Some Approximate Fixed Point Theorems - Hikari

Int. Journal of Math. Analysis, Vol. 3, 2009, no. 5, 203 - 210

Some Approximate Fixed Point Theorems Bhagwati Prasad, Bani Singh and Ritu Sahni

Department of Mathematics Jaypee Institute of Information Technology University A-10, Sector-62, Noida 201307, India [email protected]; [email protected]

Abstract The purpose of this paper is to obtain some basic approximate fixed point theorems in general settings. Few special cases are also obtained. Mathematics Subject Classifications: 54H25, 47H10, 47H15 Keywords: Fixed point, Approximate fixed point, b-metric space.

1. Introduction Let T be a self map of a metric space ( X , d ). Let us look for an approximate solution of Tx = x. If there exists a point z ∈ X such that d (Tz , z ) ≤ ε , where ε is a positive number, then z is called an approximate solution of the equation Tx = x, or equivalently, z ∈ X is an approximate fixed point (or ε -fixed point ) of T . In many situations of practical utility, the mapping under consideration may not have an exact fixed point due to some tight restriction on the space or the map, or an approximate fixed point is more than enough, an approximate solution plays an important role in such situations. The theory of fixed points and consequently of approximate fixed points finds application in mathematical economics, noncooperative game theory, dynamic programming, nonlinear analysis, variational calculus, theory of integro-differential equations and several other areas of applicable analysis (see, for instance, [5], [9], [10], [14], [15] and several references thereof).

Cromme and Diener [7] have found approximate fixed points by generalizing Brouwer’s fixed point theorem to a discontinuous map, Hou and Chen [11] have extended their results to set valued maps. Espinola and Kirk [10] obtained

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B. Prasad, B. Singh and R. Sahni

interesting results in product spaces. Tijs et al [15] have discussed approximate fixed point theorems for contractive and non-expansive maps by weakening the conditions on the spaces. R. Branzei et al [5] further extended these results to multifunctions in Banach spaces. Recently M. Berinde [4] obtained approximate fixed point theorems for operators satisfying Kannan, Chatterjea and Zamfirescu type of conditions on metric spaces. In this paper we study some basic approximate fixed point results in generalized metric spaces.

2. Preliminaries Definition 2.1 [8]. Let X be a non empty set and s ≥ 1 be a given real number. A function d : X × X → ℜ + (set of nonnegative real numbers) is said to be a b-metric iff for all x, y, z ∈ X the following conditions are satisfied: d ( x, y ) = 0 iff x = y, (i) (ii) d ( x, y ) = d ( y, x), (iii) d ( x, z ) ≤ s[d ( x, y ) + d ( y, z )]. A pair ( X , d ) is called a b-metric space.

The class of b-metric spaces is effectively larger than that of metric spaces, since a b-metric space is a metric space when s = 1 in the above condition (iii). The following example shows that a b-metric on X need not be a metric on X (see also [8, p. 264]). Example 2.1 [13]. Let X = {x1 , x 2 , x3 , x 4 } d ( x1 , x2 ) = k ≥ 2 and, d ( x1 , x3 ) = d ( x1 , x 4 ) = d ( x2 , x3 ) = d ( x2 , x4 ) = d ( x3 , x 4 ) = 1, d ( xi , x j ) = d ( x j , xi ) fo

r all i, j = 1, 2, 3, 4 and d ( xi , xi ) = 0, i = 1, 2, 3, 4. Then k d ( xi , x j ) ≤ d ( xi , x n ) + d ( x n , x j ) for n, i, j = 1, 2, 3, 4 2 and if k > 2, the ordinary triangle inequality does not hold.

[

]

Definition 2.2. Let T : X → X , ε > 0 and x0 ∈ X . Then an element x0 ∈ X is an

approximate fixed point (or ε -fixed point) of T if d (Tx0 , x0 ) < ε . T is said to satisfy approximate fixed point property (AFPP) if for every ε > 0, Fixε (T ) ≠ φ. Remark 2.1. Throughout this paper, for given ε > 0, we shall denote the set of all approximate fixed points of T by Fixε (T ) .

