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Solvability of Three-Point Boundary Value Problems at Resonance with a -Laplacian on Finite and Infinite Intervals
Three-point boundary value problems of second-order differential equation with a p-Laplacian on finite and infinite intervals are investigated in this paper. By using a new continuation theorem, sufficient conditions are given, under the resonance conditions, to guarantee the existence of solutions to such boundary value problems with the nonlinear term involving in the first-order derivative explicitly.
This paper deals with the three-point boundary value problem of differential equation with a -Laplacian where , , is a constant, , and .
Boundary value problems (BVPs) with a -Laplacian have received much attention mainly due to their important applications in the study of non-Newtonian fluid theory, the turbulent flow of a gas in a porous medium, and so on [1–10]. Many works have been done to discuss the existence of solutions, positive solutions subject to Dirichlet, Sturm-Liouville, or nonlinear boundary value conditions.
In recent years, many authors discussed, solvability of boundary value problems at resonance, especially the multipoint case [3, 11–15]. A boundary value problem of differential equation is said to be at resonance if its corresponding homogeneous one has nontrivial solutions. For (1.1), it is easy to see that the following BVP has solutions . When , they are nontrivial solutions. So, the problem in this paper is a BVP at resonance. In other words, the operator defined by is not invertible, even if the boundary value conditions are added.
For multi-point BVP at resonance without -Laplacians, there have been many existence results available in the references [3, 11–15]. The methods mainly depend on the coincidence theory, especially Mawhin continuation theorem. At most linearly increasing condition is usually adopted to guarantee the existence of solutions, together with other suitable conditions imposed on the nonlinear term.
On the other hand, for BVP at resonance with a -Laplacian, very little work has been done. In fact, when , is not linear with respect to , so Mawhin continuation theorem is not valid for some boundary conditions. In 2004, Ge and Ren [3, 4] established a new continuation theorem to deal with the solvability of abstract equation , where , are nonlinear maps; this theorem extends Mawhin continuation theorem. As an application, the authors discussed the following three-point BVP at resonance where is a constant and is a nonlinear operator. Through some special direct-sum-spaces, they proved that (1.3) has at least one solution under the following condition.
There exists a constant such that for and or for .
The above result naturally prompts one to ponder if it is possible to establish similar existence results for BVP at resonance with a -Laplacian under at most linearly increasing condition and other suitable conditions imposed on the nonlinear term.
Motivated by the works mentioned above, we aim to study the existence of solutions for the three-point BVP (1.1). The methods used in this paper depend on the new Ge-Mawhin’s continuation theorem  and some inequality techniques. To generalize at most linearly increasing condition to BVP at resonance with a -Laplacian, a small modification is added to the new Ge-Mawhin’s continuation theorem. What we obtained in this paper is applicable to BVP of differential equations with nonlinear term involving in the first-order derivative explicitly. Here we note that the techniques used in  are not applicable to such case. An existence result is also established for the BVP at resonance on a half-line, which is new for multi-point BVPs on infinite intervals [16, 17].
The paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we discuss the existence of solutions for BVP (1.1) when is a real constant, which we call the finite case. In Section 4, we establish an existence result for the bounded solutions to BVP (1.1) when , which we call the infinite case. Some explicit examples are also given in the last section to illustrate our main results.
For the convenience of the readers, we provide here some definitions and lemmas which are important in the proof of our main results. Ge-Mawhin’s continuation theorem and the modified one are also stated in this section.
Lemma 2.1. Let , . Then satisfies the properties.(1) is continuous, monotonically increasing, and invertible. Moreover with a real number satisfying ; (2)for any ,
Definition 2.2. Let be an -dimensional Euclidean space with an appropriate norm . A function is called -Carathéodory if and only if (1)for each , is measurable on ; (2)for a.e. , is continuous on ; (3)for each , there exists a nonnegative function with such that
Definition 2.3. Let , be two Banach spaces. A continuous opeartor is called quasi-linear if and only if is a closed subset of and is linearly homeomorphic to , where is an integer.
Let be the complement space of in , that is, . an open and bounded set with the origin .
Definition 2.4. A continuous operator is said to be -compact in if there is a vector subspace with and an operator continuous and compact such that for , where , are projectors such that and , , , .
Theorem 2.5 (Ge-Mawhin’s continuation theorem). Let and be two Banach spaces, an open and bounded set. Suppose is a quasi-linear operator and , is -compact. In addition, if (i), for , , (ii), for , (iii), where . Then the abstract equation has at least one solution in .
According to the usual direct-sum spaces such as those in [3, 5, 7, 11–13], it is difficult (maybe impossible) to define the projector under the at most linearly increasing conditions. We have to weaken the conditions of Ge-Mawhin continuation theorem to resolve such problem.
Definition 2.6. Let be finite dimensional subspace of . is called a semiprojector if and only if is semilinear and idempotent, where is called semilinear provided for all and .
Remark 2.7. Using similar arguments to those in , we can prove that when is a semiprojector, Ge-Mawhin’s continuation theorem still holds.
3. Existence Results for the Finite Case
Consider the Banach spaces endowed with the norm , where and with the usual Lebesgue norm denoted by . Define the operator by where . Then by direct calculations, one has
Obviously, and is close. So the following result holds.
