Uniqueness solution for bounded n-linear functional using generalized nonexpansive type

Basel Hardan; Ahmed A. Hamoud; Jayashree Patil; Alaa A. Abdallah; Homan Emadifar; Masoumeh Khademi; Kirtiwant P. Ghadle

doi:10.1515/taa-2023-0108

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Uniqueness solution for bounded n-linear functional using generalized nonexpansive type

Basel Hardan , Ahmed A. Hamoud , Jayashree Patil , Alaa A. Abdallah , Homan Emadifar , Masoumeh Khademi and Kirtiwant P. Ghadle

Published/Copyright: December 31, 2023

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From the journal Topological Algebra and its Applications Volume 11 Issue 1

Abstract

Using the class of ( μ , σ ) -nonexpansive mappings, we will effectively prove the uniqueness of the approximative fixed points set for equivalent n -linear functional spaces F , where F is nonempty and identical in every bounded case in n -Banach space.

Keywords: linear functional; normed spaces; nonexpansive mappings; fixed point; approximate fixed point

MSC 2010: 34K06; 46B50; 81Q05

1 Introduction

Functional analysis studies the spaces of functions and the processes operating on them, among which are the normed and inner product spaces. They are the basis for studying many concepts in the different branches of mathematics and measuring vector and nonvector quantities in physics. Spaces for norms and internal products serve as the foundation for examining a variety of mathematical ideas and measuring both vector and nonvector quantities in physics. Fixed point theory is one of the most important topics in the development of functional analysis. In addition, it has been used effectively in many branches of science, such as chemistry, biology, economics, computer science, and engineering. Göhler has initiated the concept of a complete two-normed space [10] and has subsequently been studied by showing the existence of fixed points of contractive mappings. As in other spaces, the fixed point theory of mappings has been developed in such spaces also. Furthermore, Iseki [18] obtained for the first time the basic results on a fixed point of mappings in a complete two-normed space. Gähler [11–13] also added the notion of n -normed spaces. Few results in n-normed spaces are available in the literature. For example, basic properties, fuzzy cases, and n -distance of n -normed space were found in [8,14,17,22,26], and sequences on n -normed space were discussed by Ekariani et al. in [9,16,27]. In addition, fixed point theory on a bounded set was introduced by Meiti and Singh [20] and Gunawan et al. [15], Batkunde et al. introduced the topological properties with regard to its quotient space of n -normed space [6,7]. Banach’s contraction mapping method [4] served as the foundation for the theory of fixed points, which has since been expanded upon and used by several researchers, see [23–25]. Several nonexpansive mapping hypotheses have received attention, and many authors have emphasized their fixed point theorems [1–3,19]. To determine whether a solution is possible and how it would apply specifically to various types of spaces in a changing environment, researchers are examining this problem.

Proving the unique solution using contractive mappings includes a big difficulty, for example, identity mapping. So, proving uniqueness using nonexpansive mappings is a new challenge, most authors avoid it. We will prove in this article that n -linear functionals have approximate fixed points in generalized normed space by the type of ( μ , σ )-nonexpansive mappings.

2 Preliminaries

Definition 2.1

[11] Let U be a real linear space dim U ≥ n , and let ‖ ⋅ , … , ⋅ ‖ : U × U → ℛ such that

‖ x 1 , … , x n ‖ = 0 if and only if x 1 , … , x n are linearly dependent for all x 1 , … , x n ∈ U ;
‖ x 1 , … , x n ‖ = ‖ x i 1 , … , x i n ‖ for all ( i 1 , … , i n ) ∈ ( 1 , … , n ) ;
‖ k x 1 , … , x n ‖ = ∣ k ∣ ‖ x 1 , … , x n ‖ , k ∈ ℛ ;
‖ x 1 + x ̀ 1 , x 2 , … , x n ‖ ≤ ‖ x 1 , … , x n ‖ + ‖ x ̀ 1 , … x n ‖ for all x 1 , x ̀ 1 , … , x n ∈ U .

Then, ( U , ‖ ⋅ , … , ⋅ ‖ ) is a linear n -normed space.

