Abstract
This paper investigates the existence of solutions for a class of variable exponent integrodifferential system with multipoint and integral boundary value condition in half line. When the nonlinearity term satisfies sub-() growth condition or general growth condition, we give the existence of solutions and nonnegative solutions via Leray-Schauder degree at nonresonance, respectively. Moreover, the existence of solutions for the problem at resonance has been discussed.
1. Introduction
In this paper, we consider the existence of solutions for the following variable exponent integrodifferential system with the following nonlinear multipoint and integral boundary value condition where and are linear operators defined by where and are uniformly bounded with exists and is called the weighted -Laplacian; satisfies , for all , and and is nonnegative, and is a positive parameter.
If and , we say the problem is nonresonant; but if and , we say the problem is resonant.
The study of differential equations and variational problems with variable exponent growth conditions is a new and interesting topic. Many results have been obtained on these problems, for example, [1–23]. We refer to [3, 19, 23], for the applied background on these problems. If and (a constant), becomes the well-known -Laplacian. If is a general function, represents a nonhomogeneity and possesses more nonlinearity, and thus is more complicated than . For example, if is a bounded domain, the Rayleigh quotient is zero in general, and only under some special conditions (see [9, 16–18]), but the fact that is very important in the study of -Laplacian problems.
Integral boundary conditions for evolution problems have been applied variously in chemical engineering, thermoelasticity, underground water flow, and population dynamics. There are many papers on the differential equations with integral boundary value conditions, for example, [24–29]. On the existence of solutions for -Laplacian systems boundary value problems, we refer to [2, 4, 7, 8, 10–12, 20–22]. In [20], the present author deals with the existence and asymptotic behavior of solutions for (1.1) with the following linear boundary value conditions when and . But results on the existence of solutions for variable exponent integrodifferential systems with nonlinear boundary value conditions are rare. In this paper, when is a general function, we investigate the existence of solutions and nonnegative solutions for variable exponent integrodifferential systems with nonlinear multipoint and integral boundary value conditions, when the problem is at nonresonance. Moreover, we discuss the existence of solutions for the problem at resonance. Since the nonlinear multipoint boundary value condition is on the derivative of solution , we meet more difficulties than [20].
Let and , the function is assumed to be Caratheodory, by this we mean,(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
Throughout the paper, we denote
The inner product in will be denoted by will denote the absolute value and the Euclidean norm on . Let denote the space of absolutely continuous functions on the interval . For , we set . For any , we denote , and . Spaces and will be equipped with the norm and , respectively. Then, and are Banach spaces. Denote with the norm .
We say a function is a solution of (1.1) if with absolutely continuous on (), which satisfies (1.1) . on .
In this paper, we always use to denote positive constants, if it cannot lead to confusion. Denote
We say satisfies sub-() growth condition, if satisfies where and . We say satisfies general growth condition, if does not satisfy sub-() growth condition.
We will discuss the existence of solutions of (1.1)-(1.2) in the following three cases.Case (i): ;Case (ii): ;Case (iii): .
This paper is divided into five sections. In the second section, we present some preliminary and give the operator equations which have the same solutions of (1.1)-(1.2) in the three cases, respectively. In the third section, we will discuss the existence of solutions of (1.1)-(1.2) when , and we give the existence of nonnegative solutions. In the fourth section, we will discuss the existence of solutions of (1.1)-(1.2) when . In the fifth section, we will discuss the existence of solutions of (1.1)-(1.2) when .
2. Preliminary
For any , denote . Obviously, has the following properties.
Lemma 2.1 (see [7]). is a continuous function and satisfies the following.(i)For any , is strictly monotone, that is (ii)There exists a function as , such that
It is well known that is a homeomorphism from to for any fixed . For any , denote by the inverse operator of , then
It is clear that is continuous and sends bounded sets into bounded sets.
Let us now consider the following problem with boundary value condition (1.2) where .
