Research | Open | Published:

# Fixed point theory for multivalued φ-contractions

## 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

## 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  for details. A mapping φ : + + is said to be a comparison function if it is increasing and φk (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 ();

• multivalued weakly Picard operators ();

• multivalued Picard operators ();

• 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 ();

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

• limit shadowing property of a multivalued operator ();

• set-to-set operatorial equations ();

• fractal operators ().

## 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 , Petruşel , Rus et al. [18, 43]. See also .

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 Tn, 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

$( 1 ) D : P ( X ) × P ( X ) → ℝ + ∪ { + ∞ } D ( A , B ) = inf { d ( a , b ) | a ∈ A , b ∈ B } , 0 , + ∞ , A ≠ ∅ ≠ B A = ∅ = B o t h e r w i s e$

The excess generalized functional

$( 2 ) ρ : P ( X ) × P ( X ) → ℝ + ∪ { + ∞ } ρ ( A , B ) = sup { D ( a , B ) | a ∈ A } , 0 , + ∞ , A ≠ ∅ ≠ B A = ∅ B = ∅ ≠ A$

The Pompeiu-Hausdorff generalized functional.

$( 3 ) H : P ( X ) × P ( X ) → ℝ + ∪ { + ∞ } H ( A , B ) = max { ρ ( A , B ) , ρ ( B , A ) } , 0 , + ∞ , A ≠ ∅ ≠ B A = ∅ = B o t h e r w i s e$

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. ) 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 . 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 ) 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 ).

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→+∞fn(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)$A T * = ⋃ n ∈ ℕ * T n ( x )$, for each x F T .

Proof. (i) This is Węgrzyk's Theorem, see .

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

$d ( x n , x n + 1 ) ≤ q φ n ( t 0 ) , f o r e a c h n ∈ ℕ * .$

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(φn (t0) ++ φn+p−1(t0)), for each n, p *. If we set s0(t) := 0 for each t +, then

$d ( x n , x n + p ) ≤ q ( s n + p - 1 ( t 0 ) - s n - 1 ( t 0 ) ) , f o r e a c h n , p ∈ ℕ * .$
(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

$d ( x n , x * ) ≤ q ( s ( t 0 ) - s n - 1 ( t 0 ) ) , f o r e a c h n ∈ ℕ * .$
(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 , Chifu and Petruşel .) 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 .) Let x F T be arbitrary. Then, x T(x) T2(x) Tn (x) Hence x Tn (x), for each n *. Moreover, $lim n → + ∞ T n ( x ) = ⋃ n ∈ ℕ * T n ( x )$. By (vii), we immediately get that $A T * = ⋃ n ∈ ℕ * T n ( x )$. Hence, $x ∈ ⋃ n ∈ ℕ * T n ( x ) = A T *$. 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 $( x n i ) i ∈ ℕ$ of $( x n ) n ∈ ℕ x n i → d x * ∈ F T$ as i → ∞.

Proof. (xxi) Let (x n )nis a sequence in X such that D(x n , T (x n )) 0 as n → ∞. Let $( x n i ) i ∈ ℕ$ 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 .

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$( T ^ m i ( A ) ) i ∈ ℕ *$of$( T ^ m ( A ) ) m ∈ ℕ *$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 . ■

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 . ■

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.

## References

1. 1.

Rus IA: Generalized Contractions and Applications. Cluj University Press; 2001.

2. 2.

Nadler SB jr: Periodic points of multi-valued ε -contractive maps. Topol Methods Nonlinear Anal 2003,22(2):399–409.

3. 3.

Covitz H, Nadler SB jr: Multivalued contraction mappings in generalized metric spaces. Israel J Math 1970, 8: 5–11. 10.1007/BF02771543

4. 4.

Frigon M: Fixed point and continuation results for contractions in metric and gauge spaces. Banach Center Publ 2007, 77: 89–114.

5. 5.

Jachymski J, Józwik I: Nonlinear contractive conditions: a comparison and related problems. In Fixed Point Theory and its Applications. Volume 77. Edited by: Jachymski J, Reich S. Polish Academy of Sciences, Institute of Mathematics, Banach Center Publications, War-saw; 2007:123–146.

6. 6.

Lazăr T, O'Regan D, Petruşel A: Fixed points and homotopy results for Ćirić-type multivalued operators on a set with two metrics. Bull Korean Math Soc 2008, 45: 67–73. 10.4134/BKMS.2008.45.1.067

7. 7.

