Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 281381, 15 pages
doi:10.1155/2010/281381
Research Article

Fixed Point Theorems on Spaces Endowed with Vector-Valued Metrics

Alexandru-Darius Filip and Adrian Petruşel

Department of Applied Mathematics, Babeş-Bolyai University, Kogǎlniceanu Street, No. 1, 400084 Cluj-Napoca, Romania

Received 2 July 2009; Revised 4 October 2009; Accepted 21 December 2009

Academic Editor: Tomas Dominguez Benavides

Copyright © 2010 Alexandru-Darius Filip and Adrian Petruşel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of this work is to present some (local and global) fixed point results for singlevalued and multivalued generalized contractions on spaces endowed with vector-valued metrics. The results are extensions of some theorems given by Perov (1964), Bucur et al. (2009), M. Berinde and V. Berinde (2007), O'Regan et al. (2007), and so forth.

1. Introduction

The classical Banach contraction principle was extended for contraction mappings on spaces endowed with vector-valued metrics by Perov in 1964 (see [1]).

Let 𝑋 be a nonempty set. A mapping 𝑑 ∶ 𝑋 × 𝑋 → ℝ 𝑚 is called a vector-valued metric on 𝑋 if the following properties are satisfied:

( 𝑑 1 ) 𝑑 ( 𝑥 , 𝑦 ) ≥ 0 for all 𝑥 , 𝑦 ∈ 𝑋 ; if 𝑑 ( 𝑥 , 𝑦 ) = 0 , then 𝑥 = 𝑦 ; ( 𝑑 2 ) 𝑑 ( 𝑥 , 𝑦 ) = 𝑑 ( 𝑦 , 𝑥 ) for all 𝑥 , 𝑦 ∈ 𝑋 ; ( 𝑑 3 ) 𝑑 ( 𝑥 , 𝑦 ) ≤ 𝑑 ( 𝑥 , 𝑧 ) + 𝑑 ( 𝑧 , 𝑦 ) for all 𝑥 , 𝑦 , 𝑧 ∈ 𝑋 .

If 𝛼 , 𝛽 ∈ ℝ 𝑚 , 𝛼 = ( 𝛼 1 , 𝛼 2 , … , 𝛼 𝑚 ) , 𝛽 = ( 𝛽 1 , 𝛽 2 , … , 𝛽 𝑚 ) , and 𝑐 ∈ ℝ , by 𝛼 ≤ 𝛽 (resp., 𝛼 < 𝛽 ) we mean that 𝛼 𝑖 ≤ 𝛽 𝑖 (resp., 𝛼 𝑖 < 𝛽 𝑖 ) for 𝑖 ∈ { 1 , 2 , … , 𝑚 } and by 𝛼 ≤ 𝑐 we mean that 𝛼 𝑖 ≤ 𝑐 for 𝑖 ∈ { 1 , 2 , … , 𝑚 } .

A set 𝑋 equipped with a vector-valued metric 𝑑 is called a generalized metric space. We will denote such a space with ( 𝑋 , 𝑑 ) . For the generalized metric spaces, the notions of convergent sequence, Cauchy sequence, completeness, open subset, and closed subset are similar to those for usual metric spaces.

If ( 𝑋 , 𝑑 ) is a generalized metric space, 𝑥 0 ∈ 𝑋 and 𝑟 = ( 𝑟 𝑖 ) 𝑚 𝑖 = 1 ∈ ℝ 𝑚 , with 𝑟 𝑖 > 0 for each 𝑖 ∈ { 1 , 2 , … , 𝑚 } , then we will denote by 𝐵 𝑥 0 𝑥 , 𝑟 ∶ = 𝑥 ∈ 𝑋 ∣ 𝑑 0 , 𝑥 < 𝑟 ( 1 . 1 ) the open ball centered in 𝑥 0 with radius 𝑟 , by 𝐵 ( 𝑥 0 , 𝑟 ) the closure (in ( 𝑋 , 𝑑 ) ) of the open ball, and by

𝐵 𝑥 0 𝑥 , 𝑟 ∶ = 𝑥 ∈ 𝑋 ∣ 𝑑 0 , 𝑥 ≤ 𝑟 ( 1 . 2 ) the closed ball centered in 𝑥 0 with radius 𝑟 .

If 𝑓 ∶ 𝑋 → 𝑋 is a singlevalued operator, then we denote by F i x ( 𝑓 ) the set of all fixed points of 𝑓 ; that is, F i x ( 𝑓 ) ∶ = { 𝑥 ∈ 𝑋 ∣ 𝑥 = 𝑓 ( 𝑥 ) } .

For the multivalued operators we use the following notations:

𝑃 𝑃 ( 𝑋 ) ∶ = { 𝑌 ⊂ 𝑋 ∣ 𝑌 ≠ ∅ } ; 𝑏 𝑃 ( 𝑋 ) ∶ = { 𝑌 ∈ 𝑃 ( 𝑋 ) ∣ 𝑌 i s b o u n d e d } ; c l ( 𝑋 ) ∶ = { 𝑌 ∈ 𝑃 ( 𝑋 ) ∣ 𝑌 i s c l o s e d } . ( 1 . 3 ) Now, if 𝐹 ∶ 𝑋 → 𝑃 ( 𝑋 ) is a multivalued operator, then we denote by F i x ( 𝐹 ) the fixed points set of 𝐹 , that is, F i x ( 𝐹 ) ∶ = { 𝑥 ∈ 𝑋 ∣ 𝑥 ∈ 𝐹 ( 𝑥 ) } .

