Existence of solutions for first order ordinary
differential equations with nonlinear boundary
conditions
Daniel Franco∗
Departamento de Matemática Aplicada
Universidad Nacional de Educación a Distancia
Apartado de Correos 60149. Madrid 28080, Spain
Juan J. Nieto
Departamento de Análisis Matemático. Facultad de Matemáticas
Universidad de Santiago de Compostela. Santiago de Compostela 15782, Spain
and
Donal O’Regan
Department of Mathematics. National University of Ireland
Galway, Ireland
Running Title: Nonlinear boundary conditions for ODEs.
∗ This
is the preprint version of the paper published in Applied Mathematics and Compu-
tation 153 (2004) 793-802.
1
Applied Mathematics and Computation 153 (2004) 793-802.
2
Abstract : We present an existence theorem for nonlinear ordinary differential
equations of first order with nonlinear boundary conditions. The result includes,
for instance, the initial value problem, the final value problem, and the antiperiodic boundary value problem. The main novelty of the method lies in that it
unifies different techniques for initial or boundary conditions.
Keywords: Nonlinear ordinary differential equation, initial value problem, antiperiodic solutions, nonlinear boundary conditions, upper and lower solutions.
1
Introduction
We are interested in solutions of the nonlinear equation
u0 (t) = F (t, u(t)) , t ∈ I = [0, T ] , T > 0
(1)
satisfying the condition
g(u(0), u(T )) = 0 ,
(2)
where F : [0, T ] × R → R and g : R2 → R are continuous functions.
If g(x, y) = x − c with c ∈ R , then (2) is the following initial condition
u(0) = c .
(3)
Similarly, if g(x, y) = y − c , then (2) is the final condition
u(T ) = c .
(4)
Finally, the antiperiodic boundary condition
u(0) = −u(T ) .
corresponds to the case g(x, y) = x + y .
(5)
Applied Mathematics and Computation 153 (2004) 793-802.
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Initial value problems are well-known and there is no doubt about their
importance. On the other hand, anti-periodic solutions are perhaps not as
widely discussed but there are some papers on this subject: First order ordinary
differential equations [1, 2, 3, 4, 5, 6], second order ordinary differential equations
[7, 8], partial differential equations and abstract differential equations [9, 10, 11,
12], antiperiodic trigonometric polynomials [13], anti-periodic wavelets [14] and
periodic quasiwavelets [15], and control problems [16]. In physics, antiperiodic
boundary conditions are considered, for instance, in [17, 18, 19, 20]. Some other
physics’ works are listed in [3].
Also, extending by symmetry the nonlinear function F to [−T, T ] × R , any
T −antiperiodic solution of (1)–(5) provides a solution of equation (1) satisfying
the 2T − periodic boundary condition
u(−T ) = u(T ) .
It is possible to define the concept of lower solution and upper solution for
equation (1) as follows.
Definition 1 We say that a function α ∈ C 1 (I) is a subsolution of equation
(1) if
α0 (t) ≤ F (t, α(t)) , t ∈ I .
(6)
Analogously, we say that β ∈ C 1 (I) is a supersolution of (1) if
β 0 (t) ≥ F (t, β(t)) , t ∈ I .
(7)
In what follows we shall assume that
α(t) ≤ β(t) , t ∈ I ,
and define the set
[α, β] = {v ∈ C(I) : α(t) ≤ v(t) ≤ β(t) , t ∈ I } .
(8)
Applied Mathematics and Computation 153 (2004) 793-802.
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Of course, to obtain a solution satisfying some initial or boundary condition and
lying between a subsolution and a supersolution we need additional conditions.
For instance, in the case of initial conditions (3), one requires
α(0) ≤ c ≤ β(0) ,
(9)
and we have the following well-known existence result.
Theorem 1 (Th. 1.2.1, [21]) Suppose that α , β ∈ C 1 (I) are subsolution
and supersolution of (1) respectively satisfying (8) and (9). Then there exists
a solution u ∈ C 1 (I) of the initial value problem (1)-(3) such that u ∈ [α, β] .