If is a solution of (2.4) with (1.2), by integrating (2.4) from 0 to , we find that
Define operator as
By solving for in (2.5) and integrating, we find that
In the following, we will give the operator equations which have the same solutions of (1.1)-(1.2) in three cases, respectively.
2.1. Case (i):
We denote in (2.7). It is easy to see that is dependent on , then we find that
The boundary value condition (1.2) implies that
For fixed , we define as
Lemma 2.2. is continuous and sends bounded sets of to bounded sets of . Moreover,
Proof. Since consists of continuous operators, it is continuous. It is easy to see that
This completes the proof.
It is clear that is continuous and sends bounded sets of to bounded sets of , and hence it is a compact continuous mapping.
If is a solution of (2.4) with (1.2), we find that
We denote
We say a set be equi-integrable, if there exists a nonnegative , such that
Lemma 2.3. The operator is continuous and sends equi-integrable sets in to relatively compact sets in .
Proof. It is easy to check that , for all . Since and
it is easy to check that is a continuous operator from to .
Let now be an equi-integrable set in , then there exists a nonnegative , such that
We want to show that is a compact set.
Let be a sequence in , then there exists a sequence such that . Since where , we have
then for any with , we have
which together with (2.17) implies
Hence, the sequence is uniformly bounded. According to the absolute continuity of Lebesgue integral, for any , there exists a such that if , then we have . Thus, (2.20) means that is equicontinuous.
Denote . Obviously, is uniformly bounded and equicontinuous on for . By Ascoli-Arzela Theorem, there exists a subsequence of being convergent in , we may assume in . Since is uniformly bounded and equicontinuous on , there exists a subsequence of such that is convergent in , we may assume in . Obviously, , for any . Repeating the process, we get a subsequence of such that is convergent in , we may assume in . Obviously, for any . Select the diagonal element, we can see that is a subsequence of which satisfies that is convergent in and in . Thus, we get a function which is defined on such that for any , and in .
From (2.20), it is easy to see that for any , exists (we denote the limit by ), and, for any , there exists an integer such that , and then
Since is uniformly bounded, then is bounded. By choosing a subsequence, we may assume that
We claim that . In fact, for any , from (2.21), we have
Since and letting , the above inequality implies
Thus,
Next, we will prove that tend to uniformly.
Suppose . From (2.21) and (2.24), we have
From (2.22), there exists a such that for . Thus, for any ,
Suppose . Since in , there exists a such that
Thus,
This means that tend to uniformly, that is, tend to in .
According to the bounded continuous of the operator , we can choose a subsequence of (which we still denote ) which is convergent in , then is convergent in .
Since
it follows from the continuity of and the integrability of in that is convergent in . Thus, is convergent in . This completes the proof.
Let us define as It is easy to see that is compact continuous.
Throughout the paper, we denote the Nemytskii operator associated to defined by
Lemma 2.4. In the Case (i), is a solution of (1.1)-(1.2) if and only if is a solution of the following abstract equation:
Proof. If is a solution of (1.1)-(1.2), by integrating (1.1) from 0 to , we find that
which implies that
From , we have
So we have
Conversely, if is a solution of (2.33), we have
which implies that
then
From (2.33), we can obtain
It follows from the condition of the mapping that
and then
thus
From (2.40) and (2.44), we obtain (1.2).
From (2.41), we have
Hence, is a solution of (1.1)-(1.2). This completes the proof.
2.2. Case (ii):
We denote in (2.7). It is easy to see that is dependent on , and we have
The boundary value condition (1.2) implies that
For fixed , we define as
Similar to Lemma 2.2, we have the following.
Lemma 2.5. is continuous and sends bounded sets of to bounded sets of . Moreover,
It is clear that is continuous and sends bounded sets of to bounded sets of , and, hence, it is a compact continuous mapping.
Let us define
Lemma 2.6. The operator is continuous and sends equi-integrable sets in to relatively compact sets in .
Proof. Similar to the proof of Lemma 2.3, we omit it here.
