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
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1 Introduction
Let X be a nonempty set. Then, we denote
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 : ℝ+→ ℝ+ 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 : ℝ+→ ℝ+ having the property that t - φ (t) → +∞, as t → +∞ is said to be a strict comparison function.
Moreover, a function φ : ℝ+→ ℝ+ is said to be a strong comparison function if it is strictly increasing and , 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 : ℝ+→ ℝ+ is a strong comparison function, and for all x1, x2 ∈ X, we have that
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 ([2–17]);
-
multivalued weakly Picard operators ([18]);
-
multivalued Picard operators ([19]);
-
sequence of multivalued operators and fixed points ([23, 24]);
-
Ulam-Hyers stability of a multivalued fixed point equation ([25]);
-
limit shadowing property of a multivalued operator ([28]);
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 [44–52].
Let X be a nonempty set. Then, we denote
Let (X, d) be a metric space. Then δ(Y ) := sup {d(a, b)| a, b ∈ Y} and
Let T : X → P(X) be a multivalued operator. Then, the operator defined by
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:
-
(a)
if T is upper semicontinuous, then T (Y) ∈ P cp (X), for every Y ∈ P cp (X);
-
(b)
the continuity of T implies the continuity of . A sequence of successive approximations of T starting from x ∈ X is a sequence (x n )n∈ℕof elements in X with x 0 = x, x n+1∈ T (x n ), for n ∈ 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 ∈ 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 , for some integer m ≥ 1, i.e., for some integer m ≥ 1.
The following (generalized) functionals are used in the main sections of the paper.
The gap functional
The excess generalized functional
The Pompeiu-Hausdorff generalized functional.
For other details and basic results concerning the above notions, see, for example, [2, 41, 44–50].
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 )n∈ℕin X such that:
-
(i)
x 0 = x, x 1 = y;
-
(ii)
x n+1∈ T (x n ), for all n ∈ ℕ;
-
(iii)
the sequence (x n )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 : ℝ+→ ℝ+ is a continuous in t = 0 and increasing function such that ψ(0) = 0, and there exists a selection t∞ of T∞ such that
In particular, if ψ(t) := ct, for each t ∈ ℝ+ (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:
-
(i)
(S F) T = F T = {x*};
-
(ii)
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 : ℝ+→ ℝ+be a strong comparison function and (b n )n∈ℕbe a sequence of non-negative real numbers, such that limn→+∞b n = 0. Then,
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 : ℝ+→ ℝ+is a comparison function and
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 )n∈ℕbe a sequence in P cl (X). By definition, we will write 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 : X → P 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) ≤ qφ(t) for every t ∈ ℝ+ (where q > 1) and t = 0 is a point of uniform convergence for the series, then T is a ψ-MWP operator, with ψ(t) := t + s(t), for each t ∈ ℝ+(where );
(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 ∈ ℝ+(where q > 1) and t = 0 is a point of uniform convergence for the series. Then, H(F S , F T ) ≤ ψ(η);
(iv) (sequence of operators) Let T, T n : X → P cl (X), n ∈ ℕ be multivalued φ-contractions such thatas n → +∞, uniformly with respect to each × ∈ X. Then, 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, is a set-to-set φ-contraction and (thus);
(vii)as n → +∞, for each × ∈ X;
(viii) and F T is compact;
(ix), for each x ∈ F T .
Proof. (i) This is Węgrzyk's Theorem, see [56].
-
(ii)
Let x 0 ∈ X and x 1 ∈ T (x 0) be arbitrarily chosen. We may suppose that x 0 ≠ x 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 1 ≠ x 2. Thus, φ(d(x 1, x 2)) < φ(qφ(t 0)). Next, there exists x 3 ∈ T(x 2) such that , . By an inductive procedure, we obtain a sequence of successive approximations for T starting from (x 0, x 1) ∈ Graph(T) such that
Denote by
Then, d(x n , xn+p) ≤ q(φn (t0) +⋯+ φn+p−1(t0)), for each n, p ∈ ℕ*. If we set s0(t) := 0 for each t ∈ ℝ+, then
By (3.1) we get that the sequence (x n )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
If we put n = 1 in (3.2), we obtain that d(x1, x*) ≤ qs(t0). Hence,
Finally, letting q ↘ 1 in (3.3), we get that
Notice that, ψ is increasing (since φ is), ψ(0) = 0 and, since t = 0 is a point of uniform convergence for the series , ψ is continuous in t = 0.
These, together with (3.4), prove that T is a ψ-MWP operator.
-
(iii)
Let x 0 ∈ F S be arbitrary chosen. Then, by (ii), we have that
Let q > 1 be arbitrary. Then, there exists x1 ∈ T (x0) such that d(x0, x1) < qH(S(x0), T (x0)). Then
By a similar procedure we can prove that, for each y0 ∈ F T , there exists y1 ∈ S(y0) such that
By the above relations and using Lemma 2.5, we obtain that
Letting q ↘ 1, we get the conclusion.
