Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation
We propose generalized forms of ultraexponential and infralogarithm functions introduced and studied by the author earlier and present two classes of special functions, namely, ultraexponential and infralogarithm f-type functions. As a result of present investigation, we obtain general solution of t...
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Нооshmаnd, M.H. 2020-02-19T04:45:45Z 2020-02-19T04:45:45Z 2011 Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation / М.Н. Нооshmаnd // Український математичний журнал. — 2011. — Т. 63, № 2. — С. 281–288. — Бібліогр.: 5 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/166346 517.9 We propose generalized forms of ultraexponential and infralogarithm functions introduced and studied by the author earlier and present two classes of special functions, namely, ultraexponential and infralogarithm f-type functions. As a result of present investigation, we obtain general solution of the Abel equation α(f(x))=α(x)+1 under some conditions on a real function f and prove a new completely different uniqueness theorem for the Abel equation stating that the infralogarithm f-type function is its unique solution. We also show that the infralogarithm f-type function is an essentially unique solution of the Abel equation. Similar theorems are proved for the ultraexponential f-type functions and their functional equation β(x)=f(β(x−1)) which can be considered as dual to the Abel equation. We also solve certain problem being unsolved before, study some properties of two considered functional equations and some relations between them. Запропоновано узагальненi форми ультраекспоненцiальних та iнфралогарифмiчних функцiй, що були введенi i вивченi автором ранiше, та наведено два класи спецiальних функцiй — ультраекспоненцiального та iнфралогарифмiчного f-типу. В результатi дослiджень отримано загальний розв’язок рiвняння Абеля α(f(x))=α(x)+1 за певних умов для реальної функцiї f i доведено нову цiлком iншу теорему єдиностi для рiвняння Абеля з твердженням про те, що функцiя iнфралогарифмiчного f-типу є єдиним розв’язком цього рiвняння. Також показано, що функцiя iнфралогарифмiчного f-типу є суттєво єдиним розв’язком рiвняння Абеля. Подiбнi теореми доведено для функцiй ультраекспоненцiального f-типу та їх функцiонального рiвняння β(x)=f(β(x−1)), яке можна вважати дуальним для рiвняння Абеля. Також розв’язано задачу, що не була розв’язана до теперiшнього часу, вивчено властивостi двох розглядуваних функцiональних рiвнянь та деякi спiввiдношення мiж ними. en Інститут математики НАН України Український математичний журнал Статті Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation Функцiї ультраекспоненцiального та iнфралогарифмiчного типiв i загальний розв’язок функцiонального рiвняння Абеля Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation |
| spellingShingle |
Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation Нооshmаnd, M.H. Статті |
| title_short |
Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation |
| title_full |
Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation |
| title_fullStr |
Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation |
| title_full_unstemmed |
Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation |
| title_sort |
functions of ultraexponential and infralogarithm types and general solution of the abel functional equation |
| author |
Нооshmаnd, M.H. |
| author_facet |
Нооshmаnd, M.H. |
| topic |
Статті |
| topic_facet |
Статті |
| publishDate |
2011 |
| language |
English |
| container_title |
Український математичний журнал |
| publisher |
Інститут математики НАН України |
| format |
Article |
| title_alt |
Функцiї ультраекспоненцiального та iнфралогарифмiчного типiв i загальний розв’язок функцiонального рiвняння Абеля |
| description |
We propose generalized forms of ultraexponential and infralogarithm functions introduced and studied by the author earlier and present two classes of special functions, namely, ultraexponential and infralogarithm f-type functions. As a result of present investigation, we obtain general solution of the Abel equation α(f(x))=α(x)+1 under some conditions on a real function f and prove a new completely different uniqueness theorem for the Abel equation stating that the infralogarithm f-type function is its unique solution. We also show that the infralogarithm f-type function is an essentially unique solution of the Abel equation. Similar theorems are proved for the ultraexponential f-type functions and their functional equation β(x)=f(β(x−1)) which can be considered as dual to the Abel equation. We also solve certain problem being unsolved before, study some properties of two considered functional equations and some relations between them.
