Skip to main content

Common fixed point results of Ciric-Suzuki-type inequality for multivalued maps in compact metric spaces

Abstract

The aim of this paper is to introduce a new type of multivalued operators similar to those of Ciric-Suzuki type. A common fixed point theorem for multivalued maps on metric spaces satisfying Ciric-Suzuki-type inequality is proved. Applications to certain functional equations arising in dynamic programming are also discussed.

MSC:54E50, 54H25, 45G05.

1 Introduction and preliminaries

In 1937, Von Neumann [1] initiated fixed point theory for multivalued mappings in the study of game theory. Indeed, the fixed point theorems for multivalued mappings are quite useful in control theory and have been frequently used in solving many problems of economics and game theory. Successively, Nadler [2] initiated the development of the geometric fixed point theory for multivalued mappings. He used the concept of the Hausdorff metric to establish the multivalued contraction principle containing the Banach contraction principle as a special case.

Consistent with Nadler ([3], p.620), (X,d) and Cl(X) will denote a metric space and the collection of all nonempty closed subsets of X, respectively. For A,BCl(X) and ϵ>0,

N ( ϵ , A ) = { x X : d ( x , a ) < ϵ  for some  a A } , E A , B = { ϵ > 0 : A N ( ϵ , B ) , B N ( ϵ , A ) } , H ( A , B ) = { inf E A , B if  E A , B , + if  E A , B = .

The hyperspace (Cl(X),H) is called the generalized Hausdorff metric space induced by the metric d on X.

Later, Doric and Lazovic [4] have extended and generalized fixed point theorems of Ciric [5], Kikkawa and Suzuki [6], Nadler [2], and others as follows.

Theorem 1.1 [4]

Define a nonincreasing function φ from [0,1) into (0,1] by

φ(r)= { 1 if  0 r < 1 2 , 1 r if  1 2 r < 1 .

Let (X,d) be a complete metric space and T:XCB(X). Assume that there exists r[0,1) such that for every x,yX,

φ(r)d(x,Tx)d(x,y)impliesH(Tx,Ty)rM(x,y),

where

M(x,y)=max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , 1 2 ( d ( x , T y ) + d ( y , T x ) ) } .

Then there exists vX such that vTv.

On the other hand, Suzuki [7] introduced the following theorem. This result involves a new type of contractive mapping and hence generalizes the well-known Edelstein fixed point theorem in [8].

Theorem 1.2 [7]

Let (X,d) be a compact metric space, and let f be a mapping on X. Assume that

1 2 d(x,fx)<d(x,y)impliesd(fx,fy)<d(x,y)

for x,yX. Then f has a unique fixed point.

Many authors have proved numerous fixed point theorems as generalization of Nadler’s theorem (see [912]).

In 2011, Haghi et al. [13] gave a very useful lemma which we need in our work.

Lemma 1.3 [13]

Let X be a nonempty set and f:XX be a function. Then there exists a subset EX such that f(E)=f(X) and f:EX is one-to-one.

In this manuscript, the well-known results of Suzuki [7], Edelstein [8] and Doric and Lazovic [4] have been merged to complement a multitude of related results from the literature. Moreover, we use Lemma 1.3 to obtain a common fixed point theorem for multivalued maps on a metric space. Finally, as an application, the existence of common solutions of certain functional equations arising in dynamic programming is proved.

2 Fixed point results

Theorem 2.1 Let (X,d) be a compact metric space, and let T:XCl(X). Assume that there exists α[0, 1 2 ) such that

1 2 d(x,Tx)<d(x,y)impliesH(Tx,Ty)<αM(x,y),

where

M(x,y)=max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , 1 2 ( d ( x , T y ) + d ( y , T x ) ) }

for all x,yX. Then T has a fixed point.

Proof Let { x n } be a sequence in X. We have that T x n is a closed set, so we can find a sequence { z n } in X such that z n T x n and d( x n , z n )=d( x n ,T x n ). Now, put β=inf{d(x,Tx):xX}=inf{ inf z T x d(x,z):xX}. X is compact, without loss of generality, we may assume that the sequences { x n } and { z n } converge to v and w in X, respectively. Let

limd( x n , z n )=limd( x n ,T x n )=limd( x n ,w)=d(v,w)=β.

