Next Article in Journal
The Tubby Torus as a Quotient Group
Previous Article in Journal
Axiomatic Approach in the Analytic Theory of Singular Perturbations
Previous Article in Special Issue
Measure of Weak Noncompactness and Fixed Point Theorems in Banach Algebras with Applications
Open AccessArticle

Viscosity Approximation Methods for * −Nonexpansive Multi-Valued Mappings in Convex Metric Spaces

1
Faculty of Mathematics, K. N. Toosi University of Technology, Tehran 16569, Iran
2
Institute of Research and Development of Processes University of the Basque Country, 48940 Leioa, Spain
3
Department of Mathematics, University of Bojnord, Bojnord 94531, Iran
*
Author to whom correspondence should be addressed.
Axioms 2020, 9(1), 10; https://doi.org/10.3390/axioms9010010
Received: 12 December 2019 / Revised: 11 January 2020 / Accepted: 12 January 2020 / Published: 17 January 2020
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)

Abstract

In this paper, we prove convergence theorems for viscosity approximation processes involving * −nonexpansive multi-valued mappings in complete convex metric spaces. We also consider finite and infinite families of such mappings and prove convergence of the proposed iteration schemes to common fixed points of them. Our results improve and extend some corresponding results.
Keywords: *−nonexpansive multi-valued mapping; viscosity approximation methods; fixed point; convex metric space *−nonexpansive multi-valued mapping; viscosity approximation methods; fixed point; convex metric space