Moreover, if there exists M > 0 such that
F (t, v) + M v ≤ F (t, u) + M u , t ∈ I , α(t) ≤ v ≤ u ≤ β(t) ,
(10)
then there exist minimal and maximal solutions of the initial problem in [α, β] .
However, the situation is totally different for the antiperiodic boundary conditions (5) and the requirement is now (see [3])
α(0) ≤ −β(T ) , β(0) ≥ −α(T ) .
(11)
The purpose of this paper is to present a new existence result for the equation
(1) with the nonlinear boundary condition (2) that includes, among others, the
case of initial value condition (3), the final condition (4) and the antiperiodic
boundary condition (5). To this end, we introduce a new concept of coupled
lower and upper solutions that allow us to obtain a pair of quasisolutions in
[α, β] . Then, under appropriate hypotheses, we show that there exists a solution
in the sector [α, β] . We point out that our method, being new, unifies the
treatment of different problems.
2
Coupled Lower and Upper Solutions
To cover different possibilities for the nonlinear boundary function g we introduce the following concept.
Applied Mathematics and Computation 153 (2004) 793-802.
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Definition 2 We say that α , β ∈ C 1 (I) are coupled lower and upper solutions
for the problem (1)-(2) if α is a subsolution and β a supersolution for the equation (1), condition (8) holds, and
g(α(0), β(T )) ≤ 0 ≤ g(β(0), α(T )) .
(12)
Definition 3 We say that v , w ∈ C 1 (I) are coupled quasisolutions for the
problem (1)-(2) if v and w are solutions of the equation (1),
α(t) ≤ v(t) ≤ w(t) ≤ β(t) , t ∈ I
(13)
g(v(0), w(T )) = 0 = g(w(0), v(T )) .
(14)
and
Theorem 2 Assume that α , β are coupled lower and upper solutions for the
problem (1)-(2) such that (10) holds. In addition, suppose that there exists
m > 0 such that for any x, x0 ∈ [α(0), β(0)] with x ≤ x0 and y, y 0 ∈ [α(T ), β(T )]
with y ≤ y 0 one has that
g(x0 , y) − mx0 ≤ g(x, y) − mx ,
(15)
g(x, y) ≤ g(x, y 0 ) .
(16)
and
Then there exist v , w coupled quasisolutions of the problem (1)-(2).
Proof
For η , ξ ∈ [α, β] , consider the functions ϕ(t) = max {η(t), ξ(t)} and
φ(t) = min {η(t), ξ(t)} and the initial value problems
v 0 (t) + M v(t) = F (t, φ(t)) + M φ(t) , t ∈ I ; v(0) = φ(0) −
g(φ(0), ϕ(T ))
, (17)
m
w0 (t)+M w(t) = F (t, ϕ(t))+M ϕ(t) , t ∈ I ; w(0) = ϕ(0)−
g(ϕ(0), φ(T ))
. (18)
m
These problems have a unique solution since they are linear. Therefore, we can
define the operator
B : [α, β] × [α, β] ⊂ C(I) × C(I) → C(I) × C(I) , B(η, ξ) = (v, w) ,
(19)
Applied Mathematics and Computation 153 (2004) 793-802.
6
where v , w are the solutions of (17)-(18). It is easy to see that B is compact
by a direct application of Ascoli-Arzelà’s theorem.
We now show that (13) is valid. Indeed, let z = v − α . Taking into account
(10), we have for any t ∈ I that
z 0 (t) + M z(t) ≥ F (t, φ(t)) + M φ(t) − F (t, α(t)) − M α(t) ≥ 0 .
Now, using conditions (12), (15) and (16) we get
z(0) = φ(0) −
g(φ(0), β(T ))
g(φ(0), ϕ(T ))
− α(0) ≥ φ(0) − α(0) −
=
m
m
g(α(0), β(T )) − g(φ(0), β(T )) g(α(0), β(T ))
−
≥
m
m
−m[φ(0) − α(0)]
= 0.
φ(0) − α(0) +
m
φ(0) − α(0) +
Thus, we conclude that z ≥ 0 on I , and α ≤ v on I . Analogously, one can show
that v ≤ β on I and that w ∈ [α, β] .
