On the stability and convergence of Mann iteration process in convex A- metric spaces
aa r X i v : . [ m a t h . GN ] J a n ON THE STABILITY AND CONVERGENCE OF MANN ITERATIONPROCESS IN CONVEX A -METRIC SPACES ISA YILDIRIM (Dedicated to Assoc. Prof. Dr. Birol Gunduz who passed away on the 3rd of April, 2019.)
Abstract.
In this paper, firstly, we introduce the concept of convexity in A -metric spacesand show that Mann iteration process converges to the unique fixed point of Zamfirescu typecontractions in this newly defined convex A -metric space. Secondly, we define the concept ofstability in convex A -metric spaces and establish stability result for the Mann iteration processconsidered in such spaces. Our results carry some well-known results from the literature toconvex A -metric spaces. Introduction and preliminaries
The Banach Fixed Point Theorem which is the one of the most important theorem in allanalysis. It plays a key role for many applications in nonlinear analysis. For example, in the areassuch as optimization, mathematical models, and economic theories. Due to this, the result has beengeneralized in various directions. As a generalization of metric space, Mustafa and Sims introduceda new class of generalized metric spaces called G -metric spaces (see [9], [10]) as a generalizationof metric spaces ( X, d ) . This was done to introduce and develop a new fixed point theory for avariety of mappings in this new setting. This helped to extend some known metric space resultsto this more general setting. The G -metric space is defined as follows: Definition 1.1. [10]
Let X be a nonempty set and let G : X × X × X → R + be a function satisfyingthe following properties:(i) G ( x, y, z ) = 0 if x = y = z (ii) < G ( x, x, y ) for all x, y ∈ X, with x = y (iii) G ( x, x, y ) ≤ G ( x, y, z ) for all x, y, z ∈ X, with z = y (iv) G ( x, y, z ) = G ( x, z, y ) = G ( y, z, x ) = ..., (symmetry in all three variables); and(v) G ( x, y, z ) ≤ G ( x, a, a ) + G ( a, y, z ) for all x, y, z, a ∈ X (rectangle inequality ) . Mathematics Subject Classification.
Primary 47H09 ; Secondary 47H10.
Key words and phrases.
Convex structure, Convex A -metric space, Mann iteration process, Stability. Then the function G is called a generalized metric or more specifically, a G -metric on X , andthe pair ( X, G ) is called a G -metric space. Mustafa et al. studied many fixed point results for a self-mapping in G -metric space. [11]-[13]can be cited for reference.On the other hand, Abbas et al. [1] introduced the concept of an A -metric space as follows: Definition 1.2.
Let X be nonempty set. Suppose a mapping A : X t → R satisfy the followingconditions: ( A ) A ( x , x , ..., x t − , x t ) ≥ , ( A ) A ( x , x , ..., x t − , x t ) = 0 if and only if x = x = ... = x t − = x t , ( A ) A ( x , x , ..., x t − , x t ) ≤ A ( x , x , ..., ( x ) t − , y ) + A ( x , x , ..., ( x ) t − , y )+ ... + A ( x t − , x t − , ..., ( x t − ) t − , y ) + A ( x t , x t , ..., ( x t ) t − , y ) for any x i , y ∈ X, ( i = 1 , , ..., t ) . Then, ( X, A ) is said to be an A -metric space. It is clear that the an A -metric space for t = 2 reduces to ordinary metric d . Also, an A -metricspace is a generalization of the G -metric space. Example 1.1. [1]
