**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-point conjugate and right-focal boundary conditions, we provide easily verifiable best possible conditions 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 equa-**
tion

*u*^{(n)}*=*
*m*
*i**=*1

*p** _{i}*(t)u

^{(i}

^{−}^{1)}+

*q(t)*(1.1)

with the conjugate boundary conditions

*u*^{(i}^{−}^{1)}(a)*=*0 (i*=*1,*. . .,m),*

*u*^{(}^{j}^{−}^{1)}(b)*=*0 (*j**=*1,. . .,n*−**m)* (1.2)
or the right-focal boundary conditions

*u*^{(i}^{−}^{1)}(a)*=*0 (i*=*1,*. . .,m),*

*u*^{(}^{j}^{−}^{1)}(b)*=*0 (*j**=**m*+ 1,. . .,n). (1.3)
Here*n**≥*2,*m*is the integer part of*n/2,**−∞**< a < b <*+*∞*,*p**i**∈**L*loc(]a,b[) (i*=*1,*. . .,n),*
*q**∈**L*loc(]a,b[), and by*u*^{(i}^{−}^{1)}(a) (by*u*^{(j}^{−}^{1)}(b)) is understood the right (the left) limit of
the function*u*^{(i}^{−}^{1)}(of the function*u*^{(j}^{−}^{1)}) at the point*a*(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, Pages1–32 DOI10.1155/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

_{b}

*a*(t*−**a)*^{n}^{−}^{1}(b*−**t)*^{2m}^{−}^{1}^{}(*−*1)^{n}^{−}^{m}*p*1(t)^{}_{+}*dt <*+*∞*,
_{b}

*a*(t*−**a)*^{n}^{−}* ^{i}*(b

*−*

*t)*

^{2m}

^{−}

^{i}^{}

*p*

*i*(t)

^{}

*dt <*+

*∞*(i

*=*2,

*. . .,m),*

_{b}*a*(t*−**a)*^{n}^{−}^{m}^{−}^{1/2}(b*−**t)*^{m}^{−}^{1/2}^{}*q(t)*^{}*dt <*+*∞*,

(1.4)

_{b}

*a*(t*−**a)*^{n}^{−}^{1}^{}(*−*1)^{n}^{−}^{m}*p*1(t)^{}_{+}*dt <*+*∞*,
_{b}

*a*(t*−**a)*^{n}^{−}^{i}^{}*p**i*(t)^{}*dt <*+*∞* (i*=*2,*. . .,m),*
_{b}

*a*(t*−**a)*^{n}^{−}^{m}^{−}^{1/2}^{}*q(t)*^{}*dt <*+*∞*

(1.5)

(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 number*x, that is,*
[x]+*=**x*+*|**x**|*

2 *.* (1.6)

*L*loc(]a,b[) (Lloc(]a,b])) is the space of functions *y*:]a,*b[**→*Rwhich are integrable on
[a+*ε,b**−**ε] (on [a*+*ε,b]) for arbitrarily smallε >*0.

*L**α,β*(]a,b[) (L^{2}* _{α,β}*(]a,b[)) is the space of integrable (square integrable) with the weight
(t

*−*

*a)*

*(b*

^{α}*−*

*t)*

*functions*

^{β}*y*:]a,b[

*→*Rwith the norm

*y**L**α,β**=*
_{b}

*a*(t*−**a)** ^{α}*(b

*−*

*t)*

^{β}^{}

*y(t)*

^{}

*dt*

*y*_{L}^{2}_{α,β}*=*
_{b}

*a*(t*−**a)** ^{α}*(b

*−*

*t)*

^{β}*y*

^{2}(t)dt

1/2
*.*

(1.7)
*L([a,b])*_{=}*L*0,0(]a,b[),*L*^{2}([a,b])_{=}*L*^{2}_{0,0}(]a,b[).

