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Volume 2009, Article ID 141959, 26 pagesdoi:10.1155/2009/141959 Research Article Maximum Principles and Boundary Value Problems for First-Order Neutral Functional Differential Equations

Volume 2009, Article ID 141959, 26 pages

doi:10.1155/2009/141959

Research Article

Maximum Principles and Boundary Value

Problems for First-Order Neutral Functional

Differential Equations

Alexander Domoshnitsky, Abraham Maghakyan,

and Roman Shklyar

Department of Mathematics and Computer Science, Ariel University Center of Samaria, Ariel 44837, Israel

Correspondence should be addressed to Alexander Domoshnitsky,adom@ariel.ac.il

Received 11 April 2009; Revised 25 August 2009; Accepted 3 September 2009

Recommended by Marta A D Garc´ıa-Huidobro

We obtain the maximum principles for the first-order neutral functional differential equation

proposed Results on existence and uniqueness of solutions for various boundary value problemsare obtained on the basis of the maximum principles

Copyrightq 2009 Alexander Domoshnitsky et al This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited

0,ωare linear continuous

Volterra operators, the spectral radius ρS of the operator S is less than one, C 0,ω is the

space of continuous functions, L∞0,ωis the space of essentially bounded functions defined on

0, ω We consider 1.1 with the following boundary condition:

where l : D 0,ω → R1 is a linear bounded functional defined on the space of absolutely

continuous functions D 0,ω By solutions of1.1 we mean functions x : 0, ω → R1from the

space D 0,ωwhich satisfy this equation almost everywhere in0, ω and such that x∈ L∞

0,ω

We mean the Volterra operators according to the classical Tikhonov’s definition

Definition 1.1 An operator T is called Volterra if any two functions x1and x2coinciding on

an interval0, a have the equal images on 0, a, that is, Tx1t  Tx2t for t ∈ 0, a and for each 0 < a ≤ ω.

Maximum principles present one of the classical parts of the qualitative theory ofordinary and partial differential equations 1 Although in many cases, speaking aboutmaximum principles, authors mean quite different definitions of maximum principles such

as e.g., corresponding inequalities, boundedness of solutions and maximum boundariesprinciples, there exists a deep internal connection between these definitions This connectionwas discussed, for example, in the recent paper2 Main results of our paper are based onthe maximum boundaries principle, that is, on the fact that the maximal and minimal values

of the solution can be achieved only at the points 0 or ω The boundaries maximum principle

in the case of the zero operator S was considered in the recent papers2,3 In this paper

we develop the maximum boundaries principle for neutral functional differential equation

1.1 and on this basis we obtain results on existence and uniqueness of solutions of variousboundary value problems

Although several assertions were presented as the maximum principles for delaydifferential equations, they can be only interpreted in a corresponding sense as analogs ofclassical ones for ordinary differential equations and do not imply important corollaries,reached on the basis of the finite-dimensional fundamental systems For example, results,associated with the maximum principles in contrast with the cases of ordinary and evenpartial differential equations, do not add so much in problems of existence and uniqueness forboundary value problems with delay differential equations The Azbelev’s definition of thehomogeneous delay differential equation 4,5 allowed his followers to consider questions

of existence, uniqueness and positivity of solutions on this basis The first results about themaximum principles for functional differential equations, which were based on the idea ofthe finite-dimensional fundamental system, were presented in the paper2

Neutral functional differential equations have their own history Equations in the form

Let us note here that the operator S : L∞0,∞ → L∞

0,∞in1.1 can be, for example, ofthe following forms:

where q j t are essentially bounded measurable functions, τ j t are measurable functions for

j  1, , m, and k i t, s are summable with respect to s and measurable essentially bounded with respect to t for i  1, , n All linear combinations of operators 1.5 and 1.6 and theirsuperpositions are also allowed

The study of the neutral functional differential equations is essentially based on thequestions of the action and estimates of the spectral radii of the operators in the spaces ofdiscontinuous functions, for example, in the spaces of summable or essentially boundedfunctions Operator 1.5, which is a linear combination of the internal superpositionoperators, is a key object in this topic Properties of this operator were studied by Drakhlin

10,11 In order to achieve the action of operator 1.5 in the space of essentially bounded

functions L∞0,∞ , we have for each j to assume that mes{t : τ j t  c}  0 for every constant c.