Запропоновано узагальненi форми ультраекспоненцiальних та iнфралогарифмiчних функцiй, що були введенi i вивченi автором ранiше, та наведено два класи спецiальних функцiй — ультраекспоненцiального та iнфралогарифмiчного f-типу. В результатi дослiджень отримано загальний розв’язок рiвняння Абеля α(f(x))=α(x)+1 за певних умов для реальної функцiї f i доведено нову цiлком iншу теорему єдиностi для рiвняння Абеля з твердженням про те, що функцiя iнфралогарифмiчного f-типу є єдиним розв’язком цього рiвняння. Також показано, що функцiя iнфралогарифмiчного f-типу є суттєво єдиним розв’язком рiвняння Абеля. Подiбнi теореми доведено для функцiй ультраекспоненцiального f-типу та їх функцiонального рiвняння β(x)=f(β(x−1)), яке можна вважати дуальним для рiвняння Абеля. Також розв’язано задачу, що не була розв’язана до теперiшнього часу, вивчено властивостi двох розглядуваних функцiональних рiвнянь та деякi спiввiдношення мiж ними.
|
| issn |
1027-3190 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/166346 |
| citation_txt |
Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation / М.Н. Нооshmаnd // Український математичний журнал. — 2011. — Т. 63, № 2. — С. 281–288. — Бібліогр.: 5 назв. — англ. |
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2025-11-25T20:31:33Z |
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| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 517.9
M. H. Hooshmand (Islamic Azad University-Shiraz Branch, Iran)
FUNCTIONS OF ULTRAEXPONENTIAL
AND INFRALOGARITHM TYPES AND GENERAL SOLUTION
OF THE ABEL FUNCTIONAL EQUATION
ФУНКЦIЇ УЛЬТРАЕКСПОНЕНЦIАЛЬНОГО
ТА IНФРАЛОГАРИФМIЧНОГО ТИПIВ I ЗАГАЛЬНИЙ
РОЗВ’ЯЗОК ФУНКЦIОНАЛЬНОГО РIВНЯННЯ АБЕЛЯ
We propose generalized forms of ultraexponential and infralogarithm functions introduced and studied by the
author earlier and present two classes of special functions, namely, ultraexponential and infralogarithm f -type
functions. As a result of present investigation, we obtain general solution of the Abel equation α(f(x)) =
= α(x) + 1 under some conditions on a real function f and prove a new completely different uniqueness
theorem for the Abel equation stating that the infralogarithm f -type function is its unique solution. We
also show that the infralogarithm f -type function is an essentially unique solution of the Abel equation.
Similar theorems are proved for the ultraexponential f -type functions and their functional equation β(x) =
= f(β(x − 1)) which can be considered as dual to the Abel equation. We also solve certain problem being
unsolved before, study some properties of two considered functional equations and some relations between
them.
Запропоновано узагальненi форми ультраекспоненцiальних та iнфралогарифмiчних функцiй, що були
введенi i вивченi автором ранiше, та наведено два класи спецiальних функцiй — ультраекспоненцiального
та iнфралогарифмiчного f -типу. В результатi дослiджень отримано загальний розв’язок рiвняння Абеля
α(f(x)) = α(x) + 1 за певних умов для реальної функцiї f i доведено нову цiлком iншу теорему
єдиностi для рiвняння Абеля з твердженням про те, що функцiя iнфралогарифмiчного f -типу є єдиним
розв’язком цього рiвняння. Також показано, що функцiя iнфралогарифмiчного f -типу є суттєво єдиним
розв’язком рiвняння Абеля. Подiбнi теореми доведено для функцiй ультраекспоненцiального f -типу
та їх функцiонального рiвняння β(x) = f(β(x − 1)), яке можна вважати дуальним для рiвняння
Абеля. Також розв’язано задачу, що не була розв’язана до теперiшнього часу, вивчено властивостi двох
розглядуваних функцiональних рiвнянь та деякi спiввiдношення мiж ними.
1. Introduction and preliminaries. In [2] we solve the following functional equation
completely and obtain its general solution
α(ax) = α(x) + 1, x ∈ R\[δ1, δ2].