1. Introduction

Many of the real world known problems that scientists are looking to solve are nonlinear. Therefore, translating linear version of such problems into their equivalent nonlinear version has a great importance. Mathematicians have tried to transfer the structure of covexity to spaces that are not linear spaces. Takahashi [1], Kirk [2,3], and Penot [4], for example, presented this notion in metric spaces. Takahashi [1] introduced the following notion of convexity in metric spaces:
Definition 1. 
([1]) Let ( X , d ) be a metric space and I = [ 0 , 1 ] . A mapping W : X × X × I X is said to be a convex structure on X if for each x , y , u X and all t I ,
d ( u , W ( x , y , t ) ) t d ( u , x ) + ( 1 t ) d ( u , y ) .
A metric space ( X , d ) together with a convex structure W is called a convex metric space and is denoted by ( X , W , d ) .
A subset C of X is called convex if W ( x , y , t ) C , for all x , y C and all t I .
Example 1.
Let X = M 2 ( R ) . For any A = a 1 a 2 a 3 a 4 and B = b 1 b 2 b 3 b 4
and t I = [ 0 , 1 ] , we define the mapping W : X × X × I X by
W ( A , B , t ) = t a 1 + ( 1 t ) b 1 t a 2 + ( 1 t ) b 2 t a 3 + ( 1 t ) b 3 t a 4 + ( 1 t ) b 4
and the metric d : X × X [ 0 , + ) by
d ( A , B ) = Σ i = 1 4 | a i b i | .
Then ( X , W , d ) is a convex metric space.
Example 2.
Let X = R 2 with the metric
d ( ( x 1 , x 2 ) , ( y 1 , y 2 ) ) = max { | x 1 y 1 | , | x 2 y 2 | } ,
for any ( x 1 , x 2 ) , ( y 1 , y 2 ) X and define the mapping W : X × X × [ 0 , 1 ] X by
W ( ( x 1 , x 2 ) , ( y 1 , y 2 ) , t ) = ( t x 1 + ( 1 t ) y 1 , t x 2 + ( 1 t ) y 2 ) ,
for each ( x 1 , x 2 ) , ( y 1 , y 2 ) X and t [ 0 , 1 ] . Then ( X , W , d ) is a convex metric space.
Example 3.
Let X = C ( [ 0 , 1 ] ) be the metric space with the metric d ( f , g ) = 0 1 | f ( x ) g ( x ) | d x and define W : X × X × [ 0 , 1 ] X by W ( f , g , t ) = t f + ( 1 t ) g , for all f , g X and t [ 0 , 1 ] . Then ( X , W , d ) is a convex metric space.
This notion of convex structure is a generalization of convexity in normed spaces and allows us to obtain results that seem to be possible only in linear spaces. One of its useful applications is the iterative approximation of fixed points in metric spaces. All of the sequences that are used in fixed point problems require linearity or convexity of the space. So, this concept of convexity helps us to define various iteration schemes and to solve fixed point problems in metric spaces. In recent years, many authors have established several results on the covergence of some iterative schemes using different contractive conditions in convex metric spaces. For more details, refer to [5,6,7,8,9,10,11,12,13,14].
Now, let us recall some definitions and concepts that will be needed to state our results:
Definition 2. 
([15]) Let ( X , d ) be a metric. A subset D is called proximinal if for each x X there exists an element y D such that d ( x , y ) = d ( x , D ) , where d ( x , D ) = inf { d ( x , z ) : z D } .
We denote the family nonempty proximinal and bounded subsets of D by P ( D ) and the family of all nonempty closed and bounded subsets of X by C B ( X ) .
For two bounded subsets A and B of a metric space ( X , d ) , the Pompeiu–Hausdorff metric between A and B is defined by
H ( A , B ) = max { sup x A d ( x , B ) , sup y B d ( A , y ) } .
Definition 3. 
([16]) Let ( X , d ) be a metric space. A multi-valued mapping T : X C B ( X ) is said to be nonexpansive if H ( T x , T y ) d ( x , y ) , for all x , y X .
An element p X is called a fixed point of T if p T ( p ) . The set of all fixed points of T are denoted by F ( T ) .
Definition 4. 
([17]) Let ( X , d ) be a metric space and D be a nonempty subset of X. A multi-valued mapping T : D C B ( D ) is called * −nonexpansive if for all x , y D and u x T ( x ) with d ( x , u x ) = inf { d ( x , z ) : z T ( x ) } , there exists u y T ( y ) with d ( y , u y ) = inf { d ( y , w ) : w T ( y ) } such that
d ( u x , u y ) d ( x , y ) .
It is clear that if T is a * −nonexpansive map, then P T is a nonexpansive map, where P T for T : D P ( D ) is defined by
P T ( x ) = { y T ( x ) : d ( x , y ) = d ( x , T ( x ) ) } ,
for all x D .
Definition 5. 
([16]) Let ( X , d ) be a metric space. A multi-valued mapping T : X C B ( X ) is said to satisfy condition (I) if there is a nondecreasing function f : [ 0 , ) [ 0 , ) with f ( 0 ) = 0 , f ( r ) > 0 for r ( 0 , ) such that d ( x , T ( x ) ) f ( d ( x , F ( T ) ) ) , for all x X .
First of all, Moudafi [18] introduced the viscosity approximation method for approximating the fixed point of nonexpansive mappings in Hilbert spaces. Since then, many authors have been extending and generalizing this result by using different contractive conditions on several spaces. For some new works in these fields, we can refer to [19,20,21,22,23,24,25,26,27]. Inspired and motivated by the research work going on in these fields, in this paper we investigate the convergence of some viscosity approximation processes for * −nonexpansive multi-valued mappings in a complete convex metric spaces. The convergence theorems for finite and infinite family of such mappings are also presented. Our results can improve and extend the corresponding main theorems in the literature.

