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

$$P\left(X\right):=\left\{Y\subset X|Y\ne \varnothing \right\},P_{cl}\left(X\right):=\left\{Y\in P\left(X\right)|Yisclosed\right\}.$$

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 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 : 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 ${\sum }_{n=1}^{\infty }{\phi }^n\left(t\right)<+\infty$, 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

$$H\left(T\left(x_1\right),T\left(x_2\right)\right)\le \phi \left(d\left(x_1,x_2\right)\right).$$

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]);

• data dependence of the fixed point set ([18, 20–22]);

• 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 ([29–31]);

• fractal operators ([32–40]).

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

$$\mathcal{P}(X):=\left\{Y\right|Y\text{is}\text{a}\text{subset}\text{of}X\},P(X):=\{Y\in \mathcal{P}(X)\left|Y\text{is}\text{nonempty}\right\}.$$

Let (X, d) be a metric space. Then δ(Y ) := sup {d(a, b)| a, b ∈ Y} and

$$\begin{array}{c}{P}_b\left(X\right):=\left\{Y\in P\left(X\right)|\delta \left(Y\right)<+\infty \right\},P_{cl}\left(X\right):=\left\{Y\in P\left(X\right)|Yisclosed\right\},\\ P_{cp}\left(X\right):=\left\{Y\in P\left(X\right)|Yiscompact\right\},P_{op}\left(X\right):=\left\{Y\in P\left(X\right)|Yisopen\right\}.\end{array}$$

Let T : X → P(X) be a multivalued operator. Then, the operator $\widehat{T}:P\left(X\right)\to P\left(X\right)$defined by

$$\widehat{T}\left(Y\right):=\bigcup _{x\in Y}T\left(x\right),forY\in P\left(X\right)$$

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 $\widehat{T}:P_{cp}\left(X\right)\to P_{cp}\left(X\right)$. 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 $p\in F_{T^m}$, for some integer m ≥ 1, i.e., $p\in {\widehat{T}}^m\left(\left\{p\right\}\right)$ for some integer m ≥ 1.

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

The gap functional

$$\begin{array}{c}\left(1\right)D:\mathcal{P}\left(X\right)\times \mathcal{P}\left(X\right)\to {ℝ}_{+}\cup \left\{+\infty \right\}\\ D\left(A,B\right)=\left\{\begin{array}{c}\hfill inf\left\{d\left(a,b\right)|a\in A,b\in B\right\},\hfill \\ \hfill 0,\hfill \\ \hfill +\infty ,\hfill \end{array}\begin{array}{c}\hfill A\ne \varnothing \ne B\hfill \\ \hfill A=\varnothing =B\hfill \\ \hfill otherwise\hfill \end{array}\right.\end{array}$$

The excess generalized functional

$$\begin{array}{c}\left(2\right)\rho :\mathcal{P}\left(X\right)\times \mathcal{P}\left(X\right)\to {ℝ}_{+}\cup \left\{+\infty \right\}\\ \rho \left(A,B\right)=\left\{\begin{array}{c}\hfill sup\left\{D\left(a,B\right)|a\in A\right\},\hfill \\ \hfill 0,\hfill \\ \hfill +\infty ,\hfill \end{array}\begin{array}{c}\hfill A\ne \varnothing \ne B\hfill \\ \hfill A=\varnothing \hfill \\ \hfill B=\varnothing \ne A\hfill \end{array}\right.\end{array}$$

The Pompeiu-Hausdorff generalized functional.

$$\begin{array}{c}\left(3\right)H:\mathcal{P}\left(X\right)\times \mathcal{P}\left(X\right)\to {ℝ}_{+}\cup \left\{+\infty \right\}\\ H\left(A,B\right)=\left\{\begin{array}{c}\hfill max\left\{\rho \left(A,B\right),\rho \left(B,A\right)\right\},\hfill \\ \hfill 0,\hfill \\ \hfill +\infty ,\hfill \end{array}\begin{array}{c}\hfill A\ne \varnothing \ne B\hfill \\ \hfill A=\varnothing =B\hfill \\ \hfill otherwise\hfill \end{array}\right.\end{array}$$

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

$$d\left(x,t^{\infty }\left(x,y\right)\right)\le \psi \left(d\left(x,y\right)\right),forall\left(x,y\right)\in Graph\left(T\right).$$

