- Research Article
- Open Access
© Rong Dong et al. 2010
- Received: 5 October 2010
- Accepted: 15 December 2010
- Published: 21 December 2010
By the Leray-Schauder's degree, the existence of solutions for a weighted -Laplacian impulsive integro-differential system with multi-point and integral boundary value conditions is considered. The sufficient results for the existence are given under the resonance and nonresonance cases, respectively. Moreover, we get the existence of nonnegative solutions at nonresonance.
- Operator Equation
- Integral Boundary
- Nonnegative Solution
- Integral Boundary Condition
- Coincidence Degree
In recent years, there has been an increasing interest in the study of differential equations with nonstandard -growth conditions. These problems have many interesting applications (see [1–4]). Many results have been obtained on these kinds of problems, for example [5–17]. If (a constant), (1.1)–(1.4) becomes the well known -Laplacian problem. If is a general function, one can see easily that in general, while , so represents a non-homogeneity and possesses more nonlinearity, thus is more complicated than . For example, we have the following.
(a)In general, the infimum of eigenvalues for the -Laplacian Dirichlet problems is zero, and only under some special conditions (see ). When ( ) is an interval, the results in  show that if and only if is monotone. But the property of is very important in the study of -Laplacian problems, for example, in , the authors use this property to deal with the existence of solutions.
(b)If and (a constant) and , then is concave, this property is used extensively in the study of one-dimensional -Laplacian problems (see ), but it is invalid for . It is another difference between and .
Recently, there are many works devoted to the existence of solutions to the Laplacian impulsive differential equation boundary value problems, for example [20–28]. Many methods had been applied to deal with these problems, for example sub-super-solution method, fixed point theorem, monotone iterative method, coincidence degree, variational principles (see ), and so forth. Because of the nonlinearity of , results about the existence of solutions for -Laplacian impulsive differential equation boundary value problems are rare (see ). In , using coincidence degree method, the present author investigate the existence of solutions for -Laplacian impulsive differential equation with multipoint boundary value conditions. Integral boundary conditions for evolution problems have various applications in chemical engineering, thermoelasticity, underground water flow and population dynamics, there are many papers on the differential equations with integral boundary value problems, for example, [32–35].
where , the impulsive condition (1.8) is called linear impulsive condition (LI for short), and (1.3) is called nonlinear impulsive condition (NLI for short). In generaly, -Laplacian impulsive problems have two kinds of impulsive conditions, that is, LI and NLI.
This paper is organized as four sections. In Section 2, we present some preliminary and give the operator equation which has the same solutions of (1.1)–(1.4). In Section 3, we give the existence of solutions and nonnegative solutions for system (1.1)–(1.4) at nonresonance. Finally, in Section 4, we give the existence of solutions for system (1.1)–(1.4) at resonance.
Lemma 2.1 (see ).
We will discuss (2.4) with (1.4) in the cases of resonance and nonresonance, respectively.
2.1. The Case of Nonresonance
Hence the sequence is uniformly bounded and equicontinuous. By Ascoli-Arzela theorem, there exists a subsequence of (which we rename the same) which is convergent in . According to the bounded continuous of the operator , we can choose a subsequence of (which we still denote by ) which is convergent in , then is convergent in .
It is easy to see from (i) and Lemma 2.2.
This completes the proof.
It follows from (2.21) that (1.3) is satisfied.
It follows from (2.22) and (2.23) that (1.4) is satisfied.
2.2. The Case of Resonance
and hence, if (2.27) has a solution, then it is unique.
It only remains to prove the continuity of . Let is a convergent sequence in and , as . Since is a bounded sequence, it contains a convergent subsequence . Suppose as . Since , letting , we have , which together with (i) implies , it means is continuous. This completes the proof.
Similar to the proof of Lemma 2.3, we have the following lemma.
In this section, we will apply Leray-Schauder's degree to deal with the existence of solutions and nonnegative solutions for system (1.1)–(1.4) at nonresonance.
then problem (1.1)−(1.4) has at least one solution.
which implies that (1.1)–(1.4) has at least one solution. This completes the proof.
There are two cases.
Thus (3.16) is valid.
Thus (3.16) is valid. Thus problem (1.1) with (1.2), (1.4), and (1.8) has at least one solution. This completes the proof.
We have the following.
As it is similar to the proof of Theorems 3.2 and 3.3, we omit it here.
Then every solution of (1.1)–(1.4) is nonnegative.
Thus every solution of (1.1)–(1.4) is nonnegative. The proof is completed.
Under the conditions of Theorem 3.1, we also assume
Then (1.1)–(1.4) has a nonnegative solution.
It is not hard to check that
In the following, we will consider the existence of solutions for system (1.1)–(1.4) at resonance.
Suppose that the following conditions hold
Then problem (4.18) with (1.2), (1.3), and (1.4) has at least one solution.
If is Caratheodory, conditions (20), (30) and (40) of Theorem 4.2 are satisfied, condition (30) of Corollary 3.6 is also satisfied, , where are positive functions satisfying ; then (4.18) with (1.2), (1.3), and (1.4) has at least one solution.
According to Theorem 4.2, we get that (4.18) with (1.2), (1.3), and (1.4) has at least a solution. This completes the proof.
From Theorem 4.2, similar to the proof of Theorem 3.3, we have the following.
then problem (4.18) with (1.2), (1.3), and (1.8) has at least one solution.
Similar to the proof of Theorem 3.2, the condition (40) of Theorem 4.2 is satisfied. Thus problem (4.18) with (1.2), (1.3) and (1.8) has at least a solution.
Similar to the proof of Theorem 3.2 and Corollary 4.3, we have the following.
If is Caratheodory, (4.45), (4.46) and conditions (20) and (30) of Theorem 4.2 are satisfied, condition (30) of Corollary 3.6 is also satisfied, , where are positive functions satisfying ; then (4.43) with (1.2), (1.3), and (1.8) has at least one solution when the parameter is small enough.
This paper is partly supported by the National Science Foundation of China (10701066, 10926075, and 10971087), China Postdoctoral Science Foundation funded project (20090460969), and the Natural Science Foundation of Henan Education Committee (2008-755-65).
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