According to Batkund et al. [5]:

(1) ‖ x 1 + k 2 x 2 + ⋯ + k n x n , x 2 , … , x n ‖ = ‖ x 1 , x 2 , … , x n ‖

for all x 1 , x 2 , … , x n ∈ U and k 2 , … , k n ∈ ℛ

Definition 2.2

[21] Let U be a real linear space dim U ≥ n + 1 , and let a function ⟨ ⋅ , ⋅ ∣ ⋅ , … , ⋅ ⟩ : U × U → ℛ such that

⟨ x 1 , x 1 ∣ x 2 , … , x n ⟩ ≥ 0 and ⟨ x 1 , x 1 ∣ x 2 , … , x n ⟩ = 0 if and only if x 1 , … , x n are linearly dependent;
⟨ x 1 , x 1 ∣ x 2 , … , x n ⟩ = ⟨ x i 1 , x i 1 ∣ x i 2 , … , x i n ⟩ , for all ( i 1 , … , i n ) ∈ ( 1 , … , n ) ;
⟨ x ̀ 1 , x 1 ∣ x 2 , … , x n ⟩ = ⟨ x 1 , x ̀ 1 ∣ x 2 , … , x n ⟩ ;
⟨ k x 1 , x 1 ∣ x 2 , … , x n ⟩ = ∣ k ∣ ⟨ x 1 , x 1 ∣ x 2 , … , x n ⟩ , k ∈ ℛ ;
⟨ x 0 + x ̀ 0 , x 1 ∣ x 2 , … , x n ⟩ = ⟨ x 0 , x 1 ∣ x 2 , … , x n ⟩ + ⟨ x ̀ 0 , x 1 ∣ x 2 , … , x n ⟩ , for all x 0 , x ̀ 0 , x 1 , … , x n ∈ U .

Then, ( U , ⟨ ⋅ , ⋅ ∣ ⋅ , … , ⋅ ⟩ ) is called an n -inner product space.

In the following part, we define the class ( μ , σ ) -nonexpansive mappings in complete n -metric space and prove that every ( μ , σ ) -nonexpansive mapping ρ : F → F with σ ≥ 0 and μ > 0 has an approximative fixed point set.

3 Main results

We will introduce concepts and ideas related to n -linear functional on n -normed space.

Definition 3.1

Let ( U , ‖ ⋅ , … , ⋅ ‖ ) be an n -normed space and x 1 , x 2 , … , x n be n -subspaces of U , and let ξ ( x 1 , x 2 , … , x n ) ⊂ F such that F : ξ ( x 1 , x 2 , … , x n ) → ℛ , then a function ξ ( x 1 , x 2 , … , x n ) = ‖ x 1 , ϱ i 1 , … ϱ i n ‖ ‖ x 2 , ς i 1 , … ς i n ‖ … ‖ x n , υ i 1 , … υ i n ‖ is called an n -linear functional on U ∀ x 1 , x 2 , … , x n ∈ U for all ( i 2 , … , i n ) ∈ ( 1 , … , n ) with ( i 2 ≤ ⋯ ≤ i n ) and i 1 ∈ { 1 , … , n } ⧹ { i 2 , … , i n } .

Definition 3.2

If ξ ( x k 1 ) ∈ F ⊂ U satisfies

(2) lim k 1 , k 2 → ∞ ‖ ξ ( x k 1 ) − ξ ( x k 2 ) 1 , x 2 , … , x n ‖ = 0 for all x 1 , … , x n ∈ U and k 1 , k 2 ∈ N

then ξ ( x k 1 ) is a Cauchy sequence.

Definition 3.3

Let ( U 1 , ‖ ⋅ , … , ⋅ ‖ ) x 1 , ( U 2 , ‖ ⋅ , … , ⋅ ‖ ) x 2 , … , ( U n , ‖ ⋅ , … , ⋅ ‖ ) x n be the n -normed space, and let ϱ = { ϱ i 1 , … , ϱ i n } ϱ , ς = { ς i 1 , … , ς i n } ς , … , and υ = { υ i 1 , … , υ i n } υ be fixed linearly independent sets. An n -linear functional ξ : U 1 × U 2 × ⋯ × U n → ℛ is said to be bounded from the first type with respect to ϱ , ς , … , υ , if there exist M ∈ ℛ + such that

(3) ∣ ξ ( x 1 , x 2 , … , x n ) ∣ ≤ M ( ‖ x 1 , ϱ i 1 , … , ϱ i n ‖ U 1 ) × ( ‖ x 2 , ς i 1 , … , ς i n ‖ U 2 ) × ⋯ × ( ‖ x n , υ i 1 , … , υ i n ‖ U n )

for all x 1 ∈ U 1 , … , x n ∈ U n .

