Unconditional convergence of difference equations

  • Daniel Franco1 and

    Affiliated with

    • Juan Peran1Email author

      Affiliated with

      Boundary Value Problems20132013:63

      DOI: 10.1186/1687-2770-2013-63

      Received: 19 December 2012

      Accepted: 20 March 2013

      Published: 28 March 2013

      Abstract

      We put forward the notion of unconditional convergence to an equilibrium of a difference equation. Roughly speaking, it means that can be constructed a wide family of higher order difference equations, which inherit the asymptotic behavior of the original difference equation. We present a sufficient condition for guaranteeing that a second-order difference equation possesses an unconditional stable attractor. Finally, we show how our results can be applied to two families of difference equations recently considered in the literature.

      MSC:39A11.

      Keywords

      difference equations global asymptotic stability

      1 Introduction

      It is somewhat frequent that the global asymptotic stability of a family of difference equations can be extended to some higher-order ones (see, for example, [14]). Consider the following simple example. If φ is the map φ ( x , y ) = 1 + ( a x / y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq1_HTML.gif, the sequence y n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq2_HTML.gif defined by y n = φ ( x , y n 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq3_HTML.gif, that is,
      y n = 1 + a x y n 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equa_HTML.gif
      with y 1 , a , x > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq4_HTML.gif, converges to F φ ( x ) = ( 1 + 1 + 4 a x ) / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq5_HTML.gif for any y 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq6_HTML.gif. Observe that F φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif is the function satisfying φ ( x , F φ ( x ) ) = F φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq8_HTML.gif. Obviously, the second-order difference equation
      y n = 1 + a x y n 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equb_HTML.gif
      also converges to F φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq9_HTML.gif for any y 1 , y 2 , a , x > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq10_HTML.gif. Let us continue to add complexity, by considering the second-order difference equations
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equ1_HTML.gif
      (1)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equ2_HTML.gif
      (2)

      For all y 1 , y 2 , a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq11_HTML.gif, the sequence defined by Equation (1) converges to the unique fixed point μ φ = a + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq12_HTML.gif of the function F φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif. However, the behavior of Equation (2) depends on the parameter a:

      • For a 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq13_HTML.gif, the odd and even index terms converge respectively to some limits, μ 1 [ 1 , + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq14_HTML.gif and μ 1 / ( μ 1 1 ) [ 1 , + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq15_HTML.gif, where μ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq16_HTML.gif may depend on y 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq6_HTML.gif, y 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq17_HTML.gif (for a = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq18_HTML.gif).

      • For 0 < a < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq19_HTML.gif, it converges to μ φ = a + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq12_HTML.gif, whatever the choice of y 1 , y 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq20_HTML.gif one makes.

      No sophisticated tools are needed to reach those conclusions: It suffices to note that the set
      A = { n : ( y n + 3 y n + 1 ) ( y n + 2 y n ) 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equc_HTML.gif
      must be either finite or equal to ℕ. As the sequences y 2 n + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq21_HTML.gif and y 2 n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq22_HTML.gif are then both eventually monotone, they converge in [ 1 , + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq23_HTML.gif to some limits, say μ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq16_HTML.gif and μ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq24_HTML.gif, satisfying
      μ 1 = 1 + a μ 1 μ 2 and μ 2 = 1 + a μ 2 μ 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equd_HTML.gif

      Therefore, one of the following statements holds: μ 2 = μ 1 / ( μ 1 1 ) [ 1 , + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq25_HTML.gif, with a = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq18_HTML.gif, or { μ 1 } { μ 2 } { { 1 , + } , { 1 + a } } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq26_HTML.gif.

      If { μ 1 } { μ 2 } = { 1 , + } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq27_HTML.gif, then that of the sequences, y 2 n + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq21_HTML.gif or y 2 n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq22_HTML.gif, which converges to +∞, has to be nondecreasing. Just look at Equation (2) to conclude that a 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq13_HTML.gif whenever { μ 1 } { μ 2 } = { 1 , + } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq27_HTML.gif.

      The case we are interested in is 0 < a < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq19_HTML.gif and we will say that μ φ = 1 + a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq28_HTML.gif is an unconditional attractor for the map φ, that is, we would consider φ ( x , y ) = 1 + ( a x / y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq29_HTML.gif with 0 < a < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq19_HTML.gif to observe that, not only (1) and (2), but all the following recursive sequences converge to μ φ = 1 + a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq28_HTML.gif, whatever the choice of y 1 , , y max { k , m } > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq30_HTML.gif we make:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Eque_HTML.gif

      In this paper, we proceed as follows. The next section is dedicated to notation and a technical result of independent interest. In Section 3, we introduce the main definition and the main result in this paper, unconditional convergence and a sufficient condition for having it in a general framework. We conclude, in Section 4, showing how the later theorem can be applied to provide short proofs for some recent convergence results on two families of difference equations and to improve them.

