- Email: [email protected]
Nonlinear Volterra integral equations and the Schröder functional equation
Nonlinear Analysis 74 (2011) 424–432
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Nonlinear Volterra integral equations and the Schröder functional equation Tomasz Małolepszy Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland
article
abstract
info
Article history: Received 1 February 2010 Accepted 29 August 2010
We show an interesting connection between a special class of Volterra integral equations and the famous Schröder equation. The basic results provide criteria for the existence of nontrivial as well as blow-up solutions of the Volterra equation, expressed in terms of the convergence of some integrals. Examples related to Volterra equations with power and exponential nonlinearities are presented. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Volterra equation Nontrivial solutions Blow-up solutions Schröder equation
1. Introduction In last 30 years many articles have been dedicated to second-kind nonlinear Volterra integral equations of a certain type with a convolution kernel, i.e. u(t ) =
t
∫
k(t − s)g (u(s))ds,
t ≥ 0,
(1)
0
where g (0) = 0. Eq. (1) appears in many physical problems, for example in the theory of water percolation from cylindrical reservoirs [1] and in the theory of nonlinear waves in tubes [2]. Just from an application point of view, the problem of the existence of nontrivial solutions and blow-up solutions is very interesting. Recall that by a nontrivial solution we mean a continuous function u, which satisfies (1), defined on the maximal interval of its existence [0, T ), where T > 0, such that u(0) = 0 and u(t ) > 0 for t ∈ (0, T ). If additionally u satisfies limt →T − u(t ) = ∞ for some T < ∞, we say that u is a blow-up solution of (1). Among the many conditions obtained for the existence of nontrivial or blow-up solutions, we can distinguish in particular two kinds of such conditions: those expressed in terms of the convergence of some series [3,4] and those expressed in terms of the convergence of some integrals [5–10]. However, there is quite a big difference between these two classes of conditions. Unlike the former, the latter are always applicable to (1) only with fixed or monotone kernel k or with fixed nonlinearity g. In this article new integral conditions for the existence of nontrivial and blow-up solutions are established without this defect. Moreover, surprisingly, they are connected with the famous Schröder functional equation. 2. Basic assumptions and facts Throughout this paper we assume that in (1) the functions k and g satisfy the following conditions: k is a locally integrable function such that k(x) > 0 for x > 0,
E-mail address: [email protected]. 0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.08.054
T. Małolepszy / Nonlinear Analysis 74 (2011) 424–432
425
g : [0, ∞) → [0, ∞) is a strictly increasing absolutely continuous function which satisfies the following conditions: g (0) = 0,
(2)
x/g (x) → 0
as x → 0 ,
(3)
x/g (x) → 0
as x → ∞.
(4)
+
Let additionally K (t ) :=
t
∫
k(s)ds
0
and the following condition hold: lim K (t ) = ∞.
t →∞
It is easy to see that the function K is strictly increasing. On the basis of the results presented in [11], we formulate the following: Fact 2.1. Under our assumptions on k and g, Eq. (1) has at most one nontrivial solution. Moreover, if there exists a nontrivial solution to (1), then it is a strictly increasing absolutely continuous function. Further in this article, we denote the iteration of two functions f and g by f ◦ g and the i-th iteration of an arbitrary function f by f i . 3. The Schröder equation Let an absolutely continuous and strictly increasing function h : [0, ∞) → [0, ∞) and c > 0, c ̸= 1 be given. We shall use the Schröder functional equation F (h(x)) = cF (x),
x ∈ I,
(5)
where F : I → R is an unknown function and I is a given interval such that h(I ) ⊂ I. In this paper we are interested only in solutions of Eq. (5), which are absolutely continuous and strictly increasing on I. For simplicity we denote a class of such solutions by S (I ), and we denote the arbitrary solution of (5) for given function h and constant c in the interval I, which belongs to S (I ), by Fh . The theory of the existence and uniqueness of such solutions is quite extended, but in most cases formulas for these solutions are not given in an elementary form. More details about the Schröder equation can be found in the classical monographs [12,13]. Remark 3.1. To ensure positivity of Fh on the given interval, in which Fh is one-signed, we use the notation ϵ Fh , where ϵ = 1 if Fh is positive on the given interval and ϵ = −1 if Fh is negative on the given interval. 4. Auxiliary results First, we present two theorems, which can be found in [4], related to the existence of nontrivial solutions of (1). Theorem 4.1. Let w be a continuous function on [0, δ], δ > 0, such that t < w(t ) < g (t ) for t ∈ (0, δ] and limt →0+ w(t t ) = 0. If the series ∞ −
K −1
i =0
(g −1 ◦ w)i (t ) w((g −1 ◦ w)i (t ))
is convergent on [0, δ], then (1) has a nontrivial solution on some interval. g (t )
Theorem 4.2. Let w be a continuous function such that w(t ) > 0 for t > 0 and lim supt →0+ w(t ) < 1. If (1) has a nontrivial solution on some interval, then the series ∞ − i =0
K −1
(g −1 )i (t ) w((g −1 )i (t ))
is convergent on [0, δ], where δ > 0. The next two theorems appeared for the first time in [3] and give us conditions for the existence of blow-up solutions.
426
T. Małolepszy / Nonlinear Analysis 74 (2011) 424–432
Theorem 4.3. Let (1) have a nontrivial solution u. If w is a strictly increasing, positive and continuous function such that w(t ) < g (t ) for t ∈ (a, ∞), w(a) = g (a), limt →∞ w(t t ) = 0 and the series ∞ −
K −1
i=0
(w −1 ◦ g )i (t ) w((w −1 ◦ g )i (t ))
is convergent for some t ∈ (a, ∞), then u is a blow-up solution to (1). Theorem 4.4. If Eq. (1) has a blow-up solution, w is a strictly increasing, positive and continuous function such that w(t ) < g (t ) w(t ) for t ∈ (a, ∞) and w(a) = g (a), and also limt →∞ t = 0, then for any ε ∈ (0, 1) there exists cε ≥ a such that the series ∞ −
K −1
i=0
((w −1 ◦ g )i (t ))1−ε g ((w −1 ◦ g )i (t ))
(6)
is convergent for any t ∈ (cε , ∞). 5. Main theorems Theorem 5.1. Let w be a continuous function on [0, δ], δ > 0, such that t < w(t ) < g (t ) for t ∈ (0, δ], limt →0+ w(t t ) = 0 and the function w(t t ) be strictly increasing on (0, δ). If there exists FΦ ∈ S (0, δ), where Φ = g −1 ◦ w , and the integral t
∫
K −1
s
w(s)
0
FΦ′ (s)
ϵ FΦ (s)
ds,
is convergent for t ∈ (0, δ), then (1) has a nontrivial solution. Proof. Take an arbitrary t < δ such that the function FΦ is either positive or negative on (0, t ) and define a sequence −1 {tn }∞ ◦ w)n (t ), this sequence is decreasing and n=0 by the following recurrence relation: t0 = t, tn+1 = Φ (tn ). Since tn = (g convergent to 0. Note that tn
∫
K
−1
s
w(s)
tn+1
FΦ′ (s)
ϵ FΦ (s)
ds ≥ K
−1
tn
∫
tn+1
w(tn+1 )
tn+1
FΦ′ (s)
ϵ FΦ (s)
ds
and, because FΦ is the solution of Schröder equation for the function Φ , tn
∫
tn+1
FΦ′ (s)
ϵ FΦ (s)
ds = ϵ ln
FΦ (tn ) FΦ (tn+1 )
= ϵ ln
1 c
=: C > 0.