To show that v ≤ w on I , set z = w − v so that
z 0 (t) + M z(t) = F (t, ϕ(t)) + M ϕ(t) − F (t, φ(t)) − M φ(t) ≥ 0 , t ∈ I ,
and
g(φ(0), ϕ(T )) − g(ϕ(0), φ(T ))
=
m
g(φ(0), ϕ(T )) − g(φ(0), φ(T )) + g(φ(0), φ(T )) − g(ϕ(0), φ(T ))
ϕ(0) − φ(0) +
≥
m
−m[ϕ(0) − φ(0)] g(φ(0), ϕ(T )) − g(φ(0), φ(T ))
ϕ(0) − φ(0) +
+
≥ 0.
m
m
z(0) = ϕ(0) − φ(0) +
This allows us to conclude that z ≥ 0 on I and that v ≤ w on I .
Hence,
B : [α, β] × [α, β] → [α, β] × [α, β]
is continuous and compact and, by Schauder’s fixed point theorem, B has a
fixed point, i.e., there exist (v, w) ∈ [α, β] × [α, β] such that
B(v, w) = (v, w)
Applied Mathematics and Computation 153 (2004) 793-802.
7
and v ≤ w (by an argument similar to the one above).
Now, by (19) we have that v , w satisfy
v 0 (t) + M v(t) = F (t, v(t)) + M v(t) , t ∈ I ; v(0) = v(0) −
g(v(0), w(T ))
,
m
and
w0 (t) + M w(t) = F (t, w(t)) + M w(t) , t ∈ I ; w(0) = w(0) −
g(w(0), v(T ))
.
m
As a result, v and w are coupled quasisolutions of (1)-(2). This concludes the
t
u
proof.
If v = w , then u = v = w is a solution of the problem (1)-(2).
Condition (15) means that g(x, y) − mx is monotone nonincreasing in x ∈
[α(0), β(0)] . The condition (16) means g is nondecreasing in the second variable.
In the case that g is decreasing in y , we could change g by −g obtaining the same
boundary condition and (16) would be valid. This is the case of the periodic
boundary conditions
u(0) = u(T ) ,
(20)
corresponding to g(x, y) = x − y . This does not verify (16) and one would be
tempted to write g(x, y) = −x + y , but unfortunately we have problems as we
will see in section 6.
We also observe that, for example, the function g corresponding to the case
of initial condition, and antiperiodic boundary conditions satisfy (15) and (16).
3
Initial and Final Value Problems
Now consider the following condition of initial type
h(u(0)) = 0
(21)
where h : R → R is continuous. Of course, when h(x) = x − c we have the
initial condition (3).
Applied Mathematics and Computation 153 (2004) 793-802.
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Corollary 1 Suppose that α , β are coupled lower and upper solutions of (1)(21) such that (10) holds. Then there exists at least one solution between the
lower and the upper solution.
Proof
By Theorem 2, we have
0 = g(v(0), w(T )) = h(v(0)) , and 0 = g(w(0), v(T )) = h(w(0)) .
As a result, v , w are solutions of equation (1) satisfying h(v(0)) = h(w(0)) = 0 .
t
u
For the condition
h(u(T )) = 0
(22)
we obtain a similar result.
Corollary 2 Suppose that α , β are coupled lower and upper solutions of (1)(22) such that (10) holds. Then there exists at least one solution between the α
and β.
Proof
We now have that 0 = g(v(0), w(T )) = h(w(T )) and 0 = g(w(0), v(T )) =
h(v(T )) . Consequently, h(w(T )) = 0 = h(v(T )) .
4
t
u
AntiPeriodic Boundary Value Problem
For the antiperiodic problem the situation is different.
Theorem 3 Suppose that α , β are coupled lower and upper solutions of (1)(5) such that (10) holds. Moreover, suppose that for U , V ∈ [α, β] with U (t) <
V (t) , t ∈ I , we have that
Z
0
T
F (s, V (s)) − F (s, U (s))
ds 6= 0.
V (s) − U (s)
(23)
Then there exists at least one solution between the lower and the upper solution.
