doi:10.1155/2010/478020
Research Article
Differential Inequalities for One Component of
Solution Vector for Systems of Linear Functional
Differential Equations
Alexander Domoshnitsky
Department of Mathematics and Computer Science, The Ariel University Center of Samaria, 44837 Ariel, Israel
Correspondence should be addressed to Alexander Domoshnitsky,[email protected]
Received 24 December 2009; Accepted 26 April 2010
Academic Editor: A ˘gacik Zafer
Copyrightq2010 Alexander Domoshnitsky. 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.
The method to compare only one component of the solution vector of linear functional differential systems, which does not require heavy sign restrictions on their coefficients, is proposed in this paper. Necessary and sufficient conditions of the positivity of elements in a corresponding row of Green’s matrix are obtained in the form of theorems about differential inequalities. The main idea of our approach is to construct a first order functional differential equation for thenth component of the solution vector and then to use assertions about positivity of its Green’s functions. This demonstrates the importance to study scalar equations written in a general operator form, where only properties of the operators and not their forms are assumed. It should be also noted that the sufficient conditions, obtained in this paper, cannot be improved in a corresponding sense and does not require any smallness of the interval0, ω, where the system is considered.
1. Introduction
Consider the following system of functional differential equations
Mixt≡xit n
j 1
Bijxj
t fit, t∈0, ω, i 1, . . . , n, 1.1
where x colx1, . . . , xn, Bij : C0,ω → L0,ω, i, j 1, . . . , n, are linear continuous
operators,C0,ωandL0,ωare the spaces of continuous and summable functionsy:0, ω →
Letl : Cn0,ω → Rn be a linear bounded functional. If the homogeneous boundary
value problemMixt 0, t∈0, ω, i 1, . . . , n, lx 0,has only the trivial solution, then
the boundary value problem
Mixt fit, t∈0, ω, i 1, lx α, 1.2
has for eachf colf1, . . . , fn,wherefi∈L0,ω, i 1, . . . , n,andα∈Rn,a unique solution,
which has the following representation1:
xt
ω
0
Gt, sfsdsXtα, t∈0, ω, 1.3
where then×nmatrixGt, sis called Green’s matrix of problem1.2, andXtis then×n
fundamental matrix of the systemMixt 0, i 1, . . . , n,such thatlX EEis the unit
n×n-matrix. It is clear from the solution representation 1.3that the matricesGt, sand
Xtdetermine all properties of solutions.
The following property is the basis of the approximate integration method by Tchaplygin2: from the conditions
Mixt≥
Miy
t, t∈0, ω, i 1, . . . , n, lx ly, 1.4
it follows that
xit≥yit, t∈0, ω, i 1, . . . , n. 1.5
Series of papers, started with the known paper by Luzin 3, were devoted to the various aspects of Tchaplygin’s approximate method. The well-known monograph by Lakshmikantham and Leela4was one of the most important in this area. The known book by Krasnosel’skii et al. 5 was devoted to approximate methods for operator equations. These ideas have been developing in scores of books on the monotone technique for approximate solution of boundary value problems for systems of differential equations. Note in this connection the important works by Kiguradze and Puza6,7and Kiguradze8.
As a particular case of system1.1, let us consider the following delay system:
xit
n
j 1
pijtxj
hijt fit, i 1, . . . , n, t∈0, ω,
xξ 0 forξ <0,
1.6
wherepij are summable functions, andhij are measurable functions such thathijt ≤ tfor
The classical Wazewskii’s theorem claims9that the condition
pij ≤0 forj /i, i, j 1, . . . , n, 1.7
is necessary and sufficient for the property1.4⇒1.5for the Cauchy problem for system of ordinary differential equations
xit
n
j 1
pijtxjt fit, i 1, . . . , n, t∈0, ω. 1.8
From formula of solution representation1.3, it is clear that property 1.4⇒1.5is true if all elements of the matricesGt, sandXtare nonnegative.