In fact this equation is a special case of the Abel’s equation (equation (7.14) in [3]),
where f = expa, 0 < a 6= 1, δ1 ≤ δ2 are the two zeros of g(x) = aa
x − x with this
assumption that δ2 = +∞ if a > 1, and
E = R\[δ1, δ2] =
R, a > e1/e,
(−∞, δ1), 1 < a ≤ e1/e,
R \ {δ1 = δ2},
(
1
e
)e
≤ a < 1,
(−∞, δ1) ∪ (δ2,+∞), 0 < a <
(
1
e
)e
.
c© M. H. HOOSHMAND, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2 281
282 M. H. HOOSHMAND
It is very important to know that the general solution does not need any conditions
on a and α, and it is new. The general solution is α = ϕ1 Ioga +[ ]a = (φ)1 Ioga +[ ]a,
where Ioga is the infralogarithm function as the dual of the ultraexponential function:
uxpa(x) = a
x
, [ ]a ultra power part function, ( )1 decimal part function, ϕ1 every 1-
periodic function and φ is any function defined on [0, 1). For this reason we call the above
equation infralogarithm functional equation. Moreover, we proved that the equation is
equivalent to f(a
x
) = f((x)1) + [x] + 1, namely co-infralogarithm functional equation,
if x is restricted to (R\[δ1, δ2]) ∩ Duxpa . Also, we prove a uniqueness theorem that
states Ioga is a unique solution of the Abel’s expa-functional equation (infralogarithm
equation), under some conditions. Of course, similar theorems for the ultraexponential
functions and their related functional equations (β(x) = aβ(x−1) that is dual of the
equation) are proved in [4].
Now, to generalize the above results for the Abel’s equation in general (for a given
real function f : R→ R), we first introduce ultraexponential and infralogarithm f -type
functions and then obtain its general solution (by using the two classes of functions),
under some conditions on f that is completely different to the previous assumptions in
the main references such as [3, 5].
If f, ϕ are real functions and µ is an integer valued function (µ : R → Z), then we
define the function fµϕ by
fµϕ (x) = fµ(x)(ϕ(x)), (1.1)
and call it µ-composition of f at ϕ or µ-iteration of f at ϕ. The domain of fµϕ is dependent
on invertibility of f and the domains of each functions f, ϕ and µ. Therefore if f is not
invertible, then Dfµϕ ⊆ µ−1([0,+∞)). Also fµϕ (x) = ϕ(x), for every x ∈ µ−1({0}).
Notice that fµI = fµ and fµf = fµ+1 (I is the identity function), also if µ = n is a
constant function, then fµ = fn (n-composition of f).
If µ(x+ k) = µ(x) + k and ϕ is k-periodic, where k is an integer, then
fµϕ (x+ k) = fµ(x+k)(ϕ(x+ k)) = fk+µ(x)(ϕ(x)) = fk(fµϕ (x)), (1.2)
for every x such that x, x + k ∈ Dfµϕ (of course if fk is defined). Especially if k = 1
and g = fµϕ , then g(x+ 1) = f(g(x)).
Denote by [x] the largest integer not exceeding x and put (x) = x − [x]. Then, for
any fixed real number r 6= 0, we set
(x)r = r
(x
r
)
, [x]r = r
[x
r
]
∀x ∈ R
and call (x)r r-decimal part of x and [x]r r-integer part of x. Since (x)1 = (x) (to
prevent any confusion between decimal and parentheses notation) sometimes we use the
symbol (x)1 instead of (x). Clearly x = [x]r + (x)r, and
[x]r ∈ 〈r〉 = rZ, (x)r ∈ r[0, 1) = [0, r) or (r, 0].
We call (x)r, [x]r r-parts of x. It is easy to see that the r-decimal part function ( )r is
r-periodic (especially the decimal part function ( ) = ( )1 is periodic of period 1).
Note. Every r-periodic function ϕ has the form ϕ = φo( )r where φ is a function
defined on r[0, 1). Since the composition of every function and a periodic function is
periodic, the r-decimal part function is the basic r-period function. In fact ϕ = φo( )r
is the general solution of the functional equation ϕ(x+ r) = ϕ(x).
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
FUNCTIONS OF ULTRAEXPONENTIAL AND INFRALOGARITHM TYPES AND GENERAL . . . 283
2. Ultraexponential f -type functions. Here we generalize the ultraexponential
function that is a unique extension of the Tetration.
Definition 2.1. We call the function f [ ]+1
( ) “semi-ultraexponential f -type function”
and denote by uptf .
Recall that if 0 < a 6= 1, then expa(x) = ax and exp(x) = expe(x) = ex (for
all x) and exp−1a = loga, ln = exp−1 . Therefore uptexpa = uxpa which is called
ultraexponential function, and uxp = uxpe is the natural ultraexponential function.