Following notions of asymptotically regular mappings is essentially due to Browder and Petryshyn [6]:

Some approximate fixed point theorems

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Definition 2.3 [6]. A map T : X → X is said to be asymptotically regular if for any x ∈ X , lim d (T n x, T n +1 x) = 0. n→∞

Consider the following conditions for all x, y ∈ X and some a ∈ (0, 1), 1 b, c ∈ (0, ) : 2 d (Tx, Ty ) ≤ ad ( x, y ) (2.1) d (Tx, Ty ) ≤ b[d ( x, Tx ) + d ( y, Ty )]

(2.2)

d (Tx, Ty ) ≤ c[d ( x, Ty ) + d ( y, Tx )]

(2.3)

As noted in Berinde [1], it is well known that the conditions (2.1) and (2.2), (2.1) and (2.3), as well as (2.2) and (2.3), respectively, are independent (see also Rhoades [12]). Zamfirescu [16] obtained some interesting results by combining conditions (2.1), (2.2) and (2.3) in metric spaces.

3. Main Results Following is the slightly extended version of the Lemma 1.1 of Berinde [4]: Lemma 3.1. Let ( X , d ) be a b-metric space and T : X → X . If T is an asymptotically regular map, then T has AFPP. Proof. It may be completed following Berinde [4]. Theorem 3.1. Let ( X , d ) be a b-metric space and T : X → X satisfies (2.1), then T has AFPP. Proof. This follows immediately from Berinde [4]. Theorem 3.2. Let ( X , d ) be a b-metric space and T : X → X satisfies (2.1). sε (1 + s ) Then for each ε > 0 , the diameter of Fixε (T ) is not larger than . 1 − as 2 Proof: We know that T has the approximate fixed point property, so we can take x and y any two ε -fixed point of T , then d ( x, y ) ≤ s[d ( x, Tx) + d (Tx, y )]

≤ sε + s 2 [d (Tx, Ty ) + d (Ty, y )] ≤ sε + s 2 d (Tx, Ty ) + s 2ε ≤ sε + s 2ε + as 2 d ( x, y ) sε (1 + s ) ∴ d ( x, y ) ≤ , for each ε > 0. 1 − as 2 This completes the proof.

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If we put s = 1 in Theorem 3.2 we obtain the result of Berinde [4]. Corollary 3.1 [4]. Let ( X , d ) be a metric space and T : X → X satisfies (2.1). 2ε Then for each ε > 0 , the diameter of Fixε (T ) is not larger than . 1− a Theorem 3.3. Let ( X , d ) be a b-metric space and T : X → X satisfies (2.2). Then T has AFPP. Proof. It follows on the same lines as in [4] for metric spaces. Theorem 3.4. Let ( X , d ) be a b-metric space and T : X → X satisfies (2.2). Then for each ε > 0 , the diameter of Fixε (T ) is not larger than sε (1 + s + 2bs ) . Proof: We know that T has the approximate fixed point property, so we can take x and y any two ε -fixed point of T , then d ( x, y ) ≤ s[d ( x, Tx) + d (Tx, y )]

≤ sε + s 2 [d (Tx, Ty ) + d (Ty, y )] ≤ sε + s 2 d (Tx, Ty ) + s 2ε ≤ sε + s 2ε + s 2 b [ d ( x, Tx) + d ( y, Ty ) ] ≤ sε + s 2ε + s 2 2bε ≤ sε (1 + s + 2bs ) ∴ d ( x, y ) ≤ sε (1 + s + 2bs ) , for each ε > 0. This completes the proof. If we put s = 1 in Theorem 3.3 we obtain the result of Berinde [4]. Corollary 3.2 [4]. Let ( X , d ) be a metric space and T : X → X a Kannan operator. Then for each ε > 0 , the diameter of Fixε (T ) is not larger than 2ε (1 + b) . Remark 3.1. A map T is called Kannan operator if Chatterjea operator if T satisfies (2.3) (see [4], [12]).