Definition 3.4

Let ( U 1 , ‖ ⋅ , … , ⋅ ‖ ) x 1 , ( U 2 , ‖ ⋅ , … , ⋅ ‖ ) x 2 , … , ( U n , ‖ ⋅ , … , ⋅ ‖ ) x n be the n -normed space and let ϱ = { ϱ i 1 , … , ϱ i n } ϱ , ς = { ς i 1 , … , ς i n } ς , … , and υ = { υ i 1 , … , υ i n } υ be fixed linearly independent sets. An n -linear functional ξ : U 1 × U 2 × ⋯ × U n → ℛ and 1 ≤ τ ≤ ∞ is said to be bounded of τ t h type with respect to a pairs of ϱ , ς , … , υ , if there exist M ∈ ℛ + such that

(4) ∣ ξ ( x 1 , x 2 , … , x n ) ∣ ≤ M ( ‖ x 1 , ϱ i 1 , … , ϱ i n ‖ U 1 τ ) 1 τ × ( ‖ x 2 , ς i 1 , … , ς i n ‖ U 2 τ ) 1 τ × ⋯ × ( ‖ x n , υ i 1 , … , υ i n ‖ U n τ ) 1 τ ,

for all x 1 ∈ U 1 , x 2 ∈ U 2 .

Definition 3.5

Let ( U , ⟨ ⋅ , ⋅ ∣ , … , ⋅ ⟩ ) be an n -inner product space, and let ξ ( x 1 , x 2 , … , x n ) ⊂ F , such that F : ξ ( x 1 , x 2 , … , x n ) → ℛ . A function

(5) ∣ ξ ( x 1 , x 2 , … , x n ) ∣ = ∑ ⟨ x 1 , ϱ i 1 ∣ ϱ i 2 , … , ϱ i n ⟩ × ∑ ⟨ x 2 , ς i 1 ∣ ς i 2 , … , ς i n ⟩ × ⋯ × ∑ ⟨ x n , υ i 1 ∣ υ i 2 , … , υ i n ⟩

is called n -linear functional on U for all x 1 , x 2 , … , x n ∈ U and for all ( i 2 , … , i n ) ∈ ( 1 , … , n ) with ( i 2 ≤ ⋯ ≤ i n ) and i 1 ∈ { 1 , … , n } ⧹ { i 2 , … , i n } .

In the next procedure, we will show that the n -linear functionals F are bounded and equivalent, forming a complete n -normed space.

Lemma 3.6

Let U be an n-normed space, and let { ξ ( h k ) } , { φ ( s k ) } ⊂ F ∈ U be two Cauchy sequences in n-linear functional space, then

ξ ( x k ) is a Cauchy Sequence on ℛ .
‖ ξ ( x k ) + φ ( s k ) , ϱ 2 , … , ϱ n ‖ is a Cauchy sequence in U ,
If { ξ ( x k ) } → x and { ξ ( x k ) } → ϰ , then x = ϰ , k → ∞ .

Proof

If { ξ ( x k ) } , { φ ( s k 1 ) } , then

‖ ξ ( x k ) , ϱ 2 , … , ϱ n ‖ = ‖ ( ξ ( x k ) − φ ( s k 1 ) ) + φ ( s k 1 ) , ϱ 2 , … , ϱ n ‖ , ≤ ‖ ( ξ ( x k ) − φ ( s k 1 ) ) , ϱ 2 , … , ϱ n ‖ + ‖ φ ( s k 1 ) , ϱ 2 , … , ϱ n ‖ ,

so we have

‖ ξ ( x k ) , ϱ 2 , … , ϱ n ‖ − ‖ φ ( s k 1 ) , ϱ 2 , … , ϱ n ‖ ≤ ‖ ( ξ ( x k ) − φ ( s k 1 ) ) , ϱ 2 , … , ϱ n ‖ + ‖ φ ( s k 1 ) , ϱ 2 , … , ϱ n ‖

and

‖ φ ( s k 1 ) , ϱ 2 , … , ϱ n ‖ − ‖ ξ ( x k ) , ϱ 2 , … , ϱ n ‖ ≤ ‖ ( ξ ( x k ) − φ ( s k 1 ) ) , ϱ 2 , … , ϱ n ‖ ,

and consolidating the above, we make sure that

∣ ‖ ξ ( x k ) , ϱ 2 , … , ϱ n ‖ − ‖ φ ( s k 1 ) , ϱ 2 , … , ϱ n ‖ ∣ ≤ ‖ ( ξ ( x k ) − φ ( s k 1 ) ) , ϱ 2 , … , ϱ n ‖ → 0 ,

as k , k 1 → ∞ . Hence, ‖ ξ ( x k ) , ϱ 2 , … , ϱ n ‖ is a Cauchy sequence in ℛ , and (i) is proven.