The set G r a p h ( 𝐹 ) = { ( 𝑥 , 𝑦 ) ∈ 𝑋 × 𝑋 ∣ 𝑦 ∈ 𝐹 ( 𝑥 ) } is called the graph of the multivalued operator 𝐹 .

In the context of a metric space ( 𝑋 , 𝑑 ) , if 𝐴 , 𝐵 ∈ 𝑃 ( 𝑋 ) , then we will use the following notations:

(a)the gap functional 𝐷 ∶ 𝑃 ( 𝑋 ) × 𝑃 ( 𝑋 ) → ℝ + : 𝐷 ( 𝐴 , 𝐵 ) ∶ = i n f { 𝑑 ( 𝑎 , 𝑏 ) ∣ 𝑎 ∈ 𝐴 , 𝑏 ∈ 𝐵 } ; ( 1 . 4 ) (b)the generalized excess functional 𝜌 ∶ 𝑃 c l ( 𝑋 ) × 𝑃 c l ( 𝑋 ) → ℝ + ∪ { + ∞ } : 𝜌 ( 𝐴 , 𝐵 ) ∶ = s u p { 𝐷 ( 𝑎 , 𝐵 ) ∣ 𝑎 ∈ 𝐴 } ; ( 1 . 5 ) (c)the generalized Pompeiu-Hausdorff functional 𝐻 ∶ 𝑃 c l ( 𝑋 ) × 𝑃 c l ( 𝑋 ) → ℝ + ∪ { + ∞ } : 𝐻 ( 𝐴 , 𝐵 ) ∶ = m a x { 𝜌 ( 𝐴 , 𝐵 ) , 𝜌 ( 𝐵 , 𝐴 ) } . ( 1 . 6 )

It is well known that 𝐻 is a generalized metric, in the sense that if 𝐴 , 𝐵 ∈ 𝑃 c l ( 𝑋 ) , then 𝐻 ( 𝐴 , 𝐵 ) ∈ ℝ + ∪ { + ∞ } .

Throughout this paper we denote by 𝑀 𝑚 , 𝑚 ( ℝ + ) the set of all 𝑚 × 𝑚 matrices with positive elements, by Θ the zero 𝑚 × 𝑚 matrix, and by 𝐼 the identity 𝑚 × 𝑚 matrix. If 𝐴 ∈ 𝑀 𝑚 , 𝑚 ( ℝ + ) , then the symbol 𝐴 𝜏 stands for the transpose matrix of 𝐴 . Notice also that, for the sake of simplicity, we will make an identification between row and column vectors in ℝ 𝑚 .

Recall that a matrix 𝐴 is said to be convergent to zero if and only if 𝐴 𝑛 → 0 as 𝑛 → ∞ (see Varga [2]).

Notice that, for the proof of the main results, we need the following theorem, part of which being a classical result in matrix analysis; see, for example, [3, Lemma 3 . 3 . 1 , page 55], [4, page 37], and [2, page 12]. For the assertion (iv) see [5].

Theorem 1.1. Let 𝐴 ∈ 𝑀 𝑚 , 𝑚 ( ℝ + ) . The following are equivalents. (i) 𝐴 is convergent towards zero.(ii) 𝐴 𝑛 → 0 as 𝑛 → ∞ .(iii)The eigenvalues of 𝐴 are in the open unit disc, that is, | 𝜆 | < 1 , for every 𝜆 ∈ ℂ with d e t ( 𝐴 − 𝜆 𝐼 ) = 0 .(iv)The matrix 𝐼 − 𝐴 is nonsingular and ( 𝐼 − 𝐴 ) − 1 = 𝐼 + 𝐴 + ⋯ + 𝐴 𝑛 + ⋯ . ( 1 . 7 ) (v)The matrix 𝐼 − 𝐴 is nonsingular and ( 𝐼 − 𝐴 ) − 1 has nonnenegative elements.(vi) 𝐴 𝑛 𝑞 → 0 and 𝑞 𝐴 𝑛 → 0 as 𝑛 → ∞ , for each 𝑞 ∈ ℝ 𝑚 .

Remark 1.2. Some examples of matrix convergent to zero are(a)any matrix 𝐴 ∶ = ( 𝑎 𝑎 𝑏 𝑏 ) , where 𝑎 , 𝑏 ∈ ℝ + and 𝑎 + 𝑏 < 1 ;(b)any matrix 𝐴 ∶ = ( 𝑎 𝑏 𝑎 𝑏 ) , where 𝑎 , 𝑏 ∈ ℝ + and 𝑎 + 𝑏 < 1 ;(c)any matrix 𝐴 ∶ = ( 𝑎 𝑏 0 𝑐 ) , where 𝑎 , 𝑏 , 𝑐 ∈ ℝ + and m a x { 𝑎 , 𝑐 } < 1 .

For other examples and considerations on matrices which converge to zero, see Rus [4], Turinici [6], and so forth.

Main result for self contractions on generalized metric spaces is Perov's fixed point theorem; see [1].

