some such that . Without loss of generality, we may assume that . Since and , we have
Denote . Obviously, , .
Since , the j th component of satisfies , . Notice that , then we can see that the j th component of satisfies
Consider . Since and , we have . Hence and have the same sign, then has no solution on for any . Thus
Thus has a solution on . Therefore , This means that is unbounded.
Similar to the proof of (30), for any , there exists a neighborhood of , and a neighborhood of such that is well defined for any and , which satisfies the requirement that, for any , has a solution . This means that the set is open in .
This completes the proof. □
We continue now with our argument previous to Lemma 2.2. Let us define
We can split as , where is the N-dimensional subspace of constant mappings. The operator Q is a continuous projection from onto . Let us consider the subset of which is given by
and define the nonlinear operator , as
We say is a closed equi-integrable set in , if there exists , such that, for any ,
Lemma 2.3
If, then the operatoris continuous and sends closed equi-integrable subsets ofinto relatively compact sets in.
Proof
It is easy to check that , . Since
it is easy to check that is a continuous operator from to .
Let now be a closed equi-integrable set in , then there exists , such that, for any ,
We want to show that is a compact set.
Let is a sequence in , then there exists a sequence such that . For any , we have
Hence the sequence is uniformly bounded and equi-continuous. By the Ascoli-Arzela theorem, there exists a subsequence of (which we rename the same) is convergent in PC. According to the bounded continuity of the operator ρ, we can choose a subsequence of (which we still denote ) which is convergent in PC, then the sequence
is convergent according to the norm in PC, by which, combined with the continuous of , we can see
is convergent according to the norm in PC.
Since
according to the continuity of , we can see is convergent in PC. Thus we conclude that is convergent in . This completes the proof. □
We denote by the Nemytskii operator associated to f defined by
Denote
and
where , , ρ is a solution of .
We assume
(H1): and , .
Lemma 2.4
If (H1) is satisfied, then u is a solution of (1)-(4) if and only if u is a solution of the following abstract operator equation:
Proof
From (4), we have
thus .
From (14), we have
where ρ satisfies
Hence u is a solution of (13).
then
thus .
By the definition of the mapping ρ, we have
then . From (13) and (15), we also have
From (16), we find that (3) is satisfied. Since , we have
Hence u is a solution of (1)-(4). This completes the proof. □
2.2 Existence of solutions
In this subsection, we will apply the Leray-Schauder degree to deal with the existence of solutions of (1)-(4).
Theorem 2.5
If (H1) is satisfied, is open bounded such that the following conditions hold:
(10):For any, the mappingbelongs to.
(20):For each, the problem
has no solution on ∂ Ω.
(30):The equation
has no solution on.
(40):The Brouwer degree.
Then (1)-(4) has a solution on.
Proof
Denote
Let us consider the following impulsive equation:
For any , observe that, if u is a solution to (17) or u is a solution to (20), we have necessarily
This means that (17) and (20) have the same solutions for .
We denote defined by
where is defined by (12). Denote
where is the solution of .
Obviously
From (10), we can see that , , . Combining (H1) and Lemma 2.2, we can see that there exists only one such that
Let
then the fixed point of is a solution for (1)-(4). Also problem (20) can be written in the equivalent form
Since f is Caratheodory, it is easy to see that is continuous and sends bounded sets into equi-integrable sets. According to Lemma 2.3, we can conclude that is compact continuous on . From Lemma 2.4, we can see that the problem (1)-(4) is equivalent to . We assume that for , (21) does not have a solution on ∂ Ω, otherwise we complete the proof. Now, from hypothesis (20), it follows that (21) has no solution for . For , (20) is equivalent to the following usual differential equation boundary value problem:
and if u is a solution to this problem, we must have
When , the problem is a usual differential equation boundary value problem. Hence , where is a constant mapping. Since , for any , there exists , such that , hence , and we have . Thus, by (22) we have
which, together with hypothesis (20), implies that . Thus we have proved that (21) has no solution on , and then we find that, for each , the Leray-Schauder degree is well defined. From the homotopy invariant property of that degree, we have
When , we have , and , . Thus
and then all the solutions of belong to . Thus
By the properties of the Leray-Schauder degree, we have
where the function ω is defined in (18) and denotes the Brouwer degree. Since, by hypothesis (40), this last degree is different from zero. Thus (1)-(4) has a solution. This completes the proof. □
In the following, we will give an application of Theorem 2.5.
Assume:
(H2):, where δ is a positive parameter, and
where , and , or , .
(H3):, , and τ satisfies , where and are positive constants.
Obviously, there exists a positive constant ε such that
where .
(H4):, .
(H5):, , , , and , .
(H6): satisfies , for any .
Denote
Obviously, is an open subset of .