Hence uxpa(x) = exp
[x]+1
a ((x)1) = a
x
(see [4]).
Now let f be a function defined on fn([0, 1)), for every non-negative integer n.
Then uptf is defined on [−1,+∞) and we have
uptf (x) = f [x]+1((x)1) =
...
x+ 1, −1 ≤ x < 0,
f(x), 0 ≤ x < 1,
f2(x− 1), 1 ≤ x < 2,
f3(x− 2), 2 ≤ x < 3,
...
Note that if f is invertible, then the domain of uptf may be larger than [−1,+∞). In
fact with the mentioned hypothesis we have
Duptf = [−1,+∞) ∪ ([−2,−1) ∩ (Df−1 − 2)) ∪ . . . = [−1,+∞) ∪ Sf−1 , (2.1)
where Sf−1 ⊆ (−∞,−1) and Sf−1 = ∅ if f is not invertible.
The function uptf satisfies the following well known functional equation
β(x) = f(β(x− 1)), (2.2)
for every x ≥ 0. For this reason we call (2.2) “ultraexponential f -type functional
equation”.
Example 2.1. If f = expa, then uptf = f
[ ]+1
( ) = uxpa . Now if a > 1, then (in
(2.1) Sf−1 = (−2,−1) so Duxpa = (−2,+∞) and we have
uxpa(x) = expa(uxpa(x− 1)), x > −1.
In this case the values of x in (2.2) are extended from x ≥ 0 to x > −1.
But if 0 < a < 1, then
Sf−1 = (−2,−1)∪(−3,−2)∪(−4+a,−3)∪(−5+a,−5+aa)∪(−6+aa
a
,−6+aa)∪. . . ,
and
uxpa(x) = expa(uxpa(x− 1)), −1 6= x > −2.
In fact the above equation holds for all x such that x, x− 1 ∈ Duxpa (see [4]).
In [4] we proved the first uniqueness theorem about the Tetration that one of its
corollaries states if a = e, then there is no any convex function on [−1,+∞) except
that f = uxpa such that satisfies the functional equation
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
284 M. H. HOOSHMAND
β(x) = expa(β(x− 1)), x > −1,
with the initial condition f(0) = 1. But for a > e it was left as an unsolved problem as
follows:
Question. Let a > e. Is there any convex function on [−1,+∞) except that
f = uxpa which satisfies the ultraexponential functional equation?
Now we claim that the answer is positive. Because, putting
φ(x) = a
2x+(−1+ln a)x2
1+ln a , f(x) = exp[x]
a (φ(x− [x])) ∀x > −2,
one can see that f satisfies the all conditions of the question (considering Corollary 2.3,
3.5 and Theorem 3.2 of [4]) and clearly f 6= uxpa . In fact there exist infinitely many
solutions for it. For if 0 < ε ≤ −1 + ln a
1 + ln a
, φ(x) = a(1−ε)x+εx
2
, then the function f
defined by the above equation satisfies the all conditions.
The following lemma gives us general solution of the ultraexponential f -type equation
for an arbitrary given function f.
Lemma 2.1. Let c be a constant real number and f a given real function. Then,
general solution of the functional equation
β(x) = f(β(x− 1)), x ≥ c, (2.3)
is
β(x) = f [x−c]+1(ϕ(x)), x ≥ c− 1, (2.4)
where ϕ is every 1-periodic real function.
Proof. Considering (1.2) if β has the form (2.4), then satisfies the equation. Converse-
ly let β satisfies the equation and fix x. Then β(x) = fn(β(x − n))), for every
non-negative integer n such that x − n ≥ c − 1. Putting n = [x − c] + 1 and
ϕ(x) = β((x− c)1 + c− 1), we have β(x) = f [x−c]+1(ϕ(x)) and ϕ is 1-periodic.
Note. If f is defined on fn([0, 1)) for every integer n ≥ 0, then uptf satisfies the
equation (2.3), when c = 0. If f is invertible, then it may be satisfied the equation for
some c < 0. For example if f = expa, then uptf satisfies the equation for c = −1
(see [4]).
But it is important to know that the function uptf (x − c) satisfies (2.3), for every
constant c (put ϕ(x) = (x− c)1), and so always it is a solution of the equation.
An important problem about the ultra f -type equation is that when uptf is its unique
solution? (under which conditions?). To answer the question, in the following theorem
we introduce a unique solution for the ultraexponential f -type equation.
