By Saber N. Elaydi

PREFACE FOREWORD the steadiness of One-Dimensional Maps creation Maps vs. distinction Equations Maps vs. Differential Equations Linear Maps/Difference Equations mounted (Equilibrium) issues Graphical generation and balance standards for balance Periodic issues and Their balance The Period-Doubling path to Chaos functions allure and Bifurcation creation Basin of allure of mounted issues Basin ofRead more...

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Semistability deﬁnition: A ﬁxed point x∗ of a map f is semistable (from the right) if for any ε > 0 there exists δ > 0 such that if 0 < x0 − x∗ < δ then |f n (x0 ) − x∗ | < ε for all n ∈ Z+ . If, in addition, lim f n (x0 ) = x∗ whenever 0 < x0 − x∗ < η for some η > 0, then x∗ n→∞ is said to be semiasymptotically stable (from the right). Semistability (semiasymptotic stability) from the left is deﬁned analogously. Suppose that f (x∗ ) = 1 and f (x∗ ) = 0. Prove that x∗ is (a) Semiasymptotically stable from the right if f (x∗ ) < 0.

Then, g(x) is also a continuous map. If f (a) = a or f (b) = b, we are done. So assume that f (a) = a and f (b) = b. Hence, f (a) > a and f (b) < b. Consequently, g(a) > 0 and g(b) < 0. By the intermediate value theorem,5 there exists a point c ∈ (a, b) with g(c) = 0. This implies that f (c) = c and c is thus a ﬁxed point of f . The above theorem says that for a continuous map f if f (I) ⊂ I, then f has a ﬁxed point in I. The next theorem gives the same assertion if f (I) ⊃ I. 5 The intermediate value theorem: Let f : I → I be a continuous map.

16). Letting f (x) = 0 and x = x1 yields x1 = x0 − f (x0 ) . f x0 ) By repeating the process, replacing x0 by x1 , x1 by x2 , . . 25). 26) we have fN (r) = r and thus r is a ﬁxed point of fN (assuming that g (r) = 0). 26) again we get g(x g (x) = 0. , x is a zero of g(x). 25) gives the next approximation x(1) of the root r. By applying the algorithm repeatedly, we obtain the sequence of approximations x0 = x(0), x(1), x(2), . . , x(n), . . (see Fig. 16). The question is whether or not this sequence converges to the root r.