Theorem 2.6
If (H1)-(H6) are satisfied, then problem (1)-(4) has at least one solution on, when the positive parameter δ is small enough.
Proof
We denote defined by
where is defined by (12).
Let us consider the following problem:
Obviously, u is a solution of (1)-(4) if and only if u is a solution of the abstract equation (23) when . We only need to prove that the conditions of Theorem 2.5 are satisfied.
(10): When the positive parameter δ is small enough, for any , we can see that the mapping belongs to .
(20): We shall prove that for each the problem
has no solution on .
If it is false, then there exists a , and is a solution of (23). We have
From the boundary value condition (4), we have
Since , there exists an such that .
-
(i)
Suppose that , then . Let , according to (2) and (H4), we have
Since , we get for any . Obviously, has constant sign on J.
The condition (H3) means that τ has constant sign on J. Thus also has constant sign on J. Assume that is positive, then we have
This is a contradiction to (24).
Assume that is negative, from (H4) and (H6), then we have
This is a contradiction to (24).
-
(ii)
Suppose that , then . This implies that for some . Without loss of generality, we may assume that , the discussion of the case of is similar.
(a0) Suppose that . From (H5), we have
Since
we can see that is increasing, and . This is a contradiction to (4).
(b0) Suppose that . Then there exists such that . Denote
We have
According to (H2) and (H4), when the positive parameter δ is small enough, we have
This is a contradiction to (4).
(c0) Suppose that . Similar to the proof of (b0), we can get a contradiction to (4).
Summarizing this argument, for each , problem (23) has no solution on .
(30) From (H2), (H5), and (H6), it is easy to see that
has no solution on .
(40) Let
According to (H6), it is easy to see that and have the same sign for any . Denote
For any , we see that does not have solutions on , then the Brouwer degree
Since (23) does not have solutions on , we have
Since
we have
This completes the proof. □
3 Dirichlet boundary value problems
In this section, we will discuss the existence of solutions of variable exponent mean curvature impulsive system Dirichlet boundary value problems, i.e. the existence of a solution of (1)-(3) and (5).
3.1 Preliminary
Let us now consider the following simple problem:
with the following impulsive boundary value conditions:
where , ; .
If u is a solution of (25) with (26), by integrating (25) from 0 to t, we find that
Denote .
From the definition of φ, we can see that
Denote
By (27), we have
The boundary value conditions imply that
Denote , . It is easy to see that ρ is dependent on a, b, and g. For fixed , , we define
Denote
If (25) with (26) has a solution in , we must have .
Denote with the norm
then W is a Banach space.
Copying the proof of Lemma 2.2, we have the following.
Lemma 3.1
The functionhas the following properties:
(10):For any fixed, , the equation
has a unique solution, where, are defined in (10) and (11).
(20):For any fixed, , contains the open ball, and then defines a mapping.
(30):The function, defined in (20), is continuous and bounded.
(40):For any fixed, , the setis open and unbounded in.
We continue now with our argument previous to Lemma 3.1.
Define the nonlinear operator , as
Similar to the proof of the Lemma 2.3, we have the following.
Lemma 3.2
If, then the operatoris continuous and sends closed equi-integrable subsets ofinto relatively compact sets in.
Denote
and
where, are defined in (19), ρ is a solution of.
Similar to the proof of Lemma 2.4, we have the following.
Lemma 3.3
If (H1) is satisfied, then u is a solution of (1)-(3) and (5), if and only if u is a solution of the following abstract operator equation:
3.2 Existence of solutions
In this subsection, we will apply the Leray-Schauder degree to deal with the existence of solutions of (1)-(3) and (5).
Theorem 3.4
If (H1) is satisfied, is open bounded such that the following conditions hold:
(10):For any, the mappingbelongs to.
(20):For each, the problem
has no solution on ∂ Ω;
(30):.
Then (1)-(3), and (5) have a solution on.
Proof
Denote
where is the solution of , is defined by (12), , are defined in (19).
From (10), we can see that , , . Combining (H1) and Lemma 3.1, we can see that there exists only one such that
Let
then the fixed point of is a solution for (1)-(3) and (5). Also problem (28) can be written in the equivalent form
Since f is Caratheodory, it is easy to see that is continuous and sends bounded sets into equi-integrable sets. According to Lemma 3.2, we can conclude that is compact continuous on . From Lemma 3.3, we can see that problem (1)-(3) and (5) is equivalent to . We assume that for , (29) does not have a solution on ∂ Ω, otherwise we complete the proof. Now, from hypothesis (20), it follows that (29) has no solution for . For , (28) is equivalent to the following usual differential equation boundary value problem:
When , the problem is a usual differential equation boundary value problem. Hence , where is a constant mapping. Since , for any , there exists , such that , hence , we have . Thus, we have
which, together with hypothesis (30), implies that . Thus we have proved that (29) has no solution on , then we see that, for each , the Leray-Schauder degree is well defined. From the condition (30) and the homotopy invariant property of that degree, we have
Thus (1)-(3) and (5) has a solution. This completes the proof. □
In the following, we will give an application of Theorem 3.4.
