Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 1083-0464 Vol. No. 02, 1996, pp. 29–33. Submitted: April 11, 1996. Published: September 9, 1996.

COMMON FIXED POINTS BY ISHIKAWA

ITERATES IN METRIC LINEAR SPACES

R.A.RASHWAN

Abstract. In this paper we obtain a common fixed point theorem for Ishikawa iterates of two self-mappings on a metric linear space under various contractive conditions.

(1991) A.M.S. Subject Classification Codes. 47H10.

Key Words and Phrases. Metric linear space, F - norm, Contractive mappings, Ishikawa iterates, Common fixed point.

1. INTRODUCTION. In [3], [4], it has shown that for a self-mapping T on a normed space or a Banach space X satisfying variuos contactive conditions, if the sequence of Ishikawa iterates associated with T converges, it coverges to a fixed point of T. These results have recently extended by L.A.Khan in [2] to the case of metric linear spaces .

In this paper, we consider two self-mappigs S and T on a metric linear space X and show that if the sequence of Ishikawa iterates associated with S and T converges, it coverges to a common fixed point of S and T.

In the sequel we assume that the topology on X is generated by an F -norm q which has the following properties (see [5], pp.28-29).

(a) q(x) ≥ 0 and q(x) = 0 iff x = 0

(b) q(x + y) ≤ q(x) + q(y)

(c) q(ax) ≤ q(x) for all (real or complex) scalars a with |a| ≤ 1

(d)If a_{n} → a and x_{n} → x, then q(a_{n}x_{n} − ax) → 0.

Note that q is continuous on X, the equation d(x, y) = q(x − y) defines a translation-invariant metric on X, and q(ax) ≤ q(bx) for all scalars a, b with |a| ≤ |b|.

Typeset by AMS-TEX

Mathematics Department, Faculty of Science, Assiut University, Assu it, EGYPT

E-mail Address: assiut@frcu.enu.eg

\begin{matrix}c \\ ◯ \end{matrix}1996 Cameron University

Let C be a nonempty covex subset of X and S, T self-maps on C. An Ishikawa scheme [1], for S, T is defined by

x_{0} ∈ C

y_{n} = (1 − β_{n})x_{n} + β_{n}Sx_{n} , n ≥ 0
x_{n+1} = (1 − α_{n})x_{n} + α_{n}T y_{n} , n ≥ 0,  (1)

where the real sequences {α_{n}}, {β_{n}} satisfy (i) 0 ≤ α_{n} ≤ 1, 0 ≤ β_{n} ≤ 1, for n ≥ 0 (ii) \underline{lim}_{n→∞} α_{n} = α > 0 From (1) and condition (i) we obtain some inequalities which will be used later

q(y_{n} − x_{n}) = q(β_{n}(Sx_{n} − x_{n}))
≤ q(Sx_{n} − T y_{n}) + q(T y_{n} − x_{n}),  (2)

q(y_{n} − T y_{n}) = q(β_{n}(Sx_{n} − T y_{n})) + (1 − β_{n})(x_{n} − T y_{n})
≤ q(Sx_{n} − T y_{n}) + q(x_{n} − T y_{n}),  (3)
q(x_{n} − Sx_{n}) ≤ q(x_{n} − T y_{n}) + q(T y_{n} − Sx_{n})  (4)

q(y_{n} − Sx_{n}) = q(1 − β_{n})(x_{n} − Sx_{n})
≤ q(x_{n} − T y_{n}) + q(T y_{n} − Sx_{n})  (5)

2. MAIN RESULT

We present our result in the form of the following theorem:

Theorem 2.1. Let C be a closed convex subset of X and let S, T be two selfmappings of C satisfying at least one of the folloing conditions

(I)q(Sx − T y) ≤ k max{q(x − y), q(x − Sx), q(y − T y), q(x − T y) + q(y − Sx)}

(II)q(Sx − T y) + q(x − Sx) + q(y − T y) ≤ c[q(x − T y) + q(y − Sx)], 0 ≤ c < 2

(III)q(Sx − T y) ≤ k max{q(x − y), q(x − Sx), q(y − T y), q(x − T y), q(y − Sx)},

for all x, y ∈ C , where 0 ≤ k < 1 and 0 ≤ c < 2. Suppose that for some x_{0} ∈ C, the sequence {x_{n}}_{n=0}^{∞} of Ishikawa iterates converges to a point u. Then u is a common fixed point of S and T. Moreover if (III) holds, then the common fixed point u is unique.

