Invariant Points and 𝜀-Simultaneous Approximation

Abstract

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 𝜀-simultaneous approximation in a metric space (𝑋,𝑑). Some results on 𝑇-invariant points for the set of 𝜀-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 𝜀>0, we define 𝑃_𝐺(𝜀)(𝐹)={𝑔₀∈𝐺∶𝑑_𝐹(𝑔₀)≤𝐷(𝐹,G)+𝜀}={𝑔₀∈𝐺∶sup_𝑦∈𝐹𝑑(𝑦,𝑔₀)≤inf_𝑔∈𝐺sup_𝑦∈𝐹𝑑(𝑦,𝑔)+𝜀}. An element 𝑔₀∈𝑃_𝐺(𝜀)(𝐹) is said to be a 𝜀-simultaneous approximation of 𝐹 with respect to 𝐺.

It can be easily seen that for 𝜀>0, the set 𝑃_𝐺(𝜀)(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(𝜀)(𝑝) are called 𝜺-approximation to 𝑝 in 𝐺.

A sequence ⟨𝑦_𝑛⟩ in 𝐺 is called a 𝜺-minimizing sequence for 𝐹, if limsup_𝑥∈𝐹𝑑(𝑥,𝑦_𝑛)≤𝐷(𝐹,𝐺)+𝜀. The set 𝐺 is said to be 𝜀-simultaneous approximatively compact with respect to 𝐹 if for every 𝑥∈𝐹, each 𝜀-minimizing sequence ⟨𝑦_𝑛⟩ in 𝐺 has a subsequence ⟨𝑦_𝑛𝑖⟩ 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],𝑑𝑢,𝑊𝑥,𝑦,𝜆≤𝜆𝑑(𝑢,𝑥)+(1−𝜆)𝑑𝑢,𝑦 (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], 𝑑𝑊𝑥,𝑝,𝜆,𝑊𝑦,𝑝,𝜆≤𝜆𝑑𝑥,𝑦. (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 𝜀-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 𝜀-simultaneous approximatively compact with respect to 𝐹, then the set 𝑃_𝐶(𝜀)(𝐹) is a nonempty compact subset of 𝐶.

Proof. Since 𝜀>0, 𝑃_𝐶(𝜀)(𝐹) is nonempty. We now show that 𝑃_𝐶(𝜀)(𝐹) is compact. Let ⟨𝑦_𝑛⟩ be a sequence in 𝑃_𝐶(𝜀)(𝐹). Then limsup_𝑥∈𝐹𝑑(𝑥,𝑦_𝑛)≤𝐷(𝐹,𝐶)+𝜀, that is, ⟨𝑦_𝑛⟩ is an 𝜀-minimizing sequence for 𝐶. Since 𝐶 is 𝜀-simultaneous approximatively compact with respect to 𝐹, there is a subsequence ⟨𝑦_𝑛𝑖⟩ such that ⟨𝑦_𝑛𝑖⟩→𝑦∈𝐶. Consider sup_𝑥∈𝐹𝑑𝑥,𝑦=sup_𝑥∈𝐹𝑑𝑥,lim𝑦_𝑛𝑖=limsup_𝑥∈𝐹𝑑𝑥,𝑦_𝑛𝑖≤𝐷(𝐹,𝐶)+𝜀. (2.1) This implies that 𝑦∈𝑃_𝐶(𝜀)(𝐹). Thus we get a subsequence ⟨𝑦_𝑛𝑖⟩ of ⟨𝑦_𝑛⟩ converging to an element 𝑦∈𝑃_𝐶(𝜀)(𝐹). Hence 𝑃_𝐶(𝜀)(𝐹) is compact.

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

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

For 𝐹={𝑥} and 𝜀=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 𝑑𝑇𝑥,𝑇𝑦≤𝑎𝑑(𝑥,𝑇𝑥)+𝑑𝑦,𝑇𝑦+𝑏𝑑𝑦,𝑇𝑥+𝑑𝑥,𝑇𝑦+𝑐𝑑𝑥,𝑦, (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 {𝑇^𝑛𝑥} 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(𝜀)(𝐹) is compact, and star shaped, then it contains a 𝑇-invariant point.