Theorem 1.3 (Perov [3]). Let ( 𝑋 , 𝑑 ) be a complete generalized metric space and the mapping 𝑓 ∶ 𝑋 → 𝑋 with the property that there exists a matrix 𝐴 ∈ 𝑀 𝑚 , 𝑚 ( ℝ ) such that 𝑑 ( 𝑓 ( 𝑥 ) , 𝑓 ( 𝑦 ) ) ≤ 𝐴 𝑑 ( 𝑥 , 𝑦 ) for all 𝑥 , 𝑦 ∈ 𝑋 .
If 𝐴 is a matrix convergent towards zero, then
(1) F i x ( 𝑓 ) = { 𝑥 ∗ } ; (2)the sequence of successive approximations ( 𝑥 𝑛 ) 𝑛 ∈ ℕ , 𝑥 𝑛 = 𝑓 𝑛 ( 𝑥 0 ) is convergent and it has the limit 𝑥 ∗ , for all 𝑥 0 ∈ 𝑋 ; (3)one has the following estimation: 𝑑 𝑥 𝑛 , 𝑥 ∗ ≤ 𝐴 𝑛 ( 𝐼 − 𝐴 ) − 1 𝑑 𝑥 0 , 𝑥 1 ; ( 1 . 8 ) (4)if 𝑔 ∶ 𝑋 → 𝑋 satisfies the condition 𝑑 ( 𝑓 ( 𝑥 ) , 𝑔 ( 𝑥 ) ) ≤ 𝜂 , for all 𝑥 ∈ 𝑋 , 𝜂 ∈ ℝ 𝑚 and considering the sequence 𝑦 𝑛 = 𝑔 𝑛 ( 𝑥 0 ) one has 𝑑 𝑦 𝑛 , 𝑥 ∗ ≤ ( 𝐼 − 𝐴 ) − 1 𝜂 + 𝐴 𝑛 ( 𝐼 − 𝐴 ) − 1 𝑑 𝑥 0 , 𝑥 1 . ( 1 . 9 )

On the other hand, notice that the evolution of macrosystems under uncertainty or lack of precision, from control theory, biology, economics, artificial intelligence, or other fields of knowledge, is often modeled by semilinear inclusion systems:

𝑥 1 ∈ 𝑇 1 𝑥 1 , 𝑥 2 , 𝑥 2 ∈ 𝑇 1 𝑥 1 , 𝑥 2 , ( 1 . 1 0 ) (where 𝑇 𝑖 ∶ 𝑋 × 𝑋 → 𝑃 ( 𝑋 ) for 𝑖 ∈ { 1 , 2 } are multivalued operators; here 𝑃 ( 𝑋 ) stands for the family of all nonempty subsets of a Banach space 𝑋 ). The system above can be represented as a fixed point problem of the form 𝑇 𝑥 ∈ 𝑇 ( 𝑥 ) w h e r e 𝑇 ∶ = 1 , 𝑇 2 ∶ 𝑋 2 𝑋 ⟶ 𝑃 2 𝑥 , 𝑥 = 1 , 𝑥 2 . ( 1 . 1 1 ) Hence, it is of great interest to give fixed point results for multivalued operators on a set endowed with vector-valued metrics or norms. However, some advantages of a vector-valued norm with respect to the usual scalar norms were already pointed out by Precup in [5]. The purpose of this work is to present some new fixed point results for generalized (singlevalued and multivalued) contractions on spaces endowed with vector-valued metrics. The results are extensions of the theorems given by Perov [1], O'Regan et al. [7], M. Berinde and V. Berinde [8], and by Bucur et al. [9].

2. Main Results

We start our considerations by a local fixed point theorem for a class of generalized singlevalued contractions.

Theorem 2.1. Let ( 𝑋 , 𝑑 ) be a complete generalized metric space, 𝑥 0 ∈ 𝑋 , 𝑟 ∶ = ( 𝑟 𝑖 ) 𝑚 𝑖 = 1 ∈ ℝ 𝑚 + with 𝑟 𝑖 > 0 for each 𝑖 ∈ { 1 , 2 , … , 𝑚 } and let 𝑓 ∶ 𝐵 ( 𝑥 0 , 𝑟 ) → 𝑋 having the property that there exist 𝐴 , 𝐵 ∈ 𝑀 𝑚 , 𝑚 ( ℝ + ) such that 𝑑 ( 𝑓 ( 𝑥 ) , 𝑓 ( 𝑦 ) ) ≤ 𝐴 𝑑 ( 𝑥 , 𝑦 ) + 𝐵 𝑑 ( 𝑦 , 𝑓 ( 𝑥 ) ) ( 2 . 1 ) for all 𝑥 , 𝑦 ∈ 𝐵 ( 𝑥 0 , 𝑟 ) . We suppose that (1) 𝐴 is a matrix that converges toward zero; (2)if 𝑢 ∈ ℝ 𝑚 + is such that 𝑢 ( 𝐼 − 𝐴 ) − 1 ≤ ( 𝐼 − 𝐴 ) − 1 𝑟 , then 𝑢 ≤ 𝑟 ; (3) 𝑑 ( 𝑥 0 , 𝑓 ( 𝑥 0 ) ) ( 𝐼 − 𝐴 ) − 1 ≤ 𝑟 . Then F i x ( 𝑓 ) ≠ ∅ .
In addition, if the matrix 𝐴 + 𝐵 converges to zero, then F i x ( 𝑓 ) = { 𝑥 ∗ } .

