Existence of Solutions and Nonnegative Solutions for Prescribed Variable Exponent Mean Curvature Impulsive System Initialized Boundary Value Problems

Abstract

This paper investigates the existence of solutions and nonnegative solutions for prescribed variable exponent mean curvature impulsive system initialized boundary value problems. The proof of our main result is based upon Leray-Schauder's degree. The sufficient conditions for the existence of solutions and nonnegative solutions have been given.

1. Introduction

The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. On the Laplacian impulsive differential equations boundary value problems, there are many results (see [1–5]). Because of the nonlinearity of -Laplacian, the results about -Laplacian impulsive differential equations boundary value problems are rare (see [6]). In [7, 8], the authors discussed the existence of solutions of -Laplacian system impulsive boundary value problems. Recently, the existence and asymptotic behavior of solutions of curvature equations have been studied extensively (see [9–15]). In [16], the authors generalized the usual mean curvature systems to variable exponent mean curvature systems. In this paper, we consider the existence of solutions and nonnegative solutions for the prescribed variable exponent mean curvature system

(11)

where , with the following impulsive initialized boundary value conditions

(12)

(13)

(14)

where

(15)

are absolutely continuous, where satisfy , is called the variable exponent mean curvature operator, and .

For any , will denote the th component of ; the inner product in will be denoted by will denote the absolute value and the Euclidean norm on . Denote that , , and , , , where ,. Denote that is the interior of ,. Let

(16)

For any , denote that . Obviously, is a Banach space with the norm ,  and is a Banach space with the norm . In the following, and will be simply denoted by and , respectively. Denote that , and the norm in is .

The study of differential equations and variational problems with variable exponent conditions is a new and interesting topic. For the applied background on this kind of problems we refer to [17–19]. Many results have been obtained on this kind of problems, for example, [20–35]. If (a constant) and (a constant), then (1.1) is the well-known mean curvature system. Since problems with variable exponent growth conditions are more complex than those with constant exponent growth conditions, many methods and results for the latter are invalid for the former; for example, if is a bounded domain, the Rayleigh quotient

(17)

is zero in general, and only under some special conditions (see [25]), but the fact that is very important in the study of -Laplacian problems.

In this paper, we investigate the existence of solutions for the prescribed variable exponent mean curvature impulsive differential system initialized boundary value problems; the proof of our main result is based upon Leray-Schauder's degree. This paper was motivated by [6, 13, 36].

Let , then the function is assumed to be Caratheodory; by this we mean that

(i)for almost every the function is continuous,

(ii)for each the function is measurable on ,

(iii)for each there is a such that, for almost every and every with , , one has

(18)

We say a function is a solution of (1.1) if with absolutely continuous on ,  , which satisfies (1.1) on .

This paper is divided into three sections; in the second section, we present some preliminary. Finally, in the third section, we give the existence of solutions and nonnegative solutions for system (1.1)–(1.4).

2. Preliminary

In this section, we will do some preparation.

Lemma 2.1 (see [16]).

is a continuous function and satisfies the following.

(i)For any is strictly monotone, that is,

(21)

(ii)For any fixed , is a homeomorphism from to

(22)

For any , denote by the inverse operator of , then

(23)

Let one now consider the following simple problem:

(24)

with the following impulsive boundary value conditions:

(25)

where ;.

Obviously, implies that If is a solution of (2.4) with (2.5), by integrating (2.4) from to , then one finds that

(26)

Define , and operator as

(27)

From the definition of , one can see that

(28)

Denote that

(29)

By (2.6), one has

(210)

Denote that with the norm

(211)

then is a Banach space.

Let one define the nonlinear operator as

(212)

Lemma 2.2.

The operator is continuous and sends closed equiintegrable subsets of into relatively compact sets in .

Proof.

It is easy to check that , for all . Since

(213)

it is easy to check that is a continuous operator from to .

Let now be a closed equiintegrable set in , then there exists , such that, for any ,

(214)

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 that

(215)

Hence the sequence is uniformly bounded and equicontinuous. By Ascoli-Arzela theorem, there exists a subsequence of (which we rename the same) that is convergent in . Then the sequence

(216)

is convergent according to the norm in . Since

(217)

according to the continuity of , we can see that is convergent in . Thus we conclude that is convergent in . This completes the proof.

