Ann. Acad. Rom. Sci.

Ser. Math. Appl.

ISSN 2066-6594

Vol. 18, No. 2/2026

SOLVING COMMON FIXED POINT

PROBLEMS WITH SUMMABLE ERRORS^∗

Simeon Reich^†

Alexander J. Zaslavski^‡

In memory of Professor Ravi P. Agarwal

Communicated by G. Moro¸sanu

DOI

10.56082/annalsarscimath.2026.2.215

Abstract

We study iterative methods for solving common ﬁxed point prob-

lems in the presence of summable computational errors.

Keywords: approximate ﬁxed point, inﬁnite product, metric space,

nonexpansive mapping.

MSC: 47H09, 47H10, 54E50.

1 Introduction

For more than sixty-ﬁve years now, there has been a lot of research activity

regarding the ﬁxed point theory of nonexpansive (that is, 1-Lipschitz) and

contractive mappings. See, for example, [1 ,6–10,13 –15,17 ,18] and references

cited therein. In the present paper, we study iterative methods for solving

common ﬁxed point problems in the presence of summable computational

errors.

Let (X, ρ) be a complete metric space. For each point x ∈ X and each

number r > 0, set

B(x, r) := {y ∈ X : ρ(x, y) ≤ r}.

^∗Accepted for publication on February 28, 2026

^†sreich@technion.ac.il, Department of Mathematics, The Technion – Israel Institute

of Technology, 32000 Haifa, Israel

^‡ajzasl@technion.ac.il, Department of Mathematics, The Technion – Israel Institute

of Technology, 32000 Haifa, Israel

215

Common ﬁxed point problems with summable errors

216

In his seminal paper [12], Ostrowski established the following result.

Theorem 1. Assume that γ ∈ (0.1), the mapping T : X → X satisﬁes

ρ(T(x), T(y)) ≤ γρ(x, y), x, y ∈ X,

and that a sequence {x_i}^∞satisﬁes

i=0

∞

X

ρ(x_i+1, T(x_i)) < ∞.

i=0

Then the sequence {x_i}^∞converges to a ﬁxed point of T.

i=0

In [3], the following generalization of Ostrowski’s theorem was obtained.

Theorem 2. Assume that a mapping T : X → X satisﬁes

ρ(T(x), T(y)) ≤ ρ(x, y), x, y ∈ X,

and that for each point x ∈ X, the sequence {Tⁱ(x)}^∞_i=1converges to a ﬁxed

i=0

point of T. Then each sequence {x_i}^∞⊂ X satisfying

∞

X

ρ(x_i+1, T(x_i)) < ∞

i=0

converges to a ﬁxed point of T.

This result became the starting point of the superiorization methodology,

where it found many applications [2,4,5]. It means that if all exact iterates

of a nonexpansive mapping converge, then the same holds true for all its

inexact iterates with summable errors.

Let T : X → X be a self-mapping of X such that

ρ(T(x), T(y)) ≤ ρ(x, y) for all x, y ∈ X.

Set T⁰(x) := x for all x ∈ X. Since the existence of a ﬁxed point of T is not

guaranteed in general, we are also interested in approximate ﬁxed points.

A point x ∈ X is called a (γ)-approximate ﬁxed point of the operator

T, where γ > 0, if ρ(x, T(x)) ≤ γ. Note that if C is a bounded, closed and

convex subset of a Banach space and T is a nonexpansive self-mapping of

C, then a (γ)-approximate ﬁxed point of T does exist for each γ > 0. As

a matter of fact, there are also unbounded sets C with this property. For

more information regarding this issue see, for instance, [16 ], [8] and [11].

S. Reich, A.J. Zaslavski

217

In the following theorem, which was obtained in [19], we assume that

for an arbitrary number ꢀ > 0 and each point x ∈ X, the iterates Tⁿ(x)

are (ꢀ)-approximate ﬁxed points of the operator T for all suﬃciently large

n. Under this assumption it is shown that for each ꢀ > 0 and each sequence

{x_n}^∞_n=0of inexact iterates of the operator T with summable errors, x_nis

an (ꢀ)-approximate ﬁxed point of T for all suﬃciently large natural numbers

n. Therefore, inexact iterates of T with summable errors have the same

asymptotic behavior as its exact iterates.

