Mem. Differential Equations Math. Phys. 36 (2005), 147–152
I. Kiguradze and N. Partsvania
ON MINIMAL AND MAXIMAL SOLUTIONS OF TWO–POINT
SINGULAR BOUNDARY VALUE PROBLEMS
(Reported on July 4, 2005)
We consider the differential equation
u00 = f (t, u, u0 )
(1)
with the boundary conditions
u(a+) = c1 ,
u(b−) = c2 ,
(21 )
u(a+) = c1 ,
u0 (b−) = c2 ,
(22 )
or
where −∞ < a < b < +∞, ci ∈ R (i = 1, 2), and f : ]a, b[ ×R2 → R satisfies the local
Carathéodory conditions.
In the case, where the function
fr∗ (t) = max{|f (t, x, y)| : |x| + |y| ≤ r}
is Lebesgue integrable on [a, b] for an arbitrary r > 0, problems (1), (21 ) and (1), (22 ) are
called regular, otherwise they are called singular.
The basis for the theory of regular problems of the type (1), (21 ) and (1), (22 ) were
laid in the classical works by S. N. Bernshtein [4], M. Nagumo [15] and H. Epheser [5].
From these works originates the tradition of formulation of theorems on solvability
of the above-mentioned problems in terms of so-called lower and upper functions of Eq.
(1). Precisely, these theorems contain sufficient conditions for existence of a solution of
problem (1), (21 ) or (1), (22 ), satisfying the inequalities
σ1 (t) ≤ u(t) ≤ σ2 (t) for a < t < b,
(3)
where σ1 and σ2 are, respectively, the lower and upper functions of Eq. (1) such that
σ1 (t) ≤ σ2 (t) for a < t < b.
(4)
Nowadays there exists a complete enough theory for solvability of the singular boundary value problems (1), (2k ), (3) (k = 1, 2). The results, obtained in this direction, are
contained, e.g., in [1]–[3], [6]–[14], [16]. However, the question for these problems to have
minimal and maximal solutions remains so far open. We made an attempt to fill to some
extent the existing gap.
To formulate our results, we use the following notations.
R = ] − ∞, +∞[ , R+ = [0, +∞[ .
u(t+) and u(t−) are, respectively, the right and the left limits of the function u at
the point t.
C̃ 1 ([t1 , t2 ]) is the space of functions u : [t1 , t2 ] → R which are absolutely continuous
together with their first derivatives.
L([t1 , t2 ]) is the space of Lebesgue integrable functions u : [t1 , t2 ] → R.
2000 Mathematics Subject Classification. 34B16, 34B18.
Key words and phrases. Two-point singular boundary value problem, minimal and
maximal solutions, lower and upper functions.
148
1 (I) and L
C̃loc
loc (I), where I ⊂ R is an open or a half-open interval, is the set of
functions u : I → R whose restrictions to any closed interval [t1 , t2 ] ⊂ I belong to the
class C̃ 1 ([t1 , t2 ]) and L([t1 , t2 ]), respectively.
1 (]a, b[) is said to be a solution of Eq. (1) if it satisfies this
The function u ∈ C̃loc
equation almost everywhere on ]a, b[ .
The solution u of Eq. (1), satisfying conditions (2k ) and (3), is said to be a solution
of problem (1), (2k ), (3).
The solution u (the solution u) of problem (1), (2k ), (3) is said to be a maximal
solution (a minimal solution) if an arbitrary solution u of this problem satisfies the
inequality
u(t) ≤ u(t) u(t) ≥ u(t) for a < t < b.
Following [6], let us introduce the definition.
Definition 1. A function σ : ]a, b[ → R is said to be a lower (resp. an upper)
function of Eq. (1) if:
(i) σ is locally absolutely continuous and σ 0 admits the representation
σ 0 (t) = γ(t) + γ0 (t),
where γ : ]a, b[ → R is a locally absolutely continuous function, while γ0 : ]a, b[ → R is a
non-decreasing (resp. non-increasing) function whose derivative is equal to zero almost
everywhere on ]a, b[ ;
(ii) the inequality
f t, σ(t), σ 0 (t) ≤ σ 00 (t)
resp. f t, σ(t), σ 0 (t) ≥ σ 00 (t)
holds almost everywhere on ]a, b[ .
