Abstract
We prove a fixed point theorem for nonlinear operators, acting on some function spaces (of set-valued maps), which satisfy suitable inclusions. We also show some applications of it in the Ulam type stability.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The question when we can replace an approximate solution to an equation by an exact solution to it (or conversely) and what error we thus commit seems to be very natural. Some convenient tools to study such issues are provided by the theory of Ulam’s (often also called the Hyers–Ulam) type stability. For some updated information and further references concerning the Ulam stability, we refer to [1, 4, 5]. Let us only mention that the investigation of that problem started with a question raised by Ulam in 1940 and an answer to it given by Hyers in [3].
It has been noticed in numerous papers that there are strict connections between some fixed point theorems and the results concerning the Ulam stability of various (differential, difference, functional, and integral) equations; for a suitable survey we refer to [2]. In this paper we continue those investigations by proving a fixed point result for a class of nonlinear operators acting on some spaces of set-valued mappings and showing several of its consequences.
Through this paper, we assume that K is a nonempty set and (Y, d) is a complete metric space. We denote by n(Y) the family of all nonempty subsets of Y, by bd(Y) the family of all nonempty and bounded subsets of Y, and by bcl(Y) the family of all closed sets from bd(Y). Moreover, h is the Hausdorff distance induced by the metric in Y and given by
It is well known that h is a metric if restricted to bcl(Y).
The number (possibly also \(\infty \))
is said to be the diameter of \(A\in n(Y)\). For \(F:K\rightarrow n(Y)\) and \(g:K\rightarrow Y\), we denote by \(\mathrm{cl\,}F\) and \(\widehat{g}\) the multifunctions defined by
We write \(a^0(x)=x\) for \(x\in K\) and \(a^{n+1}=a^n\circ a\) for \(a:K\rightarrow K\), \(n\in \mathbb {N}_0\) (\(\mathbb {N}_0\) stands for the set of nonnegative integers).
We present a theorem, concerning fixed points of some operators acting on set-valued functions, and several of its consequences. To do this, we need to introduce some notations. Namely, given functions \(a,b\in \mathbb {R}^K\) (as usually, \(B^A\) denotes the family of all functions mapping a set \(A\ne \emptyset \) into a set \(B\ne \emptyset \)) and \(F,G\in n(Y)^K\), we write \(a\le b\) provided
and \(F\subset G\) provided
moreover, we define \(F\cup G\in n(Y)^K\) by \((F\cup G)(x):=F(x)\cup G(x)\) for \(x\in K\).
We say that \(\Lambda :{\mathbb {R}_+}^K\rightarrow {\mathbb {R}_+}^K\) (where \(\mathbb {R}_+:=[0,+\infty )\)) is non-decreasing if
We always assume the Tichonoff topology (of pointwise convergence) in \(bcl(Y)^K\), with the Hausdorff metric in bcl(Y).
We write
for each sequence \((H_n)_{n\in \mathbb {N}}\) in \(bcl(Y)^K\) that is convergent in \(bcl(Y)^K\). Next, an operator \(\alpha :n(Y)^K \rightarrow n(Y)^K\) is i.p. (inclusion preserving) if
\(\alpha \) is l.p. (limit preserving) if
for each sequence \((H_n)_{n\in \mathbb {N}}\) in \(bd(Y)^K\), such that the sequences \((\mathrm{cl\,}H_n)_{n\in \mathbb {N}}\) and \((\mathrm{cl\,}(\alpha H_n))_{n\in \mathbb {N}}\) are convergent in \(bcl(Y)^K\).
We also need the following hypothesis for operators \(\alpha :bd(Y)^K \rightarrow bd(Y)^K\).
-
(H) \(\alpha \widehat{f}\) is single valued for each \(f\in Y^K\) and
$$\begin{aligned} \lim _{n \rightarrow \infty }\mathrm{cl\,}(\alpha H_n) \subset \mathrm{cl\,}\alpha \Big (\lim _{n\rightarrow \infty }\mathrm{cl\,}H_n\Big ) \end{aligned}$$for each sequence \((H_n)_{n\in \mathbb {N}}\subset bd(Y)^K\), such that the sequences \((\mathrm{cl\,}H_n)_{n\in \mathbb {N}}\) and \((\mathrm{cl\,}(\alpha H_n))_{n\in \mathbb {N}}\) are convergent in \(bcl(Y)^K\).
