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 approximations multivalued operator Picard operator weakly Picard operator fixed point strict fixed point periodic point strict periodic point multivalued weakly Picard operator multivalued Picard operator data dependence fractal operator limit shadowing set-to-set operator Ulam-Hyers stability sequence 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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