Throughout the paper it is supposed that f (·, x, y) : ]a, b[ → R is measurable for any
(x, y) ∈ R2 and f (t, ·, ·) : R2 → R is continuous for almost all t ∈ ]a, b[ . Moreover, the
functions σ1 : ]a, b[ → R and σ2 : ]a, b[ → R are, respectively, the lower and the upper
functions of Eq. (1), satisfying condition (4).
Problem (1), (21 ), (3) is investigated under the assumptions that
and
there exist finite limits σi (a+), σi (b−) (i = 1, 2),
and c1 ∈ σ1 (a+), σ2 (a+) , c2 ∈ σ1 (b−), σ2 (b−) ,
fr∗ ∈ Lloc (]a, b[) for r > 0.
(51 )
(61 )
As for problem (1), (22 ), (3), it is investigated under the assumptions that
and
there exist finite limits σi (a+), σi0 (b−) (i = 1, 2), σ10 (b−) ≤ σ20 (b−),
and c1 ∈ σ1 (a+), σ2 (a+) , c2 ∈ σ10 (b−), σ20 (b−) ,
fr∗ (·) ∈ Lloc (]a, b]) for r > 0.
(52 )
(62 )
Definition 2. A function f belongs to the class B1 (σ1 , σ2 ) if there exist numbers
a0 ∈ ]a, b[ , b0 ∈ ]a0 , b[ , and a continuous function ρ ∈ ]a, b[ → R+ such that ρ ∈ L([a, b]),
and for any t1 ∈ ]a, a0 [ , t2 ∈ ]b0 , b[ and a continuous function η : ]a, b[→ [0, 1], an arbitrary
solution u : ]t1 , t2 [ → R of the differential equation
u00 = η(t)f (t, u, u0 ),
satisfying the condition
σ1 (t) ≤ u(t) ≤ σ2 (t) for t1 < t < t2 ,
admits the estimate
|u0 (t)| ≤ ρ(t) for t1 ≤ t ≤ t2 .
(7)
149
Definition 3. A function f belongs to the class B2 (σ1 , σ2 ) if there exist a0 ∈ ]a, b[
and a continuous function ρ ∈ ]a, b[ → R+ such that ρ ∈ L([a, b]) and for any t0 ∈ ]a, a0 [
and a continuous function η : ]a, b[→ [0, 1], an arbitrary solution u : [t0 , b] → R of the
differential equation (7), satisfying the conditions
σ1 (t) ≤ u(t) ≤ σ2 (t) for t0 ≤ t ≤ b, σ1 (b−) ≤ u0 (b−) ≤ σ2 (b−),
admits the estimate
|u0 (t)| ≤ ρ(t) for t0 ≤ t ≤ b.
Everywhere below the function ω : R → ]0, +∞[ is called the Nagumo function if it
is continuous and
+∞
Z0
Z
dy
dy
= +∞,
= +∞.
ω(y)
ω(y)
0
−∞
Theorem 1. If conditions (51 ), (61 ) hold and
f ∈ B1 (σ1 , σ2 ),
then problem (1), (21 ), (3) has a minimal and a maximal solutions.
Theorem 2. If conditions (52 ), (62 ) hold and
f ∈ B2 (σ1 , σ2 ),
then problem (1), (22 ), (3) has a minimal and a maximal solutions.
From these theorems several effective conditions for the existence of extremal solutions
of problems (1), (21 ), (3) and (1), (22 ), (3) are obtained.
In particular, the following statements are valid.
Corollary 11 . Let conditions (51 ), (61 ) hold and let there exist numbers a0 ∈ ]a, b[ ,
b0 ∈ ]a0 , b[ , a non-negative function h ∈ L([a, b]) and a Nagumo function ω such that
f (t, x, y) sgn y ≥ −ω(y) h(t) + |y| for a < t < b0 , σ1 (t) ≤ x ≤ σ2 (t), y ∈ R
and
f (t, x, y) sgn y ≤ ω(y) h(t) + |y|
for a0 < t < b, σ1 (t) ≤ x ≤ σ2 (t), y ∈ R.
