Kuo, Chung-Cheng

Osaka City University Repository

Kuo, C.-C.
Osaka J. Math.
37 (2000), 345-353
PERIODIC SOLUTIONS OF A TWO-POINT
BOUNDARY VALUE PROBLEM AT RESONANCE
CHUNG-CHENG KUO*
(Received May 12, 1997)
1. Introduction
We consider the periodic boundary value problem
u + f(u)u + g(x, u) = h in (0, 2τr),
w(0) - u(2π) = α'(0) - u(2π) = 0,
where h e L!(0, 2π) is given, / : R -» R is a continuous function and g : (0, 2ττ) x
R -> R is a Caratheodory function. That is, g(x, u) is continuous in u e R for a.e.
x e (0, 2τr), is measurable in x e (0, 2ττ) for all u G R and satisfies for each r > 0,
there exists a
r
 € Ll(Q, 2π) such that
(1.2) I*(*,ι0l<β
r
(jc)
for a.e. c G (0, 2π) and all \u\ < r. Concerning the growth condition of the nonlinear
term g, we assume that either
(H) There exist a constant r0 > 0, and a, b,c,d G Ll(Q, 2π), a, b > 0 and α(jc) <
1 for a.e. x € (0, 2τr) with strict inequality on a positive measurable subset of
(0, 2τr), such that for a.e. x e (0, 2τr) and all M > r0
ΦO < g(x, u) < a(x)\u\
and for a.e. x e (0, 2π) and all u < —ΓQ
-a(x)\u\-b(x)<g(x,u)<d(x);
or
(G) There exist a constant r0 > 0, and α, b, c, d e L^O, 2τr), α, Z? > 0 and α(jc) <
1/4 for a.e. c € (0, 2ττ) with strict inequality on a positive measurable subset of
(0, 2π), such that for a.e. x e (0, 2π) and all M > r0
*Research supported in part by the National Science Council of R.O.C.
346 K. CHUNG-CHENG
and for a.e. x G (0,2π) and all u < —ΓQ
-a(x)\u\-b(x)<g(x,u)<d(x)',
holds. The solvability of the problem (1.1) has been extensively studied if the nonlin-
earity g is assumed to have either linear growth in u as \u\ -> oo (see [2,3,4,5,7,8]);
or superlinear growth in u in one of directions u —>• oo and u —> — oo, and may
be bounded in the other (see [6,9]). The purpose of this paper is to give solvability
theorems to (1.1) when g is allowed to grow superlinearly in u in one of directions
u -> oo and u -> — oo, and may grow linearly in the other. An example is given in
[7] shows that our results are almost sharp and still are new. Based on the well-known
Leray-Schauder continuation method (see [1]), we obtain solvability results under as-
sumptions either with or without a Landesman-Lazer condition (see (2.2) below).
In the following we shall make use of real Banach spaces Lp(0, 2τr), C[0, 2π],
and Sobolev spaces W2Λ(Q,2π) and Hl(Q,2π). Norms of L^(0,2π), C[0, 2π] and
Hl(Q,2π) are denoted by | | M | | L A > , I I " l i e and ||u||#ι, respectively. By a solution of (1.1),
we mean a function u e W2Λ(Q, 2π) satisfies the differential equation in (1.1) a.e.
x e (0, 2π).
2. Existence Theorems
For v G W2Λ(Q, 2π), we write v = (2π)~l /Q27Γ v(x)dx, ϋ = υ -v and so
fQπ v(x)dx = 0. We now state in the following lemmas, their proofs can be obtained
in [7], Lemmas 2,3 and 4, and so are omitted.
Lemma 1. Let m be a nonnegative function in L](0, 2ττ) and for a.e. x G
(0, 2τr), m(x) < 1 with strict inequality on a positive measurable subset of (0, 2τr),
then there exists a constant K\(ni) > 0 such that
L2π (u - u(x))(u"(x) + f(u(x))u(x) + p(x)u(x))dx
whenever p e L](0, 2π) with p(x) < m(x) for a.e. x e (0, 2π), u e W2Λ(Q, 2π) with
w(0) — u(2π) = ι/(0) — u '(2π) = 0 and f : R -^  R is a continuous function.