Hence t0
∫
K −1
tn+1
s
w(s)
FΦ′ (s)
ϵ FΦ (s)
ds =
n ∫ − i=0
ti
K −1
ti+1
s
w(s)
FΦ′ (s)
ϵ FΦ (s)
ds ≥ C
n +1 − i =1
K −1
ti
w(ti )
.
Letting n → ∞, in the last inequality we obtain t
∫
K −1 0
s
w(s)
FΦ′ (s)
ϵ FΦ (s)
ds ≥ C
∞ − i=1
K −1
ti
w(ti )
The above inequality and Theorem 4.1 imply the result.
=C
∞ − i =1
K −1
(g −1 ◦ w)i (t ) . w((g −1 ◦ w)i (t ))
g (t )
Theorem 5.2. Let w be a continuous function such that w(t ) > 0 for t > 0 and lim supt →0+ w(t ) < 1. Let additionally the function w(t t ) be strictly increasing on some interval (0, δ). If (1) has a nontrivial solution and there exists FΨ ∈ S (0, δ) where
Ψ = g −1 , then the integral ′ ∫ t K −1
0
s
w(s)
FΨ (s) ds, ϵ FΨ (s)
is convergent for any t ∈ (0, δ).
T. Małolepszy / Nonlinear Analysis 74 (2011) 424–432
427
Proof. Taking an arbitrary t < δ such that the function FΨ is either positive or negative on (0, t ) and defining a sequence −1 n {tn }∞ ) (t )), one can n=0 that is decreasing and convergent to 0 by t0 = t, tn+1 = Ψ (tn ) (the explicit formula for tn is tn = (g repeat an argument applied in the proof of Theorem 5.1. We get then respectively tn
∫
K −1
FΨ′ (s)
s
w(s)
∫
tn
FΨ′ (s)
tn
w(tn ) tn+1 ϵ FΨ (s) ′ ∫ t0 n − s FΨ (s) ti K −1 ds ≤ C K −1 w(s) ϵ FΨ (s) w(ti ) tn+1 i=0 tn+1
ϵ FΨ (s)
ds ≤ K −1
ds = CK −1
tn
w(tn )
,
and, as n goes to ∞, t
∫
K −1
0
FΨ′ (s)
s
w(s)
ϵ FΨ (s)
ds ≤ C
∞ −
K −1
i=0
(g −1 )i (t ) , w((g −1 )i (t ))
which, by using Theorem 4.2, completes the proof.
Theorem 5.3. Let w be a strictly increasing, positive and continuous function such that w(t ) < g (t ) for t ∈ (a, ∞), w(a) = g (a), limt →∞ w(t t ) = 0 and let the function w(t t ) be strictly decreasing from some t˜ ≥ a. If there exists FΦ ∈ S (t˜, ∞), where Φ = w −1 ◦ g, the integral ∞
∫
K −1
FΦ′ (s)
s
w(s)
t
ϵ FΦ (s)
ds
(7)
is convergent for some t ∈ (t˜, ∞) and (1) has a nontrivial solution, then this solution is also a blow-up solution of (1). Proof. For an arbitrary t > t˜, for which the function FΦ is either positive or negative on [t , ∞), the sequence {tn }∞ n=0 given by t0 = t, tn+1 = Φ (tn ) (and hence tn = (w −1 ◦ g )n (t )) is increasing and convergent to ∞. Because tn+1
∫
K −1
w(s)
tn
FΦ′ (s)
s
ϵ FΦ (s)
ds ≥ CK −1
tn+1
w(tn+1 )
where
∫
FΦ′ (s)
tn+1
C =
ϵ FΦ (s)
tn
ds,
we have tn+1
∫
K −1
w(s)
t0
FΦ′ (s)
s
ϵ FΦ (s)
ds ≥ C
n +1 −
K −1
ti
w(ti )
i =1
.
Letting n → ∞, we obtain ∞
∫
K −1
t
s
w(s)
FΦ′ (s)
ϵ FΦ (s)
ds ≥ C
∞ −
K −1
i =1
(w −1 ◦ g )i (t ) . w((w−1 ◦ g )i (t ))
The last inequality and Theorem 4.3 end the proof.
Theorem 5.4. Let w be a strictly increasing, positive and continuous function such that w(t ) < g (t ) for t ∈ (a, ∞) and
w(a) = g (a) and also limt →∞ w(t t ) = 0. Let additionally the function tg (t ) be strictly decreasing from some t˜ ≥ a for given ε ∈ (0, 1). If Eq. (1) has a blow-up solution and there exists FΨ ∈ S (t˜, ∞), where Ψ = w −1 ◦ g, then for any ε ∈ (0, 1) there exists cε ≥ a such that the integral 1−ε ′ ∫ ∞ FΨ (s) s K −1 ds, g (s) ϵ FΨ (s) t 1−ε
is convergent for any t > max{t˜, cε }. Proof. Fix ε ∈ (0, 1). Theorem 4.4 implies that there exists cε ≥ a such that for any t ∈ (cε , ∞) the series (6) is convergent. Take an arbitrary t > max{t˜, cε }, for which the function FΨ is either positive or negative on [t , ∞). The sequence {tn }∞ n =0 defined by the recurrence relation t0 = t, tn+1 = Ψ (tn ), which implies that tn = (w −1 ◦ g )n (t ), is increasing and convergent 1−ε
to ∞. Since the function K −1 ( sg (s) ) is decreasing for s > t˜, we have
∫
tn+1 tn
K −1
s1−ε g (s)
FΨ′ (s)
ϵ FΨ (s)
ds ≤ K −1
tn1−ε g (tn )
∫
tn+1 tn
FΨ′ (s)
ϵ FΨ (s)
ds = CK −1
tn1−ε g (tn )
,
428
T. Małolepszy / Nonlinear Analysis 74 (2011) 424–432
whence tn+1
∫
K
−1
s1−ε g (s)
t0
FΨ′ (s)
ϵ FΨ (s)
ds ≤ C
n −
K
−1
ti1−ε g (ti )
i=0
.
Letting n → ∞ in the last inequality, we obtain ∞
∫
K −1
s1−ε
g (s)
t
FΨ′ (s)
ϵ FΨ (s)
ds ≤ C
∞ −
K −1
i=0
(w −1 ◦ g )i (t )1−ε . g ((w −1 ◦ g )i (t ))
Now Theorem 4.4 together with last inequality gives the thesis.
6. Examples related to power nonlinearity Example 6.1 (Power Nonlinearity — Case I). Let the function g in Eq. (1) satisfy condition (4) and there be d > 0 and 0 < p < 1 such that g (t ) = t p ,
t ∈ [0, d).