We want to show that β=0. Suppose not, that is, β>0. Thus, we can choose n 0 N such that

d( x n ,T x n )< 3 2 βand 3 4 β<d( x n ,w)

for nN with n n 0 . Thus 1 2 d( x n ,T x n )<d( x n ,w) for n n 0 . This implies

H(T x n ,Tw)<αM( x n ,w),

where

M( x n ,w)=max { d ( x n , w ) , d ( x n , T x n ) , d ( w , T w ) , 1 2 ( d ( x n , T w ) + d ( w , T x n ) ) } .

Since z n T x n , d( z n ,Tw)H(T x n ,Tw). Thus

d( z n ,Tw)<αM( x n ,w).

Moreover, d(w,T x n )d(w, z n ). So

M( x n ,w)max { d ( x n , w ) , d ( x n , T x n ) , d ( w , T w ) , 1 2 ( d ( x n , z n ) + d ( z n , T w ) + d ( w , z n ) ) } .

Let n attend to ∞, then we have

d(w,Tw)αM(v,w),
(2.1)

where

M ( v , w ) max { d ( v , w ) , d ( v , w ) , d ( w , T w ) , 1 2 ( d ( v , w ) + d ( w , T w ) + d ( w , w ) ) } = max { d ( v , w ) , d ( w , T w ) , 1 2 ( d ( v , w ) + d ( w , T w ) ) }

but 1 2 (d(v,w)+d(w,Tw))max{d(v,w),d(w,Tw)}. Thus, M(v,w)max{d(v,w),d(w,Tw)}. From (2.1), we have

d(w,Tw)αmax { d ( v , w ) , d ( w , T w ) } αd(v,w)=αβ<β,

which is a contradiction to the definition of β. Hence, β=0.

We want to prove that T has a fixed point. Assume, on the contrary, that x n T x n for any nN. We have z n w and w=v thus z n v, then vT x n .

First, we want to show that

d(v,Tu)αmax { d ( v , u ) , d ( u , T u ) } for every uX{v}.
(2.2)

We have x n v and z n v, that is, there exist n 1 , n 2 N such that

d(v, x n ) 1 3 d(v,u)andd(v, z n ) 1 3 d(v,u)for every n>N=max{ n 1 , n 2 }

so we have, for any n>N,

d ( x n , T x n ) d ( x n , z n ) d ( x n , v ) + d ( v , z n ) 2 3 d ( v , u ) = d ( v , u ) 1 3 d ( v , u ) d ( v , u ) d ( v , x n ) d ( x n , u ) .

Thus 1 2 d( x n ,T x n )<d( x n ,u), which implies H(T x n ,Tu)<αM( x n ,u), where

M( x n ,u)=max { d ( x n , u ) , d ( x n , T x n ) , d ( u , T u ) , 1 2 ( d ( u , T x n ) + d ( x n , T u ) ) }

but vT x n , thus

d(v,Tu)<αmax { d ( x n , u ) , d ( x n , T x n ) , d ( u , T u ) , 1 2 ( d ( u , T x n ) + d ( x n , T u ) ) } .
(2.3)

We have d(u,T x n )+d( x n ,Tu)d(u,v)+d( x n ,Tu). Then (2.3) becomes

d(v,Tu)<αmax { d ( x n , u ) , d ( x n , T x n ) , d ( u , T u ) , 1 2 ( d ( u , v ) + d ( x n , T u ) ) } .

By taking n, we obtain

d(v,Tu)αmax { d ( v , u ) , d ( u , T u ) , 1 2 ( d ( u , v ) + d ( v , T u ) ) } .

If d(u,v)d(v,Tu), then

d ( v , T u ) α max { d ( v , u ) , d ( u , T u ) , d ( v , T u ) } α max { d ( v , u ) , d ( u , T u ) }

and if d(v,Tu)d(u,v), then d(v,Tu)αmax{d(v,u),d(u,Tu)}. Thus, for any uX{v},

d(v,Tu)αmax { d ( v , u ) , d ( u , T u ) } .

Second, we want to prove that for every xX,

H(Tx,Tv)<αmax { d ( x , v ) , d ( v , T v ) , d ( x , T x ) , 1 2 ( d ( x , T v ) + d ( v , T x ) ) } .
(2.4)

If x=v, it is trivial. Let xv. Then, for every nN, there exists a sequence y n Tx such that d(v, y n )d(v,Tx)+ 1 n d(v,x). For all nN, by using (2.2) we have

d ( x , T x ) d ( x , y n ) d ( x , v ) + d ( v , y n ) d ( x , v ) + d ( v , T x ) + 1 n d ( v , x ) < d ( x , v ) + α max { d ( v , x ) , d ( x , T x ) } + 1 n d ( v , x ) .