Let us suppose everywhere below that this condition is fulfilled It is known that the spectralradius of the integral operator1.6, considered on every finite interval t ∈ 0, ω, is equal to

zerosee, e.g., 4 Concerning the operator 1.5, we can note the sufficient conditions of the

fact that its spectral radius ρS is less than one Define the set κ j ε  {t ∈ 0, ∞ : t − τ j t ≤ ε} and κ ε  m

j1κ j ε If there exists such ε that mes κ ε   0, then on every finite interval

t ∈ 0, ω the spectral radius of the operator S defined by the formula 1.5 for t ∈ 0, ω

is zero In the case mesκ ε  > 0, the spectral radius of the operator S defined by 1.5 on

the finite interval t ∈ 0, ω is less than one if ess sup t ∈κ εm

j1|q j t| < 1 The inequality

ess supt ∈0,∞m

j1|q j t| < 1 implies that the spectral radius ρS of the operator S considered

on the semiaxis t ∈ 0, ∞ and defined by 1.5, satisfies the inequality ρS < 1 Usually we will also assume that τ j are nondecreasing functions for j  1, , m.

Various results on existence and uniqueness of boundary value problems for thisequation and its stability were obtained in 4, where also the basic results about therepresentation of solutions were presented Note also in this connection the papers in12–

15, where results on nonoscillation and positivity of Green’s functions for neutral functionaldifferential equations were obtained

It is known4 that the general solution of 1.1 has the representation

x t 

t

0

where the kernel Ct, s is called the Cauchy function, and Xt is the solution of the

homogeneous equationMxt  0, t ∈ 0, ω, satisfying the condition X0  1 On the

basis of representation1.7, the results about differential inequalities under correspondingconditions, solutions of inequalities are greater or less than solution of the equation can

be formulated in the following form of positivity of the Cauchy function Ct, s and the solution Xt Results about comparison of solutions for delay differential equations solved

with respect to the derivativei.e., in the case when S is the zero operator were obtained in

2,15,16, where assertions on existence and uniqueness of solutions of various boundaryvalue problems for first order functional differential equations were obtained

All results presented in the paper 15 and in the book 16 for equation with thedifference of two positive operators are based on corresponding analogs of the followingassertion15: Let the operator A and the Cauchy function Ct, s of equation

Mx t ≡ xt −Sx

be positive for 0 ≤ s ≤ t ≤ ω, then the Cauchy function Ct, s of 1.1 is also positive for 0 ≤ s ≤ t ≤

ω.

This result was extent on various boundary value problems in16 in the form: Let the

operator A and Green’s function Gt, s of problem 1.8, 1.2 be positive in the square 0, ω×0, ω

and the spectral radius of the operator Ω : C 0,ω → C 0,ω defined by the equality

The scheme of the proof was based on the reduction of problem1.8, 1.2 with c  0,

to the equivalent integral equation xt  Ωxt  ϕt, where ϕt  ω

0Gt, sfsds It

is clear that the operatorΩ is positive if the operator A and the Green’s function Gt, s are positive If the spectral radius ρΩ or, more roughly, the norm Ω of the operator Ω :

C 0,ω → C 0,ωare less than one, then there exists the inverse bounded operatorI − Ω−1 

I Ω  Ω2 · · · : C 0,ω → C 0,ω , which is of course positive This implies the positivity of

the Green’s function Gt, s of problem 1.1, 1.2 In order to get the inequality ρΩ < 1, the

classical theorems about estimates of the spectral radius of the operatorΩ : C 0,ω → C 0,ω

17 can be used All these theorems are based on a corresponding “smallness” of the operator

Ω, which is actually close to the condition Ω < 1 In order to get positivity of Ct, s and

Gt, s a corresponding smallness of B was assumed.

Below we present another approach to this problem starting with the following

question: how can one conclude about positivity of Green’s function Gt, s in the cases when the spectral radius satisfies the opposite inequality ρΩ ≥ 1 or Green’s function Gt, s changes its sign? Note, that in the case, when the operator S : L∞0,ω → L∞

0,ω is positive

and its spectral radius is less than one, the positivity of the Cauchy function Ct, s of 1.8

follows from the nonoscillation of the homogeneous equation Mx  0, and in the case of

the zero operator S, the positivity of Ct, s is even equivalent to nonoscillation 15 Thisallows us to formulate our question also in the form: how can we make the conclusions about

nonoscillation of the equation Mx  0 or about positivity of the Cauchy function Ct, s

of 1.1 without assumption about nonoscillation of the equation Mx  0? In this paper

we obtain assertions allowing to make such conclusions Our assertions are based on the

assumption that the operator A is a dominant among two operators A and B.

We assume that the spectral radius of the operator S : L∞0,ω → L∞

0,ωis less than one

In this case we can rewrite1.1 in the equivalent form

Nxt ≡ xt − I − S−1A − Bxt  I − S−1f t, t ∈ 0, ω, 1.10and its general solution can be written in the form

2 About Maximum Boundaries Principles in the Case of

Difference of Two Positive Volterra Operators

In this paragraph we consider the equation

0,∞are positive linear

continuous Volterra operators and the spectral radius ρS of the operator S is less than one These operators A and B are u-bounded operators and according to18, they can bewritten in the form of the Stieltjes integrals

respectively, where the functions a·, ξ and b·, ξ : 0, ω → R1 are measurable for ξ ∈

0, ω, at, · and bt, · : 0, ω → R1has the bounded variation for almost all t ∈ 0, ω and

t

ξ0a t, ξ, t

ξ0b t, ξ are essentially bounded.