Theorem 2.1 [A uniqueness conditions for the ultraexponential f -type functions].
Let f be a function defined on fn([0, 1)) for every integer n ≥ 0 and differentiable on
[0, 1), f(0) = 1, f ′+(0) 6= 0.
Then β = uptf is the unique solution of the ultraexponential f -type equation on
[−1,+∞) for which is increasing and differentiable on [−1, 0) and β′ is monotonic
(non-decreasing or non-increasing) on it and
β(−1) = 0, lim
x→0+
β′(x) = f ′+(0) lim
x→0−
β′(x). (2.5)
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
FUNCTIONS OF ULTRAEXPONENTIAL AND INFRALOGARITHM TYPES AND GENERAL . . . 285
Proof. Clearly uptf satisfies the conditions. Now if β = g satisfies the conditions,
then g is differentiable on [−1, 0], (0, 1) and continuous at zero
(
limx→0 g(x) = g(0) =
= f(g(−1)) = f(0) = 1
)
. Therefore 0 < g(x− 1) < 1 and g′(x) = g′(x− 1)f ′(g(x−
− 1)), for every 0 < x < 1, hence
f ′+(0) lim
x→0−
g′(x) = lim
x→0+
g′(x) = lim
x→0+
g′(x− 1)f ′(g(x− 1)) = f ′+(0) lim
t→−1+
g′(t).
So limx→0− g
′(x) = limx→−1+ g
′(x) thus g′ is constant on (−1, 0), because g′ is
monotonic on (−1, 0). Now considering limx→0− g(x) = 1, limx→−1+ g(x) = 0 we
conclude that g(x) = x+ 1 on [−1, 0] and so the Lemma 2.1 completes this proof.
Now considering the above theorem and (2.5) we have the following corollary:
Corollary 2.1. Suppose f is a function defined on fn([0, 1)) for every integer n ≥ 0
and differentiable on [0, 1) and f ′+(0) = f(0) = 1.
Then β = uptf is the only solution of the ultraexponential f -type equation on
[−1,+∞) such that β(−1) = 0, β is increasing and differentiable on [−1, 0) and β′
exists at zero and is monotonic on (−1, 0).
3. Infralogarithm f -type functions; a unique solution for the Abel functional
equation. An interesting property of the ultraexponential f -type functions is that its
inverse function (if exists) satisfies the Abel’s equation
α(f(x)) = α(x) + 1.
Let f be a function defined on fn([0, 1)) for every integer n ≥ 0 and f(0) = 1.
Then f is defined on fn([0, 1]) for every integer n ≥ 0 and
fn+1(0) = fn(1) = uptf (n), n = 0, 1, 2, . . . .
Now if f is continuous and increasing on fn([0, 1]) (∀n ≥ 0), then the sequence fn(1)
is increasing and limn→∞ fn(1) = f∞(1) = δ, where 1 < δ ≤ ∞, and
∞⋃
n=0
fn([0, 1]) = [0, δ) ⊆ Df .
Also f : [0, δ)→ [1, δ) is increasing, continuous and invertible. Therefore f−1 : [1, δ)→
→ [0, δ) is continuous, increasing and invertible too. In this case the function uptf : [−1,
+∞)→ [0, δ) is increasing, continuous and bijection too. So upt−1f : [0, δ)→ [−1,+∞)
is continuous and increasing and we have
upt−1f (x) =
...
x− 1, 0 ≤ x ≤ 1,
f−1(x), 1 ≤ x ≤ f(1),
1 + f−2(x), f(1) ≤ x ≤ f2(1),
2 + f−3(x), f2(1) ≤ x ≤ f3(1),
...
Of course the domain of upt−1f may be larger than [0, δ) if the domain of f−1 is larger
than [1, δ) (e.g. see Ioga in [2]).
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286 M. H. HOOSHMAND
Definition 3.1. Let f be a continuous and increasing real function on fn([0, 1]),
for every integer n ≥ 0, and f(0) = 1. Then we call uptf “ultraexponential f -type
function” and denote by uxpf . Also, denote uxp−1f by Iogf and call it “infralogarithm
f -type function”.
Note that if a > 1 and f = expa, then uxpf = uxpa and Iogf = uxp−1f = uxp−1a =
= Ioga.
