) Taylor & Francis Journal of Difference Equations and Applications, VOL. 10, No. 12, October 2004, pp. 1119-1127
v 3 / Taylor E, Francis croup
Open Problems and Conjectures Edited by GERRY LADAS In this section, we present some open problems and conjectures about some interesting types of difference equations. Please submit your problems and conjectures with all relevant information to G. Ladas. I pledge to donate the amount of $600 (USA) to the International Society of Difference Equations, provided that the complete solutions of the open problems and conjectures in this paper are brought to the attention of myself and the President of the Society by the end of the year 2005.
On Third-Order Rational Difference Equations: Part 3 E. CAMOUZIS^ E. CHATTERJEE^ G. LADAS""* and E.P. Q U I N N ' ' ^American College of Greece, 6 Gravias Street, Aghia Paraskevi, 15342 Athens, Greece; '"Department of Mathematics, University of Rhode tsiand, Kingston, Rl 02881-0816, USA (Received 3 April 2004; In final form 21 May 2004)
This paper is the third part in a series of manuscripts on "Open Problems and Conjectures" dealing with third-order rational difference equations of the form a + /3xn + yxn-i + Sx«-2 x«+\=TT~B TT^ TT; '
n = 0,1,...
(1)
with non-negative parameters and non-negative initial conditions. See Refs. [8,27]. Here, we will address the boundedness of all solutions of some special cases of Eq. (1) and we will pose several open problems and conjectures based on computer observations and analytic investigations. The 49 special cases of Eq. (1) which are contained in the second order rational difference equation • ^ n + i 7 ^ p
,
n = 0,1,...
were the subject of investigation in the Monograph [27]. *Corresponding author. E-mail:
[email protected] Journal of Difference Equations and Applications ISSN 1023-6198 print/ISSN 1563-5120 online © 2004 Taylor & Francis Ltd http://www.tandf.co.uk/jounials DOI: 10.1080/10236190410001726430
(2)
1120
E. CAMOUZIS
When C = 0 and B > 0, the boundedness of all solutions of Eq. (2) is completely determined by the following trichotomy result. A (SEE REFS. [18,19], OR [26]) Assume that B > 0. Then the following periodtwo trichotomy result holds for the solutions of the equation
THEOREM
« — 0, 1 , . . . .
(3)
(a) Every solution of Eq. (3) has a finite limit if and only if y< 13 + A. (b) Every solution ofEq. (3) converges to a (not necessarily prime) period-two solution of Eq. (3) if and only if (c) Equation (3) has unbounded solutions if and only if y> When C > 0, we offer the following amazing conjecture. CONJECTURE
1 Assume that C > 0.
(4)
Then every solution of Eq. (2) is bounded. A unified proof of the boundedness of all solutions of every special case of Eq. (2), when Eq. (4) holds, is a great challenge. If you wish a small taste of this challenge, try to obtain a unified proof for the boundedness of all positive solutions of the much simpler equation Ol + Xn A t X
with non-negative parameters a and A. After a thorough analysis of the existing literature on the boundedness of solutions of the 49 special cases of Eq. (2), we are happy to report that, to the best of our knowledge, there remain only two special cases to investigate for the confirmation of Conjecture 1. In the numbering system which was introduced in Ref. [27], these are equations #166 and #168. In agreement with our Conjecture 1, we also conjecture that all of their solutions are bounded. More precisely, we offer the following conjecture: CONJECTURE
2 Assume that a,l3,A,BE(0,oo).
Show that every positive solution of each of the following two equations is bounded: Eq.#166: c
4*1£O
Eq.#168 :
Xn+i =
f-
,
a+jSXn
Xn+\ =—
n = 0,1,...
lXn-i
;
,
n = 0,1,....
tSXn -r Xn-l
What is it that makes every solution of a rational equation bounded?
OPEN PROBLEMS AND CONJECTURES
1121
When can the boundedness of all solutions of a rational equation be predicted from the characteristic roots of the linearized equation about its equilibrium? Open Problem 1 Obtain necessary and sufficient conditions in terms of the parameters of Eq. {\) so that every solution of the equation is bounded. Open Problem 2 Assume that C = 0 and
y= p+S + A.
