Some local fixed point theorems and applications to open mapping principles and continuation results

The purpose of this article is to present, under weaker assumptions, some local fixed point theorems for Ćirić–Reich–Rus, Chatterjea and Berinde type generalized contractions. Then, as applications we will obtain open mapping theorems and continuation principles for these classes of mappings.


Introduction
The paper contains local fixed point theorems for three types of generalized contractions:Ćirić-Reich-Rus contractions, Chatterjea contractions and Berinde contractions. As applications, open mapping theorems and continuation principles for these classes of operators are given.
In the first section, we recall some well-known definitions and results, which are used in the main sections. In the next section, we prove local fixed point theorems for theĆirić-Reich-Rus, Chatterjea and Berinde type generalized contractions, which generalize some local theorems from [9,10,12]. In the last two sections, as applications of the previous results, we present open mapping theorems and continuation principles for the case of the above-mentioned operators, extending results presented in [9,10].
We now recall the notions of Picard and weakly Picard operators. Definition 1. 4 Let (X, d) be a metric space. a) We say that an operator f : X → X is a Picard operator if there exists a point x * ∈ X such that Fix( f ) = {x * } and the sequence ( f n (x 0 )) n∈N converges to x * for all x 0 ∈ X . b) An operator f : X → X is said to be a weakly Picard operator if the sequence ( f n (x 0 )) n∈N converges for all x 0 ∈ X and its limit belongs to the set of fixed points Fix( f ).
It is well known that anyĆirić-Reich-Rus and any Chatterjea contractions are Picard operators (P.o.), while any Berinde contraction is w.P.o. (weakly Picard operator). For other considerations on Picard operators and weakly Picard operators, see [16,18].
We also recall the notion of a field generated by an operator.

Definition 1.5
Let (X, · ) be a normed space. For an operator f : X → X , we define the field generated by f as follows: Let (X, d) be a metric space, a point x 0 ∈ X and a strictly positive r . The set B(x 0 ; r ) := {x ∈ X : d(x 0 , x) < r } is the open ball of center x 0 and radius r and also the setB(x 0 ; r ) := {x ∈ X : d(x 0 , x) ≤ r } is the closed ball of center x 0 and radius r .

Local fixed point theorems
In this section, we will present some local fixed point theorems for three types of generalized contractions. The first main result of this section is a local fixed point theorem concerningĆirić-Reich-Rus type of operators (see [12,13]). Theorem 2.1 Let (X, d) be a complete metric space, x 0 ∈ X , a positive number r and let f : then the sequence ( f n (x 0 )) n∈N of successive approximations starting from the center of the ball converges to a point x * which is a fixed point for theĆirić-Reich-Rus contraction f . Moreover, the fixed point is unique.
The considered sequence ( f n (x 0 )) n∈N has the recurrent form x n+1 = f (x n ), for all n ∈ N. Then, since We denote q := α + β 1 − β < 1. We assume holds for n ∈ N * , n ≥ 2 and we prove p(n + 1) by mathematical induction: From the above relation, we have thus, our assumption p(n) holds for all n ∈ N * , n ≥ 2.
We continue with proving that the sequence ( f n (x 0 )) n∈N is in the closed ballB(x 0 ; s), for all n ∈ N. We know that We consider an arbitrary n ∈ N * , n ≥ 2 and we compute: which proves that all elements of the sequence ( f n (x 0 )) n∈N are still in the closed ballB(x 0 ; s). Next, we prove that the sequence ( f n (x 0 )) n∈N is Cauchy inB(x 0 ; s). For n ∈ N and p ∈ N * we have Thus, we obtain that the sequence is Cauchy. By the completeness of the metric space, we also get that ( f n (x 0 )) n∈N is convergent to a point x * ∈B(x 0 ; s). We now prove that x * is a fixed point. We have Taking n −→ ∞ in the above inequality we obtain which proves that x * is a fixed point for f . Lastly, we will show by contradiction that the fixed point x * is unique in the open ball B(x 0 ; r ). We assume there exists another fixed point y * ∈ B(x 0 ; r ) such that y * = x * . Then which implies that α ≥ 1, contradicting the hypothesis and proving that x * is the unique fixed point.
In our next result we will consider Chatterjea generalized contractions and we will prove another local fixed point result for this class of mappings.

