Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 851691, 25 pages
doi:10.1155/2012/851691
Research Article
Study of Solutions to Some Functional Differential
Equations with Piecewise Constant Arguments
Juan J. Nieto1, 2 and Rosana Rodrı́guez-López1
1
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela,
15782 Santiago de Compostela, Spain
2
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203,
Jeddah 21589, Saudi Arabia
Correspondence should be addressed to Rosana Rodrı́guez-López, rosana.rodriguez.lopez@usc.es
Received 12 October 2011; Revised 6 January 2012; Accepted 9 January 2012
Academic Editor: Josef Diblı́k
Copyright q 2012 J. J. Nieto and R. Rodrı́guez-López. 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 provide optimal conditions for the existence and uniqueness of solutions to a nonlocal
boundary value problem for a class of linear homogeneous second-order functional differential
equations with piecewise constant arguments. The nonlocal boundary conditions include terms of
the state function and the derivative of the state function. A similar nonhomogeneous problem is
also discussed.
1. Introduction
In the study of second-order functional differential equations, there is a wide range of
works dealing with periodic boundary value problems and piecewise continuous functional
dependence. We mention, for instance, 1, where an equation independent of the first
derivative is analyzed, and many other works as 2–14, where existence and stability results
are provided.
Most of works on this field deal with nonconstructive existence results. However, in
7, 11, explicit solutions are found for second-order functional differential equations with
piecewise constant arguments, through the calculus of the Green’s function. Other works in
relation with first and higher-order differential equations with delay are 15–18.
In 19, a class of linear second-order differential equations with piecewise constant
arguments is considered under the nonlocal conditions x0 ϕ, xT xμ ψ, where
2
Abstract and Applied Analysis
ϕ, ψ ∈ R, that is, an initial position is assumed and the boundary conditions are independent
of the derivative of the function.
In this paper, we study the following nonlocal boundary value problem for homogeneous linear second-order functional differential equations:
x t ax t bxt cx t dxt 0,
xT x μ ϕ,
t ∈ J 0, T ,
1.1
x T x μ ψ,
for a, b, c, d, ϕ, ψ ∈ R, T > 0 and μ ∈ 0, T , where the functional dependence is given by
the greatest integer part t, and the nonlocal boundary conditions involve both the state
function and its derivative, which is the main difference from the study in 19. A discussion
for nonhomogeneous equations is also included. We consider the existence and uniqueness of
solution to this problem, providing optimal conditions and calculating the exact expression
of solutions.
To better illustrate the significant differences between the results included in this paper
and those in 19, we remark that 19 is devoted to the study of the same class of linear
second-order differential equations with piecewise constant arguments but considering a
nonlocal boundary value problem where the value of the unknown function x is fixed at
the initial instant t 0 i.e., an initial condition is imposed and the boundary condition
also involves the value of the sought solution at the right endpoint of the interval T and
an intermediate point μ. None of the conditions imposed in 19 involve the value of the
rate of change of the solution at any points. Thus, the nonlocal conditions in 19 can be
reduced, by using the expression of the solution, to a boundary value problem affecting only
the state of the solution, being independent of the rate of change of the state of the system.
On the other hand, the nonlocal problem considered in this paper does not fix a certain initial
position and, moreover, the boundary condition introduces the dependence, not only on the
state, but also on the variation of the state of the system. Indeed, the value of the unknown
function and its derivative is determined by the value of the corresponding magnitudes at an
intermediate point of the interval of interest. Moreover, since the results in these two papers
are also extensible to the special case where the intermediate point is identical to the initial
instant μ 0, we compare the consequences of both works to explain better the implications
of the study of each problem. If we consider μ 0, the results in 19 are applicable to
obtain the solution to a class of linear second-order differential equations with piecewise
constant arguments subject to the boundary conditions x0 ϕ, xT x0 ψ ϕ ψ
and, therefore, we solve a problem with fixed values of the function at the end-points of the
interval. However, again for μ 0, the results presented in this paper allow to characterize
the existence of solution and provide its explicit expression imposing boundary conditions
of the type xT x0 ϕ, x T x 0 ψ, which include, as a particular case, periodic
boundary conditions on x and x .
The main results are stated after a few preliminary results are recalled, and, finally,
examples are included to show the applicability of these results.
Abstract and Applied Analysis
3
2. Preliminaries
Consider the equation
x t ax t bxt cx t dxt 0,
t ∈ J 0, T ,
2.1
where a, b, c, d and T > 0. The following concepts and results come from 7.
Definition 2.1. 7 Let the spaces
Λ : y : J −→ R : y is continuous in J \ {1, 2, . . . , T },
and there exist y n− ∈ R, yn yn, ∀n ∈ {1, 2, . . . , T } ,
2.2
E : {x : J −→ R : x, x are continuous and x ∈ Λ}.
A solution to 2.1 is a function x ∈ E which satisfies 2.1, taking x n x n , for all n ∈
{0, 1, 2, . . . , T }.
For the constants a, b, c, d ∈ R, we define h1 s as
1−
d
d
s 2 1 − e−as ,
a
a
1−
d 2
s ,
2
if b 0, a /
0,
if b 0, a 0,
d
d a
0, a2 4b,
1 s e−a/2s − , if b /
b
2
b
2.3
αs
d βe − αeβs d
− , if b /
0, a2 > 4b,
1
b
β−α
b
⎫
⎧
a
d −a/2s ⎨
a2
a2 ⎬ d
0, a2 < 4b.