(8)
In order to examine when Eq. (1) with such g has a nontrivial solution, we take two functions: w1 (t ) = t , where p < q < 1, q
q
1
and w2 ≡ 1. Because the solutions of the Schröder equation for the functions Φ (t ) = t p and Ψ (t ) = t p (in the first case with q c = p and in the second with c = 1p ) are the same, i.e. FΦ (t ) = FΨ (t ) = ln t, Theorems 5.1 and 5.2 then take respectively the following forms: Theorem 6.2. If the integral t
∫
K −1 (s1−q ) 0
ds s(− ln s)
is convergent for t ∈ (0, δ), δ > 0, then Eq. (1) with g given by (8) has a nontrivial solution. Theorem 6.3. If Eq. (1) with g given by (8) has a nontrivial solution, then there exists δ > 0 such that the integral t
∫
K −1 ( s ) 0
ds s(− ln s)
is convergent for any t ∈ (0, δ). From Theorems 6.2 and 6.3 we immediately obtain the following well-known result (cf. [10,14]): Theorem 6.4. Eq. (1) with g satisfying (8) has a nontrivial solution if and only if the integral t
∫
K −1 ( s ) 0
ds s(− ln s)
is convergent for any t ∈ (0, δ), δ > 0. Example 6.5 (Power Nonlinearity — Case II). Now we take into consideration Eq. (1), in which the function g has the following form: g (t ) =
g1 (t ), bt p ,
t ∈ [0, d) t ≥ d,
(9)
where d > 0, p > 1 and b > 0, and the function g1 (t ) is taken such that the function g (t ) given by (9) satisfies (2)–(4). To find conditions for the existence of blow-up solution for this equation, we apply Theorems 5.3 and 5.4. For w(t ) = bap−q t q , p
where t ≥ a ≥ d and 1 < q < p, we have Φ (t ) = (w −1 ◦ g )(t ) = a( at ) q . A solution of the Schröder equation (5) for Φ with p c = q is FΦ (t ) = ln at , so from Theorem 5.3 we obtain: Theorem 6.6. If the integral ∞
∫
K −1 t
s1−q bap−q
ds s ln
s a
is convergent for some t ∈ (a, ∞) and (1) with g given by (9) has a nontrivial solution, then this solution is also a blow-up solution of (1).
T. Małolepszy / Nonlinear Analysis 74 (2011) 424–432
429 p r
Similarly, taking w(t ) = bap−r t r , where t ≥ a ≥ d and 0 < r < 1, we get Ψ (t ) = (w −1 ◦ g )(t ) = a( at ) and because a p solution of (5) for Ψ with c = r is FΨ (t ) = ln at , in this case we can write Theorem 5.4 as: Theorem 6.7. If Eq. (1) with g given by (9) has a blow-up solution, then for any ε ∈ (0, 1) there exists cε ≥ a such that the integral ∞
∫
K
−1
s1−ε−p
b
t
ds s ln
s a
is convergent for some t > cε . It can be shown (see [8]) that Theorems 6.6 and 6.7 lead to: Theorem 6.8. A nontrivial solution of Eq. (1) with g satisfying (9) is a blow-up solution if and only if the integral t
∫
K −1 (s) 0
ds s(− ln s)
is convergent for any t ∈ (0, δ). Remark 6.1. An easy consequence of Theorems 6.4 and 6.8 is that for Eq. (1) with the function g given by g (t ) =
t p1 , dp1 −p2 t p2 ,
t ∈ [0, d), t ≥ d,
where d > 0 and 0 < p1 < 1 < p2 , its nontrivial solution is always a blow-up solution. 7. Applications to other kinds of nonlinearities In the previous section we show how our results lead to well-known theorems. Now we consider examples of Eq. (1) for which, unlike in the previous cases, there are no earlier known theorems, which guarantee the existence of blow-up solutions of (1). Example 7.1 (Quasi-Exponential Nonlinearity). We consider Eq. (1) with the function g given by g (t ) =
g1 (t ), α f (γ t ),
t ∈ [0, d), t ≥ d,
(10)
where γ > 1, d > 0, α > 0, and a strictly increasing function f and a function g1 (t ) are taken such that the function g (t ) given by (10) satisfies conditions (2)–(4). Let w(t ) = α f (pt −d γ d ), where 1 < p < γ . Then the function w satisfies all assumptions of Theorem 5.3 with a = d. We have
d p
w (t ) = logp −1
−1
f
γ
t
α
,
and hence
Φ (t ) = (w −1 ◦ g )(t ) =
t logγ p
+ d logp
p
γ
,
t ≥ d.
In this case a function FΦ (t ) = t − d is a solution of the Schröder equation for Φ with c = ∞
∫
K −1
s
α f (ps−d γ d )
t
ds s−d
1 , logγ p
so the integral (7) takes the following form:
.
But because for any t > 0 t
α f (pt −d γ d )
t
<
α f (pt )
and for any fixed d there exists sufficiently large t ∗ such that t − d ≥
∫
∞
K −1 t∗
s
αf (
ps−d
γ ) d
ds s−d
∫
∞
K −1
≤2 t∗
s
αf ( ) ps
ds s
.
1 t 2
for t ∈ [t ∗ , ∞), we obtain
430
T. Małolepszy / Nonlinear Analysis 74 (2011) 424–432
This completes the proof of the following sufficient condition for the existence of blow-up solution to (1) with the function g (t ) given by (10): Theorem 7.2. If the integral ∞
∫
K
−1
s
ds
α f (ps )
t
s
is convergent for sufficiently large t and (1) with g given by (10) has a nontrivial solution, then this solution is also a blow-up solution of (1). Example 7.3 (Quasi-Power Nonlinearity). Let in (1) the function g be given by g (t ) =
g1 (t ), α f (t p ),
t ∈ [0, d), t ≥ d,
(11)
where d > 0, p > 1, and a strictly increasing function f and a function g1 (t ) are taken such that the function g (t ) given by (11) satisfies conditions (2)–(4). If we now take w(t ) = α f (dp−q t q ), where 1 < q < p, then all assumptions of Theorem 5.3 are fulfilled with a = d. Because
1q t w−1 (t ) = dq−p f −1 , α we have
Φ (t ) = (w
−1
◦ g )(t ) = d
pq t
α
,
t ≥ d.
The solution of the Schröder equation for Φ with c = FΦ (t ) = ln
p q
is
t d
and the integral (7) takes the following form: ∞
∫
K −1
s
αf (
dp−q sq
t
)
s ln
∞
∫
ds s d
K −1
= t d
sd
αf (
dp sq
)
ds s ln s
.
Finally, we obtain the following theorem: Theorem 7.4. If the integral ∞
∫
K −1
sd
α f (dp sq )
t
ds s ln s
is convergent for sufficiently large t, where p > q > 1, and (1) with g given by (11) has a nontrivial solution, then this solution is also a blow-up solution of (1). Example 7.5 (Exponential Nonlinearity). For most examples it is difficult to apply our results, particularly the necessary conditions. But here we represent an idea of how to find such conditions, even if we are not able to find a solution of the corresponding Schröder equation and thus to use theorems from Section 5. Let us consider Eq. (1) with exponential nonlinearity, i.e. with the function g of the form g (t ) =
g1 (t ),
αγ t ,
t ∈ [0, d), t ≥ d,
(12) αγ d
with γ > 1, d > 0, and the function g1 (t ) taken such that the function g (t ) given by (12) satisfies (2)–(4). Let w(t ) = dr t r , where r ∈ (0, 1) is such that r < d ln γ . The last inequality implies that assumptions of Theorem 5.4, as for the function w , are satisfied. In such a case, if Eq. (1) has a blow-up solution, we know from Theorem 4.4 that for any ε ∈ (0, 1) there exists cε ≥ d such that for any t ∈ (cε , ∞) the series ∞ − i=0
K −1
(Ψ i (t ))1−ε g (Ψ i (t ))
T. Małolepszy / Nonlinear Analysis 74 (2011) 424–432
431
is convergent, where d
Ψ (t ) = (w −1 ◦ g )(t ) =
γ
d r
γ
1 r
t
.