Theorem 2.2
Let (X, d) be a complete metric space, x 0 ∈ X , a positive number r and let f : B(x 0 ; r ) → X be a Chatterjea type contraction. If then the sequence ( f n (x 0 )) n∈N of successive approximations starting from the center of the ball converges to a point x * which is a fixed point of f . Moreover, the fixed point is unique.
The considered sequence ( f n (x 0 )) n∈N has the recurrent form x n+1 = f (x n ) for all n ∈ N. We will compute the following distance: We now denote q := γ 1 − γ < 1 and assume true for n ∈ N * , n ≥ 2. We compute p(n + 1) This implies proving p(n + 1), so by mathematical induction p(n) holds for all n ∈ N * , n ≥ 2.
We will now show that the sequence ( f n (x 0 )) n∈N is in the closed ballB(x 0 ; s), for all n ∈ N. From the hypothesis, the following relation is known: We consider an arbitrary n ∈ N * , n ≥ 2 and we compute This proves that all the elements of the sequence ( f n (x 0 )) n∈N are in the closed ballB(x 0 ; s). We now prove that the sequence ( f n (x 0 )) n∈N is Cauchy inB(x 0 ; s). Letting n ∈ N and p ∈ N * we obtain Therefore, the sequence is Cauchy and considering the completeness of the metric space, the sequence ( f n (x 0 )) n∈N is convergent to a point x * ∈B(x 0 ; s). We will next show the point x * is a fixed point. We estimate the distance We now take n −→ ∞ and we obtain which proves that x * is a fixed point for the Chatterjea contraction f . In the last part, we need to prove that x * is the unique fixed point of f , which we will do by contradiction. Assume that there exists another fixed point y * ∈ B(x 0 ; r ) such that y * = x * . Then which implies that 2γ ≥ 1, contradicting the hypothesis and proving that x * is the unique fixed point.
The last local fixed point theorem proven in this section refers to the Berinde generalized contraction (see [3,5,6]). This extension will not include the uniqueness of the fixed point, which is somehow an expected fact, since Berinde contractions are weakly Picard operators, but (in general) are not Picard operators.

Theorem 2.3
Let (X, d) be a complete metric space, x 0 ∈ X , a positive number r and let f : then the sequence ( f n (x 0 )) n∈N of successive approximations starting from the center of the ball converges to a point x * which is a fixed point of f .
The sequence ( f n (x 0 )) n∈N of successive approximations has the recurrent form We evaluate the following distance: it follows that From the above inequality, we have proven p(n + 1), so by mathematical induction p(n) holds for all n ∈ N * , n ≥ 2.
Now we show that the sequence ( f n (x 0 )) n∈N is in the closed ballB(x 0 ; s), for all n ∈ N. For this purpose, consider an arbitrary n ∈ N and we compute the following distance: proving that all the elements of the sequence ( f n (x 0 )) n∈N are in the closed ballB(x 0 ; s). Next we will show the sequence ( f n (x 0 )) n∈N is Cauchy inB(x 0 ; s). We take n ∈ N and p ∈ N * and evaluate Therefore, we get that the sequence is Cauchy and together with the completeness of the metric space, it is also convergent inB(x 0 ; s) to a point x * .
To prove x * is a fixed point, we will estimate the following distance: This leads us to which also proves that x * is a fixed point for f .

Remark 2.4
For other local or nonself fixed point theorems, see [4][5][6]9,10,[12][13][14][15]. Our results improve some corresponding local fixed point theorems, since here we do not impose conditions to get the invariance of the ball, only (weaker) assumptions assuring the convergence of the sequence of Picard iterates starting from the center of the ball to a fixed point.

