Two Point Boundary Value Problems for a Class of Second Order Ordinary Differential Equations

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Volume 2012, Article ID 794040,13pages doi:10.1155/2012/794040

Research Article

Two-Point Boundary Value Problems for a Class of

Second-Order Ordinary Differential Equations

Indranil SenGupta

1

_{and Maria C. Mariani}

2

1_{Department of Mathematical Sciences, University of Texas at El Paso, Bell Hall 316, El Paso,}

TX 79968-0514, USA

2_{Department of Mathematical Sciences, University of Texas at El Paso, Bell Hall 124, El Paso,}

TX 79968-0514, USA

Correspondence should be addressed to Indranil SenGupta,isengupta@utep.edu

Received 14 October 2011; Accepted 27 December 2011

Academic Editor: Martin Bohner

Copyrightq2012 I. SenGupta and M. C. Mariani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the general semilinear second-order ODE ugt, u, u 0 under diﬀerent two-point boundary conditions. Using the method of upper and lower solutions, we obtain an existence result. Moreover, under a growth condition on g, we prove that the set of solutions of

ugt, u, u 0 is homeomorphic to the two-dimensional real space.

1. Introduction

The Dirichlet problem for the semilinear second-order ODE

ugt, u, u0 1.1

has been studied by many authors from the pioneering work of Picard1, who proved the existence of a solution by an application of the well-known method of successive approxi-mations under a Lipschitz condition on g and a smallness condition onT. Sharper results were obtained by Hamel 2 in the special case of a forced pendulum equation see also

3,4. The existence of periodic solutions for this case has been first considered by Duﬃng

5in 1918. Variational methods have been also applied wheng gt, uby Lichtenstein6, who considered the functional

Iu

T

0

u2

2 −Gt, u

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with Gt, u ₀ugt, sds. When g depends on u, the problem is nonvariational, and diﬀerent techniques are required, for example, the shooting method introduced in 1905 by Severini7and the more general topological approach, which makes use of Leray-Schauder Degree theory. For an overview of the problem and further results, we refer the reader to

8. A diﬀerent kind of nonlinear boundary value PDEquasilinear elliptic equationswas studied extensively in9,10.

This problem is recently studied in11. Also, this problem is generalized in12–14. Several much more general forms of the problem have been studied in 15–17 via lower and upper solution method. We will study the existence of solutions of1.1under Dirichlet, periodic, and nonlinear boundary conditions of the type

u0 f1u0, uT f2uT, 1.3

where f1 and f2 are given continuous functions. Note that if fix aixbi, then 1.3

corresponds to a particular case of Sturm-Liouville conditions and Neumann conditions whena1a20.

In the second section, we impose a growth condition ong in order to obtain unique solvability of the Dirichlet problem. Furthermore, we prove that the trace mapping

Tr :u∈H20, T:ugt, u, u0 −→R2 1.4

given by Tru u0, uT is a homeomorphism, and we apply this result to obtain solutions for other boundary conditions in some specific cases.

In the third section, we construct solutions of the aforementioned problems by an iterative method based on the existence of an ordered coupleα, βof a lower and an upper solution. This method has been successfully applied to diﬀerent boundary value problems whengdoes not depend onu. For generalg, existence results have been obtained assuming that|gt, u, u| ≤Bt,|u|, whereB : 0, T×0,∞ → 0,∞is a Bernstein function for someN≥ α_∞,β_∞, namely,

iBis nondecreasing inu,

iiforα≤u≤β, if|ut| ≤Bt,|ut|for anyt, thenu_∞≤Nsee, e.g.,18.

We will assume instead a Lipschitz condition with respect to u and construct in each case a nonincreasingresp., nondecreasing sequence of upperlowersolutions that converges to a solution of the problem.

2. A Growth Condition for

g

For simplicity, let us assume thatgis continuous. We may write it as

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withr being continuous. We will assume thathsatisfies a global Lipschitz condition onu, namely,

ht, u, x−ht, u, y x−y

≤k <

π

T forx /y. 2.2

Remark 2.1. Without loss of generality, we may assume thatr∈W1,∞₀_{, T}_{. Indeed, if not, we}

may multiply1.1for any positivepsuch thatr : rp−p ∈W1,∞₀_{, T}_{in order to get the}

modified equation:

purupht, u, u0. 2.3

Note that in this case the valueπ/T in2.2must be replaced byλ1/p∞, whereλ1is the

first eigenvalue of−pufor the Dirichlet conditions.