T satisfies (2.2) and

Theorem 3.5. Let ( X , d ) be a b-metric space and T : X → X satisfies (2.3) 1 with cs < . Then T has AFPP. 2 Proof: Let ε > 0 and x ∈ X . Then d (T n x, T n +1 x) = d (T (T n −1 x), T (T n x)) ≤ c[d (T n −1 x, T (T n x)) + d (T n x, T (T n −1 x))]


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= c[d (T n −1 x, T n +1 x) + d (T n x, T n x)] = cd (T n −1 x, T n +1 x) ≤ cs[d (T n −1 x, T n x) + d (T n x, T n +1 x)] n

n

d (T x, T

n +1

This implies

cs ⎛ cs ⎞ x) ≤ d (T n −1 x, T n x) ≤ ... ≤ ⎜ ⎟ d ( x, Tx) 1 − cs ⎝ 1 − cs ⎠ d (T n x, T n +1 x) → 0, as n → ∞, ∀x ∈ X .

Now by Lemma 3.1 it follows that ∀ε > 0, Fixε (T ) ≠ φ . Theorem 3.6. Let ( X , d ) be a b-metric space and T : X → X satisfies (2.3). sε (1 + s + 2cs) . Then for each ε > 0 , the diameter of Fixε (T ) is not larger than 1 − 2s 2 c Proof: We know that T has the approximate fixed point property, so we can take x and y any two ε -fixed point of T , then d ( x , y ) ≤ s[ d ( x , Tx ) + d (Tx , y )] ≤ sε + s 2 [ d (Tx , Ty ) + d (Ty , y )] ≤ sε + s 2 d (Tx , Ty ) + s 2 ε ≤ sε + s 2ε + s 2 c [ d ( x , Ty ) + d ( y , Tx ) ] ≤ sε + s 2ε + s 2 c[ d ( x , y ) + d ( y , Ty )] + s 2 c[ d ( y , x ) + d ( x , Tx )] ≤ sε + s 2ε + 2 s 2 cd ( x , y ) + 2 s 2 cε

sε (1 + s + 2cs ) , for each ε > 0. 1 − 2s 2 c This completes the proof. ∴

d ( x, y ) ≤

If we put s = 1 in Theorem 3.6 we obtain the result of Berinde [4]. Corollary 3.3 [4]. Let ( X , d ) be a metric space and T : X → X satisfies (2.3). 2ε (1 + c) Then for each ε > 0 , the diameter of Fixε (T ) is not larger than . 1− 2c Remark 3.2. A mapping T : X → X is a Zamfirescu operator if it satisfies at least one of the conditions (2.1), (2.2) and (2.3) (cf. [1], [3], [15]).

Let ( X , d ) be a b-metric space and T : X → X a Zamfirescu operator on X. Then, for bs 2 < 1 / 2 , cs < 1 / 2 T has AFPP.

Theorem 3.7.

Proof: First we will try to concentrate the three independent conditions into a single one they all imply. Let x, y ∈ X . Suppose (2.2) holds, then we have

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d (Tx, Ty) ≤ b [ d ( x, Tx) + d ( y, Ty)] ≤ bd ( x, Tx) + bs[d ( y, x) + d ( x, Ty)] ≤ bd ( x, Tx) + bsd ( y, x) + bs 2 [d ( x, Tx) + d (Tx, Ty)] ≤ b(1 + s 2 )d ( x, Tx) + bsd ( x, y) + bs 2 d (Tx, Ty) ≤

b(1 + s 2 ) bs d ( x, Tx) + d ( x, y) 2 1 − bs 1 − bs 2

(1)