Theorem 3. Assume that for every point x ∈ X and every number ꢀ > 0,

there exists a natural number n₀such that for all natural numbers n ≥ n₀,

we have

ρ(Tⁿ⁺¹(x), Tⁿ(x)) ≤ ꢀ.

i=1

Let a sequence of numbers {ꢀ_i}^∞⊂ (0, ∞) satisfy

∞

X

ꢀ_i< ∞

i=1

and let a sequence of points {x_i}^∞⊂ X satisfy

i=0

ρ(x_i+1, T(x_i)) ≤ ꢀ_i+1for each integer i ≥ 0.

Then for each positive number ꢀ, there exists a natural number n₀such that

ρ(x_i, T(x_i)) ≤ ꢀ for each natural number i ≥ n₀.

It was shown in [19 ] that the assumption (and hence the conclusion) of

this result holds if

F := {z ∈ X : T(z) = z} = ∅

and

ρ(z, T(x))²+ c¯ρ(x, T(x))²≤ ρ(z, x)²,

for all z ∈ F and all x ∈ X, where the constant c¯ ∈ (0, 1]. (Note that these

properties hold if T is the nearest point projection onto a closed and convex

subset of a Hilbert space.)

In the present paper, we extend Theorem 3 to common ﬁxed point prob-

lems.

2 Common ﬁxed point problems

¯

Assume that (X, ρ) is a complete metric space, N ≥ m are natural numbers

and that for each i ∈ {1, . . . , m}, the operator T_i: X → X satisﬁes

ρ(T_i(x), T_i(y)) ≤ ρ(x, y), x, y ∈ X.

(1)

Common ﬁxed point problems with summable errors

218

Let the mapping

r : {0, 1, . . . , } → {1, . . . , m}

satisfy, for each integer i ≥ 0,

¯

{1, . . . , m} ⊂ r({i, . . . , i + N − 1}).

(2)

We consider the following problem:

Find x ∈ X such that T_i(x) = x, i = 1, . . . , m.

Since in general the existence of a solution to this problem is not guaranteed,

in this paper we are interested in approximate solutions to this problem, that

is, points y ∈ X which satisfy

ρ(y, T_i(y)) ≤ γ, i = 1, . . . , m,

where γ is a small positive number. In order to obtain such an approximate

solution we ﬁrst choose a point x ∈ X and then deﬁne

x₀= x,

x_i+1= T_r(i)(x_i), i = 0, 1, . . . .

We assume that this iterative process produces approximate solutions to our

common ﬁxed point problem. This fact indeed holds for many important

common ﬁxed point problems [20]. Namely, we assume that the following

property holds.

(P) For each x ∈ X and each integer k ≥ 0,

n

n+1

Y

lim ρ(

T_r(i+k)(x),

T_r(i+k)(x)) = 0.

n→∞

i=0

Note that property (P) indeed holds if for each i ∈ {1, . . . , m} and each

x ∈ X, we have

ρ(z, T_i(x))²+ c¯ρ(x, T_i(x))²≤ ρ(z, x)²,

where z is a common ﬁxed point and the constant c¯ ∈ (0, 1] [20].

In the present paper, we establish the following result.

S. Reich, A.J. Zaslavski

219

Theorem 4. Assume that {x_n}^∞_n=0⊂ X, {ꢀ_n}^∞_n=0⊂ (0, ∞),

∞

X

ꢀ_n< ∞

(3)

(4)

n=0

and that for each integer n ≥ 0, we have

ρ(x_n+1, T_r(n)(x_n)) ≤ ꢀ_n.

Then

lim ρ(x_n, x_n+1) = 0

n→∞

and for each i ∈ {1, . . . , m},

lim ρ(x_n, T_i(x_n)) = 0.

n→∞

Proof. Let ꢀ > 0 be given. By (3), there exists a natural number k such that

∞

X

ꢀ_n< ꢀ/4.

(5)

n=k

Property (P) implies that

n

n+1

Y

lim ρ(

n→∞

i=0

T_r(i+k)(x_k),

T_r(i+k)(x_k)) = 0.

(6)

i=0

In view of (6), there exists a natural number n₁such that for each integer

n ≥ n₁, we have

n

n+1

Y

ρ(

T_r(i+k)(x_k),

T_r(i+k)(x_k)) < ꢀ/4.

