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Fixed-point theorem in classes of function with values in a dq-metric space

  • Janusz Brzdęk
  • El-sayed El-hady
  • Zbigniew LeśniakEmail author
Open Access
Article
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Abstract

We prove a fixed point result for nonlinear operators, acting on some classes of functions with values in a dq-metric space, and show some applications of it. The result has been motivated by some issues arising in Ulam stability. We use a restricted form of a contraction condition.

Keywords

dq-Metric fixed point function space ulam stability difference equation 

Mathematics Subject Classification

39B82 47A63 47H10 47J99 

1 Introduction

The name of Ulam has been somehow connected with various definitions of stability (see, e.g., [1, 12, 16, 24]), but roughly speaking, the following one describes our considerations in this paper (\(A^B\) denotes the family of all functions mapping a nonempty set B into a nonempty set A, \({\mathbb {R}}\) stands for the set of all real numbers and \({\mathbb {R}}_+{:}{=}[0,\infty )\)).

Definition 1

Let (Yd) be a metric space, E be a nonempty set, \({\mathcal {D}}_0\subset {\mathcal {D}}\subset Y^{E}\) and \({\mathcal {E}}\subset {{\mathbb {R}}_+}^{E}\) be nonempty, \({\mathcal {T}}:{\mathcal {D}}\rightarrow Y^E\) and \({\mathcal {S}}:{\mathcal {E}}\rightarrow {{\mathbb {R}}_+}^{E}\). We say that the equation
$$\begin{aligned} {\mathcal {T}}(\psi )(t)=\psi (t),\qquad t\in E, \end{aligned}$$
is \({\mathcal {S}}\)-stable in \({\mathcal {D}}_0\) provided, for any \(\psi \in {\mathcal {D}}_0\) and \(\delta \in {\mathcal {E}}\) with
$$\begin{aligned} d({\mathcal {T}}(\psi )(t),\psi (t))\le \delta (t),\qquad t\in E, \end{aligned}$$
there is a solution \(\phi \in {\mathcal {D}}\) of the equation, such that
$$\begin{aligned} d(\phi (t),\psi (t))\le {\mathcal {S}}\delta (t),\qquad t\in E. \end{aligned}$$
There are some close connections between Ulam stability and fixed-point theory (see, e.g., [6]). In particular, the subsequent theorem has been presented in [7, Theorem 2] and it has been shown there how to deduce some quite general Ulam stability results from it (see also [6, 9, 14]). To formulate it, we need the following hypothesis concerning operators \(\Lambda :{{\mathbb {R}}_+}^E\rightarrow {{\mathbb {R}}_+}^E\) (E is a nonempty set):
\(({\mathcal {C}})\)
If \((\delta _n)_{n\in {\mathbb {N}}}\) is a sequence in \({{\mathbb {R}}_+}^E\) with \(\lim _{n\rightarrow \infty } \delta _n(t)=0\) for \(t\in E\), then
$$\begin{aligned} \lim _{n\rightarrow \infty } \Lambda \delta _n(t)=0,\qquad t\in E. \end{aligned}$$
Let us yet recall that \(\Lambda :{{\mathbb {R}}_+}^E\rightarrow {{\mathbb {R}}_+}^E\) is non-decreasing provided
$$\begin{aligned} \Lambda \xi (t)\le \Lambda \eta (t),\qquad t\in E, \end{aligned}$$
for every \(\xi ,\eta \in {{\mathbb {R}}_+}^E\) with \(\xi (t)\le \eta (t)\) for every \(t\in E\).

Theorem 2

Assume that (Yd) is a complete metric space, E is a nonempty set, \(\Lambda :{{\mathbb {R}}_+}^E\rightarrow {{\mathbb {R}}_+}^E\) is non-decreasing and satisfies hypothesis \(({\mathcal {C}})\), and \({\mathcal {T}}:Y^E\rightarrow Y^E\) is such that
$$\begin{aligned} d\big (({\mathcal {T}}\xi )(t),({\mathcal {T}}\mu )(t)\big )\le \Lambda \big (d(\xi ,\mu )\big )(t),\qquad \xi ,\mu \in Y^E , t\in E, \end{aligned}$$
(1)
and functions \(\varepsilon :E\rightarrow {\mathbb {R}}_+\) and \(\varphi :E\rightarrow Y\) fulfil
$$\begin{aligned} d\big (({\mathcal {T}}\varphi )(t),\varphi (t)\big )\le \varepsilon (t),\qquad t\in E, \end{aligned}$$
and
$$\begin{aligned} \sigma (t){:}{=}\sum _{n\in {\mathbb {N}}_0} (\Lambda ^{n}\varepsilon )(t)<\infty , \qquad t\in E. \end{aligned}$$
Then, for every \(t\in E\), the limit
$$\begin{aligned} \lim _{n\rightarrow \infty }({\mathcal {T}}^n\varphi )(t)=:\psi (t) \end{aligned}$$
exists and the function \(\psi \in Y^E\), defined in this way, is a fixed point of \({\mathcal {T}}\) with
$$\begin{aligned} d\big (\varphi (t),\psi (t)\big )\le \sigma (t), \qquad t\in E. \end{aligned}$$

In the next section, we present a similar fixed-point theorem for dislocated quasi-metric spaces that generalizes Theorem 2 and several similar outcomes in [5, 7, 8]. In particular, we apply a restricted version of a weaker form of condition (1) (see Remark 3).

Let us recall that a dislocated quasi-metric (dq-metric, for short), in a nonempty set Y, is a function \(d:Y\times Y\rightarrow [0,+\infty )\) that satisfies the following two conditions:
  1. (A1)

    if \(d(x,y)=d(y,x)=0\), then \(x=y\),

     
  2. (A2)

    \(d(x,y)\leqslant d(x,z)+d(z,y)\)

     
for all \(x,y,z\in Y\). The notion of a dq-metric space is a natural generalization of the usual definitions of metric, quasi-metric, partial metric, and metric-like spaces and plays crucial roles in computer science and cryptography (see, e.g., [2, 4, 11, 13, 15, 20, 21, 22, 25, 26]).