1
sin b − s − , if b /
cos b − s e
⎭
⎩
2
b
4
4
b
2 b − a /4
1
Consider also h2 s given by
1
c
0,
1 − e−as − cs 1 − e−as , if b 0, a /
a
a
c
s − s2 , if b 0, a 0,
2
c
c
a
2
1 s s − , if b e−a/2s
/ 0, a 4b,
b
2
b
4
Abstract and Applied Analysis
e−a/2s
⎧
⎨c
⎩b
βc/b − 1 eαs 1 − αc/beβs c
− ,
β−α
b
cos
b−
1 ac/2b
a
s sin
4
b − a2 /4
2
if b /
0, a2 > 4b,
⎫
a ⎬ c
b− s − ,
4 ⎭ b
2
2
if b / 0, a < 4b.
2.4
In the definitions of functions h1 and h2 , we denote, for b /
0, a2 > 4b,
a
α− 2
a
2
2
− b,
a
β− −
2
a
2
2
− b.
2.5
It is easy to check 7 that h1 0 1, h2 0 0, h1 0 0, h2 0 1.
Theorem 2.2 see 7, Theorem 2.1. The initial value problem
v t av t bvt cv t dvt 0,
t ∈ 0, ∞,
2.6
v0 v0 ,
v 0 v0 ,
for v0 , v0 ∈ R, has the solution
vt h1 t − n h2 t − n
n v0
C1 C2
C1 C2
,
v0
t ∈ n, n 1,
2.7
where n ∈ Z ,
C1 h1 1,
C2 h2 1,
C1 h1 1,
C2 h2 1.
2.8
To simplify calculus, for z ∈ 0, 1, one denotes
Hz h1 z h2 z
h1 z h2 z
C1 C2
.
C H1 C1 C2
,
2.9
Abstract and Applied Analysis
5
3. Main Results
First, we consider the nonlocal boundary value problem
x t ax t bxt cx t dxt 0,
xT x μ ϕ,
x T x μ ψ,
t ∈ J 0, T ,
3.1
for a, b, c, d, ϕ, ψ ∈ R, T > 0 and μ ∈ 0, T .
If μ 0 ϕ, the condition xT xμ ϕ is reduced to a periodic boundary condition
of Dirichlet type. On the other hand, if μ 0, we obtain x T x 0 ψ, and, moreover, if
ψ 0, we deduce the periodicity of the derivative of the function.
ϕ
Theorem 3.1. If T > 0 and μ ∈ 0, T , then problem 3.1 is solvable for each ψ in the image of the
mapping
F:
u
v
μ u
T −μ
.
−H μ− μ C
−→ HT − T C
v
Hence, there exists a unique solution to 3.1 for every
ϕ
ψ
3.2
∈ R2 if and only if the matrix
HT − T CT −μ − H μ − μ Cμ
3.3
is nonsingular. In this case, the solution is given by
vt h1 t − n h2 t − n C
n
v0
v0
,
t ∈ n, n 1, n ∈ Z ,
3.4
v
ϕ
is the inverse image of ψ by F.
If the
ϕ matrix 3.3 is singular, then there exists an infinite number of solutions to problem 3.1
slope in 3.4 any preimage of
for
every
ψ in the image of F, taking as initial position andinitial
ϕ
ϕ
2
.
by
F,
and
there
exist
no
solutions
for
the
remaining
pairs
∈
R
ψ
ψ
where
0
v0
Proof. We consider the initial value problem
v t av t bvt cv t dvt 0,
v0 v0 ,
v 0 v0 ,
t ∈ 0, ∞,
3.5
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Abstract and Applied Analysis
for v0 , v0 ∈ R, whose solution is, by Theorem 2.2 7, Theorem 2.1,
n v0
,
vt h1 t − n h2 t − n C
v0
t ∈ n, n 1,
3.6
where n ∈ Z . We analyze under which conditions the boundary conditions are fulfilled,
taking into account that
v t h1 t
− n
h2 t
− n C
n
v0
v0
,
t ∈ n, n 1,
3.7
where n ∈ Z .
First, we consider the case T ∈
/ Z, and μ ∈
/ Z. To obtain the solution to 3.1, we calculate v0 and v0 from the expressions of 3.6 and 3.7, in order to satisfy vT vμ ϕ and
v T v μ ψ. Hence the boundary conditions are written as
μ v0
C
ϕ,
− h1 μ − μ
h2 μ − μ
h1 T − T h2 T − T C
v0
T −μ μ v0
C
ψ,
h1 T − T h2 T − T C
− h1 μ − μ
h2 μ − μ
v0
T −μ
3.8
that is,
ϕ
μ v0
T −μ
HT − T C
.
−H μ− μ C
ψ
v0
3.9
The properties of functions h1 and h2 produce that the boundary conditions, in the cases
where T ∈ Z, or μ ∈ Z or both, can be derived from the expression obtained in the case
studied T ∈
/ Z, and μ ∈
/ Z.
ϕ
Therefore, this system has a solution only for ψ in the image of the mapping
u
v
−→ HT − T C
T −μ
μ u
.
−H μ− μ C
v
3.10
Therefore, the solution is unique if the matrix 3.3 is nonsingular, in which case, the initial
condition and initial slope are given as
v0
v0
HT − T CT −μ − H μ − μ Cμ
Cμ
−1
ϕ
ψ
−1 −1 ϕ
T −μ
.
