We generalize and extend Brosowski-Meinardus type results on invariant points from the set of best approximation to the set of ε-simultaneous approximation. As a consequence some results on ε-approximation and best approximation are also deduced. The results proved in this paper generalize and extend some of the known results on the subject.

1. Introduction and Preliminaries

Fixed point theory has gained impetus, due to its wide range of applicability, to resolve diverse problems emanating from the theory of nonlinear differential equations, theory of nonlinear integral equations, game theory, mathematical economics, control theory, and so forth. For example, in theoretical economics, such as general equilibrium theory, a situation arises where one needs to know whether the solution to a system of equations necessarily exists; or, more specifically, under what conditions will a solution necessarily exist. The mathematical analysis of this question usually relies on fixed point theorems. Hence finding necessary and sufficient conditions for the existence of fixed points is an interesting aspect.

Fixed point theorems have been used in many instances in best approximation theory. It is pertinent to say that in best approximation theory, it is viable, meaningful, and potentially productive to know whether some useful properties of the function being approximated is inherited by the approximating function. The idea of applying fixed point theorems to approximation theory was initiated by Meinardus [1]. Meinardus introduced the notion of invariant approximation in normed linear spaces. Brosowski [2] proved the following theorem on invariant approximation using fixed point theory by generalizing the result of Meinardus [1].

Theorem 1.1. Let T be a linear and nonexpansive operator on a normed linear space E. Let C be a T-invariant subset of E and x a T-invariant point. If the set P_C(x) of best C-approximants to x is nonempty, compact, and convex, then it contains a T-invariant point.

Subsequently, various generalizations of Brosowski′s results appeared in the literature. Singh [3] observed that the linearity of the operator T and convexity of the set P_C(x) in Theorem 1.1 can be relaxed and proved the following.

Theorem 1.2. Let T : E → E be a nonexpansive self-mapping on a normed linear space E. Let C be a T-invariant subset of E and x a T-invariant point. If the set P_C(x) is nonempty, compact, and star shaped, then it contains a T-invariant point.

Singh [4] further showed that Theorem 1.2 remains valid if T is assumed to be nonexpansive only on P_C(x)∪{x}. Since then, many results have been obtained in this direction (see Chandok and Narang [5, 6], Mukherjee and Som [7], Mukherjee and Verma [8], Narang and Chandok [9–11], Rao and Mariadoss [12], and references cited therein).

In this paper we prove some similar types of results on T-invariant points for the set of ɛ-simultaneous approximation in a metric space (X, d). Some results on T-invariant points for the set of ɛ-approximation and best approximation are also deduced. The results proved in the paper generalize and extend some of the results of [6, 8–13] and of few others.

Let G be a nonempty subset of a metric space (X, d) and let F be a nonempty bounded subset of X. For x ∈ X, let d_F(x) = sup {d(y, x) : y ∈ F}, D(F, G) = inf {d_F(x) : x ∈ G} and P_G(F) = {g₀ ∈ G : d_F(g₀) = D(F, G)}. An element g₀ ∈ P_G(F) is said to be a best simultaneous approximation of F with respect to G.

For ɛ > 0, we define P_G(ɛ)(F) = {g₀ ∈ G : d_F(g₀) ≤ D(F, G) + ɛ} = {g₀ ∈ G : sup _y∈Fd(y, g₀) ≤ inf _g∈Gsup _y∈Fd(y, g) + ɛ}. An element g₀ ∈ P_G(ɛ)(F) is said to be a ɛ-simultaneous approximation of F with respect to G.

It can be easily seen that for ɛ > 0, the set P_G(ɛ)(F) is always a nonempty bounded set and is closed if G is closed.

In case F = {p}, p ∈ X, then elements of P_G(p) are called best approximations to p in G and of P_G(ɛ)(p) are called ɛ-approximation to p in G.

A sequence 〈y_n〉 in G is called a ɛ-minimizing sequence for F, if lim sup _x∈Fd(x, y_n) ≤ D(F, G) + ɛ. The set G is said to be ɛ-simultaneous approximatively compact with respect to F if for every x ∈ F, each ɛ-minimizing sequence 〈y_n〉 in G has a subsequence converging to an element of G.

Let (X, d) be a metric space. A continuous mapping W : X × X × [0,1] → X is said to be a convex structure on X if for all x, y ∈ X and λ ∈ [0,1],

(1.1)

holds for all u ∈ X. The metric space (X, d) together with a convex structure is called a convex metric space [14].