To confirm (ii), if { ξ ( x k ) } , { φ ( s k ) } be two Cauchy sequences, then

lim k , k 1 → ∞ ‖ ξ ( x k ) − ξ ( x k 1 ) , ϱ 2 , … , ϱ n ‖ → 0 , for all ϱ 2 , … , ϱ n ∈ U

and

lim k , k 1 → ∞ ‖ φ ( x k ) − φ ( x k 1 ) , ϱ 2 , … , ϱ n ‖ → 0 , for all ϱ 2 , … , ϱ n ∈ U ,

and therefore,

‖ ( ξ ( x k ) + φ ( x k ) − ( ξ ( x k 1 ) + φ ( x k 1 ) ) ) , ϱ 2 , … , ϱ n ‖ = ‖ ( ξ ( x k ) − ξ ( x k 1 ) + ( φ ( x k ) − φ ( x k 1 ) ) ) , ϱ 2 , … , ϱ n ‖ , ≤ ‖ ( ξ ( x k ) − ξ ( x k 1 ) ) , ϱ 2 , … , ϱ n ‖ + ‖ ( φ ( x k ) − φ ( x k 1 ) ) , ϱ 2 , … , ϱ n ‖ → 0 ,

as k , k 1 → ∞ . Now, to prove (iii), we follow

‖ x − ϰ , ϱ 2 , … , ϱ n ‖ = ‖ x − ϰ + ξ ( x k ) − ξ ( x k ) , ϱ 2 , … , ϱ n ‖ , ≤ ‖ ξ ( x k ) − x , ϱ 2 , … , ϱ n ‖ + ‖ ξ ( x k ) − ϰ , ϱ 2 , … , ϱ n ‖ → 0 ,

as k → ∞ . Since dim U ≥ n , we have one choice to consider that x − ϰ is linearly dependent with ϱ 2 , … , ϱ n , which is x − ϰ = 0 . Hence, x = ϰ .□

Corollary 3.7

‖ ζ k ξ ( x k ) , ϱ 1 , … , ϱ n ‖ is a Cauchy sequence in U , such that ζ k is a Cauchy sequence in ℛ .

Corollary 3.8

Every F ∈ U is a complete n-normed space.

Theorem 3.9

That the following statements hold:

n-linear functional (5) is bounded of first type with
(6) ‖ ξ ‖ 1 = ‖ ϱ 1 , … , ϱ n ‖ U 1 ‖ ς 1 , … , ς n ‖ U 2 ⋯ ‖ υ 1 , … , υ n ‖ U n ;
n-linear functional is bounded of τ t h type with
(7) ‖ ξ ‖ τ = n n σ ‖ ϱ 1 , … , ϱ n ‖ U 1 ‖ ς 1 , … , ς n ‖ U 2 ‖ υ 1 , … , υ n ‖ U n ,
where n τ + n σ = n ;
(8) ‖ ξ ‖ 1 ≕ sup { ‖ ϱ 1 , ϱ 2 , … , ϱ n ‖ ‖ ς 1 , ς 2 , … , ς n ‖ … ‖ υ 1 , υ 2 , … , υ n ‖ , × ∑ ‖ x 1 , ϱ i 2 , … , ϱ i n ‖ U 1 ≤ 1 , ∑ ‖ x 2 , ς i 2 , … , ς i n ‖ U 2 ≤ 1 , … , ∑ ‖ x n , υ i 2 , … , υ i n ‖ U n ≤ 1 ;
(9) ‖ ξ ‖ τ ≕ sup n n σ ‖ ϱ 1 , ϱ 2 , … , ϱ n ‖ ‖ ς 1 , ς 2 , … , ς n ‖ … ‖ υ 1 , υ 2 , … , υ n ‖ , × ∑ ‖ x 1 , ϱ i 2 , … , ϱ i n ‖ U 1 τ ≤ 1 , ∑ ‖ x 2 , ς i 2 , … , ς i n ‖ U 2 τ ≤ 1 , … , ∑ ‖ x n , υ i 2 , … , υ i n ‖ U n τ ≤ 1 ;
(10) ‖ ξ ‖ 1 ≡ ‖ ξ ‖ τ .

Proof

Note that, for all x 1 ∈ U 1 , x 2 ∈ U 2 , … , x n ∈ U n , we obtain

(11) ∣ ξ ( x 1 , x 2 , … , x n ) ∣ ≤ ‖ ϱ i 1 , … , ϱ i n ‖ U 1 × ‖ ς i 1 , … , ς i n ‖ U 2 × ⋯ × ‖ υ i 1 , … , υ i n ‖ U n × ∑ ‖ x 1 , ϱ i 2 , … , ϱ i n ‖ U 1 ∑ ‖ x 2 , ς i 2 , … , ς x n ‖ U 2 ⋯ ∑ ‖ x n , υ i 2 , … , υ i n ‖ U n ,

taking

x 1 = ‖ ϱ i 2 , … , ϱ i n ‖ ϱ 1 x 2 = ‖ ς i 2 , … , ς i n ‖ ς 1 ⋮ x n = ‖ υ i 2 , … , υ i n ‖ υ 1 .