HT − T C
−H μ− μ
ψ
3.11
Abstract and Applied Analysis
7
ϕ
On the other hand, if the matrix 3.3 is singular, for ψ in the image of F, the infinitely
ϕ
many solutions are calculated from 3.6 taking as initial conditions any preimage of ψ by
F, which proves the result.
Remark 3.2. The existence and uniqueness condition 3.3 is reduced, if μ > 0, to the nonsingularity of the matrices C and
HT − T CT −μ − H μ − μ .
3.12
On the other hand, if μ 0, it is just reduced to the nonsingularity of the matrix HT −
T CT − Hμ.
Remark 3.3. In Theorem 3.1, if we consider μ 0, then the boundary conditions are xT x0ϕ, x T x 0ψ, and the condition of existence and uniqueness of solution is reduced
to the nonsingularity of
HT − T CT − I ,
3.13
which coincides with condition 15 in 7, Theorem 2.2. Moreover, if ϕ ψ 0, the abovementioned nonsingularity condition provides that the unique solution to the homogeneous
equation subject to periodic boundary value conditions is the trivial solution see 7,
Theorem 2.2.
Remark 3.4. In Theorem 3.1, the order 2 matrix 3.3 can be written in a simplified manner in
the following particular cases.
i If T ∈
/ Z and T − T μ − μ,
HT − T CT −μ − I Cμ .
3.14
HT − T CT −μ − I Cμ .
3.15
CT −μ − H μ − μ Cμ .
3.16
CT −μ − I Cμ .
3.17
ii If T ∈
/ Z and μ ∈ Z,
iii If T ∈ Z and μ ∈
/ Z,
iv If T, μ ∈ Z,
8
Abstract and Applied Analysis
Remark 3.5. Summarizing, the study of the solvability of 3.1 is reduced to the discussion of
system 3.9. In the case of existence of an infinite number of solutions, we must analyze the
rank of the matrix in 3.3 to determine whether any pair is admissible as initial position and
slope or the space of initial
is one-dimensional. If the rank of 3.3 is zero, problem
ϕconditions
3.1 is solvable only for ψ 00 , that is, problem
x t ax t bxt cx t dxt 0,
xT x μ ,
x T x μ ,
t ∈ J 0, T ,
3.18
for a, b, c, d ∈ R, T > 0 and μ ∈ 0, T , has an infinite number of solutions, given by 3.4, for
v0
any initial point v0 ∈ R2 .
ϕ
On the other hand, and denoting by G the matrix in 3.3, if rankG 1 and ψ
depends linearly on each column
vsolutions,
v of G, then we have a one-dimensional space of
ϕ
0
0
whose starting conditions V0 v0 are determined from the row of the system G v0 ψ
corresponding to a nonzero minor of G.
Next, with the purpose of extending Theorem 3.1 to the nonhomogeneous case, we
consider the following nonlocal boundary value problem for a nonhomogeneous equation:
x t ax t bxt cx t dxt σt,
xT x μ ϕ,
x T x μ ψ,
t ∈ J 0, T ,
3.19
for a, b, c, d, ϕ, ψ ∈ R, T > 0, μ ∈ 0, T , and σ ∈ Λ. Using the expression of the solution for the
corresponding initial value problem provided by 19, Theorem 5.2, we prove the following
existence and uniqueness result.
Theorem 3.6. Consider a, b, c, d ∈ R, T > 0, μ ∈ 0, T , σ ∈ Λ and ϕ, ψ ∈ R. Then problem 3.19
has a unique solution if and only if the matrix 3.3 is nonsingular. Under this assumption, the unique
solution to 3.19 is given by
xt h1 t − n h2 t − n Cn V0
n−1 k1
h1 t − n h2 t − n C
k0
t
n
k
gt − sσsds,
n−1−k
gk 1 − s
g k 1 − s
t ∈ n, n 1, n ∈ Z ,
σsds
3.20
Abstract and Applied Analysis
9
with g defined as
⎧1
−az
⎪
⎪
⎪ a 1 − e ,
⎪
⎪
⎪
⎪
⎪
z,
⎪
⎪
⎪
⎪
⎪
⎪ −a/2z
⎨
,
ze
gz ⎪ eβz − eαz
⎪
⎪
⎪
,
⎪
⎪
⎪
β−α
⎪
⎪
⎪
⎪
⎪
1
⎪
⎪
e−a/2z sin
⎩
b − a2 /4
if b 0, a /
0,
if b 0, a 0,
if b /
0, a2 4b,
3.21
if b / 0, a > 4b,
2
b−
a2
z,
4
if b /
0, a2 < 4b,
and taking, as the initial condition V0 ,
V0 HT − T C
T −1
− H μ − μ Cμ
ϕ
ψ
− MT,μ,σ
3.22
,
where
MT,μ,σ μ−1
k1 HT − T C
T
−1
k1
kμ
− H μ − μ Cμ−1−k
k
k0
T −μ
T T HT − T C
k
T −1−k
gk 1 − s
gk 1 − s
g k 1 − s
σsds
g k 1 − s
3.23
σsds
μ g μ−s
σsds −
σsds.
g T − s
μ g μ − s
gT − s
On the other hand, if the matrix 3.3 is singular, the number of solutions to 3.19 is determined by
the discussion of the linear system
HT − T CT − H μ − μ Cμ V0 ϕ
ψ
− MT,μ,σ
,
3.24
and, for each solution V0 to this system (in case it exists), the expression 3.20 provides a solution to
3.19.