      2 Preliminaries

      This section is mainly devoted to the notation we employ. In the first part, we establish some operations between subsets of real numbers and we clarify how we identify a function with a multifunction. We noticed that set-valued difference equations are not concerned with us in this paper. The reason for dealing with those set operations and notation is because it allows us to manage unboundedness and singular situations in a homogeneous way. In the second part, we introduce the families of maps Λ m k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq31_HTML.gif (a kind of averages of their variables) that we shall employ in the definition of unconditional convergence. We finish the section with a technical result on monotone sequences converging to the fixed point of a monotone continuous function.

      2.1 Basic notations

      We consider the two points compactification R ¯ = [ , + ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq32_HTML.gif of ℝ endowed with the usual order and compact topology.

      2.1.1 Operations and preorder in 2 R ¯ { } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq33_HTML.gif

      We define the operations ‘+’, ‘−’, ‘⋅’ and ‘/’ in 2 R ¯ { } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq34_HTML.gif by
      A B = { lim sup ( a n b n ) : a n , b n , a n b n R  for all  n N  and  lim a n A , lim b n B } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equf_HTML.gif

      where ∗ stands for ‘+’, ‘−’, ‘⋅’ or ‘/’. We also agree to write A = A = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq35_HTML.gif.

      Remark 1 We introduce the above notation in order to manage unboundedness and singular situations, but we point out that these are natural set-valued extensions for the arithmetic operations. Let X, Y be compact (Hausdorff) spaces, U a dense subset of X and f : U Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq36_HTML.gif. The closure Gr ( f ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq37_HTML.gif of the graph of f in X × Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq38_HTML.gif defines an upper semicontinuous compact-valued map f ¯ : X 2 Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq39_HTML.gif by f ¯ ( x ) = { y Y : ( x , y ) Gr ( f ) ¯ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq40_HTML.gif, that is, by Gr ( f ¯ ) = Gr ( f ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq41_HTML.gif (see [5]). Furthermore, as usual, one writes f ¯ ( A ) = x A f ¯ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq42_HTML.gif for A 2 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq43_HTML.gif, thereby obtaining a map f ¯ : 2 X 2 Y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq44_HTML.gif.

      To extend arithmetic operations, consider X = R ¯ × R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq45_HTML.gif, Y = R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq46_HTML.gif and U = R × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq47_HTML.gif, when f denotes addition, substraction or multiplication, and U = R × ( R { 0 } ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq48_HTML.gif, when f denotes division.

      Also define A B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq49_HTML.gif (respectively A < B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq50_HTML.gif) to be true if and only if A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq51_HTML.gif, B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq52_HTML.gif and a b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq53_HTML.gif (respectively a < b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq54_HTML.gif) for all a A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq55_HTML.gif, b B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq56_HTML.gif. Here A , B 2 R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq57_HTML.gif.

      Notice that both relations ≤ and < are transitive but neither reflexive nor symmetric.

      2.1.2 Canonical injections

      When no confusion is likely to arise, we identify a R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq58_HTML.gif with { a } 2 R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq59_HTML.gif, that is, in the sequel we consider the fixed injection a { a } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq60_HTML.gif of R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq61_HTML.gif into 2 R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq62_HTML.gif and we identify R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq61_HTML.gif with its image. We must point out that, under this convention, when a is expected to be subset of A, we understand ‘ a A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq55_HTML.gif’ as ‘there is b A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq63_HTML.gif with a = { b } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq64_HTML.gif’. For instance, one has 0 ( + ) = R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq65_HTML.gif, 1 / 0 = { , + } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq66_HTML.gif.

      2.1.3 Extension of a function as a multifunction

      Consider a map h : R ¯ m 2 R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq67_HTML.gif and denote by D ( h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq68_HTML.gif the set formed by those x R ¯ m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq69_HTML.gif for which there is b R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq70_HTML.gif with h ( x ) = { b } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq71_HTML.gif.

      If A 2 ( R ¯ m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq72_HTML.gif, then h ( A ) 2 R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq73_HTML.gif is defined to be a A h ( a ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq74_HTML.gif. Also, if B ( 2 R ¯ ) m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq75_HTML.gif, then h ( B ) 2 R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq76_HTML.gif is defined to be h ( B ) = h ( B 1 × × B m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq77_HTML.gif.

      For each function φ : U R ¯ m R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq78_HTML.gif, let φ ˆ : R ¯ m 2 R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq79_HTML.gif be defined by
      φ ˆ ( x ) = { lim sup φ ( x n ) : x n U  for all  n N  and  lim x n = x } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equg_HTML.gif
      It is obvious that φ ˆ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq80_HTML.gif if and only if x is in the closure U ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq81_HTML.gif of U in R ¯ m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq82_HTML.gif. Also notice that
      U D ( φ ˆ ) U ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equh_HTML.gif
      when φ is continuous. In this case, and if no confusion is likely to arise, we agree to denote also by φ the map φ ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq83_HTML.gif. For example, we write
      φ ( 0 ) = [ 1 , 1 ] ; φ ( ) = φ ( + ) = 0 ; D ( φ ) = R ¯ { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equi_HTML.gif

      when U = R { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq84_HTML.gif and φ ( x ) = sin ( 1 / x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq85_HTML.gif.