2. Main Results

At first, we present two lemmas that are used to prove our main result. Since the idea is similar to the one given in Lemmas 2.1 and 2.2 in [28], we only state the results without the proof:
Lemma 1.
Let { u n } and { v n } be sequences in a convex metric space ( X , W , d ) and { a n } be a sequence in [ 0 , 1 ] such that lim sup n a n < 1 . Set
d = lim sup n d ( u n , v n ) o r d = lim inf n d ( u n , v n ) .
Let u n + 1 = W ( v n , u n , a n ) for all n N . Suppose that
lim sup n ( d ( v n + 1 , v n ) d ( u n + 1 , u n ) ) 0 ,
and d < .Then
lim inf n | d ( v n + k , u n ) ( 1 + a n + a n + 1 + + a n + k 1 ) d | = 0 ,
for all k N .
Lemma 2.
Let { u n } and { v n } be bounded sequences in a convex metric space ( X , W , d ) and { a n } be a sequence in [ 0 , 1 ] with 0 < lim inf n a n lim sup n a n < 1 . Suppose that u n + 1 = W ( v n , u n , a n ) and
lim sup n ( d ( v n + 1 , v n ) d ( u n + 1 , u n ) ) 0 .
Then lim n d ( v n , u n ) = 0
Now, we state and prove the main theorem of this paper:
Theorem 1.
Let D be a nonempty, closed and convex subset of a complete convex metric space ( X , W , d ) and T : D P ( D ) be a * −nonexpansive multi-valued mapping with F ( T ) , such that T satisfies condition ( I ). Suppose that a n [ 0 , 1 ] such that 0 < lim inf n a n lim sup n a n < 1 and c n ( 0 , + ) such that lim n c n = 0 . Let { x n } be the Mann type iterative scheme defined by
x n + 1 = W ( z n , x n , a n ) ,
where d ( z n + 1 , z n ) H ( P T ( x n + 1 ) , P T ( x n ) ) + c n for z n P T ( x n ) . Then { x n } converges to a fixed point of T.
Proof. 
Take p F ( T ) . Then p P T ( p ) = { p } and we have
d ( x n + 1 , p ) = d ( W ( z n , x n , a n ) , p ) a n d ( z n , p ) + ( 1 a n ) d ( x n , p ) a n H ( P T ( x n ) , P T ( p ) ) + ( 1 a n ) d ( x n , p ) a n d ( x n , p ) + ( 1 a n ) d ( x n , p ) = d ( x n , p ) .
Hence, { d ( x n , p ) } is a decreasing and bounded below sequence and thus lim n d ( x n , p ) exists for any p F ( T ) . Therefore { x n } is bounded and so { z n } is bounded. On the other hand,
d ( z n + 1 , z n ) H ( P T ( x n + 1 ) , P T ( x n ) ) + c n d ( x n + 1 , x n ) + c n .
Thus
lim sup n ( d ( z n + 1 , z n ) d ( x n + 1 , x n ) ) 0 .
Applying Lemma 2, we get
lim n d ( z n , x n ) = 0 .
Hence, we have lim n d ( x n , T ( x n ) ) = 0 . Since T satisfies condition ( I ), we conclude that lim n d ( x n , F ( T ) ) = 0 . Next, we show that { x n } is a Cauchy sequence. Since lim n d ( x n , F ( T ) ) = 0 , thus for ε 1 > 0 , there exists n 1 N such that for all n n 1
d ( x n , F ( T ) ) ε 1 3 .