(7)

i=0

We now estimate

n−1

Y

ρ(

T_r(i+k)(x_k), x_k+n), n = 1, 2, . . . .

i=0

It follows from (4) that

ρ(x_k+1, T_r(k)(x_k)) ≤ ꢀ_k.

(8)

Common ﬁxed point problems with summable errors

220

(9)

We claim that for each integer n ≥ 1,

n−1

Y

X

ρ(

T_r(i+k)(x_k), x_k+n) ≤ ^k+n−1ꢀ_i.

i=0

i=k

In view of (8), inequality (9) holds for n = 1.

Assume that n ≥ 1 is an integer and that (9) holds. Inequalities (1), (4)

and (9) imply that

n

Y

ρ(

T_r(i+k)(x_k), x_k+n+1) ≤ ρ(x_k+n+1, T_r(n+k)(x_n+k))

i=0

n

Y

+ρ(T_r(n+k)(x_n+k),

T_r(i+k)(x_k))

i=0

n−1

T_r(i+k)(x_k)) ≤ ^k+nꢀ_i

Y

X

≤ ꢀ_k+n+ ρ(x_n+k

,

i=0

i=k

and so (9) holds for n + 1 too. Thus, we have shown by mathematical

induction that inequality (9) holds for each integer n ≥ 1, as claimed. It

now follows from (5) and (9) that for each integer n ≥ 1, we have

n−1

Y

ρ(

T_r(i+k)(x_k), x_k+n) < ꢀ/4.

(10)

i=0

By (7) and (10), for each integer n ≥ n₁+ 1,

n−1

Y

ρ(x_n+k, x_n+k+1) ≤ ρ(x_n+k

,

T_r(i+k)(x_k))

i=0

n−1

n

Y

+ρ(

T_r(i+k)(x_k),

T_r(i+k)(x_k)) + ρ(

T_r(i+k)(x_k), x_k+n+1)

i=0

n−1

n

Y

< ꢀ/4 + ρ(

T_r(i+k)(x_k),

T_r(i+k)(x_k)) + ꢀ/4 ≤ 3ꢀ/4.

i=0

Since ꢀ is an arbitrary positive number, we conclude that

lim ρ(x_n, x_n+1) = 0.

(11)

n→∞

S. Reich, A.J. Zaslavski

221

Fix p ∈ {1, . . . , m}. Next, we claim that

lim ρ(x_n, T_p(x_n)) = 0.

n→∞

Let ꢀ > 0. By (3), (4) and (11), there exists a natural number n₀such that

for each integer n ≥ n₀,

−1

¯

ꢀ_n< ꢀ(2N + 2)

(12)

and

−1

¯

ρ(x_n, x_n+1) < ꢀ(2N + 2)

.

(13)

¯

Assume that n ≥ n₀is an integer. By (13), for each i ∈ {n, . . . , n + N},

−1

¯

ρ(x_n, x_i) ≤ Nꢀ(2N + 2)

,

(14)

In view of (2), there is a natural number

¯

j ∈ {n, . . . , n + N − 1}

such that

r(j) = p.

(15)

(16)

By (4), (12), (13) and (15),

ρ(x_j, T_p(x_j)) = ρ(x_j, T_r(j)(x_j)) ≤ ρ(x_j, x_j+1) + ρ(x_j+1, T_r(j)(x_j))

−1

¯

≤ ꢀ(2N + 2)

+ ꢀ_j≤ ꢀ(N + 1)

.

In view of (1), (14) and (16), we have

ρ(x_n, T_p(x_n)) ≤ ρ(x_n, x_j) + ρ(x_j, T_p(x_j)) + ρ(T_p(x_j), T_p(x_n))

−1

¯

≤ 2ρ(x_n, x_j) + ꢀ(N + 1)

≤ 2Nꢀ(2N + 2)

+ ꢀ(N + 1)

≤ ꢀ.

Thus, for each integer n ≥ n₀, we have

ρ(x_n, T_p(x_n)) ≤ ꢀ.

Since ꢀ is an arbitrary positive number, we infer that

lim ρ(x_n, T_px_n)) = 0.

n→∞

This completes the proof of Theorem 4.

Acknowledgments. Simeon Reich was partially supported by the Is-

rael Science Foundation (Grant 820/17), by the Fund for the Promotion of

Research at the Technion (Grant 2001893) and by the Technion General

Research Fund (Grant 2016723).

Common ﬁxed point problems with summable errors

222

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