TWO-POINT BOUNDARY VALUE PROBLEMS FOR HIGHER-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH STRONG pdf

KIGURADZE Received 4 April 2004; Revised 11 December 2004; Accepted 14 December 2004 For strongly singular higher-order linear differential equations together with two-pointconjugate and

HIGHER-ORDER LINEAR DIFFERENTIAL EQUATIONS

WITH STRONG SINGULARITIES

R P AGARWAL AND I KIGURADZE

Received 4 April 2004; Revised 11 December 2004; Accepted 14 December 2004

For strongly singular higher-order linear differential equations together with two-pointconjugate and right-focal boundary conditions, we provide easily verifiable best possibleconditions which guarantee the existence of a unique solution

Copyright © 2006 Hindawi Publishing Corporation All rights reserved

1 Statement of the main results

1.1 Statement of the problems and the basic notation Consider the differential tion

Heren ≥2,m is the integer part of n/2, −∞ < a < b < + ∞, p i ∈ Lloc(]a, b[) (i =1, , n),

q ∈ Lloc(]a, b[), and by u(i−1)(a) (by u(j−1)(b)) is understood the right (the left) limit of

the functionu(i−1)(of the functionu(j−1)) at the pointa (at the point b).

Problems (1.1), (1.2) and (1.1), (1.3) are said to be singular if some or all coefficients

of (1.1) are non-integrable on [a, b], having singularities at the ends of this segment.

Hindawi Publishing Corporation

Boundary Value Problems

Volume 2006, Article ID 83910, Pages 1 32

DOI 10.1155/BVP/2006/83910

The previous results on the unique solvability of the singular problems (1.1), (1.2) and(1.1), (1.3) deal, respectively, with the cases where

(see [1,2,4,3,5,6,9–18], and the references therein)

The aim of the present paper is to investigate problem (1.1), (1.2) (problem (1.1),(1.3)) in the case, where the functions p i (i =1, , n) and q have strong singularities

at the points a and b (at the point a) and do not satisfy conditions (1.4) (conditions(1.5))

Throughout the paper we use the following notation

[x]+is the positive part of a numberx, that is,

[x]+= x + | x |

Lloc(]a, b[) (Lloc(]a, b])) is the space of functions y :]a, b[ → R which are integrable on[a + ε, b − ε] (on [a + ε, b]) for arbitrarily small ε > 0.

L α,β(]a, b[) (L2α,β(]a, b[)) is the space of integrable (square integrable) with the weight

(t − a) α(b − t) βfunctionsy :]a, b[ → Rwith the norm

α(]a, b])) is the space of functions y ∈ Lloc(]a, b[) (y ∈ Lloc(]a, b])) such

that y ∈ L2α,β(]a, b[), wherey(t) = c t y(s)ds, c =(a + b)/2 (y ∈ L2α,0(]a, b[), wherey(t) =

b

t y(s)ds).

loc (]a, b])) is the space of functions y :]a, b[ → R (y :]a, b] → R) which

are absolutely continuous together with y , , y(n−1)on [a + ε, b − ε] (on [a + ε, b]) for

C n −1,m(]a, b[) (in the space Cn −1,m(]a, b])).

Byh i:]a, b[ ×] a, b[ →[0, +∞[ (i =1, , m) we understand the functions defined by the

1.2 Fredholm type theorems Along with (1.1), we consider the homogeneous equation

loc (]a, b[) (in the space Cn −1

loc (]a, b])), then for every q ∈ L n − m,m(]a, b[)

(q ∈ L n − m,0(]a, b[)) problem (1.1), (1.2) (problem (1.1), (1.3)) is uniquely solvable in thespaceCn −1

loc (]a, b[) (in the space Cn −1

loc (]a, b])).