Applied Mathematics and Computation 153 (2004) 793-802.
Proof
9
Let z = w−v and note z ≥ 0 on I . Then, we obtain that z(0) = z(T ) ,
and for t ∈ I we have that
z 0 (t) = F (t, w(t)) − F (t, v(t)) ≥ −M z(t) .
As a result,
d Mt
[e z(t)] ≥ 0 , t ∈ I .
dt
(24)
If z(t) > 0 for each t ∈ I then
z 0 (t) = ζ(t)z(t) , t ∈ I ,
where, obviously,
ζ(t) =
F (t, w(t)) − F (t, v(t))
.
z(t)
Therefore,
Rt
ζ(s) ds
z(t) = z(0) e 0
,
and
RT
ζ(s) ds
z(0) = z(T ) = z(0)e 0
which contradicts (23).
Hence, we can suppose that there exists t0 ∈ I with z(t0 ) = 0 . This implies
(see (24)) that z(t) ≤ 0 , t ∈ [0, t0 ] . In particular, z(0) = 0 and z(T ) = 0 . As a
result from (24) we have that z(t) ≤ 0 , t ∈ I , and w = v is a solution.
t
u
Remark: Note that if F is a C 1 function, then (23) is equivalent to
Z T
∂F
(s, ζ(s)) ds 6= 0 ,
∂u
0
for any ζ ∈ [α, β] . This is the usual condition in the literature (see formula (3.5)
in [2]).
5
General Case
We now consider the general nonlinear boundary condition (2).
Applied Mathematics and Computation 153 (2004) 793-802.
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Theorem 4 Let α , β be coupled lower and upper solutions of (1)-(2) respectively such that (10), (15) and (16) hold. Moreover, suppose that there exist
nonnegative constants m0 , m00 such that for every x, x0 ∈ [α(0), β(0)] , y, y 0 ∈
[α(T ), β(T )] , x < x0 , y < y 0 the following growth conditions are satisfied
−m0 ≤
g(x0 , y) − g(x, y)
g(x, y 0 ) − g(x, y)
≤
m
,
0
≤
≤ m00 ,
x0 − x
y0 − y
(25)
and for U , V ∈ [α, β] with U (t) < V (t) , t ∈ I , we have that
RT
c 6= c0 e 0
F (s,V (s))−F (s,U (s))
V (s)−U (s)
ds
,
(26)
for any c ∈ [−m0 , m] , c0 ∈ [0, m00 ] .
Then there exists at least one solution in the sector [α, β] .
Proof
Let z = w − v . Then, 0 = g(v(0), w(T )) = g(w(0), v(T )) , and we get
g(v(0), w(T )) − g(v(0), v(T )) = g(w(0), v(T )) − g(v(0), v(T )) .
But now, using (25), we can write,
g(w(0), v(T )) − g(v(0), v(T )) = cz(0) ,
and
g(v(0), w(T )) − g(v(0), v(T )) = c0 z(T ) ,
for some c ∈ [−m0 , m] , c0 ∈ [0, m00 ] . Hence,
cz(0) = −g(v(0), v(T )) = c0 z(T ) .
As in the proofs of the previous Theorems, we have that
d Mt
[e z(t)] ≥ 0 , t ∈ I .
dt
Again, as in the proof of Theorem 3, suppose that z(t) > 0 , t ∈ I . Then,
Rt
F (t, w(t)) − F (t, v(t))
ζ(s) ds
z(t) = z(0)e 0
, ζ(s) =
, t∈I,
z(t)
Applied Mathematics and Computation 153 (2004) 793-802.
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and we get the contradiction
RT
ζ(s) ds
cz(0) = c0 z(T ) = c0 z(0)e 0
.
As a result, there exists at least one point t0 ∈ I such that z(t0 ) = 0 . From
this, we conclude that z ≤ 0 on I , and w = v .
t
u
Remark: Note that if g is a C 1 function, then (25) is valid.
Remark: In this last result, we could consider other boundary conditions such
as, for example,
g(x, y) =
x+y
,
2
g(x, y) = ax + by , a, b ∈ R ,
or
g(x, y) = x + h(y)
where h : R → R is a nondecreasing function.
