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A note on approximate fixed point property and Du-Karapinar-Shahzad’s intersection theorems

Abstract

In this note, we give new short proofs of Du-Karapinar-Shahzad’s intersection theorems for multivalued non-self-maps in complete metric spaces.

MSC:47H10, 54H25.

1 Introduction and preliminaries

Let us begin with some basic definitions and notations that will be needed in this paper. Let (X,d) be a metric space. Denote by N(X) the family of all nonempty subsets of X and by CB(X) the family of all nonempty closed and bounded subsets of X. For each xX and AX, let d(x,A)= inf y A d(x,y). A function H:CB(X)×CB(X)[0,) defined by

H(A,B)=max { sup x B d ( x , A ) , sup x A d ( x , B ) }

is said to be the Hausdorff metric on CB(X) induced by the metric d on X. The symbols and are used to denote the sets of positive integers and real numbers, respectively.

Let K be a nonempty subset of X, g:KX be a single-valued non-self-map and T:KN(X) be a multivalued non-self-map. A point v in X is a coincidence point (see, for instance, [16]) of g and T if gvTx. If g=id is the identity map, then v=gvTv and call v a fixed point of T. The set of fixed points of T and the set of coincidence points of g and T are denoted by F K (T) and COP K (g,T), respectively. In particular, if KX, we use F(T) and COP(g,T) instead of F K (T) and COP K (g,T), respectively. The map T is said to have approximate fixed point property [15] on K provided inf x K d(x,Tx)=0. It is obvious that F K (T) implies that T has approximate fixed point property.

A function φ:[0,)[0,1) is said to be an MT-function (or -function) [311] if lim sup s t + φ(s)<1 for all t[0,). Clearly, if φ:[0,)[0,1) is a nondecreasing function or a nonincreasing function, then φ is an MT-function. So, the set of MT-functions is a rich class and has the questions many of which are worth studying.

The study of fixed points for single-valued non-self-maps or multivalued non-self-maps satisfying certain contractive conditions is an interesting and important direction of research in metric fixed point theory. A great deal of such research has been investigated by several authors, see, e.g., [1119] and the references therein. Very recently, Du, Karapinar and Shahzad [11] established the following intersection existence theorem of coincidence points and fixed points of multivalued non-self-maps of Kannan type and Chatterjea type.

Theorem 1.1 [[11], Theorem 8]

Let (X,d) be a complete metric space, K be a nonempty closed subset of X, T:KCB(X) be a multivalued map and g:KX be a continuous map. Suppose that

(D1) TxK for all xK,

(D2) TxK is g-invariant (i.e., g(TxK)TxK) for each xK,

(D3) there exist a function h:K[0,) and γ[0, 1 2 ) such that

H ( T x , T y K ) γ [ d ( x , T x K ) + d ( y , T x K ) + d ( y , T y K ) ] + h ( y ) d ( g y , T x K ) for all x , y K .
(1.1)

Then COP K (g,T) F K (T).

In [11], they also gave some coincidence and fixed point theorems for multivalued non-self-maps of Mizoguchi-Takahashi type, Berinde-Berinde type and Du type.

Theorem 1.2 [[11], Theorem 19]

Let (X,d) be a complete metric space, K be a nonempty closed subset of X, T:KCB(X) be a multivalued map and g:KX be a continuous map. Suppose that conditions (D1) and (D2) as in Theorem  1.1 hold. If there exist an MT-function φ:[0,)[0,1) and a function h:K[0,) such that

H(Tx,TyK)φ ( d ( x , y ) ) d(x,y)+h(y)d(gy,TxK)for allx,yK,
(1.2)

then COP K (g,T) F K (T).

In this work, we give new short proofs of Du-Karapinar-Shahzad’s intersection theorems of COP K (g,T) and F K (T) for multivalued non-self-maps (i.e., Theorems 1.1 and 1.2) by applying an existence theorem for approximate fixed point property.