Lemma 2.7. In Case (ii), is a solution of (1.1)-(1.2) if and only if is a solution of the following abstract equation:
Proof. If is a solution of (1.1)-(1.2), by integrating (1.1) from 0 to , we find that
which implies that
From and , we obtain
then
Conversely, if is a solution of (2.51), then
Thus, , and we have
then
Thus,
Similar to the proof of Lemma 2.4, we can have
From (2.59) and (2.60), we obtain (1.2).
From (2.51), we have
then
Hence, is a solution of (1.1)-(1.2). This completes the proof.
2.3. Case (iii):
We denote in (2.7). It is easy to see that is dependent on , then we find that
The boundary value condition (1.2) implies that
For fixed , we denote
Throughout the paper, we denote
Lemma 2.8. The function has the following properties.(i)For any fixed , the equation has a unique solution .(ii)The function , defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, where the notation means
Proof. (i) From Lemma 2.1, it is immediate that
and, hence, if (2.67) has a solution, then it is unique.
Let . Since and , if , it is easy to see that there exists an such that the th component of satisfies . Thus, keeps sign on and
Obviously, , then
Thus, the th component of is nonzero and keeps sign, and then we have
Let us consider the equation
It is easy to see that all the solutions of (2.74) belong to . So, we have
and it means the existence of solutions of .
In this way, we define a function , which satisfies
(ii) By the proof of (i), we also obtain sends bounded sets to bounded sets, and
It only remains to prove the continuity of . Let be a convergent sequence in and as . Since is a bounded sequence, then it contains a convergent subsequence . Let as . Since , letting , we have . From (i), we get , it means that is continuous. This completes the proof.
It is clear that is continuous and sends bounded sets of to bounded sets of , and, hence, it is a compact continuous mapping.
Let us define where and satisfies . We denote as
Similar to Lemmas 2.3 and 2.7, we have the following
Lemma 2.9. The operator is continuous and sends equi-integrable sets in to relatively compact sets in .
Lemma 2.10. In Case (iii), is a solution of (1.1)-(1.2) if and only if is a solution of the following abstract equation:
3. Existence of Solutions in Case (i)
In this section, we will apply Leray-Schauder's degree to deal with the existence of solutions for (1.1)-(1.2) when . Moreover, we give the existence of nonnegative solutions.
Theorem 3.1. In Case (i), if satisfies sub-() growth condition, then problem (1.1)-(1.2) has at least a solution for any fixed parameter .
Proof. Denote , where is defined in (2.32). We know that (1.1)-(1.2) has the same solution of
when .
It is easy to see that the operator is compact continuous. According to Lemmas 2.2 and 2.3, we can see that is compact continuous from to .
We claim that all the solutions of (3.1) are uniformly bounded for . In fact, if it is false, we can find a sequence of solutions for (3.1) such that as and for any .
From Lemma 2.2, we have
then we have
From (3.1), we have
then
Denote , from the above inequality, we have
It follows from (2.36) and (3.3) that
For any , we have
which implies that
Thus,
It follows from (3.6) and (3.10) that is bounded.
Thus, we can choose a large enough such that all the solutions of (3.1) belong to . Thus, the Leray-Schauder degree is well defined for each , and
Let
where is defined in (2.10), thus is the unique solution of .
It is easy to see that is a solution of if and only if is a solution of the following system
Obviously, system (I ) possesses a unique solution . Note that , we have
Therefore, (1.1)-(1.2) has at least one solution when . This completes the proof.
Denote
Assume the following(A1)Let positive constant such that , and , where is defined in (3.12) and is defined in (2.10).
It is easy to see that is an open bounded domain in . We have the following.
Theorem 3.2. In the Case (i), assume that satisfies general growth condition and (A1) is satisfied, then the problem (1.1)-(1.2) has at least one solution on when the positive parameter is small enough.
Proof. Denote . According to Lemma 2.4, is a solution of
with (1.2) if and only if is a solution of the following abstract equation
From Lemmas 2.2 and 2.3, we can see that is compact continuous from to . According to Leray-Schauder's degree theory, we only need to prove that(1°) has no solution on for any ,(2°),
then we can conclude that the system (1.1)-(1.2) has a solution on .