Proof. Suppose first that Su = u for a point u in C. Then putting x = y = u into any of the inequalities (I)-(III) we easily see that T u = u. Similarly T u = u implies Su = u. Now let {x_{n}} be a sequence of Ishikawa iterates with S and T such that

lim_{n→∞} x_{n} = u.

From (1),we see that

T y_{n} − x_{n} = \frac{1}{α_{n}} α_{n}(x_{n+1} − x_{n}).

Since

\underline{lim}_{n→∞} α_{n} = α > 0,

there exists an integer N ≥ 1 such that

\frac{α}{2} ≤ α_{n}

for all n ≥ N. Hence for n ≥ N

q(T y_{n} − x_{n}) = q(\frac{α}{2}(x_{n+1} − x_{n})).

Since

lim_{n→∞} x_{n} = u,

we have

q(T y_{n} − x_{n}) → 0 as n → ∞

and hence

q(T y_{n} − u) → 0 as n → ∞.

Now for n ≥ N,

q(u − T u) ≤ q(u − T y_{n}) + q(Sx_{n} − T y_{n}) + q(Sx_{n} − T u).  (6)

Now we need to show that

q(Sx_{n} − T y_{n}) → 0 and q(Sx_{n} − T u) → o as n → ∞.

If x_{n}, y_{n} satisfy (I),then

q(Sx_{n} − T y_{n}) ≤ k max{q(Sx_{n} − T y_{n}) + q(x_{n} − T y_{n}),
q(x_{n} − T y_{n}) + q(Sx_{n} − T y_{n}),
q(Sx_{n} − T y_{n}) + q(x_{n} − T y_{n}),
2q(Sx_{n} − T y_{n}) + q(x_{n} − T y_{n})}
= k[2q(x_{n} − T y_{n}) + q(Sx_{n} − T y_{n})].

If x_{n}, y_{n} satisfy (II),then

(3 − c)q(Sx_{n} − T y_{n}) ≤ 2(c − 1)q(x_{n} − T y_{n}).

If x_{n}, y_{n} satisfy (III),then

q(Sx_{n} − T y_{n}) ≤ k max{q(x_{n} − T y_{n}) + q(Sx_{n} − T y_{n}),
q(x_{n} − T y_{n}) + q(Sx_{n} − T y_{n}),
q(Sx_{n} − T y_{n}) + q(x_{n} − T y_{n}),
q(Sx_{n} − T y_{n}) + q(x_{n} − T y_{n}),
q(Sx_{n} − T y_{n}) + q(x_{n} − T y_{n})}
= k[q(x_{n} − T y_{n}) + q(Sx_{n} − T y_{n})].

Hence in any case,

q(Sx_{n} − T y_{n}) ≤ max\{\frac{2k}{1 − k} , \frac{2(1 − c)}{3 − c}, \frac{k}{1 − k} \}q(x_{n} − T y_{n}).

Letting n → ∞, we obtain q(Sx_{n} − T y_{n}) → o. Then further q(x_{n} − Sx_{n}) and q(u − Sx_{n}) tend to zero as n → ∞. Next if x_{n}, u satisfy (I), then

q(Sx_{n} − T u) ≤ k[q(x_{n} − u) + q(x_{n} − Sx_{n}) + q(Sx_{n} − T u) + q(u − Sx_{n}].

If x_{n}, u satisfy (II),then

(2 − c)q(Sx_{n} − T u) ≤ (c − 1)[q(x_{n} − Sx_{n}) + q(u − Sx_{n})].

If x_{n}, u satisfy (III), then obviously satisfy (I) as well. Hence, we have

q(Sx_{n} − T u) ≤ max\{\frac{k}{1 − k} , \frac{(1 − c)}{2 − c}\}[q(x_{n} − u) + q(x_{n} − Sx_{n}) + q(u − Sx_{n})].