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) $T^n\left(x\right)\stackrel{H}{\to }\left\{x^{*}\right\}$ 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 lim_n→+∞b _n = 0. Then,

$$\underset{n\to +\infty }{lim}\sum _{k=0}^n{\phi }^{n-k}\left(b_k\right)=0.$$

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 : ℝ₊→ ℝ₊is a comparison function and

$$d\left(f\left(x\right),f\left(y\right)\right)\le \phi \left(d\left(x,y\right)\right)forallx,y\in X.$$

Then f is a Picard operator, i.e., f has a unique fixed point x* ∈ X and lim_n→+∞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 $A_n\stackrel{H}{\to }{A}^{*}\in P_{cl}\left(X\right)$ 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${\sum }_{n=1}^{\infty }{\phi }^n\left(t\right)$, then T is a ψ-MWP operator, with ψ(t) := t + s(t), for each t ∈ ℝ₊(where $s\left(t\right):={\sum }_{n=1}^{\infty }{\phi }^n\left(t\right)$);

(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${\sum }_{n=1}^{\infty }{\phi }^n\left(t\right)$. 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 that$T_n\left(x\right)\stackrel{H}{\to }T\left(x\right)$as n → +∞, uniformly with respect to each × ∈ X. Then, $F_{T_n}\stackrel{H}{\to }{F}_T$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, $\widehat{T}:\left(P_{cp}\left(X\right),H\right)\to \left(P_{cp}\left(X\right),H\right),\widehat{T}\left(Y\right):={\bigcup }_{x\in Y}T\left(x\right)$is a set-to-set φ-contraction and (thus)$F_{\widehat{T}}=\left\{A_T^{*}\right\}$;

(vii)$T^n\left(x\right)\stackrel{H}{\to }{A}_T^{*}$as n → +∞, for each × ∈ X;

(viii) $F_T\subset A_T^{*}$ and F _T is compact;

(ix)$A_T^{*}={\bigcup }_{n\in {\mathbb{N}}^{*}}{T}^n\left(x\right)$, 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 $d(x_2,x_3)<\frac{\phi (q\phi (t_0))}{\phi (d(x_1,x_2))}H(T(x_1)$, $T(x_2))\le \frac{\phi (q\phi (t_0))}{\phi (d(x_1,x_2))}\phi (d(x_1,x_2))\le q{\phi }^2(t_0)$. By an inductive procedure, we obtain a sequence of successive approximations for T starting from (x ₀, x ₁) ∈ Graph(T) such that

  $$d\left(x_n,x_{n+1}\right)\le q{\phi }^n\left(t_0\right),foreachn\in {\mathbb{N}}^{*}.$$

Denote by

$$s_n\left(t\right):=\sum _{k=1}^n{\phi }^k\left(t\right),foreacht>0.$$

Then, d(x _n , x_n+p) ≤ q(φⁿ (t₀) +⋯+ φ^n+p−1(t₀)), for each n, p ∈ ℕ*. If we set s₀(t) := 0 for each t ∈ ℝ₊, then

$$d\left(x_n,x_{n+p}\right)\le q\left(s_{n+p-1}\left(t_0\right)-s_{n-1}\left(t_0\right)\right),foreachn,p\in {\mathbb{N}}^{*}.$$

(3.1)

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

$$d\left(x_n,x^{*}\right)\le q\left(s\left(t_0\right)-s_{n-1}\left(t_0\right)\right),foreachn\in {\mathbb{N}}^{*}.$$

(3.2)

If we put n = 1 in (3.2), we obtain that d(x₁, x*) ≤ qs(t₀). Hence,

$$d\left(x_0,x^{*}\right)\le d\left(x_0,x_1\right)+d\left(x_1,x^{*}\right)\le t_0+qs\left(t_0\right).$$

(3.3)

Finally, letting q ↘ 1 in (3.3), we get that

$$d\left(x_0,x^{*}\right)\le t_0+s\left(t_0\right)=\psi \left(t_0\right)=\psi \left(d\left(x_0,x_1\right)\right).$$

(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 ${\sum }_{n=1}^{\infty }{\phi }^n\left(t\right)$, 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

  $$d\left(x_0,t^{\infty }\left(x_0,x_1\right)\right)\le \psi \left(d\left(x_0,x_1\right)\right),foreach{x}_1\in T\left(x_0\right).o$$

Let q > 1 be arbitrary. Then, there exists x₁ ∈ T (x₀) such that d(x₀, x₁) < qH(S(x₀), T (x₀)). Then

$$d\left(x_0,t^{\infty }\left(x_0,x_1\right)\right)\le \psi \left(qH\left(S\left(x_0\right),T\left(x_0\right)\right)\right)\le q\psi \left(H\left(S\left(x_0\right),T\left(x_0\right)\right)\right)\le q\psi \left(\eta \right).$$

By a similar procedure we can prove that, for each y₀ ∈ F _T , there exists y₁ ∈ S(y₀) such that

$$d\left(y_0,s^{\infty }\left(y_0,y_1\right)\right)\le q\psi \left(\eta \right).$$

By the above relations and using Lemma 2.5, we obtain that

$$H\left(F_S,F_T\right)\le q\psi \left(\eta \right),whereq>1.$$

Letting q ↘ 1, we get the conclusion.