Proof. We consider ( 𝑥 𝑛 ) 𝑛 ∈ ℕ the sequence of successive approximations for the mapping 𝑓 , defined by 𝑥 𝑛 + 1 𝑥 = 𝑓 𝑛 𝑥 , ∀ 𝑛 ∈ ℕ , 0 ∈ 𝑋 , b e a r b i t r a r y . ( 2 . 2 ) Using ( 3 ) , we have 𝑑 ( 𝑥 0 , 𝑥 1 ) ( 𝐼 − 𝐴 ) − 1 = 𝑑 ( 𝑥 0 , 𝑓 ( 𝑥 0 ) ) ( 𝐼 − 𝐴 ) − 1 ≤ 𝑟 ≤ ( 𝐼 − 𝐴 ) − 1 𝑟 .
Thus, by ( 2 ) we get that 𝑑 ( 𝑥 0 , 𝑥 1 ) ≤ 𝑟 and hence 𝑥 1 ∈ 𝐵 ( 𝑥 0 , 𝑟 ) . Similarly, 𝑑 ( 𝑥 1 , 𝑥 2 ) ( 𝐼 − 𝐴 ) − 1 = 𝑑 ( 𝑓 ( 𝑥 0 ) , 𝑓 ( 𝑥 1 ) ) ( 𝐼 − 𝐴 ) − 1 ≤ 𝐴 𝑑 ( 𝑥 0 , 𝑥 1 ) ( 𝐼 − 𝐴 ) − 1 + 𝐵 𝑑 ( 𝑥 1 , 𝑓 ( 𝑥 0 ) ) ( 𝐼 − 𝐴 ) − 1 ≤ 𝐴 𝑟 .
Since 𝑑 ( 𝑥 0 , 𝑥 2 ) ≤ 𝑑 ( 𝑥 0 , 𝑥 1 ) + 𝑑 ( 𝑥 1 , 𝑥 2 ) , by ( 2 ) we get
𝑑 𝑥 0 , 𝑥 2 ( 𝐼 − 𝐴 ) − 1 𝑥 ≤ 𝑑 0 , 𝑥 1 ( 𝐼 − 𝐴 ) − 1 𝑥 + 𝑑 1 , 𝑥 2 ( 𝐼 − 𝐴 ) − 1 ≤ 𝐼 𝑟 + 𝐴 𝑟 ≤ 𝐼 + 𝐴 + 𝐴 2 + ⋯ 𝑟 = ( 𝐼 − 𝐴 ) − 1 𝑟 . ( 2 . 3 ) Thus 𝑑 ( 𝑥 0 , 𝑥 2 ) ≤ 𝑟 and hence 𝑥 2 ∈ 𝐵 ( 𝑥 0 , 𝑟 ) .
Inductively, we construct the sequence ( 𝑥 𝑛 ) 𝑛 ∈ ℕ in 𝐵 ( 𝑥 0 , 𝑟 ) satisfying, for all 𝑛 ∈ ℕ , the following conditions:
(i) 𝑥 𝑛 + 1 = 𝑓 ( 𝑥 𝑛 ) ; (ii) 𝑑 ( 𝑥 0 , 𝑥 𝑛 ) ( 𝐼 − 𝐴 ) − 1 ≤ ( 𝐼 − 𝐴 ) − 1 𝑟 ; (iii) 𝑑 ( 𝑥 𝑛 , 𝑥 𝑛 + 1 ) ( 𝐼 − 𝐴 ) − 1 ≤ 𝐴 𝑛 𝑟 . From ( i i i ) we get, for all 𝑛 ∈ ℕ and 𝑝 ∈ ℕ , 𝑝 > 0 , that 𝑑 𝑥 𝑛 , 𝑥 𝑛 + 𝑝 ( 𝐼 − 𝐴 ) − 1 𝑥 = 𝑑 𝑛 , 𝑥 𝑛 + 1 ( 𝐼 − 𝐴 ) − 1 𝑥 + 𝑑 𝑛 + 1 , 𝑥 𝑛 + 2 ( 𝐼 − 𝐴 ) − 1 𝑥 + ⋯ + 𝑑 𝑛 + 𝑝 − 1 , 𝑥 𝑛 + 𝑝 ( 𝐼 − 𝐴 ) − 1 ≤ 𝐴 𝑛 𝑟 + 𝐴 𝑛 + 1 𝑟 + ⋯ + 𝐴 𝑛 + 𝑝 − 1 𝑟 ≤ 𝐴 𝑛 𝐼 + 𝐴 + 𝐴 2 + ⋯ + 𝐴 𝑝 − 1 𝑟 + ⋯ ≤ 𝐴 𝑛 ( 𝐼 − 𝐴 ) − 1 𝑟 ⟶ 0 , a s 𝑛 ⟶ ∞ . ( 2 . 4 )
Hence ( 𝑥 𝑛 ) 𝑛 ∈ ℕ is a Cauchy sequence. Using the fact that ( 𝐵 ( 𝑥 0 , 𝑟 ) , 𝑑 ) is a complete metric space, we get that ( 𝑥 𝑛 ) 𝑛 ∈ ℕ is convergent in the closed set 𝐵 ( 𝑥 0 , 𝑟 ) . Thus, there exists 𝑥 ∗ ∈ 𝐵 ( 𝑥 0 , 𝑟 ) such that 𝑥 ∗ = l i m 𝑛 → ∞ 𝑥 𝑛 .
Next, we show that 𝑥 ∗ ∈ F i x ( 𝑓 ) .
Indeed, we have the following estimation:
𝑑 𝑥 ∗ 𝑥 , 𝑓 ∗ 𝑥 ≤ 𝑑 ∗ , 𝑥 𝑛 𝑥 + 𝑑 𝑛 𝑥 , 𝑓 ∗ 𝑥 = 𝑑 ∗ , 𝑥 𝑛 𝑓 𝑥 + 𝑑 𝑛 − 1 𝑥 , 𝑓 ∗ 𝑥 ≤ 𝑑 ∗ , 𝑥 𝑛 𝑥 + 𝐴 𝑑 𝑛 − 1 , 𝑥 ∗ 𝑥 + 𝐵 𝑑 ∗ , 𝑥 𝑛 ⟶ 0 , a s 𝑛 ⟶ ∞ . ( 2 . 5 ) Hence 𝑥 ∗ ∈ F i x ( 𝑓 ) . In addition, letting 𝑝 → ∞ in the estimation of 𝑑 ( 𝑥 𝑛 , 𝑥 𝑛 + 𝑝 ) , we get 𝑑 𝑥 𝑛 , 𝑥 ∗ ≤ 𝐴 𝑛 ( 𝐼 − 𝐴 ) − 1 𝑑 𝑥 0 , 𝑥 1 . ( 2 . 6 ) We show now the uniqueness of the fixed point.
Let 𝑥 ∗ , 𝑦 ∗ ∈ F i x ( 𝑓 ) with 𝑥 ∗ ≠ 𝑦 ∗ . Then 𝑑 𝑥 ∗ , 𝑦 ∗ 𝑓 𝑥 = 𝑑 ∗ 𝑦 , 𝑓 ∗ 𝑥 ≤ 𝐴 𝑑 ∗ , 𝑦 ∗ 𝑦 + 𝐵 𝑑 ∗ 𝑥 , 𝑓 ∗ = 𝑥 ( 𝐴 + 𝐵 ) 𝑑 ∗ , 𝑦 ∗ , ( 2 . 7 ) which implies ( 𝐼 − 𝐴 − 𝐵 ) 𝑑 ( 𝑥 ∗ , 𝑦 ∗ ) ≤ 0 ∈ ℝ 𝑚 . Taking into account that 𝐼 − 𝐴 − 𝐵 is nonsingular and ( 𝐼 − 𝐴 − 𝐵 ) − 1 ∈ 𝑀 𝑚 , 𝑚 ( ℝ + ) we deduce that 𝑑 ( 𝑥 ∗ , 𝑦 ∗ ) ≤ 0 and thus 𝑥 ∗ = 𝑦 ∗ .

