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# Maximum Principles and Boundary Value Problems for First-Order Neutral Functional Differential Equations

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 141959 (2009)

## Abstract

We obtain the maximum principles for the first-order neutral functional differential equation where , and are linear continuous operators, and are positive operators, is the space of continuous functions, and is the space of essentially bounded functions defined on . New tests on positivity of the Cauchy function and its derivative are proposed. Results on existence and uniqueness of solutions for various boundary value problems are obtained on the basis of the maximum principles.

## 1. Preliminary

This paper is devoted to the maximum principles and their applications for first order neutral functional differential equation.

where are linear continuous Volterra operators, the spectral radius of the operator is less than one, is the space of continuous functions, is the space of essentially bounded functions defined on We consider (1.1) with the following boundary condition:

where is a linear bounded functional defined on the space of absolutely continuous functions . By solutions of (1.1) we mean functions from the space which satisfy this equation almost everywhere in and such that .

We mean the Volterra operators according to the classical Tikhonov's definition.

Definition 1.1.

An operator is called Volterra if any two functions and coinciding on an interval have the equal images on that is, for and for each

Maximum principles present one of the classical parts of the qualitative theory of ordinary and partial differential equations [1]. Although in many cases, speaking about maximum principles, authors mean quite different definitions of maximum principles (such as e.g., corresponding inequalities, boundedness of solutions and maximum boundaries principles), there exists a deep internal connection between these definitions. This connection was discussed, for example, in the recent paper [2]. Main results of our paper are based on the maximum boundaries principle, that is, on the fact that the maximal and minimal values of the solution can be achieved only at the points or The boundaries maximum principle in the case of the zero operator was considered in the recent papers [2, 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 various boundary value problems.

Although several assertions were presented as the maximum principles for delay differential equations, they can be only interpreted in a corresponding sense as analogs of classical 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 even partial differential equations, do not add so much in problems of existence and uniqueness for boundary value problems with delay differential equations. The Azbelev's definition of the homogeneous 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 the maximum principles for functional differential equations, which were based on the idea of the finite-dimensional fundamental system, were presented in the paper [2].

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

were considered in the known books [6–8] (see also the bibliography therein), where existence and uniqueness of solutions and especially stability and oscillation results for these equations were obtained. There exist problems in applications whose models can be written in the form [9]

This equation is a particular case of (1.1).

Let us note here that the operator in (1.1) can be, for example, of the following forms:

or

where are essentially bounded measurable functions, are measurable functions for and are summable with respect to and measurable essentially bounded with respect to for . All linear combinations of operators (1.5) and (1.6) and their superpositions are also allowed.

The study of the neutral functional differential equations is essentially based on the questions of the action and estimates of the spectral radii of the operators in the spaces of discontinuous functions, for example, in the spaces of summable or essentially bounded functions. Operator (1.5), which is a linear combination of the internal superposition operators, 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 , we have for each to assume that for every constant Let us suppose everywhere below that this condition is fulfilled. It is known that the spectral radius of the integral operator (1.6) considered on every finite interval is equal to zero (see, e.g., [4]). Concerning the operator (1.5), we can note the sufficient conditions of the fact that its spectral radius is less than one Define the set and If there exists such that ( then on every finite interval the spectral radius of the operator defined by the formula (1.5) for is zero. In the case ( the spectral radius of the operator defined by (1.5) on the finite interval is less than one if The inequality implies that the spectral radius of the operator considered on the semiaxis and defined by (1.5)satisfies the inequality Usually we will also assume that are nondecreasing functions for .

Various results on existence and uniqueness of boundary value problems for this equation and its stability were obtained in [4], where also the basic results about the representation of solutions were presented. Note also in this connection the papers in [12–15], where results on nonoscillation and positivity of Green's functions for neutral functional differential equations were obtained.

It is known [4] that the general solution of (1.1) has the representation

where the kernel is called the Cauchy function, and is the solution of the homogeneous equation satisfying the condition On the basis of representation (1.7), the results about differential inequalities (under corresponding conditions, 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 and the solution . Results about comparison of solutions for delay differential equations solved with respect to the derivative (i.e., in the case when is the zero operator) were obtained in [2, 15, 16], where assertions on existence and uniqueness of solutions of various boundary value problems for first order functional differential equations were obtained.

