Bifurcation from interval at infinity for discrete eigenvalue problems which are not linearizable
© Ma and Lu; licensee Springer. 2014
Received: 3 October 2013
Accepted: 4 April 2014
Published: 6 May 2014
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.
Keywordsdifference equation nonlinear boundary condition bifurcation from interval positive solutions
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  are not immediately applicable. However, Schmitt  and Peitgen and Schmitt  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 .
the solutions of (1.1) are a priori bounded in X for and .
there exists such that 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 .
then the component obtained by Theorem 1.1 bifurcates into the region .
This together with implies .
2 Bifurcation theorem and reduction to a compact operator equation
Lemma 2.1 ([, Theorem 1.3.3])
is unbounded in λ direction, or else
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.
, and u is increasing on ;
, and v is decreasing on .
Moreover, if and on I, then on .
Proof It is a consequence of Atici [, Section 2], so we omit it. □
Then is a linear bounded function.
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.
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 .
Since and , the strong positivity of ensures that on .
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
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).
the solutions of (3.1) are a priori bounded in X for and .
there exists a sequence such that and .
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).
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. □
The proof of Lemma 3.2 is complete. □
is unbounded in λ direction, or
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).
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. □
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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