Furthermore, assume thathsatisfies the following one-sided growth condition onu:

ht, u, x−ht, v, x

u−v ≤c, 2.4

foru /v, with

ckπ T <

_π

T

2

1

20inf≤t≤Tr

_t_. _2.5

Under these assumptions, the set S of solutions of 1.1 is homeomorphic to R2_{. More}

precisely, one has the following.

Theorem 2.2. Assume that2.2–2.4hold and leta, b∈R. Then there exists a unique solutionua,b

of 1.1satisfying the nonhomogeneous Dirichlet condition:

ua,b0 a, ua,bT b. 2.6

Furthermore, the trace mapping Tr : S, · H2 → R2 given by Tru u0, uTis a

homeo-morphism.

Proof. For anyu∈H1₀_{, T}_let_u_{be the unique solution of the linear problem:}

u−ruht, u, u,

u0 a, uT b. 2.7

It is immediate that the operatorA : H1₀_{, T} _→ _H1₀_{, T}_{given by}_A_u _u_{is compact.}

Moreover, ifSσu:uσruht, u, uwithσ∈0,1, a simple computation shows that

the following a priori bound holds for anyu, v∈H2₀_{, T}_with_u₋_v_∈_H1

00, T:

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where

μ

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

1

π/T−k, ifc≤

1 2inf r,

π/T

π/T−k 1/2infr−cT/π, otherwise.

2.9

Hence, ifu σAui.e.,Sσu 0for someσ ∈0,1, settingla,bt b−a/Tta, we

obtain

u−σl_a,b

L2≤μSσσla,bL2 ≤M 2.10

for some fixed constantM. Thus, existence follows from Leray-Schauder Theorem. Unique-ness is an immediate consequence of2.8forσ1.

Hence, Tr is bijective, and its continuity is clear. On the other hand, ifa, b → a0, b0,

applying2.8touua,b−la,bandvua0,b0−la0,b0, it is easy to see thatua,b → ua0,b0for the

H1_{-norm. As}_u

a,bandua0,b0 satisfy1.1, we conclude that alsou_a,b → u_a0,b0 for theL2-norm

and the proof is complete.

Remark 2.3. The proof ofTheorem 2.2still holds under more general assumptions forg. In fact, ifgsatisfies Caratheodory-type conditions, we may assume only that

ht, u, x−ht, v, y u−v ≤ck

x_u−₋y_v 2.11

foru /vandc, kas before, which is not equivalent to2.2–2.4whenhis noncontinuous. Thus, the result may be considered a slight extension of well-known resultssee, e.g.,19, Corollary V.2.

As a simple consequence we have the following.

Corollary 2.4. Assume that2.2and2.4hold. Further, assume that there exists a constantM >0

such that

ht, u,0sgnu≥0 for|u| ≥M,

h·, u,_u 0

L2

≤ δ

T1/2_μ for|u| ≥M ,

2.12

whereμis the constant defined by2.9andδ <1. Then1.1admits at least oneT-periodic solution, which is unique ifc <0 in2.4.

Proof. With the notations of the previous theorem, let us consider the mappingϕ : R → R given by

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FromTheorem 2.2,ϕis continuous, and it is clear thatuis a periodic solution of the problem if and only ifuua,afor someawithϕa 0.

