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Some fixed point results for multi-valued mappings in b-metric spaces

Abstract

The aim of this paper is to establish some fixed point theorems for set-valued mappings in the context of b-metric spaces. The proposed theorems expand and generalize several well-known comparable results in the literature. An example is also given to support our main result.

MSC: 46S40, 47H10, 54H25.

1 Introduction and preliminaries

The notion of metric space, introduced by Fréchet in 1906, is one of the cornerstones of not only mathematics but also several quantitative sciences. Due to its importance and application potential, this notion has been extended, improved and generalized in many different ways. An incomplete list of the results of such an attempt is the following: quasi-metric space, symmetric space, partial metric space, cone metric space, G-metric space, probabilistic metric space, fuzzy metric space and so on.

In this paper, we pay attention to the concept of b-metric space. The notion of b-metric space was introduced by Czerwik [1] in 1993 to extend the notion of metric space. In this interesting paper, Czerwik [1] observed a characterization of the celebrated Banach fixed point theorem [2] in the context of complete b-metric spaces. Following this pioneer paper, several authors have devoted their attention to research the properties of a b-metric space and have reported the existence and uniqueness of fixed points of various operators in the setting of b-metric spaces (see, e.g., [312] and some reference therein).

The aim of this paper is to generalize various known results proved by Kikkawa and Suzuki [13], Mot and Petrusel [14], Dhompongsa and Yingtaweesittikul [15] to the case of b-metric spaces and give an example to illustrate our main results.

Definition 1 Let X be any nonempty set. An element x in X is said to be a fixed point of a multi-valued mapping T:X 2 X if xTx, where 2 X denotes the collection of all nonempty subsets of X.

Let (X,d) be a metric space. Let CB(X) be the collection of all nonempty, closed and bounded subsets of X. In the sequel, we use the following notations:

d ( a , A ) = inf { d ( a , x ) : x A } , δ ( A , B ) = sup { d ( a , B ) : a A } , δ ( B , A ) = sup { d ( b , A ) : b B }

and

H(A,B)=max { δ ( A , B ) , δ ( B , A ) }

for any A,BCB(X).

Notice that H is called the Hausdorff metric induced by the metric d.

We start with recalling some basic definitions and lemmas on b-metric spaces. The definition of a b-metric space is given by Czerwik [1] (see also [4, 5]) as follows.

Definition 2 Let X be a nonempty set X and s1 be a given real number. A function d:X×X R + is called a b-metric provided that, for all x,y,zX,

(bms1) d(x,x)=0,

(bms2) d(x,y)=d(y,x),

(bms3) d(x,z)s(d(x,y)+d(y,z)).

Note that a (usual) metric space is evidently a b-metric space. However, Czerwik [1, 4] showed that a b-metric on X need not be a metric on X (see also [5, 16, 17]). The following example shows that a b-metric on X need not be a metric on X.

Example 1 (cf. [18])

Let X={a,b,c} and d(a,c)=d(2,c)=m2, d(c,b)=d(b,a)=d(b,c)=d(c,b)=1, and d(a,a)=d(b,b)=d(c,c)=0. Then d(x,y) m 2 [d(x,z)+d(z,y)] for all x,y,zX. If m>2, then the ordinary triangle inequality does not hold.

Let (X,d) be a b-metric space. We cite the following lemmas from Czerwik [1, 4, 5] and Singh et al. [18].

Lemma 1 Let (X,d) be a b-metric space. For any A,BCB(X) and any x,yX, we have the following:

  1. (1)

    d(x,B)d(x,b) for any bB,

  2. (2)

    d(x,B)H(A,B),

  3. (3)

    d(x,A)s(d(x,y)+d(y,B)).

Remark 1 Let (X,d) be a b-metric space and A be a nonempty set in (X,d) and xA, then we have

d(x,A)=0x A ¯ =A,

where A ¯ denotes the closure of A with respect to the induced metric d. Note that A is closed in (X,d) if and only if A ¯ =A.

Remark 2 The mapping d in a b-metric space (X,d) need not be jointly continuous (see, e.g., [19, 20]).

Lemma 2 Let A and B be nonempty closed and bounded subsets of a b-metric space (X,d) and q>1. Then, for all aA, there exists bB such that d(a,b)qH(A,B).