(1°) If there exists a and is a solution of (3.15) with (1.2), then satisfies
Since , there exists an such that .
(i) Suppose that , then . On the other hand, for any , we have
This implies that for each .
Note that , then (where ), holding . Since is continuous, when is small enough, from (A1), we have
It is a contradiction to for any .
(ii) Suppose that , then . This implies that
Since , it is easy to see that
Combining (3.17) and (3.21), we have
Since and is Caratheodory, it is easy to see that
thus
From Lemma 2.2, is continuous, then we have
When is small enough, from (A1) and (3.22), we can conclude that
It is a contradiction.
Summarizing this argument, for each , the problem (3.15) with (1.2) has no solution on .
(2°) Since (where is defined in (3.12)) is the unique solution of , and (A1) holds , we can see that the Leray-Schauder degree
This completes the proof.
In the following, we will deal with the existence of nonnegative solutions of (1.1)-(1.2) when . For any , the notation means for any . For any , the notation means and the notation means .
Theorem 3.3. In Case (i), we assume(10),
(20),
(30).
Then, all the solutions of (1.1)-(1.2) are nonnegative.
Proof. If is a solution of (1.1)-(1.2), then
It follows from (2.10), (10), and (30) that
Thus, for any . Holding is decreasing, namely, for any with .
According to the boundary value condition (1.2) and condition (2°), we have
then
Thus, all the solutions of (1.1)-(1.2) are nonnegative. The proof is completed.
Corollary 3.4. In Case (i), we assume(10) with ,(20),
(30),
(40).
Then, we have the following.
(a) On the conditions of Theorem 3.1, then (1.1)-(1.2) has at least a nonnegative solution .
(b) On the conditions of Theorem 3.2, then (1.1)-(1.2) has at least a nonnegative solution .
Proof. (a) Define
where
Denote
then satisfies Caratheodory condition and for any .
Obviously, we have(A2)
where , and .
Then, satisfies sub-() growth condition.
Let us consider the existence of solutions of the following system:
with boundary value condition (1.2). According to Theorem 3.1, (3.35) with (1.2) has at least a solution . From Theorem 3.3, we can see that is nonnegative. Thus, is a nonnegative solution of (1.1)-(1.2).
(b) It is similar to the proof of (a).
This completes the proof.
4. Existence of Solutions in Case (ii)
In this section, we will apply Leray-Schauder's degree to deal with the existence of solutions for (1.1)-(1.2) when .
Theorem 4.1. Assume that is an open bounded set in such that the following conditions hold.(10)For each , the problemhas no solution on .(20)The equationhas no solution on .(30)The Brouwer degree .
Then, problems (1.1)-(1.2) have a solution on .
Proof. For any , it is easy to have problem (4.1) can be written in the equivalent form
where
where .
It is easy to see that the operator is compact continuous. According to Lemmas 2.5 and 2.6, we can conclude that is continuous and compact from to . We assume that for , (4.3) does not have a solution on , otherwise we complete the proof. Now from hypothesis (10), it follows that (4.3) has no solutions for .
For , if is a solution of (4.3), we have
Thus, for , it follows from (4.3) and (4.4) that
it holds , a constant.
Therefore, when , by (4.5),
which together with hypothesis (20) implies that . Thus, we have proved that (4.3) has no solution on , then we get that for each , the Leray-Schauder degree is well defined, and, from the properties of that degree, we have
Now, it is clear that problem
is equivalent to problem (1.1)-(1.2), and (4.8) tells us that problem (4.9) will have a solution if we can show that
Since
then
From (4.4), we have . By the properties of the Leray-Schauder degree, we have
where the function is defined in (4.2) and denotes the Brouwer degree. Since by hypothesis (30), this last degree is different from zero. This completes the proof.
Our next theorem is a consequence of Theorem 4.1. As an application of Theorem 4.1, let us consider the following equation with (1.2): where is Caratheodory, is continuous and Caratheodory, and, for any fixed , if , then , for all , for all .
