Open Access

Fixed point theory for multivalued φ-contractions

Fixed Point Theory and Applications20112011:50

https://doi.org/10.1186/1687-1812-2011-50

Received: 15 March 2011

Accepted: 9 September 2011

Published: 9 September 2011

Abstract

The purpose of this paper is to present a fixed point theory for multivalued φ-contractions using the following concepts: fixed points, strict fixed points, periodic points, strict periodic points, multivalued Picard and weakly Picard operators; data dependence of the fixed point set, sequence of multivalued operators and fixed points, Ulam-Hyers stability of a multivalued fixed point equation, well-posedness of the fixed point problem, limit shadowing property of a multivalued operator, set-to-set operatorial equations and fractal operators. Our results generalize some recent theorems given in Petruşel and Rus (The theory of a metric fixed point theorem for multivalued operators, Proc. Ninth International Conference on Fixed Point Theory and its Applications, Changhua, Taiwan, July 16-22, 2009, 161-175, 2010).

2010 Mathematics Subject Classification

47H10; 54H25; 47H04; 47H14; 37C50; 37C70

Keywords

successive approximationsmultivalued operatorPicard operatorweakly Picard operatorfixed pointstrict fixed pointperiodic pointstrict periodic pointmultivalued weakly Picard operatormultivalued Picard operatordata dependencefractal operatorlimit shadowingset-to-set operatorUlam-Hyers stabilitysequence of operators

1 Introduction

Let X be a nonempty set. Then, we denote
P ( X ) : = { Y X | Y } , P c l ( X ) : = { Y P ( X ) | Y i s c l o s e d } .

If T : Y X → P(X) is a multivalued operator, then F T := {x Y | x T(x)} denotes the fixed point set T, while (S F) T := {x Y | {x} = T (x)} is the strict fixed point set of T.

Recall now two important notions, see [1] for details. A mapping φ : + + is said to be a comparison function if it is increasing and φ k (t) 0, as k → +∞. As a consequence, we also have φ(t) < t, for each t > 0, φ(0) = 0 and φ is continuous in 0.

A comparison function φ : + + having the property that t - φ (t) +∞, as t → +∞ is said to be a strict comparison function.

Moreover, a function φ : + + is said to be a strong comparison function if it is strictly increasing and n = 1 φ n ( t ) < + , for each t > 0.

If (X, d) is a metric space, then we denote by H the Pompeiu-Hausdorff generalized metric on P cl (X). Then, T : X → P cl (X) is called a multivalued φ-contraction, if φ : + + is a strong comparison function, and for all x1, x2 X, we have that
H ( T ( x 1 ) , T ( x 2 ) ) φ ( d ( x 1 , x 2 ) ) .

The purpose of this paper is to present a fixed point theory for multivalued φ-contractions in terms of the following:

  • fixed points, strict fixed points, periodic points ([217]);

  • multivalued weakly Picard operators ([18]);

  • multivalued Picard operators ([19]);

  • data dependence of the fixed point set ([18, 2022]);

  • sequence of multivalued operators and fixed points ([23, 24]);

  • Ulam-Hyers stability of a multivalued fixed point equation ([25]);

  • well-posedness of the fixed point problem ([26, 27]);

  • limit shadowing property of a multivalued operator ([28]);

  • set-to-set operatorial equations ([2931]);

  • fractal operators ([3240]).

2 Notations and basic concepts

Throughout this paper, the standard notations and terminologies in non-linear analysis are used, see for example Kirk and Sims [41], Petruşel [42], Rus et al. [18, 43]. See also [4452].

Let X be a nonempty set. Then, we denote
P ( X ) : = { Y | Y is a subset of X } , P ( X ) : = { Y P ( X ) | Y is nonempty } .
Let (X, d) be a metric space. Then δ(Y ) := sup {d(a, b)| a, b Y} and
P b ( X ) : = { Y P ( X ) | δ ( Y ) < + } , P c l ( X ) : = { Y P ( X ) | Y i s c l o s e d } , P c p ( X ) : = { Y P ( X ) | Y i s c o m p a c t } , P o p ( X ) : = { Y P ( X ) | Y i s o p e n } .
Let T : X → P(X) be a multivalued operator. Then, the operator T ^ : P ( X ) P ( X ) defined by
T ^ ( Y ) : = x Y T ( x ) , f o r Y P ( X )

is called the fractal operator generated by T.

