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Hindawi Publishing Corporation Journal of Inequalities and Applications 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 Mxt ≡ x  t − Sx  t − AxtBxtft,t∈ 0,ω, where A : C 0,ω → L ∞ 0,ω ,B : C 0,ω → L ∞ 0,ω ,andS : L ∞ 0,ω → L ∞ 0,ω are linear continuous operators, A and B are positive operators, C 0,ω is the space of continuous functions, and L ∞ 0,ω is the space of essentially bounded functions defined on 0,ω. 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. Copyright q 2009 Alexander Domoshnitsky et al. 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. 1. Preliminary This paper is devoted to the maximum principles and their applications for first order neutral functional differential equation.  Mx  t  ≡ x   t  −  Sx    t  −  Ax  t    Bx  t   f  t  ,t∈  0,ω  , 1.1 where A : C 0,ω → L ∞ 0,ω ,B: C 0,ω → L ∞ 0,ω , and S : L ∞ 0,ω → L ∞ 0,ω are linear continuous Volterra operators, the spectral radius ρS of the operator S is less than one, C 0,ω is the space of continuous functions, L ∞ 0,ω is the space of essentially bounded functions defined on 0,ω. We consider 1.1 with the following boundary condition: lx  c, 1.2 2 Journal of Inequalities and Applications where l : D 0,ω → R 1 is a linear bounded functional defined on the space of absolutely continuous functions D 0,ω . By solutions of 1.1 we mean functions x : 0,ω → R 1 from the space D 0,ω which satisfy this equation almost everywhere in 0,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 x 1 and x 2 coinciding on an interval 0,a have the equal images on 0,a, that is, Tx 1 tTx 2 t for t ∈ 0,a and for each 0 <a≤ ω. 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 0 or ω. The boundaries maximum principle in the case of the zero operator S 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  xt − q tx  τt     m  i1 b i  t  x  h i  t   f  t  ,t∈  0, ∞  , 1.3 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 x   t  − q  t  x   τ  t   m  i1 b i  t  x  h i  t   f  t  ,t∈  0, ∞  . 1.4 This equation is a particular case of 1.1. Journal of Inequalities and Applications 3 Let us note here that the operator S : L ∞ 0,∞ → L ∞ 0,∞ in 1.1 can be, for example, of the following forms:  Sy   t   m  j1 q j  t  y  τ j  t   , where τ j  t  ≤ t, y  τ j  t    0ifτ j  t  < 0,t∈  0, ∞  , 1.5 or  Sy   t   n  i1  t 0 k i  t, s  y  s  ds, t ∈  0, ∞  , 1.6 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 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 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 spectral radius of the integral operator 1.6, considered on every finite interval t ∈ 0,ω, 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 ρS is less than one. Define the set κ j ε  {t ∈ 0, ∞ : t −τ j t ≤ ε} and κ ε   m j1 κ 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 mes κ 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 ε  m j1 |q j t| < 1. The inequality ess sup t∈0,∞  m j1 |q j t| < 1 implies that the spectral radius ρS of the operator S considered on the semiaxis t ∈ 0, ∞ and defined by 1.5, satisfies the inequality ρS < 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 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 x  t    t 0 C  t, s  f  s  ds  X  t  x  0  , 1.7 where the kernel Ct, s is called the Cauchy function, and Xt is the solution of the homogeneous equation Mxt0,t∈ 0,ω, satisfying the condition X01. On the 4 Journal of Inequalities and Applications 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 Ct, s and the solution Xt. Results about comparison of solutions for delay differential equations solved with respect to the derivative i.