A convex metric space (X, d) is said to satisfy Property (I) [15] if for all x, y, p ∈ X and λ ∈ [0,1],

(1.2)

A normed linear space and each of its convex subset are simple examples of convex metric spaces with W given by W(x, y, λ) = λx + (1 − λ)y for x, y ∈ X and 0 ≤ λ ≤ 1. There are many convex metric spaces which are not normed linear spaces (see [14]). Property (I) is always satisfied in a normed linear space.

A subset K of a convex metric space (X, d) is said to be

(i)
a convex set [14] if W(x, y, λ) ∈ K for all x, y ∈ K and λ ∈ [0,1];
(ii)
p-star shaped [16] where p ∈ K, provided W(x, p, λ) ∈ K for all x ∈ K and λ ∈ [0,1];
(iii)
star shaped if it is p-star shaped for some p ∈ K.

Clearly, each convex set is star shaped but not conversely.

A self-map T on a metric space (X, d) is said to be

(i)
contraction if there exists k, 0 ≤ k < 1 such that d(Tx, Ty) ≤ kd(x, y) for all x, y ∈ X;
(ii)
nonexpansive if d(Tx, Ty) ≤ d(x, y) for all x, y ∈ X;
(iii)
quasi-nonexpansive if the set F(T) of fixed points of T is nonempty and d(Tx, p) ≤ d(x, p) for all x ∈ X and p ∈ F(T).

A nonexpansive mapping T on X with F(T) ≠ ∅ is quasi-nonexpansive, but not conversely. A linear quasi-nonexpansive mapping on a Banach space is nonexpansive. But there exist continuous and discontinuous nonlinear quasi-nonexpansive mappings that are not nonexpansive.

2. Main Results

To start with, we prove the following proposition on ɛ-simultaneous approximation which will be used in the sequel.

Proposition 2.1. Let F be a nonempty bounded subset of a metric space (X, d), and let C be a non-empty subset of X. If C is ɛ-simultaneous approximatively compact with respect to F, then the set P_C(ɛ)(F) is a nonempty compact subset of C.

Proof. Since ɛ > 0, P_C(ɛ)(F) is nonempty. We now show that P_C(ɛ)(F) is compact. Let 〈y_n〉 be a sequence in P_C(ɛ)(F). Then lim sup _x∈Fd(x, y_n) ≤ D(F, C) + ɛ, that is, 〈y_n〉 is an ɛ-minimizing sequence for C. Since C is ɛ-simultaneous approximatively compact with respect to F, there is a subsequence such that . Consider

(2.1)

This implies that y ∈ P_C(ɛ)(F). Thus we get a subsequence

of 〈y_n〉 converging to an element y ∈ P_C(ɛ)(F). Hence P_C(ɛ)(F) is compact.

For F = {x}, we have the following result on the set of ɛ-approximation.

Corollary 2.2 (see [9].)If C is an ɛ-approximatively compact set in a metric space (X, d) then P_C(ɛ)(x) is a nonempty compact set.

For F = {x} and ɛ = 0, we have the following result on the set of best approximation.

Corollary 2.3 (see [10].)Let C be a nonempty approximatively compact subset of a metric space (X, d), x ∈ X, and P_C be the metric projection of X onto C defined by P_C(x) = {y ∈ C : d(x, y)} ≡ d(x, C). Then P_C(x) is a nonempty compact subset of C.

We will be using the following result of Hardy and Rogers [17] in proving our first theorem.

Lemma 2.4. Let T be a mapping from a complete metric space (X, d) into itself satisfying

(2.2)

for any x, y ∈ X, where a, b, and c are nonnegative numbers such that 2a + 2b + c ≤ 1. Then T has a unique fixed point u in X. In fact for any x ∈ X, the sequence {Tⁿx} converges to u.

Theorem 2.5. Let T be a continuous self-map on a complete convex metric space (X, d) with Property (I) and satisfying inequality (2.2), let C be a T-invariant subset of X, and let F be a nonempty bounded subset of X such that Tx = x for all x ∈ F. If P_C(ɛ)(F) is compact, and star shaped, then it contains a T-invariant point.

Proof. Let z ∈ P_C(ɛ)(F) be arbitrary. Then by (2.2), we have for all x ∈ F

(2.3)

This gives

(2.4)

since 2a + 2b + c ≤ 1. Therefore, using definition of P_C(ɛ)(F), we get

(2.5)

Hence Tz ∈ P_C(ɛ)(F). Therefore T is a self-map on P_C(ɛ)(F).