2 Some auxiliary key results

Let (X,d) be a metric space. Recall that a function p:X×X[0,) is said to be a τ-function [35, 7, 8, 2022], first introduced and studied by Lin and Du, if the following conditions hold:

(τ 1) p(x,z)p(x,y)+p(y,z) for all x,y,zX;

(τ 2) if xX and { y n } in X with lim n y n =y such that p(x, y n )M for some M=M(x)>0, then p(x,y)M;

(τ 3) for any sequence { x n } in X with lim n sup{p( x n , x m ):m>n}=0, if there exists a sequence { y n } in X such that lim n p( x n , y n )=0, then lim n d( x n , y n )=0;

(τ 4) for x,y,zX, p(x,y)=0 and p(x,z)=0 imply y=z.

Note that with the additional condition

(τ 5) p(x,x)=0 for all xX,

a τ-function becomes a τ 0 -function [35, 7, 8] introduced by Du.

Clearly, any metric d is a τ 0 -function. Observe further that if p is a τ 0 -function, then, from (τ 4) and (τ 5), p(x,y)=0 if and only if x=y.

Example A [7]

Let X=R with the metric d(x,y)=|xy| and 0<a<b. Define the function p:X×X[0,) by

p(x,y)=max { a ( y x ) , b ( x y ) } .

Then p is nonsymmetric and hence p is not a metric. It is easy to see that p is a τ 0 -function.

Lemma 2.1 [[22], Lemma 2.1]

Let (X,d) be a metric space and p:X×X[0,) be a function. Assume that p satisfies the condition (τ 3). If a sequence { x n } in X with lim n sup{p( x n , x m ):m>n}=0, then { x n } is a Cauchy sequence in X.

Let (X,d) be a metric space and p be a τ-function. A multivalued map T:XN(X) is said to have p-approximate fixed point property on X provided

inf x X p(x,Tx)=0.

The following characterizations of MT-functions proved first by Du [6] are quite useful for proving our main results.

Theorem 2.1 [[6], Theorem 2.1]

Let φ:[0,)[0,1) be a function. Then the following statements are equivalent.

  1. (a)

    φ is an MT-function.

  2. (b)

    For each t[0,), there exist r t ( 1 ) [0,1) and ε t ( 1 ) >0 such that φ(s) r t ( 1 ) for all s(t,t+ ε t ( 1 ) ).

  3. (c)

    For each t[0,), there exist r t ( 2 ) [0,1) and ε t ( 2 ) >0 such that φ(s) r t ( 2 ) for all s[t,t+ ε t ( 2 ) ].

  4. (d)

    For each t[0,), there exist r t ( 3 ) [0,1) and ε t ( 3 ) >0 such that φ(s) r t ( 3 ) for all s(t,t+ ε t ( 3 ) ].

  5. (e)

    For each t[0,), there exist r t ( 4 ) [0,1) and ε t ( 4 ) >0 such that φ(s) r t ( 4 ) for all s[t,t+ ε t ( 4 ) ).

  6. (f)

    For any nonincreasing sequence { x n } n N in [0,), we have 0 sup n N φ( x n )<1.

  7. (g)

    φ is a function of contractive factor; that is, for any strictly decreasing sequence { x n } n N in [0,), we have 0 sup n N φ( x n )<1.

The following result was essentially proved by Du et al. in [4], but we give the proof for the sake of completeness and the readers convenience.

Lemma 2.2 [[4], Lemma 3.1]

Let (X,d) be a metric space, p be a τ 0 -function and T:XN(X) be a multivalued map. Then the following statements are equivalent.

(Q1) There exist a function ξ:[0,)[0,) and an MT-function φ:[0,)[0,1) such that for each xX, if yTx with yx, then there exists zTy such that

p(y,z)φ ( ξ ( p ( x , y ) ) ) p(x,y).

(Q2) There exist a function τ:[0,)[0,) and an MT-function κ:[0,)[0,1) such that for each xX,

p(y,Ty)κ ( τ ( p ( x , y ) ) ) p(x,y)for allyTx.

Proof If (Q1) holds, then it is easy to verify that (Q2) also holds with κφ and τξ. So it suffices to prove that ‘(Q2) (Q1)’. Suppose that (Q2) holds. Define φ:[0,)[0,1) by φ(t)= 1 + κ ( t ) 2 . Then φ is also an MT-function. Indeed, it is obvious that 0κ(t)<φ(t)<1 for all t[0,). Let { x n } n N be a strictly decreasing sequence in [0,). Since κ is an MT-function, by (g) of Theorem 2.1, we get

0 sup n N κ( x n )<1

and hence

0< sup n N φ( x n )= 1 2 [ 1 + sup n N κ ( x n ) ] <1.