• (iv) Let ε > 0. Since $T_n\left(x\right)\stackrel{H}{\to }T\left(x\right)$ as n → +∞, uniformly with respect to each x ∈ X, there exists N _ε ∈ ℕ such that

  $$\underset{x\in X}{sup}H\left(T_n\left(x\right),T\left(x\right)\right)<\epsilon ,foreachn\ge N_{\epsilon }.$$

Then, by (iii) we get that $H\left(F_{T_n},F_T\right)\le \psi \left(\epsilon \right)$, for each n ≥ N _ε . Since ψ is continuous in 0 and ψ(0) = 0, we obtain that $F_{T_n}\stackrel{H}{\to }{F}_T$.

• (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

  $$d\left(x,t^{\infty }\left(x,y\right)\right)\le \psi \left(d\left(x,y\right)\right).$$

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

  $$H\left(T\left(A\right),T\left(B\right)\right)\le \phi \left(H\left(A,B\right)\right),foreachA,B\in P_{cp}\left(X\right).$$

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

$$d\left(a,b\right)\le H\left(A,B\right).$$

(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

$$\rho \left(T\left(A\right),T\left(B\right)\right)\le \phi \left(d\left(a,b\right)\right)\le \phi \left(H\left(A,B\right)\right).$$

(3.6)

By a similar procedure, we obtain

$$\rho \left(T\left(B\right),T\left(A\right)\right)\le \phi \left(d\left(a,b\right)\right)\le \phi \left(H\left(A,B\right)\right).$$

(3.7)

Thus, (3.6) and (3.7) together imply that

$$H\left(T\left(A\right),T\left(B\right)\right)\le \phi \left(H\left(A,B\right)\right).$$

Hence, $\widehat{T}$ 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:

• (a) $F_{\widehat{T}}=\left\{A_T^{*}\right\}$

and

• (b) ${\widehat{T}}^n\left(A\right)\stackrel{H}{\to }{A}_T^{*}$ as n → +∞, for each A ∈ P _cp (X).

• (vii) By (vi)-(b) we get that $T^n\left(\left\{x\right\}\right)={\widehat{T}}^n\left(\left\{x\right\}\right)\stackrel{H}{\to }{A}_T^{*}$ as n → +∞, for each x ∈ X.

(viii)-(ix) (Chifu and Petruşel [40].) Let x ∈ F _T be arbitrary. Then, x ∈ T(x) ⊂ T²(x) ⊂ ⋯ ⊂ Tⁿ (x) ⊂ ⋯ Hence x ∈ Tⁿ (x), for each n ∈ ℕ*. Moreover, $\underset{n\to +\infty }{lim}{T}^n\left(x\right)={\bigcup }_{n\in \mathbb{N}*}{T}^n\left(x\right)$. By (vii), we immediately get that $A_T^{*}={\bigcup }_{n\in \mathbb{N}*}{T}^n\left(x\right)$. Hence, $x\in {\bigcup }_{n\in \mathbb{N}*}{T}^n\left(x\right)=A_T^{*}$. 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$F_{T^n}={\left(SF\right)}_{T^n}=\left\{x*\right\}$for n ∈ ℕ*;

(xii) If, additionally T(x) is compact for each × ∈ X, then$T^n\left(x\right)\stackrel{H}{\to }\left\{x^{*}\right\}$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 : 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 that$F_{T_n}\ne \varnothing$for each n ∈ ℕ and$T_n\left(x\right)\stackrel{H}{\to }T\left(x\right)$as n → +∞, uniformly with respect to × ∈ X. Then, $F_{T_n}\stackrel{H}{\to }{F}_T$as n → +∞.

