Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 1083-0464 Vol. No. 02, 1996, pp. 60–64. Submitted: October 17, 1996. Published: December 9, 1996.

A GENERALIZATION OF EDELSTEIN’S THEOREM

ON FIXED POINTS AND APPLICATIONS

Ioannis K. Argyros

Abstract. In this study, we approximate fixed points of nonlinear equations on topological vector spaces using implicit methods. A new Edelstein type theorem on fixed points is proved. The advantage of these methods is that they can be used to solve equations that usual explicit methods cannot.

1991 A.M.S. (MOS) Subject Classification Codes. 65J15, 47H17, 49D15.

Key Words and Phrases. Implicit methods, fixed point, topological vector space, normed space.

I. INTRODUCTION

In this study, we are concerned with the problem of approximating a fixed point z of the equation

(1)  x = F (x, x, …, x),

where F is an operator defined on a subset D^{m+1} of a topological vector space E^{m+1} with values in D, D is a compact set, and m is a nonnegative integer.

Explicit multistep methods of the form

(2)  x_{n+1} = F_{n}(x_{n}, x_{n−1}, …, x_{n−m}) (m ≥ m)

have been used extensively to approximate a fixed point z of equation (1). See for example [1], [2], [4]–[6], and the references there. Picard, Newton-like methods can be considered approximations of the form (2). With the exception of the work in [3] (for m = 1) no attention has been given to implicit multistep methods of the form

(3)  x_{n} = F (x_{n}, y_{n}^{m})
(4)  y_{n}^{m} = (y_{n−1}, y_{n−2}, …, y_{n−m}),
(5)  y_{n−i} = g_{n}^{i}(x_{n}, x_{n−1}, …, x_{n−m}),

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Cameron University, Lawton, Oklahoma

E-mail Address: ioannisa@cameron.edu

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where g_{n}^{i}: D^{m+1} → D for n ≥ m, i = 1, 2, …, m.

We provide sufficient conditions for the convergence of the sequence {x_{n}} (n ≥ m) generated by (3)–(5) to a fixed point z of equation (1) for any starting points x_{0}, x_{1}, …, x_{m−1} ∈ D. Moreover we provide a generalization of Edelstein’s theorem on fixed points [4], [2].

Using examples we finally show that the advantage of these methods is that they can be used to solve equations that explicit methods cannot.

II. CONVERGENCE ANALYSIS

The main result of our study is contained in the following convergence result for the method (3)–(5).

Theorem 1. Let m be a nonnegative integer, E a topological vector space, D a compact subset of E, F continuous operators from D^{m+1} into D, S = {x ∈ D: F (x, x, …, x) = x}, and an operator f: D × S → [0, +∞). Assume:

(a) S ≠ ∅ and f = 0, x ∈ D, z ∈ S ⇔ x = z ∈ S;

(b) f(F (z_{n}, z_{n−1}, …, z_{n−m}), z) < max{f(z_{n}, z), f(z_{n−1}, z), …, f(z_{n−m}, z)}, for all n ≥ m, z_{n}, z_{n−1}, …, z_{n−m} ∈ D, z ∈ S such that there are no relations such as z_{n} = z_{n−1} = ⋯ = z_{n−m} = z;

(c) if there exist operators h_{n}^{i}: D^{m+1} → D, i = 1, 2, …, m, n ≥ m such that z_{n−i} = h_{n}^{i}(v_{n}, v_{n−1}, …, x_{n−m}) for some v_{n}, v_{n−1}, …, v_{n−m} ∈ D then

f(F (z_{n}, z_{n−1}, …, z_{n−m}), z) < f(z_{n−i}, z) < f(v_{n−1}, z)

for all z_{n}, v_{n−1}, z_{n−i} ∈ D, n ≥ m, i = 1, 2, …, m;

(d) operator f(x, z) is continuous in x;

(e) the system (3)–(5) is solvable for arbitrary x_{0}, x_{1}, x_{2}, …, x_{m} ∈ D for all

n ≥ m. Then the sequences {x_{n}}, {y_{n−i}} (n ≥ m), i = 1, 2, …, m generated by (3)–(5) converge in the topology of E to the unique fixed point z of equation (1).

Proof: Using (3)–(5) and (c) we get

f(x_{n}, z) = f(F (x_{n}, y_{nm}), z) < f(y_{n−i}, z)
< f(x_{n−1}, z), n ≥ m, i = 1, 2, …, m,

which implies that

lim_{n→∞} f(x_{n}, z) = lim_{n→∞} f(y_{n−i}, z) for all i = 1, 2, …, m.

By the compactness of D we can find subsequences {x_{n_{j}}} ⊆ {x_{n}}, {y_{n_{j}−i}} ⊆ {y_{n−i}} and points x, y^{1}, y^{2}, …, y^{m} ∈ E such that

lim_{j→∞} x_{n_{j}} = x, lim_{j→∞} y_{n_{j}−i} = y^{i}.

Since f is continuous in x we get f(x, z) = f(y^{i}, z) for i = 1, 2, …, m. Moreover by letting j → ∞ we obtain

lim_{j→∞} F (x_{n_{j}}, y_{n_{j}−1}, …, y_{n_{j}−m}) = F (x, y^{1}, y^{2}, …, y^{m})

and by (b)

f(x, z) = f(F (x, y^{1}, y^{2}, …, y^{m}), z) < max{f(x, z), f(y^{1}, z), …, f(y^{m}, z)},

which is a contradiction. That is we deduce

x = y^{1} = y^{2} = ⋯ = y^{m} = z.