Lemma 3 Let (X,d) be a b-metric space. Let A and B be in CB(X). Then, for each α>0 and for all bB, there exists aA such that d(a,b)H(A,B)+α.

The following result was proved by Aydi et al. in [21].

Theorem 1 Let (X,d) be a complete b-metric space and let F:XCB(X) be a multi-valued mapping such that for all x,yX,

H(Fx,Fy)rM(x,y),
(1.1)

where 0r< 1 s 2 + s <1 and

M(x,y)=max { d ( x , y ) , d ( x , F x ) , d ( y , F y ) , d ( x , F y ) , d ( y , F x ) } .

Then F has a fixed point in X, that is, there exists uX such that uFu.

The following preliminary lemma will play a crucial role in the sequel.

Lemma 4 [22]

Let (X,d) be a complete b-metric space and let { x n } be a sequence in X such that d( x n + 1 , x n + 2 )βd( x n , x n + 1 ) for all n=0,1,2, , where 0β<1. Then { x n } is a Cauchy sequence in X provided that sβ<1.

2 Main results

In this section we state and prove our main results. Inspired the results of Aydi et al. [21], we establish a Kikkawa and Suzuki type fixed point theorem in the framework of b-metric spaces as follows.

Theorem 2 Let (X,d) be a complete b-metric space and let F:XCB(X) be a multi-valued mapping. Then, for s1, define a strictly decreasing function σ from [0,1) onto ( 1 2 ,1] by σ(r)= 1 ( 1 + s r ) , where r< 1 s 2 + s <1, such that

σ(r)d(x,Fx)sd(x,y)H(Fx,Fy)rd(x,y)
(2.1)

for all x,yX. Then there exists uX such that uFu.

Proof If d(x,y)=0, then by (2.1) we deduce that x=y is a fixed point of F. Hence the proof is completed. Thus, throughout the proof, we assume that d(x,y)>0 for all x,yX. Take

α= 1 2 ( 1 s 2 + s r )

and

β=r+α= 1 2 ( 1 s 2 + s + r ) .

Due to the assumption r< 1 s 2 + s , we conclude that α>0 and 0<β<1. Let x 0 X be arbitrary and x 1 F x 0 . Owing to (2.1), we have

σ(r)d( x 0 ,F x 0 )σ(r)d( x 0 , x 1 )sd( x 0 , x 1 ),

which yields that

H(F x 0 ,F x 1 )rd( x 0 , x 1 ).

By Lemma 3, there exists x 2 F x 1 . Now, by using the previous inequality, we obtain

d( x 1 , x 2 )H(F x 0 ,F x 1 )+αd( x 0 , x 1 )rd( x 0 , x 1 )+αd( x 0 , x 1 )=βd( x 0 , x 1 ),

where β=r+α. On the other hand, we have

σ ( r ) d ( x 1 , F x 1 ) σ ( r ) d ( x 1 , x 2 ) d ( x 1 , x 2 ) s d ( x 1 , x 2 ) .

Thus, we derive that

H(F x 1 ,F x 2 )rd( x 1 , x 2 )

by condition (2.1). Employing Lemma 3 again, there exists x 3 F x 2 such that

d( x 2 , x 3 )H(F x 1 ,F x 2 )rd( x 1 , x 2 )+αd( x 1 , x 2 )βd( x 1 , x 2 ).

Continuing in this way, we can construct a sequence { x n } in X such that x n + 1 F x n and

d( x n , x n + 1 ) β n d( x 0 , x 1 )
(2.2)

for all nN. Having in mind s1 together with β= 1 2 ( 1 s 2 + s +r) and r< 1 s 2 + s , one can easily obtain that sβ<1. Taking Lemma 4 into account, we conclude that the sequence { x n } is a Cauchy sequence in (X,d). Since the b-metric space (X,d) is complete, there exists uX such that lim n + d( x n ,u)=0. Due to fact that β<1, we can easily observe that

lim n + d( x n , x n + 1 )=0,

by using inequality (2.2). Notice that the condition (bms3) yields

d( x n + 1 ,u)s ( d ( x n + 1 , x n ) + d ( x n , u ) ) .

Consequently, we have

lim n + d( x n + 1 ,u)=0.