Remark: For the periodic boundary condition (20) we could still try to apply
Theorem 2 and 4 with g(x, y) = −x + y , but we do not obtain any interesting
result. Indeed, (8) and (12) imply that
α(0) = β(0) , α(T ) = β(T ) .
This means that α = β is already a solution.
Acknowledgement: First and second authors were supported in part by
D.G.E.S.I.C. (Spain), project PB97 – 0552.
Applied Mathematics and Computation 153 (2004) 793-802.
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References
[1] Y.Q. Chen, On Massera’s theorem for anti-periodic solution. Adv. Math.
Sci. Appl. 9: 125–128 (1999).
[2] D. Franco, J.J. Nieto, First-order impulsive ordinary differential equations
with anti-periodic and nonlinear boundary conditions. Nonlinear Anal. 42:
163–173 (2000).
[3] D. Franco, J.J. Nieto, D. O’Regan, Anti-periodic boundary value problem
for nonlinear first order ordinary differential equations. Math. Inequal. Appl.
(to appear).
[4] T. Jankowski, Monotone iterative technique for differential equations with
nonlinear boundary conditions. Nonlinear Stud. 8: 381–388 (2001).
[5] V. Lakshmikantham, A.S. Vatsala, Generalized quasilinearization for nonlinear problems. Kluwer Academic Publishers, Dordrecht, 1998.
[6] R. Srzednicki, On solutions of two-point boundary value problems inside
isolating segments. Topol. Methods Nonlinear Anal. 13: 73–89 (1999).
[7] W. Feng, Decomposition conditions for two-point boundary value problems.
Int. J. Math. Math. Sci. 24: 389–401 (2000).
[8] S. Gan, M. Zhang, Resonance pockets of Hill’s equations with two-step
potentials. SIAM J. Math. Anal. 32: 651–664 (2000).
[9] S. Aizicovici, S. McKibben, S. Reich, Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities. Nonlinear
Anal. Ser. A: Theory, Methods. 43: 233–251 (2001).
[10] S. Aizicovici, S. Reich, Anti-periodic solutions to a class of non-monotone
evolution equations. Discrete Contin. Dynam. Systems. 5: 35–42 (1999).
Applied Mathematics and Computation 153 (2004) 793-802.
13
[11] K.H. Hoffmann, P. Rybka, On convergence of solutions to the equation of
viscoelasticity with capilarity. Comm. Partial Differential Equations. 25:
1845–1890 (2000).
[12] G. Wang, C. Liu, The heat equation in R with anti-periodic boundary
condition. Acta Math. Sci. 19: 391–401 (1999).
[13] F.J. Delvos, L. Knoche, Lacunary interpolation by antiperiodic trigonometric polynomials. BIT. 39: 439–450 (1999).
[14] H.L. Chen, Antiperiodic wavelets. J. Comput. Math. 14: 32–39 (1996).
[15] H. Chen, Complex harmonic-splines, periodic quasiwavelets. Kluwer Academic Publishers, Dordrecht, 2000.
[16] U. Ledzewicz, A. Nowakowski, Optimality conditions for control problems
governed by abstract semilinear differential equations in complex Banach
spaces. J. Appl. Anal. 3: 67–91 (1997).
[17] T. Banks, L. Motl, A nonsupersymmetric matrix orbifold. J. High Energy
Phys. 3: Paper27, 19 pp (2000).
[18] J.A. Carrillo, A. Maia, V.M. Mostepanenko, Jacobi elliptic solutions of
λϕ4 theory in a finite domain. Internat. J. Modern Phys. A. 15: 2645–2659
(2000).
[19] G. Felder, A. Schorr, Separation of variables for quantum integrable systems on elliptic curves. J. Phys. A. 32: 8001–8022 (1999).
[20] H. Kleinert, Functional determinants from Wronski Green functions. J.
Math. Phys. 40: 6044–6051 (1999).
[21] G.S. Ladde, V. Lakshmikantham, A.S. Vatsala, Monotone iterative techniques for nonlinear differential equations. Pitman, Boston, 1985.