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Mizoguchi-Takahashi’s type common fixed point theorems without T-weakly commuting condition and invariant approximations

Abstract

The purpose of this paper is to introduce a concept of T f -orbitally lower semi-continuous mappings which is more general than the concept of T-orbitally lower semi-continuous mappings and continuous mappings and also prove Mizoguchi-Takahashi’s type coincidence point theorems by using this concept. Moreover, we show that the existence of common fixed points for Mizoguchi-Takahashi’s type multi-valued mappings do not require the condition of T-weakly commuting mappings. Finally, some invariant approximation results are obtained as applications. Our results unify, extend, and complement several well-known results.

MSC:47H10, 54H25.

1 Introduction and preliminaries

Throughout this paper, we denote by the set of all positive integers, by the set of all real numbers, and by R + the set of all nonnegative real numbers.

Let (X,d) be a metric space. We denote by 2 X the class of all nonempty subsets of X, by K(X) the class of all nonempty compact subsets of X, by CL(X) the class of all nonempty closed subsets of X, by CB(X) the class of all nonempty closed bounded subsets of X. A functional H:CL(X)×CL(X) R + {+} is said to be the Pompeiu-Hausdorff generalized metric induced by d is given by

H(A,B)= { max { sup a A d ( a , B ) , sup b B d ( b , A ) } , if the maximum exists ; + , otherwise ,

for all A,BCB(X), where d(a,B)=inf{d(a,b):bB} is the distance from a to BX.

Remark 1.1 The following properties of the Pompeiu-Hausdorff generalized metric induced by d are well known:

  1. (1)

    H is a metric on CB(X).

  2. (2)

    If A,BCB(X) and q>1, then, for all aA, there exists bB such that d(a,b)qH(A,B).

  3. (3)

    (CB(X),H) is a complete metric space provided (X,d) is a complete metric space.

Definition 1.1 Let (X,d) be a metric space, f:XX and T:X 2 X be mappings.

  1. (1)

    A point xX is said to be a fixed point of f (resp., T) if x=fx (resp., xTx). The set of all fixed points of f (resp., T) is denoted by F(f) (resp., F(T)).

  2. (2)

    A point xX is said to be a coincidence point of f and T if fxTx. The set of all coincidence points of f and T is denoted by C(f,T).

  3. (3)

    A point xX is said to be a common fixed point of f and T if x=fxTx. The set of all common fixed points of f and T is denoted by F(f,T).

Definition 1.2 Let (X,d) be a metric space, f:XX and T:X 2 X be mappings.

  1. (1)

    If, for any x 0 X, there exists a sequence { x n } in X such that x n T x n 1 for all nN, then O(T, x 0 ):={ x 0 , x 1 , x 2 ,} is said to be an orbit of T.

  2. (2)

    If, for any x 0 X, there exists a sequence {f x n } in f(X) such that f x n T x n 1 for all nN, then O f (T, x 0 ):={f x 0 ,f x 1 ,f x 2 ,} is said to be an f-orbit of T.

In 1969, Nadler [1] extended the Banach contraction principle to multi-valued mappings as follows.

Theorem 1.1 (Nadler [1])

Let (X,d) be a complete metric space and T:XCB(X) such that

H(Tx,Ty)kd(x,y)
(1.1)

for all x,yX, where k[0,1). Then T has at least one fixed point.

Since the theory of multi-valued mappings has many applications in many areas, a number of authors have focused on the topic and have published some interesting fixed point theorems in this frame. Following this trend, in 1972, Reich [2] extended Theorem 1.1 in the following way.

Theorem 1.2 (Reich [2])

Let (X,d) be a complete metric space and T:XK(X) be a mapping satisfying

H(Tx,Ty)α ( d ( x , y ) ) d(x,y)
(1.2)

for all x,yX, where α:(0,)[0,1) is R-function, that is,

lim sup x t + α(x)<1

for all t(0,). Then T has at least one fixed point.

Furthermore, Reich [2] also raised the following question in his work:

Can the range of T, that is, K(X) , be replaced by CB(X) or CL(X) ?

In 1989, Mizoguchi and Takahashi [3] gave the positive answer for the conjecture of Reich [2], when the inequality holds also for t=0, as follows.