(xv) (Well-posedness of the fixed point problem with respect to D) If (x _n )_{n ∈ ℕ}is a sequence in × such that D(x _n , T (x _n )) → 0 as n → ∞, then$x_n\stackrel{d}{\to }{x}^{*}$as 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 → ∞, then$x_n\stackrel{d}{\to }{x}^{*}$as 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(y_n+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,

$$d\left(x^{*},y\right)=D\left(T\left(x^{*}\right),y\right)\le H\left(T\left(x^{*}\right),T\left(y\right)\right)\le \phi \left(d\left(x^{*},y\right)\right).$$

Thus, y = x*. Hence, F _T ⊂ (SF) _T . Since (SF) _T ⊂ F _T , we get that (SF) _T = F _T .

• (xi) Notice first that $x^{*}\in {\left(SF\right)}_{T^n}\subset F_{T^n}$, for each n ∈ ℕ*. Consider $y\in {\left(SF\right)}_{T^n}$, for arbitrary n ∈ ℕ*. Then, by (vi) we have that

  $$d\left(x^{*},y\right)=H\left(T^n\left(x^{*}\right),T^n\left(y\right)\right)\le \phi \left(H\left(T^{n-1}\left(x^{*}\right),T^{n-1}\left(y\right)\right)\right)\le \cdots \le {\phi }^n\left(d\left(x^{*},y\right)\right).$$

Thus, y = x* and ${\left(SF\right)}_{T^n}=\left\{x^{*}\right\}$. Consider now $y\in F_{T^n}$. Then, we have

$$\begin{array}{lll}\hfill d\left(x^{*},y\right)& =D\left(T^n\left(x^{*}\right),y\right)\le H\left(T^n\left(x^{*}\right),T^n\left(y\right)\right)& \hfill \text{(1)}\\ \le \phi \left(H\left(T^{n-1}\left(x^{*}\right),T^{n-1}\left(y\right)\right)\right)\le \cdots \le {\phi }^n\left(d\left(x^{*},y\right)\right).& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

Thus, y = x* and hence $T^n\left(x\right)\stackrel{H}{\to }\left\{x^{*}\right\}$.

• (xii) Let x ∈ X be arbitrarily chosen. Then, we have

  $$\begin{array}{c}H(T^n(x),x^{*})=H(T^n(x),T^n(x^{*}))\le \phi (H(T^{n-1}(x),\\ T^{n-1}(x^{*})))\le \cdots \le \phi {(}^{n}d(x,x^{*}))\to 0\text{as}n\to +\infty .\end{array}$$

• (xiii) Let y ∈ F _S . Then,

  $$d\left(y,x^{*}\right)\le H\left(S\left(y\right),x^{*}\right)\le H\left(S\left(y\right),T\left(y\right)\right)+H\left(T\left(y\right),x^{*}\right)\le \eta +\phi \left(d\left(y,x^{*}\right)\right).$$

Thus, d(y, x*) ≤ β(η). The conclusion follows now by the following relations

$$H\left(F_S,F_T\right)=\underset{y\in F_S}{sup}d\left(y,x^{*}\right)\le \beta \left(\eta \right).$$

• (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,

  $$\begin{array}{lll}\hfill d\left(x_n,x^{*}\right)& \le D\left(x_n,T\left(x_n\right)\right)+H\left(T\left(x_n\right),T\left(x^{*}\right)\right)& \hfill \text{(1)}\\ \le D\left(x_n,T\left(x_n\right)\right)+\phi \left(d\left(x_n,x^{*}\right)\right).& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

Then

$$d\left(x_n,x^{*}\right)\le \beta \left(D\left(x_n,T\left(x_n\right)\right)\right)\to 0asn\to +\infty .$$

• (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:

$$\begin{array}{lll}\hfill d\left(x^{*},y_{n+1}\right)& \le H\left(x^{*},T\left(y_n\right)\right)+D\left(y_{n+1},T\left(y_n\right)\right)& \hfill \text{(1)}\\ \le \phi \left(d\left(x^{*},y_n\right)\right)+D\left(y_{n+1},T\left(y_n\right)\right)& \hfill \text{(2)}\\ \le \phi \left(\phi \left(d\left(x^{*},y_{n-1}\right)\right)+D\left(y_n,T\left(y_{n-1}\right)\right)\right)+D\left(y_{n+1},T\left(y_n\right)\right)& \hfill \text{(3)}\\ \le {\phi }^2\left(d\left(x^{*},y_{n-1}\right)\right)+\phi \left(D\left(y_n,T\left(y_{n-1}\right)\right)\right)+D\left(y_{n+1},T\left(y_n\right)\right)& \hfill \text{(4)}\\ \le ...\le {\phi }^{n+1}\left(d\left(x^{*},y_0\right)\right)+{\phi }^n\left(D\left(y_1,T\left(y_0\right)\right)\right)& \hfill \text{(5)}\\ +\cdots +D\left(y_{n+1},T\left(y_n\right)\right).& \hfill \text{(6)}\\ \hfill \text{(7)}\end{array}$$

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

$$d\left(y_n,x_n\right)\le d\left(y_n,x^{*}\right)+d\left(x^{*},x_n\right)\to 0asn\to +\infty .$$

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)$T^n\left(x\right)\stackrel{H}{\to }{F}_T$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 that$x_n\stackrel{d}{\to }{x}^{*}\in F_T$as n → ∞, then$T^n\left(x\right)\stackrel{H}{\to }{F}_T$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).