All results presented in the paper [15] and in the book [16] for equation with the difference of two positive operators are based on corresponding analogs of the following assertion [15]: *Let the operator**and the Cauchy function**of equation*

*be positive for*
*, then the Cauchy function*
*of*
*( 1.1 ) is also positive for*

This result was extent on various boundary value problems in [16] in the form: *Let the operator**and Green's function**of problem ( 1.8 ), ( 1.2 ) be positive in the square**and the spectral radius of the operator**defined by the equality*

*be less than one, then Green's function**of problem ( 1.1 ),* *( 1.2 ) is positive in the square*

The scheme of the proof was based on the reduction of problem (1.8), (1.2) with to the equivalent integral equation where It is clear that the operator is positive if the operator and the Green's function are positive. If the spectral radius or, more roughly, the norm of the operator are less than one, then there exists the inverse bounded operator which is of course positive. This implies the positivity of the Green's function of problem (1.1), (1.2). In order to get the inequality , the classical theorems about estimates of the spectral radius of the operator [17] can be used. All these theorems are based on a corresponding "smallness" of the operator which is actually close to the condition In order to get positivity of and a corresponding smallness of 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 in the cases when the spectral radius satisfies the opposite inequality or Green's function changes its sign? Note, that in the case, when the operator is positive and its spectral radius is less than one, the positivity of the Cauchy function of (1.8) follows from the nonoscillation of the homogeneous equation , and in the case of the zero operator the positivity of is even equivalent to nonoscillation [15]. This allows us to formulate our question also in the form: how can we make the conclusions about nonoscillation of the equation or about positivity of the Cauchy function of (1.1) without assumption about nonoscillation of the equation In this paper we obtain assertions allowing to make such conclusions. Our assertions are based on the assumption that the operator is a dominant among two operators and .

We assume that the spectral radius of the operator is less than one. In this case we can rewrite (1.1) in the equivalent form

and its general solution can be written in the form

where is the Cauchy function of (1.10) [4]. Note that this approach in the study of neutral equations was first used in the paper [14]. Below in the paper we use the fact that the Cauchy function coincides with the fundamental function of (1.10). It is also clear that

## 2. About Maximum Boundaries Principles in the Case of Difference of Two Positive Volterra Operators

In this paragraph we consider the equation

where and are positive linear continuous Volterra operators and the spectral radius of the operator is less than one.

These operators and are -bounded operators and according to [18], they can be written in the form of the Stieltjes integrals

respectively, where the functions and are measurable for and has the bounded variation for almost all and are essentially bounded.

Consider for convenience (2.1) in the following form:

We can study properties of solution of (2.3) on each finite interval since every solution of (2.1) satisfies also the equation

Consider also the homogeneous equation

and the following auxiliary equations (which are analogs of the so-called *s*-trancated equations defined first in [5])

where the operators and are defined by the formulas

and the operator is defined by the equality where for and for We have

for the operator described by formula (1.5), and

for the operator described by formula (1.6). It is clear that for every if

Functions from the space of absolutely continuous functions satisfy (2.6) almost everywhere in , we call solutions of this equation.

It was noted above that the general solution of (2.1) has the representation [4]

where the function is the Cauchy function of (2.1).We use also formula (1.12) connecting and the Cauchy function of (1.10). Note that is a solution of (2.6) as a function of the first argument for every fixed and satisfies also the equation

Let us formulate our results about positivity of the Cauchy function and the maximum boundaries principle in the case when the condition is not assumed.

Consider the equation

Theorem 2.1.

Let be a positive Volterra operator, , let the functions and be nondecreasing functions with respect to for almost every and let the following inequality be fulfilled:

then the Cauchy function of (2.12) and its derivative satisfy the inequalities for

Consider now the equation

Theorem 2.2.

Let be a positive Volterra operator, let the functions and be nondecreasing functions with respect to for almost every and let the following inequalities

be fulfilled, then the Cauchy function of (2.14) and its derivative satisfy the inequalities and for

Consider the delay equation

where

with and measurable functions and This equation is a particular case of (2.14).