Theorem 4.2. Assume that the following conditions hold:(10) for all and all , where satisfies ,(20),
(30)for large enough , the equationhas no solution on , where ,(40)the Brouwer degree for large enough , where .
Then, problem (4.14) with (1.2) has at least one solution.
Proof. Denote
At first, we consider the following problem
For any , it is easy to have problem (4.17) can be written in the equivalent form
where and are defined in Theorem 4.1.
We claim that all the solutions of (4.17) are uniformly bounded for . In fact, if it is false, we can find a sequence of solutions for (4.17) such that as and for any .
Since are solutions of (4.17), we have
It follows from Lemma 2.5 that
where means the function which is uniformly convergent to 0 (as ). According to the property of and (4.20), then there exists a positive constant such that
then we have
From (4.18), we have
then
Denote , then
Since , from (4.25), we have
Denote , then and , then possesses a convergent subsequence (which denoted by ), and then there exists a vector such that
Without loss of generality, we assume that . Since , there exist such that
and, then, from (4.25), we have
Since (as ), , and , we have
From (4.26)–(4.30), we have
So we get
where satisfies .
We denote
Since , from (4.19) and (4.32), we have
Since , according to the continuity of , we have
and it is a contradiction to (4.34). This implies that there exists a big enough such that all the solutions of (4.18) when belongs to .
For , if is a solution of (4.18), we have
For , from (4.18), we have
it holds , a constant.
Therefore, when , we have
which together with hypothesis (30) implies that . Thus, we have proved that (4.18) has no solution on , then we get that the Leray-Schauder degree is well defined for each , which implies that
Now it is clear that problem
is equivalent to problem (4.14) with (1.2), and (4.39) tells us that problem (4.40) will have a solution if we can show that
Since
then
By the properties of the Leray-Schauder degree, we have
where the function is defined in (4.15) and denotes the Brouwer degree. By hypothesis (40), this last degree is different from zero. This completes the proof.
Corollary 4.3. If is Caratheodory, which satisfies the conditions of Theorem 4.2, , where are positive functions, and satisfies and are nonnegative, then (4.14) with (1.2) has at least one solution.
Proof. Since then then it is easy to say that has only one solution in , and and, according to Theorem 4.2, we get that (4.14) with (1.2) has at least a solution. This completes the proof.
In the following, let us consider where where are Caratheodory.
We have the following.
Theorem 4.4. We assume that conditions of (10), (30), and (40) of Theorem 4.2 are satisfied, then problem (4.48) with (1.2) has at least one solution when the parameter is small enough.
Proof. Denote
Let us consider the existence of solutions of the following
with (1.2).
We know that (4.51) with (1.2) has the same solution of
where is defined in (2.32).
Obviously, . So . From the proof of Theorem 4.2, we can see that all the solutions of are uniformly bounded, then there exists a large enough such that all the solutions of belong to . Since is compact continuous from to , we have
Since are Caratheodory, we have
Thus,
Obviously, . Therefore,
Thus, when is small enough, from (4.53), we can conclude
Thus has no solution on for any , when is small enough. It means that the Leray-Schauder degree is well defined for any and
From the proof of Theorem 4.2, we can see that the right hand side is nonzero, then (4.48) with (1.2) has at least one solution, when is small enough. This completes the proof.
5. Existence of Solutions in Case (iii)
In this section, we will apply Leray-Schauder's degree to deal with the existence of solutions for (1.1)-(1.2) when .
Theorem 5.1. Assume that is an open bounded set in such that the following conditions hold.(10)For each , the problemhas no solution on .(20)The equationhas no solution on .(30)The Brouwer degree .
Then, problems (1.1)-(1.2) have a solution on .
Proof. Let us consider the following problem:
where and are defined in (2.79) and (2.80), respectively.
For any , observe that, if is a solution to (5.1) or is a solution to (5.3), we have necessarily
It means that (5.1) and (5.3) have the same solutions for .