10
Abstract and Applied Analysis
Proof. The result follows from the expression of the solution 3.20 for the corresponding
initial value problem, whose derivative is given by
x t h1 t − n h2 t − n Cn V0
n−1 k1
k0
t
k
h1 t
− n
g t − sσsds,
h2 t
− n C
n−1−k
gk 1 − s
g k 1 − s
σsds
3.25
t ∈ n, n 1, n ∈ Z ,
n
and the fact that the restrictions represented by the boundary conditions produce, respectively,
1 0 HT − T CT − H μ − μ Cμ V0 ϕ − 1 0 MT,μ,σ ,
0 1 HT − T CT − H μ − μ Cμ V0 ψ − 0 1 MT,μ,σ .
3.26
4. Examples
We present some examples where different situations are analyzed in order to decide if the
existence and uniqueness condition nonsingularity of 3.3 holds and, in case of nonuniqueness, the dimension of the space of solutions. The exact expression of the solution or
solutions is given explicitly.
We recall that, for s ∈ 0, 1, Hs is given, depending on the values of the coefficients
a, b, c, d as follows.
i If b 0, a / 0,
⎛
1
d
c
d
−as
s
1 − e−as − cs 1 − e−as −
e
1
−
1
⎜
2
a
a
a
a
⎜
Hs ⎜
⎝
d
1
−1 e−as a ce−as − c
a
a
⎞
⎟
⎟
⎟,
⎠
4.1
thus
C H1 C1 C2
C1
C2
⎛
⎜
⎜
⎝
1−
⎞
1
d d
c
2 1 − e−a 1 − e−a − c 1 − e−a ⎟
a a
a
a
⎟.
⎠
d
1
−a
−a
−1 e a ce − c
a
a
4.2
Abstract and Applied Analysis
11
Note that the determinant of the matrix C is equal to
e−a ea − a − 1d a − a ea c a2
.
detC a2
4.3
ii If b 0, a 0,
⎛
⎞
d 2
c 2
1
−
s
s
s
−
⎜
2
2 ⎟
Hs ⎝
⎠,
−ds
1 − cs
4.4
thus
C H1 C1 C2
C1 C2
⎛
⎞
d
c
1
−
1
−
⎜
2
2⎟
⎝
⎠.
−d 1 − c
4.5
The determinant of the matrix C is equal to
detC d
− c 1.
2
4.6
iii If b /
0, a2 4b,
⎛
c⎞
d a
a
d −a/2s c −a/2s
1
1
s
e
e
1
s
s
− ⎟
−
⎜
b
2
b
b
2
b⎟
⎜
$
% ⎟
Hs ⎜
2 ⎜ ⎟,
a
−a2 c
a
d
⎝
⎠
− s e−a/2s
s− s1
e−a/2s
1
b
4
4b
2
4.7
thus
⎛
c⎞
d a
a −a/2 d −a/2 c 1
e
e
1
1
− ⎟
−
1
⎜
b
2
b
b
2
b⎟
⎜
⎟.
$
%
C H1 ⎜
⎜
⎟
2
−a
a2
a
c
d
⎝
⎠
−
e−a/2
− 1
e−a/2
1
b
4
4b
2
4.8
The determinant of the matrix C is equal to
2a − 4ea/2 4 d − a2 ea/2 c 4b
.
4b
detC e
−a
4.9
12
Abstract and Applied Analysis
iv If b /
0, a2 > 4b, and denoting
a 2
a
a 2
a
− b,
β− −
− b,
2
2
2
2
⎞
⎛
d βeαs − αeβs d βc/b − 1 eαs 1 − αc/beβs c
−
−
1
⎟
⎜
b
β−α
b
β−α
b⎟
⎜
⎟,
Hs ⎜
⎟
⎜
⎠
⎝
c − αeαs β − c eβs
eαs − eβs
b d
β−α
β−α
α−
4.10
thus
⎛
⎞
d βeα − αeβ d βc/b − 1 eα 1 − αc/beβ c
−
− ⎟
⎜ 1
b
β−α
b
β−α
b⎟
⎜
⎜
⎟.
C H1 ⎜ α
β
⎟
β
α
⎝
⎠
c − αe β − c e
d b e −e
1
b
β−α
β−α
4.11
In this case, the determinant of the matrix C is equal to
detC d
1
b
βeα − αeβ
α
e − eβ βc − b eα b − αceβ
c − αeα β − c eβ
−
2
2
β−α
β−α
d eα − eβ
d c − αeα β − c eβ
c 1
−
b
β−α
b β−α
2
d α d β
d α β2 − 2b αβ
1
α
β
αe − βe ce − ce
1
2 e
b
β−α b
b
β−α
d −a
d
1
d
α
1
e αc e −
β c eβ .
b
β−α
b
b
v If b /
0, a2 < 4b, and denoting R 4.12
b − a2 /4,
&
'
⎛
⎞
d
a
1 ac/2b
c
d −a/2s −a/2s c
cos
Rs
e
sin
Rs
−
e
cos
Rs
sin
Rs
−
1
⎜
b
2R
b
b
R
b⎟
⎜
⎟
Hs ⎜
⎟,
&
'
⎝
⎠
a 2c
1
−a/2s
−a/2s
cos Rs −
− b d e
sin Rs
e
sin Rs
R
2R
4.13
Abstract and Applied Analysis
13
thus
⎛
d
⎜ 1 b
⎜
C H1 ⎜
⎝
'
&
d
1 ac/2b
c
a
e−a/2 cos R sin R −
e−a/2
cos R sin R −
2R
b
b
R
'
&
a 2c
1
sin R
e−a/2 cos R −
− b d e−a/2 sin R
R
2R
⎞
c
b⎟
⎟
⎟.