We focus our attention upon the problem of comparison for only one of the components of solution vector. Letkibe either 1 or 2. In this paper we consider the following
property: from the conditions
−1kiM ixt−
Miy
t≥0, t∈0, ω, lx ly, i 1, . . . , n, 1.9
it does follow that for a corresponding fixed component xr of the solution vector the
inequality
xrt≥yrt, t∈0, ω, 1.10
is satisfied. This property is a weakening of the property1.4⇒1.5and, as we will obtain below, leads to essentially less hard limitations on the given system. From formula of solution’s representation1.3, it follows that this property is reduced to sign-constancy of all elements standing only in therth row of Green’s matrix.
The main idea of our approach is to construct a corresponding scalar functional differential equation of the first order
xnt Bxnt f∗t, t∈0, ω, 1.11
for nth component of a solution vector, where B : C0,ω → L0,ω is a linear continuous
operator, f∗ ∈ L0,ω.This equation is built inSection 2. Then the technique of analysis of
the first-order scalar functional differential equations, developed, for example, in the works
10–12, is used. On this basis in Section 3 we obtain necessary and sufficient conditions of nonpositivity/nonnegativity of elements in nth row of Green’s matrices in the form of theorems about differential inequalities. Simple coefficient tests of the sign constancy of the elements in thenth row of Green’s matrices are proposed inSection 4for systems of ordinary differential equations and inSection 5for systems of delayed differential equations. It should be stressed that in our results a smallness of the interval0, ωis not assumed.
Note that results of this sort for the Cauchy problemi.e.,lx ≡ colx10, . . . , xn0
and Volterra operatorsBij :C0,ω → L0,ωwere proposed in the recent paper13, where the
obtained operatorB:C0,ω → L0,ωbecame a Volterra operator. In this paper we consider
other boundary conditions that imply that the operatorB:C0,ω → L0,ωis not a Volterra
2. Construction of Equation for
n
th Component of Solution Vector
In this paragraph, we consider the boundary value problem
Mixt≡xit
Together with problem 2.1, 2.2 let us consider the following auxiliary problem consisting of the system:
of the ordern−1 and the boundary conditions
lixi ci, i 1, . . . , n−1. 2.4
Let us assume that problem 2.3, 2.4 is uniquely solvable; denote by Kt, s {Kijt, s}i,j 1,...,n−1its Green’s matrix and byGt, s {Gijt, s}i,j 1,...,n Green’s matrix of the
problem2.1,2.2.
Let us start with the following assertion, explaining how the scalar functional differential equation for one of the components of the solution vector can be constructed.
whereu col{u1, . . . , un−1}is the solution of the system
mixt 0, t∈0, ω, i 1, . . . , n−1, 2.8
satisfying condition2.4.
Proof. Using Green’s matrixKt, s {Kijt, s}i,jn− 11 of problem2.3,2.4, we obtain
xit −
ω
0 n−1
j 1
Kijt, s
Bjnxn
sds
ω
0 n−1
j 1
Kijt, sfjsds
n−1
j 1
Kijt,0cj, 2.9
for everyi ∈ {1, . . . , n−1}.Substitution of these representations in the nth equation of the system2.1leads to2.5, where the operatorBand the functionf∗are described by formulas
2.6and2.7, respectively.
3. Positivity of the Elements in the Fixed
n
th Row of Green’s Matrices
Consider the boundary value problem
Mixt≡xit n
j 1
Bijxj
t fit, t∈0, ω, i 1, . . . , n,
lixi ci, i 1, . . . , n−1, xnω cn,
3.1
whereBij :C0,ω → L0,ωare linear continuous operators fori, j 1, . . . , n.
