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## Neumann and periodic boundary-value problems for quasilinear ordinary differential equations

## with a nonlinearity in the derivative ^{∗}

### Petr Girg

Abstract

We present sufficient conditions for the existence of solutions to Neu- mann and periodic boundary-value problems for some class of quasilinear ordinary differential equations. We also show that this condition is nec- essary for certain nonlinearities. Our results involve the p-Laplacian, the mean-curvature operator and nonlinearities blowing up.

### 1 Introduction

The semilinear boundary-value problems

u^{00}(t) +g(u^{0}(t)) +h(u(t)) =f(t) t∈(0, T), (1.1)
u(0) =u(T), u^{0}(0) =u^{0}(T), (1.2)
and

u^{00}(t) +g(u^{0}(t)) =f(t) t∈(0, T), (1.3)
u^{0}(0) = 0, u^{0}(T) = 0, (1.4)
have been extensively studied by many authors (see e.g. [4, 6, 16]). In this
paper we extend their results to the quasilinear boundary-value problems:

ϕ(u^{0}(t))_{0}

+g(u^{0}(t)) +h(u(t)) =f(t) t∈(0, T), (1.5)
subject to (1.2) or to (1.4).

Overall, we will assume the following:

(P) ϕis an increasing homeomorphism ofI_{1}ontoI_{2},whereI_{1}, I_{2}⊂Rare open
intervals containing zero andϕ(0) = 0,

∗Mathematics Subject Classifications: 34B15, 47H14.

Key words: p-Laplacian, Leray-Schauder degree, Landesmann-Lazer condition.

c2000 Southwest Texas State University.

Submitted July 18, 2000. Published October 16, 2000.

Supported by grants GA ˇCR # 201/97/0395, and MˇSMT ˇCR # VS 97156

1

(G) gis a continuous real function,

(H) his a continuous, bounded real function having limits in±∞:

h(−∞) := lim

ξ→−∞h(ξ)< lim

ξ→+∞h(ξ) =:h(+∞).

We also need to impose some of the following assumptions to prove several particular results.

(P’) the inverse ofϕ(denoted byϕ_{−1}) is continuously differentiable,
(P”) ϕ^{0}_{−1}(0)>0,

(PH) ϕis odd and there existc, δ >0 andp >1 such that for allz∈(−δ, δ)∩
Domϕ: c|z|^{p−1}≤ |ϕ(z)|,

(G’) g is a continuously differentiable real function.

The results presented involve blow-up-type nonlinearities such asϕ(z) = tan(z),
bounded nonlinearities of the type ϕ(z) = arctan(z), the p-Laplacian when
ϕ(z) =|z|^{p−2}z,1< p <∞, and include of course the classical results ([4, 6, 16])
where ϕ is considered to be an identity mapping on R. As for the Neumann
conditions, we obtain new results even in the semilinear case with ‘ϕ= Identity’

onR. To extend known results for (1.5)–(1.2) or (1.5)–(1.4), we mainly combine and modify methods from [4, 6, 16].

Since boundary-value problems (BVPs) of the type (1.5)–(1.2) and (1.5)–

(1.4) appear in a wide variety of applications, our extension of known results is interesting not only from the theoretical point of view but also from a practical one. For instance the p-Laplacian with p6= 2 arises in the study of nonlinear diffusion. Blow-up-type nonlinearityϕ(z) =z/√

1−z^{2}comes from the model-
ing of mechanical oscillations with relativistic correction. As an application of
bounded nonlinearity let us mention the mean curvature (or capillary surface)
operatorϕ(z) =z/√

1 +z^{2}, which occurs in problems of mathematical physics.

Related topics are the subjects of studies by many authors. Among others let us mention the recent paper [14] in which the authors study the existence of periodic solutions of systems of quasilinear ordinary differential equations (ODEs) at resonance. Let us also mention papers [9, 10]; even thought the results of those papers are of different nature, some of the techniques used are similar. The resonance problems for the p-Laplacian in partial derivatives are treated in [2] and [3].

The paper is organized as follows. We begin by stating our main results concerning the existence of the solution (Theorems 1-4) and discuss their ap- plicability to various problems arising from physics in Section 2. Proofs of Theorems 1–3 can be found in Section 3. A proof of Theorem 4 is presented in Section 4. Finally, we present some results concerning the uniqueness of the solution in Section 5.

Before proceeding to the results we shall establish the notation that is used throughout the paper:

For measurable functions defined on [0, T] we define:

L^{p} :=

n

u:kukL^{p}:=

Z _{T}

0 |u(t)|^{p}dt

!^{1}_{p}

<∞o

, 1≤p <+∞,
L^{∞} :=

n

u:kuk∞:= inf N ⊂(0, T) measN = 0

t∈(0,Tsup)\N|u(t)|<∞o .

For functions with domain [0, T], with distributional derivatives, we define:

W^{1,p}:=

u∈L^{p}: u^{0}∈L^{p} ,
W_{T}^{1,p}:=

u∈W^{1,p}: u(0) =u(T) .
For a givenk≥0, let

C_{T}^{k} :=

u:u∈C^{k}[0, T], u(0) =u(T), u^{0}(0) =u^{0}(T), . . . ,
u^{(k−1)}(0) =u^{(k−1)}(T) ,

whereC^{k}[0, T] is the standard space ofk-times continuously differentiable func-
tions defined on [0, T], endowed with the normkuk_{C}^{k}:=P_{k}

i=0ku^{(i)}k∞(C[0, T] :=

C^{0}[0, T],C_{T} :=C_{T}^{0}). ByC_{0}^{∞}(0, T) we denote the set of functions udefined on
(0, T), possessing derivatives of any order in (0, T) and such that the closure of
{t∈(0, T) : u(t)6= 0} is a subset (0, T).

To formulate our results, we use the decomposition f =fe+f , withf = 1

T
Z _{T}

0 f(t)dt (1.6)

and the following function spaces C[0, Te ] :=

n

u∈C[0, T] :R_{T}

0 u(t)dt= 0 o

,

Ce_{T} :=C_{T} ∩C[0, Te ],Ce^{1}[0, T] :=C^{1}[0, T]∩C[0, Te ],Ce_{T}^{1} :=C_{T}^{1} ∩C[0, Te ].

Since we deal with the quasilinear equation, we shall work with a more general concept of solutions of ODE than that of the classical solutions.

Definition 1.1 A function u ∈ C^{1}[0, T] is called a solution of (1.5)–(1.2) or
(1.5)–(1.4) if ϕ(u^{0})∈ C^{1}[0, T], usatisfies (1.5) pointwise in (0, T) andu(0) =
u(T),u^{0}(0) =u^{0}(T) oru^{0}(0) = 0,u^{0}(T) = 0, respectively.

### 2 Main results

In the following we will consider the range ofϕto be an interval (a, b), wherea andb could be−∞and +∞, respectively.

Let us consider (1.5)–(1.2) and (1.5)–(1.4) with h≡ 0. Substituting w =
ϕ(u^{0}) in (1.5) we obtain the semilinear first order equation:

w^{0}(t) +g ϕ_{−1} w(t)

=f(t). (2.1)

Now it remains to find how this substitution affects the boundary conditions.