⎠
4.14
The determinant of C is given by
&
'
aa 2c 2
d −a
c
e
1
sin
R
cos2 R − sin R cos R −
b
R
4R2
d
1
d
d a 2c
− e−a/2 cos R sin R 1
ce−a sin R cos R
b
b 2R
R
b
b d −a ac
d −a/2
c
2
1
sin
1
e
e
R
−
sin R
2b
R
b
R2
d −a 2
ac
aa 2c
d −a
b 1
1
e cos R 1 e
sin2 R
−
b
b
2b
4R2
R2
d
d −a/2
1 da 2c
− e−a/2 cos R −c 1
e
sin R
b
R
2b
b
d −a d −a/2
d −a/2
1 da 2c
1
e − e
−c 1
e
cos R sin R.
b
b
R
2b
b
detC 4.15
Example 4.1. Consider the problem
x t x t dxt 0,
)
(
3
t ∈ J 0, ,
2
3
1
x
ϕ,
2
2
3
1
x
ψ,
x
2
2
x
4.16
where d, ϕ, ψ ∈ R. In this case, b c 0, a 1 / 0, T 3/2 and μ 1/2. Therefore, we
d1−e−s −ds1 1−e−s
get Hs so that matrix G HT − T CT −μ − Hμ − μCμ de−s −d
e−s
H1/2C − I is reduced to
⎛
⎜−
G⎜
⎝
e−3/2
⎞
√ 2
√
e d 2ed − 2d2 2 − 2ed e−3/2 ee − 1d 2 − 2ed 2e − 2
⎟
2
2
⎟,
⎠
√ 2
√
−3/2
2
−3/2
e
ed − d 1 − ed
−e
ee − 1d 1 − ed e − 1
4.17
14
Abstract and Applied Analysis
whose determinant is calculated as
√ e 3e − 3d2 2 − 2ed 2e − 2e2 d2
detG −
2
−3/2
e
3e − 3 2e−1 − 2 d2 e−3/2 2 − 2ed
.
−
2
e−2
4.18
Hence, G is nonsingular if and only
0 and d /
2/3 − 2e1/2 . In this case, problem 4.16
ϕ if d 2/
has a unique solution for every ψ ∈ R , given by see 3.4
xt d 1 − e−t − dt 1 v0 1 − e−t v0 ,
xt 1 − e1−t
t ∈ 0, 1,
e−1 d − d v0 e−1 v0
1 − e−1 d − d 1 v0 1 − e−1 v0
d 1 − e1−t − dt − 1 1
−e−t−1
4.19
ed − d2 t 2d2 − ed − e et − ed2 ed v0
e − 1dt 2 − 2ed − eet e2 − e d e v0 ,
(
)
3
t ∈ 1, ,
2
where
v0
v0
G−1
ϕ
ψ
√
√
⎞
e−2 d2
2 e−1 d2
− √
− √
2
2
⎟ ⎜
2 e − 3 d 2d
2 e − 3 d 2d
⎟ ϕ
⎜
⎟
⎜
⎜
√
√
√
⎟ ψ ,
⎠
⎝
e − 2 d 2 ee − 2e 2
2 e − 1 d − 2e 2
− √
− √
e2e − 2 3 − 3e d 2e − 2
e2e − 2 3 − 3e d 2e − 2
4.20
⎛
that is,
√
√
− 2 e−1 d2 ϕ−
e−2 d2 ψ
v0 √
2 e − 3 d2 2d
√
√
√
−2
e − 1 d − 2e 2 ϕ −
e − 2 d 2 ee − 2e 2
.
v0 √
e2e − 2 3 − 3e d 2e − 2
4.21
Abstract and Applied Analysis
15
For the particular case where d 2, ϕ 1, and ψ −1, then
v0
v0
−1
G
1
−1
√
√ ⎞
⎞
⎛
1
2 e−1
e
1
−
−
−
√
√
⎟
⎟
⎜
⎜
4 ⎟ 1
4 e−4
⎜ 4 e − 1⎟
⎜
⎜
⎜
⎟,
⎟
√
√
⎝ 1
⎝
e 1 ⎠ −1
−e 2 e − 1
e ⎠
−
−√
√
2 e−1
e2e − 2 − 2e 2 2e − 2
⎛
4.22
and hence the unique solution to
x t x t 2xt 0,
)
(
3
t ∈ J 0, ,
2
3
1
x
1,
2
2
3
1
x
x
− 1,
2
2
x
4.23
is given by
√
e2t − 1 − 2et e
, t ∈ 0, 1,
√
−4e 8 e − 4
√ 2
e 2e 6e t − 7e2 − 21e et 4e3 12e2
xt −
4e2 24e 4et1/2 −16e2 − 16eet
)
(
−6e2 − 2e t 21e2 7e et − 12e3 − 4e2
3
−
, t ∈ 1, .