Then,

∑ ‖ x 1 , ϱ i 2 , … , ϱ i n ‖ = ∑ ‖ x 2 , ς i 2 , … , ς x n ‖ = , … , = ∑ ‖ x n , υ i 2 , … , υ i n ‖ = 1

and

(12) ∣ ξ ( x 1 , x 2 , … , x n ) ∣ = ‖ ϱ 1 , ϱ 2 , … , ϱ n ‖ − 1 ‖ ς 1 , ς 2 , … , ς n ‖ − 1 ⋯ ‖ υ 1 , υ 2 , … , υ n ‖ − 1 × ∑ ⟨ ϱ 1 , ϱ i 1 ∣ ϱ i 2 , … , ϱ i n ⟩ ∑ ⟨ ς 1 , ς i 1 ∣ ς i 2 , … , ς i n ⟩ ⋯ ∑ ⟨ υ 1 , υ i 1 ∣ υ i 2 , … , υ i n ⟩

such that i 1 ≠ 1 and i 1 ∈ { 1 , … , n } ⧹ { i 2 , … , i n } .

∣ ⟨ ϱ 1 , ϱ i 1 ∣ ϱ i 2 , … , ϱ i n ⟩ ∣ ≤ ‖ ϱ 1 , … , ϱ n ‖ ∣ ς 1 , … , ς n ‖ ⋯ ∣ υ 1 , … , υ n ‖ = 0 .

Also,

∣ ⟨ ϱ 1 , ϱ i 1 ∣ ϱ i 2 , … , ϱ i n ⟩ ∣ = ∣ ⟨ ς 1 , ς i 1 ∣ ς i 2 , … , ς i n ⟩ ∣ = ⋯ = ∣ ⟨ υ 1 , υ i 1 ∣ υ i 2 , … , υ i n ⟩ ∣ = 0 ,

since, ϱ 1 should be equal one from ϱ i 2 , … , ϱ i n , same case for ς 1 , … , υ 1 .

Then, we obtain

(13) ∣ ξ ( x 1 , x 2 , … , x n ) ∣ = ‖ ϱ 1 , … , ϱ n ‖ − 1 ‖ ς 1 , … , ς n ‖ − 1 ⋯ ‖ υ 1 , … , υ n ‖ − 1 × ⟨ ϱ 1 , ϱ 1 ∣ ϱ 2 , … , ϱ n ⟩ ⟨ ς 1 , ς 1 ∣ ς 2 , … , ς n ⟩ ⋯ ⟨ υ 1 , υ 1 ∣ υ 2 , … , υ n ⟩ = ‖ ϱ 1 , … , ϱ n ‖ ‖ ς 1 , … , ς n ‖ ⋯ ‖ υ 1 , … , υ n ‖

such that i 1 ≠ 1 and i 1 ∈ { 1 , … , n } ⧹ { i 2 , … , i n } .

Then, ξ is bounded of first type, such that

(14) ‖ ξ ‖ 1 = ‖ ϱ 1 , … , ϱ n ‖ ‖ ς 1 , … , ς n ‖ ⋯ ‖ υ 1 , … , υ n ‖ ,

condition (ii) is satisfied. We will apply Hölder inequality on equation (5) to obtain

(15) ‖ ξ ( x 1 , x 2 , … , x n ) ‖ ≤ n n σ ‖ ϱ 1 , … , ϱ n ‖ ‖ ς 1 , … , ς n ‖ ⋯ ‖ υ 1 , … , υ n ‖ × ∑ ‖ x 1 , ϱ i 2 , … , ϱ i n ‖ τ 1 τ ∑ ‖ x 2 , ς i 2 , … , ς i n ‖ τ 1 τ ⋯ ∑ ‖ x n , υ i 2 , … , υ i n ‖ τ 1 τ .

Take

x 1 = n − 1 τ ‖ ϱ 1 , … , ϱ n ‖ − 1 ( ϱ 1 + ⋯ + ϱ n ) x 2 = n − 1 τ ‖ ς 1 , … , ς n ‖ − 1 ( ς 1 + ⋯ + ς n ) ⋮ x n = n − 1 τ ‖ υ 1 , … , υ n ‖ − 1 ( υ 1 + ⋯ + υ n ) .