Remark 1

Let \(a,b\in (0,\infty )\), \(n,k\in {\mathbb {N}}\) (positive integers), \(\alpha :{\mathbb {R}}\rightarrow {\mathbb {R}}_+\) and \(\alpha ^{-1}(\{0\})=\{0\}\). Then, it is easy to check that the function \(d:{\mathbb {R}}\times {\mathbb {R}}\rightarrow [0,\infty )\), given by any of the following six formulas, is a dq-metric:
$$\begin{aligned} d(x,y)= & {} \alpha (x),\qquad x,y\in {\mathbb {R}}, \\ d(x,y)= & {} \max \,\{a|x|^k,b|y|^n\},\qquad x,y\in {\mathbb {R}}, \\ d(x,y)= & {} a|x|^k+b|y|^n,\qquad x,y\in {\mathbb {R}}, \\ d(x,y)= & {} \sqrt{a|x|^k+b|y|^n},\qquad x,y\in {\mathbb {R}}, \\ d(x,y)= & {} \root n \of {\max \,\{x-y,0\}},\qquad x,y\in {\mathbb {R}}, \\ d(x,y)= & {} \max \,\{x-[y],0\},\qquad x,y\in {\mathbb {R}}, \end{aligned}$$
where [y] denotes the integer part of y, i.e., \([y]{:}{=}\max \,\{n\in {\mathbb {Z}}: n\le y\}\) and \({\mathbb {Z}}\) stands for the set of integers. For some further examples we refer to, e.g., [2, 4, 13, 22] and the references therein.
Let d be a dq-metric in a nonempty set Y. We say that \(x\in Y\) is a limit of a sequence \((x_n)_{n=1}^{\infty }\) in Y provided
$$\begin{aligned} \lim _{n\rightarrow \infty }\max \,\{d(x_n,x),d(x,x_n)\}=0; \end{aligned}$$
then we write \(x_n\rightarrow x\) or \(x=\lim _{n\rightarrow \infty }x_n\); in view of (A2), it is easy to note that such a limit must be unique. Next, we say that a sequence \((x_n)_{n=1}^{\infty }\) in Y is Cauchy if
$$\begin{aligned} \lim _{N\rightarrow \infty } \sup _{m,n\geqslant N}\, d(x_n,x_m)=0; \end{aligned}$$
d is complete if every Cauchy sequence in Y has a limit in Y.

Remark 2

Usually, in a dq-metric space, the Cauchy sequence is defined in a somewhat different way; e.g., in a metric-like space (Yd), a sequence \((x_n)_{n=1}^{\infty }\) is said to be Cauchy if the limit \(\lim _{N\rightarrow \infty } \sup _{m,n\geqslant N}\, d(x_n,x_m)\) exists and is finite (see [3]). However, such definitions are too weak and would exclude from our considerations the metric and quasi-metric spaces. The same concerns the notion of completeness.

Our definition of a limit of a sequence is stronger than the usual, but this seems to be necessary in the proof of the main result; moreover, it actually corresponds to our definition of the Cauchy sequence and makes such limit unique (which is not the case in general) and, therefore, more useful.

2 The main result

In what follows, we always assume that (Yd) is a complete dq-metric space, i.e., d is a complete dq-metric in a nonempty set Y. Moreover, E denotes a nonempty set and \(d:{Y}^E\times {Y}^E\rightarrow {{\mathbb {R}}_+}^{E}\) is defined by
$$\begin{aligned} d(\xi ,\mu )(t){:}{=}d(\xi (t),\mu (t)),\qquad \xi ,\mu \in {Y}^E,t\in E. \end{aligned}$$
Analogously, as in the classical metric spaces, if \((\chi _n)_{n\in {\mathbb {N}}}\) is a sequence of elements of \(Y^E\), then a function \(\chi \in {Y}^E\) is a pointwise limit of \((\chi _n)_{n\in {\mathbb {N}}}\) provided
$$\begin{aligned} \lim _{n\rightarrow \infty }\max \big \{d(\chi ,\chi _n)(t),d(\chi _n,\chi )(t)\big \}=0,\qquad t\in E; \end{aligned}$$
\(\chi \in {Y}^E\) is a uniform limit of \((\chi _n)_{n\in {\mathbb {N}}}\) provided
$$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{t\in E}\max \big \{d(\chi ,\chi _n)(t),d(\chi _n,\chi )(t)\big \}=0. \end{aligned}$$
A nonempty subset \({\mathcal {F}}\) of \({Y}^E\) is called p-closed (u-closed, respectively) if every \(\chi \in {Y}^E\), which is a pointwise (uniform, resp.) limit of a sequence \((\chi _n)_{n\in {\mathbb {N}}}\) of elements of \({\mathcal {F}}\), belongs to \({\mathcal {F}}\).
Furthermore, given \(f,g\in {\mathbb {R}}^E\), we write \(f\le g\) if \(f(t)\le g(t)\) for \(t\in E\). Let \(\emptyset \ne {\mathcal {C}} \subset {Y}^E\), \(\Lambda :{\mathbb {R_+}}^{E} \rightarrow {\mathbb {R_+}}^{E}\), and \(\omega \in {\mathbb {R_+}}^{E}\). We say that \({\mathcal {T}}:{\mathcal {C}}\rightarrow {Y}^E\) is \((\omega ,\Lambda )\)—contractive provided
$$\begin{aligned} d({\mathcal {T}}\xi ,{\mathcal {T}}\mu ) \le \Lambda \delta \end{aligned}$$
for any \(\xi ,\mu \in {\mathcal {C}}\) and \(\delta \in {\mathbb {R_+}}^{E}\) with
$$\begin{aligned} \delta \le \omega ,\qquad d(\xi ,\mu ) \le \delta . \end{aligned}$$
Given a set \(A\ne \emptyset \) and \(f\in A^A\), we define \(f^n\in A^A\) (for \(n\in {\mathbb {N}}_0\)) by
$$\begin{aligned} f^0(x)=x, \qquad f^{n+1}(x)=f(f^n(x)),\qquad x\in A, n\in {\mathbb {N}}_0. \end{aligned}$$
Finally, to simplify some formulas, we denote by \(\Lambda _0\) the identity operator on \({\mathbb {R_+}}^{E}\), i.e., \(\Lambda _0 \delta =\delta \) for each \(\delta \in {\mathbb {R_+}}^{E}\).

Now, we are in a position to present the fixed-point theorem, which is the main result of this paper.

Theorem 3

Let \({\mathcal {C}} \subset {Y}^E\) be nonempty, \(\Lambda _n:{\mathbb {R_+}}^{E}\rightarrow {\mathbb {R_+}}^{E}\) for \(n\in {\mathbb {N}}\), and \({\mathcal {T}}:{\mathcal {C}}\rightarrow {\mathcal {C}}\). Assume that there exist functions \(\varepsilon _1,\varepsilon _2\in {\mathbb {R_+}}^{E}\) and \(\varphi \in {\mathcal {C}}\), such that
$$\begin{aligned}&\varepsilon _j^*(t){:}{=}\sum _{i=0}^{\infty }\Lambda _i\varepsilon _j(t)<\infty ,\qquad t\in E,j=1,2, \end{aligned}$$
(2)
$$\begin{aligned}&d({\mathcal {T}} \varphi ,\varphi )\le \varepsilon _1,\qquad d(\varphi ,{\mathcal {T}} \varphi )\le \varepsilon _2, \end{aligned}$$
(3)
$$\begin{aligned}&\liminf _{n\rightarrow \infty }\Lambda _1\Big (\sum _{i=n}^{\infty } \Lambda _{i}\varepsilon _j\Big )(t)=0,\qquad t\in E,j=1,2, \end{aligned}$$
(4)
and write \(\varepsilon ^*(t){:}{=}\max \{\varepsilon _1(t),\varepsilon _2(t)\}\) for \(t\in E\). Let \({\mathcal {T}}^n\) be \((\varepsilon ^*,\Lambda _{n})\)—contractive for \(n\in {\mathbb {N}}\) and one of the following two hypotheses be valid.
  1. (i)

    \({\mathcal {C}}\) is p-closed.

     
  2. (ii)

    \({\mathcal {C}}\) is u-closed and the sequence \(\big (\sum _{i=0}^{n}\Lambda _i\varepsilon _j\big )_{n\in {\mathbb {N}}}\) tends uniformly to \(\varepsilon _j^*\) on E for \(j=1,2\).