Let q be the star-center of P_C(ɛ)(F). Define T_n : P_C(ɛ)(F) → P_C(ɛ)(F) as T_nx = W(Tx, q, λ_n), x ∈ P_C(ɛ)(F) where 〈y_n〉 is a sequence in (0,1) such that λ_n → 1. Consider

(2.6)

where (2a + 2b + c) ≤ 1. Therefore by Lemma 2.4, each T_n has a unique fixed point z_n in P_C(ɛ)(F). Since P_C(ɛ)(F) is compact, there is a subsequence

of 〈z_n〉 such that

. We claim that Tz_∘ = z_∘. Consider

, as T is continuous. Thus

and consequently, Tz_∘ = z_∘, that is, z_∘ ∈ P_C(ɛ)(F) is a T-invariant point.

Since for an ɛ-simultaneous approximatively compact subset C of a metric space (X, d) the set of ɛ-simultaneous C-approximant is nonempty and compact (Proposition 2.1), we have the following result.

Corollary 2.6. Let T be a continuous self-map on a complete convex metric space (X, d) with Property (I) and satisfying inequality (2.2), let F be a nonempty bounded subset of X such that Tx = x for all x ∈ F, and let C be a T-invariant subset of X. If C is ɛ-simultaneous approximatively compact with respect to F and P_C(ɛ)(F) is star shaped, then it contains a T-invariant point.

Corollary 2.7 (see [8].)Let T be a continuous self-map on a Banach space X satisfying (2.2), let C be an approximatively compact and T-invariant subset of X. Let Tx_i = x_i(i = 1,2) for some x₁, x₂ not in cl(C). If the set of best simultaneous C-approximants to x₁, x₂ is star shaped, then it contains a T-invariant point.

Corollary 2.8 (see [11].)Let T be a mapping on a metric space (X, d), let C be a T-invariant subset of X and x a T-invariant point. If P_C(x) is a nonempty, compact set for which there exists a contractive jointly continuous family 𝔉 of functions and T is nonexpansive on P_C(x)∪{x} then P_C(x) contains a T-invariant point.

Corollary 2.9. Let T be a mapping on a convex metric space (X, d) with Property (I),let C be an approximatively compact, p-star shaped, T-invariant subset of X and let x be a T-invariant point. If T is nonexpansive on P_C(x)∪{x}, then P_C(x) contains a T-invariant point.

Corollary 2.10 (see [10], Theorem 4.)Let T be a quasi-nonexpansive mapping on a convex metric space (X, d) with Property (I), let C be a T-invariant subset of X, and let x be a T-invariant point. If P_C(x) is nonempty, compact, and star shaped, and T is nonexpansive on P_C(x), then P_C(x) contains a T-invariant point.

Corollary 2.11 (see [10], Theorem 5.)Let T be a quasi-nonexpansive mapping on a convex metric space (X, d) with Property (I), let C be an approximatively compact, T-invariant subset of X, and let x be a T-invariant point. If P_C(x) is star shaped and T is nonexpansive on P_C(x), then P_C(x) contains a T-invariant point.

Remark 2.12. Theorem 2.5 improves and generalizes Theorem 1 of Narang and Chandok [9] and of Rao and Mariadoss [12].

Definition 2.13. A subset K of a metric space (X, d) is said to be contractive if there exists a sequence 〈f_n〉 of contraction mappings of K into itself such that f_ny → y for each y ∈ K.

Theorem 2.14. Let T be a nonexpansive self-mapping on a metric space (X, d), let C be a T-invariant subset of X, and let F be a nonempty bounded subset of X such that Tx = x for all x ∈ F. If the set P_C(ɛ)(F) is compact and contractive, then the set P_C(ɛ)(F) contains a T-invariant point.

Proof. Proceeding as in Theorem 2.5, we can prove that T is a self-map of P_C(ɛ)(F). Since P_C(ɛ)(F) is contractive, there exists a sequence 〈f_n〉 of contraction mapping of P_C(ɛ)(F) into itself such that f_nz → z for every z ∈ P_C(ɛ)(F).

Clearly, f_nT is a contraction on the compact set P_C(ɛ)(F) for each n and so by Banach contraction principle, each f_nT has a unique fixed point, say z_n in P_C(ɛ)(F). Now the compactness of P_C(ɛ)(F) implies that the sequence 〈z_n〉 has a subsequence . We claim that z_∘ is a fixed point of T. Let ɛ > 0 be given. Since and f_nTz_∘ → Tz_∘, there exist a positive integer m such that for all n_i ≥ m

(2.7)

Again,

(2.8)

Hence

(2.9)

that is,

for all n_i ≥ m and so

. But

and therefore Tz_∘ = z_∘.