Therefore, by equation (1), we have

∣ ξ ( x 1 , x 2 , … , x n ) ∣ = n − 1 τ ‖ ϱ 1 , … , ϱ n ‖ − 1 ∑ ⟨ ϱ 1 + ⋯ + ϱ n , ϱ i 1 ∣ ϱ i 2 , … , ϱ i n ⟩ × n − 1 τ ‖ ς 1 , … , ς n ‖ − 1 ∑ ⟨ ς 1 + ⋯ + ς n , ϱ i 1 ∣ ς i 2 , … , ς i n ⟩ … × n − 1 τ ‖ υ 1 , … , υ n ‖ − 1 ∑ ⟨ υ 1 + ⋯ + υ n , υ i 1 ∣ υ i 2 , … , υ i n ⟩ = n − 1 τ ‖ ϱ 1 , … , ϱ n ‖ − 1 ∑ ⟨ ϱ i 1 , ϱ i 1 , ∣ ϱ i 2 , … , ϱ i n ⟩ × n − 1 τ ‖ ς 1 , … , ς n ‖ − 1 ∑ ⟨ ς i 1 , ς i 1 , ∣ ς i 2 , … , ς i n ⟩ … × n − 1 τ ‖ υ 1 , … , υ n ‖ − 1 ( ∑ ⟨ υ i 1 , ς i 1 , ∣ υ i 2 , … , υ i n ⟩ ) = n − 1 σ ‖ ϱ 1 , … , ϱ n ‖ − 1 n ‖ ϱ 1 , … , ϱ n ‖ 2 × n − 1 τ ‖ ς 1 , … , ς n ‖ − 1 n ‖ ς 1 , … , ς n ‖ 2 … × n − 1 τ ‖ υ 1 , … , υ n ‖ − 1 n ‖ υ 1 , … , υ n ‖ 2 = n − n σ n n ‖ ϱ 1 , … , ϱ n ‖ ‖ ς 1 , … , ς n ‖ ⋯ ‖ υ 1 , … , υ n ‖ .

Thus,

(16) ‖ ξ ‖ τ = n n τ ‖ ϱ 1 , … , ϱ n ‖ ‖ ς 1 , … , ς n ‖ ⋯ ‖ υ 1 , … , υ n ‖ ,

condition (iii) has been proven.

Now, let ξ : ( U 1 × U 2 × ⋯ × U n ) → ℛ be an n -linear functional defined on

( U 1 , ⟨ ⋅ , ⋅ ∣ ⋅ , … , ⋅ ⟩ U 1 ) , ( U 2 , ⟨ ⋅ , ⋅ ∣ ⋅ , … , ⋅ ⟩ U 2 ) , … , ( U n , ⟨ ⋅ , ⋅ ∣ ⋅ , … , ⋅ ⟩ U n ) as

(17) ξ ( x 1 , x 2 , … , x n ) = ∑ ⟨ x 1 , ϱ i 1 ∣ ϱ i 2 , … , ϱ i n ⟩ U 1 ∑ ⟨ x 2 , ς i 1 ∣ ς i 2 , … , ς i n ⟩ U 2 ⋯ ∑ ⟨ x n , υ i 1 ∣ υ i 2 , … , υ i n ⟩ U n ≤ ∑ ‖ x 1 , ϱ i 2 , … , ϱ i n ‖ U 1 ‖ ∑ ‖ ϱ i 1 , ϱ i 2 , … , ϱ i n ‖ U 1 × ∑ ‖ x 2 , ς i 2 , … , ς i n ‖ U 2 ‖ ∑ ‖ ς i 1 , ς i 2 , … , ς i n ‖ U 2 … ∑ ‖ x n , υ i 2 , … , υ i n ‖ U n ‖ ∑ ‖ υ i 1 , υ i 2 , … , υ i n ‖ U n

(18) ≤ ∑ ‖ x 1 , ϱ i 2 , … , ϱ i n ‖ U 1 ‖ ( ‖ ϱ i 1 , ϱ i 2 , … , ϱ i n ‖ U 1 ) × ∑ ‖ x 2 , ς i 2 , … , ς i n ‖ U 2 ‖ ( ‖ ς i 1 , ς i 2 , … , ς i n ‖ U 2 ) … × ∑ ‖ x n , υ i 2 , … , υ i n ‖ U n ‖ ( ‖ υ i 1 , υ i 2 , … , υ i n ‖ U n ) .