      2.2 The maps in Λ m k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq31_HTML.gif and Λ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq86_HTML.gif

      As we have announced, the unconditional convergence of a difference equation guarantees that there exists a family of difference equations that inherit its asymptotic behavior. Here, we define the set of functions that we employ to construct that family of difference equations.

      For k , m N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq87_HTML.gif, let Λ m k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq31_HTML.gif be the set formed by the maps λ : R ¯ m R ¯ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq88_HTML.gif such that
      min 1 j m x j λ i ( x ) max 1 j m x j for all  x R ¯ m , 1 i k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equ3_HTML.gif
      (3)
      Notice that λ γ Λ m k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq89_HTML.gif whenever λ Λ r k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq90_HTML.gif, γ Λ m r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq91_HTML.gif. Let Λ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq86_HTML.gif be defined as follows:
      Λ k = m N Λ m k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equj_HTML.gif

      We note that the functions in Λ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq86_HTML.gif satisfy that their behavior is enveloped by the maximum and minimum functions of its variables, which is a common hypothesis in studying higher order nonlinear difference equations.

      Some trivial examples of functions in Λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq92_HTML.gif are:

      • λ ( x 1 , , x m ) = j = 1 m α j x j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq93_HTML.gif, with α j 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq94_HTML.gif for j = 1 , , m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq95_HTML.gif, j = 1 m α j = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq96_HTML.gif.

      An important particular case is α j 0 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq97_HTML.gif, α j = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq98_HTML.gif for j j 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq99_HTML.gif.

      • λ ( x 1 , , x m ) = { j = 1 m x j α j if  min 1 j m x j > 0 , min 1 j m x j if  min 1 j m x j 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq100_HTML.gif with α j 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq94_HTML.gif for j = 1 , , m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq101_HTML.gif, j = 1 m α j = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq102_HTML.gif.

      We refer to this function simply as λ ( x 1 , , x m ) = j = 1 m x j α j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq103_HTML.gif, when it is assumed that λ Λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq104_HTML.gif.

      • λ ( x 1 , , x m ) = max j J x j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq105_HTML.gif, where J is a nonempty subset of { 1 , , m } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq106_HTML.gif.

      • λ ( x 1 , , x m ) = min j J x j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq107_HTML.gif, where J is a nonempty subset of { 1 , , m } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq106_HTML.gif.

      2.3 A technical result

      Assume a < b + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq108_HTML.gif, in the rest of this section. Recall that a continuous non-increasing function F : [ a , b ] [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq109_HTML.gif has a unique fixed point μ [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq110_HTML.gif, that is, { μ } = Fix ( F ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq111_HTML.gif.

      Lemma 1 Let F : [ a , b ] [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq112_HTML.gif be a continuous non-increasing function, { μ } = Fix ( F ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq111_HTML.gif and ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq113_HTML.gif. Define F ( x ) = F ( a ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq114_HTML.gif for x < a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq115_HTML.gif, F ( x ) = F ( b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq116_HTML.gif for x > b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq117_HTML.gif and
      a 0 = a ; b 0 = F ( a 0 ) ; a k = F ( b k 1 + ϵ k ) ; b k = F ( a k ϵ k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equk_HTML.gif

      for k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq118_HTML.gif.

      Then ( a k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq119_HTML.gif and ( b k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq120_HTML.gif are, respectively, a nondecreasing and a nonincreasing sequence in [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq121_HTML.gif. Furthermore, a k μ b k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq122_HTML.gif for all k and { lim a k , μ , lim b k } Fix ( F F ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq123_HTML.gif.

      Proof Since the map F is nonincreasing and taking into account the hypothesis a 0 a 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq124_HTML.gif, we see that ( a k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq119_HTML.gif is a nondecreasing sequence. Assume a k 1 a k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq125_HTML.gif and a k > a k + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq126_HTML.gif to reach a contradiction
      a k > a k + 1 F ( b k 1 + ϵ k ) > F ( b k + ϵ k + 1 ) b k 1 + ϵ k < b k + ϵ k + 1 b k 1 < b k F ( a k 1 ϵ k 1 ) < F ( a k ϵ k ) a k 1 ϵ k 1 > a k ϵ k a k 1 > a k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equl_HTML.gif

      Therefore, ( a k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq119_HTML.gif is a nondecreasing sequence, so by definition, ( b k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq120_HTML.gif is nonincreasing.