Theorem 3.1. The functions Iogf and [Iogf ] satisfy the Abel’s equation
α(f(x)) = α(x) + 1, 0 ≤ x < δ (3.1)
(where δ = f∞(1) and [Ioga] = [ ]o Ioga). Also, there exists an integer valued function
µ such that Iogf = µ+ f−µ−1 (on [0, δ)) and [Iogf ] = µ, (Iogf ) = f−µ−1.
Proof. Put g = uxpf and µ = [g−1]. If 0 ≤ x < δ, then x = g(y) for some y ≥ −1.
So
µ(f(x)) = [g−1(f(x))] = [g−1f(g(y))] = [g−1(g(y + 1))] = [y] + 1 = µ(x) + 1.
Therefore µ = [Iogf ] satisfies (3.1). Now putting h = µ+ f−µ−1, we have
hg(x) = µ(g(x)) + f−µ(g(x))−1(g(x)) = [x] + f−[x]−1(f [x]+1((x))1) = [x] + (x) = x,
for every x ≥ −1. Now if 0 ≤ x < δ, then µ(x) ≤ g−1(x) < µ(x) + 1 thus
g(µ(x)) ≤ x < g(µ(x) + 1)⇒ fµ(x)+1(0) ≤ x < fµ(x)+2(0)⇒ 0 ≤ f−µ(x)−1 < 1.
Hence (h) = f−µ−1, [h] = µ and if 0 ≤ x < δ, then
gh(x) = f [h(x)]+1((h(x))1) = fµ(x)+1(f−µ(x)−1(x)) = x.
Now we have
Iogf (f(x)) = h(f(x)) = µ(f(x)) + f−µ(f(x))−1(f(x)) = 1 + Iogf (x).
Theorem 3.1 is proved.
Note. The above theorem shows that we can consider the Abel’s equation as an
infralogarithm functional equation when f(0) = 1, f is continuous and increasing on
fn([0, 1]) (∀n ≥ 0). In this case we introduce general solution of the equation and proof
a uniqueness theorem about it at the end of this section.
Now we proof a theorem that states an interesting relation between the ultraexponential
f -type and Abel’s equation.
Theorem 3.2. (i) If α and β satisfy the Abel’s equation and ultra f -type functional
equation respectively, then (βα)f = f(βα) and there exists 1-periodic function Φ such
that αβ(x) = x + Φ(x) (of course it holds for all x such that the compositions are
possible).
(ii) If µ, h are two solutions of the (3.1) (Abel’s equation on [0, f∞(1))) such that
(h) = f−µ, then general solution of the equation is α = µ + ϕh, where ϕ is every
1-periodic function.
Proof. (i) Let α, β satisfy the equations, then
(βα)f(x) = β(αf(x)) = β(α(x) + 1) = f(β(α(x)) = f(βα(x)).
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FUNCTIONS OF ULTRAEXPONENTIAL AND INFRALOGARITHM TYPES AND GENERAL . . . 287
Also, we have
αβ(x) = α(f(β(x− 1)) = α(β(x− 1)) + 1.
So αβ satisfies the difference equation χ(x) = 1+χ(x−1) and so there exists 1-periodic
function Φ such that αβ(x) = x+ Φ(x).
(ii) Let α has the mentioned form, then
α(f(x)) = µ(f(x)) + ϕ(h(f(x))) = 1 + µ(x) + ϕ(h(x) + 1) = 1 + α(x).
Conversely if α is an arbitrary solution of the equation, then α(fn(x)) = α(x) + n,
for every integer n such that fn exists. Now considering the hypothesis we can put
n = −µ(x0), when x0 is fix, and we have
α(f−µ(x0)(x0)) = α(x0)− µ(x0)⇒ α((h(x0)) = α(x0)− µ(x0).
So putting ϕ = α|[0,1)o( )1 we have α(x) = µ(x) + ϕh(x) and ϕ is 1-periodic.
Theorem 3.2 is proved.
Corollary 3.1 [General solution of the Abel’s equation]. If f is continuous and inc-
reasing on fn([0, 1]), for every n ≥ 0 and f(0) = 1, then the general solution of the (3.1)
is
α = [Iogf ] + ϕ Iogf , (3.2)
where ϕ is every 1-periodic function.
Proof. Note that if h is a solution of the equation, then h + c is so (c is constant).
Now we get this result by Theorems 3.1, 3.2 (ii).