Theorem 2.3.

Let be a positive Volterra operator, and for then the Cauchy function of (2.16) and its derivative satisfy the inequalities and for

Example 2.4.

The inequality on deviating argument is essential as the following equation

demonstrates. This is a particular case of (2.16), where is the zero operator,

The function

is a nontrivial solution of (2.18) and its Cauchy function satisfies the equality

that is, for for for for We see that each interval where is a nonoscillation one for this equation, but changes its sign for

Consider the integrodifferential equation

as a particular case of (2.14).

Let us define the functions and where

Theorem 2.5.

Let be a positive Volterra operator, for and the following inequalities be fulfilled:

then the Cauchy function of (2.22) and its derivative satisfy the inequalities and for

Consider the equation

Let us define

In the following assertion the integral term is dominant.

Theorem 2.6.

Let be a positive Volterra operator, for and the following inequalities be fulfilled:

then the Cauchy function of (2.24) and its derivative satisfy the inequalities and for

Consider the equation

In the following assertion the term is a dominant one.

Theorem 2.7.

Let be a positive Volterra operator, for and the following inequalities be fulfilled:

then the Cauchy function of (2.27) and its derivative satisfy the inequalities and for

Consider now the equation

In the following assertion we do not assume inequalities or Here the sum is a dominant term.

Theorem 2.8.

Let be a positive Volterra operator, for and let the following inequalities be fulfilled:

for then the Cauchy function of (2.29) and its derivative satisfy the inequalities and for

The proofs of Theorems 2.1–2.8 are based on the following auxiliary lemmas.

Lemma 2.9 ([2]).

Let be the zero operator. Then the following two assertions are equivalent:

(1)for every positive there exists a positive function such that for

(2)the Cauchy function of (2.1) is positive for

Lemma 2.10.

Let be a positive Volterra operator, and for every positive there exist a positive function such that for then the Cauchy function of (1.10) is positive for

Proof.

Using the condition we can write (1.1) in the form (1.10). The positivity of the operator implies that the inequality follows from the inequality It is clear that the inequality *for* implies the inequality *for* Now according to Lemma 2.9, we obtain that for

Lemma 2.11.

Let and be positive Volterra operators, and for every the inequality

be fulfilled. Then for

In order to prove Lemma 2.11 we set for every in the assertion 1 of Lemma 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 and the functions and are defined by formula (2.19) respectively. The operator is the zero one for every and consequently for for and condition (2.31) is not fulfilled for .

Lemma 2.13.

Let and be positive Volterra operators, and inequality (2.31) be fulfilled for every Then and ( for

Proof.

According to Lemma 2.11, we have for The positivity of the operator and formula (1.12) implies now that for

Now let us prove that ( for We use the fact that the function , as a function of the argument for each fixed satisfies (2.11) and the condition .

The following integral equation

is equivalent to (2.11) with the condition

The spectral radius of the operator defined by the equality

is zero for every positive number [4]. Let us build the sequence

where the iterations start with the constant for .

The sequence of functions converges in the space to the unique solution of (2.33) on the interval . It is clear that this solution is absolutely continuous. It follows from the fact that all operators are Volterra ones, that the solution of (2.11) with the initial condition and the solution of (2.33) coincide for

Positivity of the operator , the inequalities and (2.31) imply nonnegativity of the derivatives

Let us prove now that the sequence of nondecreasing functions converges to the nondecreasing function Assume in the contrary that there exist two points such that Let us choose There exists a number such that for , and there exists such that for It is clear that for This contradicts to the fact that nondecreases.

We have proven that for every positive the solution of (2.33) is nondecreasing for . It means that the solution of (2.11) is nondecreasing for every and consequently for

Positivity of the operator the inequality and formula (1.12) imply now the inequality for

To prove Theorems 2.1–2.8 it is sufficient to note that the conditions of each theorem imply inequality (2.31).

Remark 2.14.