Lemma 2. Let p : R -> R be a periodic function of period 2π in Ll(Q,2π)
such that p(x) < 1/4 for a.e. x G (0, 2τr) with strict inequality on a positive measur-
able subset of (0, 2ττ). Then there exists a constant K2(p) > 0 such that
jco+2τr
for all υ e HI(XQ, XQ + 2π) with V(XQ) = υ(jc0 + 2τr) = 0 and XQ e R
TWO-POINT BOUNDARY VALUE PROBLEM 347
Lemma 3. There exists a constant K^ > 0 such that
\\U\\Hι <K3\\U+f(u)u'\\Li
for each continuous function f : R -> R and u e VΓ2>1(0, 2π) with
w(0) - w(2τr) = ι/(0) - ιι'(2π) = 0.
Theorem 1. Lef g : (0, 2π) x R -> R be a Caratheodory function satisfying
(H), then for each h e L](0, 2π) ί/ze problem (1.1) Λα^ 0 solution u e W2 l(Q, 2π),
provided that one of the following conditions holds:
(2.1) / d(x)dx < I h(x)dx < I c(x)dx\
Jo Jo Jo
flπ / 2π / 2π
(2.2) f π g°_(x)dx < f π h(x)dx < f π g°(x)dx
Jo Jo Jo
(2.3) I d(x)dx < f h(x)dx < I g°(x)dx;
Jo Jo Jo
(2.4) I g*_(x)dx < I h(x)dx < I c(x)dx\
Jo Jo Jo
where g%(x) = lim infM^oo g(x, «) and gQ_(x) = lim supM_>_00 g(x, u).
Proof. Let a e R be fixed, 0 < α < 1. We consider the boundary value problems
u" + tf(u)u + (1 - t)au + tg(x, u) = th in (0, 2π),
w(0) - u(2π) = u(0) - u'(2π) = 0
for 0 < t < 1, which becomes the original problem when t = 1. Since 0 < a < 1,
(2.5) has only a trivial solution when t = 0. To apply the Leray-Schauder continuation
method, it suffices to show that solutions to (2.5) for 0 < t < 1 have an a priori
bound in Hl(0, 2π). To this end, let θ : R -> R be a continuous function such that
for u € R, 0 < θ(u) < 1, θ(u) = 0 for \u\ < r0, and θ(u) = 1 for \u\ > 2r0. We define
e(x) =
ί min{g(Λ:, u) 4- ^(jc), a(jc)w}^(w) if u > 0
= {[ max{g(jc, M) - e(x), a(x)u}θ(u) if M < 0
and gι(x, u) = g(x, u) - g\(x, u). Then g\, g2 : (0, 2τr) x R -> R are Caratheodory
functions such that for a.e. x e (0, 2π) and u e R, M ^0
(2.6) 0 < < «(*),
348 K. CHUNG-CHENG
and for a.e. x € (0, 2ττ) and u e R
(2.7) \g2(x, ιι)| < e(x).
If u is a possible solution to (2.5) for some 0 < t < 1, then using (2.6), (2.7) and
Lemma 1, we have
2π
[(ΰ - u(x))(u"(x) + ί/(ιι(jc))ιι'(jc) + (1 - t)ctu(x)
, u(x)) + tg2(x, u(x)) - th(x))]dx
r
2π
= / [(β - ϋ(x))(u (jc) + tf(u(x))u (x) + pt(u(x))u(x)Jo
+tg2(x, u(x)) - th(x))]dx
> κ{(m)\\u\\2H> - (lkllί.1 + l l * l l L « X | β | + Pile)
if we set m(x) = max{α, a(x)} and for M(JC) ^0, pt(u(x)) = (1 — t)ct + tg\(x, u(x))/u(x),
and pt(u(x)) = (1 — t)a for M(JC) = 0. Consequently,
(2.8) Hfllβ, <|ί(|i/|
 + ||δ||^)
ΛI
for some constant C\ > 0 independent of u. It remains to show that solutions to (2.5)
for 0 < t < 1 have an a priori bound in //!(0, 2τr). We argue by contradiction, and
suppose that there exist a sequence {u
n
} in W2Λ(Q, 2π) and a corresponding sequence
{t
n
} in (0, 1) such that u
n
 is a solution to (2.5) with t = /„ and ||w
π
| |//ι > w for all n.
