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Fractional approximations of abstract semilinear parabolic problems
Extension, embedding and global stability in two dimensional monotone maps
1. | Department of Mathematics, Sultan Qaboos University, P. O. Box 36, PC 123, Al-Khod, Sultanate of Oman |
2. | Department of Mathematics and Statistics, American University of Sharjah, P. O. Box 26666, University City, Sharjah, UAE |
We consider the general second order difference equation $ x_{n+1} = F(x_n, x_{n-1}) $ in which $ F $ is continuous and of mixed monotonicity in its arguments. In equations with negative terms, a persistent set can be a proper subset of the positive orthant, which motivates studying global stability with respect to compact invariant domains. In this paper, we assume that $ F $ has a semi-convex compact invariant domain, then make an extension of $ F $ on a rectangular domain that contains the invariant domain. The extension preserves the continuity and monotonicity of $ F. $ Then we use the embedding technique to embed the dynamical system generated by the extended map into a higher dimensional dynamical system, which we use to characterize the asymptotic dynamics of the original system. Some illustrative examples are given at the end.
References:
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R. Abu-Saris, Z. AlSharawi and M. B. H. Rhouma,
The dynamics of some discrete models with delay under the effect of constant yield harvesting, Chaos Solitons Fractals, 54 (2013), 26-38.
doi: 10.1016/j.chaos.2013.05.008. |
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Z. AlSharawi, A global attractor in some discrete contest competition models with delay under the effect of periodic stocking, Abstr. Appl. Anal., (2013), Art. ID 101649, 7 pp.
doi: 10.1155/2013/101649. |
[3] |
A. M. Amleh, E. Camouzis and G. Ladas,
On second-order rational difference equation. Ⅰ, J. Difference Equ. Appl., 13 (2007), 969-1004.
doi: 10.1080/10236190701388492. |
[4] |
A. M. Amleh, E. Camouzis and G. Ladas,
On the dynamics of a rational difference equation. Ⅱ, Int. J. Difference Equ., 3 (2008), 195-225.
|
[5] |
E. Camouzis and G. Ladas, Dynamics of Third-order Rational Difference Equations with Open Problems and Conjectures, Advances in Discrete Mathematics and Applications, vol. 5, Chapman & Hall/CRC, Boca Raton, FL, 2008. |
[6] |
E. Camouzis and G. Ladas,
When does local asymptotic stability imply global attractivity in rational equations?, J. Difference Equ. Appl., 12 (2006), 863-885.
doi: 10.1080/10236190600772663. |
[7] |
W. A. Coppel,
The solution of equations by iteration, Proc. Cambridge Philos. Soc., 51 (1955), 41-43.
doi: 10.1017/S030500410002990X. |
[8] |
J.-L. Gouzé and K. P. Hadeler,
Monotone flows and order intervals, Nonlinear World, 1 (1994), 23-34.
|
[9] |
E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Advances in Discrete Mathematics and Applications, vol. 4, Chapman & Hall/CRC, Boca Raton, FL, 2005. |
[10] |
V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and its Applications, vol. 256, Kluwer Academic Publishers Group, Dordrecht, 1993.
doi: 10.1007/978-94-017-1703-8. |
[11] |
M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations, With Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, FL, 2002. |
[12] |
M. R. S. Kulenović, G. Ladas, L. F. Martins and I. W. Rodrigues, The dynamics of $x_{n+1} = \frac{\alpha+\beta x_n}{A+Bx_n+Cx_{n-1}}$: Facts and conjectures, Comput. Math. Appl., 45 (2003), 1087–1099, 2003.
doi: 10.1016/S0898-1221(03)00090-7. |
[13] |
M. R. S. Kulenović, G. Ladas and W. S. Sizer,
On the recursive sequence $x_{n+1} = (\alpha x_n+\beta x_{n-1})/(\gamma x_n+\delta x_{n-1})$, Math. Sci. Res. Hot-Line, 2 (1998), 1-16.
|
[14] |
M. R. S. Kulenović and O. Merino,
A note on unbounded solutions of a class of second order rational difference equations, J. Difference Equ. Appl., 12 (2006), 777-781.
doi: 10.1080/10236190600734184. |
[15] |
M. R. S. Kulenović and O. Merino,
Global bifurcation for discrete competitive systems in the plane, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 133-149.
doi: 10.3934/dcdsb.2009.12.133. |
[16] |
M. R. S. Kulenović and O. Merino,
Invariant manifolds for competitive discrete systems in the plane, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2471-2486.
doi: 10.1142/S0218127410027118. |
[17] |
W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. Vol. I, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1971. |
[18] |
G. Nyerges,
A note on a generalization of Pielou's equation, J. Difference Equ. Appl., 14 (2008), 563-565.
doi: 10.1080/10236190801912316. |
[19] |
H. Sedaghat, Nonlinear Difference Equations. Theory with Applications to Social Science Models, Mathematical Modelling: Theory and Applications, vol. 15, Kluwer Academic Publishers, Dordrecht, 2003.