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 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 A and the Cauchy function C  t, s of equation  M  x  t  ≡ x   t  −  Sx    t    Bx  t   f  t  ,t∈  0,ω  , 1.8 be positive for 0 ≤ s ≤ t ≤ ω, then the Cauchy function Ct, s of 1.1 is also positive for 0 ≤ s ≤ t ≤ ω. This result was extent on various boundary value problems in 16 in the form: Let the operator A and Green’s function G  t, 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  Ωx  t    ω 0 G   t, s  Ax  s  ds, 1.9 be less than one, then Green’s function G  t, s of problem 1.1, 1.2 is positive i n the square 0,ω× 0,ω. The scheme of the proof was based on the reduction of problem 1.8, 1.2 with c  0, to the equivalent integral equation xtΩxtϕt, where ϕt  ω 0 G  t, sfsds. It is clear that the operator Ω is positive if the operator A and the Green’s function G  t, s are positive. If the spectral radius ρΩ or, more roughly, the norm Ω of the operator Ω : C 0,ω → C 0,ω are less than one, then there exists the inverse bounded operator I − Ω −1  I ΩΩ 2  ··· : C 0,ω → C 0,ω , which is of course positive. This implies the positivity of the Green’s function Gt, s of problem 1.1, 1.2. In order to get the inequality ρΩ < 1, the classical theorems about estimates of the spectral radius of the operator Ω : 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 C  t, s and G  t, 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 Gt, s in the cases when the spectral radius satisfies the opposite inequality ρΩ ≥ 1 or Green’s function G  t, 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 C  t, s of 1.8 follows from the nonoscillation of the homogeneous equation M  x  0, and in the case of the zero operator S, the positivity of C  t, s 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 Mx  0 or about positivity of the Cauchy function Ct, s of 1.1 without assumption about nonoscillation of the equation M  x  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. Journal of Inequalities and Applications 5 We assume that the spectral radius of the operator S : L ∞ 0,ω → L ∞ 0,ω is less than one. In this case we can rewrite 1.1 in the equivalent form  Nx  t  ≡ x   t  −  I − S  −1  A − B  x  t    I − S  −1 f  t  ,t∈  0,ω  , 1.10 and its general solution can be written in the form x  t    t 0 C 0  t, s  I − S  −1 f  s  ds  X  t  x  0  , 1.11 where C 0 t, s 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 C 0 t, s coincides with the fundamental function of 1.10. It is also clear that  t 0 C  t, s  f  s  ds   t 0 C 0  t, s  I − S  −1 f  s  ds. 1.12 2. About Maximum Boundaries Principles in the Case of Difference of Two Positive Volterra Operators In this paragraph we consider the equation  Mx  t  ≡ x   t  −  Sx    t  −  Ax  t    Bx  t   f  t  ,t∈  0, ∞  , 2.1 where A : C 0,∞ → L ∞ 0,∞ ,B: C 0,∞ → L ∞ 0,∞ and S : L ∞ 0,∞ → L ∞ 0,∞ are positive linear continuous Volterra operators and the spectral radius ρS of the operator S is less than one. These operators A and B are u-bounded operators and according to 18, they can be written in the form of the Stieltjes integrals  Ax  t    t 0 x  ξ  d ξ a  t, ξ  ,  Bx  t    t 0 x  ξ  d ξ b  t, ξ  ,t∈  0, ∞  , 2.2 respectively, where the functions a·,ξ and b·,ξ : 0,ω → R 1 are measurable for ξ ∈ 0,ω,at, · and bt, · : 0,ω → R 1 has the bounded variation for almost all t ∈ 0,ω and  t ξ0 at, ξ,  t ξ0 bt, ξ are essentially bounded. Consider for convenience 2.1 in the following form:  Mx  t  ≡ x   t  −  Sx    t  −  t 0 x  ξ  d ξ a  t, ξ    t 0 x  ξ  d ξ b  t, ξ   f  t  ,t∈  0, ∞  . 