Using Proposition 2.1 we have the following result.

Corollary 2.15. Let T be a nonexpansive self-mapping on a metric space (X, d), let C be a T-invariant subset of X, and let F be a nonempty bounded subset of X such that Tx = x for all x ∈ F. If C is ɛ-simultaneous approximatively compact with respect to F and the set P_C(ɛ)(F) is contractive, and T-invariant, then P_C(ɛ)(F) contains a T-invariant point.

Corollary 2.16 (see [9].)Let T be a self-mapping on a metric space (X, d), let G be a T-invariant subset of X, and let x be a T-invariant point. If the set D of ɛ-approximant to x is compact, contractive and T is nonexpansive on D ∪ {x}, then D contains a T-invariant point.

Remark 2.17. Theorem 2.14 also improves and generalizes the corresponding results of Brosowski [2], Mukherjee and Verma [8, 13], Chandok and Narang [9], Rao and Mariadoss [12], and of Singh [3].

Definition 2.18. For each bounded subset G of a metric space (X, d), the Kuratowski′s measure of noncompactness of G, α[G] is defined as

(2.10)

A mapping T : X → X is called condensing if for all bounded sets G ⊂ X, α[T(G)] ≤ α[G].

We will be using the following result of [18] on fixed points of nonexpansive condensing maps.

Lemma 2.19. Let X be a complete contractive metric space with contractions {f_n}. Let C be a closed bounded subsets of X and T : C → C is nonexpansive and condensing, then T has a fixed point in C.

Using the above lemma and Theorem 2.5, we now prove the following result.

Theorem 2.20. Let (X, d) be a complete, contractive metric space with contractions f_n. Let G be a closed and bounded subset of X and let F be a nonempty bounded subset of X. If T is a nonexpansive and condensing self-map on X such that Tx = x for all x ∈ F, then P_G(ɛ)(F) has a T-invariant point.

Proof. As G is closed and bounded, P_G(ɛ)(F) is nonempty, closed and bounded. Using Theorem 2.5, we can prove that T is a self-map of P_G(ɛ)(F). Now a direct application of Lemma 2.19, gives a T-invariant point in P_G(ɛ)(F).

Corollary 2.21 (see [8], Theorem 3.1.)Let X be a complete, contractive metric space with contractions f_n. Let G be a closed and bounded subset of X. If T is a nonexpansive and condensing self-map on X such that Tx₁ = x₁ and Tx₂ = x₂ for some x₁, x₂ ∈ X, and D = P_G(x₁, x₂) is nonempty, then it has a T-invariant point.

Corollary 2.22 (see [12], Theorem 4.)Let X be a complete, contractive metric space with contractions f_n. Let G be a closed and bounded subset of X. If T is a nonexpansive and condensing self-map on X such that Tx = x for some x ∈ X, and P_G(x) is nonempty, then it has a T-invariant point.

Definition 2.23. A mapping T on a metric space (X, d) is called a Kannan mapping [19] if there exists α ∈ (0, 1/2) such that

(2.11)

for all x, y ∈ X.

Kannan [19] proved that if X is complete, then every Kannan mapping has a unique fixed point.

For ɛ-simultaneous approximation, we have the following result.

Theorem 2.24. Let G be a nonempty subset of a complete metric space (X, d) and let F be a nonempty bounded subset of X. Let T be a self-map on X with Tx = x for all x ∈ F and T^m satisfies,

(2.12)

for some positive integer m, all y, z ∈ X and some fixed 0 < α < 1/2. If D = P_G(ɛ)(F) is compact, then it has a unique fixed point of T.

Proof. As Tx = x, Tⁿx = x for all positive integers n. Let y₀ ∈ D. Then, for 0 < α < 1/2,

(2.13)

which implies that

(2.14)

Further, we have

(2.15)

Therefore, T^my₀ ∈ D, T^m(D) ⊂ D. Since T^m satisfies the conditions of Kannan map, T^m has a unique fixed point x₀ in D. Now, T^m(Tx₀) = T(T^mx₀) = Tx₀, implies that Tx₀ is a fixed point of T^m. But the fixed point of T^m is unique and equals x₀. Therefore Tx₀ = x₀ and hence x₀ is a unique fixed point of T in D.