If d(x,Tx)d(v,x), then d(x,Tx)(1+α+ 1 n )d(v,x). When n, we have

d(x,Tx)(1+α)d(v,x)<2d(v,x).

Hence, we find that 1 2 d(x,Tx)<d(v,x).

If d(v,x)d(x,Tx), then d(x,Tx)d(x,v)+αd(x,Tx)+ 1 n d(v,x). Thus,

(1α)d(x,Tx)< ( 1 + 1 n ) d(v,x)

gives when n

(1α)d(x,Tx)d(v,x).

Since 1 2 <1α, therefore 1 2 d(x,Tx)<d(v,x). Now, for every xX{v}, we have

1 2 d(x,Tx)<d(v,x),

which implies (2.4).

Finally, from (2.4) for any n>N, we get

H(T x n ,Tv)<αmax { d ( x n , v ) , d ( v , T v ) , d ( x n , T x n ) , 1 2 ( d ( x n , T v ) + d ( v , T x n ) ) } .

We have z n T x n , thus d( z n ,Tv)H(T x n ,Tv). If n, then we obtain

d(v,Tv)αd(v,Tv).

As α< 1 2 , we obtain that d(v,Tv)=0. Since Tv is closed, thus we reach the conclusion vTv. This contradicts the assumption that T has no fixed point. Hence, T has a fixed point. □

Example 2.2 Let X=[0,1] be endowed with the usual metric and T:XCl(X) be defined by

Tx= { { x 6 } if  x [ 0 , 1 ) , { 0 , 1 6 } if  x = 1 ,

with α= 1 4 . Then T satisfies the assumptions in Theorem 2.1.

Proof If x=y, it is trivial. For xy we have the following:

  1. (1)

    x,y[0,1) then

    H(Tx,Ty)=| x y 6 |< 1 4 |xy|=αd(x,y)αM(x,y);
  2. (2)

    x[0,1) and y=1 or (x=1 and y[0,1))

    H(Tx,Ty) 1 6 < 1 4 ( 5 6 ) =αd(1,T1)=αd(y,Ty)αM(x,y).

Hence, the assumptions of Theorem 2.1 are satisfied and {0} is a fixed point. □

3 Common fixed point results

Theorem 3.1 Let (X,d) be a metric space, Y be a nonempty set, g:YX and T:YCl(X) such that T(Y)g(Y) and g(Y) is a compact subspace of X. Assume that for x,yY there exists α[0, 1 2 ) such that

1 2 d(gx,Tx)<d(gx,gy)impliesH(Tx,Ty)<αM(gx,gy),

where

M(gx,gy)=max { d ( g x , g y ) , d ( g x , T x ) , d ( g y , T y ) , 1 2 ( d ( g x , T y ) + d ( g y , T x ) ) }

and gx=gy implies Tx=Ty. Then T and g have a coincidence point, that is, there exists vY such that gvTv. If Y=X, then g, T have a common fixed point provided that gTvTgv at v.

Proof By Lemma 1.3, there exists EY such that g(E)=g(Y) and g:EX is one-to-one. We define a map G:g(E)Cl(g(E)) by G(gx)=Tx.

For x,yE, Tx=Ty whenever gx=gy, i.e., G(gx)=G(gy). Also, we have TxT(E)g(E). Thus, G is a mapping. Now, for gx,gyg(E) we have

1 2 d ( g x , G ( g x ) ) <d(gx,gy)impliesH ( G ( g x ) , G ( g y ) ) <αM(gx,gy),

where

M(gx,gy)=max { d ( g x , g y ) , d ( g x , G ( g x ) ) , d ( g y , G ( g y ) ) , 1 2 ( d ( g x , G ( g y ) ) + d ( g y , G ( g x ) ) ) } .

By Theorem 2.1, G has a fixed point in g(E), that is, there exists gvg(E) such that gvG(gv)=Tv. Further, if Y=X, we have gvTv. Thus, ggvgTvTgvTTv. Now, we want to show that gv=ggv. Suppose gvggv, then we have

1 2 d(gv,Tv)=0<d(gv,ggv),which impliesH(Tv,Tgv)<αM(gv,ggv),

where

M(gv,ggv)=max { d ( g v , g g v ) , d ( g v , T v ) , d ( g g v , T g v ) , 1 2 ( d ( g v , T g v ) + d ( g g v , T v ) ) } .