2
4e2 24e 4et1/2 −16e2 − 16eet
xt 4.24
See Figure 1.
−s and the matrix
If d 0, then Hs 10 1−e
e−s
1
G HT − T CT −μ − H μ − μ Cμ H
C − I
2
is reduced to G 0 e−3/2 e−1
0 −e−3/2 e−1
4.25
, which has rank one. Since G is singular, there exists an infiϕ
nite number of solutions to problem
for every ψ in the image of the mapping given
ϕ4.16
by the matrix G, that is, for every −ϕ , where ϕ ∈ R. On each of these cases, the solutions
16
Abstract and Applied Analysis
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
t
Figure 1: Solution to problem 4.23.
are given by 3.4 taking as initial position and initial slope any preimage of
initial slope can be taken as
v0 e3/2 ϕ
,
e−1
ϕ −ϕ ,
that is, the
4.26
and the initial position can be chosen as any real number v0 .
Hence, the solutions to the problem
)
(
3
t ∈ J 0, ,
2
3
1
x
x
ϕ,
2
2
3
1
x
x
− ϕ,
2
2
x t x t 0,
4.27
where ϕ ∈ R, are given by
xt v0 1 − e−t v0 ,
t ∈ 0, 1,
4.28
and the same expression
xt 1
1−e
−t−1
v0 1 − e
v0
1 1 − e−1
e−1
v0
)
(
3
t ∈ 1, ,
2
0
−t
v0 ,
4.29
Abstract and Applied Analysis
17
2.5
2
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
t
Figure 2: Solution which generates by vertical shifts the remaining solutions to problem 4.27 with ϕ 1.
where v0 ∈ R and v0 e3/2 ϕ/e − 1. Hence, for each ϕ fixed, all the vertical shifts of the
function xt 1 − e−t e3/2 ϕ/e − 1, t ∈ 0, 3/2 are solutions to 4.27. In Figure 2, we
show the graph of a solution which generates, by vertical shifting, all the solutions to problem
4.27 for ϕ 1.
If ψ / − ϕ, there is no solution for problem 4.16 with d 0.
Finally, if d 2/3 − 2e1/2 , then
√
⎞
5 − 2 e − 2s − 2e−s
−s
1
−
e
√
⎟
⎜
3−2 e
⎟
⎜
Hs ⎜
⎟,
⎠
⎝
2e−s − 1
−s
e
√
3−2 e
1
GH
C − I
2
⎞
⎛
√
⎛
⎞
4 − 2 e − 2e−1/2
−1/2
−2e−1
1
−
e
√
⎟
⎜
−1
1−e ⎟
√
⎟⎜
⎜
3−2 e
⎟⎜ 3 − 2 e
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟⎜ −1
⎟
−
1
−1/2 ⎟⎝ 2 e
⎜
⎠
−1/2
e
−1
−1
⎠
⎝ 2 e
e −1
√
√
3−2 e
3−2 e
⎛
⎞
√
−2e−3/2 2e−1 6e−1/2 − 10 4 e −e−3/2 e−1 e−1/2 − 1
√
√ 2
⎜
⎟
3−2 e
⎜
⎟
3−2 e
⎜
⎟
⎜
⎟
⎜
⎟
−3/2
−1/2
−3/2
−1/2
2e
− 6e
4
e
−e
⎝
⎠
√
√ 2
3−2 e
3−2 e
⎛
4.30
18
Abstract and Applied Analysis
hasrank
one. Hence problem 4.16, for d 2/3 − 2e1/2 , has an infinite number of solutions
ϕ
for ψ in the image of the mapping given by the matrix G, that is, for the values of ϕ, and ψ
satisfying the system
√
√ 2
−2e−3/2 2e−1 6e−1/2 − 10 4 e v0 −3e−3/2 5e−1 e−1/2 − 5 2e1/2 v0 ϕ 3 − 2 e ,
√ 2
2e−3/2 − 6e−1/2 4 v0 3e−3/2 − 3e−1/2 − 2e−1 2 v0 ψ 3 − 2 e ,
4.31
for some values of v0 , v0 ∈ R. For the existence of these values of v0 , v0 , it is necessary and
sufficient that the rank of the matrix of the system equal to one coincides with the rank of
the matrix
⎛
√
√ 2 ⎞
−2e−3/2 2e−1 6e−1/2 − 10 4 e −3e−3/2 5e−1 e−1/2 − 5 2e1/2 ϕ 3 − 2 e
⎠
⎝
√ 2 ,
2e−3/2 − 6e−1/2 4
3e−3/2 − 3e−1/2 − 2e−1 2
ψ 3−2 e
4.32
that is, for instance,
det
−3e−3/2 5e−1 e−1/2 − 5 2e1/2 ϕ
3e−3/2 − 3e−1/2 − 2e−1 2
ψ
0,
4.33
equivalent to
ψ
3e−3/2 − 3e−1/2 − 2e−1 2
ϕ.