Taking into account u(T)−u(0) =

Z _{T}

0 u^{0}(t)dt=
Z _{T}

0 ϕ_{−1} w(t)
dt

and the fact that ϕ is an increasing homeomorphism, the periodic conditions (1.2) are transformed as follows:

Z _{T}

0 ϕ_{−1} w(t)

dt= 0, w(0) =w(T). (2.2) Analogously, an equivalent form of the Neumann conditions (1.4) reads

w(0) = 0, w(T) = 0. (2.3)

Theorem 2.1 Let the assumptions (P),(G) be satisfied and h≡0. Then, for
anyfe∈Ce_{T} :kfke L^{2} <

q3

T min{−a, b}, there exists precisely ones(fe)∈Rsuch that (2.1)–(2.2) has a solution if and only if

f =s(fe).

In this case the periodic boundary-value problem for (1.5) has a family of so-
lutions u_{c}(t) = u(t) +c, where u(t) = R_{t}

0ϕ_{−1} w(f , τe )

dτ, c ∈ R is arbitrary
andw(f ,e·) is the solution of (2.1)–(2.2). Moreover, the mappings:Ce_{T} →R,
fe7→s(fe)is continuous and the absolute value of s(fe) satisfy

|s(fe)| ≤max (

|g(ξ)| : ϕ_{−1} −
rT

3kfek_{L}^{2}

!

≤ξ≤ϕ_{−1}
rT

3kfek_{L}^{2}

!) . (2.4) Remark 2.1 Let us observe that ifϕis an homeomorphism of interval (α, β), α <0< β ontoR, then (2.4) yields an uniform bound fors(fe):

|s(fe)| ≤ max

ξ∈[α,β]|g(ξ)|.

Theorem 2.2 Suppose that (P),(P’),(G) are satisfied and h ≡ 0. Then the
assertion of Theorem 2.1 is valid and u∈C_{T}^{2}. If also(G’) holds true then the
solution w(f ,e·) of (2.1)–(2.2) is unique. The mappings w : Ce_{T} → C_{T}^{1},fe7→

w(f ,e·) and s : Ce_{T} → R,fe7→ s(fe) are continuously differentiable at any fe∈
C[0, Te ] : fe6≡0.If (P”)is satisfied then wands are continuously differentiable
also atfe≡0.

Theorem 2.3 Let the assumptions (P), (G) be satisfied and h≡0. Then, for
any fe∈C[0, Te ]: kfekL^{2} <

q3

T min{−a, b}, there exists precisely ones(fe)∈R
such that (2.1)–(2.3) has a solution if and only if f =s(fe). In this case the
Neumann boundary-value problem for (1.5) has a family of solutions u_{c}(t) =
u(t) +c,where u(t) =R_{t}

0ϕ_{−1} w(f , τe )

dτ, c∈R is arbitrary and w(f ,e·) is the
solution of (2.1)–(2.3). The mapping s:C[0, Te ]→R,fe7→s(fe)is continuous
and the absolute value of s(fe) is estimated by (2.4). If (P’) is satisfied then
u∈C_{T}^{2}. If also (G’) holds true then mappings w:Ce_{T} →C_{T}^{1}, fe7→w(f ,e·)and
s are continuously differentiable at any point fe∈C[0, Te ].

Remark 2.2 Let us see that we do not need the assumption (P”) in Theorem 2.3 in order to get that w : fe7→ w(f ,e·) and s : fe7→ s(fe) are continuously differentiable at fe≡0, as we do in Theorem 2.2. The reason is the following one: The boundary conditions (2.3) of the first order semilinear problem corre- sponding to the Neumann BVP are linear. Thus, they provide more regularity than the nonlinear ones (2.2) corresponding to the periodic BVP.

Since the results considering (1.5)–(1.2) and (1.5)–(1.4), respectively, have the same formal structure, we need to formulate them for the BVP (1.5)–(1.2) only. However, the statement, corresponding to the BVP (1.5)–(1.4), can be obtained by replacing the expressions standing in front of brackets with the bracketed ones in Theorem 2.4. We keep this notation in Section 4, when formulating Lemmas 4, 6, 7 used in the proof of Theorem 4. Note, that in the following theorem we suppose thatϕis an odd mapping, so thatb=−a.

Theorem 2.4 Assume(P),(PH),(G),(H) and r3

Tb−√ Tsup

ξ∈R|h(ξ)|>0. (2.5)
Then for any fe∈Ce_{T} ( fe∈C[0, Te ]) satisfying

kfek_{L}^{2}<

r3 Tb−√

Tsup

ξ∈R|h(ξ)| (2.6) the BVP (1.5)–(1.2) ( (1.5)–(1.4) ) has a solution if

s(fe) +h(−∞)< f < s(fe) +h(+∞), (2.7) wheres(fe)is given by Theorem 2.1 ( Theorem 2.3 ).

Suppose, moreover, that

h(−∞)< h(ξ)< h(+∞) for all ξ∈R.

Then, if (2.7) is false and fe∈ Ce_{T} ( fe∈ C[0, Te ] ) satisfies (2.6), the BVP
(1.5)–(1.2) ( (1.5)–(1.4) ) does not admit any solution.

Remark 2.3 As one can expect, an analogous result to Theorem 2.4 holds true also forhsatisfying

h(+∞)< h(−∞).

In this case, for any fe∈ Ce_{T} ( fe∈ C[0, Te ] ) satisfying (2.6), the sufficient
condition onf reads as follows:

s(fe) +h(+∞)< f < s(fe) +h(−∞). (2.8) If moreoverhsatisfies

h(+∞)< h(ξ)< h(−∞) for allξ∈R then (2.8) is also necessary.

This is in full agreement with the semilinear case for the periodic BVP studied in [6].

Remark 2.4 Notice, that if the range ofϕisRthenb= +∞and the conditions
(2.5) and (2.6) are satisfied identically. On the other hand, ifb <+∞then one
can apply Theorem 2.4 provided thatT < _{sup}^{√}^{3b}

ξ∈R|h(ξ)|. Observe that the bigger T is, the weaker condition (2.6) is.

Remark 2.5 Since we treat the problems (1.5)–(1.2) and (1.5)–(1.4) using
transformed problems (2.1)–(2.2) and (2.1)–(2.3), respectively, we need suitable
additional condition, which ensure that solutionwof (2.1)–(2.2) or (2.1)–(2.3),
respectively, satisfiesw(t)∈Imϕfor all t∈[0, T]. One of such a possible con-
dition is kfekL^{2} <

q3

T min{−a, b} (which is a consequence of (3.3)) provided h≡0.

However, we would like to point out that the restrictive condition on the

‘greatness’ of feis given not only by the specific limitation of the employed
method, but that it also arises directly from the nature of the problem. Let
us consider (1.5)–(1.2) or (1.5)–(1.4), respectively. Taking into account the
boundary conditions (1.2) or (1.4), respectively, there existst_{0}∈[0, T] :u^{0}(t_{0}) =
0. Hence integrating (1.5) we find:

ϕ(u^{0}(t)) =
Z _{t}

t0

fe(τ) +f−g(u^{0}(τ))−h(u(τ))

dτ ,

which requires a <

Z _{t}

t0

fe(τ) +f −g(u^{0}(τ))−h(u(τ))

dτ < b . (2.9) Integrating (1.5) from 0 toT we also find

f = 1 T

Z _{T}

0 (g(u^{0}(t)) +h(u(t)))dt ,

which implies

ξ∈Rinfg(ξ) + inf

ξ∈Rh(ξ)≤f ≤sup

ξ∈Rg(ξ) + sup

ξ∈Rh(ξ). (2.10) Suppose, in addition, thatg is bounded. Iff does not satisfy (2.10) or one of the following two inequalities holds:

sup

t0∈[0,T] inf

t∈[0,T]

Z _{t}

t0

fe(τ) +f−inf

ξ∈Rg(ξ)−inf

ξ∈Rh(ξ)

dτ

< a , (2.11)

t0∈[0,T]inf sup

t∈[0,T]

Z _{t}

t0

fe(τ) +f−max

ξ∈R g(ξ)−max

ξ∈R h(ξ)

dτ

> b , (2.12) then the BVP (1.5)–(1.2) or (1.5)–(1.4), respectively, does not admit any solu- tion withf =fe+f. Compare also with results in [11].