Since gvTv, ggvTgv and 1 2 (d(gv,Tgv)+d(ggv,Tv))H(Tv,Tgv), we obtain

M(gv,ggv)max { d ( g v , g g v ) , H ( T v , T g v ) } .

Hence,

H(Tv,Tgv)<αmax { d ( g v , g g v ) , H ( T v , T g v ) } αd(gv,ggv).

We have d(gv,ggv)d(gv,w)+d(w,ggv) for each wTgv. Since Tgv is a closed set, then

d(gv,ggv)d(gv,Tgv)+d(ggv,Tgv)

but ggvTgv. Thus, d(gv,ggv)d(gv,Tgv). Since gvTv, we have

d(gv,Tgv)<H(Tv,Tgv)<αd(gv,Tgv),

which is a contradiction. Therefore, gv=ggvTgv. □

Corollary 3.2 Let (X,d) be a metric space, Y be a nonempty set, f,g:YX such that f(Y)g(Y) and g(Y) is a compact subspace of X. Assume that for x,yY there exists α[0, 1 2 ) such that

1 2 d(gx,fx)<d(gx,gy)impliesd(fx,fy)<αM(gx,gy),

where

M(gx,gy)=max { d ( g x , g y ) , d ( g x , f x ) , d ( g y , f y ) , 1 2 ( d ( g x , f y ) + d ( g y , f x ) ) }

and gx=gy implies fx=fy. Then f and g have a unique coincidence point. If Y=X, then f, g have a unique common fixed point provided that f and g commute at v.

Proof The proof of this corollary follows from Theorem 3.1 by taking T:YX. We need to prove that v is a unique coincidence point of f and g. Suppose, to the contrary, that there exists zX such that zv and fz=gz. Then the inequalities d(gv,gz)>0 and 0= 1 2 d(gv,fv)<d(gv,gz) are satisfied. Thus, we have d(fv,fz)<αM(gv,gz), where

M(gv,gz)=max { d ( g v , g z ) , d ( g v , f v ) , d ( g z , f z ) , 1 2 ( d ( g v , f z ) + d ( g z , f v ) ) } ,

i.e.,

M(gv,gz)=max { d ( g v , g z ) , 0 } .

Therefore, d(fv,fz)<αd(gv,gz), that is, d(gv,gz)<αd(gv,gz), this is impossible due to α< 1 2 . Hence, v is the unique coincidence point. Moreover, if Y=X, we have fgv=gfv but fv=gv. Thus, fgv=ggv, i.e., gv is a coincidence point. Because of the uniqueness of the coincidence point, we obtain that gv=v. Then v=gv=fv. □

4 An application

Many authors have studied the existence and uniqueness of solutions of functional equations and system of functional equations in dynamic programming by using diverse fixed point theorems (cf. [1416]).

Throughout this section, we suppose that U and V are Banach spaces, WU is the state space, DV is the decision space and is the field of real numbers. Let B(W) denote the set of all the bounded real-valued functions on W. For an arbitrary hW, define h= sup x W |h(x)|. Then (B(W),) is a Banach space.

The return functions S,T:WR of the continuous decision process are defined by the functional equations

S= sup y D { q ( x , y ) + G ( x , y , S ( τ ( x , y ) ) ) } ,xW
(4.1)

and

T= sup y D { q ( x , y ) + F ( x , y , T ( τ ( x , y ) ) ) } ,xW,
(4.2)

where x and y represent the state and decision vectors, respectively, τ:W×DW represents the transformation of the process. Moreover, q, q :W×DR and G,F:W×D×RR are bounded functions.

In this article, we prove the existence and uniqueness of the common solution of functional equations (4.1) and (4.2) arising in dynamic programming, using Corollary 3.2.

Let the maps f and g be defined by

f h ( x ) = sup y D { q ( x , y ) + G ( x , y , h ( τ ( x , y ) ) ) } , x W , h B ( W ) , g h ( x ) = sup y D { q ( x , y ) + F ( x , y , h ( τ ( x , y ) ) ) } , x W , h B ( W ) .