Consider for convenience2.1 in the following form:

We can study properties of solution of2.3 on each finite interval 0, ω since every solution

x t of 2.1 satisfies also the equation

and the following auxiliary equations which are analogs of the so-called s-trancated

equations defined first in5

and the operator S s : L∞s,∞ → L∞

s,∞is defined by the equality S s y s t  Syt, where

y s t  yt for t ≥ s and yt  0 for t < s We have

s,∞ , satisfy2.6 almost everywhere in s, ∞, we call solutions of this equation.

It was noted above that the general solution of2.1 has the representation 4

x t 

t

0

where the function Ct, s is the Cauchy function of 2.1.We use also formula 1.12

connecting Ct, s and the Cauchy function C0t, s of 1.10 Note that C0t, s is a solution of

2.6 as a function of the first argument t for every fixed s and satisfies also the equation

0,∞ be a positive Volterra operator, ρ S < 1, 0 ≤ h1t ≤ h2t ≤

g1t ≤ g2t ≤ t, let the functions at, ξ and bt, ξ be nondecreasing functions with respect to ξ for

almost every t, and let the following inequality be fulfilled:

0,∞ be a positive Volterra operator, ρ S < 1, 0 ≤ h 1i t ≤

h 2i t ≤ g 1i t ≤ g 2i t ≤ t, let the functions a i t, ξ and b i t, ξ be nondecreasing functions with

respect to ξ for almost every t and let the following inequalities

be fulfilled, then the Cauchy function C t, s of 2.14 and its derivative C

t t, s satisfy the inequalities

0,∞ be a positive Volterra operator, ρ S < 1, h i t ≤ g i t ≤ t

and 0 ≤ b i t ≤ a i t for t ∈ 0, ∞, i  1, , m, then the Cauchy function Ct, s of 2.16 and

its derivative Ct t, s satisfy the inequalities Ct, s > 0 and C

that is, Ct, 0 > 0 for 0 ≤ t < 8, Ct, 0 < 0 for t > 8, Ct, s > 0 for 0 < s ≤ 2, 0 ≤

t < 4, C t, s < 0 for 0 < s ≤ 2, t > 4 We see that each interval 0, ω, where ω < 8, is a nonoscillation one for this equation, but Ct, s changes its sign for 0 < s ≤ 2, 4 < t.

Consider the integrodifferential equation

0,∞ be a positive Volterra operator, ρ S < 1, h 1i t ≤ h 2i t ≤

g 1i t ≤ g 2i t ≤ t, k i t, ξ ≥ 0, m i t, ξ ≥ 0 for t, ξ ∈ 0, ∞, i  1, , m, and the following

then the Cauchy function of 2.22 and its derivative C

t t, s satisfy the inequalities Ct, s > 0 and

0,∞ be a positive Volterra operator, ρ S < 1, ht ≤ g1t ≤

g2t ≤ t, mt, ξ ≥ 0, bt ≥ 0 for t, ξ ∈ 0, ∞ and the following inequalities be fulfilled:

b tχht, 0 ≤

g0t

then the Cauchy function of 2.24 and its derivative C

t t, s satisfy the inequalities Ct, s > 0 and

Ct t, s ≥ 0 for 0 ≤ s ≤ t < ∞.

Consider the equation

0,∞ be a positive Volterra operator, ρ S < 1, kt, ξ ≥ 0, h1t ≤

h2t ≤ gt ≤ t, at ≥ 0 for t, ξ ∈ 0, ∞ and the following inequalities be fulfilled:

h0t

then the Cauchy function of 2.27 and its derivative C

t t, s satisfy the inequalities Ct, s > 0 and

The proofs of Theorems2.1–2.8are based on the following auxiliary lemmas.

Lemma 2.9 2 Let S be the zero operator Then the following two assertions are equivalent:

1 for every positive s there exists a positive function v s ∈ D s,∞ such that M s v s t ≤ 0 for

t ∈ s, ∞,

2 the Cauchy function Ct, s of 2.1 is positive for 0 ≤ s ≤ t < ∞.

Lemma 2.10 Let S : L∞

0,∞ → L∞

0,∞ be a positive Volterra operator, ρ S < 1 and for every positive

s there exist a positive function v s ∈ D s,∞ such that M s v s t ≤ 0 for t ∈ s, ∞, then the Cauchy

function C0t, s of 1.10 is positive for 0 ≤ s ≤ t < ∞.