Remark 2.6 It is worth noting that if we consider the periodic BVP for quasi- linear ODE:

(ϕ(u^{0}))^{0}+λu^{0}+h(u) =f, t∈(0, T),

where λ∈R, then the condition (2.7) from Theorem 2.4 is reduced to

h(−∞)< f < h(+∞). (2.13) It means that we obtain the Landesman-Lazer condition as a particular result.

Here we present a proof. Ifg(z)≡λz, whereλ∈R, then integrating (2.1)
from 0 toT and using boundary conditions (2.2) we find 0 =R_{T}

0 f(t)dt= (T f).

Since (2.1)–(2.2) has a solution if and only if f =s(fe) (see Theorem 2.1), we have that s(fe) ≡ 0. Consequently, if g(z) ≡ λz then the condition (2.7) is reduced to (2.13).

Note, that in order to gets(f)e ≡0 for the Neumann boundary-value prob- lem, we have to impose g(z)≡0.

Example 2.1 Considerϕ(z) =m_{0}z/

q

1−^{z}_{c}2^{2}, m_{0} >0 and g, h∈ C(R,R), h
has finite limits h(−∞)< h(+∞). Then it follows from Theorem 2.4 that the
periodic BVP for

qm_{0}u^{0}(t)
1−^{(u}^{0}_{c}^{)}^{2}2^{(t)}

0

+g u^{0}(t)

+h u(t)

=fe(t) +f

has a solution for every couplef , fe satisfying condition (2.7):

h(−∞) +s(f)e < f < h(+∞) +s(fe),
where s:Ce_{T} →Rcomes from Theorem 2.1.

Let us remark that the conditions (2.5) and (2.6) are satisfied for anyfe∈Ce_{T}.
This follows from the fact that the range ofϕisR(i.e. b= +∞).

It is worth noting that this problem arises from the relativistic dynamics and describes damped oscillations.

Example 2.2 With respect to Theorem 2.1 for all fe∈C_{T} there exists unique
f such that the periodic boundary-value problem for

u^{00}(t) +k u^{0}(t)_{2}

=fe(t) +f

has a solution. This boundary-value problem describes a steady-state process of gas burning in jets of rockets (cf. [17]).

Example 2.3 Consider the following periodic boundary-value problem:

u^{0}(t)−(u^{0}(t))^{3}_{0}

+u^{0}(t) + arctan(u(t)) =fe(t) +f in (0, T),
u(0) =u(π), u^{0}(0) =u^{0}(π).

Let q3

T2√
93−^{π}_{2}√

T >0. Then, for allfe∈Ce_{T} : kfekL^{2} <

q3 T2√

93−^{π}_{2}√
T,
this problem has a solution if and only iff ∈Rsatisfy

−π

2 < f <π

2 . (2.14)

To prove this result, we can use Theorem 2.4. Indeed, ϕis odd homeomor-
phism of (−^{√}^{1}_{3},√^{1}

3) onto (−^{2}^{√}_{9}^{3},^{2}^{√}_{9}^{3}), so thatb=−a=^{2}^{√}_{9}^{3}. Moreover,

z±∞lim arctanz=±π

2 and −π

2 <arctanz < π

2,for allz∈R.

Hence the condition (2.14) follows from (2.7), wheres(fe)≡0, becauseg(z)≡z (see Remark 2.6).

This problem describes forced oscillations of voltage u in an electrical cir- cuit with ferro-resonance (see e.g. [12]). Such nonlinear circuits are used in radiotechnics.

Table 1 illustrates the applicability of Theorems 2.1–2.4 for some particular nonlinearities ϕ. The symbol A (applicable) indicates that the corresponding assumption is satisfied and thus using an appropriate Theorem one obtains the desired conclusion; on the other hand NA (not applicable) indicates that the corresponding assumption is not satisfied and that the selected Theorem is not applicable in that case). We also relate our results to the results already known.

Periodic,g(s) =s^{3} Neumann,g(s) =s^{3} both,h(s) = arctans

ϕ(z) Thm 1 Theorem 2 Theorem 3 Theorem 4

(P)+(G) +(P’) +(G’) +(P”) (P)+(G) +(P’) +(G’) (P)+(G)+(PH)

|z|^{p−2}z,

1< p <2 A A NA A A

z A: [6, Thm 1],

[4, Thm 3.4], [16, Thm 4]

A: [4, Thm 3.3], [16, Thm 1]

A
[6, Thm 2]^{#}

|z|^{p−2}z,

p >2 A NA A NA A

√z

1−z^{2} A A A A

√z

1+z^{2} A^{∗} A^{∗} A^{∗} T < ^{2}^{√}_{π}^{3}

A^{†}

T > ^{2}^{√}_{π}^{3}
NA
exp

1

|z|

z A A NA A NA

Table 1: Legend: ^{#} With periodic conditions only,

∗ Provided thatkfek_{L}^{2} <

q3

T, ^{†} Assuming thatkfke _{L}^{2} <

q3
T −^{π}_{2}√

T.

### 3 Proofs of Theorems 2.1 – 2.3

We start this section with the following lemma:

Lemma 3.1 Let f ∈ C[0, T], q : J_{1} → J_{2} be continuous, where J_{1}, J_{2} are
nonempty intervals and J_{1} contains zero in its interior. Let w ∈ C^{1}[0, T],
w(0) =w(T)andw satisfies

w^{0}(t) +q(w(t)) =f(t), t∈(0, T) (3.1)
then

|w(t)−w(s)| ≤ |t−s|^{1}^{2}kfek_{L}^{2}. (3.2)
Moreover, if there existst_{0}∈[0, T]such that w(t_{0}) = 0then

kwkC≤ rT

3kfek_{L}^{2}. (3.3)

Remark 3.1 The previous lemma will be used to estimate solutions of (2.1)–

(2.2) or (2.1)–(2.3), respectively. Indeed, the second of (2.2) yieldsw(0) =w(T)
directly. On the other hand, the first of (2.2), the continuity ofwand the fact,
that ϕ satisfies (P) imply that there existst_{0} ∈[0, T] such that w(t_{0}) = 0. In
the case of (2.3) we havew(0) =w(T) = 0.

Proof Using the Cauchy–Schwarz inequality it is easy to show that

|w(t)−w(s)|=
Z _{t}

s

w^{0}(τ)dτ

≤ |t−s|^{1}^{2}kw^{0}k_{L}^{2}. (3.4)

Multiplying both sides of equation (2.1) by w^{0}, integrating from 0 to T, split-
ting f as in (1.6) and using w(0) = w(T), we find that kw^{0}k^{2}_{L}2 =

f , we ^{0}

L^{2}

(where
f , we ^{0}

L^{2} :=R_{T}

0 f we ^{0} is the scalar product inL^{2}). The Cauchy-Schwarz
inequality yieldskw^{0}k_{L}^{2} ≤ kfek_{L}^{2}, which together with (3.4) establishes (3.2).

