An extension of the contraction mapping principle to Lipschitzian mappings
- Xavier A Udo-utun^{1}Email author,
- Zakawat U Siddiqui^{1} and
- Mohammed Y Balla^{1}
https://doi.org/10.1186/s13663-015-0295-4
© Udo-utun et al.; licensee Springer. 2015
Received: 27 May 2014
Accepted: 29 January 2015
Published: 7 September 2015
Abstract
Very recently, Udo-utun (Fixed Point Theory Appl. 2014:65, 2014) established a property possessed by contractive operators with nonempty fixed point sets and used it to prove the existence of fixed points of nonexpansive maps. We extend these results and prove the existence of fixed points for L-Lipschitzian maps that possess this property. Our results generalize and unify results concerning asymptotic fixed point theory of contractive mappings. An application in ordinary differential equations is illustrated via a formulation of the global existence theorem for initial value problems.
Keywords
MSC
1 Introduction
- p1.
T has a unique fixed point given by the limit \(p = \lim_{n \to\infty}x_{n} = T^{n}x_{0}\) of the Picard iteration \(\{x_{n}\}_{n = 1}^{\infty}\) where \(x_{0}\) is any initial guess in K.
- p2.
The following estimates hold: \(\|x_{n} - p\| \leq\frac{L^{n}}{1 - L}\|x_{0} - x_{1}\|\), \(\|x_{n} - p\| \leq\frac{L}{1 - L}\|x_{0} - x_{1}\|\).
- p3.
The rate of convergence of Picard iteration \(\{x_{n + 1}\} = \{Tx_{n}\}\) is given by \(\|x_{n} - p\| \leq L\|x_{n - 1} - p\|\).
Following Ciric [3], Berinde [1, 4] investigated all contractive conditions which make use of displacements of the forms \(d(x , y)\), \(d(x , Tx)\), \(d(y , Ty)\), \(d(x , Ty)\), and \(d(y , Tx)\) under the concept of a \((\delta, k)\)-weak contraction introduced by him as follows.
Definition 1
[4]
Remark 1
A current terminology for the class of \((\delta, k)\)-weak contractions is almost contractions.
The results of this paper constitute applications of this concept of \((\delta, k)\)-weak contractions to obtain existence results for all Lipschitzian maps satisfying a certain boundedness notion as stated below. It is also shown that this condition is a nontrivial extension and modification of the Banach contraction mapping principle in Banach spaces.
2 Preliminary
Our method involves proving that for arbitrary \(L(0 , \infty)\), any L-Lipschitzian map satisfying (5) possesses a fixed point by showing that a corresponding averaged operator \(S_{\lambda}x = \lambda x + (1 - \lambda)Tx\) is an example of \((\delta, k)\)-weak contractions. In other words, as follows from [1, 4], the Krasnoselskii iteration scheme \(x_{n + 1} = \lambda x_{n} + (1 - \lambda )Tx_{n}\) of T converges to a fixed point of T. Further, we demonstrate that, in Banach spaces, the hypothesis of the contraction mapping principle is a special case of (5) above.
Investigations concerning fixed points of expansive transformations include the work of Goebel and Kirk [5] on uniformly L-Lipschitzian mappings with \(L < \sqrt{\frac{5}{2}}\). This estimate for L was improved by Lifschitz [6] with \(L < \sqrt{2}\). It is worth noting that recent research trends in uniformly L-Lipschitzian mappings are concerned with the normal structure and structure of fixed point sets \(\operatorname{Fix}(T)\) of the semigroup of uniformly L-Lipschitzian mappings. Works in this direction include [7, 8] and [9]. Our results include the existence aspects of these studies as special cases.
Other examples of \((\delta, k)\)-weak contractions are given in [10]. It is shown in [1, 4] that a lot of well-known contractive conditions in the literature are special cases of the almost contraction condition (4) as it does not require that \(\delta+ k < 1\), which is assumed in almost all fixed point theorems based on the contractive conditions which involve displacements of the forms \(d(x , y)\), \(d(x , Tx)\), \(d(y , Ty)\), \(d(x , Ty)\), and \(d(y , Tx)\); see Berinde [1], Kannan [11], Rhoades [12], Zamfirescu [13] and references therein.
Examples of L-Lipschitzian operators satisfying (5) are illustrated below.