      On the other hand, as b 0 = F ( a 0 ) F ( μ ) = μ a 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq127_HTML.gif, we see by induction that a k μ b k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq122_HTML.gif for all k,
      a k = F ( b k 1 + ϵ k ) F ( μ ) = μ = F ( μ ) F ( a k ϵ k ) = b k for  k 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equm_HTML.gif
      Because of the continuity of F, we conclude that
      lim a k = F ( lim b k ) = F ( F ( lim a k ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equn_HTML.gif
      and
      lim b k = F ( lim a k ) = F ( F ( lim b k ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equo_HTML.gif

       □

      Remark 2 Suppose F not to be identically equal to +∞ and let x [ a , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq128_HTML.gif. The map
      ϵ F ( F ( x ) + ϵ ) ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equp_HTML.gif
      is decreasing in the set
      { ϵ 0 : F ( F ( x ) + ϵ ) < + } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equq_HTML.gif

      Unless F ( x ) = b < + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq129_HTML.gif, the map F verifies F ( F ( x ) ) > x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq130_HTML.gif if and only if there exists ϵ 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq131_HTML.gif such that F ( F ( x ) + ϵ ) ϵ > x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq132_HTML.gif for all ϵ [ 0 , ϵ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq133_HTML.gif.

      Therefore, if F ( F ( a ) ) > a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq134_HTML.gif, there exists ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq113_HTML.gif such that F ( F ( a ) + ϵ ) ϵ a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq135_HTML.gif and taking a = a 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq136_HTML.gif
      a F ( F ( a ) + ϵ ) ϵ = a 1 ϵ a k ϵ k μ b k 1 + ϵ k b 0 + ϵ = F ( a ) + ϵ b . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equr_HTML.gif

      As a consequence, ( a k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq119_HTML.gif, ( b k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq120_HTML.gif are well defined, without the need of extending F.

      3 Unconditional convergence to a point

      For a map h : R ¯ k 2 R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq137_HTML.gif, the difference equation
      y n = h ( y n 1 , , y n k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equ4_HTML.gif
      (4)

      is always well defined whatever the initial points y 1 , , y k R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq138_HTML.gif are, even though the y n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq2_HTML.gif are subsets of R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq61_HTML.gif, rather than points.

      A point μ R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq139_HTML.gif is said to be an equilibrium for the map h if h ( μ , , μ ) = { μ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq140_HTML.gif. The equilibrium μ is said to be stable if, for each neighborhood V of μ in R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq141_HTML.gif, there is a neighborhood W of ( μ , , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq142_HTML.gif in D ( h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq68_HTML.gif such that y n V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq143_HTML.gif for all n, whenever ( y 1 , , y k ) W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq144_HTML.gif.

      The equilibrium μ is said to be an attractor in a neighborhood V of μ in R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq141_HTML.gif, if y n R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq145_HTML.gif for all n and y n μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq146_HTML.gif in R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq147_HTML.gif, whenever y n V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq148_HTML.gif for n k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq149_HTML.gif.

      Definition 1 The point μ is said to be an unconditional equilibrium of h (respectively, unconditional stable equilibrium, unconditional attractor in V) if it is an equilibrium (respectively, stable equilibrium, attractor in V) of h λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq150_HTML.gif for all λ Λ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq151_HTML.gif.

      Definition 2 We define the equilibria, stable equilibria, attractors, unconditional equilibria, unconditional stable equilibria and unconditional attractors of a continuous function φ : U R ¯ k R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq152_HTML.gif to be those of φ ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq83_HTML.gif.

      3.1 Sufficient condition for unconditional convergence

      After giving Definitions 1 and 2 we are going to prove a result guaranteeing that a general second order difference equation as in (4) has an unconditional stable attractor.

      Let < c a < b d + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq153_HTML.gif and consider in the sequel a continuous function φ : ( a , b ) × ( c , d ) ( c , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq154_HTML.gif, satisfying the following conditions:

      (H1) φ ( x 1 , y ) < φ ( x 2 , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq155_HTML.gif, whenever a < x 2 < x 1 < b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq156_HTML.gif and c < y < d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq157_HTML.gif.

      (H2) There exists F φ : [ a , b ] [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq158_HTML.gif such that
      F φ ( x ) y φ ( x , y ) y 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equs_HTML.gif

      whenever y ( c , d ) { F φ ( x ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq159_HTML.gif.

      The functions φ ( , y ) : [ a , b ] [ c , d ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq160_HTML.gif and φ ( x , ) : ( c , d ) ( c , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq161_HTML.gif are defined in the obvious way. Notice that φ ( a , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq162_HTML.gif is the limit of a monotone increasing sequence of continuous functions, thus it is lower-semicontinuous, likewise φ ( b , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq163_HTML.gif is an upper-semicontinuous function. Remember that we denote both φ and φ ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq83_HTML.gif by φ.

      The next lemma, which we prove at the end of this section, shows that if (H1) and (H2) holds we can get some information about the behavior and properties of φ and F φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif.