In the case where condition (1.14) is violated, the question on the presence of theFredholm property for problem (1.1), (1.2) (for problem (1.1), (1.3)) in some subspace

of the spaceCn −1

loc (]a, b[) (of the space Cn −1

loc (]a, b])) remained so far open This

ques-tion is answered inTheorem 1.3(Theorem 1.5) formulated below which contains mal in a certain sense conditions guaranteeing the presence of the Fredholm property forproblem (1.1), (1.2) (for problem (1.1), (1.3)) in the spaceCn −1,m(]a, b[) (in the space

and from estimate (1.15) there respectively follow the estimates

wherer0is a positive constant independent ofq.

Theorem 1.3 Let there exist a0∈] a, b[, b0∈] a0,b[ and nonnegative numbers 1i, 2i(i =

1, , m) such that

(t − a)2m− i h i(t, τ) ≤ 1i for a < t ≤ τ ≤ a0,(b − t)2m− i h i(t, τ) ≤ 2i for b0≤ τ ≤ t < b (i =1, , m), (1.20)

Corollary 1.4 Let there exist nonnegative numbers λ1i, λ2i (i =1, , m) and functions

p0i∈ L n − i,2m − i(]a, b[) (i =1, , m) such that the inequalities

Then problem ( 1.1 ), ( 1.2 ) has the Fredholm property in the space Cn −1,m(]a, b[).

Theorem 1.5 Let there exist a0∈] a, b[ and nonnegative numbers  i(i =1, , m) such that

Corollary 1.6 Let there exist nonnegative numbers λ i(i =1, , m) and functions p0i∈

L n − i,0(]a, b[) (i =1, , m) such that the inequalities

Then problem ( 1.1 ), ( 1.3 ) has the Fredholm property in the space Cn −1,m(]a, b]).

In connection with the above-mentioned Corollary 1.1 from [10], there naturallyarises the problem of finding the conditions under which the unique solvability of prob-lem (1.1), (1.2) (of problem (1.1), (1.3)) in the space Cn −1,m(]a, b[) (in the space

p1(t) = λ

(t − a) n, p i(t) =0 (i =2, , n), (1.33)andq(t) =( n(ν) − λ)t ν − n, whereλ =0,ν > 0, then (1.1) and (1.10) have the forms

C1,1(]a, b[), and for n > 2 solutions of that equation from the space Cn −1,m(]a, b[)

consti-tute an (n − m −1)-dimensional subspace with the basis

(t − a) x1, , (t − a) x n − m −1. (1.38)Thus problem (1.340), (1.2) (problem (1.340), (1.3)) has only a trivial solution in thespaceCn −1,m(]a, b[) We show that nevertheless problem (1.34), (1.2) (problem (1.34),(1.3)) does not have a solution in the spaceCn −1,m(]a, b[) Indeed, if n =2, then (1.34)has the unique solutionu(t) =(t − a) νin the spaceC 1,1(]a, b[), and this solution does not

satisfy conditions (1.2) Ifn > 2, then an arbitrary solution of (1.34) fromCn −1,m(]a, b[)

has the form

u(t) =

n −m −1

i =1

c i(t − a) x i+ (t − a) ν, (1.39)

and this solution satisfies the boundary conditions (1.2) (the boundary conditions (1.3))

if and only ifc1, , c n − m −1are solutions of the system of linear algebraic equations

Note that in the case under consideration the functionsp i(i =1, , m) in view of

con-ditions (1.30) and (1.32) satisfy inequalities (1.22) (inequalities (1.26)), whereλ11= | λ |,

in-Now we consider the case, where

Then, in view of (1.30) and (1.33), the functionsp i(i =1, , m) satisfy all the conditions

of Corollaries1.4and1.6, but condition (1.28) inTheorem 1.7is violated On the otherhand, according to conditions (1.31) and (1.32), the characteristic equation (1.36) hassimple real rootsx1, , x nsuch that

x1> ··· > x n − m > m −1

2> x n − m+1 > ··· > x n, (1.43)

at that

So, the set of solutions of (1.340) fromCn −1,m(]a, b[) constitutes an (n − m)-dimensional

subspace with the basis

(t − a) x1, , (t − a) x n − m, (1.45)