Let X = R . Define a function A : X t → R by A ( x , x , ..., x t − , x t ) = | x − x | + | x − x | + ... + | x − x t | + | x − x | + | x − x | + ... + | x − x t | ... + | x t − − x t − | + | x t − − x t | + | x t − − x t | = t X i =1 X i Let ( X, A ) be A -metric space. Then A ( x, x, . . . , x, y ) = A ( y, y, . . . , y, x ) for all x, y ∈ X . Lemma 1.2. [1] Let ( X, A ) be A -metric space. Then for all for all x, y ∈ X we have A ( x, x, . . . , x, z ) ≤ ( t − A ( x, x, . . . , x, y )+ A ( z, z, . . . , z, y ) and A ( x, x, . . . , x, z ) ≤ ( t − A ( x, x, . . . , x, y )+ A ( y, y, . . . , y, z ) . Definition 1.3. [1] Let ( X, A ) be A -metric space.(i) A sequence { x n } in X is said to converge to a point u ∈ X if A ( x n , x n , . . . , x n , u ) → as n → ∞ . TABILITY AND CONVERGENCE IN CONVEX A -METRIC SPACES 3 (ii) A sequence { x n } in X is called a Cauchy sequence if A ( x n , x n , . . . , x n , u m ) → as n, m → ∞ .(iii) The A -metric space ( X, A ) is said to be complete if every Cauchy sequence in X isconvergent. Recently, Yildirim [19] introduced the notion of Zamfirescu mappings in A -metric space asfollows: Definition 1.4. Let ( X, A ) be A -metric space and f : X → X be a mapping. f is called a A -Zamfirescu mapping ( AZ mapping), if and only if, there are real numbers, ≤ a < , ≤ b, c < t such that for all x, y ∈ X , at least one of the next conditions is true: ( AZ ) A ( f x, f x, . . . , f x, f y ) ≤ aA ( x, x, . . . , x, y )( AZ ) A ( f x, f x, . . . , f x, f y ) ≤ b [ A ( f x, f x, . . . , f x, x ) + A ( f y, f y, . . . , f y, y )]( AZ ) A ( f x, f x, . . . , f x, f y ) ≤ c [ A ( f x, f x, . . . , f x, y ) + A ( f y, f y, . . . , f y, x )]Yildirim [19] also extended the Zamfirescu results [21] to A -metric spaces and he obtained thefollowing results on fixed point theorems for such mappings. Lemma 1.3. [19] Let ( X, A ) be A -metric space and f : X → X be a mapping. If f is a AZ mapping, then there is ≤ δ < such that (1.1) A ( f x, f x, . . . , f x, f y ) ≤ δA ( x, x, . . . , x, y ) + tδA ( f x, f x, . . . , f x, x ) and (1.2) A ( f x, f x, . . . , f x, f y ) ≤ δA ( x, x, . . . , x, y ) + tδA ( f y, f y, . . . , f y, x ) for all x, y ∈ X . Theorem 1.1. [19] Let ( X, A ) be complete A -metric space and f : X → X be an AZ mapping.Then f has a unique fixed point and Picard iteration process { x n } defined by x n +1 = f x n convergesto a fixed point of f . Studies in metric spaces are related to the existence of fixed point without approximatingthem. The reason behind is the unavailablity of convex structure in metric spaces. To solve thisproblem, Takahashi [15] introduced the notion of convex metric spaces and studied the approxi-mation of fixed points for nonexpansive mappings in this setting. Inspired by this, Yildirim andKhan [20] defined convex structure in G -metric spaces and they transformed the Mann iterativeprocess to a convex G -metric space as follows. And they also proved some fixed point theoremsdeal with convergence of Mann iteration process for some class of mappings. ISA YILDIRIM Definition 1.5. [20] Let ( X, G ) be a G -metric space. A mapping W : X × I → X is termedas a convex structure on X if G ( W ( x, y ; λ, β ) , u, v ) ≤ λG ( x, u, v ) + βG ( y, u, v ) for real