      Lemma 2 Let φ : ( a , b ) × ( c , d ) ( c , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq164_HTML.gif, where < c a < b d + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq153_HTML.gif, be a continuous function satisfying (H1) and (H2). Then the function F φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif in (H2) is unique and it is a continuous nonincreasing map, thus it has a unique fixed point μ φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq165_HTML.gif. Furthermore,
      1. (i)

        a < φ ( x , y ) < b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq166_HTML.gif for all x , y ( a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq167_HTML.gif and a φ ( x , y ) b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq168_HTML.gif for all x , y [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq169_HTML.gif.

         
      2. (ii)

        If x [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq170_HTML.gif, y ( c , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq171_HTML.gif and φ ( x , y ) = y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq172_HTML.gif, then y = F φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq173_HTML.gif.

         
      3. (iii)

        φ ( x , F φ ( x ) ) = F φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq8_HTML.gif for all x [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq170_HTML.gif.

         
      4. (iv)

        F φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif is decreasing in F φ 1 ( ( a , b ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq174_HTML.gif.

         

      We are in conditions of presenting and proving our main result.

      Theorem 1 Let φ : ( a , b ) × ( c , d ) ( c , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq164_HTML.gif, where < c a < b d + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq153_HTML.gif, be a continuous function satisfying (H1) and (H2). If μ φ ( a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq175_HTML.gif and Fix ( F φ F φ ) = Fix ( F φ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq176_HTML.gif, then μ φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq177_HTML.gif is an unconditional stable attractor of φ in ( a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq178_HTML.gif.

      Proof of Theorem 1 Consider λ Λ m 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq179_HTML.gif and denote
      y n = φ λ ( y n 1 , , y n m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equt_HTML.gif

      for some y 1 , , y m ( a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq180_HTML.gif. Notice that y n ( a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq181_HTML.gif for all n, as a consequence of (i) in Lemma 2.

      We are going to prove first that μ φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq165_HTML.gif is a stable equilibrium of φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq182_HTML.gif. By (iii) in Lemma 2, as
      λ ( μ φ , , μ φ ) = ( μ φ , μ φ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equu_HTML.gif

      we see that μ φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq165_HTML.gif is an equilibrium.

      Let ϵ ( 0 , min { μ φ a , b μ φ } ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq183_HTML.gif. Because of the continuity of F φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif, there is a ( μ φ ϵ , μ φ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq184_HTML.gif such that
      b F φ ( a ) ( μ φ , μ φ + ϵ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equv_HTML.gif
      As Fix ( F φ F φ ) = Fix ( F φ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq185_HTML.gif and F φ ( F φ ( a ) ) a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq186_HTML.gif, we have
      F φ ( F φ ( x ) ) > x for all  x [ a , μ φ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equw_HTML.gif
      If x [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq187_HTML.gif, then
      F φ ( x ) F φ ( a ) = b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equx_HTML.gif
      and
      F φ ( x ) F φ ( b ) = F φ ( F φ ( a ) ) > a . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equy_HTML.gif

      Therefore, F φ ( [ a , b ] ) [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq188_HTML.gif.

      By replacing a, b by a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq189_HTML.gif, b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq190_HTML.gif in Lemma 2(i), we see that
      y n ( a , b ) ( μ φ ϵ , μ φ + ϵ ) for all  n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equz_HTML.gif

      whenever y n ( a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq191_HTML.gif for n m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq192_HTML.gif, thus μ φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq165_HTML.gif is an unconditional stable equilibrium of φ.

      Now, if we see that
      lim F φ ( y n ) = μ φ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equaa_HTML.gif
      we are done with the whole proof. Indeed, for each accumulation point y ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq193_HTML.gif of ( y n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq194_HTML.gif, one would have
      F φ ( μ φ ) = μ φ = F φ ( y ¯ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equab_HTML.gif

      because of the continuity of F φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif. As μ φ ( a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq175_HTML.gif, this implies y ¯ = μ φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq195_HTML.gif.

      Therefore, as a consequence of Lemma 1, it suffices to find an increasing sequence n k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq196_HTML.gif of natural numbers such that
      a k F φ ( y n ) b k for all  k 0 , n n k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equac_HTML.gif
      Here, a k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq197_HTML.gif and b k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq198_HTML.gif are defined as in Lemma 1, with a 0 = a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq199_HTML.gif and ϵ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq200_HTML.gif,
      b 0 = F φ ( a ) ; a k = F φ ( b k 1 + 1 k ) ; b k = F φ ( a k 1 k ) for  k 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equad_HTML.gif
      Let n 0 = m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq201_HTML.gif, so that
      a 0 F φ ( y n ) F φ ( a ) = b 0 for  n n 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equae_HTML.gif
      Having in mind that λ Λ m 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq202_HTML.gif satisfies (3), we find n k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq196_HTML.gif from n k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq203_HTML.gif as follows. Denote
      z k = F φ 1 ( b k 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equaf_HTML.gif
      and momentarily assume n > n k 1 + m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq204_HTML.gif and b k 1 < y n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq205_HTML.gif in such a way that
      b k 1 < y n = φ ( λ 1 ( y n 1 , , y n m ) , λ 2 ( y n 1 , , y n m ) ) φ ( min { y n 1 , , y n m } , λ 2 ( y n 1 , , y n m ) ) φ ( z k , λ 2 ( y n 1 , , y n m ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equag_HTML.gif
      which implies
      φ ( z k , λ 2 ( y n 1 , , y n m ) ) λ 2 ( y n 1 , , y n m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equah_HTML.gif
      and then
      y n max { b k 1 , λ 2 ( y n 1 , , y n m ) } max { b k 1 , y n 1 , , y n m } w n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equai_HTML.gif

      for all n > n k 1 + m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq204_HTML.gif.