Proof. Let 𝑧∈𝑃_𝐶(𝜀)(𝐹) be arbitrary. Then by (2.2), we have for all 𝑥∈𝐹𝑑()()[𝑑()([𝑑()(()[𝑑([𝑑()(()[𝑑()([𝑑()(𝑥,𝑇𝑧=𝑑𝑇𝑥,𝑇𝑧≤𝑎𝑥,𝑇𝑥+𝑑𝑧,𝑇𝑧)]+𝑏𝑧,𝑇𝑥+𝑑𝑥,𝑇𝑧)]+𝑐𝑑𝑥,𝑧=𝑎𝑧,𝑇𝑧)]+𝑏𝑧,𝑇𝑥+𝑑𝑥,𝑇𝑧)]+𝑐𝑑𝑥,𝑧≤𝑎𝑧,𝑥+𝑑𝑥,𝑇𝑧)]+𝑏𝑧,𝑥+𝑑x)(()=()𝑑()+(𝑑(.,𝑇𝑥+𝑑𝑥,𝑇𝑧)]+𝑐𝑑𝑥,𝑧𝑎+𝑏+𝑐𝑥,𝑧𝑎+𝑏)[𝑥,𝑇𝑧)] (2.3) This gives ()𝑑()≤()𝑑()𝑑()()1−𝑎−𝑏𝑥,𝑇𝑧𝑎+𝑏+𝑐𝑥,𝑧𝑥,𝑇𝑧≤𝑑𝑥,𝑧 (2.4) since 2𝑎+2𝑏+𝑐≤1. Therefore, using definition of 𝑃_𝐶(𝜀)(𝐹), we get sup_𝑥∈𝐹{𝑑(𝑥,𝑇𝑧)}≤sup_𝑥∈𝐹{𝑑(𝑥,𝑧)}≤𝐷(𝐹,𝐶)+𝜀. (2.5) Hence 𝑇𝑧∈𝑃_𝐶(𝜀)(𝐹). Therefore 𝑇 is a self-map on 𝑃_𝐶(𝜀)(𝐹).
Let 𝑞 be the star-center of 𝑃_𝐶(𝜀)(𝐹). Define 𝑇_𝑛∶𝑃_𝐶(𝜀)(𝐹)→𝑃_𝐶(𝜀)(𝐹) as 𝑇_𝑛𝑥=𝑊(𝑇𝑥,𝑞,𝜆_𝑛), 𝑥∈𝑃_𝐶(𝜀)(𝐹) where ⟨𝑦_𝑛⟩ is a sequence in (0,1) such that 𝜆_𝑛→1. Consider 𝑑𝑇_𝑛𝑥,𝑇_𝑛𝑦=𝑑𝑊𝑇𝑥,𝑞,𝜆_𝑛,𝑊𝑇𝑦,𝑞,𝜆_𝑛≤𝜆_𝑛𝑑𝑇𝑥,𝑇𝑦≤𝜆_𝑛𝑎𝑑()𝑑𝑑()𝑑,𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑦+𝑏𝑦,𝑇𝑥+𝑑𝑥,𝑇𝑦+𝑐𝑑𝑥,𝑦≤𝑎𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑦+𝑏𝑦,𝑇𝑥+𝑑𝑥,𝑇𝑦+𝑐𝑑𝑥,𝑦 (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(𝜀)(𝐹). Since 𝑃_𝐶(𝜀)(𝐹) is compact, there is a subsequence ⟨𝑧_𝑛𝑖⟩ of ⟨𝑧_𝑛⟩ such that 𝑧_𝑛𝑖→𝑧_∘∈𝑃_𝐶(𝜀)(𝐹). We claim that 𝑇𝑧_∘=𝑧_∘. Consider 𝑧_𝑛𝑖=T_𝑛𝑖𝑧_𝑛𝑖=𝑊(𝑇𝑧_𝑛𝑖,𝑞,𝜆_𝑛𝑖)→𝑇𝑧_∘, as 𝑇 is continuous. Thus 𝑧_𝑛𝑖→𝑇𝑧_∘ and consequently, 𝑇𝑧_∘=𝑧_∘, that is, 𝑧_∘∈𝑃_𝐶(𝜀)(𝐹) is a 𝑇-invariant point.