*L*^{2}* _{α,β}*(]a,b[) (

^{}

*L*

^{2}

*(]a,b])) is the space of functions*

_{α}*y*

*∈*

*L*loc(]a,b[) (y

*∈*

*L*loc(]a,b])) such that

*y*

*∈*

*L*

^{2}

*(]a,*

_{α,β}*b[), where*

*y(t)*

*=*

_{t}*c**y(s)ds,c**=*(a+*b)/2 (**y**∈**L*^{2}* _{α,0}*(]a,b[), where

*y(t)*

*=*

_{b}*t* *y(s)ds).*

* · **L*^{2}* _{α,β}*and

*·*

*L*

^{2}

*denote the norms in*

_{α}^{}

*L*

^{2}

*(]a,b[) and*

_{α,β}^{}

*L*

^{2}

*(]a,b]), and are defined by the equalities*

_{α}*y*_{}_{L}^{2}

*α,β**=*max
_{t}

*a*(s*−**a)*^{α}*t*

*s* *y(τ)dτ*

2

*ds*
1/2

:*a**≤**t**≤**a*+*b*
2

+ max
_{b}

*t*(b*−**s)*^{β}_{s}

*t* *y(τ)dτ*

2

*ds*
1/2

:*a+b*
2 ^{≤}*t**≤**b*

,
*y**L*^{2}_{α}*=*max

_{t}

*a*(s*−**a)*^{α}_{t}

*s* *y(τ)dτ*

2

*ds*
1/2

:*a**≤**t**≤**b*

*.*

(1.8)

*C*^{n}_{loc}^{−}^{1}(]a,*b[) (C*^{}_{loc}^{n}^{−}^{1}(]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*arbitrarily small

*ε >*0.

*C*^{n}^{−}^{1,m}(]a,b[) (*C*^{}^{n}^{−}^{1,m}(]a,b])) is the space of functions*y**∈**C*_{loc}^{n}^{−}^{1}(]a,b[) (y*∈**C*^{n}_{loc}^{−}^{1}(]a,
*b])) such that*

_{b}

*a*

*y*^{(m)}(s)^{}^{2}*ds <*+*∞**.* (1.9)

In what follows, when problem (1.1), (1.2) is discussed, we assume that in the case*n**=*2m
the conditions

*p*_{i}*∈**L*loc

]a,*b[*^{} (i*=*1,. . .,m) (1.10)

are fulfilled, and in the case*n**=*2m+ 1 along with (1.10) the condition
lim sup

*t**→**b*

(b*−**t)*^{2m}^{−}^{1}
_{t}

*c* *p*1(s)ds^{}_{}*<*+*∞*, *c**=**a*+*b*

2 (1.11)

is also satisfied.

As for problem (1.1), (1.3), it is investigated under the assumptions
*p*_{i}*∈**L*loc

]a,b]^{} (i*=*1,. . .,m). (1.12)

A solution of problem (1.1), (1.2) (of problem (1.1), (1.3)) is sought in the space
*C*^{n}^{−}^{1,m}(]a,*b[) (in the spaceC*^{}^{n}^{−}^{1,m}(]a,b])).

By*h**i*:]a,*b[**×*]a,b[*→*[0, +*∞*[ (i*=*1,*. . .,m) we understand the functions defined by the*
equalities

*h*1(t,*τ)**=*
_{t}

*τ*(s*−**a)*^{n}^{−}^{2m}^{}(*−*1)^{n}^{−}^{m}*p*1(s)^{}_{+}*ds*^{}_{},
*h**i*(t,τ)*=*

_{t}

*τ*(s*−**a)*^{n}^{−}^{2m}*p**i*(s)ds^{}_{} (i*=*2,*. . .,m).*

(1.13)

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

*u*^{(n)}*=*
*m*
*i**=*1

*p**i*(t)u^{(i}^{−}^{1)}*.* (1.10)

From [10, Corollary 1.1] it follows that if
*p**i**∈**L**n**−**m,m*

]a,b[^{} (i*=*1,*. . .,m),*

*p*_{i}*∈**L*_{n}_{−}_{m,0}^{}]a,b[^{} (i*=*1,*. . .,m)*^{} (1.14)
and the homogeneous problem (1.10), (1.2) (problem (1.10), (1.3)) has only a trivial solu-
tion in the space*C*^{}_{loc}^{n}^{−}^{1}(]a,b[) (in the space*C*_{loc}^{n}^{−}^{1}(]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 the space

*C*

^{}

_{loc}

^{n}

^{−}^{1}(]a,b[) (in the space

*C*

^{}

_{loc}

^{n}

^{−}^{1}(]a,b])).