Lemma 3.1. Let be defined as (3.1), then is a quasi-linear operator.
Set the projector and semiprojector by where . Define the operator , by
Lemma 3.2. Let be an open and bounded set. If is a Carathéodory function, is -compact in .
Proof. Choose and define the operator by
Obviously, . Since is a Carathéodory function, we can prove that is continuous and compact for any by the standard theories.
It is easy to verify that (2.3)–(2.5) in Definition 2.3 hold. Besides, for any , So is -compact in .
Theorem 3.3. Let be a Carathéodory function. Suppose that (H1) there exist and Carathéodory functions , such that (H2) there exists such that for all and with , (H3) there exists such that for each and with either or . Then BVP (1.1) has at least one solution provided
Proof. Let , , , , , and be defined as above. Then the solutions of BVPs (1.1) coincide with those of , where . So it is enough to prove that has at least one solution.
Let . If , then . Thus, The continuity of and together with condition (H2) implies that there exists such that . So Noting that , we have
If , choose such that For this , there exists such that Set Noting (3.12)-(3.13), we have where . So And then .
Similarly, if , we can obtain , where Above all, is bounded.
Set , , where is a homeomorphism defined by for any . Next we show that is bounded if the first part of condition (H3) holds. Let , then for some and If , we can obtain that . If , then . Otherwise, which is a contraction. So and is bounded. Similarly, we can obtain that is bounded if the other part of condition (H3) holds.
Let . Then . It is obvious that for each .
Take the homotopy ( by Then for each and , , so by the degree theory Applying Theorem 2.5 together with Remark 2.7, we obtain that has a solution in . So (1.1) is solvable.
Corollary 3.4. Let be a Carathéodory function. Suppose that (H2), (H3) in Theorem 3.3 hold. Suppose further that(H1′) there exist nonnegative functions , such that
Then BVP (1.1) has at least one solution provided
If is a continuous function, we can establish the following existence result.
Theorem 3.5. Let be a continuous function. Suppose that (H1), (H3) in Theorem 3.3 hold. Suppose further that (H2′) there exist , , such that for all with , it holds that Then BVP (1.1) has at least one solution provided
Proof. If such that for some , we have . The continuity of and imply that there exists such that . From (H2'), it holds Therefore, With a similar way to those in Theorem 3.3, we can prove that (1.1) has at least one solution.
4. Existence Results for the Infinite Case
In this section, we consider the BVP (1.1) on a half line. Since the half line is noncompact, the discussions are more complicated than those on finite intervals.
Consider the spaces and defined by with the norms and , respectively, where . By the standard arguments, we can prove that and are both Banach spaces.
Similarly, we can show that is a quasi-linear operator. In order to prove that is -compact in , the following criterion is needed.
Theorem 4.1 (see ). Let . Then is relatively compact if the following conditions hold: (a)all functions from are uniformly bounded; (b)all functions from are equicontinuous on any compact interval of ; (c)all functions from are equiconvergent at infinity, that is, for any given , there exists a such that , for all , .
Lemma 4.2. Let an open and bounded set with . If is a -Carathéodory function, is -compact in .
Proof. Let and define the operator by
We just prove that is what we need. The others are similar and are omitted here.
Firstly, we show that is well defined. Let , . Because is bounded, there exists such that for any , . Noting that is a -Carathéodory function, there exists with such that Therefore where . Meanwhile, for any , we have The continuity of concludes that It is easy to see that exists and . So .
Next, we verify that is continuous. Obviously is continuous in for any . Let , in as . In fact, So by Lebesgue Dominated Convergence theorem and the continuity of , we can obtain
Finally, is compact for any . Let be a bounded set and , then there exists such that for any . Thus we have Those mean that is uniformly bounded and equiconvergent at infinity. Similarly to the proof of (4.3) and (4.6), we can show that is equicontinuous. Through Lemma 4.2, is relatively compact. The proof is complete.
Theorem 4.3. Let be a continuous and -Carathéodory function. Suppose that (H4) there exist functions such that (H5) there exists such that for all satisfying it holds ; (H6) there exist , , such that for all with , it holds (H7) there exists such that for all and with either or . Then BVP (1.1) has at least one solution provided where
Proof. Let , , , , , and be defined as above. Let , . We will prove that is bounded. In fact, for any , , that is,
The continuity of and together with conditions (H5) and (H6) implies that there exists such that
So, we have
If , it holds Therefore concludes that Meanwhile implies that where .
If , we can prove that
So is bounded. With the similar arguments to those in Theorem 3.3, we can complete the proof.
Example 5.1. Consider the three-point BVPs for second-order differential equations
where , with .
Take and . Then, we have By using Theorem 3.5, we can concluded that BVP (5.1) has at least one solution if
Example 5.2. Consider the three-point BVPs for second-order differential equations on a half line
where and continuous on with (or ) on and on .
Denote . Set , . By direct calculations, we obtain that , and . Furthermore, If there exists such that , then . Otherwise which is a contraction.
Obviously . Meanwhile, it is easy to verify that condition (H7) holds. So Theorem 4.3 guarantees that (5.5) has at least one solution.
The paper is supported by the National Natural Science Foundation of China (no. 11101385, 11226133) and by the Fundamental Research Funds for the Central Universities.
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