doi: 10.1007/978-94-017-0417-5. |
[20] |
H. L. Smith,
The discrete dynamics of monotonically decomposable maps, J. Math. Biol., 53 (2006), 747-758.
doi: 10.1007/s00285-006-0004-3. |
[21] |
H. L. Smith,
Global stability for mixed monotone systems, J. Difference Equ. Appl., 14 (2008), 1159-1164.
doi: 10.1080/10236190802332126. |
show all references
References:
[1] |
R. Abu-Saris, Z. AlSharawi and M. B. H. Rhouma,
The dynamics of some discrete models with delay under the effect of constant yield harvesting, Chaos Solitons Fractals, 54 (2013), 26-38.
doi: 10.1016/j.chaos.2013.05.008. |
[2] |
Z. AlSharawi, A global attractor in some discrete contest competition models with delay under the effect of periodic stocking, Abstr. Appl. Anal., (2013), Art. ID 101649, 7 pp.
doi: 10.1155/2013/101649. |
[3] |
A. M. Amleh, E. Camouzis and G. Ladas,
On second-order rational difference equation. Ⅰ, J. Difference Equ. Appl., 13 (2007), 969-1004.
doi: 10.1080/10236190701388492. |
[4] |
A. M. Amleh, E. Camouzis and G. Ladas,
On the dynamics of a rational difference equation. Ⅱ, Int. J. Difference Equ., 3 (2008), 195-225.
|
[5] |
E. Camouzis and G. Ladas, Dynamics of Third-order Rational Difference Equations with Open Problems and Conjectures, Advances in Discrete Mathematics and Applications, vol. 5, Chapman & Hall/CRC, Boca Raton, FL, 2008. |
[6] |
E. Camouzis and G. Ladas,
When does local asymptotic stability imply global attractivity in rational equations?, J. Difference Equ. Appl., 12 (2006), 863-885.
doi: 10.1080/10236190600772663. |
[7] |
W. A. Coppel,
The solution of equations by iteration, Proc. Cambridge Philos. Soc., 51 (1955), 41-43.
doi: 10.1017/S030500410002990X. |
[8] |
J.-L. Gouzé and K. P. Hadeler,
Monotone flows and order intervals, Nonlinear World, 1 (1994), 23-34.
|
[9] |
E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Advances in Discrete Mathematics and Applications, vol. 4, Chapman & Hall/CRC, Boca Raton, FL, 2005. |
[10] |
V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and its Applications, vol. 256, Kluwer Academic Publishers Group, Dordrecht, 1993.
doi: 10.1007/978-94-017-1703-8. |
[11] |
M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations, With Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, FL, 2002. |
[12] |
M. R. S. Kulenović, G. Ladas, L. F. Martins and I. W. Rodrigues, The dynamics of $x_{n+1} = \frac{\alpha+\beta x_n}{A+Bx_n+Cx_{n-1}}$: Facts and conjectures, Comput. Math. Appl., 45 (2003), 1087–1099, 2003.
doi: 10.1016/S0898-1221(03)00090-7. |
[13] |
M. R. S. Kulenović, G. Ladas and W. S. Sizer,
On the recursive sequence $x_{n+1} = (\alpha x_n+\beta x_{n-1})/(\gamma x_n+\delta x_{n-1})$, Math. Sci. Res. Hot-Line, 2 (1998), 1-16.
|
[14] |
M. R. S. Kulenović and O. Merino,
A note on unbounded solutions of a class of second order rational difference equations, J. Difference Equ. Appl., 12 (2006), 777-781.
doi: 10.1080/10236190600734184. |
[15] |
M. R. S. Kulenović and O. Merino,
Global bifurcation for discrete competitive systems in the plane, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 133-149.
doi: 10.3934/dcdsb.2009.12.133. |
[16] |
M. R. S. Kulenović and O. Merino,
Invariant manifolds for competitive discrete systems in the plane, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2471-2486.
doi: 10.1142/S0218127410027118. |
[17] |
W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. Vol. I, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1971. |
[18] |
G. Nyerges,
A note on a generalization of Pielou's equation, J. Difference Equ. Appl., 14 (2008), 563-565.
doi: 10.1080/10236190801912316. |
[19] |
H. Sedaghat, Nonlinear Difference Equations. Theory with Applications to Social Science Models, Mathematical Modelling: Theory and Applications, vol. 15, Kluwer Academic Publishers, Dordrecht, 2003.
doi: 10.1007/978-94-017-0417-5. |
[20] |
H. L. Smith,
The discrete dynamics of monotonically decomposable maps, J. Math. Biol., 53 (2006), 747-758.
doi: 10.1007/s00285-006-0004-3. |
[21] |
H. L. Smith,
Global stability for mixed monotone systems, J. Difference Equ. Appl., 14 (2008), 1159-1164.
doi: 10.1080/10236190802332126. |








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