Remark 2.2. By similitude to [10], a mapping 𝑓 ∶ 𝑌 ⊆ 𝑋 → 𝑋 satisfying the condition 𝑑 ( 𝑓 ( 𝑥 ) , 𝑓 ( 𝑦 ) ) ≤ 𝐴 𝑑 ( 𝑥 , 𝑦 ) + 𝐵 𝑑 ( 𝑦 , 𝑓 ( 𝑥 ) ) , ∀ 𝑥 , 𝑦 ∈ 𝑌 , ( 2 . 8 ) for some matrices 𝐴 , 𝐵 ∈ 𝑀 𝑚 , 𝑚 ( ℝ + ) with 𝐴 a matrix that converges toward zero, could be called an almost contraction of Perov type.

We have also a global version of Theorem 2.1, expressed by the following result.

Corollary 2.3. Let ( 𝑋 , 𝑑 ) be a complete generalized metric space. Let 𝑓 ∶ 𝑋 → 𝑋 be a mapping having the property that there exist 𝐴 , 𝐵 ∈ 𝑀 𝑚 , 𝑚 ( ℝ + ) such that 𝑑 ( 𝑓 ( 𝑥 ) , 𝑓 ( 𝑦 ) ) ≤ 𝐴 𝑑 ( 𝑥 , 𝑦 ) + 𝐵 𝑑 ( 𝑦 , 𝑓 ( 𝑥 ) ) , ∀ 𝑥 , 𝑦 ∈ 𝑋 . ( 2 . 9 )
If 𝐴 is a matrix that converges towards zero, then
(1) F i x ( 𝑓 ) ≠ ∅ ; (2)the sequence ( 𝑥 𝑛 ) 𝑛 ∈ ℕ given by 𝑥 𝑛 ∶ = 𝑓 𝑛 ( 𝑥 0 ) converges towards a fixed point of 𝑓 , for all 𝑥 0 ∈ 𝑋 ; (3)one has the estimation 𝑑 𝑥 𝑛 , 𝑥 ∗ ≤ 𝐴 𝑛 ( 𝐼 − 𝐴 ) − 1 𝑑 𝑥 0 , 𝑥 1 , ( 2 . 1 0 ) where 𝑥 ∗ ∈ F i x ( 𝑓 ) .
In addition, if the matrix 𝐴 + 𝐵 converges to zero, then F i x ( 𝑓 ) = { 𝑥 ∗ } .

Remark 2.4. Any matrix 𝐴 = ( 𝑎 0 0 𝑐 ) , where 𝑎 , 𝑐 ∈ ℝ + and m a x { 𝑎 , 𝑐 } < 1 , satisfies the assumptions ( 1 )-( 2 ) in Theorem 2.1.

Remark 2.5. Let us notice here that some advantages of a vector-valued norm with respect to the usual scalar norms were very nice pointed out, by several examples, in Precup in [5]. More precisely, one can show that, in general, the condition that 𝐴 is a matrix convergent to zero is weaker than the contraction conditions for operators given in terms of the scalar norms on 𝑋 of the following type: ‖ 𝑥 ‖ 𝑀 ∶ = ‖ 𝑥 1 ‖ + ‖ 𝑥 2 ‖ ‖ 𝑥 ‖ 𝐶 ∶ = m a x { ‖ 𝑥 1 ‖ , ‖ 𝑥 2 ‖ } ‖ 𝑥 ‖ 𝐸 ∶ = ( ‖ 𝑥 1 ‖ 2 + ‖ 𝑥 2 ‖ 2 ) 1 / 2 ( 𝑋 , | ⋅ | ) or 𝑓 1 , 𝑓 2 ∶ 𝑋 × 𝑋 → 𝑋 𝑎 𝑖 𝑗 , 𝑏 𝑖 𝑗 ∈ ℝ + .