Now we are going to estimatekwkC.Decompose the functionwasw=w+w,e
where R_{T}

0 w(t)dte = 0 andw∈R. Since we∈C^{1}[0, T]⊂W^{1,2}[0, T] and w(0) =
w(T), the Sobolev inequality [15, Proposition 1.3] yieldskwke C ≤q

12Tkw^{0}k_{L}^{2} ≤
qT

12kfek_{L}^{2}.Since we assume that there exists at_{0}∈[0, T] such thatw(t_{0}) = 0,
it shall be|w| ≤ kwke C. HencekwkC≤2

qT

12kfek_{L}^{2} =
qT

3kfek_{L}^{2},which is the

desired inequality (3.3). ♦

Now we proceed to proofs of Theorems 2.1–2.3.

Proof of Theorem 2.1 We divide the proof into six steps:

Step 1 Let q be a continuously differentiable and bounded real function. We
prove that for all fe∈Ce_{T} there existsf ∈Rsuch that the equation

w^{0}(t) +q w(t)

=fe(t) +f (3.5)

has a solutionw∈C^{1}[0, T] satisfying (2.2).

Let fe∈ Ce_{T} be given. Since q is continuously differentiable and bounded
function, the solution ( denoted byw(t, f , α) ) to the initial-value problem

w^{0}(t) +q w(t)

=fe(t) +f , (3.6)

w(0) =α

exists on (0, T), it is unique and is continuously differentiable with respect to
parametersα and f (see e.g [5]). Let M = sup_{y∈R}|q(y)|. Taking f > M +ε
and integrating (3.5) from 0 to T, we find w(T, f , α)> α. On the other hand,
if f <−M −ε then we get w(T, f , α) < α. Sincew depends continuously on
f, for allα∈R, there exists f_{α}∈Rsuch thatw(T, f_{α}, α) =α.Moreover, the
partial derivative with respect to the parameterf ,w_{f}(t, f , α), is a solution to
the linear initial-value problem

z^{0}(t) = 1−q^{0} w(t, f , α)

z(t), (3.7)

z(0) = 0.

The explicit formula of the solution of (3.7) yieldsw_{f}(T, f , α)>0, which means
thatw(T,·, α) is increasing andf_{α} is unique. Thus, we can define the mapping
ψ_{1} : R →[−M, M] by α7→ f_{α}. Asw_{f}(T, f , α) >0, by the abstract implicit
function theorem (see [1, Theorem 2.2.3]), we find that ψ_{1} : R→ [−M, M] is
continuous.

Rewrite (3.6) as an integral equation:

w(t) =
Z _{t}

0

fe(τ)−q(w(τ))

dτ+f t+α .
Taking α > R_{T}

0 |fe(τ)|dτ + 2T M, we see that w(t, f , α) > 0 for all t ∈ [0, T]
and f ∈ [−M, M]. Consequently R_{T}

0 ϕ_{−1} w(τ, f , α)

dτ > 0. Conversely, if
α <−R_{T}

0 |fe(τ)|dτ−2T M thenw(t, f , α)<0 for allt∈[0, T] andf ∈[−M, M].

Hence R_{T}

0 ϕ_{−1} w(τ, f , α)

dτ < 0. Since ϕ_{−1} is a continuous real function, the
mapping α7→R_{T}

0 ϕ_{−1} w(t, f , α)

dtis continuous and there exists at least one
α_{f} ∈R such thatR_{T}

0 ϕ_{−1} w(t, f , α)

dt= 0. Moreover the partial derivative of
w(t, f , α) with respect to initial condition α,w_{α}(t, f , α), is the solution of the
linear initial-value problem

v^{0}(t) +q^{0} w(t, f , α)

v(t) = 0, (3.8)

v(0) = 1.

Taking into account explicit formula of the solution of (3.8), it is easy to verify
that w_{α}(t, f , α)>0 for all t ∈[0, T] and f , α ∈R. Thus, for allf ∈R there
exists uniqueα=α_{f} such thatR_{T}

0 ϕ_{−1} w(t, f , α_{f})

dt= 0. By the same reason
as above, the mappingψ_{2}:R→Rdefined byf 7→α_{f} is continuous.

Let us consider the continuous mappingψ_{1}◦ψ_{2}: [−M, M]→[−M, M]. Due
to the Brouwer fixed point theorem there exists at least onef_{0}∈[−M, M] such
that ψ_{1} ψ_{2}(f_{0})

=f_{0}. Hence the functionw(t, f_{0}, ψ_{2}(f_{0})) is a solution of the
equation (3.5) subject to (2.2).

Step 2We show that if(P),(G)are satisfied then, for all fe∈Ce_{T}, there exists
at least onef ∈Rsuch that the boundary-value problem for the equation

w^{0}(t) +g ϕ_{−1} w(t)

=fe(t) +f (3.9)

subject to (2.2) has at least one solution.

Letwbe the solution of (3.9)–(2.2). Integrating (3.9) from 0 toT and then dividing by T, we obtain

1 T

Z _{T}

0 g ϕ_{−1}(w(t))

d t=f .

With respect to (3.3) (considerq:=g◦ϕ_{−1}in Lemma 3.1) the following a priori
bound onwresults: kwkC≤q

T3kfek_{L}^{2}. Hence we estimatef as follows:

|f| ≤sup (

|g(ξ)|:ϕ_{−1} −
rT

3kfekL^{2}

!

≤ξ≤ϕ_{−1}
rT

3kfekL^{2}

!)

. (3.10)

As kwkC ≤ q

T3kfke _{L}^{2}, the restriction of g◦ ϕ_{−1} on I =
h−q

T3kfek_{L}^{2} ,
q

T3kfekL^{2}

i

is essential in further considerations. Note that the assumption

kfke L^{2} ≤ q

T3min{|a|, b} imply I ⊂ (a, b) (= Domϕ_{−1}); hence g◦ϕ_{−1} is well
defined on I. Let us introduce the following sequence of functions γ_{n}(x) :=

nR

I% ^{x−y}_{n}

g ϕ_{−1}(y)

dy,where%:R→Ris the regularization kernel given by

%(x) :=

(

c_{0}e^{|x|}^{2}^{1}^{−1} for|x|<1,

0 for|x| ≥1, (3.11)
wherec_{0}is a constant such thatR_{1}

−1%(x)dx= 1.It is easy to verify that
γ_{n} ^{∞}_{n=1}
converges tog◦ϕ_{−1}uniformly onI,thatγ_{n}is continuously differentiable for any
n∈Nand thatkγnk_{C(I)}≤ kg◦ϕ_{−1}k_{C(I}_{)}(C(I) denotes the space of continuous
real functions defined onI).

From the first step we know that there existw_{n} andf_{n} such that
w^{0}_{n}(t) +γ_{n} w_{n}(t)

=fe(t) +f_{n}

and Z _{T}

0

ϕ_{−1} w_{n}(t)

dt= 0, w_{n}(0) =w_{n}(T).