In the case where condition (1.14) is violated, the question on the presence of the
Fredholm property for problem (1.1), (1.2) (for problem (1.1), (1.3)) in some subspace
of the space*C*^{}^{n}_{loc}^{−}^{1}(]a,*b[) (of the space* *C*^{}_{loc}^{n}^{−}^{1}(]a,b])) remained so far open. This ques-
tion is answered inTheorem 1.3(Theorem 1.5) formulated below which contains opti-
mal in a certain sense conditions guaranteeing the presence of the Fredholm property for
problem (1.1), (1.2) (for problem (1.1), (1.3)) in the space*C*^{}^{n}^{−}^{1,m}(]a,b[) (in the space
*C*^{n}^{−}^{1,m}(]a,*b])).*

*Definition 1.1. We say that problem (1.1), (1.2) (problem (1.1), (1.3)) has the Fredholm*
property in the space*C*^{}^{n}^{−}^{1,m}(]a,b[) (in the space*C*^{}^{n}^{−}^{1,m}(]a,b])) if the unique solvability
of the corresponding homogeneous problem (1.10), (1.2) (problem (1.10), (1.3)) in this
space implies the unique solvability of problem (1.1), (1.2) (problem (1.1), (1.3)) in the
space*C*^{}^{n}^{−}^{1,m}(]a,b[) (in the space*C*^{}^{n}^{−}^{1,m}(]a,b])) for every*q**∈**L*^{2}_{2n}_{−}_{2m}_{−}_{2,2m}* _{−}*2(]a,b[) (for
every

*q*

*∈*

*L*

^{2}

_{2n}

_{−}_{2m}

_{−}_{2}(]a,b])) and for its solution the following estimate

*u*^{(m)}^{}* _{L}*2

*≤*

*r*

*q*

*L*

^{2}

_{2n}

_{−}_{2m}

_{−}_{2,2m}

*2*

_{−}*u*^{(m)}^{}* _{L}*2

*≤*

*r*

*q*

*L*

^{2}

_{2n}

_{−}_{2m}

*2*

_{−}

(1.15)
is valid, where*r*is a positive constant independent of*q.*

*Remark 1.2. If*

*q**∈**L*^{2}_{2n}_{−}_{2m,2m}^{}]a,b[^{} ^{}*q**∈**L*^{2}_{2n}_{−}_{2m,0}^{}]a,b[^{} (1.16)
or

*q**∈**L**n**−**m**−*1/2,m*−*1/2

]a,b[^{} ^{}*q**∈**L**n**−**m**−*1/2,0

]a,b[^{}, (1.17)

then

*q**∈**L*^{2}_{2n}_{−}_{2m}_{−}_{2,2m}_{−}_{2}^{}]a,*b[*^{} ^{}*q**∈**L*^{2}_{2n}_{−}_{2m}_{−}_{2}^{}]a,b]^{} (1.18)

and from estimate (1.15) there respectively follow the estimates
*u*^{(m)}^{}* _{L}*2

*≤*

*r*0

*q*

_{L}^{2}

_{2n+2m,2m}

*u*

^{(m)}

^{}

*2*

_{L}*≤*

*r*0

*q*

_{L}^{2}

_{2n}

_{−}_{2m,0}

,
*u*^{(m)}^{}* _{L}*2

*≤*

*r*0

*q*

*L*

*n*

*−*

*m*

*−*1/2,m

*−*1/2

*u*^{(m)}^{}* _{L}*2

*≤*

*r*0

*q*

*L*

*n*

*−*

*m*

*−*1/2,0

, (1.19)

where*r*0is a positive constant independent of*q.*

*Theorem 1.3. Let there exista*0*∈*]a,b[,*b*0*∈*]a0,b[ and nonnegative numbers_{1i}*,*_{2i}(i*=*
1,*. . .,m) such that*

(t*−**a)*^{2m}^{−}^{i}*h** _{i}*(t,τ)

*≤*

_{1i}

*fora < t*

*≤*

*τ*

*≤*

*a*0,

(b*−**t)*^{2m}^{−}^{i}*h**i*(t,*τ)**≤*2i *forb*0*≤**τ**≤**t < b*(i*=*1,. . .,m), (1.20)
*m*

*i**=*1

(2m*−**i)2*^{n}^{−}^{i+1}

(2m*−*2i+ 1)!!_{1i}*<*(2n*−*2m*−*1)!!,
*m*

*i**=*1

(2m*−**i)2*^{n}^{−}^{i+1}

(2m*−*2i+ 1)!!2i*<*(2n*−*2m*−*1)!!,

(1.21)

*where (2n**−*2i*−*1)!!*=*1.3*···*(2n*−*2i*−*1).*Then problem (1.1), (1.2) has the Fredholm*
*property in the spaceC*^{}^{n}^{−}^{1,m}(]a,b[).