−3e−3/2 5e−1 e−1/2 − 5 2e1/2
4.34
Under this assumption, there exists an infinite number of solutions to problem
x t x t (
)
3
2
xt
0,
t
∈
J
0,
,
2
3 − 2e1/2
1
3
x
ϕ,
x
2
2
3
1
x
ψ,
x
2
2
4.35
Abstract and Applied Analysis
19
which are given by
√
5 − 2 e − 2t − 2e−t
xt v0 1 − e−t v0 ,
√
3−2 e
t ∈ 0, 1,
⎞
√
3 − 2 e − 2e−1
−1
1
−
e
√
⎟ √
⎜
⎟ v0
⎜
3−2 e
5 − 2 e − 2t − 1 − 2e−t−1
−t−1
⎟
⎜
xt 1−e
√
−1
⎟
⎜
3−2 e
⎠ v0
⎝ 2 e −1
−1
e
√
3−2 e
√
√
−2e−t 4e1/2−t 4 et 4e−1 t − 6t 4e − 16 e − 8e−1 15
v0
√ 2
3−2 e
√
(
)
−e−t − 2e1−t 2e1/2−t 2e−1 t − 2t − 2 e − 4e−1 7 3
v0 , t ∈ 1, ,
√
2
3−2 e
⎛
4.36
where v0 and v0 satisfy one of the equations in 4.31.
On the other hand, if 4.34 fails, there is no solution to 4.35.
Example 4.2. Next, consider the problem
x t cx t dxt 0,
)
(
3
t ∈ J 0, ,
2
3
1
x
ϕ,
2
2
3
1
x
x
ψ,
2
2
x
4.37
where c, d, ϕ, ψ ∈ R. In this case, a b 0, T 3/2, and μ 1/2. Hence, the matrix
1
G HT − T CT −μ − H μ − μ Cμ H
C − I
2
4.38
⎞
d2 2c − 16d c − 2d 2c2 − 16c 16
⎟
⎜
16
16
⎟
⎜
G⎜
⎟,
⎝ d2 2c − 4d
2
c − 2d 2c − 4c ⎠
4
4
4.39
is reduced to
⎛
20
Abstract and Applied Analysis
whose determinant is detG dd 8 − 4c/8. If d /
0 and d /
4c − 8, then G is invertible
ϕ
and problem 4.37 has a unique solution for every ψ ∈ R2 . Since
⎛
G−1
⎞
2c − 4d 4c2 − 8c c − 2d 2c2 − 16c 16
−
⎜
⎟
1
d
2d
⎜
⎟
⎜
⎟,
⎠
d − 4c 8 ⎝
d 2c − 16
−2d 4c − 8
2
the unique solution to 4.37, for
vt ϕ
ψ
4.40
fixed, is given by
d
c
1 − t2 v0 t − t2 v0 ,
2
2
t ∈ 0, 1,
⎛
⎞
d
c v 0
1
−
1
−
d
c
vt 1 − t − 12 t − 1 − t − 12 ⎝
2
2⎠
2
2
v0
−d 1 − c
d
d
c
t − 1 − t − 12 −d v0
1 − t − 12
1−
2
2
2
d
c
c
t − 1 − t − 12 1 − c v0 ,
1 − t − 12 1 −
2
2
2
(
)
3
t ∈ 1, ,
2
4.41
where
v0 v0
c − 2d 2c2 − 16c 16
2c − 4d 4c2 − 8c
ϕ−
ψ,
dd 8 − 4c
2dd 8 − 4c
2d 4c − 8
d 2c − 16 v0 v .
−
d − 4c 8
2d − 4c 8 0
4.42
2
−1/2
s−s2
and G −11/16
,
For instance, for c 2 and d 1, then Hs 1−s−s/2 1−2s
1/4
0
0
4
−1
which is invertible
ϕ with inverse G −2 −11/2 . Then the unique solution to 4.37, for c 2
and d 1 and ψ fixed, is given by
1 2
7t − 11t 8 ψ, t ∈ 0, 1,
vt 2 t2 − t ϕ 2
(
)
1
1 2
3
vt − 4t2 − 12t 8 ϕ −5t 13t − 4 ψ, t ∈ 1, .
2
2
2
4.43
See Figure 3 for the solution to 4.37 with c 2, d 1, ϕ 1, and ψ −1:
⎧
1
⎪
⎪
⎪ −3 t2 7t − 8 ,
⎨
2
vt ⎪1 2
⎪
⎪
⎩ t −t−4 ,
2
t ∈ 0, 1,
)
3
t ∈ 1, .
2
(
4.44
Abstract and Applied Analysis
21
−1.5
−2
−2.5
−3
−3.5
−4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
t
Figure 3: Solution to 4.37 for c 2, d 1, ϕ 1, and ψ −1.
On the other hand, if d 0, then
⎛
Hs ⎞
cs2
2 ⎠,
0 1 − cs
⎝1 s −
⎛
⎞
c2 − 8c 8
⎜0
⎟
1
8
⎟
GH
C − I ⎜
2
⎝
c − 2c ⎠
2
0
2
4.45
√
is singular with rank 1, since G12 0 if and only if c 4 ± 2 2 and G22 0 if and only if c 0
or c 2.
there exists an infinite number of solutions to problem 4.37 with
ϕ d 20, for
Hence,
ϕ
every ψ in the image of the mapping given by the matrix G, that is, for every ψ ∈ R such
that 4c2 − 2cϕ c2 − 8c 8ψ. Under this assumption, the solutions to 4.37 with d 0 are
given by
c
vt v0 t − t2 v0 ,
2
t ∈ 0, 1,
⎛
c ⎞ 1 1−
v
c
2⎠ 0
vt 1 t − 1 − t − 12 ⎝
2
v0
0 1−c
c
c
v0 1 − 1 − c t − 1 − t − 12
2
2
v0 ,
4.46
)
3
t ∈ 1, ,
2
(
22
Abstract and Applied Analysis
where the initial position v0 can be chosen as any real number and the initial slope must
satisfy
v0 c2
8ϕ
,
− 8c 8
2ψ
,
v0 2
c − 2c
√
if c / 4 ± 2 2,
4.47
if c /
0, c /
2.