From2.8, if|a| ≥ M, we takev aandσ 1. Observe thatS1uaa 0. Therefore it

follows that

_u_a,a

L2≤μh·, a,0L2≤

δ

T1/2|a|, 2.14

where in the last inequality we used2.6. Hence|ua,a−a| ≤

_T

0 |ua,a| ≤δ|a|or, equivalently:

1−δ|a| ≤ |ua,a| ≤1δ|a|. 2.15

Letpbe the unique solutionin distributional senseof the following problem:

p−rξp0, p0 pT 1, 2.16

whereξ∈L∞0, Tis given by

ξt h

t, ua,a, ua,a

−ht, ua,a,0

ua,a

, 2.17

withξt 0 ifu_a,at 0. A simple computation shows thatpis positive, and multiplying the equation byp, we obtain

ϕa

T

0

pu_a,a−

T

0

pht, ua,a,0. 2.18

Hence by2.12ϕa≤0≤ϕ−afor1−δa≥Mand existence follows from the continuity ofϕ. On the other hand, ifuandvare periodic solutions of the problem, then

u−vrψu−vht, u, v−ht, v, v0, 2.19

where

ψt ht, u, u

₋_h_{t, u, v}

u−v ∈L

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Now takep >0 as the unique solution of the problemp−rψp 0, p0 pT 1. Multiplying the previous equality bypu−vand applying the boundary conditions forp, u, andv, we observe

T

0

pu−vu−v −

T

0

pu−v pu−vu−v

−

_T

0

pu−vu−v

_T

0

pu−v2,

T

0

prψu−vu−v−

T

0

prψu−v

2

−

_T

0

pu−v

2

T

0

u−vu−vp.

2.21

Thus we obtain

0

_T

0

pu−v2−

_T

0

pht, u, v−ht, v, vu−v

≥

T

0

pu−v2−c

T

0

pu−v2.

2.22

Ifc <0, we conclude thatuv.

Remark 2.5. In the previous proof, note that the sign condition onhis only used for1−δ|a| ≤

|u| ≤1δ|a|. Thus,2.12may be replaced by the weaker condition

ht,·,0|I1≥0≥ht,·,0|I2 2.23

whereIj aj−δj|aj|, ajδj|aj|for someaj∈R,δj<1 with

h·, u,_u 0

L2

≤ δj

T1/2_μ foru∈Ij. 2.24

Remark 2.6. As a particular case of Corollary 2.4, we deduce the existence of T-periodic solutions under the following Landesman-Lazer type conditionssee, e.g.,20:

lim inf

|u| → ∞ ht, u,0sgnu≥0,

lim

|u| → ∞

h·, u,_u 0

L2

0.

2.25

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Corollary 2.7. Assume that2.2and2.4hold. Further, assume that there exists a constantM >0

such that

ht, u,0sgnu<0 for|u| ≥M. 2.26

Then1.1is solvable under periodic or Sturm-Liouville conditions:

u0 a1u0 b1, uT a2uT b2, a1≥0≥a2. 2.27

Furthermore, ifc <0 in2.4, then the respective solutions are unique.

Proof. For the periodic problem, defineϕas in the previous corollary. Fora≥M, ifua,at0> a

for somet0, we may assume thatt0is maximum, and hence

ut0 −gt0, ut0,0 −ht0, ut0,0>0, 2.28

a contradiction. Thus,u≤ a, which implies thatϕa ≥0. In the same way, we deduce that

ϕa≤0 fora≤ −M. Uniqueness follows as inCorollary 2.4.

For2.27conditions, let us first note that ifλ >0, the linear problem

vrv−λv0, v0 a1v0 b1, vT a2vT b2 2.29

is uniquely solvable, and settingwu−vproblem1.1–2.27is equivalent to

wgt, w, w0, w0 a1w0, wT a2w0, 2.30

where gt, w, w : gt, wv, wv−λv satisfies the hypothesis. Hence, it suﬃces to consider only the homogeneous caseb1 b2 0. In the same way as before, defineϕ:R2 →

R2_by

ϕa, b u_a,b0, u_a,bT. 2.31

Fora ≥ M ≥ |b|, we obtain that u_a,b0 ≤ 0 ≤ u₋_a,b0, and forb ≥ M ≥ |a|, it holds that

u_a,bT≥0≥u_a,₋_bT. By the generalized intermediate value theorem, we deduce thatϕhas a zero in−M, M×−M, M. Uniqueness can be proved as in the periodic case.

3. Iterative Sequences of Upper and Lower Solutions

In this section, we construct solutions of 1.1 under the mentioned two-point boundary conditions by an iterative method. As before, considergt, u, u rtuht, u, uwhere

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We will need the following auxiliary lemmas.