Furthermore we note that S is a singleton set. If a ≠ b, a, b ∈ S, then by (b) we obtain

f(a, b) = f(F (a, b, …, b), b) < f(a, b)

which is a contradiction. Hence we get a = b.

That completes the proof of the Theorem.

Remark 1. Let m = 1, g_{n}^{1} = g, g(x_{n}, x_{n−1}) = F_{k}(x_{n}, x_{n−1}), n ≥ 1, k ≥ 0, where F_{0}(v, w) = w, F_{1}(v, w) = F (v, w), F_{i}(v, w) = F (v, F_{i−1}(v, w)), for v, w ∈ D, i = 2, 3, …, k, y_{n}^{1} = g(x_{n}, x_{n−1}) (n ≥ 1), then Theorem 1 reduces to Theorem 1 in [3]. Theorem 1 cannot be used for the important class of multistep Newton-like methods where m > 1 [1], [2].

The following theorem generalizes Edelstein’s Theorem on fixed points [4], [2].

Theorem 2. Let m be a nonnegative integer, D a closed bounded subset of a normed space E, and F an operator from D^{m+1} into D.

Assume:

(a)

∥F (v_{1}, v_{2}, …, v_{m+1}) − F (w_{1}, w_{2}, …, w_{m+1}∥

< max{∥v_{i} − w_{i}∥}, i = 1, 2, …, m + 1
if (v_{1}, v_{2}, …, v_{m+1}) ≠ (w_{1}, w_{2}, …, w_{m+1})
and v_{i} ≠ v_{j}; or w_{i} ≠ w_{j} for i ≠ j

≤ ∥v_{i} − w_{i}∥, i = 1, 2, …, m + 1
if v_{1} = v_{2} = ⋯ = v_{m+1} and w_{1} = w_{2} = ⋯ = w_{m+1}.

(b) Either D is compact or F maps D × D into a compact subset of D. Then the following are true:

(i) Equation

F (x, x, …, x) = x

has a unique z solution in D. (ii) Every equation

(6)  x_{n} = F (x_{n}, x_{n−1}, …, x_{n−m}) (n ≥ m)

has also a unique solution x_{n} and lim_{n→∞} x_{n} = z for any x_{0} ∈ D.

Proof: It follows immediately by using the operator F_{y}(x) = F (x, y) for all x ∈ D, y ∈ D^{m} and Edelstein’s Theorem on fixed points [4], [2].

Remark 2. Theorem 2 generalizes Edelstein’s Theorem on fixed points [4], [2], and reduces to Theorem 6 in [3] for m = 1. Obviously earlier results [1], [2], [4]–[6] cannot be used to assert existence or the convergence of approximate solutions for equation (1) if instead of (6) Picard’s iteration for G(x) = F (x, x, …, x) is used. Moreover the results in [3] cannot be used for multistep Newton-like methods where m > 1 [1], [2].

In the following example we provide an application for Theorem 2.

Example 1. Let E = R, D = [a, b] with a ≤ 0 ≤ b, ∥x − y∥ = |x − y| and F (x, y) = −\frac{1}{6}x + \frac{2}{3}y. Then we have

∥F (x, y) − F (v, w)∥ = |−\frac{1}{6}(x − v) + \frac{2}{3}(y − w)∥

≤ \frac{1}{6}|x − v| + \frac{2}{3}|y − w| (7)

for all x, y, v, w ∈ D. It now follows from (7) that hypothesis (a) of Theorem 2 is satisfied. Moreover (6) (for m = 1) becomes

x_{n} = \frac{4}{7}x_{n−1} = (\frac{4}{7})^{n} x_{0} → 0 = z as n → ∞

for any x_{0} ∈ R.

In the example that follows we provide an explicit sequence that does not converge and an implicit sequence that converges to a fixed point of an operator.

Example 2. Let E, D be as in Example 1 and consider the problem of approximating the unique fixed point z = 0 of the operator F on D given by F (x, y) = cx for all x, y ∈ D with c ≠ 0. We use the approximations

(8)  F (a_{n}) + d(a_{n+1} − a_{n}) = 0 (n ≥ 0),
(9)  F (b_{n+1}) + d(b_{n+1} − b_{n}) = 0 (n ≥ 0)

for a_{0}, b_{0} ∈ D. Approximation (8) is explicit whereas (9) is an implicit method. With the above choices (8) and (9) become respectively

a_{n+1} = \frac{d − c}{d}a_{n} (n ≥ 0)

and

b_{n+1} = \frac{d}{c + d}b_{n} (n ≥ 0).

For c = 3 and d = 1 we obtain

a_{n+1} = −2a_{n} = (−2)^{n+1}a_{0} ↛ 0 = z for any a_{0} ≠ 0

whereas

b_{n+1} = \frac{1}{4}b_{n} = (\frac{1}{4})^{n+1} b_{0} → 0 = z for any b_{0}.

REFERENCES

[1] Argyros, I.K. and Szidarovszky, F. Convergence of general iteration schemes, J. Math. Anal. Appl. Vol. 168, No. 1 (1992), 42–62.

[2] Argyros, I.K. and Szidarovszky, F. The theory and applications of iteration methods, C.R.C. Press, Inc. Boca Raton, Florida, 1993.

[3] Binh, T.Q., Chuong, N.M., and Ton, B.A. Approximation of nonlinear operator equations, submitted.

[4] Edelstein, M. On fixed and periodic points under contractive mappings, J. London Math. Soc. 37, (1962), 74–79.

[5] Kirk, W.A. A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72, (1965), 1004–1006.

[6] Petryshin, W.V. and Williamson, T.E. Strong and weak convergence of the sequence of successive approximation for quasi-nonexpansive mappings, J. Math. Anal. Appl. 43, (1973), 459–497.