As an application of the previous results we present an existence theorem for a system of operatorial equations.

Theorem 2.6. Let 𝑖 , 𝑗 ∈ { 1 , 2 } be a Banach space and let 𝑥 ∶ = ( 𝑥 1 , 𝑥 2 ) , 𝑦 ∶ = ( 𝑦 1 , 𝑦 2 ) ∈ 𝑋 × 𝑋 be two operators. Suppose that there exist | 𝑓 1 ( 𝑥 1 , 𝑥 2 ) − 𝑓 1 ( 𝑦 1 , 𝑦 2 ) | ≤ 𝑎 1 1 | 𝑥 1 − 𝑦 1 | + 𝑎 1 2 | 𝑥 2 − 𝑦 2 | + 𝑏 1 1 | 𝑥 1 − 𝑓 1 ( 𝑦 1 , 𝑦 2 ) | + 𝑏 1 2 | 𝑥 2 − 𝑓 2 ( 𝑦 1 , 𝑦 2 ) | , , | 𝑓 2 ( 𝑥 1 , 𝑥 2 ) − 𝑓 2 ( 𝑦 1 , 𝑦 2 ) | ≤ 𝑎 2 1 | 𝑥 1 − 𝑦 1 | + 𝑎 2 2 | 𝑥 2 − 𝑦 2 | + 𝑏 2 1 | 𝑥 1 − 𝑓 1 ( 𝑦 1 , 𝑦 2 ) | + 𝑏 2 2 | 𝑥 2 − 𝑓 2 ( 𝑦 1 , 𝑦 2 ) | . such that, for each 𝐴 ∶ = ( 𝑎 1 1 𝑎 1 2 𝑎 2 1 𝑎 2 2 ) , one has: (1) 0 (2) 𝑢 1 = 𝑓 1 𝑢 1 , 𝑢 2 , 𝑢 2 = 𝑓 1 𝑢 1 , 𝑢 2 ( 2 . 1 1 ) In addition, assume that the matrix 𝑥 ∗ ∈ 𝑋 × 𝑋 converges to 𝐴 + 𝐵 .
Then, the system
𝐸 ∶ = 𝑋 × 𝑋 has at least one solution 𝑓 ∶ 𝐸 → 𝑃 c l ( 𝐸 ) . Moreover, if, in addition, the matrix 𝑓 ( 𝑥 1 , 𝑥 2 ) ∶ = ( 𝑓 1 ( 𝑥 1 , 𝑥 2 ) , 𝑓 2 ( 𝑥 1 , 𝑥 2 ) ) converges to zero, then the above solution is unique.

Proof. Consider 𝑥 = 𝑓 ( 𝑥 ) and the operator 𝑥 ∈ 𝐸 given by the expression ( 1 ) + ( 2 ) . Then our system is now represented as a fixed point equation of the following form: ‖ 𝑓 ( 𝑥 ) − 𝑓 ( 𝑦 ) ‖ ≤ 𝐴 ⋅ ‖ 𝑥 − 𝑦 ‖ + 𝐵 ⋅ ‖ 𝑥 − 𝑓 ( 𝑦 ) ‖ , f o r e a c h 𝑥 , 𝑦 ∈ 𝐸 ∶ = 𝑋 × 𝑋 . ( 2 . 1 2 ) , ( 𝐸 , 𝑑 ) . Notice also that the conditions 𝑑 ( 𝑢 , 𝑣 ) ∶ = ‖ 𝑢 − 𝑣 ‖ ∶ = ( | 𝑢 1 − 𝑣 1 | | 𝑢 2 − 𝑣 2 | ) can be jointly represented as follows: 𝑋 Hence, Corollary 2.3 applies in 𝑑 , 𝜌 , with 𝑋 .

We present another result in the case of a generalized metric space but endowed with two metrics.

Theorem 2.7. Let 𝑓 ∶ 𝑋 → 𝑋 be a nonempty set and let 𝐶 ∈ 𝑀 𝑚 , 𝑚 ( ℝ + ) be two generalized metrics on 𝑑 ( 𝑓 ( 𝑥 ) , 𝑓 ( 𝑦 ) ) ≤ 𝜌 ( 𝑥 , 𝑦 ) ⋅ 𝐶 ; . Let ( 𝑋 , 𝑑 ) be an operator. We assume that (1)there exists 𝑓 ∶ ( 𝑋 , 𝑑 ) → ( 𝑋 , 𝑑 ) such that 𝐴 , 𝐵 ∈ 𝑀 𝑚 , 𝑚 ( ℝ + ) (2) 𝑥 , 𝑦 ∈ 𝑋 is a complete generalized metric space; (3) 𝜌 ( 𝑓 ( 𝑥 ) , 𝑓 ( 𝑦 ) ) ≤ 𝐴 𝜌 ( 𝑥 , 𝑦 ) + 𝐵 𝜌 ( 𝑦 , 𝑓 ( 𝑥 ) ) . ( 2 . 1 3 ) is continuous;(4)there exists 𝐴 such that for all F i x ( 𝑓 ) ≠ ∅ . one has 𝐴 + 𝐵 If the matrix F i x ( 𝑓 ) = { 𝑥 ∗ } . converges towards zero, then ( 𝑥 𝑛 ) 𝑛 ∈ ℕ
In addition, if the matrix 𝑥 𝑛 + 1 = 𝑓 ( 𝑥 𝑛 ) converges to zero, then 𝑥 0 ∈ 𝑋