The space of solutions of the homogeneous equation

in the case is one dimensional. All nontrivial solutions of (2.37) are proportional to One of the assertions of Lemma 2.11 claims that for that is, all nontrivial positive solutions do not decrease. This allows us to consider Lemma 2.13 as the maximum boundaries principle for (2.1). Theorems 2.1–2.8 present the sufficient conditions of this maximum principle for the equations (2.12), (2.14), (2.16), (2.22), (2.24), (2.27) and (2.29) respectively.

Remark 2.15.

The condition about the spectral radius of the operator is essential as the following example demonstrates.

Example 2.16.

Consider the equation

The spectral radius of the operator , defined by the formula is equal to one. All other conditions of Theorems 2.1–2.8 for the zero operators and are fulfilled. The space of solutions of this neutral homogeneous equation is infinitely dimensional. Every linear functions satisfy the homogeneous equation

If , the solutions are decreasing.

## 3. About Nondecreasing Solutions of Neutral Equations

Let us consider the equation

where and are positive linear continuous Volterra operators, and the spectral radius of the operator is less than one. If the operator is positive, then is not generally speaking a positive operator. This is the main difficulty in the study of positivity of the solution and its derivative . All previous results about the positivity of solutions for this equation assumed the negativity of the operator (see, e.g., [12, 13, 15]). In this paragraph we propose results about positivity of solutions in the case of the positive operator defined by the equality

Let us start with the equation

Theorem 3.1.

Assume that the spectral radius of the operator defined by equality (3.2) is less than one, and are nondecreasing functions, and the coefficients satisfy the inequalities and for then the solution of the equation

such that satisfies the inequalities for and in the case, when there exists such that the solution of (3.3) is nonnegative and nondecreasing for every positive nondecreasing function

Consider the equation

Theorem 3.2.

Let the spectral radius of the operator defined by equality (3.2) be less than one, be a nondecreasing function and the functions and be nondecreasing functions with respect to , and the following inequalities be fulfilled

for then the solution of the equation

such that satisfies the inequalities for and in the case, when there exists such that the solution of (3.5) is nonnegative and nondecreasing for every positive nondecreasing function

Consider the integrodifferential equation

Theorem 3.3.

Let the spectral radius of the operator defined by equality (3.2) be less than one, be a nondecreasing function and , for and the following inequalities be fulfilled

then the solution of the equation

such that satisfies the inequalities for and in the case, when there exists such that the solution of (3.8) is nonnegative and nondecreasing for every positive nondecreasing function

Consider the equation

In the following assertion the integral term is dominant.

Theorem 3.4.

Let the spectral radius of the operator defined by equality (3.2) be less than one, be a nondecreasing function, for and the following inequality be fulfilled

then the solution of the equation

such that satisfies the inequalities for and in the case, when there exists such that the solution of (3.11) is nonnegative and nondecreasing for every positive nondecreasing function

Consider the equation

In the following assertion the term is dominant.

Theorem 3.5.

Let the spectral radius of the operator defined by equality (3.2) be less than one, be a nondecreasing function, for and the following inequality

be fulfilled, then the solution of the equation

such that satisfies the inequalities for and in the case, when there exists such that the solution of (3.14) is nonnegative and nondecreasing for every positive nondecreasing

Consider now the equation

In the following assertion we do not assume inequalities or Here the sum is a dominant term.

Theorem 3.6.

Let the spectral radius of the operator defined by equality (3.2) be less than one, be a nondecreasing function, for and the following inequalities be fulfilled

for then the solution of the equation

such that satisfies the inequalities for and in the case, when there exists such that the solution of (3.17) is nonnegative and nondecreasing for every positive nondecreasing

Let us write (3.1) in the form

The spectral radius of the operator is less than one*,* then there exists the bounded operator and we can write (3.21) in the form

Denote by the Cauchy function of the equation which is also the fundamental function of (3.1).

Proofs of Theorems 3.1–3.6 are based on the following auxiliary assertions.

Lemma 3.7.

Let and be positive Volterra operators, the spectral radius of the operator be less than one and

for every nonnegative Then and for

Proof.

