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Some results in fixed point theory and application to the convergence of some iterative processes
 Najeh Redjel^{1, 2}Email author and
 Abdelkader Dehici^{1, 2}
https://doi.org/10.1186/s1366301504151
© Redjel and Dehici 2015
 Received: 20 March 2015
 Accepted: 28 August 2015
 Published: 24 September 2015
Abstract
In this paper, we study the existence and uniqueness of fixed points for a class of selfmappings satisfying certain rational expressions on closed, bounded and convex subsets with normal structures in reflexive Banach spaces. We show that, in particular, this class extends that introduced by Ray and Singh (Indian J. Pure Appl. Math. 9:216221, 1978). As an application, we give an investigation of the convergence and stability of some iterative processes associated to these mappings.
Keywords
 Banach space
 normal structure
 rational expression
 fixed point
 Picard iteration
 Mann iteration
 Ishikawa iteration
 Kirk iteration
MSC
 47H09
 47H10
 54H25
1 Introduction
It is well known that the Banach contraction principle was the starting point for the development of fixed point theory allowing the advancement of the analysis and nonlinear analysis in particular. During more than fifty years, there has been a lot of production in this area and many wellknown fixed theorems have been established and investigated by several authors (see, for example, [1–7] and the references therein).
Definition 1.1
 (i)
Φ is monotone nondecreasing;
 (ii)
\(\lim_{n \longrightarrow+ \infty} \Phi^{n}(t) = 0\) for all \(t > 0\) (\(\Phi^{n}\) stands for the nth iterate of Φ).
Remark 1.2
Every comparison function satisfies \(\Phi(0) = 0\) and \(\Phi(t) < t\), \(\forall t > 0\).
Fixed point theory has been and will remain an important tool in the study of the existence and uniqueness of solutions of differential and integral equations. Among the extensions of Banach’s principle, we quote the fixed point result of Boyd and Wong [8].
Theorem 1.3
[8]
Other rational expressions generalizing the same principle have been investigated by several authors. Among the more classical ones, those that have been established by Jaggi, Dass and Gupta.
Theorem 1.4
[9]
Theorem 1.5
[10]
Definition 1.6
Let \((X, \\cdot\)\) be a normed space and K be a nonempty subset of X. A selfmapping T on K is called a nonexpansive mapping if \(\T(x)  T(y)\ \leq\ x  y \\) for all \(x, y \in K\).
The set of nonexpansive selfmappings on K will be denoted by \(\Delta_{K}\).
We note that the existence and uniqueness of fixed points for this type of mappings are not in general ensured. However, some fixed point theorems for nonexpansive mappings have been derived by several authors (we quote, for example, [11, 12] and Chapter 3 of [13]).
Definition 1.7
Remark 1.8
If T is orbitally controlled, then for each \(x \in K\) and \(n \geq2\), there exist \(1 \leq \beta_{1}^{x} \leq n\) and \(2 \leq \beta _{2}^{x} \leq n\) such that \(\delta(O(x, n)) = \x  T^{\beta_{1}^{x}}(x)\\) and \(\delta((O(T(x), n1))) = \T(x)  T^{\beta_{2}^{x}}(x)\\). We denote by \(r_{1}^{x}(T)\) (resp. \(r_{2}^{x}(T)\)) the smallest integer \(\beta_{1}^{x}\) (resp. \(\beta_{2}^{x}\)) such that \(\delta(O(x, n)) = \x  T^{\beta_{1}^{x}}(x)\\) (resp. \(\delta(O(T(x), n  1)) = \T(x)  T^{\beta_{2}^{x}}(x)\\). We note that \(\delta(O(x, n)) \geq\delta(O(T(x), n  1))\).
The set of orbitally controlled selfmappings on K will be denoted by \(\Xi_{K}\).
2 Main results
First of all, we will show that \(\Xi_{K}\) contains in particular the set of all nonexpansive selfmappings \(\Delta_{K}\).
Proposition 2.1
Let \((X, \\cdot\)\) be a normed space and K be a nonempty bounded subset of X. Then \(\Delta _{K} \subseteq\Xi_{K}\).
Proof
Let us give now the definition of quasicontraction selfmappings.
Definition 2.2
The set of quasicontraction selfmappings on K will be denoted by \(\Theta_{K}\).
Proposition 2.3
Let \((X, \\cdot\)\) be a normed space and K be a nonempty bounded subset of X. Then \(\Theta _{K} \subseteq\Xi_{K}\).