(13)
Fix ε ∈ (0, 1). The sequence {tn }∞ n=0 defined by the recurrence relation t0 = t ∗ ,
tn+1 = Ψ (tn )
(14)
is, of course, increasing and convergent to ∞ for every t > d. Moreover, for that sequence the following lemma holds: ∗
Lemma 7.6. There exists t0 > cε such that ti+2 − ti+1 > pi+1 (t1 − t0 ) for all i ∈ N, where p > 2. Proof. Obviously ti+1
ti+2 − ti+1
=
ti+1 − ti
ti
Ψ (ti+1 )
Ψ (ti )
ti+1 ti
−1
−1
for i ∈ N. Because it is easy to check that for r ∈ (0, 1) such that r < d ln γ the function Ψ1 , where Ψ1 (t ) := defined by (13), is strictly increasing for t > d, we obtain ti+2 − ti+1 Ψ (ti ) Ψ (t0 ) > > , ti+1 − ti ti t0
i ∈ N.
Ψ (t ) t
and Ψ is
(15)
From assumptions about functions g and w it follows that limt →∞ Ψ1 (t ) = ∞; therefore there exists t ∗ > cε such that Ψ (t ∗ ) =: p > 2. Taking t0 = t ∗ , we finally get from (15) t∗ ti+2 − ti+1 > p(ti+1 − ti ) = pi+1 (t1 − t0 ).
∗ Now take a sequence {tn }∞ n=0 defined by (14) with the starting point t0 = t as in the above lemma and define an absolutely ∗ continuous and strictly increasing function FΨ (t ) : [t , ∞) → R in the following way:
FΨ (t ) =
2i+1 ti+1 − ti
(t − ti ) + 2i+1 for t ∈ [ti , ti+1 ).
1−ε Without loss of generality, we assume that the function tg (t ) is strictly decreasing for t ≥ t ∗ . Then
∫
tn+1
K
−1
s1−ε g (s)
tn
FΨ (s)
′
FΨ (s)
−1
ds ≤ K
tn1−ε
tn+1
∫
g (tn )
tn
1−ε tn FΨ′ (s) −1 ds = ln 2 K g (tn ) FΨ (s)
and
∫
tn+1
K
−1
s1−ε g (s)
t0
FΨ (s)
′
FΨ (s)
ds ≤ ln 2
n −
K
−1
ti1−ε
.
g (ti )
i=0
Letting n → ∞ in the last inequality, we obtain
∫
∞
K
−1
s1−ε
FΨ (s)
′
g (s)
t∗
FΨ (s)
ds ≤ ln 2
∞ −
K
−1
i=0
(w −1 ◦ g )i (t ∗ )1−ε g ((w −1 ◦ g )i (t ∗ ))
and this means that if Eq. (1) has a blow-up solution, then the integral
∫
∞
K −1
s1−ε
FΨ (s) ds FΨ (s)
′
g (s)
t∗
is convergent. Now let us note that our choice of starting point t0 allows us to apply Lemma 7.6 to estimate a derivative of
FΨ and thus we get FΨ′ (t ) =
2i+1 ti+1 − ti
<
2i+1 pi (t1 − t0 )
=
2
i
t1 − t0
p
2
for t ∈ (ti , ti+1 ),
which implies that limt →∞ FΨ′ (t ) = limi→∞ FΨ′ (t ) = 0. From that fact and the form of FΨ (t ), which is just a linear spline, it ∗ follows that there exists t ≥ t such that t ≥ FΨ (t ) for t ≥ t. Moreover, for h(t ) := Ψ 1(t ) and t ∈ (ti , ti+1 ) we have h( t ) =
1
Ψ (t )
<
1
Ψ (ti )
=
1 ti+1
<
1 ti+1 − ti
< FΨ′ (t ),
432
T. Małolepszy / Nonlinear Analysis 74 (2011) 424–432
and hence
FΨ′ (t ) h(t ) a.e. in [t , ∞) > t FΨ (t ) and finally ∞
∫
K −1
s1−ε
FΨ (s)
′
g (s) FΨ (s)
t
ds >
∫ t
∞
K −1
s1−ε g (s)
ds sΨ ( s)
.
Since the above analysis can be repeated for any other fixed ε ∈ (0, 1), we prove the following result: Theorem 7.7. If Eq. (1) with the function g given by (12) has a blow-up solution, then for any ε ∈ (0, 1), ∞
∫
K t
−1
s1−ε
αγ s
ds
1 s s γr
is convergent for sufficiently large t, where r ∈ (0, 1) is such that r < d ln γ . Acknowledgments The author is greatly indebted to the referees for their valuable comments and helpful suggestions. References [1] J. Goncerzewicz, H. Marcinkowska, W. Okrasiński, K. Tabisz, On the percolation of water from a cylindrical reservoir into the surrounding soil, Zastos. Mat. 16 (1978) 249–261. [2] J. Keller, Propagation of simple nonlinear waves in gas filled tubes with friction, Z. Angew. Math. Phys. 32 (1981) 170–181. [3] T. Małolepszy, W. Okrasiński, Conditions for blow-up of solutions of some nonlinear Volterra integral equations, J. Comput. Appl. Math. 205 (2007) 744–750. [4] W. Okrasiński, Nontrivial solutions for a class of nonlinear Volterra equations with convolution kernel, J. Integral Equations Appl. 3 (1991) 399–409. [5] P. Bushell, W. Okrasiński, Nonlinear Volterra equations and the Apéry identities, Bull. London Math. Soc. 24 (1992) 478–484. [6] P. Bushell, W. Okrasiński, On the maximal interval of existence for solutions to some nonlinear Volterra integral equations with convolution kernel, Bull. London Math. Soc. 28 (1996) 59–65. [7] G. Gripenberg, Unique solutions of some Volterra integral equations, Math. Scand. 48 (1981) 59–67. [8] T. Małolepszy, W. Okrasiński, Blow-up conditions for nonlinear Volterra integral equations with power nonlinearity, Appl. Math. Lett. 21 (2008) 307–312. [9] W. Mydlarczyk, A condition for finite blow-up time for a Volterra integral equation, J. Math. Anal. Appl. 181 (1994) 248–253. [10] W. Mydlarczyk, A Volterra inequality with the power type nonlinear kernel, J. Inequal. Appl. 6 (2001) 625–631. [11] W. Okrasiński, Note on kernels to some nonlinear Volterra integral equations, Bull. London Math. Soc. 24 (1992) 373–376. [12] M. Kuczma, Functional Equations in a Single Variable, PWN, Warsaw, 1968. [13] M. Kuczma, B. Choczewski, R. Ger, Iterative Functional Equations, Cambridge Univ. Press, Cambridge, 1990. [14] M. Niedziela, W. Okrasiński, A note on Volterra integral equations with power nonlinearity, J. Integral Equations Appl. 18 (2006) 509–519.