Letting n → ∞, we have q(Sx_{n} − T u) → 0. Thus it follows from (6) that T u = u. In view our remark at the beginning of the proof u is a fixed point of S as well. In order to show the uniqueness of u in the case (III), let v(v ≠ u) be another common fixed point of S and T, then using (III) we have

q(u − v) = q(Su − T v) ≤ max{q(u − v), 0, 0, q(u − v)},

whence u = v.

Finally we give some examples of a metric linear space and two mappings which satisfy the contactive conditions of Theorem 2.1.

Example 2.1. Let X = R, the set of real numbers and q be the F -norm defined by

q(x) = \frac{|x|}{1 + |x|}

Let C = [0, 1] and S, T : C → C be defined by:

Sx = \{ \begin{matrix}0 & , 0 ≤ x < 1 \\ \frac{1}{4} & , x = 1. \end{matrix} , T x = \{ \begin{matrix}0 & , 0 ≤ x < 1 \\ \frac{1}{2} & , x = 1. \end{matrix} .

Then S, T satisfy condition (III) of Theorem 2.1 with k = \frac{2}{3} as follows

(i) If x = y = 1 , then

q(Sx − T y) = \frac{1}{5},

\frac{2}{3}q(y − Sx) = \frac{2}{3}q(\frac{3}{4}) = \frac{2}{7} > \frac{1}{5},

\frac{2}{3}q(x − T y) = \frac{2}{3}q(\frac{1}{2}) = \frac{2}{9} > \frac{1}{5},

(ii)If 0 ≤ x, y ≤ 1, then (III) is trivially satisfied. (iii)If x = 1, and 0 ≤ y < 1, then

q(Sx − T y) = q(\frac{1}{4}) = \frac{1}{5}

\frac{2}{3}q(x − T y) = \frac{2}{3}q(1) = \frac{1}{3} > \frac{1}{5}.

\frac{2}{3}q(x − Sx) = \frac{2}{3}q(\frac{3}{4}) = \frac{2}{7} > \frac{1}{5}.

(iv) If 0 ≤ x ≤ 1, y = 1, then (III) (Similarly as in (iii)). Note that 0 is the unique common fixed point of S and T.

Example 2.2. Let X, q, C be as in example 2.1 and let S, T : C → C be defined by:

Sx = \{ \begin{matrix}1 − x & , 0 ≤ x < 1 \\ 0 & ,  x = 1. \end{matrix} ,  T x = \{ \begin{matrix}\frac{1}{2} & , 0 ≤ x < 1 \\ 0 & , x = 1. \end{matrix} .

Then S, T satisfy condition (II) of Theorem 2.2 with c = \frac{3}{2} as follows If x = 1 and 0 ≤ y < 1, then

\begin{aligned}L & = q(Sx − T y) + q(x − Sx) + q(y − T y) \\ & = q(\frac{1}{2}) + q(1) + q(y − \frac{1}{2}) = q(1) + q(y) = \frac{1}{2} + q(y) \end{aligned}

R = q(x − T y) + q(y − Sx) = q(\frac{1}{2}) + q(y) = \frac{1}{3} + q(y),

So that

\frac{L}{R} = \frac{\frac{1}{2} + q(y)}{\frac{1}{3} + q(y)} ≤ \frac{3}{2}.

In this case x = \frac{1}{2} is a common fixed point of S and T.

REFERENCES

1. S. Ishikawa, Fixed point by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147 - 150.

2. L.A.Khan, Fixed points by Ishikawa in metric linear spaces, Math.Rep.Toyama univ 12 (1989), 57-63.

3. S.A.Naimpally and K.L.Singh, Extensions of some fiexd point theorems of Rhoades, J.Math.Anal.appl. 96 (1983), 437-446.

4. B.E.Rhoades, Comments on two fixed point iteration methods, J.Math.Anal.Appl. 56 (1976), 741-750.

5. H.H.Schaefer, Topological vector spaces, Spriner-Verlag,Berlin 1971.