Clearly, (H) is somewhat complementary to (1).
Finally, \(\widetilde{\delta } :bd(Y)^K\rightarrow {\mathbb {R}_+}^K\) is given by the formula
and, for every \(t\in \mathbb {R}_+\) and \(a\in \mathbb {R}^K_+\), we define the mapping \(ta\in \mathbb {R}^K_+\) by \((ta)(x):=ta(x)\) for \(x\in K\).
2 Main results
In the sequel \(\alpha :bd(Y)^K \rightarrow bd(Y)^K\), \(\mathcal {G}:bd(Y)^K\rightarrow bd(Y)^K\) and \(\Lambda :\mathbb {R}^K_+\rightarrow \mathbb {R}^K_+\) are given. We consider functions \(F\in bd(Y)^K\) that satisfy the equation:
\(\mathcal {G}\)–approximately, i.e., such that
We use the following contraction condition on \(\alpha \):
Now, we are in a position to present the main result of this paper.
Theorem 1
Assume that \(\Lambda \) is non-decreasing, \(\alpha \) is i.p. and satisfies (3), \(F\in bd(Y)^K\), \(\mathcal {G}:bd(Y)^K\rightarrow bd(Y)^K\), (2) holds, and
Suppose that \(\alpha \) is l.p. or (H) is valid. Then, there exists a function \(f:K\rightarrow Y\), such that \(\widehat{f}\) is a fixed point of the operator \(\alpha \) (i.e., \(\alpha \widehat{f}=\widehat{f}\)) and
Moreover, if \(G\in bd(Y)^K\) satisfies the conditions
with some \(\mu :K\rightarrow \mathbb {R}_+\) such that
then \(G=\widehat{f}\).
Proof
Fix \(x\in K\). Since \(\alpha \) is i.p., by (2), we get
for every \(n\in \mathbb {N}_0\) (nonnegative integers). Hence
Therefore, for \(k\in \mathbb {N}\), \(n\in \mathbb {N}_0\), we have
Furthermore, by (4), we get
Moreover,
whence \((\mathrm{cl\,}\alpha ^nF(x))_{n\in \mathbb {N}_0}\) is a Cauchy sequence of closed and bounded sets and, as the space (bcl(Y), h) is complete, there exists the limit
Furthermore, by (3) and (7), we have
and \((\Lambda ^{n}(\widetilde{\delta }F)(x))_{n\in \mathbb {N}_0}\) is convergent to 0 as \(n \rightarrow \infty \). Therefore, the set \(\rho (x)\) has exactly one element for each \(x\in K\) and we denote that element by f(x).
If \(\alpha \) is l.p., it is clear that
Thus, \(\alpha \widehat{f}=\widehat{f}\).
If (H) holds, then
whence again, \(\alpha \widehat{f}=\widehat{f}\). Next, by (6), we have
for \(n\in \mathbb {N}\), and consequently, with \(n\rightarrow \infty \), we obtain \(h(\widehat{f}(x), F(x))\le \kappa (x)\).
It remains to show the statement on the uniqueness of \(\widehat{f}\). Therefore, fix \(G\in bd(Y)^K\) and \(\mu \in \mathbb {R}_+^K\), such that (5) holds, \(G\subset \alpha G \), and \(h(G(x),F(x))\le \mu (x)\) for \(x\in K\). Define the multifunction \(\mathcal {B}_F:K\rightarrow n(Y)\) by
Then, it is easily seen that \(F,G\subset \mathcal {B}_F\), and consequently
Next, for each \(n\in \mathbb {N}\), we have \(G\subset \alpha ^n G\), whence
Note that for every \(x\in K\), \(y,z\in \mathcal {B}_F(x)\) and \(w_1,w_2\in F(x)\), we have
This means that \(\delta (\mathcal {B}_F(x))\le \kappa (x)+2\mu (x)\) for each \(x\in K\). Therefore, we get
This completes the proof in view of (5). \(\square \)
3 Some consequences
The next simple theorems show some direct applications of Theorem 1; they correspond to the results on stability of functional equations (for the set-valued mappings) in [6,7,8,9,10].