      As a consequence, the nonincreasing sequence w n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq206_HTML.gif is bounded below by b k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq207_HTML.gif. It cannot be the case that lim w n > b k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq208_HTML.gif, because in such a case there is a subsequence y n j > b k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq209_HTML.gif converging to lim w n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq210_HTML.gif and such that λ 2 ( y n j 1 , , y n j m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq211_HTML.gif converges to a point w lim w n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq212_HTML.gif.

      Since
      y n j φ ( z k , λ 2 ( y n j 1 , , y n j m ) ) w n j , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equaj_HTML.gif
      one has
      φ ( z k , w ) = lim w n > b k 1 = F ( z k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equak_HTML.gif
      and then
      φ ( z k , w ) w > F ( z k ) w . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equal_HTML.gif
      By applying (H2), we see that
      lim w n = φ ( z k , w ) < w , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equam_HTML.gif

      a contradiction.

      Therefore, lim w n = b k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq213_HTML.gif and there exists m k n k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq214_HTML.gif such that
      y n < b k 1 + 1 k for all  n m k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equan_HTML.gif
      that is,
      F φ ( y n ) a k for all  n m k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equao_HTML.gif
      Analogously, we see that there exists n k m k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq215_HTML.gif such that
      F φ ( y n ) b k for all  n n k . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equap_HTML.gif

       □

      Proof of Lemma 2

      • Uniqueness of F φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif: Let y 1 < y 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq216_HTML.gif and x in [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq217_HTML.gif such that
        y i y φ ( x , y ) y 1 > 0 for  i = 1 , 2 , y ( y 1 , y 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equaq_HTML.gif
      Then
      0 < φ ( x , y ) y < 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equar_HTML.gif

      a contradiction.

      • (i): It suffices to prove the first assertion, because [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq121_HTML.gif is a closed set and, by definition,
        φ ( x , y ) = { lim sup φ ( x n , y n ) : x n x , y n y , x n ( a , b ) , y n ( c , d ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equas_HTML.gif

      for all ( x , y ) [ a , b ] × [ c , d ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq218_HTML.gif. Assume now that ( x , y ) ( a , b ) × ( a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq219_HTML.gif. We consider the following three possible situations. If φ ( x , y ) = y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq172_HTML.gif, it is obvious that φ ( x , y ) ( a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq220_HTML.gif.On the other hand, if φ ( x , y ) > y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq221_HTML.gif and x ( a , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq222_HTML.gif, then

      a y < φ ( x , y ) < φ ( x , y ) F φ ( x ) b . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equat_HTML.gif
      Finally, if φ ( x , y ) < y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq223_HTML.gif and x ( x , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq224_HTML.gif, then
      b y > φ ( x , y ) > φ ( x , y ) F φ ( x ) a . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equau_HTML.gif
      • (ii): Suppose y F φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq225_HTML.gif. Since φ ( x , y ) = y R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq226_HTML.gif, then φ ( x , y ) y = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq227_HTML.gif or φ ( x , y ) y = R ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq228_HTML.gif. In any event, it cannot be the case that
        F φ ( x ) y φ ( x , y ) y 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equav_HTML.gif

      which contradicts hypothesis (H2), thus y = F φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq229_HTML.gif.

      • (iii): Since F φ ( [ a , b ] ) [ a , b ] [ c , d ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq230_HTML.gif, it is worth considering the following three cases for each x [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq231_HTML.gif: first, x ( a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq232_HTML.gif, F φ ( x ) ( c , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq233_HTML.gif and then (after probing continuity, monotonicity and statement (iv)), we proceed with the case x { a , b } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq234_HTML.gif, F φ ( x ) ( c , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq233_HTML.gif and finally with x [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq231_HTML.gif, F φ ( x ) { c , d } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq235_HTML.gif.Case x ( a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq232_HTML.gif and F φ ( x ) ( c , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq236_HTML.gif: Since φ ( x , y ) > y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq221_HTML.gif when y < F φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq237_HTML.gif and φ ( x , y ) < y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq223_HTML.gif when y > F φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq238_HTML.gif, we see that
        φ ( x , F φ ( x ) ) = F φ ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equaw_HTML.gif

      because of the continuity of φ ( x , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq239_HTML.gif.