For the continuity of concepts with respect to multivalued operators, we refer to [44, 45], etc.

It is known that if (X, d) is a metric spaces and T : X → P cp (X), then the following conclusions hold:
  1. (a)

    if T is upper semicontinuous, then T (Y) P cp (X), for every Y P cp (X);

     
  2. (b)

    the continuity of T implies the continuity of T ^ : P c p ( X ) P c p ( X ) . A sequence of successive approximations of T starting from x X is a sequence (x n )nof elements in X with x 0 = x, x n+1 T (x n ), for n .

     

If T : Y X → P(X), then F T := {x Y | x T (x)} denotes the fixed point set T, while (SF) T := {x Y | {x} = T (x)} is the strict fixed point set of T. By Graph(T) := {(x, y) Y × × : y T(x)}, we denote the graphic of the multivalued operator T.

If T : X → P(X), then T0 := 1 X , T1 := T,..., Tn+1= T T n , n denote the iterate operators of T.

By definition, a periodic point for a multivalued operator T : X → P cp (X) is an element p X such that p F T m , for some integer m ≥ 1, i.e., p T ^ m ( { p } ) for some integer m ≥ 1.

The following (generalized) functionals are used in the main sections of the paper.

The gap functional
( 1 ) D : P ( X ) × P ( X ) + { + } D ( A , B ) = inf { d ( a , b ) | a A , b B } , 0 , + , A B A = = B o t h e r w i s e
The excess generalized functional
( 2 ) ρ : P ( X ) × P ( X ) + { + } ρ ( A , B ) = sup { D ( a , B ) | a A } , 0 , + , A B A = B = A
The Pompeiu-Hausdorff generalized functional.
( 3 ) H : P ( X ) × P ( X ) + { + } H ( A , B ) = max { ρ ( A , B ) , ρ ( B , A ) } , 0 , + , A B A = = B o t h e r w i s e

For other details and basic results concerning the above notions, see, for example, [2, 41, 4450].

We recall now the notion of multivalued weakly Picard operator.

Definition 2.1. (Rus et al. [18]) Let (X, d) be a metric space. Then, T : X → P (X) is called a multivalued weakly Picard operator (briefly MWP operator) if for each x X and each y T(x) there exists a sequence (x n )nin X such that:
  1. (i)

    x 0 = x, x 1 = y;

     
  2. (ii)

    x n+1 T (x n ), for all n ;

     
  3. (iii)

    the sequence (x n )nis convergent and its limit is a fixed point of T.

     

Definition 2.2. Let (X, d) be a metric space and T : X → P (X) be a MWP operator. Then, we define the multivalued operator T : Graph(T) → P(F T ) by the formula T(x, y) = { z F T | there exists a sequence of successive approximations of T starting from (x, y) that converges to z }.

Definition 2.3. Let (X, d) be a metric space and T : X → P (X) a MWP operator. Then, T is said to be a ψ-multivalued weakly Picard operator (briefly ψ-MWP operator) if and only if ψ : + + is a continuous in t = 0 and increasing function such that ψ(0) = 0, and there exists a selection t of T such that
d ( x , t ( x , y ) ) ψ ( d ( x , y ) ) , f o r a l l ( x , y ) G r a p h ( T ) .

In particular, if ψ(t) := ct, for each t + (for some c > 0), then T is called c-MWP operator, see Petruşel and Rus [26]. See also [53, 54].

We recall now the notion of multivalued Picard operator.

Definition 2.4. Let (X, d) be a complete metric space and T : X → P (X). By definition, T is called a multivalued Picard operator (briefly MP operator) if and only if:
  1. (i)

    (S F) T = F T = {x*};

     
  2. (ii)

    T n ( x ) H { x * } as n → ∞, for each x X.

     

For basic notions and results on the theory of weakly Picard and Picard operators, see [42, 43, 53, 54].

The following lemmas will be useful for the proof of the main results.

Lemma 2.5. ([1, 18]) Let (X, d) be a metric space and A, B P cl (X). Suppose that there exists η > 0 such that for each a A there exists b B such that d(a, b) ≤ η] and for each b B there exists a A such that d(a, b) ≤ η]. Then, H(A, B) ≤ η.

Lemma 2.6. ([1, 18]) Let (X, d) be a metric space and A, B P cl (X). Then, for each q > 1 and for each a A there exists b B such that d(a, b) < qH(A, B).

