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Bifurcation from interval at infinity for discrete eigenvalue problems which are not linearizable
Advances in Difference Equations volume 2014, Article number: 125 (2014)
Abstract
In this paper, we are concerned with the bifurcation from infinity for a class of discrete eigenvalue problems with nonlinear boundary conditions which are not linearizable and give a description of the behavior of the bifurcation components.
MSC:34B10, 34B15.
1 Introduction
For over a decade, there has been significant interest in positive solutions and multiple positive solutions for boundary value problems for finite difference equations; see, for example, [1–11]. Much of this interest has been spurred on by the applicability of the topological method such as the upper and lower solutions technique [2], a number of new fixed point theorems and multiple fixed point theorems [3–7] as applied to certain discrete boundary value problems. Quite recently, RodrÃguez [9] and Ma [10] have given a topological proof and used a bifurcation theorem to study the structure of positive solutions of a difference equation. In this paper, we demonstrate a bifurcation technique that takes advantage of dealing with the discrete eigenvalue problem with nonlinear boundary conditions
which has different asymptotic linearizations at infinity. Here , Δ is a forward difference operator with , are constants, is a parameter, the functions , with on and functions f, g satisfy and .
Since problem (1.1) has different linearizations at infinity, the standard global bifurcation results [12] are not immediately applicable. However, Schmitt [13] and Peitgen and Schmitt [14] obtained a theorem on bifurcation from intervals at infinity. We can use this theorem to discuss the bifurcation from infinity for problem (1.1) and obtain further information on the location and behavior of the bifurcating sets of solutions.
We will make the following assumptions:
-
(H1) , with on , and .
-
(H2) and there exist constants and functions such that
(1.2)(1.3)with
-
(H3) and there exist constants and functions such that
(1.4)with
Let and be the space of all real-valued functions on . Then it is a Banach space with the norm .
Definition 1.1 ([[13], p.450], [14])
A solution set of (1.1) is said to bifurcate from infinity in the interval if
-
(i)
the solutions of (1.1) are a priori bounded in X for and .
-
(ii)
there exists such that and .
Let and be the first eigenvalues of
and
respectively, and be the corresponding eigenfunctions of and , respectively, and be positive and normalized as and .
The main results are the following.
Theorem 1.1 Assume that (H1)-(H3) hold. Then, for any , the interval is a bifurcation interval from infinity of (1.1), and there exists no bifurcation interval from infinity of (1.1) in the set . More precisely, there exists an unbounded, closed and connected component in , consisting of positive solutions of (1.1) and bifurcating from .
Theorem 1.2 Assume that (H1)-(H3) hold.
-
(i)
If
(1.7)
then the component obtained by Theorem 1.1 bifurcates into the region .
-
(ii)
If
(1.8)
then the component obtained by Theorem 1.1 bifurcates into the region .
Remark 1.1 Notice that . Indeed, let and satisfy (1.5) and (1.6), respectively. Multiplying (1.5) by and (1.6) by , summing from to and subtracting, we have that
This together with implies .
2 Bifurcation theorem and reduction to a compact operator equation
Our main tools in the proof of Theorems 1.1-1.2 are topological arguments [15] and the global bifurcation theorem for mappings which are not necessary smooth [13, 14].
Let V be a real Banach space and be completely continuous. Let us consider the equation
Lemma 2.1 ([[13], Theorem 1.3.3])
Let V be a Banach space, be completely continuous and () be such that the solutions of (2.1) are a priori bounded in V for and , i.e., there exists such that
Furthermore, assume that
for large. Then there exists a closed connected set of solutions of (2.1) that is unbounded in , and either
-
(i)
is unbounded in λ direction, or else
-
(ii)
there exists an interval such that and bifurcates from infinity in .
To establish Theorem 1.1, we begin with the reduction of (1.1) to a suitable equation for a compact operator and give some preliminary results.
Let , be the solutions of the initial value problems
and
respectively, here . It is easy to compute and show that
-
(i)
, and u is increasing on ;
-
(ii)
, and v is decreasing on .
Lemma 2.2 Let . Then the linear boundary value problem
has a solution
where
Moreover, if and on I, then on .
Proof It is a consequence of Atici [[5], Section 2], so we omit it. □
Let . Then the linear boundary value problem
has a solution
From the properties of , , it follows that
Define the operator as follows:
By a standard argument of compact operator, it is easy to show that T is a compact operator and it is strong positive, meaning that on for any with the condition that and on I; see [2, 4, 5].
Let be defined as
Then is a linear bounded function.
From Lemma 2.2, let denote the resolvent of the linear boundary value problems
and
respectively. Taking into account (), (), one can repeat the argument of the operator T with some obvious changes. It follows that , are linear mappings of X compactly into X and they are strong positive.
Let , be the solutions of linear boundary value problem (2.4) with , in place of , and , , respectively. Repeating the argument of with some minor changes, we have that is a linear, bounded mapping and
Lemma 2.3 Let (H1)-(H3) hold. If is a bifurcation interval from infinity of the set of nonnegative solutions of (1.1), then we have . Moreover, there exist constants small enough and large enough such that any nonnegative solution u of (1.1) is positive on whenever and .