and consequently, both problem (1.340), (1.2) and problem (1.340), (1.3) in the tioned space have only trivial solutions Hence in view of Corollaries 1.4 and 1.6 theunique solvability of problems (1.34), (1.2) and (1.34), (1.3) follows inCn −1,m(]a, b[) Let

men-us show that these problems inCn −1

loc (]a, b]) have infinite sets of solutions Indeed, for any

for anyc n − m+1 ∈ R However, this system has a unique solution for an arbitrarily fixed

c n − m+1 Thus problem (1.34), (1.2) (problem (1.34), (1.3)) has a one-parameter family ofsolutions in the spaceCn −1

loc (]a, b]).

1.3 Existence and uniqueness theorems.

Theorem 1.9 Let there exist t0∈] a, b[ and nonnegative numbers 1i, 2i(i =1, , m) such that along with ( 1.21 ) the conditions

(t − a)2m− i h i(t, τ) ≤ 1i for a < t ≤ τ ≤ t0,(b − t)2m− i h i(t, τ) ≤ 2i for t0≤ τ ≤ t < b (1.49)hold Then for every q ∈ L2n −2m−2,2m−2(]a, b[) problem ( 1.1 ), ( 1.2 ) is uniquely solvable in the space Cn −1,m(]a, b[).

Corollary 1.10 Let there exist t0∈] a, b[ and nonnegative numbers λ1i, λ2i(i =1, , m) such that conditions ( 1.23 ) are fulfilled, the inequalities

(−1)n − m(t − a) n p1(t) ≤ λ11, (t − a) n − i+1p i(t)  ≤ λ1i (i =2, , m) (1.50)

hold almost everywhere on ]a, t0[, and the inequalities

(−1)n − m(t − a) n −2m(b − t)2mp1(t) ≤ λ21,(t − a) n −2m(b − t)2m− i+1p i(t)  ≤ λ2i (i =2, , m) (1.51)hold almost everywhere on ]t0,b[ Then for every q ∈ L2

n −2m−2,2m−2(]a, b[) problem ( 1.1 ), ( 1.2 ) is uniquely solvable in the space Cn −1,m(]a, b[).

Theorem 1.11 Let there exist nonnegative numbers  i(i =1, , m) such that along with ( 1.25 ) the conditions

Remark 1.13 The above-given conditions on the unique solvability of problems (1.1),(1.2) and (1.1), (1.3) are optimal since, asExample 1.8shows, in Theorems1.9,1.11andCorollaries1.10,1.12none of strict inequalities (1.21), (1.23), (1.25), and (1.27) can bereplaced by nonstrict ones

Remark 1.14 If along with the conditions of Theorem 1.9 (of Theorem 1.11) tions (1.28) are satisfied as well, then for everyq ∈ L2n −2m−2,m−2(]a, b[) (for every q ∈

2.1 Lemmas on integral inequalities Throughout this section, we assume that−∞ <

t0< t1< + ∞, and for any function u :]t0,t1[→ R, byu(t0) andu(t1) we understand theright and the left limits of that function at the pointst0andt1

Lemma 2.1 Let u ∈ Cloc(]t0,t1]) and

where α = −1 If, moreover, either

If conditions (2.2) are fulfilled, then in view of (2.1), (2.7) results in (2.4)

It remains to consider the case when conditions (2.3) hold Then due to (2.1) we have

On the other hand, from (2.7) we have

If in this inequality we pass to the limit ass → t0, then we get inequality (2.4) 

Lemma 2.2 Let u ∈ Cloc(]t0,t1]) and

Consequently, inequality (2.12) is valid

Now we consider the case where conditions (2.3) hold Then, taking into account(2.11), we obtain

Hence it is obvious thatu satisfies equality (2.9) On the other hand, (2.5) yields

If in this inequality we pass to the limit ass → t0, then we obtain inequality (2.12) 