Lazăr TA, Petruşel A, Shahzad N: Fixed points for non-self operators and domain invariance theorems. Nonlinear Anal 2009, 70: 117–125. 10.1016/j.na.2007.11.037

8. 8.

Meir A, Keeler E: A theorem on contraction mappings. J Math Anal Appl 1969, 28: 326–329. 10.1016/0022-247X(69)90031-6

9. 9.

Mizoguchi N, Takahashi W: Fixed point theorems for multivalued mappings on complete metric spaces. J Math Anal Appl 1989, 141: 177–188. 10.1016/0022-247X(89)90214-X

10. 10.

Petruşel A: Generalized multivalued contractions. Nonlinear Anal 2001, 47: 649–659. 10.1016/S0362-546X(01)00209-7

11. 11.

Petruşel A, Rus IA: Fixed point theory for multivalued operators on a set with two metrics. Fixed Point Theory 2007, 8: 97–104.

12. 12.

Rhoades BE: Some theorems on weakly contractive maps. Nonlinear Anal 2001, 47: 2683–2693. 10.1016/S0362-546X(01)00388-1

13. 13.

Smithson RE: Fixed points for contractive multifunctions. Proc Am Math Soc 1971, 27: 192–194. 10.1090/S0002-9939-1971-0267564-4

14. 14.

Tarafdar E, Yuan GXZ: Set-valued contraction mapping principle. Appl Math Letter 1995, 8: 79–81.

15. 15.

Xu HK: ε -chainability and fixed points of set-valued mappings in metric spaces. Math Japon 1994, 39: 353–356.

16. 16.

Xu HK: Metric fixed point theory for multivalued mappings. Diss Math 2000, 389: 39.

17. 17.

Yuan GXZ: KKM Theory and Applications in Nonlinear Analysis. Marcel Dekker, New York; 1999.

18. 18.

Rus IA, Petruşel A, Sîntămărian A: Data dependence of the fixed point set of some multivalued weakly Picard operators. Nonlinear Anal 2003,52(8):1947–1959. 10.1016/S0362-546X(02)00288-2

19. 19.

Petruşel A, Rus IA: Multivalued Picard and weakly Picard operators. In Fixed Point Theory and Applications. Edited by: Llorens Fuster E, Garcia Falset J, Sims B. Yokohama Publ; 2004:207–226.

20. 20.

Lim TC: On fixed point stability for set-valued contractive mappings with applications to generalized differential equations. J Math Anal Appl 1985, 110: 436–441. 10.1016/0022-247X(85)90306-3

21. 21.

Markin JT: Continuous dependence of fixed points sets. Proc Am Math Soc 1973, 38: 545–547. 10.1090/S0002-9939-1973-0313897-4

22. 22.

Saint-Raymond J: Multivalued contractions. Set-Valued Anal 1994, 2: 559–571. 10.1007/BF01033072

23. 23.

Fraser RB, Nadler SB jr: Sequences of contractive maps and fixed points. Pac J Math 1969, 31: 659–667.

24. 24.

Papageorgiou NS: Convergence theorems for fixed points of multifunctions and solutions of differential inclusions in Banach spaces. Glas Mat Ser III 1988, 23: 247–257.

25. 25.

Rus IA: Remarks on Ulam stability of the operatorial equations. Fixed Point Theory 2009, 10: 305–320.

26. 26.

Petruşel A, Rus IA: Well-posedness of the fixed point problem for multivalued operators. In Applied Analysis and differential Equations. Edited by: Cârjă O, Vrabie II. World Scientific; 2007:295–306.

27. 27.

Petruşel A, Rus IA, Yao JC: Well-posedness in the generalized sense of the fixed point problems. Taiwan J Math 2007,11(3):903–914.

28. 28.

Glăvan V, Guţu V: On the dynamics of contracting relations. In Analysis and Optimization of differential Systems. Edited by: Barbu V, Lasiecka I, Tiba D, Varsan C. Kluwer; 2003:179–188.

29. 29.

Nadler SB jr: Multivalued contraction mappings. Pac J Math 1969, 30: 475–488.

30. 30.

Andres J: Some standard fixed-point theorems revisited. Atti Sem Mat Fis Univ Modena 2001, 49: 455–471.

31. 31.

De Blasi FS: Semifixed sets of maps in hyperspaces with applications to set differential equations. Set-Valued Anal 2006, 14: 263–272. 10.1007/s11228-005-0011-3

32. 32.

Andres J, Fišer J: Metric and topological multivalued fractals. Internat J Bifur Chaos Appl Sci Engrg 2004, 14: 1277–1289. 10.1142/S021812740400979X

33. 33.