Suppose that the following conditions hold.

(Q1) For any hB(W), there exists kB(W) such that

fh(x)=gk(x),xW.

(Q2) There exists hB(W) such that

fh(x)=gh(x)impliesgfh(x)=fgh(x).

Theorem 4.1 Suppose that conditions (Q1) and (Q2) are satisfied and g(B(W)) is a closed and bounded subspace of B(W). Assume that there exists α[0, 1 2 ) such that for every (x,y)W×D, h,kB(W) and tW, the inequality

1 2 | f h ( t ) g h ( t ) | < | g h ( t ) g k ( t ) |

implies

| G ( x , y , h ( t ) ) G ( x , y , k ( t ) ) | <αM(gh,gk),

where

M ( g h , g k ) = max { | g h ( t ) g k ( t ) | , | g h ( t ) f h ( t ) | , | g k ( t ) f k ( t ) | , | g h ( t ) f k ( t ) | + | g k ( t ) f h ( t ) | 2 } ,

then functional equations (4.1) and (4.2) have a unique common bounded solution in W.

Proof Note that, due to q, q , G and F are bounded, f and g are self-maps of B(W). (B(W),d) is a complete metric space, where d is the metric defined by the supremum norm on B(W). Since g(B(W)) is a closed and bounded subspace of B(W), then g(B(W)) is complete and bounded. That is, g(B(W)) is compact. Conditions (Q1) and (Q2) imply that f(B(W))g(B(W)) and f and g commute at their coincidence points.

Let ε be an arbitrary positive real number, and h 1 , h 2 B(W). For xW, choose y 1 , y 2 D such that

f h 1 (x)<q(x, y 1 )+G ( x , y 1 , h 1 ( τ 1 ) ) +ε
(4.3)

and

f h 2 (x)<q(x, y 2 )+G ( x , y 2 , h 2 ( τ 2 ) ) +ε,
(4.4)

where τ 1 =τ(x, y 1 ) and τ 2 =τ(x, y 2 ).

Furthermore, from the definition of f, we have

f h 1 (x)q(x, y 2 )+G ( x , y 2 , h 1 ( τ 2 ) )
(4.5)

and

f h 2 (x)q(x, y 1 )+G ( x , y 1 , h 2 ( τ 1 ) ) .
(4.6)

If we suppose that the following inequality holds

1 2 | f h 1 ( x ) g h 1 ( x ) | < | g h 1 ( x ) g h 2 ( x ) | ,

then

| G ( x , y 1 , h 1 ( τ 1 ) ) G ( x , y 1 , h 2 ( τ 1 ) ) | <αM(g h 1 ,g h 2 ),
(4.7)

where

M ( g h 1 , g h 2 ) = max { | g h 1 ( x ) g h 2 ( x ) | , | g h 1 ( x ) f h 1 ( x ) | , | g h 2 ( x ) f h 2 ( x ) | , | g h 1 ( x ) f h 2 ( x ) | + | g h 2 ( x ) f h 1 ( x ) | 2 } .

From (4.3), (4.6) and (4.7), we obtain that

f h 1 ( x ) f h 2 ( x ) < G ( x , y 1 , h 1 ( τ 1 ) ) G ( x , y 1 , h 2 ( τ 1 ) ) + ε | G ( x , y 1 , h 1 ( τ 1 ) ) G ( x , y 1 , h 2 ( τ 1 ) ) | + ε < α M ( g h 1 , g h 2 ) + ε .
(4.8)

Similarly, (4.4), (4.5) and (4.7) imply

f h 2 ( x ) f h 1 ( x ) < G ( x , y 1 , h 2 ( τ 1 ) ) G ( x , y 1 , h 1 ( τ 1 ) ) + ε | G ( x , y 1 , h 1 ( τ 1 ) ) G ( x , y 1 , h 2 ( τ 1 ) ) | + ε < α M ( g h 1 , g h 2 ) + ε .
(4.9)

Thus, from (4.8) and (4.9) we have

| f h 1 ( x ) f h 2 ( x ) | <αM(g h 1 ,g h 2 )+ε.