Thus, there exists p 1 F ( T ) such that for all n n 1 ,
d ( x n , p 1 ) ε 1 2 .
It follows that
d ( x n + m , x n ) d ( x n + m , p 1 ) + d ( p 1 , x n ) d ( x n , p 1 ) + d ( p 1 , x n ) ε 1 2 + ε 1 2 = ε 1 ,
for all m , n n 1 . Therefore { x n } is a Cauchy sequence and hence it is convergent. Let lim n x n = p * . We will show that p * is a fixed point of T.
Since lim n x n = p * , thus for given ε 2 > 0 , there exists n 2 N such that for all n n 2 ,
d ( x n , p * ) ε 2 4 .
Moreover, lim n d ( x n , F ( T ) ) = 0 implies that there exists a natural number n 3 n 2 such that for all n n 3 ,
d ( x n , F ( T ) ) ε 2 12 ,
and thus there exists p 2 F ( T ) such that for all n n 3 ,
d ( x n , p 2 ) ε 2 8 .
Therefore
d ( T ( p * ) , p * ) d ( T ( p * ) , p 2 ) + d ( p 2 , T ( x n 3 ) ) + d ( T ( x n 3 ) , p 2 ) + d ( p 2 , x n 3 ) + d ( x n 3 , p * ) H ( P T ( p * ) , P T ( p 2 ) ) + 2 H ( P T ( p 2 ) , P T ( x n 3 ) ) + d ( p 2 , x n 3 ) + d ( x n 3 , p * ) d ( p * , p 2 ) + 2 d ( p 2 , x n 3 ) + d ( p 2 , x n 3 ) + d ( x n 3 , p * ) d ( p * , x n 3 ) + d ( x n 3 , p 2 ) + 2 d ( p 2 , x n 3 ) + d ( p 2 , x n 3 ) + d ( x n 3 , p * ) = 2 d ( x n 3 , p * ) + 4 d ( x n 3 , p 2 ) ε 2 2 + ε 2 2 = ε 2 .
Thus, p * T ( p * ) and therefore p * is a fixed point of T. □
As a result of Theorem 1, Corollaries 1 and 2 are obtained:
Corollary 1.
Let D be a nonempty, closed and convex subset of a complete convex metric space ( X , W , d ) , T : D P ( D ) be * −nonexpansive multi-valued mapping with F ( T ) such that T satisfies condition ( I ) and f : D D be a contractive mapping with a contractive constant k ( 0 , 1 ) . Then the iterative sequence { x n } defined by
x n + 1 = W ( z n , f ( x n ) , a n )
where z n P T ( x n ) and 0 < lim inf n a n lim sup n a n < 1 , converges to a fixed point of T.
Corollary 2.
Let D be a nonempty, closed, and convex subset of a complete convex metric space ( X , W , d ) and T : D P ( D ) be * −nonexpansive multi-valued mapping with F ( T ) . Let { x n } be the Ishikawa type iterative scheme defined by
x n + 1 = W ( z n , x n , a n ) y n = W ( z n , x n , b n )
where z n P T ( y n ) , z n P T ( x n ) , and { a n } , { b n } [ 0 , 1 ] . Then { x n } converges to a fixed point of T if and only if lim n d ( x n , F ( T ) ) = 0 .
The above result can be generalized to the finite and infinite family of * −nonexpansive multi-valued mappings:
Theorem 2.
Let D be a nonempty, closed, and convex subset of a complete convex metric space ( X , W , d ) and { T i : D P ( D ) : i = 1 , , k } be a finite family of * −nonexpansive multi-valued mappings such that F : = i = 1 k F ( T i ) . Consider the iterative process defined by