      • Monotonicity and (iv): Suppose

      F φ ( x 1 ) F φ ( x 2 ) for  a x 1 < x 2 b . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equax_HTML.gif
      If y [ F φ ( x 1 ) , F φ ( x 2 ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq240_HTML.gif, then y = a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq241_HTML.gif, y = b http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq242_HTML.gif or y φ ( x 2 , y ) < φ ( x 1 , y ) y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq243_HTML.gif, thus
      [ F φ ( x 1 ) , F φ ( x 2 ) ] { a , b } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equay_HTML.gif
      • Continuity: If x [ a , b ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq244_HTML.gif and
        w I = ( lim inf z x F φ ( z ) , lim sup y x F φ ( y ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equaz_HTML.gif
      then there exist two sequences y n , z n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq245_HTML.gif with
      F φ ( z n ) < w < F φ ( y n ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equba_HTML.gif
      Thus,
      φ ( z n , w ) < w < φ ( y n , w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbb_HTML.gif

      and, by (ii), one has φ ( x , w ) = w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq246_HTML.gif. Since w ( c , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq247_HTML.gif, this would imply F φ ( x ) = w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq248_HTML.gif for all w I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq249_HTML.gif, which is impossible.

      • (iii) Case x { a , b } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq250_HTML.gif and F φ ( x ) ( c , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq236_HTML.gif: Since
        F φ ( t ) φ ( t , F φ ( a ) ) F φ ( a ) for all  t ( a , b ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbc_HTML.gif
      and because of the continuity of F φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq7_HTML.gif, we have
      F φ ( a ) = sup t F φ ( t ) sup t φ ( t , F φ ( a ) ) F φ ( a ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbd_HTML.gif

      but

      sup t φ ( t , F φ ( a ) ) = φ ( a , F φ ( a ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Eqube_HTML.gif
      Analogously, it can be seen that φ ( b , F φ ( b ) ) = F φ ( b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq251_HTML.gif.
      • (iii) Case F φ ( x ) { c , d } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq252_HTML.gif: First assume F φ ( x ) = c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq253_HTML.gif and recall that, by definition,
        φ ( x , c ) = { lim sup φ ( x n , y n ) : x n x , y n c , x n ( a , b ) , y n ( c , d ) } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbf_HTML.gif
      Suppose φ ( x n , y n ) c > c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq254_HTML.gif for all n. Then
      1 F φ ( x n ) y n φ ( x n , y n ) y n F φ ( x n ) y n c y n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbg_HTML.gif

      which implies F φ ( x n ) c > c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq255_HTML.gif, eventually for all n.Since F φ ( x n ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq256_HTML.gif, we reach a contradiction. Therefore,

      φ ( x , F φ ( x ) ) = { c } = { F φ ( x ) } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbh_HTML.gif
      Analogously, we see that φ ( x , F φ ( x ) ) = { d } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq257_HTML.gif when F φ ( x ) = d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq258_HTML.gif. □

      4 Examples and applications

      4.1 The difference equation y n = A + ( y n k y n q ) p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq259_HTML.gif with 0 < p < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq260_HTML.gif

      The paper [6] is devoted to prove that every positive solution to the difference equation
      y n = A + ( y n k y n q ) p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbi_HTML.gif
      converges to the equilibrium A + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq261_HTML.gif, whenever
      A ( 0 , + ) and p ( 0 , min { 1 , ( A + 1 ) / 2 } ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbj_HTML.gif

      Here, k , q { 1 , 2 , 3 , } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq262_HTML.gif are fixed numbers.

      Although paper [6] complements [7], where the case p = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq263_HTML.gif had been considered, it should be noticed that the case p = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq263_HTML.gif, A 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq264_HTML.gif is not dealt with in [6]. Furthermore, we cannot assure the global attractivity in this case.

      The results in [6] can be easily obtained by applying Theorem 1 above. Furthermore, we slightly improve the results in [6] by establishing the unconditional stability of the equilibrium A + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq261_HTML.gif, whenever A ( 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq265_HTML.gif, p ( 0 , min { 1 , ( A + 1 ) / 2 } ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq266_HTML.gif. We may assume without loss of generality that the initial values y 1 , , y m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq267_HTML.gif are greater than A. Here, and in the sequel m = max { k , q } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq268_HTML.gif.

      Let
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbk_HTML.gif
      and
      λ ( x 1 , , x m ) = ( x q , x k ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbl_HTML.gif
      Define F φ ( + ) = A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq269_HTML.gif, consider for the moment a fixed x [ A , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq270_HTML.gif and define F φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq9_HTML.gif to be the unique positive zero of the function f x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq271_HTML.gif given by
      f x ( y ) = φ ( x , y ) y . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbm_HTML.gif

      Notice that f x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq271_HTML.gif is concave, f x ( 0 ) = A > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq272_HTML.gif, and f x ( + ) = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq273_HTML.gif.

