Construction of lower and upper solutions for first-order periodic problem
Ma and Zhang Boundary Value Problems
Construction of lower and upper solutions for first-order periodic problem
Ruyun Ma 0 1 2
Lu Zhang 0 1 2
0 Lanzhou , 730070 , P.R. China
1 Northwest Normal University
2 Department of Mathematics
In this paper, we construct nonconstant lower and upper solutions for the periodic boundary value problem x + f (t, x) = e(t), x(0) = x(T ) and find their estimates. We prove the existence of positive solutions for the singular problem x + g(x) = e(t), x(0) = x(T ) by using these results.
solutions; periodic boundary value problem; singular problem; lower and upper
1 Introduction
The method of lower and upper solutions is an elementary but powerful tool in the
existence theory of solutions to initial value problems and for periodic boundary value
problems, even in cases where no special structure is assumed on the nonlinearity. Starting with
the pioneering work of Moretto [] for locally Lipschitzian ordinary differential equations,
the method of upper and lower solutions for the periodic boundary value problem
x + f (t, x) = ,
x() = x(T )
has been extended to the case of a continuous right-hand side f : [, T ] × R
In Franco et al. [], the method of lower and upper solutions was applied to the periodic
x = f (t, x),
x() = x(T ),
and they also obtained a similar result to Lemma . below.
In order to apply these results, finding upper and lower solutions is very important. But
the problem of construction of lower and upper solutions has been solved very rarely (see,
e.g., [, ]). In this paper we fill this gap and present conditions ensuring the existence of
nonconstant lower and upper solutions to the first-order periodic boundary value problem
(.) and find their estimates. This enables us to prove the existence result for the periodic
problem with strong singularity.
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In order to explain the main results of the paper, let us introduce some notation: C[, T ]
stands for the set of functions continuous on [, T ]. For x ∈ C[, T ], we denote
T
x =
E =
E± =
If x ∈ C[, T ], then we write x+(t) = max{x(t), } and x–(t) = max{–x(t), }. For e ∈
C[, T ], we put
and note that E = E+ – E–.
Let us recall first some classical definitions and results. They are taken from [].
Definition . A lower solution α (respectively, an upper solution β) of problem (.) is a
function α ∈ C[, T ] such that
The basic existence theorem of the method of upper and lower solutions for (.) can be
stated as follows.
Lemma . If problem (.) has a lower solution α and upper solution β such that α(t) ≤
β(t) for all t ∈ [, T ] (resp. β(t) ≤ α(t) for all t ∈ [, T ]), then problem (.) has at least
one solution x such that α(t) ≤ x(t) ≤ β(t) for all t ∈ [, T ] (resp. β(t) ≤ x(t) ≤ α(t) for all
t ∈ [, T ]).
The paper is organized as follows. In Section we develop a method to construct lower
and upper solutions for (.). As the application, in Section , we establish an existence
result for a nonlinear first-order periodic problem with strong singularity.
2 Construction of lower and upper solutions
In this section we consider
x + f (t, x) = e(t),
x() = x(T ),
where f : [, T ] × R → R and e : [, T ] → R are continuous.
Let us consider an auxiliary boundary value problem
R
where δ ∈ C[, T ] satisfies δ¯ = and ζ ∈ .
f (t, x) ≤ c(t) for all t ∈ [, T ] and x ∈ B, B + η ,
where η(t) = φ–(t) – φ+(t) –+ and φ = c – e. If
c¯ – e¯ ≤ ,
B ≤ β ≤ B + η in [, T ].
Taking x = + and
ω (t) + φ+(t) – – φ–(t) + = ,
it is easy to see that β() = β(T ). Since | – t x[φ+(s) – – φ–(s) +] ds| = | η(s) ds| ≤
t
η , choosing ζ = B + η , we obtain
which means that (.) holds.
Now, using (.), it follows that + ≤
–, implying that
β (t) = xω (t) = –x φ+(s) – – φ–(s) + ≤ –φ(t) = e(t) – c(t).
From (.) and (.) we deduce that
and the proof is completed.
β (t) + f (t, β) ≤ β (t) + c(t) ≤ e(t) for all t ∈ [, T ],
Using similar arguments, one can prove the following proposition.
f (t, x) ≥ c(t) for all t ∈ [, T ] and x ∈ B, B + η ,
where η(t) = φ–(t) – φ+(t) –+ and φ = c – e. If
c¯ – e¯ ≥ ,
B ≤ α ≤ B + η in [, T ].
Theorem . and Theorem . are simple examples of existence results which follow
immediately from Lemma ., Propositions . and ..
(–)i+f (t, x) ≤ (–)i+ci(t),
(–)i+(c¯i – e¯) ≤
for all t ∈ [, T ], x ∈ [Bi, Bi + ηi ] and all i ∈ {, },
Then problem (.) possesses a solution x such that
Theorem . Assume that there are B, B ∈ R, c, c ∈ C[, T ] such that the assumptions
of Theorem . (...truncated)