In what follows, we shall show that

d(u,Fx)srd(u,x)

for all xX{u}. Since d( x n ,u)0 as n+, there exists n 0 N such that

d( x n ,u) 1 3 d(u,x)

for all nN with n n 0 . Then we have

σ ( r ) d ( x n , F x n ) d ( x n , F x n ) d ( x n , x n + 1 ) s ( d ( x n , u ) + d ( u , x n + 1 ) ) 2 s 3 d ( u , x ) s d ( u , x ) s d ( x n , u ) s d ( x n , x ) ,

and hence by assumption (2.1) we get H(F x n ,Fx)rd( x n ,x). Further, we have

d ( u , F x ) s ( d ( u , x n + 1 ) + d ( x n + 1 , F x ) ) s ( d ( u , x n + 1 ) + H ( F x n , F x ) ) s ( d ( u , x n + 1 ) + r d ( x n , x ) ) .

Letting n+ in the inequality above, we obtain

d(u,Fx)rsd(u,x)
(2.3)

for all xX{u}.

Next, we prove that

H(Fx,Fu)rd(x,u)

for all xX with xu. For all nN, we choose v n Fx such that

d(u, v n )d(u,Fx)+ 1 n d(x,u).

Then, using (2.3) and the previous inequality, we get

d ( x , F x ) d ( x , v n ) s ( d ( x , u ) + d ( u , v n ) ) s ( d ( x , u ) + d ( u , F x ) + 1 n d ( x , u ) ) ( d ( x , u ) + s r d ( u , x ) + 1 n d ( x , u ) ) = s ( 1 + s r + 1 n ) d ( x , u ) .

Hence, for all nN, we obtain σ(r)d(x,Fx)sd(x,u). So, we have

H(Fx,Fu)rd(x,u).

Finally, if for some nN we have x n = x n + 1 , then x n is a fixed point of F. Consequently, throughout the proof we assume that x n x n + 1 for all nN. This implies that there exists an infinite subset J of such that x n u for all nJ. By Lemma 1, we have

d ( u , F u ) s ( d ( u , x n + 1 ) + d ( x n + 1 , F u ) ) s ( d ( u , x n + 1 ) + H ( F x n , F u ) ) s ( d ( u , x n + 1 ) + r d ( x n , u ) ) .

Letting n+ in the inequality above, with nJ, we find that

d(u,Fu)=0.

By Remark 1, we deduce that uFu and hence u is a fixed point of F. □

Remark 3 Taking s=1 in Theorem 2 (it corresponds to the case of metric spaces), the condition on r< 1 2 , σ(r)= 1 1 + r , we find Theorem 1.2 of Kikkawa and Suzuki. Hence, Theorem 2 is an extension of the result of Kikkawa et al. [13], which itself improves the theorem of Nadler [7].

In the case where T:XX is a single-valued mapping on a b-metric space, we have the following corollary (it is a consequence of Theorem 2).

Corollary 1 Let (X,d) be a complete b-metric space and let F:XX be a single-valued mapping. Define a strictly decreasing function σ from [0,1) onto ( 1 2 ,1] by σ(rs)= 1 1 + s r , r< 1 s 2 + s <1 such that

σ(rs)d(x,Fx)sd(x,y)d(Fx,Fy)rd(x,y)
(2.4)

for all x,yX. Then there exists uX such that u=Fu.

Proof It follows by applying Theorem 2 and the fact that H(Fx,Fy)=d(Fx,Fy). □

Remark 4 Corollary 1 implies the corresponding result of Suzuki [23] if we take s=1.

The following theorem is a result of Reich type [8] as well as a generalization of Kikkawa and Suzuki type in the framework of b-metric spaces.

Theorem 3 Let (X,d) be a complete b-metric space and let F:XCB(X) be a multi-valued mapping. If for s1 there exist nonnegative numbers a, b, c with s(a+b+c)[0,1) and θ= 1 s b s c 1 + s a such that

θd(x,Fx)sd(x,y)H(Fx,Fy)ad(x,y)+bd(x,Fx)+cd(y,Fy)
(2.5)

for all x,yX, then F has a fixed point.

Proof Let x 0 X be arbitrary and x 1 F x 0 , then we have

θd( x 0 ,F x 0 )θd( x 0 , x 1 )sd( x 0 , x 1 ).