Lemma 2.7. (Generalized Cauchy's Lemma) (Rus and Şerban [55]) Let φ : + +be a strong comparison function and (b n )nbe a sequence of non-negative real numbers, such that limn→+∞b n = 0. Then,
lim n + k = 0 n φ n - k ( b k ) = 0 .

The following result is known in the literature as Matkowski-Rus's theorem (see [1]).

Theorem 2.8 Let (X, d) be a complete metric space and f : X → × be a φ-contraction, i.e., φ : + +is a comparison function and
d ( f ( x ) , f ( y ) ) φ ( d ( x , y ) ) f o r a l l x , y X .

Then f is a Picard operator, i.e., f has a unique fixed point x* X and limn→+∞f n (x) = x*, for all × X.

Finally, let us recall the concept of H-convergence for sets. Let (X, d) be a metric space and (A n )nbe a sequence in P cl (X). By definition, we will write A n H A * P c l ( X ) as n → ∞ if and only if H(A n , A*) 0 as n → ∞.

3 A fixed point theory for multivalued generalized contractions

Our first result concerns the case of multivalued φ-contractions.

Theorem 3.1. Let (X, d) be a complete metric space and T : XP cl (X) be a multivalued φ-contraction. Then, we have:

(i) (Existence of the fixed point) T is a MWP operator;

(ii) If additionally φ(qt) ≤ (t) for every t + (where q > 1) and t = 0 is a point of uniform convergence for the series n = 1 φ n ( t ) , then T is a ψ-MWP operator, with ψ(t) := t + s(t), for each t +(where s ( t ) : = n = 1 φ n ( t ) );

(iii) (Data dependence of the fixed point set) Let S : X → P cl (X) be a multivalued φ-contraction and η > 0 be such that H(S(x), T(x)) ≤ η, for each × X. Suppose that φ(qt) ≤ qφ (t) for every t +(where q > 1) and t = 0 is a point of uniform convergence for the series n = 1 φ n ( t ) . Then, H(F S , F T ) ≤ ψ(η);

(iv) (sequence of operators) Let T, T n : XP cl (X), n be multivalued φ-contractions such that T n ( x ) H T ( x ) as n → +∞, uniformly with respect to each × X. Then, F T n H F T as n → +∞.

If, moreover T(x) P cp (X), for each × X, then we additionally have:

(v) (generalized Ulam-Hyers stability of the inclusion × T(x)) Let ε > 0 and × X be such that D(x, T(x)) ε. Then there exists x* F T such that d(x, x*) ≤ ψ(ε);

(vi) T is upper semicontinuous, T ^ : ( P c p ( X ) , H ) ( P c p ( X ) , H ) , T ^ ( Y ) : = x Y T ( x ) is a set-to-set φ-contraction and (thus) F T ^ = { A T * } ;

(vii) T n ( x ) H A T * as n → +∞, for each × X;

(viii) F T A T * and F T is compact;

(ix) A T * = n * T n ( x ) , for each x F T .

Proof. (i) This is Węgrzyk's Theorem, see [56].
  1. (ii)
    Let x 0 X and x 1 T (x 0) be arbitrarily chosen. We may suppose that x 0x 1. Denote t 0 := d(x 0, x 1) > 0. Then, for any q > 1 there exists x 2 T(x 1) such that d(x 1, x 2) < qH(T (x 0), T (x 1)) ≤ qφ(t 0). We may again suppose that x 1x 2. Thus, φ(d(x 1, x 2)) < φ((t 0)). Next, there exists x 3 T(x 2) such that d ( x 2 , x 3 ) < φ ( q φ ( t 0 ) ) φ ( d ( x 1 , x 2 ) ) H ( T ( x 1 ) , T ( x 2 ) ) φ ( q φ ( t 0 ) ) φ ( d ( x 1 , x 2 ) ) φ ( d ( x 1 , x 2 ) ) q φ 2 ( t 0 ) . By an inductive procedure, we obtain a sequence of successive approximations for T starting from (x 0, x 1) Graph(T) such that
    d ( x n , x n + 1 ) q φ n ( t 0 ) , f o r e a c h n * .
     