Since for an 𝜀-simultaneous approximatively compact subset 𝐶 of a metric space (𝑋,𝑑) the set of 𝜀-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 𝜀-simultaneous approximatively compact with respect to 𝐹 and 𝑃_𝐶(𝜀)(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(𝜀)(𝐹) is compact and contractive, then the set 𝑃_𝐶(𝜀)(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(𝜀)(𝐹). Since 𝑃_𝐶(𝜀)(𝐹) is contractive, there exists a sequence ⟨𝑓_𝑛⟩ of contraction mapping of 𝑃_𝐶(𝜀)(𝐹) into itself such that 𝑓_𝑛𝑧→𝑧 for every 𝑧∈𝑃_𝐶(𝜀)(𝐹).
Clearly, 𝑓_𝑛𝑇 is a contraction on the compact set 𝑃_𝐶(𝜀)(𝐹) for each 𝑛 and so by Banach contraction principle, each 𝑓_𝑛𝑇 has a unique fixed point, say 𝑧_𝑛 in 𝑃_𝐶(𝜀)(𝐹). Now the compactness of 𝑃_𝐶(𝜀)(𝐹) implies that the sequence ⟨𝑧_𝑛⟩ has a subsequence ⟨𝑧_𝑛𝑖⟩→𝑧_∘∈𝐷. We claim that 𝑧_∘ is a fixed point of 𝑇. Let 𝜀>0 be given. Since 𝑧_𝑛𝑖→𝑧_∘ and 𝑓_𝑛𝑇𝑧_∘→𝑇𝑧_∘, there exist a positive integer 𝑚 such that for all 𝑛_𝑖≥𝑚𝑑𝑧_𝑛𝑖,𝑧_∘<𝜀2,𝑑𝑓_𝑛𝑖𝑇𝑧_∘,𝑇𝑧_∘<𝜀2. (2.7) Again, 𝑑𝑓_𝑛𝑖𝑇𝑧_𝑛𝑖,𝑓_𝑛𝑖𝑇𝑧_∘≤𝑑𝑧_𝑛𝑖,𝑧_∘<𝜀2. (2.8) Hence 𝑑𝑓_𝑛𝑖𝑇𝑧_𝑛𝑖,𝑇𝑧_∘≤𝑑𝑓_𝑛𝑖𝑇𝑧_𝑛𝑖,𝑓_𝑛𝑖𝑇𝑧_∘+𝑑𝑓_𝑛𝑖𝑇𝑧_∘,𝑇𝑧_∘<𝜀2+𝜀2, (2.9) that is, 𝑑(𝑓_𝑛𝑖𝑇𝑧_𝑛𝑖,𝑇𝑧_∘)<𝜀 for all 𝑛_𝑖≥𝑚 and so 𝑓_𝑛𝑖𝑇𝑧_𝑛𝑖→𝑇𝑧_∘. But 𝑓_𝑛𝑖𝑇𝑧_𝑛𝑖=𝑧_𝑛𝑖→𝑧_∘ 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 𝜀-simultaneous approximatively compact with respect to 𝐹 and the set 𝑃_𝐶(𝜀)(𝐹) is contractive, and 𝑇-invariant, then 𝑃_𝐶(𝜀)(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 𝜀-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 𝛼[𝐺]=inf𝜀>0∶𝐺iscoveredbyafinitenumberofclosedballscenteredatpointsof𝑋ofradius≤𝜀. (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(𝜀)(𝐹) has a 𝑇-invariant point.

Proof. As 𝐺 is closed and bounded, 𝑃_𝐺(𝜀)(𝐹) is nonempty, closed and bounded. Using Theorem 2.5, we can prove that 𝑇 is a self-map of 𝑃_𝐺(𝜀)(𝐹). Now a direct application of Lemma 2.19, gives a 𝑇-invariant point in 𝑃_𝐺(𝜀)(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 𝑑𝑇𝑥,𝑇𝑦≤𝛼𝑑(𝑥,𝑇𝑥)+𝑑𝑦,𝑇𝑦 (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 𝜀-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, 𝑑𝑇^𝑚𝑦,𝑇^𝑚𝑧≤𝛼𝑑𝑦,𝑇^𝑚𝑦+𝑑𝑧,𝑇^𝑚𝑧, (2.12) for some positive integer 𝑚, all 𝑦,𝑧∈𝑋 and some fixed 0<𝛼<1/2. If 𝐷=𝑃_𝐺(𝜀)(𝐹) is compact, then it has a unique fixed point of 𝑇.