Then problem (1), (21 ), (3) has a minimal and a maximal solutions.
Corollary 12 . Let conditions (52 ), (62 ) hold and let there exist a non-negative
function h ∈ L([a, b]) and a Nagumo function ω such that
f (t, x, y) sgn y ≥ −ω(y) h(t) + |y| for a < t < b, σ1 (t) ≤ x ≤ σ2 (t), y ∈ R.
Then problem (1), (22 ), (3) has a minimal and a maximal solutions.
Corollary 21 . Let conditions (51 ), (61 ) hold and let there exist numbers a0 ∈ ]a, b[ ,
b0 ∈ ]a0 , b[ , λ ∈ ]0, b − a[ , h2 ∈ R+ , and non-negative functions h0 ∈ Lloc (]a, b[) and
h1 ∈ L([a, b]) such that
Zb
(t − a)(b − t)h0 (t) dt < +∞
a
and the inequalities
f (t, x, y) sgn y ≥ −h0 (t) −
"
#
λ
+ h1 (t) |y| − h2 y 2
(t − a)(b − t)
for a < t < b0 , σ1 (t) ≤ x ≤ σ2 (t), y ∈ R
150
and
f (t, x, y) sgn y ≤ h0 (t) +
"
#
λ
+ h1 (t) |y| + h2 y 2
(t − a)(b − t)
for a0 < t < b, σ1 (t) ≤ x ≤ σ2 (t), y ∈ R
are fulfilled. Then problem (1), (21 ), (3) has a minimal and a maximal solutions.
Corollary 22 . Let conditions (52 ), (62 ) hold and let there exist numbers λ ∈ ]0, 1[ ,
h2 ∈ R+ , and non-negative functions h0 ∈ Lloc (]a, b]) and h1 ∈ L([a, b]) such that
"
#
λ
f (t, x, y) sgn y ≥ −h0 (t) −
+ h1 (t) |y| − h2 y 2
t−a
for a < t < b, σ1 (t) ≤ x ≤ σ2 (t), y ∈ R.
Let, moreover,
Zb
(t − a)h0 (t) dt < +∞.
a
Then problem (1), (22 ), (3) has a minimal and a maximal solutions.
As an example, we consider the differential equation
u00 = f0 (t, u, u0 ) + f1 (t, u, u0 )u0 ,
(10 )
where f0 : ]a, b[ ×R2 → R and f1 : ]a, b[ ×R2 → R are functions satisfying the local
Carathéodory conditions, and there exists a positive constant r0 such that
f0 (t, x, y)x ≥ 0 for a < t < b, |x| ≥ r0 , y ∈ R.
Then arbitrary constants
σ1 ∈ ] − ∞, −r0 [ and σ2 ∈ ]r0 , +∞[
are, respectively, the lower and the upper functions of Eq. (10 ). Moreover, it is obvious
that
if c1 ∈ [σ1 , σ2 ], c2 ∈ [σ1 , σ2 ] if c1 ∈ [σ1 , σ2 ] ,
then an arbitrary solution of problem (10 ), (21 ) (of problem (10 ), (22 )) admits the estimate
σ1 ≤ u(t) ≤ σ2 for a < t < b.
(30 )
Therefore, in the sequel we consider not problems (10 ), (21 ), (30 ) and (10 ), (22 ), (30 ) but
problems (10 ), (21 ) and (10 ), (22 ).
Set
∗
(·) = sup{|f0 (·, x, y)| : |x| ≤ r, y ∈ R} for r > 0 (i = 0, 1).
fir
Corollaries 21 and 22 imply the following propositions.
Corollary 31 . Let
Zb
∗
(t − a)(b − t)f0r
(t) dt < +∞,
∗
∈ Lloc (]a, b[) for r > 0
f1r
a
and there exist numbers a0 ∈ ]a, b[ , b0 ∈ ]0, b[ , continuous functions λ : R → [0, b − a[ ,
` : R → R+ and a non-negative function h ∈ L([a, b]) such that the inequalities
f1 (t, x, y) ≥ −
and
f1 (t, x, y) ≤
λ(x)
− `(x) h(t) + |y| for a < t < b0 , (x, y) ∈ R2
(t − a)(b − t)
λ(x)
+ `(x) h(t) + |y| for a0 < t < b, (x, y) ∈ R2
(t − a)(b − t)
151
are fulfilled. Then problem (10 ), (21 ) has a minimal and a maximal solutions.