     
Then, for each \(t\in E\), there exists the limit
$$\begin{aligned} \psi (t){:}{=}\lim _{n\rightarrow \infty }{\mathcal {T}}^n\varphi (t) \end{aligned}$$
(5)
and the function \(\psi \in {\mathcal {C}}\), defined in this way, is a fixed point of \({\mathcal {T}}\) with
$$\begin{aligned} d({\mathcal {T}}^n\varphi ,\psi )\le \sum _{i=n}^{\infty } \Lambda _{i}\varepsilon _1,\qquad d(\psi ,{\mathcal {T}}^n\varphi )\le \sum _{i=n}^{\infty } \Lambda _{i}\varepsilon _2,\qquad n\in {\mathbb {N}}_0. \end{aligned}$$
(6)
Moreover, the following two statements are valid:
  1. (a)
    for every sequence \((k_n)_{n\in {\mathbb {N}}}\) of positive integers with \(\lim _{n\rightarrow \infty } k_n=\infty \), \(\psi \) is the unique fixed point of \({\mathcal {T}}\), such that
    $$\begin{aligned} d({\mathcal {T}}^{k_n}\varphi ,\psi )\le \sum _{i=k_n}^{\infty } \Lambda _{i}\varepsilon _j,\qquad d(\psi ,{\mathcal {T}}^{k_n}\varphi )\le \sum _{i=k_n}^{\infty } \Lambda _{i}\varepsilon _l,\qquad n\in {\mathbb {N}}, \end{aligned}$$
    with some \(j,l\in \{1,2\}\);
     
  2. (b)
    if
    $$\begin{aligned} \liminf _{n\rightarrow \infty }\Lambda _n\varepsilon _j^*(t)=0,\qquad j=1,2,t\in E, \end{aligned}$$
    (7)
    then \(\psi \) is the unique fixed point of \({\mathcal {T}}\) with
    $$\begin{aligned} d(\varphi ,\psi )\le \varepsilon _1^*,\qquad d(\psi ,\varphi )\le \varepsilon _2^*, \end{aligned}$$
    and for every \(j,l\in \{1,2\}\)
    $$\begin{aligned} \psi (t)= \lim _{n \rightarrow \infty } {\mathcal {T}}^{k_n} \xi (t), \qquad \xi \in {\mathcal {C}},d(\xi ,\psi ) \le \varepsilon _j^*,d(\psi , \xi ) \le \varepsilon _l^*,t\in E, \end{aligned}$$
    (8)
    for every sequence \((k_n)_{n\in {\mathbb {N}}}\) of positive integers with \(\lim _{n\rightarrow \infty } \Lambda _{k_n}\varepsilon _m^*(t)=0\) for \(t\in E\) and \(m\in \{j,l\}\).
     

Proof

Clearly, (3) implies that, for any \(k,l\in {\mathbb {N}}\) and \(n\in {\mathbb {N}}_0\)
$$\begin{aligned} d({\mathcal {T}}^{n+k}\varphi ,{\mathcal {T}}^n\varphi )\le & {} \sum _{i=0}^{k-1} d({\mathcal {T}}^{n+i+1}\varphi ,{\mathcal {T}}^{n+i}\varphi )\nonumber \\\le & {} \sum _{i=n}^{n+k-1} \Lambda _{i}\varepsilon _1\le \sum _{i=n}^{\infty } \Lambda _{i}\varepsilon _1, \end{aligned}$$
(9)
$$\begin{aligned} d({\mathcal {T}}^n\varphi ,{\mathcal {T}}^{n+l}\varphi )\le & {} \sum _{i=0}^{l-1} d({\mathcal {T}}^{n+i}\varphi ,{\mathcal {T}}^{n+i+1}\varphi )\nonumber \\\le & {} \sum _{i=n}^{n+l-1} \Lambda _{i}\varepsilon _2 \le \sum _{i=n}^{\infty } \Lambda _{i}\varepsilon _2, \end{aligned}$$
(10)
whence
$$\begin{aligned} d({\mathcal {T}}^{n+k}\varphi ,{\mathcal {T}}^{n+l}\varphi )\le & {} d({\mathcal {T}}^{n+k}\varphi ,{\mathcal {T}}^n\varphi )+d({\mathcal {T}}^n\varphi ,{\mathcal {T}}^{n+l}\varphi )\\\le & {} \sum _{i=n}^{\infty } \Lambda _{i}\varepsilon _1 +\sum _{i=n}^{\infty } \Lambda _{i}\varepsilon _2. \end{aligned}$$
Therefore, by (2), \(({\mathcal {T}}^n\varphi (t))_{n\in {\mathbb {N}}}\) is a Cauchy sequence in Y for each \(t\in E\). Since Y is complete, this sequence is convergent. Consequently, (5) defines a function \(\psi \in {\mathcal {C}}\).
Letting \(k\rightarrow \infty \) in (9) and \(l\rightarrow \infty \) in (10), on account of (5), we get
$$\begin{aligned} d({\mathcal {T}}^n\varphi ,\psi )\le \sum _{i=n}^{\infty } \Lambda _{i}\varepsilon _1,\qquad d(\psi ,{\mathcal {T}}^n\varphi )\le \sum _{i=n}^{\infty } \Lambda _{i}\varepsilon _2,\qquad n\in {\mathbb {N}}_0, \end{aligned}$$
(11)
which is (6). Next, using (11), we get
$$\begin{aligned} d({\mathcal {T}}^{n+1}\varphi ,{\mathcal {T}}\psi ) \le \Lambda _1\Big (\sum _{i=n}^{\infty } \Lambda _{i}\varepsilon _1\Big ),\qquad d({\mathcal {T}}\psi ,{\mathcal {T}}^{n+1}\varphi ) \le \Lambda _1\Big (\sum _{i=n}^{\infty } \Lambda _{i}\varepsilon _2\Big ) \end{aligned}$$
for \(n\in {\mathbb {N}}_0\). Hence, for each \(n\in {\mathbb {N}}_0\)
$$\begin{aligned} d(\psi ,{\mathcal {T}}\psi )&\le d(\psi ,{\mathcal {T}}^{n+1}\varphi )+d({\mathcal {T}}^{n+1}\varphi ,{\mathcal {T}}\psi ) \le d(\psi ,{\mathcal {T}}^{n+1}\varphi )+\Lambda _1\Big (\sum _{i=n}^{\infty } \Lambda _{i}\varepsilon _1\Big ),\\ d({\mathcal {T}}\psi ,\psi )&\le \Lambda _1\Big (\sum _{i=n}^{\infty } \Lambda _{i}\varepsilon _2\Big )+d({\mathcal {T}}^{n+1}\varphi ,\psi ), \end{aligned}$$
which with \(n\rightarrow \infty \) yields \(d( \psi ,{\mathcal {T}}\psi )=0\) and \(d({\mathcal {T}}\psi ,\psi )=0\) [in view of (4)], and consequently \({\mathcal {T}}\psi =\psi \).
Let \((k_n)_{n\in {\mathbb {N}}}\) be a sequence of positive integers with \(\lim _{n\rightarrow \infty } k_n=\infty \) and \(\xi \in Y^E\) be a fixed point of \({\mathcal {T}}\) with
$$\begin{aligned}&d({\mathcal {T}}^{k_n}\varphi (x),\xi (x))\le \sum _{i=k_n}^{\infty } \Lambda _{i}\varepsilon _j(t),\qquad d(\xi (x),{\mathcal {T}}^{k_n}\varphi (x))\le \sum _{i=k_n}^{\infty } \Lambda _{i}\varepsilon _l(t),\qquad \\&n\in {\mathbb {N}}, t\in E, \end{aligned}$$
with some \(j,l\in \{1,2\}\). Then, by (6)
$$\begin{aligned} d(\xi (t),\psi (t))&\le \; d(\xi (t),{\mathcal {T}}^{k_n}\varphi (t))+ d({\mathcal {T}}^{k_n}\varphi (t),\psi (t))\\&\le \;\sum _{i=k_n}^{\infty } \Lambda _{i}\varepsilon _l(t)+\sum _{i=k_n}^{\infty } \Lambda _{i}\varepsilon _1(t),\qquad n\in {\mathbb {N}}_0, t\in E, \end{aligned}$$
$$\begin{aligned} d(\psi (t),\xi (t))&\le \; d(\psi (t),{\mathcal {T}}^{k_n}\varphi (t))+d({\mathcal {T}}^{k_n}\varphi (t),\xi (t))\\&\le \;\sum _{i=k_n}^{\infty } \Lambda _{i}\varepsilon _2(t)+\sum _{i=k_n}^{\infty } \Lambda _{i}\varepsilon _j(t),\qquad n\in {\mathbb {N}}_0, t\in E, \end{aligned}$$
whence letting \(n\rightarrow \infty \), we get \(\xi =\psi \).
It remains to prove statement (b). Therefore, assume that (7) holds and \(\xi \in Y^E\) is a fixed point of \({\mathcal {T}}\) with
$$\begin{aligned} d(\varphi ,\xi )\le \varepsilon _1^*,\qquad d(\xi ,\varphi )\le \varepsilon _2^*. \end{aligned}$$
Then, for any \(n\in {\mathbb {N}}_0\), we have
$$\begin{aligned} d(\psi ,\xi )&\le d(\psi ,{\mathcal {T}}^n\varphi )+d({\mathcal {T}}^n\varphi ,{\mathcal {T}}^n\xi )\nonumber \\&\le d(\psi ,{\mathcal {T}}^n\varphi )+\Lambda _{n}\varepsilon _1^*, \end{aligned}$$
$$\begin{aligned} d(\xi ,\psi )&\le d({\mathcal {T}}^n\xi ,{\mathcal {T}}^n\varphi )+d({\mathcal {T}}^n\varphi ,\psi )\nonumber \\&\le \Lambda _{n}\varepsilon _2^*+d(\psi ,{\mathcal {T}}^n\varphi ), \end{aligned}$$
whence letting \(n\rightarrow \infty \), we can easily see that \(\xi =\psi \).
Now, let \(j,l \in \{1,2\}\) and \((k_n)_{n\in {\mathbb {N}}}\) be a sequence of positive integers with
$$\begin{aligned} \lim _{n\rightarrow \infty } \Lambda _{k_n}\varepsilon _m^*(t)=0,\qquad t\in E,m\in \{j,l\}. \end{aligned}$$
Let \(\xi \in {\mathcal {C}}\) be a function such that \(d(\xi ,\psi ) \le \varepsilon _j^*\) and \(d(\psi , \xi ) \le \varepsilon _l^*\). Then
$$\begin{aligned} d({\mathcal {T}}^{k_n} \xi , \psi )= & {} d({\mathcal {T}}^{k_n} \xi , {\mathcal {T}}^{k_n} \psi ) \le \Lambda _{k_n} \varepsilon _j^*,\qquad n \in {\mathbb {N}}, \\ d(\psi ,{\mathcal {T}}^{k_n} \xi )= & {} d({\mathcal {T}}^{k_n} \psi ,{\mathcal {T}}^{k_n} \xi ) \le \Lambda _{k_n} \varepsilon _l^*,\qquad n \in {\mathbb {N}}. \end{aligned}$$
Letting \(n \rightarrow \infty \), we get (8). \(\square \)