Proof. As 𝑇𝑥=𝑥, 𝑇^𝑛𝑥=𝑥 for all positive integers 𝑛. Let 𝑦₀∈𝐷. Then, for 0<𝛼<1/2, 𝑑𝑥,𝑇^𝑚𝑦₀=𝑑𝑇^𝑚𝑥,𝑇^𝑚𝑦₀≤𝛼𝑑𝑥,𝑇^𝑚𝑥+𝑑𝑦₀,𝑇^𝑚𝑦₀=𝛼𝑑𝑦₀,𝑇^𝑚𝑦₀≤𝛼𝑑𝑦₀,𝑥+𝑑𝑥,𝑇^𝑚𝑦₀, (2.13) which implies that 𝑑𝑥,𝑇^𝑚𝑦₀≤𝛼𝑑𝑦1−𝛼⁰,𝑥. (2.14) Further, we have sup_𝑥∈𝐹𝑑𝑥,𝑇^𝑚𝑦₀≤sup_𝑥∈𝐹𝑑𝑦₀,𝑥≤𝐷(𝐹,𝐺)+𝜀. (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 𝜀-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 𝑑𝑇^𝑚𝑦,𝑇^𝑚𝑧≤𝛼𝑑𝑦,𝑇^𝑚𝑦+𝑑𝑧,𝑇^𝑚𝑧, (2.16) for some positive integer 𝑚, all 𝑦,𝑧∈𝑋 and some fixed 0<𝛼<1/2 then 𝐷=𝑃_𝐺(𝜀)(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 𝜀-simultaneous approximation.

For 𝜀>0, we define 𝑅_𝐺(𝜀)(𝐹)={𝑔₀∈𝐺∶sup_𝑔∈𝐺𝑑(𝑔,𝑔₀)+𝜀≤inf_𝑔∈𝐺sup_𝑦∈𝐹𝑑(𝑦,𝑔)}. An element 𝑔₀∈𝑅_𝐺(𝜀)(𝐹) is said to be a 𝜀-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 𝜀-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(𝜀)(𝐹) is compact and star shaped. Then 𝑅_𝐺(𝜀)(𝐹) contains a 𝑇-invariant point.

Proof. Let 𝑔_∘∈𝑅_𝐺(𝜀)(𝐹). Consider 𝑑𝑇𝑔_∘,𝑔+𝜀≤𝑑𝑔_∘,𝑔+𝜀≤inf_𝑔∈𝐺sup_𝑦∈𝐹𝑑𝑦,𝑔, (2.17) and so 𝑇𝑔_∘∈𝑅_𝐺(𝜀)(𝐹), that is, 𝑇∶𝑅_𝐺(𝜀)(𝐹)→𝑅_𝐺(𝜀)(𝐹). Since 𝑅_𝐺(𝜀)(𝐹) is star shaped, there exists 𝑝∈𝑅_𝐺(𝜀)(𝐹) such that 𝑊(𝑧,𝑝,𝜆)∈𝑅_𝐺(𝜀)(𝐹) for all 𝑧∈𝑅_𝐺(𝜀)(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(𝜀)(𝐹). Since 𝑇 is a self-map on 𝑅_𝐺(𝜀)(𝐹) and 𝑅_𝐺(𝜀)(𝐹) is star shaped, each 𝑇_𝑛 is a well defined and maps 𝑅_𝐺(𝜀)(𝐹) into 𝑅_𝐺(𝜀)(𝐹). Moreover, 𝑑𝑇_𝑛𝑦,𝑇_𝑛𝑧=𝑑𝑊𝑇𝑦,𝑝,𝑘_𝑛,𝑊𝑇𝑧,𝑝,𝑘_𝑛≤𝑘_𝑛𝑑𝑇𝑦,𝑇𝑧≤𝑘_𝑛𝑎𝑑𝑦,𝑇𝑦+𝑑(𝑧,𝑇𝑧)+𝑏𝑑𝑧,𝑇𝑦+𝑑𝑦,𝑇𝑧+𝑐𝑑𝑦,𝑧, (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(𝜀)(𝐹), that is, 𝑇_𝑛𝑥_𝑛=𝑥_𝑛 for each 𝑛. Since 𝑅_𝐺(𝜀)(𝐹) is compact, ⟨𝑥_𝑛⟩ has a subsequence 𝑥_𝑛𝑖→𝑥∈𝑅_𝐺(𝜀)(𝐹). Now, we claim that 𝑇𝑥=𝑥. As 𝑑(𝑥_𝑛𝑖,𝑇𝑥)≤𝑑(𝑥_𝑛𝑖,𝑥). On letting 𝑛→∞, we have 𝑥_𝑛𝑖→𝑇𝑥. 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 𝜀-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(𝜀)(𝑥) is nonempty compact, star shaped, and let 𝑇 be nonexpansive on 𝑅_𝐺(𝜀)(𝑥). Then there exists a 𝑔_∘∈𝑅_𝐺(𝜀)(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 𝜀-simultaneous approximation.
(ii) By taking 𝐹={𝑥₁,𝑥₂}, 𝑥₁,𝑥₂∈𝑋, the set 𝑃_𝐺(𝜀)(𝐹) (respectively, 𝑅_𝐺(𝜀)(𝐹)) is the set of 𝜀-simultaneous approximation (respectively, 𝜀-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.