Dynamics of Third-Order Rational Difference Equations with Open Problems and ConjecturesExtending and generalizing the results of rational equations, Dynamics of Third Order Rational Difference Equations with Open Problems and Conjectures focuses on the boundedness nature of solutions, the global stability of equilibrium points, the periodic character of solutions, and the convergence to periodic solutions, including their p |
Contents
| 1 | |
| 3 | |
Equations with Bounded Solutions | 29 |
Existence of Unbounded Solutions | 75 |
Periodic Trichotomies | 105 |
Known Results for Each of the 225 Special Cases | 133 |
Appendix A | 461 |
Appendix B | 513 |
| 535 | |
| 553 | |
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Common terms and phrases
2-H Ean Appl arbitrary nonnegative initial arbitrary positive initial Assume boundedness character Camouzis Can_1 change of variables character of solutions characteristic equation Clearly Determine the set equation has unbounded equation is bounded equation was investigated equation xn+1 ESCP2 established eventually exist bounded solutions exist solutions Global Attractivity global character globally asymptotically holds initial conditions x−2 Investigate the global investigated in 69 L.A.S. Conjecture Ladas Lemma linearized equation locally asymptotically stable necessarily prime nonnegative initial conditions normalized form xn+1 Open Problem P2-solution P2-Tricho periodic solution positive constant positive equilibrium point positive initial conditions positive numbers positive parameters positive solution prime period prime period-two solution proof is complete rational difference equation rational equations result follows Riccati equations sake of contradiction set G Show strictly increasing Theorem Thrm unique positive equilibrium unique prime period-two unstable written x n+1 zero equilibrium
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Page 2 - The art of doing mathematics consists in finding that special case which contains all the germs of generality.
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Page 5 - The following three theorems state necessary and sufficient conditions for all the roots of a real polynomial of degree two, three, or four, respectively, to have modulus less than one.
Page 7 - A negative semicycle of {xn} consists of a "string" of terms {x*+i , Xfc+2, . . . ,x/}, all less than x, with k > —2 and / < oo and such that either k = —2 or k > — 1 and x* > x and either / = oo or / < oo and x/+i > x. The first semicycle of a solution starts with the term x_i and is positive if...
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Page 4 - Eq. (1.1.1) is called globally asymptotically stable if x is locally stable, and x is also a global attractor of Eq.
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