Utilizing (3.3) and kγk_{C(I)} ≤ kg◦ϕ_{−1}k_{C(I)} < +∞ we find that

f_{n} ^{∞}_{n=1} is
a bounded sequence. By (3.3), w_{n} are equibounded and, by (3.2), w_{n} are
equicontinuous (we apply Lemma 3.1 with q :=γ_{n} for each n∈N, employing
the fact that resulting inequalities do not contain q). Hence we can select
subsequencesw_{n}_{k}, f_{n}_{k} such thatw_{n}_{k}→winC_{T} andf_{n}_{k}→f. Since

w_{n}_{k}(t) =w_{n}_{k}(0) +
Z _{t}

0

fe(τ) +f_{n}_{k}−γ_{n}_{k}(w_{n}_{k}(τ))

dτ ,

passing to the limit we obtain w(t) =w(0) +

Z _{t}

0

fe(τ) +f−g ϕ_{−1}(w(τ))
dτ .

As the integrand in the right-hand-side is continuous,w∈C_{T}^{1} and satisfies (3.9)
in (0, T). Thus Step 2 is over.

Step 3We prove that iff_{1}, f_{2}∈Rand the equation (2.1) with f =fe+f_{i}, i=
1,2 has a solution satisfying (2.2), then f_{1} = f_{2}. Conversely, assume that
f_{1} > f_{2} and there exists w_{i} ∈C_{T}^{1}, i= 1,2 such that w_{i}^{0}(t) +g(ϕ_{−1}(w_{i}(t))) =
fe(t) +f_{i}, i= 1,2 subject to (2.2). Then from (1.2) it follows that the function
ϕ_{−1} w_{1}(t)

−ϕ_{−1} w_{2}(t)

is a T-periodic function with mean value zero, so
that there exist t_{0} and δ_{1} >0 such that ϕ_{−1} w_{1}(t_{0})

−ϕ_{−1} w_{2}(t_{0})

= 0 and
ϕ_{−1} w_{1}(t)

−ϕ_{−1} w_{2}(t)

<0 for allt_{0}< t < t_{0}+δ_{1} (note that as we consider
the periodic problem, we can shift the problem suitably int if it is necessary).

Taking into account that ϕis an increasing homeomorphism, we get w_{1}(t_{0}) =
w_{2}(t_{0}) andw_{1}(t)−w_{2}(t)<0 for allt_{0}< t < t_{0}+δ_{1}.Sincew_{1}(t_{0}) =w_{2}(t_{0}),f_{1}>

f_{2} and since the functions g(ϕ_{−1}(w_{1}(·))) and g(ϕ_{−1}(w_{2}(·))) are continuous,

there exists δ_{2} > 0 such that |g(ϕ_{−1}(w_{1}(t)))−g(ϕ_{−1}(w_{2}(t)))| <(f_{1}−f_{2})/2
for anyt_{0}< t < t_{0}+δ_{2}. Takingδ= min{δ1, δ_{2}}, we arrive at

(w_{1}−w_{2})^{0}(t) =f_{1}−f_{2}+g(ϕ_{−1}(w_{1}(t)))−g(ϕ_{−1}(w_{2}(t)))>0

for all t_{0} < t < t_{0}+δ. However,w_{1}(t)−w_{2}(t)< 0 for allt_{0} < t < t_{0}+δ, a
contradiction.

Step 4Estimate (2.4).

The estimate (2.4) is a direct consequence of (3.10).

Step 5Continuity of s.

The proof of the continuity ofs is analogous to that one in [6] (page 256, step 5) and so is omitted.

Step 6Completion of the proof of Theorem 2.1.

Taking into account the relation between the solutionuof the problem (1.5)–

(1.2) with h≡0 and the solutionw of the problem (2.1)–(2.2): u^{0} =ϕ_{−1}(w),

the assertion of Theorem 2.1 follows. ♦

Proof of Theorem 2.2 Sinceu^{0}=ϕ_{−1}(w), using (P’) we get u∈C_{T}^{2}.
Assumptions (P’) and (G’) imply thatg◦ϕ_{−1}: (a, b)→Ris a continuously
differentiable real function. Letqbe a continuously differentiable and bounded
real function satisfyingq(y) =g ϕ_{−1}(y)

for all|y| ≤q

T3kfekL^{2}.

Taking into account a priori bound (3.3) and our definition ofq, any solution
of (2.1)–(2.2) satisfies (3.5)–(2.2) and vice versa. Thus we are reduced to prove
the uniqueness of the solution to (3.5)–(2.2). Due to Theorem 2.1, for each
fe∈Ce_{T} :kfekCT <

q3

T min{|a|, b}, there exists precisely ones(fe)∈Rsuch that
(2.1)–(2.2) possesses solution; s(fe) is also the unique value corresponding tofe
such that (3.5)–(2.2) has solution. Sinceqsatisfies assumptions of the Step 1 of
the proof of Theorem 2.1, the initial valuew(0) =αis uniquely determined by
α=φ_{2}(s(fe)). Now, asq is a continuously differentiable function, the solution
of the initial value problem (3.6), withf =s(fe), α=φ_{2}(f), is unique (see [5])
and so is that of (3.5)–(2.2) withf =s(fe).

What remains to prove is thatw andsare continuously differentiable with
respect to fe. Following the idea of the proof of Theorem 3.4 in [4] we define
G:C_{T} ×R×R×Ce_{T} →C_{T} ×R×Rby the formula

G(w, f , α,fe) :=

w(t)−f t−R_{t}

0fe(τ)dτ +R_{t}

0q w(τ)
dτ −α
R_{T}

0 ϕ_{−1} w(τ)
dτ
w(T)−α

. (3.12)

As in [4] one can check that the operator equation

G(w, f , α,fe) = 0 (3.13)

is equivalent to the boundary-value problem (3.5)–(2.2). As a consequence of
this fact we obtain that for allfe_{0}∈Ce_{T} there exists a unique triple (w_{0}, f_{0}, α_{0})∈
C_{T}^{1} ×R×Rsuch thatG(w_{0}, f_{0}, α_{0},fe_{0}) = 0.

Now applying the abstract implicit function theorem we conclude the proof.

To do this we shall prove:The assumptions of the implicit function theorem (see
[1, Theorem 2.2.3]) are satisfied at any point(w_{0}, f_{0}, α_{0},fe_{0})∈C_{T}^{1}×R×R×Ce_{T},
fe_{0}6≡0 (and if(P”) also forfe_{0}≡0),at which the equation (3.13) holds.