Proof
Lemma 2.4
Let \((X, \\cdot\)\) be a normed space and K be a nonempty subset of X. Let \(n \in\mathbb{N}\), \(n \geq2\), if \(T \in\Delta_{K}\) and \(x \in K\) such that \(\delta(O(x, n)) = \delta (O(T(x), n  1))\), then \(r_{2}^{x}(T) > r_{1}^{x}(T)\).
Proof
Assume that \(r_{2}^{x}(T) \leq r_{1}^{x}(T)\). By hypothesis, we obtain that \(\delta(O(x, n)) = \delta(O(T(x), n  1)) = \T(x)  T^{r_{2}^{x}(T)}(x)\ = \x T^{r_{1}^{x}(T)}(x)\\). The fact that \(T \in \Delta_{K}\) implies that \(\delta(O(x, n)) = \delta(O(T(x), n  1)) = \ T(x)  T^{r_{2}^{x}(T)}(x)\ \leq\x  T^{r_{2}^{x}(T)  1}(x)\\), which is a contradiction with \(r_{1}^{x}(T)\) is the smallest integer \(\beta _{1}^{x}\) such that \(\delta(O(x, n)) = \x  T^{\beta_{1}^{x}}(x)\\). □
Proposition 2.5
If K is a nonempty, bounded and convex subset in X and \(T \in\Xi_{K}\) such that \(r_{2}^{x}(T) > r_{1}^{x}(T)\) for all \(x \in K\), then \(F(T) = F(P(T))\).
Proof
Remark 2.6
We note also that the condition \(\lambda_{1} > 0\) is crucial as the following example shows.
Example 2.7
By the same techniques as in the proof of Proposition 2.5, we prove the following proposition.
Proposition 2.8
If K is a nonempty, bounded and convex subset in a normed space X and \(T \in\Xi_{K}\) with \(k = 1\) for all \(n \in\mathbb{N}\), \(n \geq2\) and all \(x \in K\), then \(F(T) = F(P(T))\).
Proof
Now, we give the following important preparatory result.
Proposition 2.9
 (i)
\(\Phi_{1}, \Phi_{2} : [0,+\infty[\, \times\,[0,+\infty[\, \longrightarrow [0,+\infty[\) nondecreasing mappings relative to the first variable and \(\Phi_{3}: [0,+\infty[\, \times\,[0,+\infty[\, \longrightarrow [0,+\infty[\) such that \(\Phi_{3}(t_{2}, t_{3}) = 0 \Longleftrightarrow(t_{2}, t_{3}) = (0, 0)\);
 (ii)
\(\Phi_{2}(t, 0) = 0\) for all \(t \geq0\);
 (iii)
\(\frac{\Phi_{1}(t_{1}, t_{2}) + \Phi_{2}(t_{1}, t_{3})}{\Phi_{3}(t_{2}, t_{3})}\leq t_{1}\) for all \(t_{1} \geq 0\) and \((t_{2}, t_{3}) \in[0, + \infty[ \,\times\,[0, + \infty[ \,\backslash\{ (0, 0)\}\).
Proof
Definition 2.10
Under the assumptions of Proposition 2.9, in the following theorem we take \(\Phi_{1}(t_{1}, t_{2}) = \psi_{1}(t_{1}) \psi_{2}(t_{2})\), \(\Phi _{2}(t_{1}, t_{2}) = \psi_{1}(t_{1}) \psi_{3}(t_{2})\), \(\Phi_{3}(t_{1}, t_{2}) = \psi_{2}(t_{1}) + \psi_{3}(t_{2})\) (\(t_{1}, t_{2} \geq0\)), where \(\psi_{1}\), \(\psi_{2}\) and \(\psi_{3}\) are selfmappings defined on \([0, + \infty[\).
Theorem 2.11
Then T has a unique fixed point in K.
Proof

If \(\delta(K_{1}) = 0\), the problem is solved since in this case \(K_{1} = \{x_{0}\}\), and thus \(T(x_{0}) = x_{0}\).

Assume that \(\delta(K_{1}) > 0\).
Second case: \(g = \sum_{i = 1}^{n}\lambda_{i} T(h_{i})\), \(h_{i} \in H\), \(\lambda_{i} \geq0\), \(\sum_{i = 1}^{n}\lambda_{i} = 1\).
Remark 2.12
If \(\psi_{1}(t) = \psi_{2}(t) = \psi _{3}(t) = t\), we find the main result of [14].
3 Applications
We start this section by giving the concept of φquasinonexpansive mappings.
Definition 3.1
It is easy to observe that every contraction mapping is φquasinonexpansive with \(\varphi(t) = \alpha t\), \(0\leq\alpha <1\) and \(t \in[0, +\infty[\), but the converse is false in general as the following example shows.