Proof Using the condition ρ S < 1, we can write 1.1 in the form 1.10 The positivity of

the operator S : L∞0,∞ → L∞

0,∞ implies that the inequalityI − S−1f ≥ 0 follows from

the inequality f ≥ 0 It is clear that the inequality M s v s t ≤ 0 for t ∈ s, ∞ implies

the inequalityN s v s t ≤ 0 for t ∈ s, ∞ Now according toLemma 2.9, we obtain that

be fulfilled Then C0t, s > 0 for 0 ≤ s ≤ t < ∞.

In order to proveLemma 2.11we set v s t ≡ 1, t ∈ s, ∞ for every s ∈ 0, ∞ in the

assertion 1 ofLemma 2.10

Remark 2.12 The condition

cannot be set instead of condition 2.31 in Lemma 2.11 as Example 2.4 demonstrates It

is clear that this inequality is fulfilled for 2.18, where Axt  x0 and Bxt 

b1txh1t, the functions h1and b1are defined by formula2.19 respectively The operator

A s is the zero one for every s > 0 and consequently A s1t  0 for t ∈ s, ∞, B21t 

1/2 for t ∈ 2, ∞ and condition 2.31 is not fulfilled for s  2.

Lemma 2.13 Let A : C 0,∞ → L∞

0,∞ , B : C 0,∞ → L∞

0,∞ and S : L∞0,∞ → L∞

0,∞ be positive Volterra operators, ρ S < 1 and inequality 2.31 be fulfilled for every s ∈ 0, ∞ Then

The following integral equation

is equivalent to2.11 with the condition xs  1.

The spectral radius of the operator T : C s,ω → C s,ω , defined by the equality

where the iterations start with the constant x0t ≡ 1 for t ∈ s, ω.

The sequence of functions x m t converges in the space C s,ω to the unique solution

x t of 2.33 on the interval s, ω It is clear that this solution is absolutely continuous It follows from the fact that all operators are Volterra ones, that the solution yt of 2.11 with

the initial condition ys  1 and the solution xt of 2.33 coincide for t ∈ s, ω.

Positivity of the operator S, the inequalities ρS < 1 and 2.31 imply nonnegativity

Let us prove now that the sequence x m of nondecreasing functions converges to the

nondecreasing function x Assume in the contrary that there exist two points t1 < t2, such

that xt1 > xt2 Let us choose ε < xt1 − xt2/2 There exists a number N1ε such that

|xt1 − x m t1| < ε for m ≥ N1ε, and there exists N2ε such that |x m t2 − xt2| < ε for

m ≥ N2ε It is clear that x m t1 > x m t2 for m ≥ max{N1ε, N2ε} This contradicts to the fact that x m t nondecreases.

We have proven that for every positive ω, the solution x of2.33 is nondecreasing for

t ∈ s, ω It means that the solution x of 2.11 is nondecreasing for every t ∈ s, ∞ and

in the case ρS < 1 is one dimensional All nontrivial solutions of 2.37 are proportional to

C0t, 0 One of the assertions ofLemma 2.11claims that∂/∂tC0t, 0 ≥ 0 for 0 ≤ t < ∞, that

is, all nontrivial positive solutions do not decrease This allows us to considerLemma 2.13

as the maximum boundaries principle for 2.1 Theorems 2.1–2.8 present the sufficientconditions of this maximum principle for the equations2.12, 2.14, 2.16, 2.22, 2.24,

2.27 and 2.29 respectively

Remark 2.15 The condition ρ S < 1 about the spectral radius of the operator S : L∞

0,∞ →

L∞0,∞is essential as the following example demonstrates

Example 2.16 Consider the equation

The spectral radius of the operator S : L∞0,∞ → L∞

0,∞, defined by the formulaSyt 

y t/2, is equal to one All other conditions of Theorems 2.1–2.8for the zero operators A and B are fulfilled The space of solutions of this neutral homogeneous equation is infinitely dimensional Every linear functions x  1 − ct satisfy the homogeneous equation

If c > 0, the solutions x are decreasing.

3 About Nondecreasing Solutions of Neutral Equations

Let us consider the equation

Mxt ≡ xt Sx

t − Axt  Bxt  ft, t ∈ 0, ∞, 3.1

where A : C 0,∞ → L∞

0,∞ and B : C 0,∞ → L∞

0,∞ are positive linear continuous Volterra

operators, and the spectral radius ρS of the operator S : L∞

0,∞ → L∞

0,∞ is less than one

If the operator S is positive, then I  S−1 I − S  S2− S3 · · · is not generally speaking apositive operator This is the main difficulty in the study of positivity of the solution x and its

derivative x All previous results about the positivity of solutions for this equation assumed

the negativity of the operator Ssee, e.g., 12,13,15 In this paragraph we propose results

about positivity of solutions in the case of the positive operator S defined by the equality