Open mapping results
In this section, we present an application of the local fixed point theorems to open mapping principles. First, we consider the case ofĆirić-Reich-Rus type operators.  (u; r )).
For this, let u ∈ V such that B(u; r ) ⊂ V, let y 0 ∈ B g(u); 1 − α − 2β 1 − β r , and consider the operator Then, h is aĆirić-Reich-Rus type contraction, since for x 1 , x 2 ∈ B(u; r ) arbitrarily chosen, we have Now, we compute the following distance: By the local fixed point theorem forĆirić-Reich-Rus generalized contractions, we obtain that there exists a unique fixed point x * ∈ B(u; r ), such that for the chosen y 0 ∈ B g(u); We denote the set W := B g(u); 1 − α − 2β 1 + β r . Since B(u; r ) ⊂ V and also W ⊂ g(B(u; r )) we obtain that W ⊂ g(B(u; r )) ⊂ g(V To show the above inclusion, let u ∈ V such that B(u; r ) ⊂ V , and let y 0 ∈ B g(u); 1 − 2γ 1 − γ r . We consider the same operator h defined at (7), which now is a Chatterjea contraction, because for any arbitrary x 1 and x 2 in B(u; r ), we get We now compute Having this estimation, in view of the local fixed point theorem for the Chatterjea operator, we obtain that there exists a unique fixed point x * ∈ B(u; r ) such that for the chosen y 0 , we have g(B(u; r )).
We denote W := B g(u); 1 − 2γ 1 + γ r . We already know that B(u; r ) ⊂ V which implies that g(B(u; r )) ⊂ g(V ), and because W ⊂ g(B(u; r )), we can conclude that the field operator g is open in the Banach space (E, · ).
The final case for this application is the one of Berinde type operators, where we obtain the same conclusion. (1 − α)r ). Again, we consider the operator h defined at (7), and we state that, in this case, it is a Berinde operator, since for any arbitrary x 1 and We still need to compute the following distance We consider W := B(g(u); (1 − α)r ). We already know that B(u; r ) ⊂ V , which means that g(B(u; r )) ⊂ g(V ), and since the inclusion W ⊂ g(B(u; r )) happens, then W ⊂ g(V ) concluding that g is an open operator.
Remark 3.4 Similar results as presented above have been obtained in [10]. Our results complement some theorems from the above mentioned paper, by considering other classes of generalized contractions.

Continuation theorems
In the last section of the paper, we will present some continuation results for three classes of generalized contractions:Ćirić-Reich-Rus contractions, Chatterjea contractions and Berinde type contractions. We denote C R(Y, X ) the family of all contractions from Y to X and by We begin by defining the concept of (α, β)-contractive family. (X, d) be a metric space and (J, ρ) be a connected metric space. We say that the sequence (H λ ) λ∈J ⊂ C R(Y, X ) is an (α, β)-contractive family if there exist α ∈ (0, 1), p ∈ (0, 1] and M > 0 such that

Definition 4.1 Let
Below we have the continuation principle corresponding to theĆirić-Reich-Rus generalized contractions.

Theorem 4.2 Let (X, d) be a complete metric space and Y a closed subset such that intY = ∅. Let (J, ρ) be a connected metric space and (H λ ) λ∈J be an (α, β)-contractive family from C R δY (Y, X ). The following conclusions occur:
(i) If there exists a point λ * 0 ∈ J , such that the equation H λ * 0 (x) = x has a solution, then the equation H λ (x) = x has a unique solution for any λ ∈ J ; (ii) If H λ (x λ ) = x λ for any λ ∈ J , then the operator is continuous.
Proof Let x λ and x μ be two fixed points of H λ and H μ , respectively. Then, This inequality implies that for any λ and μ. Let We will show that the set Q is both closed and open. First, for proving that Q is closed, let (λ n ) n∈N ⊂ Q such that λ n −→ λ * . We now show that λ * ∈ Q.
We also know the fact that the sequence (λ n ) n∈N is Cauchy, which implies The last two inequalities tell us that proving that the sequence (x λ n ) is Cauchy, and since the space X is complete, we get that it is also convergent, and so, the set Q is closed. For proving that Q is also open, let λ 0 ∈ Q. This implies that there exists a point x λ 0 ∈ intY such that x λ 0 = H λ 0 (x λ 0 ). We will show that there exist ε > 0 and an open ball B(λ 0 ; ε) ⊂ Q. Since intY is open and x λ 0 ∈ intY , we obtain that there exists an open ball B(x λ 0 ; r ) ⊆ intY . Now, let ε > 0 such that , and let an arbitrary λ ∈ B(λ 0 ; ε). We will now show that λ ∈ Q. We begin by estimating the distance between H λ (x λ 0 ) and x λ 0 : From the above relations, we have that is aĆirić-Reich-Rus contraction. By the local fixed point theorem for theĆirić-Reich-Rus operators, we obtain that Fix(H λ ) = ∅, which implies that λ ∈ Q. From what we have proven until now, we get that the operator j is single-valued and with an arbitrary ε > 0. This immediately implies that which proves the fact that j is continuous.
Next, we introduce the notion of a γ -contractive family Below we have the application for the Chatterjea generalized contractions. (ii) If H λ (x λ ) = x λ , for any λ ∈ J , then the operator defined at (8) is continuous.
Proof Following the example of Theorem 4.2, we start by estimating the distance between two arbitrary fixed points x λ and x μ of the operators H λ and H μ , respectively: Now, we consider the set Q defined at (10) and again we try to prove that it is both closed and open, to prove that Q is the whole space J . We begin by proving Q is closed.
Let the sequence (λ n ) n∈N ⊂ Q such that λ n −→ λ * . We show that λ * ∈ Q. Arbitrarily taking two fixed points x λ and x μ of the operators H λ and H μ , respectively, we have Since the sequence (λ n ) n∈N is Cauchy, then there exists an ε > 0 such that Taking into account the two previous inequalities, we get that thus the sequence (x λ n ) λ n ∈J is Cauchy in Y , and because the space is complete, it is also convergent. In view of the Hölder continuity, the sequence (λ n ) n∈N is also convergent, showing that Q is closed. For obtaining that Q is also open, let λ 0 ∈ Q arbitrarily, meaning that there exists a point x λ 0 ∈ intY such that x λ 0 = H λ 0 (x λ 0 ). We will show that there exists an open ball B(λ 0 ; ε) ⊂ Q, with ε > 0. Since intY is open and x λ 0 ∈ intY , we get the existence of an open ball B(x λ 0 ; r ) ⊆ intY , with r > 0. We take ε > 0 such that an arbitrary λ ∈ B(λ 0 ; ε), and we prove that λ ∈ Q. For this, we compute: We know the operator is a Chatterjea contraction. Considering the estimation for the distance between H λ (x λ 0 ) and x λ 0 , we are in the terms of the local fixed point theorem for the Chatterjea operator, thus getting that Fix(H λ ) = ∅, which proves that λ is indeed in Q. We already know that the operator j is single-valued, and also To prove the continuity of this operator, we consider with an arbitrary ε > 0. Considering the two previous inequalities, we get which proves the desired conclusions.
We also define the notion of an (α, L)-contractive family and introduce the continuation principle in regards of the Berinde type of operators.