Corollary 2.25. Let F be a nonempty bounded subset of a complete metric space (X, d) and G a subset of X. Let G be ɛ-simultaneous approximatively compact with respect to F, and T a self map on X with T_x = x for all x ∈ F, and T^m satisfies

(2.16)

for some positive integer m, all y, z ∈ X and some fixed 0 < α < 1/2 then D = P_G(ɛ)(F) has a unique fixed point of T.

Remarks 2.26. Theorem 2.24 extends and generalizes Theorem 3.2 of Mukherjee and Verma [8] and Theorem 5 of Rao and Mariadoss [12] from the set of best simultaneous approximation and best approximation, respectively, to ɛ-simultaneous approximation.

For ɛ > 0, we define R_G(ɛ)(F) = {g₀ ∈ G : sup _g∈Gd(g, g₀) + ɛ≤inf _g∈Gsup _y∈Fd(y, g)}. An element g₀ ∈ R_G(ɛ)(F) is said to be a ɛ-simultaneous coapproximation of F with respect to G.

A mapping T : X → X satisfies condition (A) (see [13]) if d(Tx, y) ≤ d(x, y) for all x, y ∈ X.

We now prove a result for T-invariant points from the set of ɛ-simultaneous coapproximations.

Theorem 2.27. Let T be a self-map satisfying condition (A) and inequality (2.2) on a convex metric space (X, d) satisfying Property (I), let G be a subset of X, and let F be a nonempty bounded subset of X such that R_G(ɛ)(F) is compact and star shaped. Then R_G(ɛ)(F) contains a T-invariant point.

Proof. Let g_∘ ∈ R_G(ɛ)(F). Consider

(2.17)

and so Tg_∘ ∈ R_G(ɛ)(F), that is, T : R_G(ɛ)(F) → R_G(ɛ)(F). Since R_G(ɛ)(F) is star shaped, there exists p ∈ R_G(ɛ)(F) such that W(z, p, λ) ∈ R_G(ɛ)(F) for all z ∈ R_G(ɛ)(F), λ ∈ [0,1]. Let 〈k_n〉, 0 ≤ k_n < 1, be a sequence of real numbers such that k_n → 1 as n → ∞. Define T_n as T_n(z) = W(Tz, p, k_n), z ∈ R_G(ɛ)(F). Since T is a self-map on R_G(ɛ)(F) and R_G(ɛ)(F) is star shaped, each T_n is a well defined and maps R_G(ɛ)(F) into R_G(ɛ)(F). Moreover,

(2.18)

where k_n[2a + 2b + c] ≤ 1. So by Lemma 2.4 each T_n has a unique fixed point x_n ∈ R_G(ɛ)(F), that is, T_nx_n = x_n for each n. Since R_G(ɛ)(F) is compact, 〈x_n〉 has a subsequence

. Now, we claim that Tx = x. As

. On letting n → ∞, we have

. Therefore Tx = x, that is, x is T-invariant, hence the result.

For F = {x}, x ∈ X, we have the following result on the set of ɛ-coapproximation.

Corollary 2.28 . (see [9], Theorem 4.)Let T be a self-map satisfying condition (A) on a convex metric space (X, d) satisfying Property (I), let G be a subset of X such that R_G(ɛ)(x) is nonempty compact, star shaped, and let T be nonexpansive on R_G(ɛ)(x). Then there exists a g_∘ ∈ R_G(ɛ)(x) such that Tg_∘ = g_∘.

Remarks 2.29. (i) Theorem 2.27 also improves and generalizes Theorem 4.1 of Mukherjee and Verma [13] from the set of best approximation to ɛ-simultaneous approximation.

(ii) By taking F = {x₁, x₂}, x₁, x₂ ∈ X, the set P_G(ɛ)(F) (respectively, R_G(ɛ)(F) ) is the set of ɛ-simultaneous approximation (respectively, ɛ-simultaneous coapproximation) to the pair of points x₁, x₂ and so the results of this paper generalize and extend the corresponding results proved in [6].

Acknowledgments

The authors are thankful to the learned referee for careful reading and very valuable suggestions leading to an improvement of the paper. The research work of the second author has been supported by the UGC India under the Emeritus Fellowship.