Theorem 2
Let \(F, G :K\rightarrow bd(Y)\), \(\Psi :Y\rightarrow Y\), \(\xi :K\rightarrow K\), \(\lambda \in \mathbb {R}_+\),
Then, there exists a unique function \(f:K\rightarrow Y\), such that \(\Psi \circ f\circ \xi =f\) and
Proof
Define \(\alpha :bd(Y)^K \rightarrow bd(Y)^K\) by
Then, it is easily seen that it is i.p. Next, let \((H_n)_{n\in \mathbb {N}}\) be a sequence in \(bd(Y)^K\), such that there exist \(H_L:=\lim _{n\rightarrow \infty }\mathrm{cl\,}H_n \in bcl(Y)^K\) and \(\lim _{n \rightarrow \infty }\mathrm{cl\,}(\alpha H_n)\in bcl(Y)^K.\) Clearly, on account of (9),
for every \(x\in K\) and \(n\in \mathbb {N}\), which implies that
Consequently \(\alpha \) is l.p. Let \(\Lambda :\mathbb {R}^K_+\rightarrow \mathbb {R}^K_+\) be given by
Then, it is non-decreasing and (3) holds. Define \(\mathcal {G}:bd(Y)^K\rightarrow bd(Y)^K\) by
Then, in view of (10), (2) is valid, too. Hence, according to Theorem 1, there exists a function \(f:K\rightarrow Y\), such that \(\widehat{f}\) is a fixed point of the operator \(\alpha \) (i.e., \(\Psi \circ f\circ \xi =f\)) and
Moreover, by (8)
thus (5) holds with \(\mu =\kappa \), and consequently, such f must be unique. \(\square \)
Theorem 3
Assume that \((Y,\cdot )\) is a group with the neutral element e and d is invariant (i.e., \(d(xz,yz)=d(x,y)=d(zx,zy)\) for \(x,y,z\in Y\)). Let \(F, G :K\rightarrow bd(Y)\), \(e\in G(x)\) for \(x\in K\), \(\Psi :Y\rightarrow Y\), \(\xi :K\rightarrow K\), \(\lambda \in \mathbb {R}_+\), (9) holds,
where \(AB:=\{ab:\ a\in A,\, b\in B\}\) for nonempty \(A,B\subset Y\). Then, there exists a unique function \(f:K\rightarrow Y\), such that \(\Psi \circ f\circ \xi =f\) and
Proof
It is sufficient to argue analogously as in the proof of Theorem 2 with function \(\mathcal {G}:bd(Y)^K\rightarrow bd(Y)^K\) given by
Then, in view of (13), (2) is valid and, according to Theorem 1, there exists a function \(f:K\rightarrow Y\), such that \(\widehat{f}\) is a fixed point of \(\alpha \) and
Since
and
we get (14). Furthermore, since \(\kappa (x)\le \mu (x):=\nu (x)+\gamma (x)\) for \(x\in K\), (11) and (12) imply that
Therefore, (5) is valid whence f is unique in view of Theorem 1. \(\square \)
Clearly, in the particular case where \(\lambda \in (0,1)\) and
estimation (14) can be replaced by the following one:
References
Brillouët-Belluot, N., Brzdęk, J., Ciepliński, K.: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal. 2012, 716936-1–716936-41 (2012)
Brzdęk, J., Cădariu, L., Ciepliński, K.: Fixed point theory and the Ulam stability. J. Funct. Sp. 2014, 829419-1–829419-16 (2014)
Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)
Hyers, D.H., Isac, G., Rassias, ThM: Stability of Functional Equations in Several Variables. Birkhäuser, Boston (1998)
Jung, S.M.: Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Optimization and Its Applications, vol. 48. Springer, New York (2011)
Nikodem, K., Popa, D.: On selections of general linear inclusions. Publ. Math. Debrecen 75, 239–249 (2009)
Piszczek, M.: The properties of functional inclusions and Hyers–Ulam stability. Aequ. Math. 85, 111–118 (2013)
Popa, D.: A stability result for a general linear inclusion. Nonlinear Funct. Anal. App. 3, 405–414 (2004)
Popa, D.: Functional inclusions on square-symetric grupoid and Hyers–Ulam stability. Math. Inequal. Appl. 7, 419–428 (2004)
Popa, D.: A property of a functional inclusion connected with Hyers–Ulam stability. J. Math. Inequal. 4, 591–598 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
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.
About this article
Cite this article
Brzdęk, J., Piszczek, M. Fixed points of some nonlinear operators in spaces of multifunctions and the Ulam stability. J. Fixed Point Theory Appl. 19, 2441–2448 (2017). https://doi.org/10.1007/s11784-017-0441-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-017-0441-1