Assume the following.
(H7):, where δ is a positive parameter, g is defined in (H2).
Denote
Theorem 3.5
If (H1) and (H7) are satisfied, then problem (1)-(3) and (5) has at least one solution on, when the positive parameter δ is small enough.
Proof
Let us consider the following problem:
Obviously, u is a solution of (1)-(3) and (5) if and only if u is a solution of the abstract equation (30) when . We only need to prove that the conditions of Lemma 3.1 are satisfied.
(10) When the positive parameter δ is small enough, for any , we can see that the mapping belongs to .
(20) We shall prove that for each the problem
has no solution on .
If it is false, then there exists a , and is a solution of (30). We have
From the proof of Lemma 2.2, we can see that . Obviously
It is easy to verify that .
This is a contradiction.
Thus, for each , problem (30) has no solution on .
(30) Obviously .
This completes the proof. □
4 Neumann boundary value problems
In this section, we will discuss the existence of solutions of variable exponent mean curvature impulsive system Neumann boundary value problems, i.e. the existence of a solution of (1)-(3) and (6).
4.1 Preliminary
Let us now consider the following simple problem:
with the following impulsive boundary value conditions:
where , ; and it satisfies .
Denote , , and
Let , then . implies .
If u is a solution of (31) with (32), by integrating (31) from 0 to t, we find that
From the definition of φ, we can see that
Denote
By (33), we have
We can split as , where is the N-dimensional subspace of constant mappings. The operator Q is a continuous projection from onto . Let us consider the subset of which is given by
and the nonlinear operator , as
Copying the proof of Lemma 2.3, we have the following.
Lemma 4.1
The operatoris continuous and sends closed equi-integrable subsets ofinto relatively compact sets in.
Denote
whereis defined in (19).
Similar to the proof of Lemma 2.4, we have the following.
Lemma 4.2
u is a solution of (1)-(3) and (6) if and only if u is a solution of the following abstract equation:
where, are defined in (19).
4.2 Existence of solutions
In this subsection, we will apply the Leray-Schauder degree to deal with the existence of solutions for (1)-(3) and (6).
Assume the following.
(H8):.
Theorem 4.3
If (H8) is satisfied, Ω is an open bounded set insuch that the following conditions hold:
(10):For any, the mappingbelongs to.
(20):For each, the problem
has no solution on ∂ Ω.
(30):The equation
has no solution on;
(40):The Brouwer degree.Then (1)-(3) and (6) has a solution on.
Proof
It is similar to the proof of Theorem 2.6, we omit it here. □
In the following, we will give an application of Theorem 4.3.
Assume the following.
(H9):, .
(H10): satisfies , for any , where is defined before (H4).
Denote
Obviously, is an open subset of .
Theorem 4.4
If (H2)-(H4) and (H8)-(H10) are satisfied, then problem (1)-(3) and (6) has at least one solution on, when the positive parameter δ is small enough.
Proof
We denote defined by
Let us consider the problem
where , are defined in (19).
It is easy to see u is a solution of (1)-(3) and (6) if and only if u is a solution of the abstract equation (34) when . We only need to prove that the conditions of Theorem 4.3 are satisfied.
(10): When the positive parameter δ is small enough, for any , we can see that the mapping belongs to .
(20): We shall prove that for each the problem
has no solution on , i.e. (34) has no solution on .
If it is false, then there exists a , and is a solution of (34). Integrating (35) from 0 to t, we have
From the boundary value condition (6), we have
Since , there exists an such that .
-
(i)
Suppose that , .
Similar to the proof of (i) of (20) of Theorem 2.6, we get a contradiction to (36).
-
(ii)
Suppose that , . This implies that for some .
Denote
Since , we have
According to (H10), when the positive parameter δ is small enough, we have
This is a contradiction.
Summarizing this argument, for each , problem (34) has no solution on .
(30) and (40) similar to the proof of (30) and (40) of Theorem 2.6. □
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The authors are highly grateful to the referees for careful reading and comments on this paper. The research is partly supported by the National Natural Science Foundation of China (11326161) and the key projects of Science and Technology Research of the Henan Education Department (14A110011).
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Yin, L., Liu, X., Liu, J. et al. On the prescribed variable exponent mean curvature impulsive system boundary value problems. Bound Value Probl 2014, 139 (2014). https://doi.org/10.1186/s13661-014-0139-x
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DOI: https://doi.org/10.1186/s13661-014-0139-x