Theorem 1.3 (Mizoguchi and Takahashi [3])

Let (X,d) be a complete metric space and T:XCB(X). Assume that

H(Tx,Ty)α ( d ( x , y ) ) d(x,y)
(1.3)

for all x,yX, where α:[0,)[0,1) is MT-function, that is,

lim sup x t + α(x)<1

for all t[0,). Then T has at least one fixed point.

Remark 1.2 It is well known that, if α:[0,)[0,1) is a nondecreasing function or a nonincreasing function, then α is a MT-function. Therefore, the class of MT-functions is a rich class and so this class has been investigated heavily by many authors.

In 2007, Eldred et al. [4] claimed that Theorem 1.3 is equivalent to Theorem 1.1 in the following sense:

If a mapping T:XCB(X) satisfies (1.3), then there exists a nonempty complete subset M of X satisfying the following:

  1. (1)

    M is T-invariant, that is, TxM for all xM.

  2. (2)

    T satisfies (1.1) for all x,yM.

In the same year, Suzuki [5] produced an example which shows that Mizoguchi-Takahashi’s fixed point theorem for multi-valued mappings is a real generalization of Nadler’s contraction principle. Since the primitive proof of Mizoguchi-Takahashi’s fixed point theorem is quite difficult, Suzuki gave a very simple proof of Mizoguchi-Takahashi’s theorem. Several authors devoted their attention to investigate its generalizations in various different directions of the Mizoguchi-Takahashi’s fixed point theorem (see [614] and references therein).

In 2009, Kamran [15] extended the result of Mizoguchi and Takahashi [3] for closed multi-valued mappings and proved a fixed point theorem by using the concept of T-orbitally lower semi-continuous mappings as follows:

Definition 1.3 ([16])

Let (X,d) be a metric space, T:XCL(X) be a mapping multi-valued, and let ξX.

  1. (1)

    A mapping g:XR is said to be lower semi-continuous at ξ if, for any sequence { x n } in X such that x n ξ as n,

    gξ lim inf n g x n .
  2. (2)

    A mapping g:XR is said to be T-orbitally lower semi-continuous at ξ if, for any sequence { x n } in O(T, x 0 ) such that x 0 X and x n ξ as n,

    gξ lim inf n g x n .

The following result is a main result of Kamran [15].

Theorem 1.4 (Kamran [15])

Let (X,d) be a complete metric space and T:XCL(X) be a mapping satisfying

d(y,Ty)α ( d ( x , y ) ) d(x,y)
(1.4)

for all xX and yTx, where α:[0,)[0,1) is MT-function. Then:

(K1) For each x 0 X, there exist an orbit { x n } of T and ξX such that lim n x n =ξ.

(K2) ξ is a fixed point of T if and only if the function g:XR defined by g(x):=d(x,Tx) for all xX is T-orbitally lower semi-continuous at ξ.

Recently, Ali [17] extended the above result to common fixed point theorem by using the concept of T-weakly commuting as follows:

Definition 1.4 ([18])

Let (X,d) be a metric space, f:XX and T:XCL(X) be mappings. The mapping f is said to be T-weakly commuting at xX if ffxTfx.

The following result is a main result of Ali [17].

Theorem 1.5 (Ali [17])

Let (X,d) be a metric space, f:XX and T:XCL(X) be two mappings such that T(X)f(X) and

d(fy,Ty)α ( d ( f x , f y ) ) d(fx,fy)
(1.5)

for all xX and fyTx, where α:[0,)[0,1) is MT-function. If (f(X),d) is a complete metric space, then

(A1) For any x 0 X, there exists an f-orbit {f x n } of T and fξf(X) such that lim n f x n =fξ.

(A2) ξ is a coincidence point of f and T if and only if the function h:XR defined by h(x):=d(fx,Tx) for all xX is lower semi-continuous at ξ.

(A3) If ffξ=fξ and f is T-weakly commuting at ξ, then f and T have a common fixed point.

In this paper, we introduce the concept of T f -orbitally lower semi-continuous mappings and, using this concept, prove Mizoguchi-Takahashi’s type coincidence point theorems. Also, we show that the condition of ‘T-weakly commuting of f’ can be omit to prove Mizoguchi-Takahashi’s type common fixed point theorems. By the same procedure, we can improve Theorem 1.5 by dropping the condition of ‘f is T-weakly commuting at ξ’ in (A3). As applications, we derive the invariant approximation results.