Take

x 1 = ϱ 1 ‖ ϱ 1 , ϱ 2 , … , ϱ n ‖ , x 2 = ς 1 ‖ ς 1 , ς 2 , … , ς n ‖ , … , x n = υ 1 ‖ υ 1 , υ 2 , … , υ n ‖ .

Then, we obtain

(19) ∑ ‖ x 1 , ϱ i 2 , … , ϱ i n ‖ U 1 ≤ 1 , ∑ ‖ x 2 , ς i 2 , … , ς i n ‖ U 2 ≤ 1 , … , ∑ ‖ x n , υ i 2 , … , υ i n ‖ U n ‖ ≤ 1 .

From equations (18) and (19), we obtain (8). The same proof of method (iv) can be used to find equation (9) using condition (iii). To show condition (v), we suppose that equation (8) is satisfying, which implies that each term of the total is less than 1, i.e.,

∑ ‖ x 1 , ϱ i 2 , … , ϱ i n ‖ τ ∑ ‖ x 2 , ς i 2 , … , ς i n ‖ τ ⋯ ∑ ‖ x n , υ i 2 , … , υ i n ‖ τ ≤ 1 .

Then,

‖ ξ ‖ 1 ≤ ‖ ξ ‖ τ .

Letting equation (9) be satisfied, then

∑ ‖ x 1 , ϱ i 2 , … , ϱ i n ‖ τ ∑ ‖ x 2 , ς i 2 , … , ς i n ‖ τ ⋯ ∑ ‖ x n , υ i 2 , … , υ i n ‖ τ ≤ 1 ,

this leads to

∑ ‖ x 1 , ϱ i 2 , … , ϱ i n ‖ ∑ ‖ x 2 , ς i 2 , … , ς i n ‖ ⋯ ∑ ‖ x n , υ i 2 , … , υ i n ‖ ≤ n n σ ,

and

∑ 1 n x 1 σ n n σ , ϱ i 2 , … , ϱ i n ∑ 1 n x 2 σ n n σ , ς i 2 , … , ς i n ⋯ ∑ 1 n x n σ n n σ , υ i 2 , … , υ i n ≤ 1 .

Then,

ξ x 1 n 1 σ , x 2 n 1 σ , … , x n n 1 σ , τ ≤ ‖ ξ ( x 1 , x 2 , … , x n ) ‖ 1 .

Thus,

‖ ξ ( x 1 , x 2 , … , x n ) ‖ τ ≤ n n σ ‖ ξ ( x 1 , x 2 , … , x n ) ‖ 1 .

We obtain that ‖ ξ ( x 1 , x 2 , … , x n ) ‖ 1 and ‖ ξ ( x 1 , x 2 , … , x n ) ‖ τ are equivalent by

(20)□ ‖ ξ ( x 1 , x 2 , … , x n ) ‖ 1 ≤ ‖ ξ ( x 1 , x 2 , … , x n ) ‖ τ ≤ n n σ ‖ ξ ( x 1 , x 2 , … , x n ) ‖ 1 .

Definition 3.10

Suppose that F ≠ ϕ be a complete n -normed space, we say ρ : F → U is ( μ , σ ) -nonexpansive mapping, and if, for all x i , s i , ϰ 2 , … , ϰ n ∈ U , ξ ( x i ) , φ ( s i ) ⊂ F ∈ U and μ , σ ∈ ℛ , i ∈ ( 1 , … , n ) , we have

(21) ‖ ρ ξ ( x i ) − ρ φ ( s i ) , ϰ 2 , … , ϰ n ‖ 2 ≤ μ ‖ ρ ξ ( x i ) − φ ( s i ) , ϰ 2 , … , ϰ n ‖ 2 + μ ‖ ρ φ ( s i ) − ξ ( x i ) , ϰ 2 , … , ϰ n ‖ 2 + σ ‖ ρ ξ ( x i ) − ξ ( x i ) , ϰ 2 , … , ϰ n ‖ 2 + σ ‖ ρ φ ( s i ) − φ ( s i ) , ϰ 2 , … , ϰ n ‖ 2 + 2 ( μ − σ ) ‖ ξ ( x i ) − φ ( s i ) , ϰ 2 , … , ϰ n ‖ 2 .

Remark 3.1

Using (10) in Definition 3.10 to get

(22) ‖ ξ ( x i ) , ϰ 2 , … , ϰ n ‖ ≤ ‖ ρ ξ ( x i ) , ϰ 2 , … , ϰ n ‖ .