One can see that the operatorGand the partial Fr´echet derivativesG_{(w,f,α)}
andG_{f}_{e}(the first partial Fr´echet derivatives ofGwith respect to (w, f , α) andf,
respectively) are continuous in the neighbourhood of (w_{0}, f_{0}, α_{0},fe_{0}). By a direct
calculation, we get that the partial Fr´echet derivativeG_{(w,f,α)}(w_{0}, f_{0}, α_{0},fe_{0}) :
C_{T}^{1}×R×R→ C_{T}^{1} ×R×R,is given by the formula:

(ω, σ, κ)7→

ω(t) +R_{t}

0q^{0} w_{0}(τ)

ω(τ)dτ −σt−κ
R_{T}

0 ϕ_{−1} w_{0}(τ)
ω(τ)dτ
ω(T)−κ

. (3.14)

Let (φ, β, r)e ∈Ce_{T} ×R×R. Then the operator equation
G_{(w,f,α)}

w_{0}, f_{0}, α_{0},fe_{0}

(ω, σ, κ) =
Z _{t}

0

φ(τ)e dτ , β , r T

is equivalent to the initial-value problem
ω^{0}(t) +q^{0} w_{0}(t)

ω(t) =σ+φ(t)e , (3.15) ω(0) =κ ,

subjected to the additional conditions
Z _{T}

0 ϕ^{0}_{−1} w_{0}(τ)

ω(τ)dτ =β , ω(T) =κ+r. (3.16)
LetA(t) :=e^{R}^{0}^{t}^{q}^{0}^{(w}^{0}^{(τ))dτ} >0. The solution of the linear initial value prob-
lem (3.15) has the explicit form:

ω(t) =κ 1 A(t)+ 1

A(t)
Z _{t}

0

φ(s)A(s)dse +σ 1 A(t)

Z _{t}

0

A(s)ds . (3.17) Substituting (3.17) into (3.16) we obtain system of two linear equations for σ and κ. This system can be uniquely solved if corresponding determinant D is different from zero. By a straightforward calculation we obtain:

D = 1

A(T) h Z T

0 A(s)ds
Z _{T}

0

ϕ^{0}_{−1}(w_{0}(t))

A(t) dt (3.18)

−
Z _{T}

0

ϕ^{0}_{−1}(w_{0}(t))
A(t)

Z _{t}

0 A(s)ds dt i

+
Z _{T}

0

ϕ^{0}_{−1}(w_{0}(t))
A(t)

Z _{t}

0 A(s)ds dt

= 1

A(T)
Z _{T}

0 A(s)
Z _{s}

0

ϕ^{0}_{−1}(w_{0}(t))
A(t) dt ds+

Z _{T}

0

ϕ^{0}_{−1} w_{0}(t)
A(t)

Z _{t}

0 A(s)ds dt .

Sinceϕ_{−1}:I_{2}→I_{1}is an increasing homeomorphism, it follows thatϕ^{0}_{−1}(z)>0
a.e. in I_{2}. By a contradiction, one can show that if fe6≡0 then w_{0} satisfying
(2.1)–(2.2) is not a constant function. Thusϕ^{0}_{−1}(w_{0}(t))>0 on some subset of
[0, T] of positive measure, which taking into account (3.18) implies D>0.

Now let us considerfe≡0. We have to imposeϕ^{0}(0)>0 to concludeD >0.

The reason consists in the fact thatw_{0}≡0 is the unique solution of (2.1)–(2.2)
withfe≡0.

We proved thatG_{(w,f,α)}(w_{0}, f_{0}, α_{0}) :C_{T}^{1} ×R×R→C_{T}^{1}×R×Ris bijective
linear mapping. Then the inverse of G_{(w,f,α)}(w_{0}, f_{0}, α_{0}) is continuous due to
the Banach open mapping theorem (see [18]). Thus G_{(w,f,α)}(w_{0}, f_{0}, α_{0}) is an
isomorphism of C_{T}^{1} ×R×R onto itself and the assumptions of the implicit
function theorem [1, Theorem 2.2.3] are satisfied. This completes the proof of

Theorem 2.2. ♦

Proof of Theorem 2.3 At first we perform the proof under the assumptions
(P), (P’) and (G’). Considering (3.3) we can define continuously differentiable
and bounded functionqsatisfyingq(y) =g ϕ_{−1}(y)

for all|y| ≤q

T3kfek_{L}^{2}.
Since the related first order problem (2.1)–(2.3) is the same as that one
considered in [4], the existence of the solution and the differentiability ofw(fe)
ands(fe) follows from [4, Theorem 3.4].

The assumption (P’) can be omitted and the assumption (G’) can be replaced by (G) using the same argument as in the Step 2 of the proof of Theorem 2.1.

From [16, Theorem 3] we obtain the uniqueness of f corresponding to a fixed fe. The continuity ofs(fe) can be proved in the same manner as in Step 5 of the proof of Theorem 2.1. A priori bound (2.4) is a consequence of (3.3) (cf. Step

4 in the proof of Theorem 2.1). ♦

### 4 Proof of Theorem 2.4

To prove Theorem 2.4, we need the following comparison results:

Lemma 4.1 Letv^{0}(t)+g

ϕ_{−1} v(t)

≤α+fe(t)on[0, T]andvsatisfies (2.2).

Then α≥s(fe)(wheres(fe)comes from Theorem 2.1).

Proof Assume conversely thatα < s(fe). Letwsatisfies (2.1) and (2.2). Since
ϕ_{−1}(v(0))−ϕ_{−1}(w(0)) =ϕ_{−1}(v(T))−ϕ_{−1}(w(T))

and Z _{T}

0 ϕ_{−1} v(t)

−ϕ_{−1} w(t)
dt= 0,

there exists a t_{0} ∈ [0, T) and δ > 0 such that ϕ_{−1} v(t_{0})

−ϕ_{−1} w(t_{0})

= 0
and ϕ_{−1} v(t)

−ϕ_{−1} w(t)

>0 for allt_{0} < t < t_{0}+δ (note that we consider
periodic problem so if necessary we can shift suitably the problem int). Taking
into account thatϕis an increasing homeomorphism, we getv(t_{0}) =w(t_{0}) and
v(t)> w(t) for all t_{0}< t < t_{0}+δ. On the other hand, there existsδ^{0} >0 such
that

(v^{0}−w^{0})(t)≤α−s(fe)−g

ϕ_{−1} v(t)
+g

ϕ_{−1} w(t)

<0

for allt : |t−t_{0}| < δ^{0}. Since v(t_{0}) = w(t_{0}), we obtainv(t)−w(t) <0 for all
t_{0}< t < t_{0}+δ^{0}, which is a contradiction. ♦

Lemma 4.2 Letv^{0}(t)+g

ϕ_{−1} v(t)

≤α+f(t)e on[0, T]andvsatisfies (2.3), thenα≥s(f)e (heres(fe)is taken from Theorem 2.3).

Proof Suppose the contrary, i.e. α < s(fe). Letwsatisfies (2.1)–(2.3), then
v^{0}(0)≤ −g

ϕ_{−1} v(0)

+fe(0) +α <−g

ϕ_{−1} w(0)

+fe(0) +s(fe) =w^{0}(0).

From here and (2.3) we get that there existsε >0, such thatv(t)< w(t) for all t∈(0, ε).

Ifv(t)< w(t) for allt∈(0, T] thenv(T)< w(T) and eitherv orwcan not
satisfy (2.3) atT. Thusε≤T and there existst_{0}∈(0, T] such that

v(t)< w(t) for allt∈(0, t_{0}) andv(t_{0}) =w(t_{0}).
Fort∈(0, t_{0}) we have

v(t)−v(t_{0})

t−t_{0} > w(t)−w(t_{0})
t−t_{0} ,
i.e. v^{0}(t_{0})≥w^{0}(t_{0}). On the other hand, we have

w^{0}(t_{0}) = −g ϕ_{−1} w(t_{0})

+fe(t_{0}) +s(fe)

= −g ϕ_{−1} v(t_{0})

+fe(t_{0}) +s(fe)

> −g ϕ_{−1} v(t_{0})

+fe(t_{0}) +α≥v^{0}(t_{0}),

which is a contradiction. ♦

Remark 4.1 It is possible to show that the assertions of Lemmas 4.1 and 4.2 hold true also with inverted inequalities. This is in agreement with the semilinear periodic problem studied in [6]. These inequalities are used to prove the ‘dual’ version of Theorem 2.4 withh(+∞)< h(−∞).