By condition (2.5) we get

H(F x 0 ,F x 1 )ad( x 0 , x 1 )+bd( x 0 ,F x 0 )+cd( x 1 ,F x 1 ).

Let h(1, 1 s ( a + b + c ) ), then by Lemma 2 there exists x 2 F x 1 such that

d( x 1 , x 2 )hH(F x 0 ,F x 1 ),

which yields

d ( x 1 , x 2 ) h H ( F x 0 , F x 1 ) h ( a d ( x 0 , x 1 ) + b d ( x 0 , F x 0 ) + c d ( x 1 , F x 1 ) ) h ( a + b ) d ( x 0 , x 1 ) + h c d ( x 1 , x 2 ) h ( a + b ) 1 h c d ( x 0 , x 1 ) .

Now, we have

θd( x 1 ,F x 1 )θd( x 1 , x 2 )sd( x 1 , x 2 ).

Due to assumption (2.1), we get

H(F x 1 ,F x 2 )ad( x 1 , x 2 )+bd( x 1 ,F x 1 )+cd( x 2 ,F x 2 ).

Taking Lemma 2 into account, we conclude that there exists x 3 F x 2 such that

d( x 2 , x 3 )hH(F x 1 ,F x 2 ).

Consequently, we have

d ( x 2 , x 3 ) h H ( F x 1 , F x 2 ) h ( a d ( x 1 , x 2 ) + b d ( x 1 , F x 1 ) + c d ( x 2 , F x 2 ) ) h ( a + b ) d ( x 1 , x 2 ) + h c d ( x 2 , x 3 ) h ( a + b ) 1 h c d ( x 1 , x 2 ) .

Continuing in a similar way, we can obtain a sequence { x n } of successive approximations for F, starting from x 0 , satisfying the following:

  1. (a)

    x n + 1 F x n for all nN;

  2. (b)

    d( x n , x n + 1 ) k n d( x 0 , x 1 ) for all nN,

where k= h ( a + b ) 1 h c <1. Now, following the lines in the proof of Theorem 2, we deduce that the sequence { x n } converges to some uX with respect to the metric d, that is, lim n + d( x n ,u)=0.

For this purpose, we first claim that

d(u,Fx)s ( a + b θ ) d(u,x)+scd(x,Fx)

for all xX{u}. Since d( x n ,u)0 as n+ under the metric d, there exists n 0 N such that

d( x n ,u) 1 3 d(u,x)

for each n n 0 . Then we have

θ d ( x n , F x n ) d ( x n , F x n ) d ( x n , x n + 1 ) s ( d ( x n , u ) + d ( u , x n + 1 ) ) s ( 2 3 d ( u , x ) ) s ( d ( u , x ) d ( x n , u ) ) s d ( x n , x ) ,

which implies that

H ( F x n , F x ) a d ( x n , x ) + b d ( x n , F x n ) + c d ( x , F x ) a d ( x n , x ) + b θ d ( x n , x ) + c d ( x , F x ) = ( a + b θ ) d ( x n , x ) + c d ( x , F x )

for all n n 0 . Thus we have

d ( u , F x ) s ( d ( u , x n + 1 ) + d ( x n + 1 , F x ) ) s ( d ( u , x n + 1 ) + H ( F x n , F x ) ) s ( d ( u , x n + 1 ) + ( a + b θ ) d ( x n , x ) + c d ( x , F x ) )

for all n n 0 . Letting n+, we get

d(u,Fx)s ( a + b θ ) d(u,x)+scd(x,Fx)

for all xX{u}.

Next, we show that

H(Fx,Fu) ( a + b s θ ) d(x,u)+cd(u,Fu)

for all xX with xu. Now, for all nN, there exists y n Fx such that

d(u, y n )d(u,Fx)+ 1 n d(x,u).

On the other hand, we have

d ( x , F x ) d ( x , y n ) s ( d ( x , u ) + d ( u , y n ) ) = s ( d ( x , u ) + d ( u , y n ) ) s ( d ( x , u ) + d ( u , F x ) + 1 n d ( x , u ) ) s ( d ( x , u ) + s ( a + b θ ) d ( u , x ) + c d ( x , F x ) + 1 n d ( x , u ) ) = s ( 1 + s a + s b θ + s n ) d ( x , u ) + s c d ( x , F x )

for all nN. Letting n+ in the inequality above, we derive that

(1sc)d(x,Fx)s ( 1 + s a + s b θ ) d(x,u).