Corollary 32 . Let
Zb
∗
(t − a)f0r
(t) dt < +∞,
∗
f1r
∈ Lloc (]a, b]) for r > 0
a
and there exist continuous functions λ : R → [0, 1[ and ` : R → R+ and a non-negative
function h ∈ L([a, b]) such that on ]a, b[ ×R2 the inequality
λ(x)
− `(x) h(t) + |y|
t−a
hold. Then problem (10 ), (22 ) has a minimal and a maximal solutions.
f1 (t, x, y) ≥ −
Acknowledgement
Supported by INTAS (Grant # 03-51-5007).
References
1. R. P. Agarwal and D. O’Regan, Nonlinear superlinear singular and nonsingular
second order boundary value problems. J. Differential Equations 143 (1998), No. 1,
60-95.
2. R. P. Agarwal and D. O’Regan, Second-order boundary value problems of
singular type. J. Math. Anal. Appl. 226 (1998), 414-430.
3. R. P. Agarwal and D. O’Regan, Singular differential and integral equations with
applications. Kluwer Academic Publishers, The Netherlands, 2003.
4. S. N. Bernshtein, On the equations on the calculus of variations. (Russian) Usp.
Mat. Nauk 8 (1940), No. 1, 32-74.
5. H. Epheser, Über die Existenz der Lösungen von Randwertaufgaben mit gewöhnlichen nichtlinearen Differentialgleichungen zweiter Ordnung. Mat. Z. 61 (1955), No. 4,
435-454.
6. I. Kiguradze, On some singular boundary value problems for nonlinear second
order ordinary differential equations. (Russian) Differentsial’nye Uravneniya 4 (1968),
No. 10, 1753-1773; English transl.: Differ. Equations 4 (1968), 901-910.
7. I. Kiguradze, On a singular two-point boundary value problem. (Russian) Differentsial’nye Uravneniya 5 (1969), No. 11, 2002-2016; English transl.: Differ. Equations
5 (1969), 1493-1504.
8. I. Kiguradze, On a singular boundary value problem. J. Math. Anal. Appl. 30
(1970), No. 3, 475-489.
9. I. Kiguradze, Some singular boundary value problems for ordinary differential
equations. (Russian) Tbilisi University Press, Tbilisi, 1975.
10. I. Kiguradze, Some optimal conditions for the solvability of two-point singular
boundary value problems. Funct. Differ. Equ. 10 (2003), No. 1-2, 259-281.
11. I. Kiguradze and B. Půža, On two-point boundary value problems for second
order singular functional differential equations. Funct. Differ. Equ. 12 (2005), No. 3-4,
271-294.
12. I. Kiguradze, B. Půža, and I. P. Stavroulakis, On singular boundary value
problems for functional differential equations of higher order. Georgian Math. J. 8
(2001), No. 4, 791-814.
13. I. T. Kiguradze and B. L. Shekhter, Singular boundary value problems for
second order ordinary differential equations. (Russian) Itogi Nauki i Tekhniki 30 (1987),
105-201; English transl.: J. Sov. Math. 43 (1988), No. 2, 2340-2417.
152
14. A. G. Lomtatidze, On positive solutions of boundary value problems for second order ordinary differential equations with singularities. (Russian) Differentsial’nye
Uravneniya 23 (1987), No. 10, 1685-1692.
15. M. Nagumo, Über die Differentialgleichung y 00 = f (x, y, y 0 ). Proc. Phys.-Math.
Soc. Japan 19 (1937), 861-866.
16. N. I. Vasil’ev and Yu. A. Klokov, Foundations of the theory of boundary-value
problems for ordinary differential equations. (Russian) Zinatne, Riga, 1978.
Authors’ address:
A. Razmadze Mathematical Institute
Georgian Academy of Sciences
1, M. Aleksidze St., Tbilisi 0193
Georgia