Denote by
s n ( t ) : = k = 1 n φ k ( t ) , f o r e a c h t > 0 .
Then, d(x n , xn+p) ≤ q(φ n (t0) ++ φn+p−1(t0)), for each n, p *. If we set s0(t) := 0 for each t +, then
d ( x n , x n + p ) q ( s n + p - 1 ( t 0 ) - s n - 1 ( t 0 ) ) , f o r e a c h n , p * .
(3.1)
By (3.1) we get that the sequence (x n )nis Cauchy and hence it is convergent in (X, d) to some x* X. Notice that, by the φ-contraction condition, we immediately get that Graph(T) is closed in X × X. Hence, x* F T . Then, by (3.1) letting p → + , we obtain that
d ( x n , x * ) q ( s ( t 0 ) - s n - 1 ( t 0 ) ) , f o r e a c h n * .
(3.2)
If we put n = 1 in (3.2), we obtain that d(x1, x*) ≤ qs(t0). Hence,
d ( x 0 , x * ) d ( x 0 , x 1 ) + d ( x 1 , x * ) t 0 + q s ( t 0 ) .
(3.3)
Finally, letting q 1 in (3.3), we get that
d ( x 0 , x * ) t 0 + s ( t 0 ) = ψ ( t 0 ) = ψ ( d ( x 0 , x 1 ) ) .
(3.4)

Notice that, ψ is increasing (since φ is), ψ(0) = 0 and, since t = 0 is a point of uniform convergence for the series n = 1 φ n ( t ) , ψ is continuous in t = 0.

These, together with (3.4), prove that T is a ψ-MWP operator.
  1. (iii)
    Let x 0 F S be arbitrary chosen. Then, by (ii), we have that
    d ( x 0 , t ( x 0 , x 1 ) ) ψ ( d ( x 0 , x 1 ) ) , f o r e a c h x 1 T ( x 0 ) . o
     
Let q > 1 be arbitrary. Then, there exists x1 T (x0) such that d(x0, x1) < qH(S(x0), T (x0)). Then
d ( x 0 , t ( x 0 , x 1 ) ) ψ ( q H ( S ( x 0 ) , T ( x 0 ) ) ) q ψ ( H ( S ( x 0 ) , T ( x 0 ) ) ) q ψ ( η ) .
By a similar procedure we can prove that, for each y0 F T , there exists y1 S(y0) such that
d ( y 0 , s ( y 0 , y 1 ) ) q ψ ( η ) .
By the above relations and using Lemma 2.5, we obtain that
H ( F S , F T ) q ψ ( η ) , w h e r e q > 1 .
Letting q 1, we get the conclusion.
  1. (iv)
    Let ε > 0. Since T n ( x ) H T ( x ) as n+∞, uniformly with respect to each x X, there exists N ε such that
    sup x X H ( T n ( x ) , T ( x ) ) < ε , f o r e a c h n N ε .
     
Then, by (iii) we get that H ( F T n , F T ) ψ ( ε ) , for each n ≥ N ε . Since ψ is continuous in 0 and ψ(0) = 0, we obtain that F T n H F T .
  1. (v)
    Let ε > 0 and x X be such that D(x, T(x)) ≤ ε. Then, since T(x) is compact, there exists y T(x) such that d(x, y) ≤ ε. By the proof of (i), we have that
    d ( x , t ( x , y ) ) ψ ( d ( x , y ) ) .
     
Since x* := t (x, y) F T , we get the desired conclusion d(x, x*) ≤ ψ(ε).
  1. (vi)
    (Andres-Górniewicz [39], Chifu and Petruşel [40].) By the φ-contraction condition, one obtain that the operator T is H-upper semicontinuos. Since T(x) is compact, for each x X, we know that T is upper semicontinuous if and only if T is H-upper semicontinuous. We will prove now that
    H ( T ( A ) , T ( B ) ) φ ( H ( A , B ) ) , f o r e a c h A , B P c p ( X ) .
     