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Nonlinear Volterra integral equations and the Schröder functional equation Tomasz Małolepszy Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland
article
abstract
info
Article history: Received 1 February 2010 Accepted 29 August 2010
We show an interesting connection between a special class of Volterra integral equations and the famous Schröder equation. The basic results provide criteria for the existence of nontrivial as well as blow-up solutions of the Volterra equation, expressed in terms of the convergence of some integrals. Examples related to Volterra equations with power and exponential nonlinearities are presented. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Volterra equation Nontrivial solutions Blow-up solutions Schröder equation
1. Introduction In last 30 years many articles have been dedicated to second-kind nonlinear Volterra integral equations of a certain type with a convolution kernel, i.e. u(t ) =
t
∫
k(t − s)g (u(s))ds,
t ≥ 0,
(1)
0
where g (0) = 0. Eq. (1) appears in many physical problems, for example in the theory of water percolation from cylindrical reservoirs [1] and in the theory of nonlinear waves in tubes [2]. Just from an application point of view, the problem of the existence of nontrivial solutions and blow-up solutions is very interesting. Recall that by a nontrivial solution we mean a continuous function u, which satisfies (1), defined on the maximal interval of its existence [0, T ), where T > 0, such that u(0) = 0 and u(t ) > 0 for t ∈ (0, T ). If additionally u satisfies limt →T − u(t ) = ∞ for some T < ∞, we say that u is a blow-up solution of (1). Among the many conditions obtained for the existence of nontrivial or blow-up solutions, we can distinguish in particular two kinds of such conditions: those expressed in terms of the convergence of some series [3,4] and those expressed in terms of the convergence of some integrals [5–10]. However, there is quite a big difference between these two classes of conditions. Unlike the former, the latter are always applicable to (1) only with fixed or monotone kernel k or with fixed nonlinearity g. In this article new integral conditions for the existence of nontrivial and blow-up solutions are established without this defect. Moreover, surprisingly, they are connected with the famous Schröder functional equation. 2. Basic assumptions and facts Throughout this paper we assume that in (1) the functions k and g satisfy the following conditions: k is a locally integrable function such that k(x) > 0 for x > 0,
E-mail address: [email protected]. 0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.08.054
T. Małolepszy / Nonlinear Analysis 74 (2011) 424–432
425
g : [0, ∞) → [0, ∞) is a strictly increasing absolutely continuous function which satisfies the following conditions: g (0) = 0,
(2)
x/g (x) → 0
as x → 0 ,
(3)
x/g (x) → 0
as x → ∞.
(4)
+
Let additionally K (t ) :=
t
∫
k(s)ds
0
and the following condition hold: lim K (t ) = ∞.
t →∞
It is easy to see that the function K is strictly increasing. On the basis of the results presented in [11], we formulate the following: Fact 2.1. Under our assumptions on k and g, Eq. (1) has at most one nontrivial solution. Moreover, if there exists a nontrivial solution to (1), then it is a strictly increasing absolutely continuous function. Further in this article, we denote the iteration of two functions f and g by f ◦ g and the i-th iteration of an arbitrary function f by f i . 3. The Schröder equation Let an absolutely continuous and strictly increasing function h : [0, ∞) → [0, ∞) and c > 0, c ̸= 1 be given. We shall use the Schröder functional equation F (h(x)) = cF (x),
x ∈ I,
(5)
where F : I → R is an unknown function and I is a given interval such that h(I ) ⊂ I. In this paper we are interested only in solutions of Eq. (5), which are absolutely continuous and strictly increasing on I. For simplicity we denote a class of such solutions by S (I ), and we denote the arbitrary solution of (5) for given function h and constant c in the interval I, which belongs to S (I ), by Fh . The theory of the existence and uniqueness of such solutions is quite extended, but in most cases formulas for these solutions are not given in an elementary form. More details about the Schröder equation can be found in the classical monographs [12,13]. Remark 3.1. To ensure positivity of Fh on the given interval, in which Fh is one-signed, we use the notation ϵ Fh , where ϵ = 1 if Fh is positive on the given interval and ϵ = −1 if Fh is negative on the given interval. 4. Auxiliary results First, we present two theorems, which can be found in [4], related to the existence of nontrivial solutions of (1). Theorem 4.1. Let w be a continuous function on [0, δ], δ > 0, such that t < w(t ) < g (t ) for t ∈ (0, δ] and limt →0+ w(t t ) = 0. If the series ∞ −
K −1
i =0
(g −1 ◦ w)i (t ) w((g −1 ◦ w)i (t ))
is convergent on [0, δ], then (1) has a nontrivial solution on some interval. g (t )
Theorem 4.2. Let w be a continuous function such that w(t ) > 0 for t > 0 and lim supt →0+ w(t ) < 1. If (1) has a nontrivial solution on some interval, then the series ∞ − i =0
K −1
(g −1 )i (t ) w((g −1 )i (t ))
is convergent on [0, δ], where δ > 0. The next two theorems appeared for the first time in [3] and give us conditions for the existence of blow-up solutions.
426
T. Małolepszy / Nonlinear Analysis 74 (2011) 424–432
Theorem 4.3. Let (1) have a nontrivial solution u. If w is a strictly increasing, positive and continuous function such that w(t ) < g (t ) for t ∈ (a, ∞), w(a) = g (a), limt →∞ w(t t ) = 0 and the series ∞ −
K −1
i=0
(w −1 ◦ g )i (t ) w((w −1 ◦ g )i (t ))
is convergent for some t ∈ (a, ∞), then u is a blow-up solution to (1). Theorem 4.4. If Eq. (1) has a blow-up solution, w is a strictly increasing, positive and continuous function such that w(t ) < g (t ) w(t ) for t ∈ (a, ∞) and w(a) = g (a), and also limt →∞ t = 0, then for any ε ∈ (0, 1) there exists cε ≥ a such that the series ∞ −
K −1
i=0
((w −1 ◦ g )i (t ))1−ε g ((w −1 ◦ g )i (t ))
(6)
is convergent for any t ∈ (cε , ∞). 5. Main theorems Theorem 5.1. Let w be a continuous function on [0, δ], δ > 0, such that t < w(t ) < g (t ) for t ∈ (0, δ], limt →0+ w(t t ) = 0 and the function w(t t ) be strictly increasing on (0, δ). If there exists FΦ ∈ S (0, δ), where Φ = g −1 ◦ w , and the integral t
∫
K −1
s
w(s)
0
FΦ′ (s)
ϵ FΦ (s)
ds,
is convergent for t ∈ (0, δ), then (1) has a nontrivial solution. Proof. Take an arbitrary t < δ such that the function FΦ is either positive or negative on (0, t ) and define a sequence −1 {tn }∞ ◦ w)n (t ), this sequence is decreasing and n=0 by the following recurrence relation: t0 = t, tn+1 = Φ (tn ). Since tn = (g convergent to 0. Note that tn
∫
K
−1
s
w(s)
tn+1
FΦ′ (s)
ϵ FΦ (s)
ds ≥ K
−1
tn
∫
tn+1
w(tn+1 )
tn+1
FΦ′ (s)
ϵ FΦ (s)
ds
and, because FΦ is the solution of Schröder equation for the function Φ , tn
∫
tn+1
FΦ′ (s)
ϵ FΦ (s)
ds = ϵ ln
FΦ (tn ) FΦ (tn+1 )
= ϵ ln
1 c
=: C > 0.