• (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 $x_n\stackrel{d}{\to }{x}^{*}\in F_T$ as n → +∞.

Then, we have:

$$H\left(T\left(x_n\right),F_T\right)=H\left(T\left(x_n\right),T\left(x^{*}\right)\right)\le \phi \left(d\left(x_n,x^{*}\right)\right)\to 0asn\to +\infty .$$

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 ${\left(x_{n_i}\right)}_{i\in \mathbb{N}}$ of ${\left(x_n\right)}_{n\in \mathbb{N}}{x}_{n_i}\stackrel{d}{\to }{x}^{*}\in F_T$ 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 ${\left(x_{n_i}\right)}_{i\in \mathbb{N}}$ be a subsequence of (x _n )_n∈ℕsuch that $x_{n_i}\stackrel{d}{\to }{x}^{*}$ as i → ∞. Then, there exists $y_{n_i}\in T\left(x_{n_i}\right)$, i ∈ ℕ such that $y_{n_i}\stackrel{d}{\to }{x}^{*}$ 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 ∈ 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

$$x,y\in Xwithx\ne yandd\left(x,y\right)<\epsilon impliesH\left(T\left(x\right),T\left(y\right)\right)\le \phi \left(d\left(x,y\right)\right).$$

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)${\widehat{T}}^m:P_{cp}\left(X\right)\to P_{cp}\left(X\right)$is a continuous (ε, φ)-contraction, for each m ∈ ℕ*;

(ii) if, additionally, there exists some A ∈ P _cp (X) such that a sub-sequence${\left({\widehat{T}}^{m_i}\left(A\right)\right)}_{i\in \mathbb{N}*}$of${\left({\widehat{T}}^m\left(A\right)\right)}_{m\in \mathbb{N}*}$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 $\widehat{T}$ given by $\widehat{T}\left(Y\right):={\bigcup }_{x\in Y}T\left(x\right)$ maps P _cp (X) to P _cp (X) and it is continuous. By induction we get that ${\widehat{T}}^m:P_{cp}\left(X\right)\to P_{cp}\left(X\right)$ and it is continuous. We will prove that $\widehat{T}$ 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 $H\left(\widehat{T}\left(A\right),\widehat{T}\left(B\right)\right)\le \phi \left(H\left(A,B\right)\right)$. Notice first that, by the symmetry of the Pompoiu-Hausdorff metric we only need to prove that

$$\underset{u\in \widehat{T}\left(A\right)}{sup}D\left(u,\widehat{T}\left(B\right)\right)\le \phi \left(H\left(A,B\right)\right).$$

Let $u\in \widehat{T}\left(A\right)$. Then, there exists a₀ ∈ A such that u ∈ T (a₀). It follows that

$$D\left(u,T\left(b\right)\right)\le H\left(T\left(a_0\right),T\left(b\right)\right),foreveryb\in B.$$

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

$$H\left(T\left(a_0\right),T\left(b_0\right)\right)\le \phi \left(d\left(a_0,b_0\right)\right)\le \phi \left(H\left(A,B\right)\right).$$

Hence

$$D\left(u,T\left(b\right)\right)\le \phi \left(H\left(A,B\right)\right).$$

Moreover, by the compactness of $\widehat{T}\left(A\right)$ we get the conclusion, namely

$$\underset{u\in \widehat{T}\left(A\right)}{sup}D\left(u,\widehat{T}\left(B\right)\right)\le \phi \left(H\left(A,B\right)\right).$$

For the case of arbitrary m ∈ ℕ*, the proof of the fact that ${\widehat{T}}^m$is a (ε, φ)-contraction easily follows by induction.

• (ii) By (i) and the properties of the function φ, we get that ${\widehat{T}}^m$ 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 $H\left({\widehat{T}}^m\left(A\right),{\widehat{T}}^m\left(B\right)\right)<H\left(A,B\right)$. 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.