Lemma 4.3 Letϕbe an increasing homeomorphism ofRonto itself and there
exist c, C > 0 and p > 1: c|z|^{p−1} ≤ |ϕ(z)| ≤ C(|z|^{p−1}+ 1) for all z ∈ R.

Then, for any y∈C_{T} (y∈C[0, T]), there exists exactly oneu∈C^{1}[0, T]with
ϕ(u^{0})∈C^{1}[0, T] and satisfying

ϕ(u^{0}(t))_{0}

−ϕ(u(t)) =y(t) t∈(0, T) (4.1) subject to (1.2) ( (1.4) ).

Proof For the sake of brevity we present the proof only for the periodic con- ditions. In the case of the Neumann conditions, the proof is analogous.

At first we prove that, for any giveny∈C_{T}, there exists precisely one weak
solution of (4.1)–(1.2), where the weak solution of (4.1)–(1.2) is any function
u∈W_{T}^{1,p} such that the following identity

Z _{T}

0 {ϕ(u^{0})v^{0}+ϕ(u)v}=−
Z _{T}

0 yv (4.2)

is satisfied for eachv∈W_{T}^{1,p}. Then, using an regularity argument, we show that
this functionuis smooth enough and satisfies (4.1)–(1.2) in the sense indicated
in the assertion of the lemma.

Existence and uniqueness of the weak solution. Let us define ψ:W_{T}^{1,p} →
L^{p}^{0} (wherep^{0}:=p/(p−1)) byu7→ψ(u) if and only ifhψ(u), vi=R_{T}

0 ϕ(u^{0})v^{0},
u, v ∈W_{T}^{1,p} for all v ∈W_{T}^{1,p}. Since |ϕ(z)| ≤ C(|z|^{p−1}+ 1) for all z ∈R, it is
easy to verify thatψ is a continuous operator; this follows from the Nemitskii
theorem (see [1, Theorem 1.2.2]). From the fact thatϕis an increasing function
we get strict monotonicity of ψ. Since any monotone continuous operator is
also hemicontinuous (see [18]), we get that ψ is a hemicontinuous one. The
assumption |ϕ(z)| ≥ c|z|^{p−1} for all z ∈ R implies that ψ is weakly coercive.

Now the existence and uniqueness of the weak solution of (4.1)–(1.2) follows from [18, Theorem 32.H].

Regularity. Suppose that uis a weak solution of (4.1)–(1.2). We show that
u∈C_{T}^{1}, ϕ(u^{0})∈ C^{1}[0, T] and that (4.1) holds pointwise in (0, T). Integrating
by parts we can rewrite the equation (4.2) into the form:

Z _{T}

0

ϕ(u^{0}(t))−
Z _{t}

0

ϕ u(τ)

+y(τ) dτ

v^{0}(t)dt+ (4.3)

Z _{t}

0 ϕ u(τ)

+y(τ) dτ v(t)

T 0

= 0. Let us define a functionM : [0, T]→R,

t7→ϕ(u^{0})−
Z _{t}

0

ϕ u(τ)

+y(τ) dτ.

It is easy to see thatM ∈L^{p}^{0} and from (4.3) we get
Z _{T}

0 M(t)v^{0}(t)dt= 0 for allv∈C_{0}^{∞}(0, T).

Hence Z _{T}

0

δM

δt v= 0 for allv∈C_{0}^{∞}(0, T),

where ^{δM}_{δt} denotes the distributional derivative ofM. SinceM ∈L^{p}^{0} ,→L^{1}and

δM

δt = 0, we obtain thatM(t) =k a.e. in [0, T] andk∈R. Thus
ϕ(u^{0}(t))−

Z _{t}

0

ϕ u(τ)

+y(τ)

dτ−k= 0 a.e. in [0, T]. (4.4) Since ϕ is an increasing homeomorphism of R onto itself, we can rewrite the previous equation into the following form

u^{0}(t)−ϕ_{−1}
Z _{t}

0

ϕ u(τ)

+y(τ) dτ−k

= 0. (4.5)

Now let us define a functionF :R×[0, T]→R,
(z, t)7→z−ϕ_{−1}

Z _{t}

0

ϕ u(τ)

+y(τ) dτ−k

.

It is possible to show thatF is continuous onR×[0, T]. Moreover,F(·, t0) is
an increasing function for allt_{0}∈[0, T], and

z→−∞lim F(z, t_{0}) =−∞, lim

z→+∞F(z, t_{0}) = +∞.
Hence for eacht∈[0, T] there exists exactly onez(t)∈R, such that

F(z(t), t) = 0.

Since ^{∂F}_{∂z} is continuous and ^{∂F}_{∂z} = 1 6= 0, we can apply the implicit function
theorem to show thatz(·)∈C[0, T]. From (4.5) we get

F(u^{0}(t), t) = 0 a.e. in [0, T].
Thus we arrive at

z(t) =u^{0}(t) a.e. in [0, T].

Sinceu∈W_{T}^{1,p}is absolutely continuous, this identity holds true for allt∈[0, T]
and thusu∈C^{1}[0, T]∩W_{T}^{1,p}.

Now it remains to prove thatϕ(u^{0})∈C^{1}[0, T].Let us defineG:R×[0, T]→
R,

(z, t)7→z−
Z _{t}

0

ϕ(u(τ))−y(τ)

dτ −k .

The functionGis continuous onR×[0, T]. Moreover, for allt_{0}∈[0, T],G(·, t0)
is an increasing function and lim_{z→±∞}G(z, t_{0}) =±∞. Hence for eacht∈[0, T]
there exists exactly one z(t) such that

G z(t), t

= 0.

Since u ∈ C^{1}[0, T] and y ∈ C[0, T], the partial derivatives ^{∂G}_{∂z} and ^{∂G}_{∂t} are
continuous; moreover ^{∂G}_{∂z} = 16= 0. Then the implicit function theorem yields
z(t) ∈C^{1}[0, T]. Taking into account (4.4) and the fact that u∈ C^{1}[0, T], we
see that z(t) = ϕ(u^{0}) for all t ∈ [0, T]; thus ϕ(u^{0}) ∈ C^{1}[0, T]. Now, since
u ∈C^{1}[0, T]∩W_{T}^{1,p} and ϕ(u^{0})∈ C^{1}[0, T], integrating (4.2) by parts we show
thatusatisfies (4.1) in (0, T) and thatu^{0}(0) =u^{0}(T). This concludes the proof.

♦

Now we can define a solution operatorK:C_{T} →C_{T}^{1},

y7→K(y), (4.6)

whereK(y) is the solution of (4.1)–(1.2). Analogously we can define a solution
operatorK^{0}:C[0, T]→C^{1}[0, T] corresponding to the Neumann problem (4.1)–

(1.4).

Lemma 4.4 LetK andK^{0} be defined as above. ThenK is compact as a map-
ping between C_{T} and C_{T}^{1} andK^{0} is compact as a mapping betweenC[0, T]and
C^{1}[0, T].

Proof We prove the compactness ofK. The proof of the compactness ofK^{0}
is analogous.