numbers λ and β in I = [0 , satisfying λ + β = 1 and x, y, u and v ∈ X. A G -metric space ( X, G ) with a convex structure W is called a convex G -metric space anddenoted as ( X, G, W ) . A nonempty subset C of a convex G -metric space ( X, G, W ) is said to be convex if W ( x, y ; a, b ) ∈ C for all x, y ∈ C and a, b ∈ I. Definition 1.6. [20] Let ( X, G, W ) be convex G -metric space with convex structure W and f : X → X be a mapping. Let { α n } be a sequence in [0 , for n ∈ N . Then for any given x ∈ X, theiterative process defined by the sequence { x n } as (1.3) x n +1 = W ( x n , f x n ; 1 − α n , α n ) , n ∈ N , is called Mann iterative process in the convex metric space ( X, G, W ) . The iterative approximation of a fixed point for certain classes of mappings is one of themain tools in the fixed point theory. Many authors ([3, 4, 5, 6, 7, 8, 14, 16, 17, 18]) discussed theexistence of fixed points and convergence of different iterative processes for various mappings inconvex metric spaces.Keeping the above in mind, in this paper, we first define the concept of convexity in A -metricspaces. Then, we use Mann iteration in this newly defined convex A -metric space to prove someconvergence results for approximating fixed points of some classes of mappings. We also discussstability result for the Mann iteration process. Results in this paper show that different iterationmethods can be used to approximate fixed points of different class of mappings in A -metric spaces.Our results are just new in the setting.Now, we define convex structure in A -metric spaces as follows. Definition 1.7. Let ( X, A ) be a A -metric space and I = [0 , . A mapping W : X t × I t → X istermed as a convex structure on X if A ( u , u , ..., u t − , W ( x , x , ..., x t − , x t ; a , a , ..., a t ))(1.4) ≤ a A ( u , u , ..., u t − , x ) + a A ( u , u , ..., u t − , x )+ ... + a t A ( u , u , ..., u t − , x t )= t X i =1 a i A ( u , u , ..., u t − , x i ) for real numbers a , a , ..., a t in I = [0 , satisfying P ti =1 a i = 1 and u i , x i ∈ X for all i = 1 , , ..., t . TABILITY AND CONVERGENCE IN CONVEX A -METRIC SPACES 5 An A -metric space ( X, A ) with a convex structure W is called a convex A -metric space anddenoted as ( X, A, W ) . A nonempty subset C of a convex A -metric space ( X, A, W ) is said to be convex if W ( x , x , ..., x t − , x t ; a , a , ..., a t ) ∈ C for all x i ∈ C and a i ∈ I , i = 1 , , ..., t . Next, we transform the Mann iteration process to a convex A -metric space as follows. Definition 1.8. Let ( X, A, W ) be convex A -metric space with convex structure W and f : X → X be a mapping. Let { α ni } be sequences in [0 , for all i = 1 , , ..., t and n ∈ N . Then for any given x ∈ X, the iteration process defined by the sequence { x n } as (1.5) x n +1 = W ( x n , x n , ..., x n , f x n ; α n , α n , ..., α nt ) , is called Mann iteration process in the convex metric space ( X, A, W ) . It follows from the structure of convex A -metric space that A ( x n +1 , u , u , ..., u t − ) = A ( W ( x n , x n , ..., x n , f x n ; α n , α n , ..., α nt ) , u , u , ..., u t − )(1.6) ≤ α n A ( x n , u , u , ..., u t − ) + α n A ( x n , u , u , ..., u t − )+ ... + α nt A ( f x n , u , u , ..., u t − ) . If we take t = 2 in (1 . 