Theorem 3 implies at once the following.

Theorem 4

Let \({\mathcal {C}} \subset {Y}^E\) be nonempty, \({\mathcal {T}}:{\mathcal {C}}\rightarrow {\mathcal {C}}\) and \(\Lambda :{\mathbb {R_+}}^{E}\rightarrow {\mathbb {R_+}}^{E}\). Assume that there exist functions \(\varepsilon _1,\varepsilon _2\in {\mathbb {R_+}}^{E}\) and \(\varphi \in {\mathcal {C}}\), such that
$$\begin{aligned}&\varepsilon _j^*(x){:}{=}\sum _{i=0}^{\infty }\Lambda ^i\varepsilon _j(t)<\infty ,\qquad t\in E,j=1,2, \nonumber \\&\quad d({\mathcal {T}} \varphi ,\varphi )\le \varepsilon _1,\qquad d( \varphi ,{\mathcal {T}}\varphi )\le \varepsilon _2, \end{aligned}$$
(12)
$$\begin{aligned}&\liminf _{n\rightarrow \infty }\Lambda \Big (\sum _{i=n}^{\infty } \Lambda ^{i}\varepsilon _j\Big )(t)=0,\qquad t\in E,j=1,2, \end{aligned}$$
(13)
and \({\mathcal {T}}\) is \((\varepsilon ^*,\Lambda )\)-contractive, where \(\varepsilon ^*(t){:}{=}\max \{\varepsilon _1(t),\varepsilon _2(t)\}\) for \(t\in E\). Next, let one of the following two hypotheses hold.
  1. (i)

    \({\mathcal {C}}\) is p-closed.

     
  2. (ii)

    \({\mathcal {C}}\) is u-closed and the sequence \(\big (\sum _{i=0}^{n}\Lambda ^i\varepsilon _j\big )_{n\in {\mathbb {N}}}\) tends uniformly to \(\varepsilon _j^*\) on E for \(j=1,2\).

     
Then, for each \(t\in E\), there exists the limit
$$\begin{aligned} \psi (t){:}{=}\lim _{n\rightarrow \infty }{\mathcal {T}}^n\varphi (t) \end{aligned}$$
and the function \(\psi \in {\mathcal {C}}\), defined in this way, is a fixed point of \({\mathcal {T}}\) with
$$\begin{aligned} d({\mathcal {T}}^n\varphi ,\psi )\le \sum _{i=n}^{\infty } \Lambda ^{i}\varepsilon _1,\qquad d(\psi ,{\mathcal {T}}^n\varphi )\le \sum _{i=n}^{\infty } \Lambda ^{i}\varepsilon _2,\qquad n\in {\mathbb {N}}_0. \end{aligned}$$
Moreover, the following two statements are valid:
  1. (a)
    For every sequence \((k_n)_{n\in {\mathbb {N}}}\) of positive integers with \(\lim _{n\rightarrow \infty } k_n=\infty \), \(\psi \) is the unique fixed point of \({\mathcal {T}}\) with
    $$\begin{aligned} d({\mathcal {T}}^{k_n}\varphi ,\psi )\le \sum _{i=k_n}^{\infty } \Lambda ^{i}\varepsilon _1,\qquad d(\psi ,{\mathcal {T}}^{k_n}\varphi )\le \sum _{i=k_n}^{\infty } \Lambda ^{i}\varepsilon _2,\qquad n\in {\mathbb {N}}. \end{aligned}$$
     