Let υ
n
 = u
n
/\\u
n
\\H\, then |K||//ι = 1 for all n € N, and by (2.8) we have \\vn\\H\ -> 0
as n -> oo. Because ||ϋ
n
| |//ι < ||υ
n
| |//ι + | |ft
π
| |//ι is bounded, we may assume with-
out loss of generality that {£>„} converges to β in R. In this case, we have v
n
 -> β
in Hl(Q,2π) with β ^ 0 because ||υ
n
||#ι for all e N. We consider only the case
β > 0, for the case β < 0 can be treated similarly. Using the compact imbedding
of Hl(Q, 2π) -> C[0, 2π], we have υ
n
 -> ^ in C[0, 2π], and so there exists n0 e N
such that υ
n
 > β/2 on [0, 2π] for all n > n$. Hence, we may assume that ιι
n
 > ΓQ on
[0, 2τr] independent of n. Clearly, u
n
(x) -> oo for each x e [0, π]. Integrating (2.5)
over (0, 2ττ) when u = u
n
 and t = ί
n
, we have
TWO-POINT BOUNDARY VALUE PROBLEM 349
Γ
2π
 Γ
2π
 Γ
2π
I g(x, u
n
(x))dx < (1 - t
n
)a I u
n
(x)dx + t
n
 I g(x, u
n
(x))dx
Jo Jo Jo
='J2Jo
<2 9>
h(x)dx.
We first assume that h satisfies either (2.2) or (2.3). It follows from (H) and the
fact that u
n
 > ΓQ on [0, 2π] independent of n, g(x, u
n
) is bounded from below by
a function in L[(0, 2π) independent of n. Applying Fatou's lemma to the inequality
/0
2π
 g(x, u
n
(x))dx < /027Γ h(x)dx, we have /02π g+(x)dx < f*π h(x)dx, which contra-
dicts the second inequality in either (2.2) or (2.3). In order, we assume that h satisfies
(2.1) or (2.4). It follows from (H), (2.9) and the fact that u
n
 > r0 on [0, TT] indepen-
dent of n, we have
ί c(x)dx<[ g(x,u
n
(x))dx< ί h(x)dx,
Jo Jo Jo
which contradicts the second inequality in either (2.1) or (2.4). This completes the
proof of the theorem.
In order, to obtain an existence theorem to (1.1) in which the condition (H) is
replaced by (G). An example is given in [7] shows that the condition (G) in the fol-
lowing theorem is almost sharp, that is, when the number 1/4 in (G) is replaced by
any number larger than 1/4, the assertion is false. D
Theorem 2. Let g : (0, 2π) x R -> R be a Caratheodory function satisfying
(G). Then for each h e L^O, 2π) the problem (1.1) has a solution u G W2'!(0, 2π),
provided that one of conditions (2.1), (2.2), (2.3) and (2.4) holds.
Proof. Let α e R be fixed, 0 < α < 1/4. We consider the boundary value prob-
lems (2.5) for 0 < t < 1, and define
,
 Λg(jc, u) = {
[ ma\{g(x, u) - e(x), α(x)u}θ(u) if u < 0
and £2 = g — £ι» where θ : R -> R is a continuous function and e e L!(0, 2π) which
are defined as in the proof of Theorem 1. It follows that for a.e. x e (0, 2ττ) and all
u € R
(2.10) 0 < #!(*, u)u and \g2(x, u)\ < φc),
and for a.e. x e (0, 2π) and all u < 0
(2.11) -α(x)\u\<gl(x,u)<0.
350 K. CHUNG-CHENG
As in the proof of Theorem 1, it suffices to show that solutions to (2.5) for 0 <
t < 1 have an a priori bound in Hl(0,2π). Indeed, if u is a possible solution to
(2.5), then we can write « as u = u+ — «~, where «+ = max{w,0} and u~ =
max{— «,0}. We will first show that u~ is bounded in //!(0, π). To this end, let us
extend «(*), a(x), g\(x, u), g2(x, u) and h(x) 2π -periodically in x to all of R, and
then using the same notations for the periodic extensions as for the original functions.