Hence t0
∫
K −1
tn+1
s
w(s)
FΦ′ (s)
ϵ FΦ (s)
ds =
n ∫ − i=0
ti
K −1
ti+1
s
w(s)
FΦ′ (s)
ϵ FΦ (s)
ds ≥ C
n +1 − i =1
K −1
ti
w(ti )
.
Letting n → ∞, in the last inequality we obtain t
∫
K −1 0
s
w(s)
FΦ′ (s)
ϵ FΦ (s)
ds ≥ C
∞ − i=1
K −1
ti
w(ti )
The above inequality and Theorem 4.1 imply the result.
=C
∞ − i =1
K −1
(g −1 ◦ w)i (t ) . w((g −1 ◦ w)i (t ))
g (t )
Theorem 5.2. Let w be a continuous function such that w(t ) > 0 for t > 0 and lim supt →0+ w(t ) < 1. Let additionally the function w(t t ) be strictly increasing on some interval (0, δ). If (1) has a nontrivial solution and there exists FΨ ∈ S (0, δ) where
Ψ = g −1 , then the integral ′ ∫ t K −1
0
s
w(s)
FΨ (s) ds, ϵ FΨ (s)
is convergent for any t ∈ (0, δ).
T. Małolepszy / Nonlinear Analysis 74 (2011) 424–432
427
Proof. Taking an arbitrary t < δ such that the function FΨ is either positive or negative on (0, t ) and defining a sequence −1 n {tn }∞ ) (t )), one can n=0 that is decreasing and convergent to 0 by t0 = t, tn+1 = Ψ (tn ) (the explicit formula for tn is tn = (g repeat an argument applied in the proof of Theorem 5.1. We get then respectively tn
∫
K −1
FΨ′ (s)
s
w(s)
∫
tn
FΨ′ (s)
tn
w(tn ) tn+1 ϵ FΨ (s) ′ ∫ t0 n − s FΨ (s) ti K −1 ds ≤ C K −1 w(s) ϵ FΨ (s) w(ti ) tn+1 i=0 tn+1
ϵ FΨ (s)
ds ≤ K −1
ds = CK −1
tn
w(tn )
,
and, as n goes to ∞, t
∫
K −1
0
FΨ′ (s)
s
w(s)
ϵ FΨ (s)
ds ≤ C
∞ −
K −1
i=0
(g −1 )i (t ) , w((g −1 )i (t ))
which, by using Theorem 4.2, completes the proof.
Theorem 5.3. Let w be a strictly increasing, positive and continuous function such that w(t ) < g (t ) for t ∈ (a, ∞), w(a) = g (a), limt →∞ w(t t ) = 0 and let the function w(t t ) be strictly decreasing from some t˜ ≥ a. If there exists FΦ ∈ S (t˜, ∞), where Φ = w −1 ◦ g, the integral ∞
∫
K −1
FΦ′ (s)
s
w(s)
t
ϵ FΦ (s)
ds
(7)
is convergent for some t ∈ (t˜, ∞) and (1) has a nontrivial solution, then this solution is also a blow-up solution of (1). Proof. For an arbitrary t > t˜, for which the function FΦ is either positive or negative on [t , ∞), the sequence {tn }∞ n=0 given by t0 = t, tn+1 = Φ (tn ) (and hence tn = (w −1 ◦ g )n (t )) is increasing and convergent to ∞. Because tn+1
∫
K −1
w(s)
tn
FΦ′ (s)
s
ϵ FΦ (s)
ds ≥ CK −1
tn+1
w(tn+1 )
where
∫
FΦ′ (s)
tn+1
C =
ϵ FΦ (s)
tn
ds,
we have tn+1
∫
K −1
w(s)
t0
FΦ′ (s)
s
ϵ FΦ (s)
ds ≥ C
n +1 −
K −1
ti
w(ti )
i =1
.
Letting n → ∞, we obtain ∞
∫
K −1
t
s
w(s)
FΦ′ (s)
ϵ FΦ (s)
ds ≥ C
∞ −
K −1
i =1
(w −1 ◦ g )i (t ) . w((w−1 ◦ g )i (t ))
The last inequality and Theorem 4.3 end the proof.
Theorem 5.4. Let w be a strictly increasing, positive and continuous function such that w(t ) < g (t ) for t ∈ (a, ∞) and
w(a) = g (a) and also limt →∞ w(t t ) = 0. Let additionally the function tg (t ) be strictly decreasing from some t˜ ≥ a for given ε ∈ (0, 1). If Eq. (1) has a blow-up solution and there exists FΨ ∈ S (t˜, ∞), where Ψ = w −1 ◦ g, then for any ε ∈ (0, 1) there exists cε ≥ a such that the integral 1−ε ′ ∫ ∞ FΨ (s) s K −1 ds, g (s) ϵ FΨ (s) t 1−ε
is convergent for any t > max{t˜, cε }. Proof. Fix ε ∈ (0, 1). Theorem 4.4 implies that there exists cε ≥ a such that for any t ∈ (cε , ∞) the series (6) is convergent. Take an arbitrary t > max{t˜, cε }, for which the function FΨ is either positive or negative on [t , ∞). The sequence {tn }∞ n =0 defined by the recurrence relation t0 = t, tn+1 = Ψ (tn ), which implies that tn = (w −1 ◦ g )n (t ), is increasing and convergent 1−ε
to ∞. Since the function K −1 ( sg (s) ) is decreasing for s > t˜, we have
∫
tn+1 tn
K −1
s1−ε g (s)
FΨ′ (s)
ϵ FΨ (s)
ds ≤ K −1
tn1−ε g (tn )
∫
tn+1 tn
FΨ′ (s)
ϵ FΨ (s)
ds = CK −1
tn1−ε g (tn )
,
428
T. Małolepszy / Nonlinear Analysis 74 (2011) 424–432
whence tn+1
∫
K
−1
s1−ε g (s)
t0
FΨ′ (s)
ϵ FΨ (s)
ds ≤ C
n −
K
−1
ti1−ε g (ti )
i=0
.
Letting n → ∞ in the last inequality, we obtain ∞
∫
K −1
s1−ε
g (s)
t
FΨ′ (s)
ϵ FΨ (s)
ds ≤ C
∞ −
K −1
i=0
(w −1 ◦ g )i (t )1−ε . g ((w −1 ◦ g )i (t ))
Now Theorem 4.4 together with last inequality gives the thesis.
6. Examples related to power nonlinearity Example 6.1 (Power Nonlinearity — Case I). Let the function g in Eq. (1) satisfy condition (4) and there be d > 0 and 0 < p < 1 such that g (t ) = t p ,
t ∈ [0, d).