Proof Let be a nonnegative solution of (1.1) with such that
If
then we have
here is a linear operator and
From conditions (1.3) and (1.4), for any , there exist constants such that
Those imply that both and are bounded. By the compactness of and , it follows from (2.9) that there exist a function and a subsequence of , still denoted by , such that
By (2.7), it follows from (2.10)-(2.11) that
Since ϵ is arbitrary, it follows that
Let , and as . Then, in view of (2.9),
We claim that
Since
it follows from (2.15) that
Moreover, we have
Since and , the strong positivity of ensures that on .
Obviously, satisfies the following boundary value problem:
This combined with , satisfying (1.5) and (1.6) can get that
and
Thus
Since on , (2.12) implies that on for j large enough, and so is from (2.8). This leads to the latter part of assertions of this proposition. □
3 Existence of a bifurcation interval from infinity
This section is devoted to studying the existence of a bifurcation interval from infinity for (1.1). To do this, we associate with (1.1) a nonlinear mapping as follows:
We note that a nonnegative attains (1.1) if and only if .
In this section, we shall apply Lemma 2.1 to show that for any , the interval is a bifurcation interval from infinity for (3.1) and, consequently, is a bifurcation interval from infinity for the nonnegative solutions of (1.1).
In fact, if is a bifurcation interval from infinity for (3.1), then, according to Definition 1.1, we have
-
(i)
the solutions of (3.1) are a priori bounded in X for and .
-
(ii)
there exists a sequence such that and .
Let be any convergent subsequence of , and let
We claim that
Indeed, as in the proof of Lemma 2.3, we have the same conclusion that there exist some and such that
Since
it follows from the strong positivity of and the positivity of that
This together with the strong positivity of implies that
By using (2.18) and (2.19) with obvious changes, it follows that
From (3.5), it follows that on for j large enough and so is from (2.8). Therefore is actually an interval of bifurcation from infinity for (1.1).
In what follows, we shall apply Lemma 2.1 to show that is a bifurcation interval from infinity for (3.1), two lemmas on the nonexistence of solutions will be first shown. Let be defined as
Here is a smooth cut-off function such that
Lemma 3.1 Let (H1)-(H3) hold and be a compact interval with . Then there exists a constant such that
Proof Assume on the contrary that there exist , and such that
The same argument as in the proof of Lemma 2.3 gives a contradiction that . This is a contradiction. The proof of Lemma 3.1 is complete. □
Lemma 3.2 Let (H1)-(H3) hold. Then for any fixed, there exists a constant such that
Proof Assume on the contrary that there exist , , and can be taken such that
Using the same argument as in the proof of Lemma 2.3, we can obtain a subsequence of , still denoted by , which may satisfy that on for all . It follows that
Thus
Moreover, it follows from satisfies (1.5) and (3.12) that
This implies that
Hence assertion (2.10) gives
Now use again for (3.12) the same procedure as in the proof of Lemma 2.3, then we see that some subsequence of , still denoted by , tends to a positive function in X. Take so small that . Then combining (3.14) with (2.15) leads to a contradiction that
The proof of Lemma 3.2 is complete. □
Lemma 3.3 Let and , where is an integer. Assume that (H1)-(H3) hold. Then there exists a constant satisfying as such that for any n large enough,
Proof First we show assertion (3.15). From Lemma 3.1, there exists such that as satisfying
Since and for n large enough from (3.8), by the homotopy invariance and normalization of the topology degree, it follows that for any n large enough,
Next, we show assertion (3.16). We may derive from Lemma 3.2 that
So for any n large enough, by the homotopy invariance, it follows that
 □
Proof of Theorem 1.1 For any fixed with , set , . It is easy to verify that for any fixed n large enough, there exists such that as satisfying that for any , it follows from Lemmas 3.1-3.3 that all conditions of Lemma 2.1 are satisfied. So there exists a closed connected component of solutions (3.1) such that is unbounded in and either
-
(i)
is unbounded in λ direction, or
-
(ii)
there exists an interval such that and bifurcates from infinity in .
By Lemma 3.1, the case (ii) cannot occur. Thus is unbounded bifurcated from in . Furthermore, set for n large enough, we have from Lemma 3.1 that for any closed interval , if , then in X is impossible. So must be bifurcated from . □
Next, we are devoted to the proof of Theorem 1.2, which characterized the bifurcation components of (1.1).
Proof of Theorem 1.2 Under condition (1.7), assume to the contrary that there exists a positive solution of (1.1) with , and
If , then the same argument as in the proof of Lemma 2.3 shows the existence of a positive function such that a subsequence of , still denoted by , tends to in X. It follows that for any j large enough, we have
which implies that
Set
Note that we consider only the cases and . Either the case or the case can be dealt with in a similar way with a minor modification. It follows from (3.18) that, for any , there exists such that for any ,
Thus, for any ,
On the other hand, we have
These two assertions combined, we obtain that for any ,
On the right-hand side, we see from (1.7) that
This means that for any j large enough,
which contradicts the assumption . Case (1.7) has been proved. Case (1.8) can be also verified by the same arguments, and the proof of Theorem 1.2 is complete. □
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Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (No. 11361054, No. 11201378), SRFDP (No. 20126203110004), Gansu provincial National Science Foundation of China (No. 1208RJZA258).
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RM completed the main study, carried out the results of this article and YL drafted the manuscript, checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
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Ma, R., Lu, Y. Bifurcation from interval at infinity for discrete eigenvalue problems which are not linearizable. Adv Differ Equ 2014, 125 (2014). https://doi.org/10.1186/1687-1847-2014-125
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DOI: https://doi.org/10.1186/1687-1847-2014-125