Then u : R -> R is a periodic solution of period 2π to (2.5). Clearly, | |M~||#I =
\\u~\\ H*(x,x+2π) fc>Γ all * € R. If first suppose that u~ has a zero in [0, 2π] and set
v(x) = -u~(x) for all x e R. Let JCG = max{jc|w~(jc) = 0, 0 < x < 2τr}, and let [c0, dQ]
be a component of the support of u~ in [JCQ, XQ + 2π]. Then
v" + tf(υ)υ + (1 - t)av + tgι(x, v) + tg2(x, v) = th in (c0, d0),
Multiplying each side of (2.12) by —υ and integrating over [co»^o]» and then using
V(CQ) = v(do) = 0 and the decomposition of g, g = g\ + #2, we obtain
° - [(I - t)<x + ta(x)]v2(x)}dx
(2.13) < Γ {(„'(,))« - [(1 -
Λo I L
-t Γ
JCQ
tg2(x, v(x»v(x)dx t h(x)v(x)dx.
Thus (2.13) holds for all components of the support of v in [JCQ, XQ + 2π], and also
holds for the complement in [JCQ, XQ + 2π] of the support of v because υ is identically
zero on that set. In this case, we obtain
x0+2π
{(ι/(jc))2 — p(x)v2(x)}dxL
< I {(v'OO)2 - [(1 - t)a + ta(x)]υ2(x)}dx
J*o
for some constant €2 > 0 independent of w, where /?(*) = max{α, a(x)}. Combining
(2.14) with Lemma 2, we have
K2\\u-\\2Hι = K2\\υ\\
2
Hl = K2\\v\\
2
Hl(xo^^ < C2||ι|-|^ι.
Consequently, there exists a constant Ca > 0 such that for each 0 < f < 1, we have
(2.15) l l iT l l t f , <C3
TWO-POINT BOUNDARY VALUE PROBLEM 351
for all possible solutions u to (2.5) with zeros in [0,2π]. In order, we consider the
case that u~ has no zero in [0, 2π]. In this case, we have u = — u~ and u < 0 on
[0, 2π]. Since a(x) < 1/4 < 1 and u is a periodic solution of period 2π to (2.5), one
may now proceed as in the proof of Theorem 1 to show the existence of a bound on
H W | | / / ' = l l«~ΊI//' Integrating (2.5) over [0, 2τr], we have
r
2π
 r
2
" r
2
"(2.16) (1 - t)a I u(x)dx +1 I g(x, u(x))dx = / / h(x)dx,
Jo Jo Jo
and then using g = g\ + g2, the boundedness of w~, the boundary condition of (2.5)
and (2.16), we also have
(1 - t)a I u(x)dx +1 I g\(x, u(x))dx
Ju(x)>0 Ju(x)>0
= t I h(x)dx — t I g\(x,u(x))dx
Jθ Ju(x)<0(2.17)
 f2π
-t I g2(x, u(x))dx - (1 - t)a I u(x)dx
Jθ Ju(x)<0
i l l w ' l l c + Ikllu +2πα||ιι-||c
< C4
for some constant €4 > 0 independent of u. It follows that there exists a constant
C5 > 0 such that
f
/
Jo
(2.18)  |(1 - f)αιι(jc) + tg(x, «(*)) - th(x)\dx < C5
for all possible solutions u to (2.5) and 0 < t < 1. But
u" + tf(u)u(2.19)
= -(1 - t)au - tg(x, u) + th in (0, 2π).