Example 3.2
Definition 3.3
A function \(\Phi: [0, + \infty[ \,\longrightarrow[0, + \infty[\) is called subadditive if for all \(t_{1}, t_{2} \in[0, + \infty[\), we have \(\Phi(t_{1} + t_{2}) \leq\Phi (t_{1}) + \Phi(t_{2})\).
Theorem 3.4
Under assumptions of Theorem 2.11, if \(\psi_{1}\) is a subadditive function with \(\psi_{1} \leq\min\{ \psi_{2}, \psi_{3}\}\), then T is a \(\psi_{2}\)quasinonexpansive selfmapping on K.
Proof
In the case where \(\beta_{n} = 0\), the Ishikawa iteration reduces to the Mann iteration. In general, there is no dependence between convergence results for Picard, Mann and Ishikawa iterations. However, some partial results on the equivalence of these processes have been given by Rhoades and Soltuz (see [21–26]).
By using Theorem 3.4 together with ([28], Theorem 3.7), we obtain the following result for the convergence of the iterative schemes of Mann and Ishikawa.
Proposition 3.5
Let \((X, \\cdot\)\) be a Banach space. Let \(\{\alpha_{n}\}_{n}\) and \(\{\beta_{n}\}_{n}\) be two real sequences in \([0, 1]\) such that \(\{\alpha_{n} \beta_{n}\}_{n}\) converges to some positive real number, let \(x_{0} \in X\). Under the assumptions of Theorem 3.4 with \(\psi_{2}\) a comparison continuous function, the Ishikawa sequence given by (6) converges to the unique fixed point of T. Moreover, if \(\{\beta_{n}\}_{n}\) is the constant sequence equal to 0, the Mann iteration given by (5) converges to the same unique fixed point of T.
Definition 3.6
Let A be a bounded subset of a normed space \((X, \\cdot\)\). The Kuratowski measure of noncompactness of A denoted by \(\alpha(A)\) is defined as the infimum of all \(\epsilon> 0\) such that A admits a finite covering consisting of subsets with diameter less than ϵ. A continuous mapping \(T: X \longrightarrow X\) is said to be densifying if for every bounded subset A of X such that \(\alpha(A) > 0\), we have \(\alpha(T(A)) < \alpha(A)\).
 (a)
regularity: \(\alpha(\Omega) = 0\) if and only if Ω is totally bounded;
 (b)
nonsingularity: α is equal to zero on every oneelement set;
 (c)
monotonicity: \(\Omega_{1} \subseteq\Omega_{2}\) implies \(\alpha(\Omega_{1}) \leq\alpha(\Omega_{2})\);
 (d)
semiadditivity: \(\alpha(\Omega_{1} \cup\Omega _{2}) = \max\{\alpha(\Omega_{1}),\alpha(\Omega_{2}) \}\);
 (e)
Lipschitzianity: \( \alpha(\Omega_{1})  \alpha (\Omega_{2}) \leq2 \rho(\Omega_{1}, \Omega_{2}) \), where ρ denotes the Hausdorff semimetric \(\rho(\Omega_{1}, \Omega_{2}) = \inf\{\epsilon> 0: \Omega_{1} + \epsilon\overline{B}_{X} \supset\Omega_{2},\Omega_{2} + \epsilon \overline{B}_{X} \supset\Omega_{1} \}\);
 (f)
continuity: For any \(\Omega\in P(X)\) and any ϵ, there exists \(\delta> 0\) such that \(\alpha(\Omega)  \alpha(\Omega_{1}) < \epsilon\) for all \(\Omega_{1}\) satisfying \(\rho(\Omega , \Omega_{1}) < \delta\);
 (g)
semihomogeneity: \(\alpha(t \Omega) = t \alpha (\Omega)\) for any number t;
 (h)
algebraic semiadditivity: \(\alpha(\Omega_{1} + \Omega_{2}) \leq\alpha(\Omega_{1}) + \alpha(\Omega_{2})\);
 (i)
invariance under translations: \(\alpha(\Omega+ x_{0}) = \alpha(\Omega)\) for any \(x_{0} \in X\).
The following two theorems are omitted.
Theorem 3.7
([29], Theorem 1.1.5)
The Kuratowski measure of noncompactness is invariant under passage to the closure and to the convex hull: \(\alpha(\Omega) = \alpha(\overline{\Omega}) = \alpha(\operatorname{co}(\Omega))\).
Let us recall the following theorem due to Diaz and Metcalf [11].