*Corollary 1.4. Let there exist nonnegative numbersλ*_{1i}*,λ*_{2i} (i*=*1,*. . .,m) and functions*
*p*0i*∈**L**n**−**i,2m**−**i*(]a,*b[) (i**=*1,*. . .,m) such that the inequalities*

(*−*1)^{n}^{−}^{m}*p*1(t)*≤* *λ*11

(t*−**a)** ^{n}*+

*λ*21

(t*−**a)*^{n}^{−}^{2m}(b*−**t)*^{2m}+*p*01(t),
*p**i*(t)^{}*≤* *λ*_{1i}

(t*−**a)*^{n}^{−}* ^{i+1}*+

*λ*

_{2i}

(t*−**a)*^{n}^{−}^{2m}(b*−**t)*^{2m}^{−}* ^{i+1}*+

*p*0i(t) (i

*=*2,

*. . .,m)*

(1.22)

*hold almost everywhere on ]a,b[, and*
*m*

*i**=*1

2^{n}^{−}^{i+1}

(2m*−*2i+ 1)!!*λ*_{1i}*<*(2n*−*2m*−*1)!!,
*m*

*i**=*1

2^{n}^{−}^{i+1}

(2m*−*2i+ 1)!!*λ*2i*<*(2n*−*2m*−*1)!!.

(1.23)

*Then problem (1.1), (1.2) has the Fredholm property in the spaceC*^{}^{n}^{−}^{1,m}(]a,b[).

*Theorem 1.5. Let there exista*0*∈*]a,b[ and nonnegative numbers* _{i}*(i

*=*1,. . .,m) such that (t

*−*

*a)*

^{2m}

^{−}

^{i}*h*

*i*(t,τ)

*≤*

*i*

*fora < t*

*≤*

*τ*

*≤*

*a*0(i

*=*1,

*. . .,m),*(1.24)

*m*
*i**=*1

(2m*−**i)2*^{n}^{−}^{i+1}

(2m*−*2i+ 1)!!_{i}*<*(2n*−*2m*−*1)!!. (1.25)
*Then problem (1.1), (1.3) has the Fredholm property in the spaceC*^{}^{n}^{−}^{1,m}(]a,b]).

*Corollary 1.6. Let there exist nonnegative numbersλ**i*(i*=*1,. . .,m) and functions*p*0i*∈*
*L**n**−**i,0*(]a,b[) (i*=*1,*. . .,m) such that the inequalities*

(*−*1)^{n}^{−}^{m}*p*1(t)*≤* *λ*1

(t*−**a)** ^{n}*+

*p*01(t),

*p*

*i*(t)

^{}

*≤*

*λ*

*i*

(t*−**a)*^{n}^{−}* ^{i+1}*+

*p*0i(t) (i

*=*2,

*. . .,m)*

(1.26)

*hold almost everywhere on ]a,b[, and*
*m*

*i**=*1

2^{n}^{−}^{i+1}

(2m*−*2i+ 1)!!*λ**i**<*(2n*−*2m*−*1)!!. (1.27)
*Then problem (1.1), (1.3) has the Fredholm property in the spaceC*^{}^{n}^{−}^{1,m}(]a,b]).

In connection with the above-mentioned Corollary 1.1 from [10], there naturally
arises 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 *C*^{}^{n}^{−}^{1,m}(]a,b[) (in the space
*C*^{n}^{−}^{1,m}(]a,*b])) guarantees the unique solvability of that problem in the spaceC*^{}_{loc}^{n}^{−}^{1}(]a,b[)
(in the space*C*^{}^{n}_{loc}^{−}^{1}(]a,b])).

The following theorem is valid.

*Theorem 1.7. If*

*p**i**∈**L**n**−**i,2m**−**i*

]a,b[^{} (i*=*1,*. . .,m),*
*p**i**∈**L**n**−**i,0*

]a,b[^{} (i*=*1,*. . .,m)*^{}, (1.28)
*and problem (1.1), (1.2) (problem (1.1), (1.3)) is uniquely solvable in the spaceC*^{}^{n}^{−}^{1,m}(]a,
*b[) (in the spaceC*^{}^{n}^{−}^{1,m}(]a,b])), then this problem is uniquely solvable in the space*C*^{}^{n}_{loc}^{−}^{1}(]a,
*b[) (in the spaceC*^{}^{n}_{loc}^{−}^{1}(]a,*b])) as well.*

If condition (1.28) is violated, then, as it is clear from the example below, problem
(1.1), (1.2) (problem (1.1), (1.3)) may be uniquely solvable in the space*C*^{}^{n}^{−}^{1,m}(]a,b[)
(in the space*C*^{}^{n}^{−}^{1,m}(]a,b])) and this problem may have an infinite set of solutions in the
space*C*^{}_{loc}^{n}^{−}^{1}(]a,b[) (in the space*C*^{}_{loc}^{n}^{−}^{1}(]a,b])).