  2. (b)
    If
    $$\begin{aligned} \liminf _{n\rightarrow \infty }\Lambda ^n\varepsilon _j^*(t)=0,\qquad t\in E,j=1,2, \end{aligned}$$
    (14)
    then \(\psi \) is the unique fixed point of \({\mathcal {T}}\) with
    $$\begin{aligned} d(\varphi ,\psi )\le \varepsilon _1^*,\qquad d(\psi ,\varphi )\le \varepsilon _2^*. \end{aligned}$$
    and for every \(j,l\in \{1,2\}\),
    $$\begin{aligned} \psi (t)= \lim _{n \rightarrow \infty } {\mathcal {T}}^{k_n} \xi (t), \qquad \xi \in {\mathcal {C}},d(\xi ,\psi ) \le \varepsilon _j^*,d(\psi , \xi ) \le \varepsilon _l^*,t\in E, \end{aligned}$$
    for every sequence \((k_n)_{n\in {\mathbb {N}}}\) of positive integers with \(\lim _{n\rightarrow \infty } \Lambda _{k_n}\varepsilon _m^*(t)=0\) for \(t\in E\) and \(m\in \{j,l\}\).
     

Proof

It is enough to notice that \({\mathcal {T}}^n\) is \((\varepsilon ^*,\Lambda ^{n})\)—contractive for each \(n\in {\mathbb {N}}\) and use Theorem 3. \(\square \)

Remark 3

There arises a natural question whether, in some situations, assumption (2) can be weaker than (12) with \(\Lambda {:}{=}\Lambda _{1}\). Below, we provide a somewhat trivial example that this is the case.

Let \(Y={\mathbb {R}}^3\) be endowed with the euclidean norm, \(c\in {\mathbb {R}}\) and \(E={\mathbb {R}}\). Define the operator \({\mathcal {T}}:Y^{{\mathbb {R}}}\rightarrow Y^{{\mathbb {R}}}\) by
$$\begin{aligned} {\mathcal {T}}\phi (x)=(0,\phi _1(x),\phi _2(x)+c),\qquad x\in {\mathbb {R}}, \end{aligned}$$
for every \(\phi =(\phi _1,\phi _2,\phi _3)\in Y^{{\mathbb {R}}}\). Then
$$\begin{aligned} \Vert {\mathcal {T}}\phi (x)-{\mathcal {T}}\mu (x)\Vert&=\;\Vert (0,\phi _1(x)-\mu _1(x),\phi _2(x)-\mu _2(x))\Vert \\&\le \; \Vert \phi (x)-\mu (x)\Vert ,\qquad x\in {\mathbb {R}}, \end{aligned}$$
for every \(\phi =(\phi _1,\phi _2,\phi _3),\mu =(\mu _1,\mu _2,\mu _3)\in Y^{{\mathbb {R}}}\). This shows that \(\Lambda _1\) and \(\Lambda _2\) exist, because it is enough to take any \(\Lambda _1\) and \(\Lambda _2\) with \(\Lambda _i\delta \ge \delta \) for \(\delta \in {{\mathbb {R}}_+}^{{\mathbb {R}}}\) and \(i=1,2\). Next
$$\begin{aligned} {\mathcal {T}}^n\phi (x)=(0,0,nc),\qquad x\in {\mathbb {R}},n\in {\mathbb {N}}, n\ge 3, \end{aligned}$$
for every \(\phi =(\phi _1,\phi _2,\phi _3)\in Y^{{\mathbb {R}}}\) and we can take \(\Lambda _n\delta (x)=0\) for \(\delta \in {{\mathbb {R}}_+}^{{\mathbb {R}}}\), \(x\in {\mathbb {R}}\) and \(n\in {\mathbb {N}}\), \(n\ge 3\). Clearly, in such a case, (2) holds for any \(\varepsilon _1,\varepsilon _2\in {\mathbb {R}}_+^{{\mathbb {R}}}\).
We show that, for any such \(\Lambda _1\), we must have
$$\begin{aligned} \Lambda _1\delta \ge \delta ,\qquad \delta \in {{\mathbb {R}}_+}^{{\mathbb {R}}}. \end{aligned}$$
(15)
Therefore, take arbitrary \(\delta \in {{\mathbb {R}}_+}^{{\mathbb {R}}}\) and define \(\phi ,\psi \in Y^{{\mathbb {R}}}\) by
$$\begin{aligned} \phi (x)=(\delta (x),0,0),\qquad \psi (x)=(0,0,0),\qquad x\in {\mathbb {R}}. \end{aligned}$$
Then
$$\begin{aligned} {\mathcal {T}}\phi (x)= & {} (0,\delta (x),c),\qquad {\mathcal {T}}\psi (x)=(0,0,c),\qquad x\in {\mathbb {R}}, \\ \Vert \phi (x)-\psi (x)\Vert= & {} \Vert (\delta (x),0,0)\Vert =\delta (x),\qquad x\in {\mathbb {R}}, \end{aligned}$$
and
$$\begin{aligned} \Vert {\mathcal {T}}\phi (x)-{\mathcal {T}}\psi (x)\Vert =\Vert (0,\delta (x),0)\Vert =\delta (x)\le \Lambda _1 \delta (x),\qquad x\in {\mathbb {R}}. \end{aligned}$$
This shows that (15) holds, whence, by induction, we obtain that for each \(n\in {\mathbb {N}}\)
$$\begin{aligned} {\Lambda _1}^n\delta \ge \delta , \end{aligned}$$
and therefore
$$\begin{aligned} \sum _{i=0}^{\infty }{\Lambda _1}^i\delta (x)=\infty ,\qquad x\in {\mathbb {R}},\delta (x)\ne 0. \end{aligned}$$

3 Some comments

We need yet the following hypothesis concerning operators \(\Lambda :{\mathbb {R_+}}^{E}\rightarrow {\mathbb {R_+}}^{E}\).
\(({\mathcal {C}})\)
If \((\delta _n)_{n\in {\mathbb {N}}}\) is a sequence of elements of \({{\mathbb {R}}_+}^{E}\) with
$$\begin{aligned} \lim _{n\rightarrow \infty } \delta _n(t)=0,\qquad t\in E, \end{aligned}$$
then
$$\begin{aligned} \liminf _{n\rightarrow \infty } \Lambda \delta _n(t)=0,\qquad t\in E. \end{aligned}$$

Remark 4

Note that if \(\Lambda _1\) fulfils hypothesis \(({\mathcal {C}})\), then (4) results at once from (2). Analogously, (12) yields (13) if \(\Lambda \) fulfils \(({\mathcal {C}})\).