Theorem 3.8
 (i)
\(F(T)\) is nonempty;
 (ii)for each \(y \in X\) such that \(y \notin F(T)\) and for each \(z \in F(T)\), we have$$d\bigl(T(y), z \bigr) < d( y, z ). $$
 (a)
for each \(x_{0} \in X\), the Picard sequence \(\{T^{n}(x_{0})\}\) contains no convergent subsequences;
 (b)
for each \(x_{0} \in X\), the sequence \(\{T^{n}(x_{0})\}\) converges to a point belonging to \(F(T)\).
Theorem 3.9
([30])
Let \(T: K \longrightarrow K\) be a densifying mapping defined on a closed, bounded, convex subset K of a Banach space X. Then T has at least one fixed point.
Let us define the geometric property \(P (f_{1}, f_{2}, f_{3})\) given as follows.
Let \((X, \\cdot\)\) be a normed space, we say that X has the property \(P (f_{1}, f_{2}, f_{3})\) if for all \(x, y \in X\), \(f_{1}(\ x + y\) = f_{2}(\ x\) + f_{3}(\ y\)\) and \(x \neq0\), \(y \neq0\), then \(x = cy \) (\(c > 0\)).
In the case where \(f_{1} = f_{2} = f_{3} = I_{{\mathbb{R}}^{+}}\), this property is satisfied by strictly convex Banach spaces.
Theorem 3.10
Let \(T: K \longrightarrow K\) be a densifying mapping defined on a closed, bounded, convex subset of a Banach space X having the property \(P (\psi_{1}, \psi_{2}, \psi_{3})\), where \(\psi_{2}(t) \leq t \) for all \(t \geq0\) and \(\psi_{1}\) is a nondecreasing, subadditive function with \(\psi_{1} \leq\min\{\psi_{2}, \psi_{3}\}\). Assume that T satisfies (2) and \(F(T) \cap\sum_{T} = \emptyset\). Then, for each \(x \in K\), the sequence \(\{S^{n}(x)\}\) converges to a fixed point of T.
Proof
As an immediate consequence of the proof of Theorem 3.10, we have the following.
Theorem 3.11
Let \((X, \\cdot\)\) be a reflexive Banach space having the property \(P (\psi_{1}, \psi_{2}, \psi_{3})\), where \(\psi_{2}(t) \leq t \) for all \(t \geq0\) and \(\psi_{1}\) is a nondecreasing, subadditive function with \(\psi_{1} \leq\min\{\psi_{2}, \psi_{3}\}\). Let \(T: K \longrightarrow K\) be a selfmapping defined on a closed, bounded, convex subset K with normal structure in X. Assume that T satisfies assumptions of Theorem 2.11. If \(x_{0} \notin\sum_{T}\) (where \(x_{0}\) is the unique fixed point of T in K) and S is a densifying mapping, then, for each \(x \in K\), the sequence \(\{S^{n}(x)\} \) converges to \(x_{0}\).
Recall that for the case of numerical stability we say that a fixed point iteration process is numerically stable if small perturbation in the initial data induces a small influence of the computed value of the fixed point. For the remainder of our study, we need the following two definitions about stability of general iterative processes.
Definition 3.12
 (i)
the iteration process (7) is said to be Tstable if \(\lim_{n \longrightarrow\infty}\epsilon_{n} = 0\) implies \(\lim_{n \longrightarrow\infty} y_{n} = z_{0}\);
 (ii)
the iteration process (7) is said to be almost Tstable if \(\sum_{n \in\mathbb{N}}\epsilon_{n} < \infty\) implies \(\lim_{n \longrightarrow\infty} y_{n} = z_{0}\).
For more information and interesting comments on these notions of stability, we can see [31]. On the other hand, it is easy to observe that an iterative process (7) which is Tstable is almost Tstable but the converse is not true in general (see the counter example given in [32]). Furthermore, the iterative processes can converge without being stable. Indeed, the following example given in [32, 33] confirms this.
Example 3.13
Let \((X, \\cdot\) = (\mathbb {R}, \cdot)\) and \(T: [0, 1]\longrightarrow[0, 1]\), \(T(x) = x\). It is easy to observe that \(F(T) = [0, 1]\).
In the following result, we establish almost stability of Picard’s iterative process for our context of selfmappings.
Corollary 3.14
Proof
The result is established by combining the fact that T is \(\psi_{2}\)quasinonexpansive together with Theorem 4.5 in [28]. □
Declarations
Acknowledgements
The authors would like to thank the anonymous referees and the editor for their valuable comments which helped to improve this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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