-
(iv)
Let ε > 0. Since as n → +∞, uniformly with respect to each x ∈ X, there exists N ε ∈ ℕ such that
Then, by (iii) we get that , for each n ≥ N ε . Since ψ is continuous in 0 and ψ(0) = 0, we obtain that .
-
(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
Since x* := t∞ (x, y) ∈ F T , we get the desired conclusion d(x, x*) ≤ ψ(ε).
-
(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
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
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
By a similar procedure, we obtain
Thus, (3.6) and (3.7) together imply that
Hence, 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:
-
(a)
and
-
(b)
as n → +∞, for each A ∈ P cp (X).
-
(vii)
By (vi)-(b) we get that as n → +∞, for each x ∈ X.
(viii)-(ix) (Chifu and Petruşel [40].) Let x ∈ F T be arbitrary. Then, x ∈ T(x) ⊂ T2(x) ⊂ ⋯ ⊂ Tn (x) ⊂ ⋯ Hence x ∈ Tn (x), for each n ∈ ℕ*. Moreover, . By (vii), we immediately get that . Hence, . 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, thenfor n ∈ ℕ*;
(xii) If, additionally T(x) is compact for each × ∈ X, thenas 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 : ℝ+→ ℝ+is given by β(η) := sup{t ∈ ℝ+| t - φ(t) ≤ η};
(xiv) Let T n : X → P cl (X), n ∈ ℕ be a sequence of multivalued operators such thatfor each n ∈ ℕ andas n → +∞, uniformly with respect to × ∈ X. Then, as n → +∞.
(xv) (Well-posedness of the fixed point problem with respect to D) If (x n )n ∈ ℕis a sequence in × such that D(x n , T (x n )) → 0 as n → ∞, thenas n → ∞;
(xvi) (Well-posedness of the fixed point problem with respect to H) If (x n )n∈ℕis a sequence in × such that H(x n , T (x n )) → 0 as n → ∞, thenas n → ∞;
(xvii) (Limit shadowing property of the multivalued operator) Suppose additionally that φ is a sub-additive function. If (y n )n∈ℕis a sequence in × such that D(yn+1, T(y n )) → 0 as n → ∞, then there exists a sequence (x n )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 y ≠ x*, 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,
Thus, y = x*. Hence, F T ⊂ (SF) T . Since (SF) T ⊂ F T , we get that (SF) T = F T .
-
(xi)
Notice first that , for each n ∈ ℕ*. Consider , for arbitrary n ∈ ℕ*. Then, by (vi) we have that
Thus, y = x* and . Consider now . Then, we have
Thus, y = x* and hence .
-
(xii)
Let x ∈ X be arbitrarily chosen. Then, we have
-
(xiii)
Let y ∈ F S . Then,
Thus, d(y, x*) ≤ β(η). The conclusion follows now by the following relations
-
(xiv)
follows by (xiii).
-
(xv)
([26, 27]) Let (x n )n∈ℕbe a sequence in X such that D(x n , T (x n )) → 0 as n → ∞. Then,
Then
-
(xvi)
follows by (xv).
-
(xvii)
Let (y n )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 ∈ 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:
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 )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∈ℕ, we have
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)as n → +∞, for each × ∈ X;
(xix) T(x) = F T , for each × ∈ F T ;
(xx) If (x n )n∈ℕ⊂ X is a sequence such thatas n → ∞, thenas n → +∞.
Proof. (xviii) By T(F T ) = F T and Theorem 3.1 (vi), we have that . The conclusion follows by Theorem 3.1 (vii).
-
(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 .
-
(xx)
Let (x n )n∈ℕ⊂ X is a sequence such that as n → +∞.
Then, we have:
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 )n∈ℕis a sequence in × such that D(x n , T (x n )) → 0 as n → ∞, then there exists a subsequence of as i → ∞.
Proof. (xxi) Let (x n )n∈ℕis a sequence in X such that D(x n , T (x n )) → 0 as n → ∞. Let be a subsequence of (x n )n∈ℕsuch that as i → ∞. Then, there exists , i ∈ ℕ such that 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 ∈ ℝ+ 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 : ℝ+→ ℝ+ is a strong comparison function and
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)is a continuous (ε, φ)-contraction, for each m ∈ ℕ*;
(ii) if, additionally, there exists some A ∈ P cp (X) such that a sub-sequenceofconverges 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 given by maps P cp (X) to P cp (X) and it is continuous. By induction we get that and it is continuous. We will prove that 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 . Notice first that, by the symmetry of the Pompoiu-Hausdorff metric we only need to prove that
Let . Then, there exists a0 ∈ A such that u ∈ T (a0). It follows that
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
Hence
Moreover, by the compactness of we get the conclusion, namely
For the case of arbitrary m ∈ ℕ*, the proof of the fact that is a (ε, φ)-contraction easily follows by induction.
-
(ii)
By (i) and the properties of the function φ, we get that 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 . 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.
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Lazăr, V.L. Fixed point theory for multivalued φ-contractions. Fixed Point Theory Appl 2011, 50 (2011). https://doi.org/10.1186/1687-1812-2011-50
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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
- limit shadowing
- set-to-set operator
- Ulam-Hyers stability
- sequence of operators