Remark 5

Let \(j\in {\mathbb {N}}\) and \({\mathbb {K}}\) be either the set of reals \({\mathbb {R}}\) or the set of complex numbers \({\mathbb {C}}\). Fix \(f_{i}:E\rightarrow E\) and \(L_i :E\rightarrow {\mathbb {K}}\) for \(i=1,\ldots ,j\). Then, the operator \({\mathcal {T}}:{{\mathbb {K}}}^E\rightarrow {{\mathbb {K}}}^E\), given by
$$\begin{aligned} {\mathcal {T}}\phi (t){:}{=} \sum _{i=1}^{j}L_i(t)\phi (f_i(t)),\qquad \phi \in {\mathbb {K}}^E, t\in E, \end{aligned}$$
is \((\omega ,\Lambda )\) contractive, with any \(\omega \in {\mathbb {R}}_+^E\) and \(\Lambda :{\mathbb {R_+}}^{E}\rightarrow {\mathbb {R_+}}^{E}\) defined by the formula
$$\begin{aligned} \Lambda \delta (t){:}{=}&\;\sum _{i=1}^{j}|L_i(t)|\delta (f_i(t)),\qquad \delta \in {{\mathbb {R}}_+}^{E}, t\in E. \end{aligned}$$
Moreover, \(({\mathcal {C}})\) holds.
Next, for any function \(\varepsilon _0:E\rightarrow {\mathbb {R}}_+\) with \(\varepsilon _0^*\) given by [see (12)]
$$\begin{aligned} \varepsilon _0^*(x){:}{=}\sum _{i=0}^{\infty }\Lambda ^i\varepsilon _0(t)<\infty ,\qquad t\in E,j=1,2, \end{aligned}$$
we have
$$\begin{aligned} \Lambda \varepsilon _0^*(t)&=\;\sum _{i=1}^{j} |L_i(t)| \sum _{k=0}^{\infty } (\Lambda ^{k}\varepsilon _0)(f_i(t))=\sum _{k=0}^{\infty } \sum _{i=1}^{j} |L_i(t)| (\Lambda ^{k}\varepsilon _0)(f_i(t))\\&=\;\sum _{k=1}^{\infty } (\Lambda ^{k}\varepsilon _0)(t),\qquad t\in E, \end{aligned}$$
and analogously, by induction, we get
$$\begin{aligned} \Lambda ^n\varepsilon _0^*(t)=\sum _{k=n}^{\infty } (\Lambda ^{k}\varepsilon _0)(t),\qquad t\in E,n\in {\mathbb {N}}_0. \end{aligned}$$
This means that (12) yields (14). Therefore, [9, Theorem 1] can be derived from Theorem 4.

Remark 6

Let \(F:E \times \mathbb {R_+}\rightarrow \mathbb {R_+}\) be subadditive and non-decreasing with respect to the second variable (i.e., \(F(x,a+b)\le F(x,a)+F(x,b)\) and \(F(x,a)\le F(x,c)\) for \(a,b,c\in \mathbb {R_+}\) with \(a\le c\) and \(x\in E\)). Let \(f:E \rightarrow E\) be given and \(\Lambda :{\mathbb {R_+}}^{E}\rightarrow {\mathbb {R_+}}^{E}\) be defined by
$$\begin{aligned} \Lambda \varepsilon (x)= F(x,\varepsilon (f(x))),\qquad x\in E,n\in {\mathbb {N}}_0,\varepsilon \in {\mathbb {R_+}}^{E}. \end{aligned}$$
We show that for such \(\Lambda \), condition (12) yields (13) and (14).
Therefore, assume that (12) holds for some suitable \(\varepsilon _j\) with \(j=1,2\). Fix \(x\in E\) and define a function \(F_0:\mathbb {R_+}\rightarrow \mathbb {R_+}\) by
$$\begin{aligned} F_0(a)=F(x,a),\qquad a\in \mathbb {R_+}. \end{aligned}$$
Since \(F_0\) is non-decreasing and \(\Lambda ^{n}\varepsilon _1(f(x))\ge 0\) for each \(n\in {\mathbb {N}}_0\), we have
$$\begin{aligned} \Lambda ^{n+1}\varepsilon _1(x)= F_0\big (\Lambda ^{n}\varepsilon _1(f(x))\big )\ge F_0(0). \end{aligned}$$
Hence, by (12), we get \(F_0(0)=0\).
Fix \(j\in \{1,2\}\). Next, we prove that \(F_0\) is continuous at 0 or there exists \(l_0\in {\mathbb {N}}\) with
$$\begin{aligned} \Lambda ^{n}\varepsilon _j(f(x))=0,\qquad n\in {\mathbb {N}},n>l_0. \end{aligned}$$
To this end suppose that \(F_0\) is not continuous at 0 and there is a strictly increasing sequence \(\big (k_n\big )_{n\in {\mathbb {N}}}\) of positive integers, such that \(\Lambda ^{k_n} \varepsilon _j(f(x))\ne 0\) for \(n\in {\mathbb {N}}\). Since \(F_0\) is non-decreasing and \(F(0)=0\), there exists \(d>0\) with \(F_0(c)>d\) for every \(c>0\), whence
$$\begin{aligned} \Lambda ^{k_n+1} \varepsilon _j(x)=F_0\big (\Lambda ^{k_n} \varepsilon _j(f(x))\big )\ge d,\qquad n\in {\mathbb {N}}, \end{aligned}$$
which contradicts to (12).
Thus, we have proved that
$$\begin{aligned} \lim _{j\rightarrow \infty }F_0\Big (\sum _{n=j}^{\infty } \Lambda ^{n} \varepsilon _j(f(x))\Big )=0,\qquad j=1,2. \end{aligned}$$
Furthermore, by subadditivity of \(F_0\), for every \(k,l\in {\mathbb {N}}_0\), \(l>k\), we get
$$\begin{aligned} F_0\Big (\sum _{n=k}^{\infty } \Lambda ^{n} \varepsilon _j(f(x))\Big )\le \sum _{n=k}^{l} \Lambda ^{n+1} \varepsilon _j(x)+F_0\Big (\sum _{n=l+1}^{\infty } \Lambda ^{n} \varepsilon _j(f(x))\Big ) \end{aligned}$$
whence letting \(l\rightarrow \infty \), we obtain
$$\begin{aligned} \Lambda \Big (\sum _{n=k}^{\infty } \Lambda ^{n} \varepsilon _j(x)\Big )=F_0\Big (\sum _{n=k}^{\infty } \Lambda ^{n} \varepsilon _j(f(x))\Big )\le \sum _{n=k+1}^{\infty } \Lambda ^{n} \varepsilon _j(x) \end{aligned}$$
and consequently, by induction (with \(k=0\))
$$\begin{aligned} \Lambda ^l\Big (\sum _{n=0}^{\infty } \Lambda ^{n} \varepsilon _j(x)\Big )\le \sum _{n=l}^{\infty } \Lambda ^{n} \varepsilon _j(x),\qquad l\in {\mathbb {N}}. \end{aligned}$$
Clearly, using those inequalities, we can easily deduce (13) and (14) from (12).

Now, consider a very special situation when the set E has only one element, \(E=\{s\}\). Then, actually, each \({\mathcal {C}}\subset Y^E\) can be considered as a subset of Y of the form \(C{:}{=}\{\phi (s): \phi \in {\mathcal {C}}\}\).