For this purpose, let A, B P cp (X) and let u T (A). Then, there exists a A such that u T(a). For a A, by the compactness of the sets A, B there exists b B such that
d ( a , b ) H ( A , B ) .
(3.5)
Then, we have D(u, T(B)) ≤ D(u, T(b)) ≤ H(T(a), T(b)) ≤ φ(d(a, b)). Hence, by the above relation and by (3.5) we get
ρ ( T ( A ) , T ( B ) ) φ ( d ( a , b ) ) φ ( H ( A , B ) ) .
(3.6)
By a similar procedure, we obtain
ρ ( T ( B ) , T ( A ) ) φ ( d ( a , b ) ) φ ( H ( A , B ) ) .
(3.7)
Thus, (3.6) and (3.7) together imply that
H ( T ( A ) , T ( B ) ) φ ( H ( A , B ) ) .
Hence, T ^ is a self-φ-contraction on the complete metric space (P cp (X), H)). By the φ-contraction principle for singlevalued operators (see Theorem 2.8), we obtain:
  1. (a)

    F T ^ = { A T * }

     
and
  1. (b)

    T ^ n ( A ) H A T * as n → +, for each A P cp (X).

     
  2. (vii)

    By (vi)-(b) we get that T n ( { x } ) = T ^ n ( { x } ) H A T * as n → +, for each x X.

     

(viii)-(ix) (Chifu and Petruşel [40].) Let x F T be arbitrary. Then, x T(x) T2(x) T n (x) Hence x T n (x), for each n *. Moreover, lim n + T n ( x ) = n * T n ( x ) . By (vii), we immediately get that A T * = n * T n ( x ) . Hence, x n * T n ( x ) = A T * . The proof is complete. ■

A second result for multivalued φ-contractions is as follows.

Theorem 3.2. Let (X, d) be a complete metric space and T : X → P cl (X) be a multivalued φ-contraction with (SF) T . Then, the following assertions hold:

(x) F T = (SF) T = {x*};

(xi) If, additionally T(x) is compact for each × X, then F T n = ( S F ) T n = { x * } for n *;

(xii) If, additionally T(x) is compact for each × X, then T n ( x ) H { x * } as n → +∞, for each x X;

(xiii) Let S : X → P cl (X) be a multivalued operator and η > 0 such that F S and H(S(x), T(x)) ≤ η, for each × X. Then, H(F S , F T ) ≤ β(η), where β : + +is given by β(η) := sup{t +| t - φ(t) ≤ η};

(xiv) Let T n : X → P cl (X), n be a sequence of multivalued operators such that F T n for each n and T n ( x ) H T ( x ) as n → +∞, uniformly with respect to × X. Then, F T n H F T as n → +.

(xv) (Well-posedness of the fixed point problem with respect to D) If (x n )n is a sequence in × such that D(x n , T (x n )) 0 as n → ∞, then x n d x * as n → ∞;

(xvi) (Well-posedness of the fixed point problem with respect to H) If (x n )nis a sequence in × such that H(x n , T (x n )) 0 as n → ∞, then x n d x * as n → ∞;

(xvii) (Limit shadowing property of the multivalued operator) Suppose additionally that φ is a sub-additive function. If (y n )nis a sequence in × such that D(yn+1, T(y n )) 0 as n → ∞, then there exists a sequence (x n )n X of successive approximations for T, such that d(x n , y n ) 0 as n → ∞.

Proof. (x) Let x* (SF) T . Notice first that (SF) T = {x*}. Indeed, if y (SF) T with yx*, then d(x*, y) = H(T(x*), T(y)) ≤ φ(d(x*, y)). By the properties of φ, we immediately get that y = x*. Suppose now that y F T . Then,
d ( x * , y ) = D ( T ( x * ) , y ) H ( T ( x * ) , T ( y ) ) φ ( d ( x * , y ) ) .
Thus, y = x*. Hence, F T (SF) T . Since (SF) T F T , we get that (SF) T = F T .
  1. (xi)
    Notice first that x * ( S F ) T n F T n , for each n *. Consider y ( S F ) T n , for arbitrary n *. Then, by (vi) we have that
    d ( x * , y ) = H ( T n ( x * ) , T n ( y ) ) φ ( H ( T n - 1 ( x * ) , T n - 1 ( y ) ) ) φ n ( d ( x * , y ) ) .
     
Thus, y = x* and ( S F ) T n = { x * } . Consider now y F T n . Then, we have
d ( x * , y ) = D ( T n ( x * ) , y ) H ( T n ( x * ) , T n ( y ) ) (1) φ ( H ( T n - 1 ( x * ) , T n - 1 ( y ) ) ) φ n ( d ( x * , y ) ) . (2) (3)
Thus, y = x* and hence T n ( x ) H { x * } .
  1. (xii)
    Let x X be arbitrarily chosen. Then, we have
    H ( T n ( x ) , x * ) = H ( T n ( x ) , T n ( x * ) ) φ ( H ( T n 1 ( x ) , T n 1 ( x * ) ) ) φ ( n d ( x , x * ) ) 0 as n + .
     