Proof. We consider the sequence of successive approximations 𝜌 𝑥 1 , 𝑥 2 𝑓 𝑥 = 𝜌 0 𝑥 , 𝑓 1 𝑥 ≤ 𝐴 𝜌 0 , 𝑥 1 𝑥 + 𝐵 𝜌 1 𝑥 , 𝑓 0 𝑥 = 𝐴 𝜌 0 , 𝑥 1 , 𝜌 𝑥 2 , 𝑥 3 𝑓 𝑥 = 𝜌 1 𝑥 , 𝑓 2 𝑥 ≤ 𝐴 𝜌 1 , 𝑥 2 𝑥 + 𝐵 𝜌 2 𝑥 , 𝑓 1 ≤ 𝐴 2 𝜌 𝑥 0 , 𝑥 1 , ⋮ 𝜌 𝑥 𝑛 , 𝑥 𝑛 + 1 ≤ 𝐴 𝑛 𝜌 𝑥 0 , 𝑥 1 , ∀ 𝑛 ∈ ℕ , 𝑛 ≥ 1 . ( 2 . 1 4 ) defined recurrently by 𝑝 ∈ ℕ , 𝑝 > 0 being arbitrary. The following statements hold: 𝜌 𝑥 𝑛 , 𝑥 𝑛 + 𝑝 𝑥 ≤ 𝜌 𝑛 , 𝑥 𝑛 + 1 𝑥 + 𝜌 𝑛 + 1 , 𝑥 𝑛 + 2 𝑥 + ⋯ + 𝜌 𝑛 + 𝑝 − 1 , 𝑥 𝑛 + 𝑝 ≤ 𝐴 𝑛 𝜌 𝑥 0 , 𝑥 1 + 𝐴 𝑛 + 1 𝜌 𝑥 0 , 𝑥 1 + ⋯ + 𝐴 𝑛 + 𝑝 − 1 𝜌 𝑥 0 , 𝑥 1 ≤ 𝐴 𝑛 𝐼 + 𝐴 + 𝐴 2 + ⋯ + 𝐴 𝑝 − 1 𝜌 𝑥 + ⋯ 0 , 𝑥 1 = 𝐴 𝑛 ( 𝐼 − 𝐴 ) − 1 𝜌 𝑥 0 , 𝑥 1 . ( 2 . 1 5 ) Now, let 𝑛 → ∞ , 𝜌 ( 𝑥 𝑛 , 𝑥 𝑛 + 𝑝 ) → 0 ∈ ℝ 𝑚 . We estimate ( 𝑥 𝑛 ) 𝑛 ∈ ℕ Letting 𝜌 we obtain that ( 1 ) . Thus 𝑑 𝑥 𝑛 , 𝑥 𝑛 + 𝑝 𝑓 𝑥 = 𝑑 𝑛 − 1 𝑥 , 𝑓 𝑛 + 𝑝 − 1 𝑥 ≤ 𝜌 𝑛 − 1 , 𝑥 𝑛 + 𝑝 − 1 ⋅ 𝐶 ≤ 𝐴 𝑛 − 1 ( 𝐼 − 𝐴 ) − 1 𝜌 𝑥 0 , 𝑥 1 𝐶 ⟶ 0 , a s 𝑛 ⟶ ∞ . ( 2 . 1 6 ) is a Cauchy sequence with respect to ( 𝑥 𝑛 ) 𝑛 ∈ ℕ .
On the other hand, using the statement 𝑑 , we get
( 𝑋 , 𝑑 )
Hence, 𝑥 ∗ ∈ 𝑋 is a Cauchy sequence with respect to 𝑥 ∗ = l i m 𝑛 → ∞ 𝑥 𝑛 . Since 𝑑 is complete, one obtains the existence of an element 𝑥 ∗ = 𝑓 ( 𝑥 ∗ ) such that F i x ( 𝑓 ) ≠ ∅ with respect to 𝑥 𝑛 + 1 = 𝑓 ( 𝑥 𝑛 ) .
We prove next that 𝑛 ∈ ℕ , that is, 𝑛 → ∞ . Indeed, since 𝑓 , for all 𝑑 , letting 𝑥 ∗ = 𝑓 ( 𝑥 ∗ ) and taking into account that 𝑥 ∗ is continuous with respect to 𝑥 ∗ , 𝑦 ∗ ∈ F i x ( 𝑓 ) , we get that 𝑥 ∗ ≠ 𝑦 ∗ .
The uniqueness of the fixed point 𝜌 𝑥 ∗ , 𝑦 ∗ 𝑓 𝑥 = 𝜌 ∗ 𝑦 , 𝑓 ∗ 𝑥 ≤ 𝐴 𝜌 ∗ , 𝑦 ∗ 𝑦 + 𝐵 𝜌 ∗ 𝑥 , 𝑓 ∗ = 𝑥 ( 𝐴 + 𝐵 ) 𝜌 ∗ , 𝑦 ∗ . ( 2 . 1 7 ) is proved below.
Let 𝐴 + 𝐵 such that 𝑥 ( 𝐼 − 𝐴 − 𝐵 ) 𝜌 ∗ , 𝑦 ∗ 𝑥 ≤ 0 ⟹ 𝜌 ∗ , 𝑦 ∗ ≤ 0 ⟹ 𝑥 ∗ = 𝑦 ∗ . ( 2 . 1 8 ) . We estimate ( 𝑋 , 𝑑 ) Thus, using the additional assumption on the matrix 𝑥 0 ∈ 𝑋 , we have that 𝑟 ∶ = ( 𝑟 𝑖 ) 𝑚 𝑖 = 1 ∈ ℝ 𝑚 +

In what follows, we will present some results for the case of multivalued operators.