Given \(e\in \mathbb {R_+}\), \(\lambda :\mathbb {R_+}\rightarrow \mathbb {R_+}\) and \(C\subset Y\), analogously as before, we say that \(T:C\rightarrow C\) is \((e,\lambda )\)—contractive provided
$$\begin{aligned} d(Ty,Tz)\le \lambda ( \delta ), \end{aligned}$$
for every \(y,z\in Y\) and \(\delta \in {\mathbb {R}}_+\), such that \(d(y,z) \le \delta \le e\).
Next, for \(\lambda _1:\mathbb {R_+}\rightarrow \mathbb {R_+}\), hypothesis \(({\mathcal {C}})\) takes the following form:
\(({\mathcal {C}}_0)\)
If \((\delta _n)_{n\in {\mathbb {N}}}\) is a sequence in \({\mathbb {R}}_+\) with
$$\begin{aligned} \lim _{n\rightarrow \infty } \delta _n=0, \end{aligned}$$
then
$$\begin{aligned} \liminf _{n\rightarrow \infty } \lambda _1(\delta _n)=0. \end{aligned}$$

Theorem 3, with \(y_0=\varphi (s)\) and \(z_0=\psi (s)\), takes the following form (we write \(\lambda _{0}(\varepsilon ){:}{=}\varepsilon \) for each \(\varepsilon \in \mathbb {R_+}\)).

Theorem 5

Let \(T:Y\rightarrow Y\), \(\lambda _n:\mathbb {R_+}\rightarrow \mathbb {R_+}\) for \(n\in {\mathbb {N}}\), and \(\lambda _1\) satisfy hypothesis \(({\mathcal {C}}_0)\). Suppose that there exist \(y_0\in Y\) and \(\varepsilon _1,\varepsilon _2\in \mathbb {R_+}\), such that
$$\begin{aligned}&d(T(y_0),y_0)\le \varepsilon _1,\qquad d(y_0,T(y_0))\le \varepsilon _2, \nonumber \\&\varepsilon _j^*{:}{=}\sum _{i=0}^{\infty }\lambda _i(\varepsilon _j)<\infty ,\qquad j=1,2, \end{aligned}$$
(16)
and \(T^n\) is \((\varepsilon ^*,\lambda _n)\)—contractive for \(n\in {\mathbb {N}}\) with \(\varepsilon ^*{:}{=}\max \,\{\varepsilon ^*_1,\varepsilon ^*_2\}\). Then, the limit
$$\begin{aligned} z_0{:}{=}\lim _{n\rightarrow \infty }T^n(y_0) \end{aligned}$$
exists and \(z_0\) is a unique fixed point of T with
$$\begin{aligned} d(T^n(y_0),z_0)\le \sum _{i=n}^{\infty } \lambda _{i}(\varepsilon _1),\qquad d(z_0,T^n(y_0))\le \sum _{i=n}^{\infty } \lambda _{i}(\varepsilon _2),\qquad n\in {\mathbb {N}}_0. \end{aligned}$$
Moreover, the following two statements are valid:
  1. (a)
    for every sequence \((k_n)_{n\in {\mathbb {N}}}\) of positive integers with \(\lim _{n\rightarrow \infty } k_n=\infty \), \(z_0\) is the unique fixed point of T with
    $$\begin{aligned} d(T^{k_n}(y_0),z_0)\le \sum _{i=k_n}^{\infty } \lambda ^{i}(\varepsilon _1),\qquad d(z_0,T^{k_n}(y_0))\le \sum _{i=k_n}^{\infty } \lambda ^{i}(\varepsilon _2),\qquad n\in {\mathbb {N}}; \end{aligned}$$
     
  2. (b)
    if
    $$\begin{aligned} \liminf _{n\rightarrow \infty }\lambda _n(\varepsilon _j^*)=0,\qquad j=1,2, \end{aligned}$$
    then \(z_0\) is the unique fixed point of T, such that
    $$\begin{aligned} d(y_0,z_0)\le \varepsilon _1^*,\qquad d(z_0,y_0)\le \varepsilon _2^*. \end{aligned}$$
     

Clearly, if there is \(\lambda \in {\mathbb {R}}_+\), such that \(\lambda _n(a)=\lambda ^n a\) for \(a\in {\mathbb {R}}_+\) and \(n\in {\mathbb {N}}\), then Theorem 5 becomes a natural modification of the Banach Contraction Principle (with a local contraction condition) and (16) means that \(\lambda <1\).

4 Ulam stability

Now, we show how we can derive some simple Ulam stability outcomes from the results of the previous section. To this end, given \(e>0\) or \(e=\infty \), we need the subsequent hypothesis.
  1. (H1)
    \(j\in {\mathbb {N}}\), \(L_i:E \rightarrow {\mathbb {R}}_+\) for \(i=1,\ldots ,j\), \(\Phi :E\times Y^{j}\rightarrow Y\), and
    $$\begin{aligned} d(\Phi (t,w_1,...,w_j),\Phi (t,z_1,...,z_j))\le \sum _{k=1}^{j}L_k(t)d(w_k,z_k) \end{aligned}$$
    for any \(t\in E\) and \((w_1,...,w_j),(z_1,...,z_j)\in Y^j\), such that \(d(z_i,w_i)\le e\) for \(i=1,\ldots ,j\).
     
The following corollary also can be easily deduced from Theorem  2.

Corollary 6

Assume that \(\varepsilon _1,\varepsilon _2:E\rightarrow {\mathbb {R}}_+\), hypothesis (H1) is valid with \(e{:}{=}\sup \,\{\varepsilon _j^*(t): t\in E, j=1,2\}\), where
$$\begin{aligned} \varepsilon _j^*(t){:}{=}\sum _{i=0}^{\infty }\Lambda ^i\varepsilon _j(t)<\infty ,\qquad t\in E,j=1,2, \end{aligned}$$
and \(\Lambda :{\mathbb {R}}_+^{E}\rightarrow {\mathbb {R}}_+^{E}\) is given by
$$\begin{aligned} \Lambda \delta (t)=\sum _{k=1}^{j}L_k(t)\delta (f_k(t)),\qquad \delta \in {\mathbb {R}}_+^{E},t\in E, \end{aligned}$$
with some \(f_1,\dots ,f_j:E\rightarrow E\), and \(\varphi :E\rightarrow Y\) is such that
$$\begin{aligned}&d(\Phi (t,\varphi (f_1(t)),...,\varphi (f_j(t))),\varphi (t))\le \varepsilon _1(t), \qquad t\in E, \end{aligned}$$
(17)
$$\begin{aligned}&d(\varphi (t),\Phi (t,\varphi (f_1(t)),...,\varphi (f_j(t))))\le \varepsilon _2(t), \qquad t\in E. \end{aligned}$$
(18)
Then, the limit
$$\begin{aligned} \psi (t){:}{=}\lim _{n\rightarrow \infty }{\mathcal {T}}^n\varphi (t) \end{aligned}$$
(19)
exists for each \(t\in E\), with \({\mathcal {T}}\) given by
$$\begin{aligned} {\mathcal {T}}\varphi (t){:}{=}\Phi (t,\varphi (f_1(t)),...,\varphi (f_j(t))), \qquad \varphi \in Y^E,\,t\in E, \end{aligned}$$
and the function \(\psi :E\rightarrow Y\), defined by (19), is the unique solution of the functional equation:
$$\begin{aligned} \Phi (t,\psi (f_1(t)),...,\psi (f_j(t)))=\psi (t), \qquad t\in E, \end{aligned}$$
(20)
such that
$$\begin{aligned} d(\varphi (t),\psi (t))\le \varepsilon _1^*(t),\qquad d(\psi (t),\varphi (t))\le \varepsilon _2^*(t), \qquad t\in E. \end{aligned}$$
(21)