(8)
In order to examine when Eq. (1) with such g has a nontrivial solution, we take two functions: w1 (t ) = t , where p < q < 1, q
q
1
and w2 ≡ 1. Because the solutions of the Schröder equation for the functions Φ (t ) = t p and Ψ (t ) = t p (in the first case with q c = p and in the second with c = 1p ) are the same, i.e. FΦ (t ) = FΨ (t ) = ln t, Theorems 5.1 and 5.2 then take respectively the following forms: Theorem 6.2. If the integral t
∫
K −1 (s1−q ) 0
ds s(− ln s)
is convergent for t ∈ (0, δ), δ > 0, then Eq. (1) with g given by (8) has a nontrivial solution. Theorem 6.3. If Eq. (1) with g given by (8) has a nontrivial solution, then there exists δ > 0 such that the integral t
∫
K −1 ( s ) 0
ds s(− ln s)
is convergent for any t ∈ (0, δ). From Theorems 6.2 and 6.3 we immediately obtain the following well-known result (cf. [10,14]): Theorem 6.4. Eq. (1) with g satisfying (8) has a nontrivial solution if and only if the integral t
∫
K −1 ( s ) 0
ds s(− ln s)
is convergent for any t ∈ (0, δ), δ > 0. Example 6.5 (Power Nonlinearity — Case II). Now we take into consideration Eq. (1), in which the function g has the following form: g (t ) =
g1 (t ), bt p ,
t ∈ [0, d) t ≥ d,
(9)
where d > 0, p > 1 and b > 0, and the function g1 (t ) is taken such that the function g (t ) given by (9) satisfies (2)–(4). To find conditions for the existence of blow-up solution for this equation, we apply Theorems 5.3 and 5.4. For w(t ) = bap−q t q , p
where t ≥ a ≥ d and 1 < q < p, we have Φ (t ) = (w −1 ◦ g )(t ) = a( at ) q . A solution of the Schröder equation (5) for Φ with p c = q is FΦ (t ) = ln at , so from Theorem 5.3 we obtain: Theorem 6.6. If the integral ∞
∫
K −1 t
s1−q bap−q
ds s ln
s a
is convergent for some t ∈ (a, ∞) and (1) with g given by (9) has a nontrivial solution, then this solution is also a blow-up solution of (1).
T. Małolepszy / Nonlinear Analysis 74 (2011) 424–432
429 p r
Similarly, taking w(t ) = bap−r t r , where t ≥ a ≥ d and 0 < r < 1, we get Ψ (t ) = (w −1 ◦ g )(t ) = a( at ) and because a p solution of (5) for Ψ with c = r is FΨ (t ) = ln at , in this case we can write Theorem 5.4 as: Theorem 6.7. If Eq. (1) with g given by (9) has a blow-up solution, then for any ε ∈ (0, 1) there exists cε ≥ a such that the integral ∞
∫
K
−1
s1−ε−p
b
t
ds s ln
s a
is convergent for some t > cε . It can be shown (see [8]) that Theorems 6.6 and 6.7 lead to: Theorem 6.8. A nontrivial solution of Eq. (1) with g satisfying (9) is a blow-up solution if and only if the integral t
∫
K −1 (s) 0
ds s(− ln s)
is convergent for any t ∈ (0, δ). Remark 6.1. An easy consequence of Theorems 6.4 and 6.8 is that for Eq. (1) with the function g given by g (t ) =
t p1 , dp1 −p2 t p2 ,
t ∈ [0, d), t ≥ d,
where d > 0 and 0 < p1 < 1 < p2 , its nontrivial solution is always a blow-up solution. 7. Applications to other kinds of nonlinearities In the previous section we show how our results lead to well-known theorems. Now we consider examples of Eq. (1) for which, unlike in the previous cases, there are no earlier known theorems, which guarantee the existence of blow-up solutions of (1). Example 7.1 (Quasi-Exponential Nonlinearity). We consider Eq. (1) with the function g given by g (t ) =
g1 (t ), α f (γ t ),
t ∈ [0, d), t ≥ d,
(10)
where γ > 1, d > 0, α > 0, and a strictly increasing function f and a function g1 (t ) are taken such that the function g (t ) given by (10) satisfies conditions (2)–(4). Let w(t ) = α f (pt −d γ d ), where 1 < p < γ . Then the function w satisfies all assumptions of Theorem 5.3 with a = d. We have
d p
w (t ) = logp −1
−1
f
γ
t
α
,
and hence
Φ (t ) = (w −1 ◦ g )(t ) =
t logγ p
+ d logp
p
γ
,
t ≥ d.
In this case a function FΦ (t ) = t − d is a solution of the Schröder equation for Φ with c = ∞
∫
K −1
s
α f (ps−d γ d )
t
ds s−d
1 , logγ p
so the integral (7) takes the following form:
.
But because for any t > 0 t
α f (pt −d γ d )
t
<
α f (pt )
and for any fixed d there exists sufficiently large t ∗ such that t − d ≥
∫
∞
K −1 t∗
s
αf (
ps−d
γ ) d
ds s−d
∫
∞
K −1
≤2 t∗
s
αf ( ) ps
ds s
.
1 t 2
for t ∈ [t ∗ , ∞), we obtain
430
T. Małolepszy / Nonlinear Analysis 74 (2011) 424–432
This completes the proof of the following sufficient condition for the existence of blow-up solution to (1) with the function g (t ) given by (10): Theorem 7.2. If the integral ∞
∫
K
−1
s
ds
α f (ps )
t
s
is convergent for sufficiently large t and (1) with g given by (10) has a nontrivial solution, then this solution is also a blow-up solution of (1). Example 7.3 (Quasi-Power Nonlinearity). Let in (1) the function g be given by g (t ) =
g1 (t ), α f (t p ),
t ∈ [0, d), t ≥ d,
(11)
where d > 0, p > 1, and a strictly increasing function f and a function g1 (t ) are taken such that the function g (t ) given by (11) satisfies conditions (2)–(4). If we now take w(t ) = α f (dp−q t q ), where 1 < q < p, then all assumptions of Theorem 5.3 are fulfilled with a = d. Because
1q t w−1 (t ) = dq−p f −1 , α we have
Φ (t ) = (w
−1
◦ g )(t ) = d
pq t
α
,
t ≥ d.
The solution of the Schröder equation for Φ with c = FΦ (t ) = ln
p q
is
t d
and the integral (7) takes the following form: ∞
∫
K −1
s
αf (
dp−q sq
t
)
s ln
∞
∫
ds s d
K −1
= t d
sd
αf (
dp sq
)
ds s ln s
.
Finally, we obtain the following theorem: Theorem 7.4. If the integral ∞
∫
K −1
sd
α f (dp sq )
t
ds s ln s
is convergent for sufficiently large t, where p > q > 1, and (1) with g given by (11) has a nontrivial solution, then this solution is also a blow-up solution of (1). Example 7.5 (Exponential Nonlinearity). For most examples it is difficult to apply our results, particularly the necessary conditions. But here we represent an idea of how to find such conditions, even if we are not able to find a solution of the corresponding Schröder equation and thus to use theorems from Section 5. Let us consider Eq. (1) with exponential nonlinearity, i.e. with the function g of the form g (t ) =
g1 (t ),
αγ t ,
t ∈ [0, d), t ≥ d,
(12) αγ d
with γ > 1, d > 0, and the function g1 (t ) taken such that the function g (t ) given by (12) satisfies (2)–(4). Let w(t ) = dr t r , where r ∈ (0, 1) is such that r < d ln γ . The last inequality implies that assumptions of Theorem 5.4, as for the function w , are satisfied. In such a case, if Eq. (1) has a blow-up solution, we know from Theorem 4.4 that for any ε ∈ (0, 1) there exists cε ≥ d such that for any t ∈ (cε , ∞) the series ∞ − i=0
K −1
(Ψ i (t ))1−ε g (Ψ i (t ))
T. Małolepszy / Nonlinear Analysis 74 (2011) 424–432
431
is convergent, where d
Ψ (t ) = (w −1 ◦ g )(t ) =
γ
d r
γ
1 r
t
.