So, by Theorem 2.1 again, we prove that φ is an MT-function.

For each xX, let yTx with yx. Then p(x,y)>0. By (Q2), we have

p(y,Ty)<φ ( τ ( p ( x , y ) ) ) p(x,y).

Since φ(t)>0 for all t[0,), there exists zTy such that

p(y,z)<φ ( τ ( p ( x , y ) ) ) p(x,y),

which shows that (Q1) holds with ξτ. So, by above, we prove ‘(Q1) (Q2)’. □

Now, we present an existence theorem for p-approximate fixed point property and approximate fixed point property, which is indeed a somewhat generalized form of [[4], Theorem 3.3] and is one of the key technical devices in the new short proofs of Theorems 1.1 and 1.2.

Theorem 2.2 Let (X,d) be a metric space, p be a τ 0 -function and T:XN(X) be a multivalued map. Assume that one of (L1) and (L2) is satisfied, where

(L1) there exist a nondecreasing function ξ:[0,)[0,) and an MT-function φ:[0,)[0,1) such that for each xX, if yTx with yx, then there exists zTy such that

p(y,z)φ ( ξ ( p ( x , y ) ) ) p(x,y);

(L2) there exist a nondecreasing function τ:[0,)[0,) and an MT-function κ:[0,)[0,1) such that for each xX,

p(y,Ty)κ ( τ ( p ( x , y ) ) ) p(x,y)for allyTx.

Then the following statements hold.

  1. (a)

    There exists a Cauchy sequence { x n } n N in X such that

    1. (i)

      x n + 1 T x n for all nN,

    2. (ii)

      inf n X p( x n , x n + 1 )= lim n p( x n , x n + 1 )= lim n d( x n , x n + 1 )= inf n N d( x n , x n + 1 )=0.

  2. (b)

    inf x X p(x,Tx)= inf x X d(x,Tx)=0; that is, T has p-approximate fixed point property and approximate fixed point property on X.

Proof By Lemma 2.2, it suffices to prove that the conclusions hold under assumption (L1). Let uX be given. If uTu, then

inf x X p(x,Tx)p(u,Tu)p(u,u)=0,

and

inf x X d(x,Tx)d(u,u)=0,

which implies that inf x X p(x,Tx)= inf x X d(x,Tx)=0. Let w n =u for all nN. Thus we have

w n + 1 = u T u = T w n for all  n N , lim n p ( w n , w n + 1 ) = inf n N p ( w n , w n + 1 ) = p ( u , u ) = 0 ,

and

lim n d( w n , w n + 1 )= inf n N d( w n , w n + 1 )=d(u,u)=0.

Clearly,

p( w n + 1 , w n + 2 )=0=φ ( ξ ( p ( w n , w n + 1 ) ) ) p( w n , w n + 1 )for all nN.

So, conclusions (a) and (b) hold in this case uTu, no matter what condition one begins with. Suppose that uTu. Put x 1 =u and x 2 T x 1 . Then x 2 x 1 and hence p( x 1 , x 2 )>0. Assume that condition (L1) is satisfied. Then there exists x 3 T x 2 such that

p( x 2 , x 3 )φ ( ξ ( p ( x 1 , x 2 ) ) ) p( x 1 , x 2 ).

If x 2 = x 3 T x 2 , then, following a similar argument as above, the conclusions are also proved. If x 3 x 2 , then there exists x 4 T x 3 such that

p( x 3 , x 4 )φ ( ξ ( p ( x 2 , x 3 ) ) ) p( x 2 , x 3 ).

By induction, we can obtain a sequence { x n } in X satisfying x n + 1 T x n and

p( x n + 1 , x n + 2 )φ ( ξ ( p ( x n , x n + 1 ) ) ) p( x n , x n + 1 )for all nN.
(2.1)

Since φ(t)<1 for all t[0,), inequality (2.1) implies that the sequence { p ( x n , x n + 1 ) } n N is strictly decreasing in [0,). Hence

lim n p( x n , x n + 1 )= inf n N p( x n , x n + 1 )0 exists.
(2.2)

Since ξ is nondecreasing, { ξ ( p ( x n , x n + 1 ) ) } n N is a nonincreasing sequence in [0,). Since φ is an MT-function, by (g) of Theorem 2.1, we have

0 sup n N φ ( ξ ( p ( x n , x n + 1 ) ) ) <1.