Theorem 3.1. Let problem2.3,2.4be uniquely solvable, all elements of its (n−1×n−1Green’s matrixKt, snonnegative, and the operatorsBin,−BniandBnnpositive operators fori 1, . . . , n−1. Then the following 2 assertions are equivalent:
1there exists an absolutely continuous vector functionvsuch thatvnt>0, Mivt≤0, fort ∈0, ω, i 1, . . . , n,and the solution of the homogeneous equation (miut 0 fort ∈0, ω, i 1, . . . , n−1,satisfying the conditionsliui livi, i 1, . . . , n−1, is nonpositive;
2the boundary value problem 3.1 is uniquely solvable for every summable f
colf1, . . . , fnandc colc1, . . . , cn ∈ Rn and elements of the nth row of its Green’s matrix satisfy the inequalities: Gnjt, s ≤ 0 for j 1, . . . , n, t, s ∈ 0, ω, while
Gnnt, s<0for0≤t < s≤ω.
Proof. Let us start with the implication1⇒2.By virtue ofLemma 2.1, the componentxn
of the solution vector of problem3.1satisfies2.5. Condition1by virtue of Theorem 1 of the paper14implies that Green’s function GNt, sof the boundary value problem
exists and satisfies the inequalitiesGNt, s≤0 fort, s∈0, ω,whileGNt, s<0 for 0≤t < s≤ω.Lemma 2.1, the representations of solutions of boundary value problem3.1and the scalar one-point problem3.2imply the equality
xnt Then the following 2 assertions are equivalent:
1∗there exists an absolutely continuous vector functionvsuch thatvnt > 0,Mnvt ≤
0, Mivt ≥ 0 fort ∈ 0, ω, i 1, . . . , n−1, and the solution of the homogeneous equation (miut 0 fort ∈ 0, ω, i 1, . . . , n−1, satisfying the conditionsliui
livi, i 1, . . . , n−1,is nonnegative;
2∗the boundary value problem 3.1 is uniquely solvable for every summable f
colf1, . . . , fnandc colc1, . . . , cn ∈ Rn and elements of the nth row of its Green’s matrix satisfies the inequalities: Gnjt, s ≥ 0 for j 1, . . . , n−1, Gnnt, s ≤ 0 for
t, s∈0, ωwhileGnnt, s<0for0≤t < s≤ω.
4. Sufficient Conditions of Nonpositivity of the Elements in
the
n
th Row of Green’s Matrices for System of Ordinary
Differential Equations
In this paragraph, we consider the system of the ordinary differential equations
xit
Theorem 4.1. Let the following conditions be fulfilled:
1pij ≤0fori /j, i, j 1, . . . , n−1;
2pjn≥0, pnj≤0forj 1, . . . , n−1, pnn ≥0; 3there exists a positive numberαsuch that
pnnt−
Proof. Let us prove that all elements of Green’s matrixKt, sof the auxiliary boundary value problem
are nonnegative. The conditions1,2, and the inequality
Now according to equivalence of assertionsaandbin Theorem 3.1 of the paper13, we get the nonnegativity of all elements of its Green’s matrixKt, s.
Let us setvit −e−αt fori 1, . . . , n−1,and vnt e−αt in the condition1 of
Theorem 3.1. We obtain that this condition is satisfied ifαsatisfies the following system of the inequalities:
α≤ −pint
n−1
j 1
pijt, i 1, . . . , n−1, t∈0, ω,
pnnt−
n−1
j 1
pnjt≤α, t∈0, ω.
4.6
Now by virtue ofTheorem 3.1, all elements of thenth row of Green’s matrix satisfy the inequalities Gnjt, s≤0 forj 1, . . . , n−1,and, using14, we can conclude thatGnnt, s<
0 for 0≤t < s≤ω.
Consider now the following ordinary differential system of the second order;
x1t p11tx1t p12tx2t f1t,
x2t p21tx1t p22tx2t f2t,
t∈0, ω, 4.7
with the conditions
x10 x1ω c1, x2ω c2. 4.8
FromTheorem 4.1as a particular case forn 2, we obtain the following assertion.