2 Mizoguchi-Takahashi’s type coincidence and common fixed point theorems

In this section, we start with the following concept.

Definition 2.1 Let (X,d) be a metric space, f:XX, T:XCL(X) be mappings, and let x 0 ,ξX.

  1. (1)

    A mapping h:f(X)R is said to be lower semi-continuous at if, for any sequence {f x n } in f(X) such that f x n fξ as n,

    h(fξ) lim inf n h(f x n ).
  2. (2)

    A mapping h:f(X)R is said to be T f -orbitally lower semi-continuous at if, for any sequence {f x n } in O f (T, x 0 ) such that f x n fξ as n,

    h(fξ) lim inf n h(f x n ).

Next, we apply the following useful lemma due to Haghi et al. [19] and Theorem 1.4 to obtain new Mizoguchi-Takahashi’s type common fixed point theorem.

Lemma 2.1 ([19])

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

The following result is a main result in this paper.

Theorem 2.2 Let (X,d) be a metric space, f:XX and T:XCL(X) be two mappings such that T(X)f(X) and

d(fy,Ty)α ( d ( f x , f y ) ) d(fx,fy)
(2.1)

for all xX and fyTx, where α:[0,)[0,1) is MT-function. If (f(X),d) is a complete metric space, then

(S1) For each x 0 X, there exist an f-orbit {f x n } of T and fξf(X) such that lim n f x n =fξ.

(S2) ξ is a coincidence point of f and T if and only if the function h:f(S)R defined by h(fx):=d(fx,Tx) for all fxf(S) is T f -orbitally lower semi-continuous at , where SX and f | S is one-to-one.

(S3) If ξ is a coincidence point of f and T such that ffξ=fξ, then f and T have a common fixed point.

Proof Let f:XX be a mapping. Using Lemma 2.1, there exists EX such that f(E)=f(X) and f | E is one-to-one. Now, we can define a mapping G:f(E)CL(X) by

G(fx)=Tx
(2.2)

for all xE. Since f | E is one-to-one, it follows that G is well defined. Since T satisfies the contractive condition (2.1), we have

d(fy,Ty)α ( d ( f x , f y ) ) d(fx,fy)
(2.3)

for all xX and fyTx. By the construction of G, we get

d ( f y , G ( f y ) ) α ( d ( f x , f y ) ) d(fx,fy)
(2.4)

for all fxf(E) and fyG(fx). This implies that G is satisfies the contractive condition (1.4). From (S1), it follows that, for each x 0 X, there exist an orbit {f x n } of G and fξf(E) such that lim n f x n =fξ. This implies that (K1) in Theorem 1.4 holds.

Again, by the construction of G, it follows that (S2) is equivalent to the following condition:

is a fixed point of G, that is, fξG(fξ) if and only if the function g:f(E)R defined by g(fx)=d(fx,G(fx)) for all fxf(E) is G-orbitally lower semi-continuous at .’

Thus (S2) holds. Let ξ is a coincidence point of f and T, that is, fξTξ. Next, we suppose that ffξ=fξ. Let z:=fξ and so z=fξ=ffξ=fzTξ. Since fzTξ, it follows from the contractive condition (2.1) that

d ( f z , T z ) α ( d ( f ξ , f z ) ) d ( f ξ , f z ) = α ( 0 ) 0 = 0 ,

which shows that fzTz. Therefore, z=fzTz, that is, z is a common fixed point of f and T. This completes the proof. □

Remark 2.1 Theorem 2.2 generalizes many results in the following sense:

  1. (1)

    The inequality (2.1) is weaker than some kinds of the contractive conditions such as Mizoguchi-Takahashi’s contractive condition [3], Nadler’s contractive condition [1], Kamran’s contractive condition [15], etc.

  2. (2)

    The range of T is CL(X) which is more general than CB(X).

  3. (3)

    For the existence of coincidence point, we merely require that the condition in (S2), whereas other result demands stronger than this condition.

  4. (4)

    For the existence of common fixed point, we only requires the condition ffξ=fξ, whereas Theorem 1.5 requires both of this condition and the ‘T-weakly commuting at ξ’ condition.