*Example 1.8. Suppose*

*g**n*(x)*=**x(x**−*1)*···*(x*−**n*+ 1). (1.29)
Then

(*−*1)^{n}^{−}^{m}*g**n*

*m**−*1

2 * ^{=}*2

^{−}*(2m*

^{n}*−*1)!!(2n

*−*2m

*−*1)!!, (1.30)

*g*

_{n}

^{}
*m**−*1

2 * ^{=}*0 for

*n*

*=*2m,

*g*

_{n}

^{}*m**−*1
2 *g*_{n}

*m**−*1

2 *<*0 for*n**=*2m+ 1, (1.31)
(*−*1)^{n}^{−}^{m}*g**n*

*k**−*1

2 *>*(*−*1)^{n}^{−}^{m}*g**n*

*m**−*1

2 for*k**∈ {*0,. . .,n*}*and*m**−**k*is even. (1.32)

If

*p*1(t)*=* *λ*

(t*−**a)** ^{n}*,

*p*

*i*(t)

*=*0 (i

*=*2,

*. . .,n),*(1.33) and

*q(t)*

*=*(g

*(ν)*

_{n}*−*

*λ)t*

^{ν}

^{−}*, where*

^{n}*λ*

*=*0,

*ν>*0, then (1.1) and (1.10) have the forms

*u*^{(n)}*=* *λ*

(t*−**a)*^{n}*u*+^{}*g**n*(*ν*)*−**λ*^{}(t*−**a)*^{ν}^{−}* ^{n}*, (1.34)

*u*

^{(n)}

*=*

*λ*

(t*−**a)*^{n}*u.* (1.340)

First we consider the case where

*λ**=**g*_{n}

*m**−*1

2 *.* (1.35)

Then from (1.31) and (1.32) it easily follows that the characteristic equation

*g** _{n}*(x)

*=*

*λ*(1.36)

has only real roots*x** _{i}*(i

*=*1,. . .,

*n) such that*

*x*1

*=*

*x*2

*=*1

2 for*n**=*2,
*x*1*>**···**> x**m**−*1*> m**−*1

2^{=}*x**m**=**x**m+1**>**···**> x*2m for*n**=*2m,
*x*1*>**···**> x*_{m}*> m**−*1

2*> x*_{m+1}*>**···**> x*_{2m+1} for*n**=*2m+ 1.

(1.37)

Hence it is evident that for*n**=*2 (1.340) does not have a solution belonging to the space
*C*^{1,1}(]a,b[), and for*n >*2 solutions of that equation from the space*C*^{}^{n}^{−}^{1,m}(]a,b[) consti-
tute an (n*−**m**−*1)-dimensional subspace with the basis

(t*−**a)*^{x}^{1},. . ., (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 the
space*C*^{}^{n}^{−}^{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 space*C*^{}^{n}^{−}^{1,m}(]a,*b[). Indeed, ifn**=*2, then (1.34)
has the unique solution*u(t)**=*(t*−**a)** ^{ν}*in the space

*C*

^{}

^{1,1}(]a,b[), and this solution does not satisfy conditions (1.2). If

*n >*2, then an arbitrary solution of (1.34) from

*C*

^{}

^{n}

^{−}^{1,m}(]a,b[) has the form

*u(t)**=*

*n**−**m**−*1
*i**=*1

*c** _{i}*(t

*−*

*a)*

^{x}*+ (t*

^{i}*−*

*a)*

*, (1.39)*

^{ν}and this solution satisfies the boundary conditions (1.2) (the boundary conditions (1.3))
if and only if*c*1,. . .,c*n**−**m**−*1are solutions of the system of linear algebraic equations

*n**−**m**−*1
*i**=*1

*g*_{k}^{}*x*_{i}^{}(b*−**a)*^{x}^{i}*c*_{i}*= −**g** _{k}*(

*ν*)(b

*−*

*a)*

*(k*

^{ν}*=*0,. . .,n

*−*

*m*

*−*1)