Example 1
Let \(E = (-\infty, \infty)\) and \(K= [0 , \infty)\), the operator \(T : K \rightarrow K\) defined for any fixed \(\tau> 0\), defined by \(Tx = \tau e^{\tau- x}\), \(x \in K\) is L-Lipschitzian, since \(\|Tx - Ty\| = |\tau e^{\tau- x} - \tau e^{\tau- y}|\leq\tau e^{\tau}|x - y|\), \(x,y \in K\) and has the fixed point \(x = \tau\). It is remarkable that T satisfies the condition (5) viz.: \(\frac{\|y - Tx\|}{\|x - y\| }= \frac{|y - Tx|}{|x - y|}\leq1 + \frac{|xe^{\tau} - \tau e^{\tau}|}{|x - y|}\). By the mean value theorem we obtain \(\frac{\|y - Tx\|}{\|x - y\|}\leq1 + |x - \tau|\frac{|M_{1} - M_{2}|}{|x - y|}\) for some \(M_{1}\), \(M_{2}\) lying between x and τ. This shows that there exists \(M \geq1\) such that \(\|y - Tx\| \leq M\|x - y\|\) for all x and y in an appropriate closed neighborhood of the fixed point τ with \(x \neq y\) and \(x,y \notin \operatorname{Fix}(T)\).
Similarly, on the other hand the operator \(T : K \rightarrow K\) defined by \(Tx = e^{\tau- x}\) is L-Lipschitzian with Lipschitz constant \(L = e^{\tau}\) and has no fixed point. As a counterexample, it is interesting that T does not satisfy property (5). This follows from the fact that \(\frac{\|y - Tx\|}{\|x - y\|}= \frac{|y - Tx|}{|x - y|}\leq1 + \max\{e^{-x} , e^{-\tau}\}\frac{|1 - \tau|}{|x - y|} > M\) for all \(M > 0\), i.e. when x and y are very close.
In [4] Berinde proved the following theorem, the first almost contraction mapping principle.
Theorem 1
[1]
- (1)
\(\operatorname{Fix}(T) = \{x \in X : Tx = x\} \neq\emptyset\).
- (2)
For any \(x_{0} \in X\) the Picard iteration \(\{x_{n}\}\) given by \(x_{n + 1} = T^{n}x_{0}\), \(n = 0, 1, 2, \ldots \) , converges to some \(x^{*} \in \operatorname{Fix}(T)\).
- (3)The estimateshold, where δ is the constant appearing in (4).$$\begin{aligned}& d\bigl(x_{n} , x^{*}\bigr) \leq \frac{\delta^{n}}{1 - \delta}d(x_{0} , x_{1}),\quad n = 0, 1, 2, \ldots, \\& d\bigl(x_{n} , x^{*}\bigr) \leq \frac{\delta}{1 - \delta}d(x_{n - 1} , x_{n}),\quad n = 0, 1, 2, \ldots, \end{aligned}$$
- (4)Under the additional condition that there exist \(\theta\in (0 , 1)\) and some \(k_{1} \geq0\) such thatthe fixed point \(x^{*}\) is unique and the Picard iteration converges at the rate \(d(x_{n} , x^{*}) \leq \theta d(x_{n -1} , x^{*})\), \(n \in\mathbb{N}\).$$ d(Tx , Ty) \leq \theta d(x , y) + k_{1}d(x , Tx)\quad \textit{for all } x, y \in X , $$(6)
- (a)
to establish our claim it suffices to prove the averaged operator \(S_{\lambda}\) given by \(S_{\lambda}x = \lambda x + (1 - \lambda )Tx\) is a \((\delta, k)\)-weak contraction and then apply Theorem 1 to obtain fixed points of \(S_{\lambda}\);
- (b)
for \(\lambda\in[0 , 1)\) the fixed point set \(\operatorname{Fix}(S_{\lambda })\) of \(S_{\lambda} = \lambda I + (1 - \lambda)T\) coincides with \(\operatorname{Fix}(T)\).
In the sequel we shall make use of the Archimedean property and a recent result in [2] below.
Lemma 1
[2]
Let V be a normed linear space and \(T : V \rightarrow V\) a map. If \(\|x - y\| \leq\|y - Tx\|\) then \(Tx \neq Ty\) for any distinct \(x,y \in V\) satisfying \(x, y \notin \operatorname{Fix}(T)\).