References

1 Meinardus G., Invarianz bei linearen approximationen, Archive for Rational Mechanics and Analysis. (1963) 14, 301–303, 0156143, ZBL0122.30801.
10.1007/BF00250708
Web of Science® Google Scholar
2 Brosowski B., Fixpunktsätze in der approximationstheorie, Mathematica. (1969) 11, 195–220, 0277979, ZBL0207.45502.
Google Scholar
3 Singh S. P., An application of a fixed-point theorem to approximation theory, Journal of Approximation Theory. (1979) 25, no. 1, 89–90, https://doi.org/10.1016/0021-9045(79)90036-4, 526280, ZBL0399.41032.
10.1016/0021-9045(79)90036-4
Web of Science® Google Scholar
4 Singh S. P., V. Lakshmikantham, Application of fixed point theorems in approximation theory, Applied Nonlinear Analysis, 1979, Academic Press, New York, NY, USA, 389–394, 537550, ZBL0448.47045.
10.1016/B978-0-12-434180-7.50038-3
Google Scholar
5 Chandok S. and Narang T. D., Common fixed points of nonexpansive mappings with applications to best and best simultaneous approximation, to appear in Journal of Applied Analysis, vol. 18, 2012.
Google Scholar
6 Chandok S. and Narang T. D., ε-simultaneous approximation and invariant points, to appear in The Bulletin of the Belgian Mathematical Society, vol. 19, no. 1, 2012.
Google Scholar
7 Mukherjee R. N. and Som T., A note on an application of a fixed point theorem in approximation theory, Indian Journal of Pure and Applied Mathematics. (1985) 16, no. 3, 243–244, 785288, ZBL0606.41048.
Web of Science® Google Scholar
8 Mukherjee R. N. and Verma V., Some fixed point theorems and their applications to best simultaneous approximations, Publications de l′Institut Mathématique. (1991) 49, 111–116, 1127387, ZBL0744.41025.
Google Scholar
9 Narang T. D. and Chandok S., On ε-approximation and fixed points of nonexpansive mappings in metric spaces, Matematichki Vesnik. (2009) 61, no. 2, 165–171, 2511233.
Google Scholar
10 Narang T. D. and Chandok S., Fixed points of quasi-nonexpansive mappings and best approximation, Selçuk Journal of Applied Mathematics. (2009) 10, no. 2, 75–80, 2589085, ZBL1183.41030.
Google Scholar
11 Narang T. D. and Chandok S., Fixed points and best approximation in metric spaces, Indian Journal of Mathematics. (2009) 51, no. 2, 293–303, 2537948, ZBL1180.41024.
Google Scholar
12 Rao G. S. and Mariadoss S. A., Applications of fixed point theorems to best approximations, Serdica. Bulgaricae Mathematicae Publicationes. (1983) 9, no. 3, 244–248, 744154, ZBL0553.41034.
Google Scholar
13 Mukherjee R. N. and Verma V., Best approximations and fixed points of nonexpansive maps, Bulletin of the Calcutta Mathematical Society. (1989) 81, no. 3, 191–196, 1008610.
Google Scholar
14 Takahashi W., A convexity in metric space and nonexpansive mappings I, Kodai Mathematical Seminar Reports. (1970) 22, 142–149, 0267565, https://doi.org/10.2996/kmj/1138846111, ZBL0268.54048.
10.2996/kmj/1138846111
Google Scholar
15 Guay M. D., Singh K. L., and Whitfield J. H. M., S. P. Singh and J. H. Bury, Fixed point theorems for nonexpansive mappings in convex metric spaces, 80, Proceedings of the International Conference on Nonlinear Analysis, 1982, Marcel Dekker, 179–189.
Google Scholar
16 Itoh S., Some fixed-point theorems in metric spaces, Fundamenta Mathematicae. (1979) 102, no. 2, 109–117, 525934, ZBL0412.54054.
10.4064/fm-102-2-109-117
Google Scholar
17 Hardy G. E. and Rogers T. D., A generalization of a fixed point theorem of Reich, Canadian Mathematical Bulletin. (1973) 16, 201–206, 0324495, ZBL0266.54015.
10.4153/CMB-1973-036-0
Web of Science® Google Scholar
18 Chandler E. and Faulkner G., A fixed point theorem for nonexpansive, condensing mappings, Journal of Australian Mathematical Society A. (1980) 29, no. 4, 393–398, 578698, https://doi.org/10.1017/S1446788700021546, ZBL0402.47034.
10.1017/S1446788700021546
Google Scholar
19 Kannan R., Some results on fixed points. II, The American Mathematical Monthly. (1969) 76, 405–408, 0257838, https://doi.org/10.2307/2316437, ZBL0179.28203.
10.2307/2316437
Web of Science® Google Scholar

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Invariant Points and ε-Simultaneous Approximation

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Invariant Points and ε-Simultaneous Approximation

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