5) and (1 . . 3) and (1 . Lemma 1.4. [2] If δ is a real number such that ≤ δ < and { ε n } is a sequence of positivenumbers such that lim n →∞ ε n = 0 , then for any sequence of positive numbers { u n } satisfying u n +1 ≤ δu n + ε n , n = 0 , , . . . we have lim n →∞ u n = 0 . Main Results2.1 Convergence Result: In this section, we prove the Mann iteration process convergesto fixed point of Zamfirescu mappings in complete convex metric space ( X, A, W ). Theorem 2.1. Let ( X, A, W ) be a complete convex A -metric space with a convex structure W and, f : X → X be an AZ mapping . Let { x n } be defined iteratively by (1 . and x ∈ X, with { α nt } ⊂ [0 , , t P i =1 a i = 1 satisfying ∞ P n =0 α nt = ∞ for all n ∈ N and i = 1 , , ..., t. Then { x n } convergesto a unique fixed point of f. ISA YILDIRIM Proof. From Theorem 1.1, we know that an AZ mapping has a unique fixed point in X . Call it u and consider x i ∈ X , i = 1 , , ..., t .At least one of ( AZ ), ( AZ ) and ( AZ ) is satisfied. If ( AZ ) , ( AZ ) or ( AZ ) holds, we knowthat the following inequality from Lemma 1.3(2.1) A ( f x, f x, . . . , f x, f y ) ≤ δA ( x, x, . . . , x, y ) + tδA ( f x, f x, . . . , f x, x )for all x, y ∈ X .Let { x n } be the Mann iteration process (1 . x ∈ X arbitrary. Then A ( u, u, ..., u, x n +1 ) ≤ A ( u, u, ..., u, W ( x n , x n , ..., x n , f x n ; α n , α n , ..., α nt )) ≤ α n A ( u, u, ..., u, x n ) + α n A ( u, u, ..., u, x n )+ ... + α nt A ( u, u, ..., u, f x n )= (1 − α nt ) A ( u, u, ..., u, x n ) + α nt A ( u, u, ..., u, f x n ) . Take x = u and y = x n in (2.1) to obtain A ( u, u, ..., u, f x n ) = A ( f u, f u, ..., f u, f x n )(2.2) ≤ δA ( u, u, ..., u, x n ) + tδA ( f u, f u, . . . , f u, u )= δA ( u, u, ..., u, x n )which together with (2.2) yields A ( u, u, ..., u, x n +1 ) ≤ (1 − α nt ) A ( u, u, ..., u, x n ) + α nt δA ( u, u, ..., u, x n )(2.3) = [1 − (1 − δ ) α nt ] A ( u, u, ..., u, x n ) . Inductively we get A ( u, u, ..., u, x n +1 ) ≤ [1 − (1 − δ ) α nt ] A ( u, u, ..., u, x n )(2.4) ≤ [1 − (1 − δ ) α nt ] (cid:2) − (1 − δ ) α n − t (cid:3) A ( u, u, ..., u, x n − )... ≤ n Q k =0 (cid:2) − (1 − δ ) α kt (cid:3) A ( u, u, ..., u, x )As 0 ≤ δ < , (cid:8) α kt (cid:9) ⊂ [0 , 1] and ∞ P k =0 α kt = ∞ , we havelim n →∞ n Q k =0 (cid:2) − (1 − δ ) α kt (cid:3) = 0 , which by (2.4) implieslim n →∞ A ( u, u, ..., u, x n +1 ) = lim n →∞ A ( x n +1 , x n +1 , ..., x n +1 , u ) = 0 . Hence the sequence { x n } defined iteratively by (1 . 5) converges to the fixed point of f . (cid:3) TABILITY AND CONVERGENCE IN CONVEX A -METRIC SPACES 7 Now, we will give stability result for the Mann iteration (1 . 5) incomplete convex A -metric space. Definition 2.1. Let ( X, A, W ) be a convex A -metric space with a convex structure W and, f : X → X be a mapping, x ∈ X and let us assume that the iteration process (1 . , that is, thesequence { x n } defined by (1 . , converges to a fixed point u of f .Let { y n } be an arbitrary sequence in X and set ǫ n = A ( y n +1 , y n +1 , ..., y n +1 , g ( f, y n )) for n = 0 , , , ... where g ( f, y n ) = W ( y n , y n , ..., y n , f y n ; α n , α n , ..., α nt ) and { α ni } are real sequences in [0 , for i = 1 , , ..., t .We say that the Mann iteration process (1 . is