Theorem 2.8. Let 𝑟 𝑖 > 0 be a complete generalized metric space and let 𝑖 ∈ { 1 , 2 , … , 𝑚 } , 𝐹 ∶ 𝐵 ( 𝑥 0 , 𝑟 ) → 𝑃 c l ( 𝑋 ) with 𝐴 , 𝐵 ∈ 𝑀 𝑚 , 𝑚 ( ℝ + ) for each 𝑥 , 𝑦 ∈ 𝐵 ( 𝑥 0 , 𝑟 ) . Consider 𝑢 ∈ 𝐹 ( 𝑥 ) a multivalued operator. One assumes that (i)there exist 𝑣 ∈ 𝐹 ( 𝑦 ) such that for all 𝑑 ( 𝑢 , 𝑣 ) ≤ 𝐴 𝑑 ( 𝑥 , 𝑦 ) + 𝐵 𝑑 ( 𝑦 , 𝑢 ) ; ( 2 . 1 9 ) and 𝑥 1 ∈ 𝐹 ( 𝑥 0 ) there exists 𝑑 ( 𝑥 0 , 𝑥 1 ) ( 𝐼 − 𝐴 ) − 1 ≤ 𝑟 ; with 𝑢 ∈ ℝ 𝑚 + (ii)there exists 𝑢 ( 𝐼 − 𝐴 ) − 1 ≤ ( 𝐼 − 𝐴 ) − 1 𝑟 such that 𝑢 ≤ 𝑟 (iii)if 𝐴 is such that F i x ( 𝐹 ) ≠ ∅ , then ( i i ) .
If ( i i i ) is a matrix convergent towards zero, then 𝑥 1 ∈ 𝐹 ( 𝑥 0 ) .

Proof. By 𝑑 𝑥 0 , 𝑥 1 ( 𝐼 − 𝐴 ) − 1 ≤ 𝑟 ≤ ( 𝐼 − 𝐴 ) − 1 𝑥 𝑟 ⟹ 𝑑 0 , 𝑥 1 ≤ 𝑟 ⟹ 𝑥 1 ∈ 𝐵 𝑥 0 , 𝑟 . ( 2 . 2 0 ) and 𝑥 1 ∈ 𝐹 ( 𝑥 0 ) , there exists 𝑥 2 ∈ 𝐹 ( 𝑥 1 ) such that 𝑑 𝑥 1 , 𝑥 2 ( 𝐼 − 𝐴 ) − 1 𝑥 ≤ 𝐴 𝑑 0 , 𝑥 1 ( 𝐼 − 𝐴 ) − 1 𝑥 + 𝐵 𝑑 1 , 𝑥 1 ( 𝐼 − 𝐴 ) − 1 ≤ 𝐴 𝑟 . ( 2 . 2 1 ) For 𝑑 𝑥 0 , 𝑥 2 ( 𝐼 − 𝐴 ) − 1 𝑥 ≤ 𝑑 0 , 𝑥 1 ( 𝐼 − 𝐴 ) − 1 𝑥 + 𝑑 1 , 𝑥 2 ( 𝐼 − 𝐴 ) − 1 ≤ 𝐼 𝑟 + 𝐴 𝑟 ≤ 𝐼 + 𝐴 + 𝐴 2 + ⋯ 𝑟 = ( 𝐼 − 𝐴 ) − 1 𝑟 𝑥 ⟹ 𝑑 0 , 𝑥 2 ≤ 𝑟 ⟹ 𝑥 2 ∈ 𝐵 𝑥 0 . , 𝑟 ( 2 . 2 2 ) , there exists 𝑥 2 ∈ 𝐹 ( 𝑥 1 ) with 𝑥 3 ∈ 𝐹 ( 𝑥 2 ) Hence 𝑑 𝑥 2 , 𝑥 3 ( 𝐼 − 𝐴 ) − 1 𝑥 ≤ 𝐴 𝑑 1 , 𝑥 2 ( 𝐼 − 𝐴 ) − 1 𝑥 + 𝐵 𝑑 2 , 𝑥 2 ( 𝐼 − 𝐴 ) − 1 ≤ 𝐴 2 𝑟 , ( 2 . 2 3 ) Next, for 𝑑 𝑥 0 , 𝑥 3 ( 𝐼 − 𝐴 ) − 1 𝑥 ≤ 𝑑 0 , 𝑥 1 ( 𝐼 − 𝐴 ) − 1 𝑥 + 𝑑 1 , 𝑥 2 ( 𝐼 − 𝐴 ) − 1 𝑥 + 𝑑 2 , 𝑥 3 ( 𝐼 − 𝐴 ) − 1 ≤ 𝐼 𝑟 + 𝐴 𝑟 + 𝐴 2 𝑟 ≤ 𝐼 + 𝐴 + 𝐴 2 + ⋯ 𝑟 = ( 𝐼 − 𝐴 ) − 1 𝑟 𝑥 ⟹ 𝑑 0