Lemma 2.9 is true for (3.22). Let us set in the assertion 1 of Lemma 2.9. Condition (3.23) implies, according to Lemma 2.9, that for

Now let us prove that for We use the fact that the function as a function of argument for each fixed positive satisfies the equation

and the condition

The following integral equation

is equivalent to (3.24) with the condition

The spectral radius of the operator defined by the equality

is zero for every positive number [4]. Let us build the sequence

where the iterations start with the constant for .

The sequence of functions converges in the space to the unique solution of (3.25) on the interval . It is clear that this solution is absolutely continuous. It follows from the fact that all operators are Volterra ones, that the solution of (3.24) with the initial condition and the solution of (3.25) coincide for

Positivity of the operator , the inequalities and (3.23) imply nonnegativity of the derivatives

Repeating the argumentation used in the proof of Lemma 2.13, we obtain that this sequence of nondecreasing functions converges to the nondecreasing solution , that is, for

Concerning nonhomogeneous (3.1) we propose the following assertion.

Lemma 3.8.

Let and be positive Volterra operators, the spectral radius of the operator be less than one and inequality (3.23) be fulfilled for every nonnegative Then the solution of the homogeneous equation

such that satisfies inequalities for If in addition the nonnegative function satisfies the inequality for then the solution of (3.1) is nonnegative and nondecreasing for every positive nondecreasing

Remark 3.9.

The inequality *for* is fulfilled if a nonnegative function is nondecreasing and the norm of the operator is less than one.

Proof of Lemma 3.8.

Assertions about nonnegativity of solution of the homogeneous equation and its derivative follows from the equalities and and Lemma 3.7. From the representation of solutions of (3.22) we can write

It is clear now that the inequality for positivity of and nonnegativity of for imply the inequalities for

The proofs of Theorems 3.1–3.6 follows from the fact that conditions of every theorem imply the conditions of Lemma 3.8 for corresponding equations.

Remark 3.10.

The condition

cannot be set instead of condition (3.23) as Example 2.4 demonstrates. In this example, the operator is the zero one, and for and for condition (3.31) fulfilled, the Cauchy function of (2.18) and its derivative change their signs. We noted that the inequality on delays avoids this situation.

Remark 3.11.

In the case of the neutral equation

where

the inequality for does not avoid the changes of signs of and its derivative:

and for and for

This example demonstrates that we cannot set very natural inequality

instead of

in Theorem 3.1 even in the case when for .

## 4. Maximum Boundaries Principles in Existence and Uniqueness of Boundary Value Problems

Consider the boundary value problems of the following type

where is a linear bounded functional and

It was explained in Remark 2.14 that Lemma 2.13 can be considered as the maximum boundaries principle for (4.1), and Theorems 2.1–2.8 present sufficient conditions of the maximum boundaries principles for equations (2.12), (2.14), (2.16), (2.22), (2.24), (2.27) and (2.29) respectively (i.e., under these conditions the modulus of nontrivial solutions of the corresponding homogeneous equations does not decrease). This allows us to obtain various results about existence and uniqueness of solutions of boundary value problems for these equations without the standard assumption about smallness of the norms of the operators and .

The assertions about existence and uniqueness are based on the known Fredholm alternative for functional differential equations.

Lemma 4.1 ([4]).

Let the spectral radius of the operator be less than one, then boundary value problem (4.1), (4.2) is uniquely solvable for each if and only if the homogeneous problem

has only the trivial solution.

Theorem 4.2.

Let be a positive Volterra operator and its spectral radius satisfy the inequality and for each the inequality

be fulfilled. Then the following assertions are true:

(1)If is a linear nonzero positive functional, then boundary value problem (4.1), (4.2) is uniquely solvable for each

(2)The boundary value problem (4.1), (4.5), where

and the norm of the linear functional is less than one is uniquely solvable for each

(3)The boundary value problem (4.1), (4.6), where

with , and there exists an index such that is uniquely solvable for each

(4)The boundary value problem (4.1), (4.7), where

in the case when for for , , and there exists such that is uniquely solvable for each

Proof.