Lemma 2.3 Let α > −1 and

The following lemma easily follows fromLemma 2.3

Lemma 2.4 Let α > −1, β > −1, and

Lemma 2.5 Let u ∈ Clocm −1(]t0,t1[),

Remark 2.6 Inequality (2.23) cannot be replaced by the inequality

loc (]t0,t1[) be a function satisfying conditions ( 2.22 ), and p ∈

Lloc(]t0,t1]) be such that

where j ∈ {1, , m } and 0> 0 Then

The following lemma can be proved similarly toLemma 2.7

Lemma 2.6 Let u ∈ C mloc−1(]t0,t1[) be a function satisfying conditions ( 2.27 ), and p ∈

Lloc([t0,t1[) be such that

where j ∈ {1, , m } and 0> 0 Then

2.2 A lemma on the properties of functions from the spaceCn −1,m(]a, b[) In this

sec-tion, as above, we assume thatm is the integral part of the number n/2.

The proof of this lemma is given in [12]

2.3 Lemmas on the sequences of solutions of auxiliary problems Suppose

a < t0k< t1k< b (k =1, 2, .), lim

k →+∞ t0k= a, lim

k →+∞ t1k= b. (2.43)For the differential equation

we consider the auxiliary boundary conditions

Throughout this section, when problems (1.1), (1.2) and (2.44), (2.45) are discussed,

(That is, uniformly on [a + δ, b − δ] for an arbitrarily small δ > 0).

Proof For an arbitrary (m −1)-times continuously differentiable function v :]a,b[→ R,

andg i(t) (i =1, , n) are the polynomials of (n −1)th degree, satisfying the conditions

whereI(t0)=[t0, (a + b)/2] for t0< (a + b)/2 and I(t0)=[(a + b)/2, t0] fort0> (a + b)/2.

In view of inequalities (2.50), the identities

u(ik −1)(t) = 1

(m − i)!

t

t (t − s) m − i u(m)k (s)ds (j =0, 1;i =1, , m; k =1, 2, .) (2.61)

By (2.68) and (2.70), (2.50) results in (2.52) Therefore,u ∈ C n −1,m([a, b[) On the other

hand, from (2.66) it is obvious thatu is a solution of (1.1) In the case, wheren =2m,

from (2.67) equalities (1.2) follow, that is,u is a solution of problem (1.1), (1.2)

Let us show thatu is a solution of that problem in the case n =2m + 1 as well In view

of (2.67), it suffices to prove that u(m)(b) =0 First we find an estimate for the sequence(u(m+1)k )+∞

k =1 For this, without loss of generality we assume that

Form > 1, due to condition (2.73) we have

whereρ ∗ = ρ0(b − t1)1/2+ 2(ρ + ρ0) Ifm =1, then by virtue of inequalities (2.72), (2.77),and (2.81), from (2.82) we find

that is, again estimate (2.83) is valid

By virtue of (2.43), (2.68) and (2.70), (2.83) implies

u(m)(t)  ≤ ρ ∗(b − t)1/2 fort1≤ t < b, (2.85)

and consequently,u(m)(b) =0 Thus we proved thatu is a solution of problem (1.1), (1.2)also in the casen =2m + 1 In the space Cn −1,m(]a, b[) problem (1.1), (1.2) does not haveanother solution since in that space the homogeneous problem (1.10), (1.2) has only atrivial solution

To complete the proof of the lemma, it remains to show that condition (2.53) is isfied Assume the contrary Then there existδ ∈]0, ( b − a)/2[, ε > 0, and an increasing

sat-sequence of natural numbers (k )+ = ∞1such that

Analogously we can prove the following lemma

Lemma 2.10 Let for every natural k, problem ( 2.44 ), ( 2.46 ) have a solution u k ∈ C n −1

loc (]a, b]), and there exist a nonnegative constant r0such that inequalities ( 2.50 ) are fulfilled Let, moreover,

lim

k →+∞ u(ik −1)(t) = u(i−1)(t) (i =1, , n) uniformly in ]a, b]. (2.88)