Let us consider the following sequence of equations:

(ϕ(u^{0}_{n}(t)))^{0}−ϕ(u_{n}(t)) =y_{n}(t), (4.7)
subject to (1.2), where {y_{n}}^{∞}_{n=1} ⊂ C[0, T] is bounded. We are going to show
that one can select a convergent subsequence from{u_{n}}^{∞}_{n=1}⊂C_{T}^{1}. Multiplying
the equation (4.7) by u_{n}, integrating from 0 to T, integrating the first term in
the left-hand-side by parts and using the periodic conditions (1.2), we obtain

−
Z _{T}

0

ϕ(u^{0}_{n}(t))u^{0}_{n}(t) +ϕ(u_{n}(t))u_{n}(t) dt=
Z _{T}

0 y_{n}(t)u_{n}(t)dt . (4.8)
Since we assume that|ϕ(z)| ≥c|z|^{p−1}for allz∈R, we find that

kunk^{p}

W_{T}^{1,p} ≤1
c

Z _{T}

0

ϕ(u^{0}_{n}(t))u^{0}_{n}(t) +ϕ(u_{n}(t))u_{n}(t) dt .

Using this we estimate the terms on the left-hand side of (4.8). The right hand- side of (4.8) is estimated by the H¨older inequality. Therefore we find that

kunk^{p}_{W}1,p

T ≤1

ckynk_{L}p0kunkL^{p}≤ 1

ckynk_{L}p0kunk_{W}^{1,p}

T ,

which implies

kunk_{W}^{1,p}

T ≤ 1

c(kynk_{L}p0)^{p−1}^{1} . (4.9)
Thus{un}^{∞}_{n=1} is bounded inW_{T}^{1,p}. Since W_{T}^{1,p} is compactly imbedded inC_{T},
we can select {unk}^{∞}_{k=1} such that u_{n}_{k} → w in C_{T}. Let h_{k}(t) = y_{n}_{k}(t)−
ϕ u_{n}_{k}(t)

. One can see that there existsβ > 0 such that khkkC ≤β. Hence
k ϕ u^{0}_{n}_{k}_{0}

k_{C} ≤ β. It follows from (1.2) that there exists t^{k}_{0} ∈ [0, T], such
that u^{0}_{n}_{k}(t^{k}_{0}) = 0. From (4.7) we obtain ϕ(u^{0}_{n}_{k}(t)) = R_{t}

t^{k}_{0}h_{k}(τ)dτ and conse-
quentlykϕ(u^{0}_{n}_{k}(t))kC ≤T β. Thus ϕ(u^{0}_{n}_{k}(t)) is bounded inC_{T}^{1} norm. Due to
the compact imbedding ofC_{T}^{1} intoC_{T} we can select ϕ(u^{0}_{n}

kj)→v in C_{T}. Since
u^{0}_{n}

kj = ϕ_{−1} ϕ(u^{0}_{n}

kj)

and ϕ_{−1} is continuous, u^{0}_{n}

kj → ϕ_{−1}(v) in C_{T}. On the
other hand,u_{n}_{kj} can be written in the form

u_{n}_{kj}(t) =u_{n}_{kj}(0) +
Z _{t}

0 ϕ_{−1}

ϕ(u^{0}_{n}

kj(τ))

dτ .

Sinceu_{n}_{kj} →w asn_{k}_{j} → inC_{T}, we find thatw(t) =w(0) +R_{t}

0ϕ_{−1}(v(τ))dτ.
As the integrand is continuous, differentiating the former equation we getw^{0}=
ϕ_{−1}(v). Thusu_{n}_{kj} →winC_{T}^{1}. This ends the proof. ♦
Remark 4.2 The proof of the previous lemma is based on the ideas from [7].

Letg, h,f , Te be as in Theorem 2.4 and letz_{0}:= 2 sup_{ξ∈R}|g(ξ)|+|h(+∞)|+

|h(−∞)|+ 2 sup_{ξ∈R}|h(ξ)|+kfek_{C}. We define a functionl:R→Rby
l(z) :=

z for 0≤z≤z_{0},

z_{0} forz > z_{0}, (4.10)

forz≥0 andl(z) =−l(−z) forz <0. We also define:

Γ(p, c, F) :=

rT 3

F c

^{p−1} (4.11)

for anyp >1,c >0,F ≥0.

Lemma 4.5 Letϕbe an increasing homeomorphism ofRontoRand there exist
c, C >0 andp >1: c|z|^{p−1}≤ |ϕ(z)| ≤C |z|^{p−1}+ 1

. Let(G) be satisfied and
g be bounded. Then, for anyλ∈[0,1] and|f| ≤sup_{ξ∈R}|h(ξ)|+ sup_{ξ∈R}|g(ξ)|,
all solutions of

ϕ(u^{0})_{0}

+λg(u^{0}) +l(u) =λ(fe+f) (4.12)
subject to (1.2) ( (1.4) ) are a priori bounded by

kuk_{C}^{1} ≤(1 + 2T) Γ

p, c,kfek_{L}^{2}+√
T z_{0}

+z_{0}.

Proof We rewrite the equation (4.12) as
ϕ(u^{0})_{0}

+λg(u^{0}) =λ(fe+f)−l(u).

Then it follows from (3.3) (consider q:=λgin Lemma 3.1) that any solutionu of (4.12)–(1.2) satisfy

kϕ(u^{0})kC≤
rT

3

λfe−l(u)

L^{2} .

Taking into account the assumption|ϕ(z)| ≥c|z|^{p−1}, the following inequality
c|u^{0}|^{p−1}≤ |ϕ(u^{0})| ≤

rT 3

λfe−l(u)

L^{2}

is satisfied for anyt∈[0, T]. This implies
ku^{0}kC ≤ 1

c rT

3

λfe−l(u)

L^{2}

!_{p−1}^{1}
.

Since kλfe−l(u)k_{L}^{2} ≤ kfek_{L}^{2} +√

T z_{0} (recall that kλfek_{L}^{2} = |λ|kfek_{L}^{2}, where
λ∈[0,1] andkl(u)k_{L}^{2}≤√

Tsup_{ξ∈R}|l(ξ)|=√

T z_{0}), we get
ku^{0}kC≤Γ

p, c,kfke L^{2}+√
T z_{0}

. (4.13)

Let us split the function u as u = ue+u, where u = _{T}^{1} R_{T}

0 u(t)dt. As
keuk_{C}≤R_{T}

0 |u^{0}| ≤Tku^{0}k_{C}, from (4.13) we have
keukC≤TΓ

p, c,kfek_{L}^{2}+√
T z_{0}

. (4.14)

Now rewrite (4.12) as

ϕ(u^{0})_{0}

=λ

fe+f−g(u^{0})

−l(u)

and suppose thatu > z_{0}+TΓ

p, c,kfek_{L}^{2}+√
T z_{0}

. Then (4.14) impliesu(t)>

z_{0} for all t ∈ [0, T] and, by the definition of l, we have that l(u) = z_{0} for all
t∈[0, T]. Hence we can rewrite (4.12) as

(ϕ(u^{0}))^{0}=λ

fe+f −g(u^{0})
−z_{0}.

Integrating from 0 to T, usingg(u^{0})<sup_{ξ∈R}|g(ξ)|and f <e kfekC we get that
ϕ(u^{0}(T))< λ(kfekC+f+ sup

ξ∈R|g(ξ)|)−z_{0}

!

T+ϕ(u^{0}(0)).