Lemma 3.1. Assume that2.2holds and letλ >0 be large enough. Then for anyz, θ∈C0, T,

uruht, z, u−λuθt 3.1

is uniquely solvable under Dirichlet, periodic, or2.27conditions. Furthermore, the applicationT :

C0, T2 → C0, Tgiven byTz, θ uis compact.

Proof. Takingλ > kπ/T−π/T2−1/2infr, existence and uniqueness follow as a particular case ofTheorem 2.2andCorollary 2.7for

gt, u, uruht, z, u−λu−θt. 3.2

Letz, θ → z0, θ0, and setuTz, θ,u0Tz0, θ0. Then

u−u0rψu−u0−λu−u0 ht, z, u₀−ht, z0, u0

θ−θ0, 3.3

whereψt ht, z, u−ht, z, u₀/u−u₀. Hence, it suﬃces to prove that the following a priori bound holds for anywsatisfying periodic or homogeneous Dirichlet or2.27conditions:

w_H1 ≤cw

rψw−λw_L2, 3.4

where the constantcdepends only onk. For Dirichlet and2.27conditions, apply Cauchy-Schwartz inequality to the integral−₀TpLw·wwhereLwwrψw−λwandpe0Trψ,

and observe that 0 < m≤ p≤ Mfor somemandMdepending only onknote that under homogeneous2.27conditions, it holds that−pww|T₀ p0a1w02−pTa2wT2 ≥0. For

periodic conditions, takepsuch thatp rψp−rwithrconstant andp0 pT 1 and the proof follows.

Lemma 3.2. Letφ ∈ L∞0, Tand assume thatwφw−λw ≥ 0 a.e. forλ > 0. Thenw ≤ 0,

provided thatwsatisfies one of the boundary conditions:

iw0, wT≤0,

iiwT−w0 0≥wT−w0,

iiiw0−a1w0≥0≥wT−a2wT, a1 ≥0≥a2.

Proof. For w0, wT ≤ 0, the result is the well-known maximum principle for Dirichlet conditions. Ifii holds andw0 wT > 0, asw cannot achieve a positive maximum on0, T, we have thatw0 wT 0 andw, w ≥ 0 over a maximal intervalt0, T.

Takingpe0tφ, we deduce thatpwis nondecreasing ont₀, T, a contradiction. Ifiiiholds

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In order to prove the main result of this section, we recall thatα, β is an ordered couple of a lower and an upper solution for1.1ifα≤βand

αg·, α, α≥0≥βg·, β, β, 3.5

under the following boundary conditions.

For the Dirichlet problem,

α0≤a≤β0, αT≤b≤βT. 3.6

For the periodic problem,

αT−α0 0βT−β0, αT−α0≤0≤βT−β0. 3.7

For the problem1.3,

α0−f1α0≥0≥β0−f1

β0, αT−f2αT≤0≤βT−f2

βT. 3.8

We make the following extra assumption. There exists a constantR >0 such that

ht, u, α−ht, v, α u−v ≤R,

ht, u, β−ht, v, β

u−v ≥ −R, 3.9

for anyu, vsuch thatαt≤ut, vt≤βt, and for1.3: there exists a constantR >0 such that

f1x−f1

y

x−y ≤R forα0≤x, y≤β0, f2x−f2

y

x−y ≥ −R forαT≤x, y≤βT.

3.10

Then we have the following.

Theorem 3.3. Assume that there exists an ordered coupleα, βof a lower and an upper solution for

Dirichlet, periodic, or1.3conditions. Further, assume that2.2and3.9hold (and also3.10, for the1.3case). Then the respective boundary value problem admits at least one solution uwith α≤u≤β.

Remark 3.4. Observe that a Lipschitz condition is a particular case of a Nagumo condition.

This result can also be obtained as a Corollary of Theorem 3.2 in21.