If we suppose in the contrary that the assertions 1–4 are not true, then according to Lemma 4.1, the nontrivial solution of homogeneous problem (4.3) exists. According to Lemma 2.13 (see also Remark 2.14), the maximum boundaries principle is true, moreover, solutions of the homogeneous equation does not decrease on Conditions of each of the assertions 1–4 lead us to the inequality which contradicts to the existence of the nontrivial solution of the homogeneous problem .

Theorem 4.3.

Conditions of each of Theorems 2.1–2.8 imply assertions 1–4 of Theorem 4.2 for equations (2.12), (2.14), (2.16), (2.22), (2.24), (2.27) and (2.29) respectively.

In order to prove Theorem 4.3 we have only to note that conditions of Theorem 4.2 for corresponding equations follow from the conditions of each of Theorems 2.1–2.8.

Remark 4.4.

The condition

cannot be set instead of condition (4.4) as the following example demonstrates.

Example 4.5.

Consider the boundary value problem

where and are defined by formula (2.19) respectively. Here the operator is defined as ( and inequality (4.8) is fulfilled, but the operator is a zero operator for and inequality (4.4) is not true. Formula (2.20) defines the nontrivial solution of the homogeneous problem

According to Lemma 4.1, problem (4.9) cannot be uniquely solvable for each

Remark 4.6.

The periodic problem

where demonstrates that the condition in the assertion 2 of Theorem 4.2 is essential: the function for is a nontrvial solution of the homogeneous boundary value problem

and reference to Lemma 4.1 completes this example.

Remark 4.7.

According to Lemma 4.1, the problem

demonstrates that the condition about existence of such that in the assertion 3 is essential, and the problem

demonstrates that the condition about existence of such that in the assertion 4 of Theorem 4.2 is essential. If we suppose that such does not exist, then the function for is a nontrvial solution of each of the homogeneous boundary value problems

Remark 4.8.

The condition for in the assertion 4 of Theorem 4.2 cannot be omitted that follows from the example of one of the reviewers: the function is a nontrivial solution of the boundary value problem

where for for for In this case and consequently all other conditions of the assertion 4 of Theorem 4.2 are fulfilled.

Remark 4.9.

Let us define the set

and the following condition:

(a)there exists a set of nonzero measure (i.e.,

Instead of the condition in the assertion 2 of Theorem 4.2 we can assume that the inequality and the condition (a) are fulfilled.

Remark 4.10.

Instead of the condition about existence of such that in the assertion 3 of Theorem 4.2 we can assume that

Condition (4.18) is essential as the following example demonstrates.

Example 4.11.

The homogeneous boundary value problem

where

has a nontrivial solution

In this case we have

Remark 4.12.

Let us denote as the following set: Instead of the condition about existence of such that in the assertion 4, we can assume the following:

Example 4.13.

Consider (4.19), where the coefficient is defined by (4.21), with the boundary condition

Homogeneous problem (4.19), (4.23) has a nontrivial solution defined by (4.22). In this case we have

Consider now (3.1) with the opposite sign near the neutral term (

Theorem 4.14.

Let the conditions of Lemma 3.8 be fulfilled for (3.1), then assertions 1–4 of Theorem 4.2 are true for (3.1).

Proof follows from the fact that according to Lemma 3.8, solution of the homogeneous equation

does not decrease. Conditions of each of the assertions 1–4 of Theorem 4.2 lead us to the maximum boundaries principle and consequently to the inequality which contradicts to the existence of the nontrivial solution of the homogeneous problem .

Theorem 4.15.

Conditions of each of Theorems 3.1–3.6 imply assertions 1–4 of Theorem 4.2 for equations (3.3), (3.5), (3.8), (3.11), (3.14) and (3.17) respectively.

Proof follows from the fact that conditions of each of Theorems 3.1–3.6 imply that the conditions of Lemma 3.8 are fulfilled.

Note that Remarks 4.4–4.12 are relevant also for Theorems 4.14–4.15.

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## Acknowlegments

This research was supported by The Israel Science Foundation (Grant no. 828/07). The authors thank the reviewers for their valuable suggestions.

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### Keywords

- Maximum Principle
- Nontrivial Solution
- Spectral Radius
- Homogeneous Equation
- Functional Differential Equation