Definition 3.11

Let ξ ( x i ) , φ ( s i ) ⊂ F ∈ U , a point ( ‖ ξ ( x 1 ) , x 2 , … , x n ‖ → ℛ ) ∈ F is called fixed point for ρ when ‖ ρ ξ ( x 1 ) , x 2 , … , x n ‖ = ‖ ξ ( x 1 ) , x 2 , … , x n ‖ . A set F ⊃ ‖ ξ ( x i ) , x 2 , … , x n ‖ → ℛ is called an approximate fixed point set for ρ when ‖ ρ ξ ( x i ) , x 2 , … , x n ‖ = ‖ ξ ( x i ) , x 2 , … , x n ‖ , for all x 2 , … , x n ∈ U , and i 1 ∈ ( 1 , … , n ) \ ( i 2 , … , i n ) .

Theorem 3.12

Let ρ : F → U be ( μ , σ ) -nonexpansive type. Thus, ρ ( ξ ( x i ) ) = ξ ( x i ) for all ξ ( x i ) ∈ U , i ∈ ( 1 , … , n ) .

Proof

Suppose

(23) ρ ( ξ ( x 1 , x 2 , … , x n ) ) = ∑ i 1 , i 2 , … , i n ∈ { 1 , 2 , … , n } ξ ( x i 1 , x i 2 , … , x i n ) → ℛ ,

for all x 1 , x 2 , … , x n ∈ U . Then, ∀ ξ , φ ⊂ F ∈ U and x i 1 , … , x i n ∈ x 1 , … , x n , s i 1 , … s i n ∈ s 1 , … s n , where { r i 1 , r i 2 , … , r i n } are linear independent subsets of U such that r takes the following values sequentially, ϱ , ς , υ , , b , c , and ϱ ˆ :

⟨ ρ ξ ( x i ) − ρ φ ( s i ) ⟩ ≤ ‖ x i 1 − s i 1 , ϱ ˆ i 2 , … , ϱ ˆ i n ‖ ‖ x i 2 − s i 2 , ς ˆ i 2 , … , ς ˆ i n ‖ ⋯ ‖ x i n − s i n , υ ˆ i 2 , … , υ ˆ i n ‖ , ≤ ‖ ϱ ˆ i 2 , … , ϱ ˆ i n ‖ ‖ ς ˆ i 2 , … , ς ˆ i n ‖ ‖ υ ˆ i 2 , … , υ ˆ i n ‖ ⋯ ‖ x i n − s i n , x i 1 − s i 1 , … , x i n − s i n ‖ .

We obtain

(24) ‖ ρ ξ ( x i ) − ρ ξ ( s i ) ‖ 2 ≤ ‖ ϱ ˆ i 2 , … , ϱ ˆ i n ‖ ‖ ς ˆ i 2 , … , ς ˆ i n ‖ ‖ υ ˆ i 2 , … , υ ˆ i n ‖ ⋯ ∣ ‖ ξ ( x ) − φ ( s ) ‖ 2 .

Hence,

(25) ‖ ρ ξ ( x i ) − ρ ξ ( s i ) ‖ 2 ≤ σ ‖ ρ ξ ( x i ) − ξ ( x i ) ‖ 2 + σ ‖ ρ φ ( s i ) − φ ( s i ) ‖ 2 − 2 σ ‖ ξ ( x i ) − φ ( s i ) ‖ 2 ,

where

σ ‖ ρ ξ ( x i ) − ξ ( x i ) ‖ 2 + σ ‖ ρ φ ( s i ) − φ ( s i ) ‖ 2 − 2 σ = ‖ ϱ ˆ i 2 , … , ϱ ˆ i n ‖ ‖ ς ˆ i 2 , … , ς ˆ i n ‖ ‖ υ ˆ i 2 , … , υ ˆ i n ‖ .

Then, ρ has an approximate fixed point set. Hence, equation (10) showed that all n -linear functional spaces are identical in every bounded case. Hence, they all have the same solution, which entails the same approximation of fixed point sets in an n -Banach space.□

4 Conclusion

Proving the uniqueness of the study relies on the properties of the space of generalized linear functionals rather than the properties of nonexpansive mappings and introduces an open problem. A new approach proves in this article the uniqueness of any approximate solution in all n -linear functionals using generalized nonexpansive mappings, by proving that the n -linear functionals are bounded, equivalent and form an n -Banach space. Studying the open problem will be in the next articles.

Conflict of interest: Authors state no conflict of interest.

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Received: 2023-08-22

Revised: 2023-12-04

Accepted: 2023-12-29

Published Online: 2023-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/taa-2023-0108

Keywords for this article

linear functional; normed spaces; nonexpansive mappings; fixed point; approximate fixed point

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