Let λ:= sup n N φ(ξ(p( x n , x n + 1 ))). So λ[0,1) and we get from (2.1) that

p( x n + 1 , x n + 2 )λp( x n , x n + 1 ) λ n p( x 1 , x 2 )for each nN.
(2.3)

Since λ[0,1), lim n λ n =0 and hence the last inequality implies

lim n p( x n , x n + 1 )=0.
(2.4)

By (2.2) and (2.4), we obtain

inf n N p( x n , x n + 1 )= lim n p( x n , x n + 1 )=0.
(2.5)

Now, we claim that { x n } is a Cauchy sequence in X. Let α n = λ n 1 1 λ p( x 1 , x 2 ), nN. For m,nN with m>n, by (2.3), we have

p( x n , x m ) j = n m 1 p( x j , x j + 1 )< α n .

Since λ[0,1), lim n α n =0 and hence

lim n sup { p ( x n , x m ) : m > n } =0.

Applying Lemma 2.1, we show that { x n } is a Cauchy sequence in X. Hence lim n d( x n , x n + 1 )=0. Since inf n N d( x n , x n + 1 )d( x m , x m + 1 ) for all mN and lim m d( x m , x m + 1 )=0, one also obtains

lim n d( x n , x n + 1 )= inf n N d( x n , x n + 1 )=0.
(2.6)

So conclusion (a) is proved. To see (b), since x n + 1 T x n for each nN, we have

inf x X p(x,Tx)p( x n ,T x n )p( x n ,f x n + 1 )
(2.7)

and

inf x X d(x,Tx)d( x n ,T x n )d( x n ,f x n + 1 )
(2.8)

for all nN. Combining (2.6), (2.7) and (2.8), we get

inf x X p(x,Tx)= inf x X d(x,Tx)=0.

The proof is completed. □

The following existence theorem is obviously an immediate result from Theorem 2.2.

Theorem 2.3 Let (X,d) be a metric space, p be a τ 0 -function and T:XN(X) be a multivalued map. Assume that one of (H1) and (H2) is satisfied, where

(H1) there exists an MT-function α:[0,)[0,1) such that for each xX, if yTx with yx, then there exists zTy such that

p(y,z)α ( p ( x , y ) ) p(x,y);

(H2) there exists an MT-function β:[0,)[0,1) such that for each xX,

p(y,Ty)β ( p ( x , y ) ) p(x,y)for allyTx.

Then the following statements hold.

  1. (a)

    There exists a Cauchy sequence { x n } n N in X such that

    1. (i)

      x n + 1 T x n for all nN,

    2. (ii)

      inf n X p( x n , x n + 1 )= lim n p( x n , x n + 1 )= lim n d( x n , x n + 1 )= inf n N d( x n , x n + 1 )=0.

  2. (b)

    inf x X p(x,Tx)= inf x X d(x,Tx)=0; that is, T has p-approximate fixed point property and approximate fixed point property on X.

Lemma 2.3 Let τ:[0,)[0,) be a nondecreasing function and κ:[0,)[0,1) be an MT-function. Then κτ is an MT-function.

Proof Let { x n } n N be a strictly decreasing sequence in [0,). Since τ is a nondecreasing function, { τ ( x n ) } n N is a nonincreasing sequence in [0,). Since κ is an MT-function, by (f) of Theorem 2.1, we get

0 sup n N κ ( τ ( x n ) ) <1,

or, equivalently,

0 sup n N (κτ)( x n )<1.

So, by Theorem 2.1 again, we prove that κτ is an MT-function. □

Applying Lemma 2.3, we conclude that Theorem 2.2 is also a special case of Theorem 2.3. Therefore we obtain the following important fact.

Theorem 2.4 Theorem  2.2 and Theorem  2.3 are equivalent.

3 Short proofs of Theorems 1.1 and 1.2

Let us see how we can utilize Theorem 2.3 to prove Theorem 1.1.