y 1 n = W ( z 1 n , x n , a 1 n ) , y 2 n = W ( z 2 n , x n , a 2 n ) , y ( k 1 ) n = W ( z ( k 1 ) n , x n , a ( k 1 ) n ) , x n + 1 = W ( z k n , x n , a k n ) ,
where a i n [ 0 , 1 ] and z i n P T i ( y ( i 1 ) n ) ( y 0 n = x n ) , for all n N and i = 1 , 2 , , k . Then { x n } converges to a point in F if and only if lim n d ( x n , F ) = 0 .
Proof. 
The necessity of conditions is obvious and we will only prove the sufficiency. Let p F . we have
d ( y 1 n , p ) = d ( W ( z 1 n , x n , a 1 n ) , p ) a 1 n d ( z 1 n , p ) + ( 1 a 1 n ) d ( x n , p ) a 1 n H ( P T 1 ( x n ) , P T 1 ( p ) ) + ( 1 a 1 n ) d ( x n , p ) a 1 n d ( x n , p ) + ( 1 a 1 n ) d ( x n , p ) = d ( x n , p ) , d ( y 2 n , p ) = d ( W ( z 2 n , x n , a 2 n ) , p ) a 2 n d ( z 2 n , p ) + ( 1 a 2 n ) d ( x n , p ) a 2 n H ( P T 2 ( y 1 n ) , P T 2 ( p ) ) + ( 1 a 2 n ) d ( x n , p ) a 2 n d ( y 1 n , p ) + ( 1 a 2 n ) d ( x n , p ) a 2 n d ( x n , p ) + ( 1 a 2 n ) d ( x n , p ) = d ( x n , p ) , d ( y ( k 1 ) n , p ) = d ( W ( z ( k 1 ) n , x n , a ( k 1 ) n ) , p ) a ( k 1 ) n d ( z ( k 1 ) n , p ) + ( 1 a ( k 1 ) n ) d ( x n , p ) a ( k 1 ) n H ( P T k 1 ( y ( k 2 ) n ) , P T k 1 ( p ) ) + ( 1 a ( k 1 ) n ) d ( x n , p ) a ( k 1 ) n d ( y ( k 2 ) n , p ) + ( 1 a ( k 1 ) n ) d ( x n , p ) a ( k 1 ) n d ( x n , p ) + ( 1 a ( k 1 ) n ) d ( x n , p ) = d ( x n , p ) .
Thus
d ( x n + 1 , p ) = d ( W ( z k n , x n , a k n ) , p ) a k n d ( z k n , p ) + ( 1 a k n ) d ( x n , p ) a k n H ( P T k ( y ( k 1 ) n ) , P T k ( p ) ) + ( 1 a k n ) d ( x n , p ) a k n d ( y ( k 1 ) n , p ) + ( 1 a k n ) d ( x n , p ) a k n d ( x n , p ) + ( 1 a k n ) d ( x n , p ) = d ( x n , p ) .
Therefore, { d ( x n , p ) } is a decreasing sequence and so d ( x n + m , p ) d ( x n , p ) , for all n , m N . As in the proof of Theorem 1, { x n } is a Cauchy sequence and thus lim n x n exists and equals to some p * D . Again, with a similar process as in the proof of Theorem 1, we conclude that p * P T i ( q ) for all i = 1 , , k . Hence p * F and this completes the proof of theorem. □
Theorem 3.
Let D be a nonempty, closed, and convex subset of a complete convex metric space ( X , W , d ) and { T i : D P ( D ) : i = 1 , } be an infinite family of * −nonexpansive multi-valued mappings such that F : = i = 1 F ( T i ) . Consider the iterative process defined by
x n + 1 = W ( z n , x n , a n ) y n = W ( z n , x n , b n )
where z n P T n ( y n ) , z n P T n ( x n ) and { a n } , { b n } [ 0 , 1 ] . Then { x n } converges to a point in F if and only if lim n d ( x n , F ) = 0 .