Theorem 4.2. Let the following two conditions be fulfilled:
1p11 ≥0, p12≥0, p21≤0, p22 ≥0; 2there exists a positiveαsuch that
p22t−p21t≤α≤p11t−p12t, t∈0, ω. 4.9
Then problem4.7,4.8is uniquely solvable for every summablef colf1, f2andc
{c1, c2} ∈R2,and the elements of the second row of Green’s matrix of problem4.7,4.8satisfy the inequalities:G2it, s≤0for1 1,2, t∈0, ω, G22t, s<0for 0≤t < s < ω.
Remark 4.3. If coefficientspijare constants, the second condition inTheorem 4.2is as follows:
Remark 4.4. Let us demonstrate that inequality4.10is best possible in a corresponding case and the condition
p22−p21≤p11−p12ε, p11−p12ε >0 4.11
cannot be set instead of4.10. The characteristic equation of the system
x1t p11x1t p12x2t 0,
x2t p21x1t p22x2t 0,
t∈0, ω, 4.12
with constant coefficients is as follows:
λ2p11p22
λp11p22−p12p21 0. 4.13
If we setp11 p22 0, p21 < 0, p12 > 0, p12 −p21 < ε,then the roots areλ1 i√−p12p21,
λ2 −i√−p12p21,and the problem
x1t p12x2t 0,
x2t p21x1t 0,
t∈0, ω,
x10 x1ω, x2ω 0
4.14
has nontrivial solution forω 2π/√−p12p21.
5. Sufficient Conditions of Nonpositivity of the Elements in
the
n
th Row of Green’s Matrices for Systems with Delay
Let us consider the system of the delay differential equations
xit
n
j 1
pijtxj
t−τijt
fit, i 1, . . . , n, t∈0, ω. 5.1
xiξ 0 for ξ <0, i 1, . . . , n, 5.2
with the boundary conditions
xi0 xiω ci, i 1, . . . , n−1, xnω cn. 5.3
We introduce the denotations: p∗ij ess suppijt, pij∗ ess infpijt,τij∗
Theorem 5.1. Let the following conditions be fulfilled: 1pij ≤0fori /j, i, j 1, . . . , n−1;
2pjn≥0, pnj≤0forj 1, . . . , n−1, pnn ≥0; 3τii 0fori 1, . . . , n−1;
4there exists a positive numberαsuch that
pnnteατnnt−
n−1
j 1
pnjteατnjt
≤α≤ min
1≤i≤n−1
⎧ ⎨
⎩−pinteατintpiit
n−1
j 1,i /j
pijteατijt
⎫ ⎬
⎭, t∈0, ω.
5.4
Then problem5.1,5.3is uniquely solvable for every summablef colf1, . . . , fnand
c {c1, . . . , cn} ∈ Rn, and the elements of thenth row of Green’s matrix of problem5.1,5.3 satisfy the inequalities:Gnjt, s≤0fort, s∈0, ω, j 1, . . . , n, Gnnt, s<0for0≤t < s≤ω.
Proof. Repeating the explanations in the beginning of the proof ofTheorem 4.1, we can obtain on the basis of Theorem 3.1 of the paper13that all the elements ofn−1×n−1Green’s matrixKt, sof the auxiliary problem, consisting of the system
xit n−1
j 1
pijtxj
t−τijt fit, i 1, . . . , n−1, t∈0, ω, 5.5
and the boundary conditionsxi0 xiω ci, i 1, . . . , n−1,are nonnegative.
Let us setvit −e−αt fori 1, . . . , n−1,and vnt e−αt in the condition1 of
Theorem 3.1. We obtain that the condition 1 of Theorem 3.1is satisfied if αsatisfies the following system of the inequalities:
α≤ −pinteατintpiit
n−1
j 1,i /j
pijteατijt, i 1, . . . , n−1, t∈0, ω, 5.6
pnnteατnnt−
n−1
j 1
pnjteατnjt≤α, t∈0, ω. 5.7
Now by virtue ofTheorem 3.1, all elements of thenth row of Green’s matrix of problem
5.1,5.3satisfy the inequalities Gnjt, s≤0fort, s ∈0, ω, j 1, . . . , n,whileGnnt, s<
0 for 0≤t < s≤ω.