We denote that is the Nemytski operator associated to defined by

(218)

Define as

(219)

where .

Lemma 2.3.

is a solution of (1.1)–(1.4) if and only if is a solution of the following abstract equation:

(220)

where , .

Proof.

• (i) If is a solution of (1.1)–(1.4), since implies that , by integrating (1.1) from to , then we find that

  

  (221)

Thus

(222)

Hence is a solution of (2.20).

• (ii) If is a solution of (2.20), then it is easy to see that (1.2) is satisfied. Let , then we have

  

  (223)

From (2.20) we also have

(224)

(225)

From (2.24), we can see that (1.3) is satisfied. Let , from (2.24), then we have

(226)

Since we must have ; thus,

(227)

Hence is a solution of (1.1)–(1.4). This completes the proof.

3. Main Results and Proofs

In this section, we will apply Leray-Schauder's degree to deal with the existence of solutions for (1.1)–(1.4).

We assume that

(H) for all

Theorem 3.1.

If (H) is satisfied, then is an open bounded set in such that the following conditions hold.

(1) For any the mapping belongs to .

(2) For each , the problem

(31)

has no solution on .

Then (1.1)–(1.4) has at least one solution on .

Proof.

Denote that

(32)

For any , define as

(33)

Denote that

We know that (1.1)–(1.4) has the same solutions of

(34)

Since is Caratheodory, it is easy to see that is continuous and sends bounded sets into equiintegrable sets. According to Lemma 2.2, we can conclude that is compact continuous on for any . We assume that for (3.4) does not have a solution on ; otherwise, we complete the proof. Now from hypothesis (2), it follows that (3.4) has no solution for ,.

When , (3.1) is equivalent to the following usual problem:

(35)

where obviously is the unique solution to this problem.

Since , we have proved that (3.4) has no solution ,on , then we get that, for each , Leray-Schauder's degree is well defined. From the homotopy invariant property of that degree, we have

(36)

This completes the proof.

In the following, we will give an application of Theorem 3.1.

Denote that and

(37)

Obviously, is an open subset of .

Assume that

(H), where is a positive parameter, and is Caratheodory.

(H),   for all.

Theorem 3.2.

If are satisfied, then problem (1.1)–(1.4) has at least one solution on , when positive parameter is small enough.

Proof.

Let one consider the problem

(38)

where is defined in (3.3).

Obviously, is a solution of (1.1)–(1.4) if and only if is a solution of the abstract equation (3.8) when . We only need to prove that the conditions of Theorem 3.1 are satisfied.

(1) When positive parameter is small enough, for any , we can see that the mapping belongs to .

(2) We shall prove that for each the problem

(39)

has no solution on .

If it is false, then there exists a and is a solution of (3.8). Then there exists an such that .

• (i) Suppose that , then . For any since according to (H) and (1.2), we have

  

  (310)

It is a contradiction.

• (ii) Suppose that , . This implies that for some .

Denote that

(311)

Since we have

(312)

According to (H)-(H), when positive parameter is small enough, we have

(313)

It is a contradiction.

Summarizing this argument, for each the problem (3.8) with (1.4) has no solution on .

Since and is the unique solution of , then the Leray-Schauder's degree

(314)

This completes the proof.

In the following, we will discuss the existence of nonnegative solutions of (1.1)–(1.4). For any , the notation means that for every .

Assume the following

(H) where is a positive parameter, and

(315)

where , and for all .

(H)and for all ,.

(H).

(H) satisfies for all .

Theorem 3.3.

If HH are satisfied, then the problem (1.1)–(1.4) has a nonnegative solution, when positive parameter is small enough.

Proof.

From Theorem 3.2, we can get the existence of solutions of (1.1)–(1.4). If is a solution of (1.1)–(1.4), according to (1.4) and (H), then we have

(316)

Obviously

(317)

When , we have

(318)

then we can see that there exists a such that when . Thus for any . Thus is increasing in , that is, for any with . Since , it is easy to see that for any . From (3.17) and (H), we can easily see that

(319)

From (H), we can see that

(320)

Similarly, we can see that

(321)

Repeating the step, we can see that

(322)

Hence is nonnegative. This completes the proof.