  2. (xiii)
    Let y F S . Then,
    d ( y , x * ) H ( S ( y ) , x * ) H ( S ( y ) , T ( y ) ) + H ( T ( y ) , x * ) η + φ ( d ( y , x * ) ) .
     
Thus, d(y, x*) ≤ β(η). The conclusion follows now by the following relations
H ( F S , F T ) = sup y F S d ( y , x * ) β ( η ) .
  1. (xiv)

    follows by (xiii).

     
  2. (xv)
    ([26, 27]) Let (x n )nbe a sequence in X such that D(x n , T (x n )) → 0 as n → ∞. Then,
    d ( x n , x * ) D ( x n , T ( x n ) ) + H ( T ( x n ) , T ( x * ) ) (1) D ( x n , T ( x n ) ) + φ ( d ( x n , x * ) ) . (2) (3)
     
Then
d ( x n , x * ) β ( D ( x n , T ( x n ) ) ) 0 a s n + .
  1. (xvi)

    follows by (xv).

     
  2. (xvii)

    Let (y n )nbe a sequence in X such that D(y n+1, T (y n )) → 0 as n → ∞. Then, there exists u n T (y n ), n such that d(y n+1, u n ) → 0 as n → +∞.

     
We shall prove that d(y n , x*) → 0 as n → +∞. We successively have:
d ( x * , y n + 1 ) H ( x * , T ( y n ) ) + D ( y n + 1 , T ( y n ) ) (1) φ ( d ( x * , y n ) ) + D ( y n + 1 , T ( y n ) ) (2) φ ( φ ( d ( x * , y n - 1 ) ) + D ( y n , T ( y n - 1 ) ) ) + D ( y n + 1 , T ( y n ) ) (3) φ 2 ( d ( x * , y n - 1 ) ) + φ ( D ( y n , T ( y n - 1 ) ) ) + D ( y n + 1 , T ( y n ) ) (4) . . . φ n + 1 ( d ( x * , y 0 ) ) + φ n ( D ( y 1 , T ( y 0 ) ) ) (5) + + D ( y n + 1 , T ( y n ) ) . (6) (7)

By the generalized Cauchy's Lemma, the right-hand side tends to 0 as n → +∞. Thus, d(x*, yn+1) → 0 as n → +∞.

On the other hand, by the proof of Theorem 3.1 (i)-(ii), we know that there exists a sequence (x n )nof successive approximations for T starting from arbitrary (x0, x1) Graph(T ) which converge to a fixed point x* X of the operator T. Since the fixed point is unique, we get that d(x n , x*) → 0 as n → +∞. Hence, for such a sequence (x n )n, we have
d ( y n , x n ) d ( y n , x * ) + d ( x * , x n ) 0 a s n + .

The proof is complete. ■

A third result for multivalued φ-contraction is the following.

Theorem 3.3. Let (X, d) be a complete metric space and T : X → P cp (X) be a multivalued φ-contraction such that T(F T ) = F T . Then, we have:

(xviii) T n ( x ) H F T as n → +∞, for each × X;

(xix) T(x) = F T , for each × F T ;

(xx) If (x n )n X is a sequence such that x n d x * F T as n → ∞, then T n ( x ) H F T as n → +.

Proof. (xviii) By T(F T ) = F T and Theorem 3.1 (vi), we have that F T = A T * . The conclusion follows by Theorem 3.1 (vii).
  1. (xix)

    Let x F T be arbitrary. Then, x T(x) and thus F T T(x). On the other hand T(x) T(F T ) F T . Thus, T(x) = F T , for each x F T .

     
  2. (xx)

    Let (x n )n X is a sequence such that x n d x * F T as n → +∞.

     
Then, we have:
H ( T ( x n ) , F T ) = H ( T ( x n ) , T ( x * ) ) φ ( d ( x n , x * ) ) 0 a s n + .

The proof is complete. ■

For compact metric spaces, we have:

Theorem 3.4. Let (X, d) be a compact metric space and T : X → P cl (X) be a multivalued φ-contraction. Then, we have:

(xxi) (Generalized well-posedness of the fixed point problem with respect to D) If (x n )nis a sequence in × such that D(x n , T (x n )) 0 as n → ∞, then there exists a subsequence ( x n i ) i of ( x n ) n x n i d x * F T as i → ∞.