Proof. Forλ≥Rlarge enough andu∈C0, TdefineTuuto be the unique solution of the following problem:

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satisfying, respectively, Dirichlet, periodic, or the Sturm-Liouville condition:

u0−Ru0 f1u0−Ru0, uT RuT f2uT RuT. 3.12

Compactness ofTfollows easily fromLemma 3.1. Moreover, ifu≤β, then

uruht, u, uRu−λu R−λu≥R−λβ

≥R−λββrβht, β, β. 3.13

Hence, setting

ψt ht, u, u

₋_h_{t, u, β}

u−β , 3.14

we deduce that

u−βrψu−β−λu−β≥ht, β, βRβ−ht, u, βRu≥0. 3.15

For Dirichlet and periodic cases, it follows thatu≤β. For1.3, note that

u0−Ru0 f1u0−Ru0≥f1

β0−Rβ0,

uT RuT f2uT RuT≤f2

βTRβT.

3.16

Hence,

u−β0−Ru−β0≥0≥u−βT−Ru−βT, 3.17

and fromLemma 3.2, we also obtain thatu≤β. In the same way, ifu≥α, we obtain thatu≥α

and the proof follows from Schauder Fixed Point Theorem.

Remark 3.5. Existence conditions in Corollary 2.7 are easily improved by applying

Theorem 3.3. Indeed, under condition 2.26, it is immediate that −M, M is an ordered

couple of a lower and an upper solution.

Remark 3.6. In the context of the previous theorem, fromLemma 3.1, we deduce the existence of a constantKsuch that ifuTuforα≤u≤β, then

_u

L∞ ≤K. 3.18

Example 3.7. It is easy to see that the problem

u−u0,

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has no solution, although α −1 is a lower solution. From the previous theorem, we deduce that no upper solutionβ≥ −1 exists. This can be proved directly from the following conditions:

β−β≤0, β0≤β0, βT≥βT 1. 3.20

Indeed, as no negative minimum exists, ifβ0, βT≥0, we may taket0maximum such that

β>0 overt0, T. Hence,β2T−β2T≤β2t0−β2t0≤0, a contradiction. On the other

hand, ifβ0 <0, thenβ < 0 on0, TandβT 0, a contradiction sinceβ ≤ β. The case

β0≥0> βTcan be easily reduced to the previous one.

In order to construct solutions by iteration, we need a stronger assumption onh. There exists a constantRsuch that

|ht, u, x−ht, v, x| ≤R|u−v| 3.21

foru, vsuch thatαt≤ut, vt≤βt, andx∈R.

Corollary 3.8. Assume that there exists an ordered coupleα, βof a lower and an upper solution for

Dirichlet, periodic, or1.3conditions. Further, assume that2.2and3.21hold (and also3.10, for the1.3case). Setλ≥Rlarge enough, and define the sequences{u_n}and{un}given by

u₀α, u0β, 3.22

andun1,u_n₁are the (unique) solutions of the following problems:

u_n₁ru_n₁ht, un, un1

−λun1−λun,

u_n₁ru_n₁ht, u_n, u_n₁−λu_n₁−λu_n, 3.23

under the respective boundary conditions. Thenu_n unis an ordered couple of a lower and an upper solution. Furthermore,{un}(resp.,{u_n}) is nonincreasing (nondecreasing) and converges to a solution of the problem.

Remark 3.9. Observe that this is also a classical result that can be found in the works of Adje

22or Cabada23, for example.

Proof. From the previous theorem, we know thatα≤u1≤β. Moreover,

u₁ru₁ht, u1, u1

λ−Ru1−β

ht, u1, u1

Ru1

−ht, β, u₁Rβ≤0.

3.24

Hence,u1 is an upper solution of the problem. Inductively, it follows thatun is an upper

solution for everyn, withα≤un1≤un. Henceunconverges pointwise to a functionu.

From

u_n₁ru_n₁ht, un, un1

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pointwise. Moreover, by Lemma 3.1, we know that {un}is bounded in H1₀_{, T}_{; hence in}

H2₀_{, T}_{, it follows easily that}

u ruht, u, u0. 3.26

Thus,uis a solution of the problem. The proof foru_n is analogous. Moreover, if we assume thatu_n ≤un, it is immediate thatu_n₁≤un1.

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