2.4 Lemmas on a priori estimates.

Lemma 2.11 Let conditions ( 1.20 ) and ( 1.21 ) be fulfilled, where h i(i =1, , m) are tions given by equalities ( 1.13 ), a0∈] a, b[, b0∈] a0,b[, and 1i, 2i(i =1, , m) are non- negative numbers Then there exists a positive constant r0 such that for any t0∈] a, a0[,

Proof of Lemma 2.11 By virtue of inequalities (1.21), there existsγ ∈]0, 1[ such that

r0=22m+2(1 +b − a)2γ −2. (2.94)

Assume now that for somet0∈] a, a0[,t1∈] b0,b[, and q ∈ L2

n −2m−2,2m−2(]a, b[) problem

(1.1), (2.89) has a solutionu Multiplying (1.1) by (−1)n − m(t − a) n −2mu(t) and then

inte-grating fromt0tot1, byLemma 2.12we obtain

On the other hand, if we putc =(a + b)/2, then again on the basis of Lemmas2.7and

The proof of the following lemma is analogous to that ofLemma 2.11

Lemma 2.13 Let conditions ( 1.12 ), ( 1.24 ), and ( 1.25 ) hold, where h i(i =1, , m) are tions given by equalities ( 1.13 ), a0∈] a, b[, and  i(i =1, , m) are nonnegative numbers Then there exists a positive constant r0such that for any t0∈] a, a0[ and q ∈ L2

Lemma 2.14 Let conditions ( 1.10 ), ( 1.20 ), and ( 1.21 ) hold, and in the case, where n is odd, in addition condition ( 1.11 ) be fulfilled, where h i(i =1, , m) are functions given by equalities ( 1.13 ), a0∈] a, b[, b0∈] a0,b[, and 1i, 2i(i =1, , m) are nonnegative numbers Let, moreover, the homogeneous problem ( 1.10), ( 1.2 ) in the space Cn −1,m(]a, b[) have only a trivial solution Then there exist δ ∈]0, ( b − a)/2[ and r > 0 such that for any t0∈] a, a + δ],

n −2m−2,2m−2(]a, b[) such that problem (2.44), (2.45) has a solution

u k ∈ Clocn −1(]a, b[) satisfying the inequality

On the other hand, by Lemmas2.9and2.11, we have

wherer0is a positive constant independent ofk Thus if we pass to the limit in the last

inequality ask →+∞, then we obtain the contradiction 1≤0, which proves the lemma



Analogously we can prove the following lemma if we apply Lemmas2.10and2.13

instead of Lemmas2.9and2.11

Lemma 2.15 Let conditions ( 1.12 ), ( 1.24 ), and ( 1.25 ) hold, where h i(i =1, , m) are tions given by equalities ( 1.13 ), a0∈] a, b[, and  i(i =1, , m) are nonnegative numbers Let, moreover, the homogeneous problem ( 1.10), ( 1.3 ) in the space Cn −1,m(]a, b]) have only a trivial solution Then there exist δ ∈]0, b − a[ and r > 0 such that for any t0∈] a, a + δ] and

func-q ∈ L2

n −2m−2(]a, b]) problem ( 1.1 ), ( 2.99 ) is uniquely solvable in the space Cn −1

loc (]a, b]) and its solution admits the estimate

3 Proof of the main results

Proof of Theorem 1.3 ( Theorem 1.5 ) Suppose problem (1.10), (1.2) (problem (1.10),(1.3)) has only a trivial solution, andr and δ are the numbers appearing inLemma 2.14

loc (]a, b[) (in the space Cn −1

loc (]a, b])) has a unique solution u kand

(That is, uniformly on [a + δ, b − δ] for an arbitrarily small δ > 0).

Proof For an arbitrary... the boundary conditions (1.2) (the boundary conditions (1.3))

if and only ifc1, , c n − m −1are solutions of the system of linear. .. infinite sets of solutions Indeed, for any

for anyc n − m+1 ∈ R However, this system has a unique solution for an arbitrarily fixed