Hence, we have θd(x,Fx)sd(x,u), which implies

H ( F x , F u ) a d ( x , u ) + b d ( x , F x ) + c d ( u , F u ) ( a + b s θ ) d ( x , u ) + c d ( u , F u )

for all xX{u}.

Finally, if for some nN we have x n = x n + 1 , then x n is a fixed point of F. Assume that x n x n + 1 for all nN. Thus, there exists an infinite subset J of such that x n u for all nJ. Now, for all nJ, we have

d ( u , F u ) s ( d ( u , x n + 1 ) + d ( x n + 1 , F u ) ) s ( d ( u , x n + 1 ) + H ( F x n , F u ) ) s ( d ( u , x n + 1 ) + ( a + s b θ ) d ( x n , u ) + c d ( u , F u ) ) .

Letting n+ with nJ, we get

d(u,Fu)=0.

By Remark 1, we deduce that uFu and hence u is a fixed point of F. □

Remark 5 Taking s=1 in Theorem 3 (it corresponds to the case of metric spaces), with a+b+c[0,1), θ= 1 b c 1 + a , we get Theorem 6.6 of Mot and Petrusel [14] which itself is an extension of the theorem given in Reich [8], p.5, as well as a generalization of Kikkawa-Suzuki’s Theorem 1.1.

If T:XX is a single-valued mapping on a b-metric space, we have the following corollary which is a consequence of Theorem 3.

Corollary 2 Let (X,d) be a complete b-metric space and let F:XX be a single-valued mapping. If for s1 there exist nonnegative numbers a, b, c with s(a+b+c)[0,1) and θ= 1 s b s c 1 + s a such that

θd(x,Fx)sd(x,y)d(Fx,Fy)ad(x,y)+bd(x,Fx)+cd(y,Fy)
(2.6)

for all x,yX, then F has a fixed point.

Remark 6 If we take s=1 in Corollary 2, we immediately get a Kikkawa-Suzuki type fixed point theorem for a Reich-type single-valued operator, see [8, 24].

Example 2 Let X=[1,) and d(x,y)= | x y | 2 for all x,yX. Then d is a b-metric on X with s=2 and (X,d) is complete. Also, d is not a metric on X. Define F:XCB(X) by

Fx= [ 2 , 2 + x 3 ]

for all x,yX. Consider H(Fx,Fy)= 1 9 ( x y ) 2 = 1 9 d(x,y), where r= 1 9 < 1 6 = 1 s 2 + s <1. So all the conditions of Theorem 2 are satisfied. Moreover, 2 and 3 are the two fixed points of F.

References

  1. Czerwik S: Contraction mappings in b -metric spaces. Acta Math. Inform. Univ. Ostrav. 1993, 1: 5–11.

    MATH  MathSciNet  Google Scholar 

  2. Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3: 133–181.

    MATH  Google Scholar 

  3. Azam A, Mehmood N, Ahmad J, Radenović S: Multivalued fixed point theorems in cone b -metric spaces. J. Inequal. Appl. 2013., 2013: Article ID 582

    Google Scholar 

  4. Czerwik S, Dlutek K, Singh SL: Round-off stability of iteration procedures for operators in b -metric spaces. J. Natur. Phys. Sci. 1997, 11: 87–94.

    MATH  MathSciNet  Google Scholar 

  5. Czerwik S: Nonlinear set-valued contraction mappings in b -metric spaces. Atti Semin. Mat. Fis. Univ. Modena 1998,46(2):263–276.

    MATH  MathSciNet  Google Scholar 

  6. Hussain N, Shah MH: KKM mappings in cone b -metric spaces. Comput. Math. Appl. 2011,62(4):1677–1684. 10.1016/j.camwa.2011.06.004

    MATH  MathSciNet  Article  Google Scholar 

  7. Nadler S: Multi-valued contraction mappings. Pac. J. Math. 1969, 20: 475–488.

    MathSciNet  Article  Google Scholar 

  8. Reich S: Fixed points of contractive functions. Boll. Unione Mat. Ital. 1972, 5: 26–42.

    MATH  Google Scholar 

  9. Aydi H, Bota M-F, Karapınar E, Moradi S: A common fixed point for weak- ϕ -contractions on b -metric spaces. Fixed Point Theory 2012,13(2):337–346.