Consequently, Theorem 2.2 extends and improves Nadler’s contraction principle [1], Mizoguchi-Takahashi’s theorem [3], Theorem 2.1 of Kamran [15], Theorem 2.2 of Ali [17], and several results in the literature. Moreover, for the single-valued case, Theorem 2.2 also unifies Banach’s contraction principle [20] and many well-known results.

Corollary 2.3 Let (X,d) be a metric space, f:XX and T:XCL(X) be two mappings such that T(X)f(X) and

H(Tx,Ty)α ( d ( f x , f y ) ) d(fx,fy)
(2.5)

for all xX and fyTx, where α:[0,)[0,1) is an MT-function. If (f(X),d) is a complete metric space, then

(S1) For each x 0 X, there exist an f-orbit {f x n } of T and fξf(X) such that lim n f x n =fξ.

(S2) ξ is a coincidence point of f and T if and only if the function h:f(S)R defined by h(fx):=d(fx,Tx) for all fxf(S) is T f -orbitally lower semi-continuous at , where SX and f | S is one-to-one.

(S3) If ξ is a coincidence point of f and T such that ffξ=fξ, then f and T have a common fixed point.

Proof Since d(fy,Ty)H(Tx,Ty) for all fyTx, it follows from the contractive condition (2.5) that the inequality (2.1) holds. Therefore, we get the result. □

Corollary 2.4 Let (X,d) be a metric space, f:XX and T:XCL(X) be two mappings such that T(X)f(X) and

H(Tx,Ty)α ( d ( f x , f y ) ) d(fx,fy)
(2.6)

for all x,yX, where α:[0,)[0,1) is an MT-function. If (f(X),d) is a complete metric space, then:

(S1) For each x 0 X, there exist an f-orbit {f x n } of T and fξf(X) such that lim n f x n =fξ.

(S2) ξ is a coincidence point of f and T if and only if the function h:f(S)R defined by h(fx):=d(fx,Tx) for all fxf(S) is T f -orbitally lower semi-continuous at , where SX and f | S is one-to-one.

(S3) If ξ is a coincidence point of f and T such that ffξ=fξ, then f and T have a common fixed point.

Proof Since the condition (2.6) implies the condition (2.5), we get the result. □

If we take α(t)=k for all t[0,), where k is constant number with k[0,1), then we get the following result.

Corollary 2.5 Let (X,d) be a metric space, f:XX and T:XCL(X) be two mappings such that T(X)f(X) and satisfying

d(fy,Ty)kd(fx,fy),
(2.7)

for each xX and fyTx, where k[0,1). If (f(X),d) is a complete metric space, then:

(S1) For each x 0 X, there exist an f-orbit {f x n } of T and fξf(X) such that lim n f x n =fξ.

(S2) ξ is a coincidence point of f and T if and only if the function h:f(S)R defined by h(fx):=d(fx,Tx) for all fxf(S) is T f -orbitally lower semi-continuous at , where SX and f | S is one-to-one.

(S3) If ξ is a coincidence point of f and T such that ffξ=fξ, then f and T have a common fixed point.

Corollary 2.6 Let (X,d) be a metric space, f:XX and T:XCL(X) be two mappings such that T(X)f(X) and

H(Tx,Ty)kd(fx,fy)
(2.8)

for all xX and fyTx, where k[0,1). If (f(X),d) is a complete metric space, then:

(S1) For each x 0 X, there exist an f-orbit {f x n } of T and fξf(X) such that lim n f x n =fξ;

(S2) ξ is a coincidence point of f and T if and only if the function h:f(S)R defined by h(fx):=d(fx,Tx) for all fxf(S) is T f -orbitally lower semi-continuous at , where SX and f | S is one-to-one;

(S3) If ξ is a coincidence point of f and T such that ffξ=fξ, then f and T have a common fixed point.

Corollary 2.7 Let (X,d) be a metric space, f:XX and T:XCL(X) be two mappings such that T(X)f(X) and

H(Tx,Ty)kd(fx,fy)
(2.9)

for all x,yX, where k[0,1). If (f(X),d) is a complete metric space, then:

(S1) For each x 0 X, there exist an f-orbit {f x n } of T and fξf(X) such that lim n f x n =fξ.

(S2) ξ is a coincidence point of f and T if and only if the function h:f(S)R defined by h(fx):=d(fx,Tx) for all fxf(S) is T f -orbitally lower semi-continuous at , where SX and f | S is one-to-one.