We denote defined by
where is defined by (2.32). Let
where
where satisfies
and the fixed point of is a solution for (5.3). Also problem (5.3) can be written in the equivalent form
Since is Caratheodory, it is easy to see that is continuous and sends bounded sets into equi-integrable sets. It is easy to see that is compact continuous. According to Lemmas 2.8 and 2.9, we can conclude that is continuous and compact from to . We assume that for , (5.9) does not have a solution on , otherwise we complete the proof. Now, from hypothesis (10), it follows that (5.9) has no solutions for . For , (5.3) is equivalent to the problem
If is a solution to this problem, we must have
Hence,
where is a constant.
It is easy to see that keeps the same sign of . From , we have . From the continuity of , there exist , such that . Hence, , it holds , a constant. Thus, by (5.11), we have
which together with hypothesis (20) implies that . Thus, we have proved that (5.9) has no solution on , then we get that the Leray-Schauder degree is well defined for , and from the properties of that degree, we have
Now it is clear that problem
is equivalent to the problem (1.1)-(1.2), and (5.14) tells us that problem (5.15) will have a solution if we can show that
Since
then
Similar to Lemma 2.8, we have
Thus, , then . From (5.18), we have
By the properties of the Leray-Schauder degree, we have
where the function is defined in (5.2) and denotes the Brouwer degree. By hypothesis (30), this last degree is different from zero. This completes the proof.
Our next theorem is a consequence of Theorem 5.1. As an application of Theorem 5.1. Let us consider the following equation with (1.2) where is Caratheodory, is continuous and Caratheodory, and, for any fixed , if , then , for all , for all .
Theorem 5.2. Assume that the following conditions hold.(10) for all and all , where satisfies ,(20) for uniformly, (30)for large enough , the equation has no solution on , where ,(40)the Brouwer degree for large enough , where .
Then, problem (5.22) with (1.2) has at least one solution.
Proof. Denote
At first, we consider the following problem
According to Lemma 2.10, we know problem (5.25) with (1.2) has the same solution of
Similar to the proof of Theorem 4.2, we obtain that all the solutions of (5.26) are uniformly bounded for . Then, there exists a big enough such that all the solutions of (5.26) belong to , and then we have
If we prove that , then we obtain the existence of solutions for (5.22) with (1.2).
Now, we consider the following equation
where .
We denote defined by
Similar to the proof of Theorem 5.1, we know that (5.28) has the same solution of
Similar to the discussions of Theorem 4.2, we can obtain that all the solutions of (5.28) are uniformly bounded for each . When , similar to the proof of Theorem 5.1, we can prove that (5.28) has no solution on . Then, we get that the Leray-Schauder degree is well defined for , which implies that
Now it is clear that . So . If we prove that , then we obtain the existence of solutions for (5.22) with (1.2). Similar to the proof of Theorem 5.1, we have
According to hypothesis (40), this last degree is different from zero. We obtain that (5.22) with (1.2) has at least one solution. This completes the proof.
Similar to Corollary 4.3 and Theorem 4.4, we have the following.
Corollary 5.3. If is Caratheodory, which satisfies the conditions of Theorem 5.2, , where are positive functions, and satisfies and are nonnegative, then (5.22) with (1.2) has at least one solution.
Theorem 5.4. We assume that conditions of (10), (30), and (40) of Theorem 5.2 are satisfied, then problem (4.48) with (1.2) has at least one solution when the parameter is small enough.
6. Examples
Example 6.1. Consider the following problem
where , , .
Obviously, is Caratheodory, , , the conditions of Corollary 4.3 are satisfied, then (S1) has a solution.
Example 6.2. Consider the following problem:
where is Caratheodory and
Obviously, is Caratheodory, , , the conditions of Theorem 3.2 are satisfied, then (S2) has a solution when is small enough.
Acknowledgments
The authors would like to appreciate the referees for their helpful comments and suggestions. Partly supported by the National Science Foundation of China (10701066 & 10926075 & 10971087) and China Postdoctoral Science Foundation funded project (20090460969) and the Natural Science Foundation of Henan Education Committee (2008-755-65).