Remark 3.1. Theorems 3.1, 3.2 and Corollary 3.1 state some important facts for
the Abel’s equation (for real functions) that one of them says the general solution can be
gotten from the mentioned essential solutions (h and µ in Theorem 3.2 (ii)). Theorem 3.1
grantees that Iogf and [Iogf ] (infralogarithm f -type function and its bracket) are the
essential real solutions for the Abel’s equation (under the conditions on f). On the other
hand, replacing ϕ by α and putting c = 1 in the equation (1.48) of [3] we get (7.1) of
[3] that is the Abel’s equation α(f(x)) = α(x) + 1. Now comparing Theorem 1.9 of
[3] about this equation and our results, we can see some similarities between the given
form α = ϕh + µ of Theorem 3.2 (ii) and α = ϕ0a + d of [3] (the original form is
α = ϕ0[a(x)] + d(x)c). Indeed, in these two similar forms both µ and d are integer
valued functions and α = ϕh+µ gives general solution of the equation but α = ϕ0a+d
gives a unique corresponded solution for the given function ϕ0 with some conditions.
Of course the conditions of the theorems are completely different (see [3] and [5]).
The above theorem states some interesting relations between the general solutions of
the Abel’s equation and the ultraexponential f -type equation specially when we consider
the (3.1) (0 ≤ x < f∞(1)). If φ = 0 in (3.1), then it may α = β−1, e.g., α = Iogf and
β = uxpf . Also, putting ϕ = ( )1 in the general solution (part (ii)) implies µ+ f−µ is
a solution of the equation.
Now we are ready to introduce a uniqueness conditions for the infralogarithm f -type
function regarding to the (3.1) and similar to Theorem 2.1.
Theorem 3.3 [A uniqueness theorem for Abel’s equation]. Let f be continuous and
increasing on fn([0, 1]), for every n ≥ 0, and differentiable on [0, 1), (1, f(1)) and
f(0) = 1.
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288 M. H. HOOSHMAND
If h is a solution of (3.1) such that h′ is monotonic on (0, 1) and
h(0) = −1, lim
x→1−
h′(x) = f ′+(0) lim
x→1+
h′(x), (3.3)
then h = Iogf (on [0, f∞(1))).
Proof. First note that Iogf satisfies the conditions, clearly. Now if 0 < x < 1, then
1 < f(x) < f(1) and h′(x) = f ′(x)h′(f(x)). Since f is increasing and continuous on
[0, 1) and f(0) = 1 we have
lim
x→0+
h′(x) = lim
x→0+
f ′(x)h′(f(x)) = f ′+(0) lim
t→1+
h′(t) = lim
x→1−
h′(x).
Therefore limx→0+ h
′(x) = limx→1− h
′(x) so h′ is constant on (0, 1), because it is
monotonic on (0, 1). Now considering −1 = h(0) = limx→0+ h(x) and 0 = h(1) =
= limx→1− h(x) we conclude that h(x) = x− 1 on [0, 1].
Finally since f : [0, f∞(1)) → [1, f∞(1)) =
⋃∞
n=0[fn(1), fn+1(1)] is continuous
and increasing and h(fn(x)) = h(x) + n for n ≥ 0, we conclude h = Iogf on
[0, f∞(1)).
Theorem 3.3 is proved.
Now considering the above theorem and (3.3), we have the following corollary:
Corollary 3.2. Let f be continuous and increasing on fn([0, 1]), for every n ≥ 0,
and differentiable on [0, 1), (1, f(1)) and f ′+(0) = f(0) = 1. Then α = Iogf is the
only solution of the Abel’s equation such that α(0) = −1 and α′ exists at x = 1 and
monotonic on (0, 1).
Moreover, in this case Iogf is differentiable on [0, f(1)) and if f is differentiable
overall [0, δ), then Iogf is so.
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2. Hooshmand M. H. Infra logarithm and ultra power part functions // Integral Transforms Spec. Funct. –
2008. – 19, № 7. – P. 497 – 507.
3. Kuczma M. Functional equations in a single variable. – Warszawa: PWN-Polish Sci. Publ., 1968.
4. Hooshmand M. H. Ultra power and ultraexponential functions // Integral Transforms Spec. Funct. –
2006. – 17, № 8. – P. 549 – 558.
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Received 21.04.10,
after revision — 23.11.10
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 2
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