Proof

Let us note that inequalities (17) and (18) imply (3). Next
$$\begin{aligned}&\liminf _{n\rightarrow \infty }\Lambda ^n\varepsilon _j^*(t)=0,\qquad t\in E,j=1,2,\\&\liminf _{n\rightarrow \infty }\Lambda \Big (\sum _{i=n}^{\infty } \Lambda ^{i}\varepsilon _j\Big )(t)=0,\qquad t\in E,j=1,2, \end{aligned}$$
in view of Remarks 4 and 5. Therefore, by Theorem  4, the function \(\psi \) defined by (19) is the unique fixed point of \({\mathcal {T}}\) (that is a solution of (20)) satisfying (21). \(\square \)
Stability of functional equations of form (20) (or related to it) has been already studied by several authors, and for further information, we refer to the survey papers [1, 8]. A particular case of (20) is the linear functional equation of the form
$$\begin{aligned} \phi (t){:}{=}\sum _{i=1}^j L_i(t)\phi (f_i(t)), \qquad \varphi \in Y^E,\,t\in E, \end{aligned}$$
under the assumptions as in Remark 5; some recent results concerning stability of less general cases of it can be found in [10, 18, 19, 23].
As an example of applications of Corollary 6 consider stability of the difference equation:
$$\begin{aligned} \psi (i)=\Phi (i,\psi (i+1)),\qquad i\in {\mathbb {N}}, \end{aligned}$$
(22)
where \(\Phi :{\mathbb {N}}\times Y\rightarrow Y\) is given and \(\psi :{\mathbb {N}}\rightarrow Y\) is unknown. Clearly, (22) is a very simple particular case of (20), with \(E={\mathbb {N}}\), \(j=1\) and \(f_1(i)=i+1\) for \(i\in X\).
Let \((a_n)_{n\in {\mathbb {N}}}\) be a sequence of positive reals, such that
$$\begin{aligned} \sum _{k=1}^{\infty }\prod _{l=0}^{k-1}a_{i+l}<\infty ,\qquad i\in {\mathbb {N}}. \end{aligned}$$
(23)
For instance, we can take \(\rho \in (0,1)\) and write
$$\begin{aligned} a_{2n}=\frac{1}{\rho },\qquad a_{2n-1}=\rho ^2,\qquad n\in {\mathbb {N}}. \end{aligned}$$
Then
$$\begin{aligned} \prod _{l=0}^{2k}a_{i+l}=\rho \prod _{l=0}^{2k-2}a_{i+l},\qquad \prod _{l=0}^{2k+1}a_{i+l}=\rho \prod _{l=0}^{2k-1}a_{i+l},\qquad k\in {\mathbb {N}}, \end{aligned}$$
whence (23) is valid and
$$\begin{aligned} \sum _{k=1}^{\infty }\prod _{l=0}^{k-1}a_{i+l}=\frac{a_i(1+a_{i+1})}{1-\rho },\qquad i\in {\mathbb {N}}. \end{aligned}$$
Let operator \(\Lambda :{\mathbb {R}}_+^{{\mathbb {N}}}\rightarrow {\mathbb {R}}_+^{{\mathbb {N}}}\) be defined by
$$\begin{aligned} \Lambda \delta (i)=a_i\delta (i+1),\qquad \delta \in {\mathbb {R}}_+^{{\mathbb {N}}},i\in {\mathbb {N}}. \end{aligned}$$
Note that
$$\begin{aligned} \Lambda ^k\delta (i)=\delta (i+k)\prod _{l=0}^{k-1}a_{i+l},\qquad k\in {\mathbb {N}},\delta \in {\mathbb {R}}_+^{{\mathbb {N}}}, \end{aligned}$$
whence
$$\begin{aligned} \sum _{k=1}^{n}\Lambda ^k\delta (i)=\sum _{k=1}^{n}\delta (i+k)\prod _{l=0}^{k-1}a_{i+l},\qquad n\in {\mathbb {N}},\delta \in {\mathbb {R}}_+^{{\mathbb {N}}}. \end{aligned}$$
(24)
Take \(\gamma >0\) and let \(\phi :{\mathbb {N}}\rightarrow Y\) fulfil inequalities (17) and (18) with some \(\varepsilon _1,\varepsilon _2:{\mathbb {N}}\rightarrow [0,\gamma ]\), that is
$$\begin{aligned} d(\Phi (i,\phi (i+1)),\phi (i))\le \varepsilon _1(i), \qquad d(\phi (i),\Phi (i,\phi (i+1)))\le \varepsilon _2(i), \qquad i\in {\mathbb {N}}. \end{aligned}$$
Then, (24) implies that, for each \(j\in \{1,2\}\)
$$\begin{aligned} \varepsilon _j^*(i){:}{=}&\,\sum _{k=0}^{\infty }\Lambda ^k\varepsilon _j (i)\le \gamma \Big (1+\sum _{k=1}^{\infty }\prod _{l=0}^{k-1}a_{l+i}\Big )<\infty ,\qquad i\in {\mathbb {N}}. \end{aligned}$$
Next, if
$$\begin{aligned} d(\Phi (i,z),\Phi (i,w))\le a_id(z,w),\qquad w,z\in Y,i\in {\mathbb {N}},d(z,w)\le e, \end{aligned}$$
where \(e{:}{=}\sup \,\{\varepsilon _j^*(i): i\in {\mathbb {N}}, j=1,2\}\), then \(\Phi \) is as in (H1) with \(j=1\), and the assumptions of Corollary 6 are satisfied with
$$\begin{aligned} L_1(i)= a_i,\qquad f_1(i)=i+1,\qquad i\in {\mathbb {N}}. \end{aligned}$$
Hence, the limit
$$\begin{aligned} \psi (i){:}{=}\lim _{n\rightarrow \infty }{\mathcal {T}}^n\phi (i) \end{aligned}$$
(25)
exists for each \(i\in {\mathbb {N}}\), with
$$\begin{aligned} {\mathcal {T}}\xi (i){:}{=}\Phi (i,\xi (i+1)), \qquad \xi \in Y^E,\,i\in {\mathbb {N}}, \end{aligned}$$
and \(\psi :E\rightarrow Y\), given by (25), is the unique solution of (22), such that
$$\begin{aligned} d(\phi (i),\psi (i))\le \varepsilon _1^*(i), \qquad d(\psi (i),\phi (i))\le \varepsilon _2^*(i), \qquad i\in {\mathbb {N}}. \end{aligned}$$

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Open AccessThis 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 and Affiliations

  1. 1.Faculty of Applied MathematicsAGH University of Science and TechnologyKrakówPoland
  2. 2.Mathematics DepartmentCollege of Science, Jouf UniversitySakakaKingdom of Saudi Arabia
  3. 3.Basic Science Department, Faculty of Computers and InformaticsSuez Canal UniversityIsmailiaEgypt
  4. 4.Department of MathematicsPedagogical UniversityKrakówPoland

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