(13)
Fix ε ∈ (0, 1). The sequence {tn }∞ n=0 defined by the recurrence relation t0 = t ∗ ,
tn+1 = Ψ (tn )
(14)
is, of course, increasing and convergent to ∞ for every t > d. Moreover, for that sequence the following lemma holds: ∗
Lemma 7.6. There exists t0 > cε such that ti+2 − ti+1 > pi+1 (t1 − t0 ) for all i ∈ N, where p > 2. Proof. Obviously ti+1
ti+2 − ti+1
=
ti+1 − ti
ti
Ψ (ti+1 )
Ψ (ti )
ti+1 ti
−1
−1
for i ∈ N. Because it is easy to check that for r ∈ (0, 1) such that r < d ln γ the function Ψ1 , where Ψ1 (t ) := defined by (13), is strictly increasing for t > d, we obtain ti+2 − ti+1 Ψ (ti ) Ψ (t0 ) > > , ti+1 − ti ti t0
i ∈ N.
Ψ (t ) t
and Ψ is
(15)
From assumptions about functions g and w it follows that limt →∞ Ψ1 (t ) = ∞; therefore there exists t ∗ > cε such that Ψ (t ∗ ) =: p > 2. Taking t0 = t ∗ , we finally get from (15) t∗ ti+2 − ti+1 > p(ti+1 − ti ) = pi+1 (t1 − t0 ).
∗ Now take a sequence {tn }∞ n=0 defined by (14) with the starting point t0 = t as in the above lemma and define an absolutely ∗ continuous and strictly increasing function FΨ (t ) : [t , ∞) → R in the following way:
FΨ (t ) =
2i+1 ti+1 − ti
(t − ti ) + 2i+1 for t ∈ [ti , ti+1 ).
1−ε Without loss of generality, we assume that the function tg (t ) is strictly decreasing for t ≥ t ∗ . Then
∫
tn+1
K
−1
s1−ε g (s)
tn
FΨ (s)
′
FΨ (s)
−1
ds ≤ K
tn1−ε
tn+1
∫
g (tn )
tn
1−ε tn FΨ′ (s) −1 ds = ln 2 K g (tn ) FΨ (s)
and
∫
tn+1
K
−1
s1−ε g (s)
t0
FΨ (s)
′
FΨ (s)
ds ≤ ln 2
n −
K
−1
ti1−ε
.
g (ti )
i=0
Letting n → ∞ in the last inequality, we obtain
∫
∞
K
−1
s1−ε
FΨ (s)
′
g (s)
t∗
FΨ (s)
ds ≤ ln 2
∞ −
K
−1
i=0
(w −1 ◦ g )i (t ∗ )1−ε g ((w −1 ◦ g )i (t ∗ ))
and this means that if Eq. (1) has a blow-up solution, then the integral
∫
∞
K −1
s1−ε
FΨ (s) ds FΨ (s)
′
g (s)
t∗
is convergent. Now let us note that our choice of starting point t0 allows us to apply Lemma 7.6 to estimate a derivative of
FΨ and thus we get FΨ′ (t ) =
2i+1 ti+1 − ti
<
2i+1 pi (t1 − t0 )
=
2
i
t1 − t0
p
2
for t ∈ (ti , ti+1 ),
which implies that limt →∞ FΨ′ (t ) = limi→∞ FΨ′ (t ) = 0. From that fact and the form of FΨ (t ), which is just a linear spline, it ∗ follows that there exists t ≥ t such that t ≥ FΨ (t ) for t ≥ t. Moreover, for h(t ) := Ψ 1(t ) and t ∈ (ti , ti+1 ) we have h( t ) =
1
Ψ (t )
<
1
Ψ (ti )
=
1 ti+1
<
1 ti+1 − ti
< FΨ′ (t ),
432
T. Małolepszy / Nonlinear Analysis 74 (2011) 424–432
and hence
FΨ′ (t ) h(t ) a.e. in [t , ∞) > t FΨ (t ) and finally ∞
∫
K −1
s1−ε
FΨ (s)
′
g (s) FΨ (s)
t
ds >
∫ t
∞
K −1
s1−ε g (s)
ds sΨ ( s)
.
Since the above analysis can be repeated for any other fixed ε ∈ (0, 1), we prove the following result: Theorem 7.7. If Eq. (1) with the function g given by (12) has a blow-up solution, then for any ε ∈ (0, 1), ∞
∫
K t
−1
s1−ε
αγ s
ds
1 s s γr
is convergent for sufficiently large t, where r ∈ (0, 1) is such that r < d ln γ . Acknowledgments The author is greatly indebted to the referees for their valuable comments and helpful suggestions. References [1] J. Goncerzewicz, H. Marcinkowska, W. Okrasiński, K. Tabisz, On the percolation of water from a cylindrical reservoir into the surrounding soil, Zastos. Mat. 16 (1978) 249–261. [2] J. Keller, Propagation of simple nonlinear waves in gas filled tubes with friction, Z. Angew. Math. Phys. 32 (1981) 170–181. [3] T. Małolepszy, W. Okrasiński, Conditions for blow-up of solutions of some nonlinear Volterra integral equations, J. Comput. Appl. Math. 205 (2007) 744–750. [4] W. Okrasiński, Nontrivial solutions for a class of nonlinear Volterra equations with convolution kernel, J. Integral Equations Appl. 3 (1991) 399–409. [5] P. Bushell, W. Okrasiński, Nonlinear Volterra equations and the Apéry identities, Bull. London Math. Soc. 24 (1992) 478–484. [6] P. Bushell, W. Okrasiński, On the maximal interval of existence for solutions to some nonlinear Volterra integral equations with convolution kernel, Bull. London Math. Soc. 28 (1996) 59–65. [7] G. Gripenberg, Unique solutions of some Volterra integral equations, Math. Scand. 48 (1981) 59–67. [8] T. Małolepszy, W. Okrasiński, Blow-up conditions for nonlinear Volterra integral equations with power nonlinearity, Appl. Math. Lett. 21 (2008) 307–312. [9] W. Mydlarczyk, A condition for finite blow-up time for a Volterra integral equation, J. Math. Anal. Appl. 181 (1994) 248–253. [10] W. Mydlarczyk, A Volterra inequality with the power type nonlinear kernel, J. Inequal. Appl. 6 (2001) 625–631. [11] W. Okrasiński, Note on kernels to some nonlinear Volterra integral equations, Bull. London Math. Soc. 24 (1992) 373–376. [12] M. Kuczma, Functional Equations in a Single Variable, PWN, Warsaw, 1968. [13] M. Kuczma, B. Choczewski, R. Ger, Iterative Functional Equations, Cambridge Univ. Press, Cambridge, 1990. [14] M. Niedziela, W. Okrasiński, A note on Volterra integral equations with power nonlinearity, J. Integral Equations Appl. 18 (2006) 509–519.