_{n}

_{−}

_{m}

_{−}_{1}

*i**=*1

*g**k*

*x**i*

(b*−**a)*^{x}^{i}*c**i**= −**g**k*(ν)(b*−**a)** ^{ν}* (k

*=*

*m,. . .*,n

*−*1)

,

(1.40)

where*g*0(x)*≡*1,*g**k*(x)*=**x(x**−*1)*···*(x*−**k*+ 1) for*x**≥*1. However, this system does not
have a solution for large*ν.*

Note that in the case under consideration the functions*p**i*(i*=*1,*. . .,m) in view of con-*
ditions (1.30) and (1.32) satisfy inequalities (1.22) (inequalities (1.26)), where*λ*11*= |**λ**|*,
*λ*1i*=**λ*21*=**λ*2i*=*0 (i*=*2,. . .,m) (λ1*= |**λ**|*,*λ**i**=*0 (i*=*2,. . .,m)), *p*0i(t)*≡*0 (i*=*1,*. . .,m),*
and

*m*
*i**=*0

2^{n}^{−}^{i+1}

(2m*−*2i+ 1)!!*λ*1i*=*(2n*−*2m*−*1)!!

_{m}

*i**=*0

2^{n}^{−}^{i+1}

(2m*−*2i+ 1)!!*λ**i**=*(2n*−*2m*−*1)!!

*.*

(1.41)

Therefore we showed that in Theorems1.3,1.5 and their corollaries none of strict in- equalities (1.21), (1.23), (1.25), and (1.27) can be replaced by nonstrict ones, and in this sense the above-given conditions on the presence of the Fredholm property for problems (1.1), (1.2) and (1.1), (1.3) are the best possible.

Now we consider the case, where

0*<*(*−*1)^{n}^{−}^{m}*λ <*(*−*1)^{n}^{−}^{m}*g**n*

*m**−*1

2 *.* (1.42)

Then, in view of (1.30) and (1.33), the functions*p**i*(i*=*1,*. . .,m) satisfy all the conditions*
of Corollaries1.4and1.6, but condition (1.28) inTheorem 1.7is violated. On the other
hand, according to conditions (1.31) and (1.32), the characteristic equation (1.36) has
simple real roots*x*1,. . .,x*n*such that

*x*1*>**···**> x**n**−**m**> m**−*1

2*> x**n**−**m+1**>**···**> x**n*, (1.43)
at that

*x*_{n}_{−}_{m+1}*> m** _{−}*1. (1.44)

So, the set of solutions of (1.340) from*C*^{}^{n}^{−}^{1,m}(]a,b[) constitutes an (n*−**m)-dimensional*
subspace with the basis

(t*−**a)*^{x}^{1},*. . ., (t**−**a)*^{x}^{n}^{−}* ^{m}*, (1.45)

and consequently, both problem (1.340), (1.2) and problem (1.340), (1.3) in the men-
tioned space have only trivial solutions. Hence in view of Corollaries 1.4 and 1.6 the
unique solvability of problems (1.34), (1.2) and (1.34), (1.3) follows in*C*^{}^{n}^{−}^{1,m}(]a,b[). Let
us show that these problems in*C*^{}^{n}_{loc}^{−}^{1}(]a,*b]) have infinite sets of solutions. Indeed, for any*
*c*_{i}*∈*R(i*=*1,. . .,n*−**m*+ 1), the function

*u(t)**=*

*n**−**m+1*
*i**=*1

*c**i*(t*−**a)*^{x}* ^{i}*+ (t

*−*

*a)*

*(1.46) is a solution of (1.34) from*

^{ν}*C*

^{}

_{loc}

^{n}

^{−}^{1}(]a,b]), satisfying the conditions

*u*^{(i}^{−}^{1)}(a)*=*0 (i*=*1,. . .,m). (1.47)
This function satisfies the boundary conditions (1.2) (the boundary conditions (1.3)) if
and only if*c*1,. . .,c*n**−**m*are solutions of the system of algebraic equations

*n**−**m*
*i**=*1

*g**k*

*x**i*

(b*−**a)*^{x}^{i}*c**i**=*

*−**g**k*

*x**n**−**m+1*

(b*−**a)*^{x}^{n}^{−}^{m+1}*c**n**−**m+1**−**g**k*(ν)(b*−**a)** ^{ν}*(k

*=*0,

*. . .,n*

*−*

*m*

*−*1)

_{n}

_{−}

_{m}

*i**=*1

*g**k*
*x**i*

(b*−**a)*^{x}^{i}*c**i**=*

*−**g**k*
*x**n**−**m+1*

(b*−**a)*^{x}^{n}^{−}^{m+1}*c**n**−**m+1**−**g**k*(*ν*)(b*−**a)** ^{ν}*(k

*=*

*n*

*−*

*m,. . .*,m)

(1.48)

for any*c*_{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 of
solutions in the space*C*^{}_{loc}^{n}^{−}^{1}(]a,b]).