Find all special cases of Eq. (1) which have unbounded solutions. See Refs. [3,7,11]. CONJECTURE
3 Assume that
C = 0 and y> P + 8 + A. Show that Eq. (1) has unbounded solutions. Open Problem 3 Assume that C = 0,
8 + A+B>0,
p + A>0,
and
y=l3
Is every solution of Eq. (1) bounded? Open Problem 4
Find necessary and sufficient conditions on the parameters a,p,y,8
of the equation Xn+\=
a + j8x« + yxn-i + 8x^-2 Xn-\
•
,
« = 0, 1 , . . .
SO that every solution is bounded. See Ref. [6]. Open Problem 5 Find necessary and sufficient conditions on the parameters a,l3,y,8 of the equation Xn+\=
,
n = 0, 1,...
SO that every solution is bounded. In particular address the case 5 = 0 . Next, we focus on the boundedness character of solutions of all special cases of the equation a + l3X + 8Xn-2 Xn+\r^TTT
'
« = 0,1,....
(5)
Equation (5) contains 49 special cases with positive parameters of which 21 are trivial, linear, Riccati, or reducible to linear or Riccati-type equations.
1122
E. CAMOUZIS ef a/.
Fifteen special cases of Eq. (5) have been partially investigated. In the notation which was introduced in Ref. [27], they are: #21 ([14,15]),
#25 ([25]), #27 ([25]), #35 ([22]),
#39 ([22]),
#50 ([4]),
#67 ([25]),
#77 ([22]), #81 ([13]), #93 ([24]),
#102 ([25]),
#58 ([12]),
#60 ([1]),
#122 ([22]), #124 ([25])
Finally, the following thirteen special cases of Eq. (5) have not yet been thoroughly investigated, as far as we know: #44,
#69,
#89,
#91,
#106, #114,
#142
#150, #158, #171, #173, #175,
#206 We conjecture that the boundedness character of solutions of Eq. (5) is as follows: 4 The only special cases of Eq. (5) which contain unbounded solutions are the six linear equations: #5, #13, #41, #49, #57, and #121, and the following five non-linear equations: #8, #14, #35, #44, and #50. The proofs that equations #8 and #14 contain unbounded solutions are trivial. The proof for equation #50 is given in Ref. [4] where it is shown that when CONJECTURE
a 1 is not clear and requires further investigation. Therefore, there remains only Eq. #44, for which we conjecture that it has unbounded solutions, but we cannot yet confirm it analytically.
OPEN PROBLEMS AND CONJECTURES
1123
More precisely, we offer the following conjecture: CONJECTURE
5 Show that the equation #44:
x«+, = ^ ^ t i ^ ,
n = O,l,...
Xn-2
has unbounded solutions for every value of the parameter a. Now, we look at the boundedness character of all special cases of the equation = 0,1,....
A + CXn-l -\-DXn-2
(6)
Equation (6) contains 49 special cases with positive parameters of which 21 are trivial, linear, Riccati, or reducible to linear or Riccati-type equations. Thirteen special cases have already been investigated to some extent. They are: #22 ([15,28]),
#31 ([9,10]),
#34 ([9]),
#48 ([10]),
#64 ([9]),
#73 ([2,9]),
#76 ([16]),
#82 ([17]),
#103 ([25]),
#128 ([9]),
#97 ([9,20,21]), #100 ([23]), #179 ([9,21])
Finally, as far as we know, the following 15 special cases of Eq. (6) have not yet been thoroughly investigated: #36,
#40,
#51,
#63,
#78,
#96,
#111,
#115, #127, #147, #151, #163,
#178,
#182,
#211
The boundedness of solutions of Eq. (6) when C= 0
and
D> 0
is completely determined by the following trichotomy result for the equation ""-^' THEOREM B (SEE REF.
A+DXn-2
'
"
'^'^'--
^^>
[9]) Assume that D>0
and
y -f S 4- A > 0.