    MATH  MathSciNet  Google Scholar 

  10. Boriceanu M: Strict fixed point theorems for multivalued operators in b -metric spaces. Int. J. Mod. Math. 2009,4(2):285–301.

    MATH  MathSciNet  Google Scholar 

  11. Bota M-F, Karapınar E, Mlesnite O: Ulam-Hyers stability results for fixed point problems via alpha-psi-contractive mapping in b -metric space. Abstr. Appl. Anal. 2013., 2013: Article ID 825293

    Google Scholar 

  12. Bota M-F, Karapınar E: A note on ‘Some results on multi-valued weakly Jungck mappings in b -metric space’. Cent. Eur. J. Math. 2013, 11: 1711–1712. 10.2478/s11533-013-0272-2

    MATH  MathSciNet  Google Scholar 

  13. Kikkawa M, Suzuki T: Three fixed point theorems for generalized contractions with constants in complete metric spaces. Nonlinear Anal. 2008, 69: 2942–2949. 10.1016/j.na.2007.08.064

    MATH  MathSciNet  Article  Google Scholar 

  14. Mot G, Petrusel A: Fixed point theory for a new type of contractive multi-valued operators. Nonlinear Anal. 2009, 70: 3371–3377. 10.1016/j.na.2008.05.005

    MATH  MathSciNet  Article  Google Scholar 

  15. Dhompongsa S, Yingtaweesittikul H: Fixed points for multi-valued mappings and the metric completeness. Fixed Point Theory Appl. 2009., 2009: Article ID 972395

    Google Scholar 

  16. Boriceanu M: Fixed point theory for multivalued generalized contraction on a set with two b -metrics. Stud. Univ. Babeş-Bolyai, Math. 2009,LIV(3):1–14.

    MathSciNet  Google Scholar 

  17. Bota M, Molnar A, Varga C: On Ekeland’s variational principle in b -metric spaces. Fixed Point Theory 2011,12(2):21–28.

    MATH  MathSciNet  Google Scholar 

  18. Singh SL, Prasad B: Some coincidence theorems and stability of iterative procedures. Comput. Math. Appl. 2008, 55: 2512–2520. 10.1016/j.camwa.2007.10.026

    MATH  MathSciNet  Article  Google Scholar 

  19. Păcurar M: Sequences of almost contractions and fixed points in b -metric spaces. An. Univ. Vest. Timiş., Ser. Mat.-Inform. 2010,XLVIII(3):125–137.

    Google Scholar 

  20. Pacurar M: A fixed point results for φ -contractions on b -metric spaces without the boundedness assumption. Fasc. Math. 2010, 43: 127–137.

    MATH  MathSciNet  Google Scholar 

  21. Aydi H, Bota MF, Karapınar E, Mitrović S: A fixed point theorem for set-valued quasicontractions in b -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 88

    Google Scholar 

  22. Singh SL, Czerwik S, Krol K, Singh A: Coincidences and fixed points of hybrid contractions. Tamsui Oxford Univ. J. Math. Sci. 2008, 24: 401–416.

    MATH  MathSciNet  Google Scholar 

  23. Suzuki T: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 2008, 136: 1861–1869.

    MATH  Article  Google Scholar 

  24. Reich S: Kannan’s fixed point theorem. Boll. Unione Mat. Ital. 1971, 4: 1–11.

    MATH  Google Scholar 

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Acknowledgements

First author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. The third author gratefully acknowledges the support from the Higher Education Commission of Pakistan. The authors thank the anonymous referees for their remarkable comments, suggestions and ideas that helped to improve this paper.

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Kutbi, M.A., Karapınar, E., Ahmad, J. et al. Some fixed point results for multi-valued mappings in b-metric spaces. J Inequal Appl 2014, 126 (2014). https://doi.org/10.1186/1029-242X-2014-126

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Keywords

  • Hausdorff metric
  • set-valued mapping
  • fixed point
  • b-metric space