(S3) If ξ is a coincidence point of f and T such that ffξ=fξ, then f and T have a common fixed point.

3 Invariant approximation results

Several problems concerning invariant approximations for self-mappings were obtained as applications of fixed point, coincidence point, and common fixed point results (see [2127] and references therein). Also, Kamran [18], Latif and Bano [28], and O’Regan and Shahzad [29, 30] obtained invariant approximation results for multi-valued mappings.

In this section, we study invariant approximation results for nonlinear single-valued mapping and multi-valued mapping as applications of main results in Section 2.

Let M be a subset of a normed space E and pE. The set

Best M (p):= { x M : x p = d ( p , M ) }

is called the set of best M-approximants to pX out of M, where d(p,M)= inf y M yp.

Here, we derive some invariant approximation results.

Theorem 3.1 Let M be subset of normed space (E,), pE, f:MM be a mapping and T:MCL(M) be a multi-valued mappings such that

d(fy,Ty)α ( f x f y ) fxfy
(3.1)

for each x B M (p) and fyTx, where α:[0,)[0,1) is an MT-function. Suppose that the following conditions hold:

  1. (1)

    T( Best M (p))f( Best M (p))= Best M (p).

  2. (2)

    f( Best M (p)) is a complete subspace of M.

  3. (3)

    f | Best M ( p ) is one-to-one.

  4. (4)

    sup y T x ypfxp for all x Best M (p).

Then we have the following:

(S1) For each x 0 Best M (p), there exists an f-orbit {f x n } of T and fξf( Best M (p)) such that lim n f x n =fξ.

(S2) ξC(f,T) Best M (p) if and only if the function h:f( Best M (p))R defined by h(fx):=d(fx,Tx) for all fxf( Best M (p)) is T f -orbitally lower semi-continuous at .

(S3) If ξC(f,T) Best M (p) such that ffξ=fξ, then fξF(f,T) Best M (p).

Proof From the assumption, it follows that f | Best M ( p ) is a single-valued mapping from Best M (p) to Best M (p). Now, we show that T | Best M ( p ) is a multi-valued mapping from Best M (p) to CL( Best M (p)). First, we claim that Tx Best M (p) for all x Best M (p). Let x Best M (p) and zTx. Since f( Best M (p))= Best M (p), we have fx Best M (p) and hence fxp=d(p,M).

Now, we obtain

d(p,M)zp sup y T x ypfxp=d(p,M).

This implies that zp=d(p,M) and thus z Best M (p). Therefore, Tx Best M (p) for all x Best M (p). Since Tx is closed for all xM, it follows that Tx is closed for all x Best M (p). Hence T | Best M ( p ) is a multi-valued mapping from Best M (p) to CL( Best M (p)). It is easy to obtain that

F(f | Best M ( p ) ,T | Best M ( p ) )=F(f,T) Best M (p).

Thus the result follows from Theorem 2.2 with X= Best M (p). This completes the proof. □

Theorem 3.2 Let M be subset of normed space (E,), pE, and T:MCL(M) be a multi-valued mapping such that

d(y,Ty)α ( x y ) xy
(3.2)

for all x Best M (p) and yTx, where α:[0,)[0,1) is an MT-function. Suppose that the following conditions hold:

  1. (1)

    T( Best M (p)) Best M (p);

  2. (2)

    Best M (p) is complete subspace of M;

  3. (3)

    sup y T x ypxp for all x Best M (p).

Then we have the following:

(S1) For each x 0 Best M (p), there exists an orbit { x n } of T and ξ Best M (p) such that lim n x n =ξ;

(S2) ξF(T) Best M (p) if and only if the function g: Best M (p)R, defined by g(x):=d(x,Tx) for all xf( Best M (p)), is T-orbitally lower semi-continuous at ξ.

Proof Take f as the identity mapping from M into M in Theorem 3.1, we get the result. □

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Acknowledgements

The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (KRF-2013053358).

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Correspondence to Wutiphol Sintunavarat or Yeol Je Cho.

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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 manuscript.

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Keywords

  • Mizoguchi-Takahashi’s type common fixed point theorem
  • T-weakly commuting mapping
  • -function
  • MT-function
  • invariant approximation