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

• Janusz Brzdęk
• Zbigniew Leśniak
Open Access
Article

## 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}

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}
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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## Authors and Affiliations

• Janusz Brzdęk
• 1