Proof. (xxi) Let (x n )nis a sequence in X such that D(x n , T (x n )) 0 as n → ∞. Let ( x n i ) i be a subsequence of (x n )nsuch that x n i d x * as i → ∞. Then, there exists y n i T ( x n i ) , i such that y n i d x * as i → ∞. By the φ-contraction condition, we have that T has closed graph. Hence, x* F T . ■

Remark 3.1. For the particular case φ(t) = at (with a [0, 1[), for each t + see Petruşel and Rus [57].

Recall now that a self-multivalued operator T : X → P cl (X) on a metric space (X, d) is called (ε, φ)-contraction if ε > 0, φ : + + is a strong comparison function and
x , y X w i t h x y a n d d ( x , y ) < ε i m p l i e s H ( T ( x ) , T ( y ) ) φ ( d ( x , y ) ) .

Then, for the case of periodic points we have the following results.

Theorem 3.5. Let (X, d) be a metric space and T : X → P cp (X) be a continuous (ε, φ)-contraction. Then, the following conclusions hold:

(i) T ^ m : P c p ( X ) P c p ( X ) is a continuous (ε, φ)-contraction, for each m *;

(ii) if, additionally, there exists some A P cp (X) such that a sub-sequence ( T ^ m i ( A ) ) i * of ( T ^ m ( A ) ) m * converges in (P cp (X), H) to some X* P cp (X), then there exists x* X* a periodic point for T.

Proof. (i) By Theorem 3.1 (vi) we have that the operator T ^ given by T ^ ( Y ) : = x Y T ( x ) maps P cp (X) to P cp (X) and it is continuous. By induction we get that T ^ m : P c p ( X ) P c p ( X ) and it is continuous. We will prove that T ^ is a (ε, φ)-contraction., i.e., if ε > 0 and A, B P cp (X) are two distinct sets such that H(A, B) < ε, then H ( T ^ ( A ) , T ^ ( B ) ) φ ( H ( A , B ) ) . Notice first that, by the symmetry of the Pompoiu-Hausdorff metric we only need to prove that
sup u T ^ ( A ) D ( u , T ^ ( B ) ) φ ( H ( A , B ) ) .
Let u T ^ ( A ) . Then, there exists a0 A such that u T (a0). It follows that
D ( u , T ( b ) ) H ( T ( a 0 ) , T ( b ) ) , f o r e v e r y b B .
Since A, B P cp (X), there exists b0 B such that d(a0, b0) ≤ H(A, B) < ε. Thus, by the (ε, φ)-contraction condition, we get
H ( T ( a 0 ) , T ( b 0 ) ) φ ( d ( a 0 , b 0 ) ) φ ( H ( A , B ) ) .
Hence
D ( u , T ( b ) ) φ ( H ( A , B ) ) .
Moreover, by the compactness of T ^ ( A ) we get the conclusion, namely
sup u T ^ ( A ) D ( u , T ^ ( B ) ) φ ( H ( A , B ) ) .
For the case of arbitrary m *, the proof of the fact that T ^ m is a (ε, φ)-contraction easily follows by induction.
  1. (ii)

    By (i) and the properties of the function φ, we get that T ^ m is an ε-contractive operator, i.e., if ε > 0 and A, B P cp (X) are two distinct sets such that H(A, B) < ε, then H ( T ^ m ( A ) , T ^ m ( B ) ) < H ( A , B ) . Now the conclusion follows from Theorem 3.2 in [2]. ■

     

Theorem 3.6. Let (X, d) be a compact metric space and T : X → P cp (X) be a continuous (ε; φ)-contraction. Then, there exists x* X a periodic point for T.

Proof. The conclusion follows by Theorem 3.5 (ii) and Corollary 3.3. in [2]. ■

Remark 3.2. We also refer to [58, 59] for some results of this type for multivalued operators of Reich's type.

The author declares he has no competing interests.

Authors’ Affiliations

(1)
Department of Applied Mathematics, Babeş-Bolyai University Cluj-Napoca
(2)
Vasile Goldiş Western University Arad, Satu-Mare Branch

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© Lazăr; licensee Springer. 2011

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