Barnsley MF: Fractals Everywhere. Academic Press, Boston; 1988.

34. 34.

Hutchinson JE: Fractals and self-similarity. Indiana Univ Math J 1981, 30: 713–747. 10.1512/iumj.1981.30.30055

35. 35.

Jachymski J: Continuous dependence of attractors of iterated function systems. J Math Anal Appl 1996, 198: 221–226. 10.1006/jmaa.1996.0077

36. 36.

Lasota A, Myjak J: Attractors of multifunctions. Bull Polish Acad Sci Math 2000, 48: 319–334.

37. 37.

Petruşel A: Singlevalued and multivalued Meir-Keeler type operators. Revue D'Analse Num et de Th de l'Approx Tome 2001, 30: 75–80.

38. 38.

Yamaguti M, Hata M, Kigani J: Mathematics of Fractals. In Translations Math.Monograph. Volume 167. AMS Providence, RI; 1997.

39. 39.

Andres J, Górniewicz L: On the Banach contraction principle for multivalued mappings. Approximation, Optimization and Mathematical Economics (Pointe-à-Pitre, 1999), Physica, Heidelberg 2001, 1–23.

40. 40.

Chifu C, Petruşel A: Multivalued fractals and multivalued generalized contractions. Chaos Solit Fract 2008, 36: 203–210. 10.1016/j.chaos.2006.06.027

41. 41.

Kirk WA, Sims B, (eds): Handbook of Metric Fixed Point Theory. Kluwer, Dordrecht; 2001.

42. 42.

Petruşel A: Multivalued weakly Picard operators and applications. Sci Math Japon 2004, 59: 169–202.

43. 43.

Rus IA, Petruşel A, Petruşel G: Fixed Point Theory. Cluj University Press, Cluj-Napoca; 2008.

44. 44.

Aubin JP, Frankowska H: Set-Valued Analysis. Birkhauser, Basel; 1990.

45. 45.

Beer G: Topologies on Closed and Closed Convex Sets. Kluwer, Dordrecht; 1993.

46. 46.

Ayerbe Toledano YM, Dominguez Benavides T, López Acedo L: Measures of Noncompactness in Metric Fixed Point Theory. Birkhäuser Verlag, Basel; 1997.

47. 47.

Goebel K, Kirk WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.

48. 48.

Górniewicz L: Topological Fixed Point Theory of Multivalued Mappings. Kluwer, Dordrecht; 1999.

49. 49.

Hu S, Papageorgiou NS: Handbook of Multivalued Analysis, Vol. I and II. Kluwer, Dordrecht; 1997.

50. 50.

Rus IA, Petruşel A, Petruşel G: Fixed Point Theory 1950–2000: Romanian Contributions. House of the Book of Science, Cluj-Napoca; 2002.

51. 51.

Granas A, Dugundji J: Fixed Point Theory. Springer, Berlin; 2003.

52. 52.

Takahashi W: Nonlinear Functional Analysis. In Fixed Point Theory and its Applications. Yokohama Publishers, Yokohama; 2000.

53. 53.

Rus IA: Weakly Picard mappings. Comment Math University Carolinae 1993, 34: 769–773.

54. 54.

Rus IA: Picard operators and applications. Sci Math Japon 2003, 58: 191–219.

55. 55.

Rus IA, Şerban MA: Some generalizations of a Cauchy lemma and applications. In Topics in Mathematics, Computer Science and Philosophy-A Festschrift for Wolfgang. Edited by: Breckner W. Cluj University Press; 2008:173–181.

56. 56.

Węgrzyk R: Fixed point theorems for multifunctions and their applications to functional equations. Dissertationes Math (Rozprawy Mat.) 1982, 201: 28.

57. 57.

Petruşel A, Rus IA: The theory of a metric fixed point theorem for multivalued operators. In Proc Ninth International Conference on Fixed Point Theory and its Applications, Changhua, Taiwan, July 16–22, 2009. Edited by: Lin LJ, Petruşel A, Xu HK. Yokohama Publ; 2010:161–175.

58. 58.

Reich S: Fixed point of contractive functions. Boll Un Mat Ital 1972, 5: 26–42.

59. 59.

Reich S: A fixed point theorem for locally contractive multivalued functions. Rev Roumaine Math Pures Appl 1972, 17: 569–572.

## Author information

Correspondence to Vasile L Lazăr.

## Rights and permissions

Reprints and Permissions

• #### DOI

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

### 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 