Author Contributions

Data curation, A.G.; Formal analysis, A.G.; Software, A.G.; Writing—original draft, A.G.; Conceptualization, H.P.M.; Project administration, H.P.M.; Supervision, M.D.L.S.; Funding acquisition, M.D.L.S.; Writing—review and editing, M.D.L.S. and M.R.; Validation, M.R. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Basque Government through grant IT1207-19.

Acknowledgments

The authors are grateful to the referees for valuable suggestions and to the Basque Government for Grant IT1207-19.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Takahashi, W. A convexity in metric spaces and nonexpansive mappings. Kodai Math. Sem. Rep. 1970, 22, 142–149. [Google Scholar] [CrossRef]
  2. Kirk, W.A. An abstract fixed point theorem for nonexpansive mappings. Proc. Am. Math. Soc. 1981, 82, 640–642. [Google Scholar] [CrossRef]
  3. Kirk, W.A. Fixed point theory for nonexpansive mappings II. Contemp. Math. 1983, 18, 121–140. [Google Scholar]
  4. Penot, J.P. Fixed point theorems without convexity. Bull. Soc. Math. France Mem. 1979, 60, 129–152. [Google Scholar] [CrossRef]
  5. Chang, S.S.; Kim, J.K. Convergence theorems of the Ishikawa type iterative sequences with errors for generalized quasi-contractive mappings in convex metric spaces. Appl. Math. Lett. 2003, 16, 535–542. [Google Scholar] [CrossRef]
  6. Chang, S.S.; Kim, J.K.; Jin, D.S. Iterative sequences with errors for asymptotically quasi-nonexpansive type mappings in convex metric spaces. Arch. Inequal. Appl. 2004, 2, 365–374. [Google Scholar]
  7. Ding, X.P. Iteration processes for nonlinear mappings in convex metric spaces. J. Math. Anal. Appl. 1988, 132, 114–122. [Google Scholar] [CrossRef]
  8. Khan, A.R.; Ahmed, M.A. Convergence of a general iterative scheme for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces and applications. Comput. Math. Appl. 2010, 59, 2990–2995. [Google Scholar] [CrossRef]
  9. Kim, J.K.; Kim, K.H.; Kim, K.S. Three-step iterative sequences with errors for asymptotically quasi-nonexpansive mappings in convex metric spaces. Nonlinear Anal. Convex Anal. 2004, 1365, 156–165. [Google Scholar]
  10. Rafiq, A. Fixed point of Ciric quasi-contractive operators in generalized convex metric spaces. Gen. Math. 2006, 14, 79–90. [Google Scholar]
  11. Saluja, G.S.; Nashine, H.K. Convergence of implicit iteration process for a finite family of asymptotically Quasi-nonexpansive mappings in convex metric spaces. Opuscula Math. 2010, 30, 331–340. [Google Scholar] [CrossRef]
  12. Tian, Y.X. Convergence of an Ishikawa type Iterative scheme for asymptotically quasi- nonexpansive mappings. Comput. Math. Appl. 2005, 49, 1905–1912. [Google Scholar] [CrossRef]
  13. Wang, C.; Zhu, J.H.; Damjanovic, B.; Hu, L.G. Approximating fixed points of a pair of contractive type mappings in generalized convex metric spaces. Appl. Math. Comput. 2009, 215, 1522–1525. [Google Scholar] [CrossRef]
  14. Wang, C.; Liu, L.W. Convergence theorems of fixed points of uniformly quasi-Lipschitzian mappings in convex metric spaces. Nonlinear Anal. 2009, 70, 2067–2071. [Google Scholar] [CrossRef]
  15. Roshdi, K. Best approximation in metric spaces. Proc. Amer. Math. Soc. 1988, 103, 579–586. [Google Scholar]
  16. Shahzad, N.; Zegeye, H. On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces. Nonlinear Anal. 2009, 71, 838–844. [Google Scholar] [CrossRef]
  17. Hussain, T.; Latif, A. Fixed points of multivalued nonexpansive maps. Math. Japon. 1988, 33, 385–391. [Google Scholar]
  18. Moudafi, A. Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 2000, 241, 46–55. [Google Scholar] [CrossRef]
  19. Deng, W.Q. A new viscosity approximation method for common fxed points of a sequence of nonexpansive mappings with weakly contractive mappings in Banach spaces. J. Nonlinear Sci. Appl. 2016, 9, 3920–3930. [Google Scholar] [CrossRef]
  20. Khan, A.R.; Yasmin, N.; Fukhar-ud-din, H.; Shukri, S.A. Viscosity approximation method for generalized asymptotically quasi-nonexpansive mappings in a convex metric space. Fixed Point Theory Appl. 2015, 2015, 196. [Google Scholar] [CrossRef]
  21. Lin, Y.C.; Sharma, B.K.; Kumar, A.; Gurudwan, N. Viscosity approximation method for common fixed point problems of a finite family of nonexpansive mappings. J. Nonlinear Convex Anal. 2017, 18, 949–966. [Google Scholar]
  22. Liu, X.; Chen, Z.; Xiao, Y. General viscosity approximation methods for quasi-nonexpansive mappings with applications. J. Inequal. Appl. 2019, 2019, 71. [Google Scholar] [CrossRef]
  23. Liu, C.; Song, M. The new viscosity approximation methods for nonexpansive nonself-mappings. Int. J. Mod. Nonlinear Theory Appl. 2016, 5, 104–113. [Google Scholar] [CrossRef]
  24. Naqvi, S.F.A.; Khan, M.S. On the viscosity rule for common fixed points of two nonexpansive mappings in Hilbert spaces. Open J. Math. Sci. 2017, 1, 111–125. [Google Scholar] [CrossRef]
  25. Thong, D.V. Viscosity approximation methods for solving fixed-point problems and split common fixed-point problems. J. Fixed Point Theory Appl. 2016. [Google Scholar] [CrossRef]
  26. Xiong, T.; Lan, H. Strong convergence of new two-step viscosity iterative approximation methods for set-valued nonexpansive mappings in CAT(0) spaces. J. Funct. Spaces 2018, 2018. [Google Scholar] [CrossRef]
  27. Khan, S.H.; Fukhar-ud-din, H. Approximating fixed points of ρ-nonexpansive mappings by RK-iterative process in modular function spaces. J. Nonlinear Var. Anal. 2019, 3, 107–114. [Google Scholar]
  28. Suzuki, T. Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005, 1, 103–123. [Google Scholar] [CrossRef]
Back to TopTop