f -stable or stable with respect to f if and onlyif lim n →∞ ǫ n = 0 ⇐⇒ lim n →∞ y n = u. Theorem 2.2. Let ( X, A, W ) be a complete convex A -metric space with a convex structure W and, f : X → X be an AZ mapping . Let { x n } be defined iteratively by (1 . and x ∈ X, with { α nt } ⊂ [0 , , t P i =1 a i = 1 satisfying < α ≤ α n and ∞ P n =0 α nt = ∞ for all n ∈ N and i = 1 , , ..., t. Then the Mann iteration process (1 . is f -stable.Proof. From Theorem 1.1, we know that f has a unique fixed point. Suppose that u ∈ X . Fromfrom Lemma 1.3, we also know that(2.5) A ( f x, f x, . . . , f x, f y ) ≤ δA ( x, x, . . . , x, y ) + tδA ( f x, f x, . . . , f x, x ) . ISA YILDIRIM Let { y n } ⊂ X and ǫ n = A ( y n +1 , y n +1 , ..., y n +1 , g ( f, y n )). Assume that lim n →∞ ǫ n = 0. Then,we will show that lim n →∞ y n = u . From (2.5) and triangle inequality, we get A ( y n +1 , y n +1 , . . . , y n +1 , , u )(2.6) ≤ ( t − A ( y n +1 , y n +1 , . . . , y n +1 , , g ( f, y n ))+ A ( g ( f, y n ) , g ( f, y n ) , . . . , g ( f, y n ) , u )= ( t − A ( y n +1 , y n +1 , . . . , y n +1 , , g ( f, y n ))+ A ( u, u, . . . , u, g ( f, y n )) ≤ ( t − ε n + A ( u, u, . . . , u, g ( f, y n ))= ( t − ε n + A ( u, u, . . . , u, W ( y n , y n , . . . , y n , f y n ; α n , α n , . . . , α nt )) ≤ ( t − ε n + α n A ( u, u, . . . , u, y n ) + α n A ( u, u, . . . , u, y n )+ · · · + α nt A ( u, u, . . . , u, f y n )= ( t − ε n + (1 − α nt ) A ( u, u, . . . , u, y n ) + α nt A ( u, u, . . . , u, f y n ) ≤ ( t − ε n + (1 − α nt ) A ( u, u, . . . , u, y n )+ α nt [ δA ( u, u, . . . , u, y n ) + tδA ( f u, f u, . . . , f u, u )]= [1 − (1 − δ ) α nt ] A ( y n , y n , . . . , y n , u ) + ( t − ε n . Since 0 ≤ − (1 − δ ) α nt < − (1 − δ ) α < 1, using Lemma 1.4 in (2.6) yieldslim n →∞ A ( y n , y n , . . . , y n , u ) = 0 , that is, lim n →∞ y n = u . TABILITY AND CONVERGENCE IN CONVEX A -METRIC SPACES 9 Conversely, let lim n →∞ y n = u . Then, ε n = A ( y n +1 , y n +1 , . . . , y n +1 , , g ( f, y n )) ≤ ( t − A ( y n +1 , y n +1 , . . . , y n +1 , , u ) + A ( u, u, . . . , u, g ( f, y n ))= ( t − A ( y n +1 , y n +1 , . . . , y n +1 , , u )+ A ( u, u, . . . , u, W ( y n , y n , . . . , y n , f y n ; α n , α n , . . . , α nt )) ≤ ( t − A ( y n +1 , y n +1 , . . . , y n +1 , , u ) + α n A ( u, u, . . . , u, y n )+ α n A ( u, u, . . . , u, y n ) + · · · + α nt A ( u, u, . . . , u, f y n )= ( t − A ( y n +1 , y n +1 , . . . , y n +1 , , u ) + (1 − α nt ) A ( u, u, . . . , u, y n )+ α nt A ( u, u, . . . , u, f y n ) ≤ ( t − A ( y n +1 , y n +1 , . . . , y n +1 , , u ) + (1 − α nt ) A ( u, u, . . . , u, y n )+ α nt [ δA ( u, u, . . . , u, y n ) + tδA ( f u, f u, . . . , f u, u )] ≤ ( t − A ( y n +1 , y n +1 , . . . , y n +1 , , u ) + (1 − α nt ) A ( u, u, . . . , u, y n )+ α nt δA ( u, u, . . . , u, y n )Letting n → ∞ in the above inequality, we have lim n →∞ ǫ n = 0. 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H., Convexity in G -metric spaces and approximation of fixed points by Mann iterativeprocess, arXiv:1911.04867 [math.GN], 2019.[21] Zamfirescu T., Fix point theorems in metric spaces, Arch. Math., 23 (1972), 292–298. Department of Mathematics, Faculty of Science, Ataturk University, Erzurum, 25240, Turkey. Email address ::