3 Main results
Theorem 2
Proof
In this case \(\|y - S_{\lambda}y\|\) takes the form \(\|x_{m} - x_{m + 1}\|\) where \(x_{k} = S^{k}_{\lambda}x_{0}\), while \(\|x - y\| \leq\|y - Tx\|\) takes the form \(\|x_{n} - x_{m}\| \leq\|x_{m} - x_{n + 1}\|\). Clearly, \(\|x_{m} - x_{m + 1}\| \leq\|x_{n} - x_{m}\|\) since the condition \(\|x_{n} - x_{m}\| \leq\|x_{m} - x_{n + 1}\|\) implies that \(n \geq m\) for n and m large enough. This means that \(\|y - S_{\lambda}y\| \leq\|x - y\|\) whenever \(\|x - y\| \leq\|y - S_{\lambda }x\|\) in \(K_{0}\). This yields \(\|y - Ty - \lambda(y - Ty)\| \leq\|x - y\|\) for all \(\lambda\in[0 , \infty)\). Therefore, since λ can be made as small as we please, also \(\|y - Ty\| \leq\|x - y\|\) in \(K_{0}\), so (9) yields \(\|y - Tx\| \leq3\|x - y\|\) in \(K_{0}\). Let \(K_{1}\) be the smallest open set in K containing \(K_{0}\) and, considering the continuity of T, we conclude that condition (5) is satisfied by nonexpansive mappings T for which \(\operatorname{Fix}(T) \neq \emptyset\). □
Theorem 3
Further, condition (10) generalizes contraction condition (i.e. (1) with \(L \in(0 , 1)\)) in Banach spaces.
Proof
On the other hand if (or when) \(\|x - y\| \leq\|y - Tx\|\) then, by (10), (11) yields \(\|S_{\lambda}x - S_{\lambda}y\| \leq (1 - \lambda)(L + M)\|x - y\| + \|y - S_{\lambda}x\|\). So choosing \(\lambda\in(0 , 1)\) such that \((1 - \lambda) < \min \{\frac{1}{1 + L} , \frac{1}{M + L} \}\) and \(k = 1\) and by Theorem 1 we conclude that \(S_{\lambda}\) has a fixed point in K. Therefore T has a fixed point in K.
To complete the proof we need to establish that condition (10) generalizes, in the Banach space context, the contraction condition, viz.: \(\|Tx - Ty\| \leq L\|x - y\|\); \(L \in(0 , 1)\). We note that since the collection of all contractions is a proper subclass of the class of nonexpansive mappings with nonempty fixed point set then by Theorem 2 all contractions satisfy condition (10) since the fixed point set of contractions is nonempty, which ends the proof. □
Next, using Theorem 2 and putting \(L = 1\) in Theorem 3 we obtain the following fixed point results for nonexpansive mappings in arbitrary Banach spaces.
Corollary 1
The proof follows from the proof of Theorem 3 by putting \(L = 1\). An alternative proof of sufficiency was given in Udo-utun [2].
4 Application to global existence theory
In this section we illustrate applicability of our main results by formulating the global existence condition which constitutes a nontrivial improvement of the well-known Picard-Lindelof result, see for example [14, 15], the theorem for the initial value problem \(x^{\prime} = f(t , x)\); \(x(t_{0}) = x_{0}\) where \(f : G \rightarrow\mathbb{R}\), G is the rectangular region G: \(a \leq t \leq b\), \(\alpha\leq x \leq\beta\) and \(t_{0} \in[a , b]\).
Theorem 4
Proof
5 Conclusion
- 1.
Our fixed point results do not guarantee the uniqueness of fixed points, however, an appropriate application of condition (6) of Theorem 1 yields the uniqueness of fixed points.
- 2.
We observe that Theorem 2 can be investigated for a characterization of Lipschitzian mappings T with nonempty fixed point sets \(\operatorname{Fix}(T)\).
- 3.
It is obvious that condition (5) can be invaluable in ordinary differential and integral equations for investigations of continuation of solutions, asymptotic properties of solutions, absolute stability, and instability of solutions.
Declarations
Acknowledgements
We acknowledge Professor AU Afuwape’s support, for provision of a rich literature and for pointing out current issues in the qualitative theory of ordinary differential equations. The friendliness of the editorial team of Springer-open and the reviewer’s suggestions are extremely appreciated and herewith acknowledged.
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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