Remark 5.2. It was explained in the previous paragraph that in the case of ordinary system
τij 0, i, j 1, . . . , nwith constant coefficients pij, inequality 5.4 is best possible in a
Let us consider the second-order scalar differential equation
yt p11tyt p12tyt−τ12t f1t, t∈0, ω, 5.8
whereyξ yξ 0 forξ <0,with the boundary conditions
y0 yω c1, yω c2, 5.9
and the corresponding differential system of the second order
x1t p11tx1t p12tx2t−τ12t f1t,
x2t−x1t 0,
t∈0, ω, 5.10
wherex1ξ x2ξ 0 forξ <0,with the boundary conditions
x10 x1ω c1, x2ω c2. 5.11
It should be noted that the elementG21t, sof Green’s matrix of system5.10,5.11
coincides with Green’s function Wt, sof the problem5.8,5.9for scalar second-order equation.
Theorem 5.3. Assume thatp12≥0and there exists a positive numberαsuch that
α2p12teατ12t≤αp11t, t∈0, ω. 5.12
Then problem5.10, 5.11 is uniquely solvable for every summable f colf1, f2and
c colc1, c2∈R2,and the elements of the second row of Green’s matrix of this problem satisfy the inequalities:G2jt, s≤0, j 1,2, t, s∈0, ω, whileG22t, s<0 for 0≤t < s < ω.
In order to proveTheorem 5.3, we setv1t −αe−αt, v2t e−αtin the assertion1
ofTheorem 3.1.
Remark 5.4. Inequality5.12is best possible in the following sense. Let us addεin its right hand side. We get that the inequality
α2p12teατ12t≤αp11t ε, t∈0, ω, 5.13
and the assertion ofTheorem 5.3is not true. Let us set that coefficients are constants:p11 0
and 0 < p12 < ε.It is clear that the inequality 5.13is fulfilled if we set αsmall enough.
Consider the following homogeneous boundary value problem:
x1t p12x2t 0,
x2t−x1t 0,
x10 x1ω, x2ω 0.
The componentsx1, x2of the solution vector are periodic and forω 2π/√p12the boundary
value problem5.14has a nontrivial solution.
Let us prove the following assertions, giving an efficient test of nonpositivity of the elements in the nth row of Green’s matrix in the case when the coefficients|pnj|are small
enough forj 1, . . . , n−1.
Theorem 5.5. Let the following conditions be fulfilled: 1pij ≤0fori /j, i, j 1, . . . , n−1;
In the left-hand side, we have the inequality
pnnteατnn−
Substituting thisαinto5.19and the right part of5.17, we obtain inequalities5.15and
Remark 5.6. It can be stressed that we do not require a smallness of the interval 0, ω in Theorems5.1–5.5.
Remark 5.7. It can be noted that inequality5.15 is best possible in the following sense. If
pnj 0 forj 1, . . . , n−1, pnn const>0,then system5.1and inequality5.15become of
the following forms:
xit fit, i 1, . . . , n−1, xnt pnnxnt−τnn fnt, t∈0, ω. 5.20
pnnτnn≤ 1
e, t∈0, ω, 5.21
respectively. The opposite to5.21inequalitypnnτnn>1/eimplies oscillation of all solutions 15of the equation
xnt pnnxt−τnn 0, t∈0, ω. 5.22
It implies that the homogeneous problem
xit 0, i 1, . . . , n−1, xnt pnnxnt−τnn 0, t∈0, ω,
xi0 xiω, i 1, . . . , n−1, xnω 0 5.23
has nontrivial solutions for correspondingω.Now it is clear that we cannot substitute
pnntτnnexp
⎧ ⎨ ⎩τnn
n−1
j 1
pnj∗
⎫ ⎬ ⎭≤
1ε
e , t∈0, ω, 5.24
whereεis any positive number instead of inequality5.15.
Acknowledgments
The author thanks the referees for their available remarks. This research was supported by The Israel Science FoundationGrant no. 828/07.
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