Since ε is arbitrary, therefore for any xW, we have

1 2 | f h 1 ( x ) g h 1 ( x ) | < | g h 1 ( x ) g h 2 ( x ) | implies | f h 1 ( x ) f h 2 ( x ) | <αM(g h 1 ,g h 2 ),

where

M ( g h 1 , g h 2 ) = max { | g h 1 ( x ) g h 2 ( x ) | , | g h 1 ( x ) f h 1 ( x ) | , | g h 2 ( x ) f h 2 ( x ) | , | g h 1 ( x ) f h 2 ( x ) | + | g h 2 ( x ) f h 1 ( x ) | 2 } .

Therefore, by Corollary 3.2, f and g have a unique common fixed point, and hence functional equations (4.1) and (4.2) have a unique bounded common solution. □

Remark 4.2 Fukhar-ud-din et al. [17] have established fixed point results on a non-compact domain in uniformly convex metric spaces for a single-valued map satisfying a contractive condition closely related to the Suzuki condition employed in Theorem 1.2. It will be interesting to extend the results of this paper in this general setup (cf. Open problem 1 on p.4759 in [17]).

References

  1. 1.

    Von Neuman J: Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes. Ergeb. Math. Kolloq. 1937, 8: 73-83.

    MATH  Google Scholar 

  2. 2.

    Nadler SB Jr.: Multi-valued contraction mappings. Pac. J. Math. 1969, 30: 475-488. 10.2140/pjm.1969.30.475

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Nadler SB Jr. Monographs and Textbooks in Pure and Applied Mathematics 49. In Hyperspaces of Sets: A Text with Research Questions. Dekker, New York; 1978.

    Google Scholar 

  4. 4.

    Doric D, Lazovic R: Some Suzuki-type fixed point theorems for generalized multivalued mappings and applications. Fixed Point Theory Appl. 2011., 2011: Article ID 40

    Google Scholar 

  5. 5.

    Ciric LB: Fixed points for generalized multi-valued contractions. Mat. Vesn. 1972,9(24):265-272.

    MathSciNet  MATH  Google Scholar 

  6. 6.

    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

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Suzuki T: A new type of fixed point theorem in metric spaces. Nonlinear Anal. 2009, 71: 5313-5317. 10.1016/j.na.2009.04.017

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Edelstein M: On fixed and periodic points under contractive mappings. J. Lond. Math. Soc. 1962, 37: 74-79.

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Abtahi, M: A fixed point theorem for contractive mappings that characterizes metric completeness. J. Math. Anal. Appl. arXiv:1207.6207v1 (2012)

  10. 10.

    Bari CD, Vetro P: Common fixed points for self mappings on compact metric spaces. Appl. Math. Comput. 2013, 219: 6804-6808. 10.1016/j.amc.2013.01.022

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Doric D, Kadelburg Z, Radenović S: Edelstein-Suzuki-type fixed point results in metric and abstract metric spaces. Nonlinear Anal. 2012, 75: 1927-1932. 10.1016/j.na.2011.09.046

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Singh SL, Mishra SN: Remarks on recent fixed point theorems. Fixed Point Theory Appl. 2010., 2010: Article ID 452905

    Google Scholar 

  13. 13.

    Haghi RH, Rezapour S, Shahzad N: Some fixed point generalizations are not real generalizations. Nonlinear Anal. 2011, 74: 1799-1803. 10.1016/j.na.2010.10.052

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Khan AR: Common fixed point and solution of nonlinear functional equations. Fixed Point Theory Appl. 2013., 2013: Article ID 290

    Google Scholar 

  15. 15.

    Popescu O: A new type of contractive multivalued operators. Bull. Sci. Math. 2013, 137: 30-44. 10.1016/j.bulsci.2012.07.001

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Singh SL, Mishra SN: Fixed point theorems for single-valued and multi-valued maps. Nonlinear Anal. 2011, 74: 2243-2248. 10.1016/j.na.2010.11.029

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Fukhar-ud-din H, Khan AR, Akhtar Z: Fixed point results for a generalized nonexpansive map in uniformly convex metric spaces. Nonlinear Anal. 2012, 75: 4747-4760. 10.1016/j.na.2012.03.025

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research grant UKM-DIP-2012-31.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Fawzia Shaddad.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final version.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Shaddad, F., Noorani, M.S.M. & Alsulami, S.M. Common fixed point results of Ciric-Suzuki-type inequality for multivalued maps in compact metric spaces. J Inequal Appl 2014, 7 (2014). https://doi.org/10.1186/1029-242X-2014-7

Download citation

Keywords

  • metric space
  • common fixed point
  • Urysohn integral equations