**1.3. Existence and uniqueness theorems.**

*Theorem 1.9. Let there existt*0*∈*]a,b[ and nonnegative numbers1i*,*2i(i*=*1,. . .,m) such
*that along with (1.21) the conditions*

(t*−**a)*^{2m}^{−}^{i}*h**i*(t,τ)*≤*1i *fora < t**≤**τ**≤**t*0,

(b*−**t)*^{2m}^{−}^{i}*h** _{i}*(t,τ)

*≤*

_{2i}

*fort*0

*≤*

*τ*

*≤*

*t < b*(1.49)

*hold. Then for everyq*

*∈*

*L*

^{2}

_{2n}

_{−}_{2m}

_{−}_{2,2m}

*2(]a,b[) problem (1.1), (1.2) is uniquely solvable in*

_{−}*the spaceC*

^{}

^{n}

^{−}^{1,m}(]a,b[).

*Corollary 1.10. Let there existt*0*∈*]a,b[ and nonnegative numbers*λ*1i*,λ*2i(i*=*1,*. . .,m)*
*such that conditions (1.23) are fulfilled, the inequalities*

(*−*1)^{n}^{−}* ^{m}*(t

*−*

*a)*

^{n}*p*1(t)

*≤*

*λ*11, (t

*−*

*a)*

^{n}

^{−}

^{i+1}^{}

*p*

*i*(t)

^{}

*≤*

*λ*1i (i

*=*2,. . .,m) (1.50)

*hold almost everywhere on ]a,t*0*[, and the inequalities*

(*−*1)^{n}^{−}* ^{m}*(t

*−*

*a)*

^{n}

^{−}^{2m}(b

*−*

*t)*

^{2m}

*p*1(t)

*≤*

*λ*21,

(t*−**a)*^{n}^{−}^{2m}(b*−**t)*^{2m}^{−}^{i+1}^{}*p**i*(t)^{}*≤**λ*2i (i*=*2,*. . .,m)* (1.51)
*hold almost everywhere on ]t*0,b[. Then for every*q**∈**L*^{2}_{2n}_{−}_{2m}_{−}_{2,2m}_{−}_{2}(]a,b[) problem (1.1),
*(1.2) is uniquely solvable in the spaceC*^{}^{n}^{−}^{1,m}(]a,b[).

*Theorem 1.11. Let there exist nonnegative numbers** _{i}*(i

*=*1,. . .,m) such that along with

*(1.25) the conditions*

(t*−**a)*^{2m}^{−}^{i}*h** _{i}*(t,τ)

*≤*

_{i}*fora < t*

*≤*

*τ*

*≤*

*b*(i

*=*1,

*. . .,m)*(1.52)

*hold. Then for every*

*q*

*∈*

*L*

^{2}

_{2n}

_{−}_{2m}

*2(]a,b]) problem (1.1), (1.3) is uniquely solvable in the*

_{−}*spaceC*

^{}

^{n}

^{−}^{1,m}(]a,b]).

*Corollary 1.12. Let there exist nonnegative numbersλ**i*(i*=*1,. . .,m) such that condition
*(1.27) holds, and the inequalities*

(*−*1)^{n}^{−}* ^{m}*(t

*−*

*a)*

^{n}*p*1(t)

*≤*

*λ*1, (t

*−*

*a)*

^{n}

^{−}

^{i+1}^{}

*p*

*i*(t)

^{}

*≤*

*λ*1i (i

*=*2,

*. . .,m)*(1.53)

*are fulfilled almost everywhere on ]a,b[. Then for everyq*

*∈*

*L*

^{2}

_{2n}

_{−}_{2m}

_{−}_{2}(]a,b]) problem (1.1),

*(1.3) is uniquely solvable in the spaceC*

^{}

^{n}

^{−}^{1,m}(]a,b]).

*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 Theorems1.9,1.11and
Corollaries1.10,1.12none of strict inequalities (1.21), (1.23), (1.25), and (1.27) can be
replaced by nonstrict ones.