      Clearly, F φ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq9_HTML.gif is also the unique zero of the increasing function
      j x ( y ) = φ ( x , y ) F φ ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbn_HTML.gif
      Since
      j x ( A ) = A p ( F φ ( x ) ) p x p 0 and j x ( + ) = + > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbo_HTML.gif

      we see that condition (H2) holds and μ φ = A + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq274_HTML.gif.

      As for condition
      Fix ( F φ F φ ) = Fix ( F φ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equ5_HTML.gif
      (5)
      if A x < y < + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq275_HTML.gif with
      A + ( y x ) p = y , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbp_HTML.gif
      then y > A + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq276_HTML.gif and
      φ ( y , x ) x = A + ( x y ) p x = A + 1 y A y ( y A ) 1 / p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbq_HTML.gif
      Since the function
      h ( z ) = A + 1 z A z ( z A ) 1 / p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbr_HTML.gif

      has a unique critical point in ( A , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq277_HTML.gif and h ( A + 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq278_HTML.gif, h ( + ) = A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq279_HTML.gif, the necessary and sufficient condition for (5) to hold is that h ( A + 1 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq280_HTML.gif, that is, p ( A + 1 ) / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq281_HTML.gif.

      By this reasoning, we also get for free, unconditional stable convergence for several difference equations as, for instance:
      y n = A + ( y n q y n r y n s y n t ) p with  0 < p < min { 1 / 2 , ( A + 1 ) / 4 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbs_HTML.gif
      or
      y n = A + ( y n q + y n r y n s + y n t ) p with  0 < p < min { 1 , ( A + 1 ) / 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbt_HTML.gif
      just considering respectively
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbu_HTML.gif

      where m = max { q , r , s , t } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq282_HTML.gif.

      4.2 The difference equation y n = α + β y n 1 A + B y n 1 + C y n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq283_HTML.gif

      Here, α , β , A , B , C , y 0 , y 1 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq284_HTML.gif. In 2003, three conjectures on the above equation were posed in [8]. In all three cases ( B = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq285_HTML.gif, α , β , A , C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq286_HTML.gif; A = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq287_HTML.gif, α , β , B , C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq288_HTML.gif; and α , β , A , B , C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq289_HTML.gif, respectively) it was postulated the global asymptotic stability of the equilibrium. These conjectures have resulted in several papers since then (see [912]). Let us see when there is unconditional convergence.

      Consider a = c = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq290_HTML.gif, b = c = + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq291_HTML.gif, and
      φ ( x , y ) = α + β y A + D x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbv_HTML.gif
      with α , β , A 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq292_HTML.gif, D > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq293_HTML.gif. We solve in y the equation y = α + β y A + D x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq294_HTML.gif to obtain
      y = α A β + D x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbw_HTML.gif
      so we consider A > β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq295_HTML.gif, α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq296_HTML.gif to define
      F φ ( x ) = α A β + D x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbx_HTML.gif
      A simple calculation shows that (H2) holds, μ φ ( a , b ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq175_HTML.gif and Fix ( F φ F φ ) = Fix ( F φ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq185_HTML.gif:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equby_HTML.gif

      Therefore, μ φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq165_HTML.gif is an unconditional stable attractor of φ in ( a , b ) = ( 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq297_HTML.gif whenever A > β 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq298_HTML.gif, D > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq293_HTML.gif and α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq296_HTML.gif.

      If we choose
      λ ( x 1 , x 2 ) = ( B x 1 + C x 2 B + C , x 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equbz_HTML.gif
      and D = B + C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq299_HTML.gif, we obtain unconditional stable convergence for the equation
      y n = α + β y n 1 A + B y n 1 + C y n 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equca_HTML.gif

      whenever A > β 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq298_HTML.gif, B + C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq300_HTML.gif, C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq301_HTML.gif and α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq296_HTML.gif.

      Other choices of λ result on the unconditional stable convergence of difference equations such as
      y n = α + β y n 1 + γ y n 2 A + B y n 1 + C y n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equcb_HTML.gif
      with A > β + γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq302_HTML.gif, B + C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq300_HTML.gif, γ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq303_HTML.gif, α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq296_HTML.gif, β 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq304_HTML.gif, C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq301_HTML.gif. Or
      y n = α + β max { y n 1 , y n 2 } A + D min { y n 1 , y n 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_Equcc_HTML.gif

      with A > β 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq298_HTML.gif, D > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq293_HTML.gif, α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-63/MediaObjects/13661_2012_Article_316_IEq296_HTML.gif.

      Declarations

      Acknowledgements

      Dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.

      We are grateful to the anonymous referees for their helpful comments and suggestions. This research was supported in part by the Spanish Ministry of Science and Innovation and FEDER, grant MTM2010-14837.

      Authors’ Affiliations

      (1)
      Departamento de Matemática Aplicada, Universidad Nacional de Educación a Distancia (UNED)

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      Copyright

      © Franco and Peran; licensee Springer. 2013

      This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.