Definition 4.5
Let (X, d) be a metric space and (J, ρ) be a connected metric space. We define the (α, L)contractive family as a sequence (H λ ) λ∈J included in C R(Y, X ) such that there exist α ∈ (0, 1), M > 0 and p ∈ (0, 1] with The next theorem has similar conclusions as above, but also requires an additional condition for the considered (α, L) -contractive family. (ii) If H λ (x λ ) = x λ , for any λ ∈ J and also α + L < 1, where α ∈ (0, 1) and L > 0, then the operator is single valued and continuous.
Proof As before, let x λ be a fixed point of H λ and x μ a fixed point of the operator H μ . Then, which implies that By the same rationale as before, we consider the set Q = {λ ∈ J | ∃x λ ∈ intY such that x λ = H λ (x λ )} , and we will prove that it is both open and closed. For this, let (λ n ) n∈N be a Cauchy sequence in Q. Then, We know that the sequence is Cauchy, so ρ(λ m , λ n ) < ε := ε (1 − α − L) which also implies that d(x λ m , x λ n ) < ε , meaning the sequence (x λ n ) n∈N is Cauchy in a complete metric space, thus convergent. By the Hölder continuity, we also get that the sequence (λ n ) n∈N is convergent to a point λ * in Q. This proves that the set Q is closed.
To prove that the set Q is open, we consider λ 0 in Q, thus there exists a point x λ 0 in intY such that x λ 0 = H λ 0 (x λ 0 ). Again, we prove that there exist ε > 0 and an open ball B(λ 0 ; ε) ⊂ Q. Since intY is open and x λ 0 ∈ intY , we obtain the existence of an open ball B(x λ 0 ; r ) ⊆ intY . We consider ε > 0 such that , and an arbitrary λ in B(λ 0 ; ε). We will prove that λ is in Q.
Next, we define the operator H λ : B(x λ 0 ; r ) → X, as a Berinde contraction. Considering the previous estimation for the distance between H λ (x λ 0 ) and x λ 0 , we are in the terms of the local fixed point theorem for the Berinde operator, and so, we get that Fix(H λ ) = ∅, thus proving that λ is indeed in Q.
Remark 4.1 Further research for generalized contractions in various metric type spaces can be considered following [1,2,13,15].