2
/2 , so that G 0 1/8 . Therefore,
For instance, if d 0 and c 1, then Hs 10 s−s
0 −1/2
1−s
there
exists
an
infinite
number
of
solutions
to
problem
4.37
with
d
0
and
c 1, for every
ϕ
ψ such that ψ −4ϕ. In this case, the solutions to 4.37 with d 0 and c 1 are given by
⎧ 1 2
⎪
⎪
t
ϕ,
8
t
−
⎨
2
vt v0 ⎪
⎪
⎩4ϕ,
t ∈ 0, 1,
(
)
3
t ∈ 1, ,
2
4.48
where v0 is any real number. Note that v0 8ϕ. All the solutions are obtained as vertical shifts
of the function
⎧ 1 2
⎪
⎪
t
ϕ,
8
t
−
⎨
2
ft ⎪
⎪
⎩4ϕ,
t ∈ 0, 1,
(
)
3
t ∈ 1, .
2
4.49
See Figure 4 for the graph of function f for ϕ 1, which generates by vertical shifting
the one-dimensional space of solutions to problem 4.37 with d 0, c 1, ϕ 1, and ψ −4.
Finally, if d 4c − 8, then
c ⎞
1 − 2c − 4s2 s − s2
2 ⎠,
Hs ⎝
⎛
−4c 8s
⎛
1 − cs
⎞
3c2 − 18c 24 3c2 − 16c 16
⎜
⎟
2
8
⎜
⎟
G⎜
⎟,
⎝
⎠
2
−
10c
8
3c
6c2 − 24c 24
2
4.50
which has rank 1. Note that G11 /
0 for c /
2 and c /
4; G12 /
0 for c /
4/3 and c /
4; G21 /
0 for
0 for c /
2 and c /
4/3.
c/
2; and G22 /
Abstract and Applied Analysis
23
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
t
Figure 4: Solution which generates by vertical shifting the remaining solutions to problem 4.37 with
d 0, c 1, ϕ 1, and ψ −4.
Therefore, problem 4.37 for d 4c − 8 has solution an infinite number of solutions
if and only if
3c2 − 18c 24
ψ 6c2 − 24c 24 ϕ,
2
4.51
3c2 − 10c 8
3c2 − 16c 16
ψ
ϕ,
8
2
4.52
in which case the expressions of the solutions are given by 4.41, replacing the value of d,
where v0 and v0 satisfy
3c2 − 18c 24
3c2 − 16c 16 v0 v0 ϕ,
2
8
3c2 − 10c 8 6c2 − 24c 24 v0 v0 ψ.
2
4.53
4.54
Indeed,
, and the problem has an infinite number of solutions for
i if c 2, then G 00 −1/2
0
ψ 0 see 4.52 since 4.51 is trivially satisfied, where v0 ∈ R and v0 −2ϕ from
4.53 since 4.54 is trivially satisfied,
0
ii if c 4/3, then G 8/3
8/3 0 , and the problem has an infinite number of solutions
for ϕ ψ see 4.51 since 4.52 is trivially satisfied, where v0 3/8ϕ 3/8ψ
from 4.53 or 4.54 and v0 ∈ R,
iii if c /
2 and c /
4/3, then the elements in the second row of G are nonzero and, hence,
the problem has an infinite number of solutions for ϕ 3c2 −18c24/26c2 −24c
24ψ from 4.51 or the same expression ϕ 3c2 − 16c 16/43c2 − 10c 8ψ
from 4.52. The initial position and slope are taken satisfying 4.54.
24
Abstract and Applied Analysis
0 0
If we take c 4 see the last case distinguished, then G 24
16 , and conditions 4.51 and
4.52 are reduced to ϕ 0 and 4.54 is written as 24v0 16v0 ψ.
5. Conclusions
In this work, we have analyzed the existence and uniqueness of solutions to the nonlocal
boundary value problem for linear second-order functional differential equations with piecewise constant arguments:
x t ax t bxt cx t dxt σt,
xT x μ ϕ,
x T x μ ψ,
t ∈ J 0, T ,
5.1
where a, b, c, d, ϕ, ψ ∈ R, T > 0, μ ∈ 0, T and σ ∈ Λ, by reducing the study to the discussion
of a linear system of order 2. The nonlocal boundary conditions considered involve terms
of the state function and the derivative of the state function, and the problem of interest
also includes, as a particular case, periodic boundary value problems for second-order linear
differential equations with functional dependence given by the greatest integer part.
Besides providing conditions on the existence and uniqueness of solutions, this
approach also allows to obtain the expression of solutions explicitly, process which is illustrated with some examples.
The differences between the results in this paper and those in 19 have been included
in the introduction.
Acknowledgments
The authors thank the editor and the anonymous referees for their helpful suggestions towards the improvement of the paper. This research is partially supported by Ministerio de
Ciencia e Innovación and FEDER, Project MTM2010-15314.
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