Short proof of Theorem 1.1 Since K is a nonempty closed subset of X and X is complete, (K,d) is also a complete metric space. Let xK. Put k= γ 1 γ and λ= 1 + k 2 . So, 0k<λ<1. Let yTxK be arbitrary. So, d(y,TxK)=0. By (D2), we have d(gy,TxK)=0. Hence inequality (1.1) implies

H(Tx,TyK)γ [ d ( x , T x K ) + H ( T x , T y K ) ] for all yTxK.
(3.1)

Inequality (3.1) shows that

d(y,TyK)H(Tx,TyK)kd(x,TxK)<λd(x,y)for all yTxK.
(3.2)

Define G:KCB(K) by

Gx=TxKfor all xK,

and let μ:[0,)[0,1) be defined by

η(t)=λfor all t[0,).

Then μ is an MT-function. By (3.2), we obtain

d(y,Gy)μ ( d ( x , y ) ) d(x,y)for all yGx.

Applying Theorem 2.3 with pd, there exists a Cauchy sequence { x n } n N in K such that

x n + 1 G x n =T x n Kfor all nN
(3.3)

and

lim n d( x n , x n + 1 )= inf n N d( x n , x n + 1 )=0.
(3.4)

By the completeness of K, there exists vK such that x n v as n. By (3.3) and (D2), we have

g x n + 1 T x n Kfor each nN.
(3.5)

Since g is continuous and lim n x n =v, we have

lim n g x n =gv.
(3.6)

Since the function xd(x,Tv) is continuous, by (1.1), (3.3), (3.4), (3.5) and (3.6), we get

d ( v , T v K ) = lim n d ( x n + 1 , T v K ) lim n H ( T x n , T v K ) lim n { γ [ d ( x n , T x n K ) + d ( v , T x n K ) + d ( v , T v K ) ] + h ( v ) d ( g v , T x n K ) } lim n { γ [ d ( x n , x n + 1 ) + d ( v , x n + 1 ) + d ( v , T v K ) ] + h ( v ) d ( g v , g x n + 1 ) } = γ d ( v , T v K ) ,

which implies d(v,TvK)=0. By the closedness of Tv, we have vTvK. From (D2), gvTvKTv. Hence we verify v COP K (g,T) F K (T). The proof is complete. □

In order to finish off our work, let us prove Theorem 1.2 by applying Theorem 2.3.

Short proof of Theorem 1.2 Since K is a nonempty closed subset of X and X is complete, (K,d) is also a complete metric space. Note first that for each xK, by (D2), we have d(gy,TxK)=0 for all yTxK. So, for each xK, by (1.2), we obtain

d(y,TyK)φ ( d ( x , y ) ) d(x,y)for all yTxK.
(3.7)

Define G:KCB(K) by

Gx=TxKfor all xK.

From (3.7), we obtain

d(y,Gy)φ ( d ( x , y ) ) d(x,y)for all yGx.

By using Theorem 2.3, there exists a Cauchy sequence { x n } n N in K such that

x n + 1 G x n =T x n Kfor all nN
(3.8)

and

lim n d( x n , x n + 1 )= inf n N d( x n , x n + 1 )=0.
(3.9)

By the completeness of K, there exists vK such that x n v as n. Thanks to (3.8) and (D2), we have

g x n + 1 T x n Kfor each nN.
(3.10)

Since g is continuous and lim n x n =v, we have

lim n g x n =gv.
(3.11)

Since the function xd(x,Tv) is continuous, by (1.2), (3.8), (3.10) and (3.11), we get

d ( v , T v K ) = lim n d ( x n + 1 , T v K ) lim n H ( T x n , T v K ) lim n { φ ( d ( x n , v ) ) d ( x n , v ) + h ( v ) d ( g v , T x n K ) } lim n { φ ( d ( x n , v ) ) d ( x n , v ) + h ( v ) d ( g v , g x n + 1 ) } = 0 ,

which implies d(v,TvK)=0. By the closedness of Tv, we have vTvK. By (D2), gvTvKTv and hence v COP K (g,T) F K (T). The proof is complete. □

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Acknowledgements

In this research, the author was supported by grant No. NSC 102-2115-M-017-001 of the National Science Council of the Republic of China.

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Correspondence to Wei-Shih Du.

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Keywords

  • τ-function
  • MT-function (-function)
  • coincidence point
  • fixed point
  • approximate fixed point property