*Remark 1.14. If along with the conditions of* Theorem 1.9 (of Theorem 1.11) condi-
tions (1.28) are satisfied as well, then for every*q**∈**L*^{2}_{2n}_{−}_{2m}_{−}_{2,m}* _{−}*2(]a,b[) (for every

*q*

*∈*

*L*

^{2}

_{2n}

_{−}_{2m}

_{−}_{2}(]a,b])) problem (1.1), (1.2) (problem (1.1), (1.3)) is uniquely solvable in the space

*C*

^{}

_{loc}

^{n}

^{−}^{1}(]a,b[) (in the space

*C*

^{}

_{loc}

^{n}

^{−}^{1}(]a,b])).

*Remark 1.15. Corollaries*1.10and 1.12are more general than the results of paper [7]

concerning unique solvability of problems (1.1), (1.2) and (1.1), (1.3).

**2. Auxiliary statements**

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

*t*0*< t*1*<*+*∞*, and for any function*u*:]t0,*t*1[*→*R, by*u(t*0) and*u(t*1) we understand the
right and the left limits of that function at the points*t*0and*t*1.

*Lemma 2.1. Letu**∈**C*loc(]t0,t1*]) and*
_{t}_{1}

*t*0

*t**−**t*0

_{α+2}

*u*^{}^{2}(t)dt <+*∞*, (2.1)

*whereα** = −**1. If, moreover, either*

*α >**−*1, *u*^{}*t*1

*=*0 (2.2)

*or*

*α <**−*1, *u*^{}*t*0

*=*0, (2.3)

*then*

_{t}_{1}

*t*0

*t**−**t*0

_{α}

*u*^{2}(t)dt*≤* 4
(1 +*α)*^{2}

_{t}_{1}

*t*0

*t**−**t*0

_{α+2}

*u*^{}^{2}(t)dt. (2.4)
*Proof. According to the formula of integration by parts, we have*

_{t}_{1}

*s*

*t**−**t*0

*α*

*u*^{2}(t)dt*=* 1
1 +*α*

*t*1*−**t*0

1+α

*u*^{2}^{}*t*1

*−*
*s**−**t*0

1+α

*u*^{2}(s)^{}

*−* 2
1 +*α*

_{t}_{1}

*s*

*t**−**t*01+α

*u(t)u** ^{}*(t)dt for

*t*0

*< s < t*1

*.*

(2.5)

However,

*−* 2
1 +*α*

*t**−**t*0

1+α

*u(t)u** ^{}*(t)

*=*

*−* 2
1 +*α*

*t**−**t*0

1+α/2

*u** ^{}*(t)

*t*

*−*

*t*0

*α/2*

*u(t)*^{}

*≤* 2
(1 +*α)*^{2}

*t**−**t*0

*α+2*

*u*^{}^{2}(t) +1
2

*t**−**t*0

*α*

*u*^{2}(t).

(2.6)

Thus identity (2.5) implies
_{t}_{1}

*s*

*t**−**t*0*α*

*u*^{2}(t)dt*≤* 2
1 +*α*

*t*1*−**t*01+α

*u*^{2}^{}*t*1

*−*

*s**−**t*01+α

*u*^{2}(s)^{}

+ 4

(1 +*α)*^{2}
_{t}_{1}

*s*

*t**−**t*0

*α+2*

*u*^{}^{2}(t)dt for*t*0*< s < t*1*.*

(2.7)

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
_{t}_{1}

*t*0

*u** ^{}*(t)

^{}

*dt <*+

*∞*,

*u(s)*

^{}

*≤*

_{s}

*t*0

*u** ^{}*(t)

^{}

*dt*

*=*

_{s}*t*0

*t**−**t*0

_{−}*α/2**−*1
*t**−**t*0

1+α/2*u** ^{}*(t)

^{}

*dt*

*≤*
_{s}

*t*0

*t**−**t*0

_{−}*α**−*2

*dt*

1/2_{s}

*t*0

*t**−**t*0

2+α

*u*^{}^{2}(t)dt

1/2

*≤ |*1 +*α**|*^{−}^{1/2}

*s**−**t*0*−*(α+1)/2_{s}

*t*0

*t**−**t*02+α

*u*^{}^{2}(t)dt

1/2

for*t*0*< s < t*1

(2.8)

and, consequently,

lim*s**→**t*0

*s**−**t*0*α+1*

*u*^{2}(s)*=*0. (2.9)