Then the following period-two trichotomy result holds for the solutions of Eq. (7): (a) Every solution of Eq. (7) has a finite limit if and only if y< 8 + A. (b) Every solution of Eq. (7) converges to a {not necessarily prime) period-two solution of Eq. (7) if and only if
1124
E. CAMOUZIS e/a/.
(c) Equation (7) has unbounded solutions if and only if y> 8-hA. For the remaining cases, we offer the following conjecture. CONJECTURE
6 Assume that
OO. Then the only special cases ofEq. (6) which have unbounded solutions are Eqs. #15, #36 and #51. The proofs that Eqs. #15 and #36 have unbounded solutions are straightforward. On the other hand, the proof that Eq. #51 has unbounded solutions is still not clear to us. CONJECTURE
7 Show that the equation Eq.#51:
Xn+i = ^-^^^^^,
n = O,l,...
X
has unbounded solutions. CONJECTURE
8 Find necessary and sufficient conditions on the parameters
of the equation a-\- l3Xn-\- yXn-\ + bXn-2 ,
X Xn+\
n — U, I , . . .
A -hXn-\
so that every solution is bounded. CONJECTURE
9 Find necessary and sufficient conditions on the parameters
a,/3, y, S,A of the equation ,
A -f- Xn-2
n = 0,1,...
SO that every solution is bounded. CONJECTURE
10 Show that every solution of the equation a + Pxn-\-yXn-] Xn+l=
,
n = 0, 1,...
Xn-2
is bounded if and only if
Finally, we look at the boundedness nature of solutions of all special cases of the equation l3Xn -I- yXn-l
-h 8Xn-2
Xn+i = p , „ —7; , BXn-^CXn~l +DXn-2
^ ,
n = 0,1,....
,ON
(8)
Equation (8) contains 49 special cases with positive parameters of which 3 are trivial and 6 are reducible to linear equations.
OPEN PROBLEMS AND CONJECTURES
1125
The following seventeen special cases have already been investigated to some extent: #26 ([26]),
#27 ([25]), #32 ([26]),
#33 ([10]),
#34 ([9,10]),
#39 ([22]), #54 ([26]),
#55 ([26]),
#58 ([12]),
#60 ([1]),
#62 ([11]),
#64 ([9]),
#86 ([26])
#93 ([24]),
#99 ([5]),
#100 ([23]),
#130 ([7]) Finally, as far as we know, the following 23 special cases of Eq. (8) have not yet been thoroughly investigated: #28,
#38,
#40,
#56,
#59,
#63,
#87,
#88,
#92,
#94,
#98,
#108,
#112,
#116, #131, #132, #156, #160,
#164,
#186, #187, #188, #216.
11 Prove that the complete list of special cases of Eq. (8) which possesses unbounded solutions is as follows: CONJECTURE
(1) All special cases with C = 0 and
y > 0.
C = 0 and
y = 0.
(2) Two equations with Namely, Eq.#8:
jc«+i=-^,
n = 0,1,...
Xn-2
and Eq.#14: (3)
Xn+\=^^^^,
n = 0,1,....
The following seven equations with OO. Eq.#15:
x«+i=^^,
E q . # 2 8 : Xn+\ Eq.#38:
x«+i =
n = O,l,...
CXn-] -\rXn-2
"""'^
,
DXn -\- Xn-\
Eq.#59:
x»^. = ^"" + ^ " - ^ , X
Eq.#92:
^«+i = ^ - —
,
n = 0,1,...
tSX + X
Eq.#98 :
J:«+I =
-— Xn -\- CATn-i
,
n = 0,1,.
1126
E. CAMOUZIS e( a/,
and Eq.#186:
x«+i =
;
,
n = 0,1,....
Open Problem 6 Determine all special cases ofEq. (1) with C = 0 which have unbounded solutions when
CONJECTURE
12 Show that the answer to Open Problem 6 is Eq. #120.
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OPEN PROBLEMS AND CONJECTURES
1127
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