Suppose (2.3) holds, then we obtain d (Tx, Ty ) ≤ c [ d ( x, Ty ) + d ( y , Tx ) ] ≤ cs[ d ( x, y ) + d ( y , Ty )] + cs[ d ( y , Ty ) + d (Ty , Tx )] ≤ csd ( x, y ) + 2csd ( y , Ty ) + csd (Tx, Ty ) 2cs cs d (Tx, Ty ) ≤ d ( y , Ty ) + d ( x, y ) 1 − cs 1 − cs Similarly, we have d (Tx, Ty ) ≤ c [ d ( x, Ty ) + d ( y, Tx) ]

(2 a )

≤ cs[d ( x, Tx) + d (Tx, Ty )] + cs[d ( y, x) + d ( x, Tx)] ≤ csd ( x, y ) + 2csd ( x, Tx) + csd (Tx, Ty ) 2cs cs d (Tx, Ty ) ≤ d ( x, Tx) + d ( x, y ) 1 − cs 1 − cs bs b(1 + s 2 ) cs , , }. It is easy to see that δ ∈ [0, 1) . Let δ = max{a, 1 − bs 2 2(1 − bs 2 ) 1 − cs If T satisfies at least one of the conditions (2.1), (2.2) and (2.3), then

(2b)

and

d (Tx, Ty ) ≤ 2δ d ( x, Tx ) + δ d ( x, y )

(3a)

d (Tx, Ty ) ≤ 2δ d ( y, Ty ) + δ d ( x, y )

(3b)

hold. Using these conditions implied by (2.1)-(2.3), we obtain d (T n x, T n +1 x) = d (T (T n −1 x), T (T n x))

≤ 2δ d (T n −1 x, T (T n −1 x)) + δ d (T n −1 x, T n x) = (3δ )d (T n −1 x, T n x) ⇒ d (T n x, T n +1 x) ≤ ...(3 δ ) n d ( x, Tx) ∴ d (T n x, T n +1 x) → 0 as n → ∞, ∀x ∈ X . Now by Lemma 3.1 it follows that ∀ε > 0, Fixε (T ) ≠ φ . Theorem 3.8. Let ( X , d ) be a b-metric space and T : X → X a Zamfirescu operator. Then for each ε > 0 , the diameter of Fixε (T ) is not larger


than

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⎧ sε (1 + s + 2δ s ) bs b(1 + s 2 ) cs ⎫ , where δ = max ⎨a, , , ⎬. 2 2 1−δ ⎩ 1 − bs 2(1 − bs ) 1 − cs ⎭

Proof: In the proof of Theorem 3.7 we have already shown that if T satisfies at least one of the conditions (2.1), (2.2) or (2.3), then d (Tx, Ty ) ≤ 2δ d ( x, Tx) + δ d ( x, y ) and d (Tx, Ty ) ≤ 2δ d ( y, Ty ) + δ d ( x, y ) hold. We know that T has the approximate fixed point property, so we can take x and y any two ε -fixed point of T , then d ( x , y ) ≤ s [ d ( x , Tx ) + d ( Tx , y )] ≤ s ε + s 2 [ d ( Tx , Ty ) + d ( Ty , y )] ≤ s ε + s 2 d ( Tx , Ty ) + s 2 ε ≤ s ε + s 2 ε + s 2 2 δ d ( x , Tx ) + δ d ( x , y ) ≤ s ε + s 2 ε + s 2 2 δε + δ d ( x , y ) sε (1 + s + 2δ s ) d ( x, y ) ≤ , for each ε > 0. ∴ 1−δ This completes the proof.

If we put s = 1 in Theorem 3.3 we obtain the result of Berinde [4]. Corollary 3.4 [4]. Let ( X , d ) be a metric space and T : X → X a Zamfirescu operator. Then for each ε > 0 , the diameter of Fixε (T ) is not larger b c 2ε (1 + δ ) , where δ = max{a, than , }. 1− b 1− c 1−δ

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