Existence of solutions for first order ordinary

differential equations with nonlinear boundary

conditions

Daniel Franco∗

Departamento de Matematica Aplicada

Universidad Nacional de Educacion a Distancia

Apartado de Correos 60149. Madrid 28080, Spain

Juan J. Nieto

Departamento de Analisis Matematico. Facultad de Matematicas

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 antiperi-

odic 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, an-

tiperiodic solutions, nonlinear boundary conditions, upper and lower solutions.

1 Introduction

We are interested in solutions of the nonlinear equation

u′(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 ) . (5)

corresponds to the case g(x, y) = x + y .

Applied Mathematics and Computation 153 (2004) 793-802. 3

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 ∈ C1(I) is a subsolution of equation

(1) if

α′(t) ≤ F (t, α(t)) , t ∈ I . (6)

Analogously, we say that β ∈ C1(I) is a supersolution of (1) if

β′(t) ≥ F (t, β(t)) , t ∈ I . (7)

In what follows we shall assume that

α(t) ≤ β(t) , t ∈ I , (8)

and define the set

[α, β] = {v ∈ C(I) : α(t) ≤ v(t) ≤ β(t) , t ∈ I } .

Applied Mathematics and Computation 153 (2004) 793-802. 4

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 ∈ C1(I) are subsolution

and supersolution of (1) respectively satisfying (8) and (9). Then there exists

a solution u ∈ C1(I) of the initial value problem (1)-(3) such that u ∈ [α, β] .

Moreover, if there exists M > 0 such that

F (t, v) + Mv ≤ F (t, u) + Mu , 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 con-

ditions (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 intro-

duce the following concept.

Applied Mathematics and Computation 153 (2004) 793-802. 5

Definition 2 We say that α , β ∈ C1(I) are coupled lower and upper solutions

for the problem (1)-(2) if α is a subsolution and β a supersolution for the equa-

tion (1), condition (8) holds, and

g(α(0), β(T )) ≤ 0 ≤ g(β(0), α(T )) . (12)

Definition 3 We say that v , w ∈ C1(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)

and

g(v(0), w(T )) = 0 = g(w(0), v(T )) . (14)

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, x′ ∈ [α(0), β(0)] with x ≤ x′ and y, y′ ∈ [α(T ), β(T )]

with y ≤ y′ one has that

g(x′, y)−mx′ ≤ g(x, y)−mx , (15)

and

g(x, y) ≤ g(x, y′) . (16)

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′(t) + Mv(t) = F (t, φ(t)) + Mφ(t) , t ∈ I ; v(0) = φ(0)− g(φ(0), ϕ(T ))m

, (17)

w′(t)+Mw(t) = F (t, ϕ(t))+Mϕ(t) , t ∈ I ; w(0) = ϕ(0)− g(ϕ(0), φ(T ))m

. (18)

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-Arzela’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′(t) + Mz(t) ≥ F (t, φ(t)) + Mφ(t)− F (t, α(t))−Mα(t) ≥ 0 .

Now, using conditions (12), (15) and (16) we get

z(0) = φ(0)− g(φ(0), ϕ(T ))m

− α(0) ≥ φ(0)− α(0)− g(φ(0), β(T ))m

=

φ(0)− α(0) +g(α(0), β(T ))− g(φ(0), β(T ))

m− g(α(0), β(T ))

m≥

φ(0)− α(0) +−m[φ(0)− α(0)]

m= 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′(t) + Mz(t) = F (t, ϕ(t)) + Mϕ(t)− F (t, φ(t))−Mφ(t) ≥ 0 , t ∈ I ,

and

z(0) = ϕ(0)− φ(0) +g(φ(0), ϕ(T ))− g(ϕ(0), φ(T ))

m=

ϕ(0)− φ(0) +g(φ(0), ϕ(T ))− g(φ(0), φ(T )) + g(φ(0), φ(T ))− g(ϕ(0), φ(T ))

m≥

ϕ(0)− φ(0) +−m[ϕ(0)− φ(0)]

m+

g(φ(0), ϕ(T ))− g(φ(0), φ(T ))m

≥ 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′(t) + Mv(t) = F (t, v(t)) + Mv(t) , t ∈ I ; v(0) = v(0)− g(v(0), w(T ))m

,

and

w′(t) + Mw(t) = F (t, w(t)) + Mw(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

proof. ut

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. 8

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 .

ut

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 )) . ut

4 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∫ T

0

F (s, V (s))− F (s, U(s))V (s)− U(s)

ds 6= 0. (23)

Then there exists at least one solution between the lower and the upper solution.

Applied Mathematics and Computation 153 (2004) 793-802. 9

Proof 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′(t) = F (t, w(t))− F (t, v(t)) ≥ −Mz(t) .

As a result,d

dt[eMtz(t)] ≥ 0 , t ∈ I . (24)

If z(t) > 0 for each t ∈ I then

z′(t) = ζ(t)z(t) , t ∈ I ,

where, obviously,

ζ(t) =F (t, w(t))− F (t, v(t))

z(t).

Therefore,

z(t) = z(0) e

∫ t

0ζ(s) ds

,

and

z(0) = z(T ) = z(0)e∫ T

0ζ(s) ds

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. ut

Remark: Note that if F is a C1 function, then (23) is equivalent to∫ T

0

∂F

∂u(s, ζ(s)) ds 6= 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. 10

Theorem 4 Let α , β be coupled lower and upper solutions of (1)-(2) respec-

tively such that (10), (15) and (16) hold. Moreover, suppose that there exist

nonnegative constants m′ , m′′ such that for every x, x′ ∈ [α(0), β(0)] , y, y′ ∈

[α(T ), β(T )] , x < x′ , y < y′ the following growth conditions are satisfied

−m′ ≤ g(x′, y)− g(x, y)x′ − x

≤ m , 0 ≤ g(x, y′)− g(x, y)y′ − y

≤ m′′ , (25)

and for U , V ∈ [α, β] with U(t) < V (t) , t ∈ I , we have that

c 6= c′e

∫ T

0

F (s,V (s))−F (s,U(s))V (s)−U(s) ds

, (26)

for any c ∈ [−m′,m] , c′ ∈ [0,m′′] .

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 )) = c′z(T ) ,

for some c ∈ [−m′,m] , c′ ∈ [0,m′′] . Hence,

cz(0) = −g(v(0), v(T )) = c′z(T ) .

As in the proofs of the previous Theorems, we have that

d

dt[eMtz(t)] ≥ 0 , t ∈ I .

Again, as in the proof of Theorem 3, suppose that z(t) > 0 , t ∈ I . Then,

z(t) = z(0)e∫ t

0ζ(s) ds

, ζ(s) =F (t, w(t))− F (t, v(t))

z(t), t ∈ I ,

Applied Mathematics and Computation 153 (2004) 793-802. 11

and we get the contradiction

cz(0) = c′z(T ) = c′z(0)e∫ T

0ζ(s) ds

.

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 . ut

Remark: Note that if g is a C1 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. 12

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