Hence, from (2.18), (2.19) and Lemma 3 that there exists a constant C^ > 0 such that
(2.20) H U H * . <*3C5<C6
for all possible solutions u to (2.5) and 0 < t < 1. We may now use (2.20) as (2.8) in
the proof of Theorem 1 to show the existence | |M||WI < Cη for some constant C-j > 0
independent of u. Hence the proof of this theorem is complete. D
By slightly modifying proofs of Theorems 1 and 2, we can obtain new solvability
conditions in which the nonlinearity g satisfies the following condition:
352 K. CHUNG-CHENG
(F) There exist constants ?£, γ, 8 > 0 and ΣΓ, d e Ll(Q, 2π) such that for a.e. x €
(0, 2π) and all u > f^
g(x,u)u>c(x)\u\l-y,
and for a.e. x € (0, 2π) and all u < — fζ
and conditions (2.1)-(2.4) may be replaced by one of the following conditions:
(2.21) / * gδ_(x)dx < I π h(x)dx = 0 < / π gl(x)dχ-,
Jo Jo Jo
(2.22) f gδ_(x)dx<0=[ h(x)dx<[ c(x)dx;
Jo Jo Jo
(2.23) / * d(x)dx < f " h(x)dx = 0 < /* * βϊ(*)Λc;
Jo Jo Jo
where g8_(x) - limsuρM^_00 g(x, u)\u\δ and g$(x) = liminfα-.oo g(x, u)\u\Y .
Theorem 3. Let g : (0, 2π) x R -> R be a Caratheodory function satisfy-
ing (F) and (H), then for each h e Ll(Q,2π) the problem (1.1) /ww a solution u 6
W2J(0, 2π\ provided that one of (2.21)-(2.23) holds.
Proof. In proving Theorem 1, conditions (2.1)-(2.4) are used only to produce
contradiction in the final part of the proof. Thus we can proceed exactly the same way
as in the proof of Theorem 1 up to the point where β > 0 is considered and (2.9) is
satisfied. In this case, we may assume similarly that u
n
 > rj on [0, 2π] independent of
n. It follows from (F) and the fact that υ
n
 > β/2 on [0, 2π] for all n > WQ* we have
( β\-y2J
for a.e. x € (0, 2π) and all n > n$. Applying Fatou's lemma to the left hand side of
the inequality
\\U
n
\\YHl f π
Jo
* h(X)dx =0,
o
we have
β-
y
 I * gl(x)dx < 0,
Jo
TWO-POINT BOUNDARY VALUE PROBLEM 353
or equivalently
which contradicts the second inequality in either (2.21) or (2.23). Hence the proof is
complete. D
Similarly we can obtain the following theorem in which the condition (H) may be
replaced by (G).
Theorem 4. Let g : (0, 2π) x R -» R be a Caratheodory function satisfy-
ing (F) and (G), then for each h e L^O, 2π) the problem (1.1) has a solution u e
W2Λ(Q, 2π), provided that one 0/(2.21)-(2.23) holds.
References
[1] K. Deimling: Nonlinear Functional Analysis, Springer-Verlag, New York, 1985.
[2] P. Drabek and S. Invernizzi: On the periodic BVP for the forced duffing equation with jumping
nonlinearity, Nonlinear Anal. 10 (1986), 643-650.
[3] C.P. Gupta, J.J. Nieto and L. Sanchez: Periodic solutions of some lineard and duffing equationsr,
J. Math. Anal. Appl. 140 (1989), 67-82.
[4] M.P. Gossez and P. Oman: Nonresonance with respect to the Fucik spectrum for periodic so-
lutions of second order ordinary differential equations, Nonlinear Anal. 14 (1990), 1079-1104.
[5] R. lannacci and M.N. Nkashama: Nonlinear two point boundary value problems at resonance
without Landesman-Lazer condition, Proc. Amer. Math. Soc. 106 (1989), 943-952.
[6] R. Kannan and R. Ortega: Periodic solutions of pendulum-type equations, J. Differential Equa-
tions, 59 (1985), 123-144.
[7] J. Mawhin and J.R. Ward: Periodic solutions of some forced Lineard differential equations at
resonance, Arch. Math. 41 (1983), 337-351.
[8] M. Martelli and J.D. Schuur: Periodic solutions of linear type second order ordinary differential
equations, Tδhoku Math. J. 32 (1980), 201-207.
[9] J.R. Ward: Periodic solutions for systems of second order ordinary differential equations, J.
Math. Anal. Appl. 81 (1981), 92-98.
Department of Mathematics
Fu Jen University
Taipei, Taiwan
e-mail: cckuo@math.fju.edu.tw



Periodic solutions of a two-point boundary value problem at resonance

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Periodic solutions of a two-point boundary value problem at resonance

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