2.3 6 Journal of Inequalities and Applications We can study properties of solution of 2.3 on each finite interval 0,ω since every solution xt of 2.1 satisfies also the equation  Mx  t  ≡ x   t  −  Sx    t  −  t 0 x  ξ  d ξ a  t, ξ    t 0 x  ξ  d ξ b  t, ξ   f  t  ,t∈  0,ω  . 2.4 Consider also the homogeneous equation  Mx  t  ≡ x   t  −  Sx    t  −  Ax  t    Bx  t   0,t∈  0, ∞  , 2.5 and the following auxiliary equations which are analogs of the so-called s-trancated equations defined first in 5  M s x  t  ≡ x   t  −  S s x    t  −  A s x  t    B s x  t   0,t∈  s, ∞  ,s≥ 0, 2.6 where t he operators A s : C s,∞ → L ∞ s,∞ and B s : C s,∞ → L ∞ s,∞ are defined by the formulas  A s x  t    t s x  ξ  d ξ a  t, ξ  ,  B s x  t    t s x  ξ  d ξ b  t, ξ  ,t∈  s, ∞  , 2.7 and the operator S s : L ∞ s,∞ → L ∞ s,∞ is defined by the equality S s y s tSyt, where y s tyt for t ≥ s and yt0fort<s.We have  S s y   t   m  j1 q j  t  y  τ j  t   , where τ j  t  ≤ t, y  τ j  t    0ifτ j  t  <s, t∈  s, ∞  , 2.8 for the operator described by formula 1.5,and  S s y   t   n  i1  t s k i  t, s  y  s  ds, t ∈  s, ∞  , 2.9 for the operator described by formula 1.6. It is clear that ρS s  < 1 for every s ∈ 0, ∞ if ρS < 1. Functions from the space D s,∞ of absolutely continuous functions x : s, ∞ → R 1 ,x  ∈ L ∞ s,∞ , satisfy 2.6 almost everywhere in s, ∞, we call solutions of this equation. It was noted above that the general solution of 2.1 has the representation 4 x  t    t 0 C  t, s  f  s  ds  X  t  x  0  , 2.10 Journal of Inequalities and Applications 7 where the function Ct, s is the Cauchy function of 2.1.We use also formula 1.12 connecting Ct, s and the Cauchy function C 0 t, s of 1.10.NotethatC 0 t, s is a solution of 2.6 as a function of the first argument t for every fixed s and satisfies also the equation  N s x  t  ≡ x   t    I − S s  −1  −  A s x  t    B s x  t   0,t∈  s, ∞  . 2.11 Let us formulate our results about positivity of the Cauchy function Ct, s and the maximum boundaries principle in the case when the condition C  t, s > 0 is not assumed. Consider the equation x   t  −  Sx    t  −  g 2  t  g 1  t  x  ξ  d ξ a  t, ξ    h 2  t  h 1  t  x  ξ  d ξ b  t, ξ   f  t  ,t∈  0, ∞  . 2.12 Theorem 2.1. Let S : L ∞ 0,∞ → L ∞ 0,∞ be a positive Volterra operator, ρS < 1, 0 ≤ h 1 t ≤ h 2 t ≤ g 1 t ≤ g 2 t ≤ t, let the functions at, ξ and bt, ξ be nondecreasing functions with respect to ξ for almost every t, and let the following inequality be fulfilled: h 2  t   ξh 1  t  b  t, ξ  ≤ g 2  t   ξg 1  t  a  t, ξ  ,t∈  0, ∞  , 2.13 then the Cauchy function Ct, s of 2.12 and its derivative satisfy the inequalities Ct, s > 0,C  t t, s ≥ 0 for 0 ≤ s ≤ t<∞. Consider now the equation x   t  −  Sx    t   m  i1  −  g 2i  t  g 1i  t  x  ξ  d ξ a i  t, ξ    h 2i  t  h 1i  t  x  ξ  d ξ b i  t, ξ    f  t  ,t∈  0, ∞  . 2.14 Theorem 2.2. Let S : L ∞ 0,∞ → L ∞ 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 h 2i  t   ξh 1i  t  b i  t, ξ  ≤ g 2i  t   ξg 1i  t  a i  t, ξ  ,t∈  0, ∞  ,i 1, ,m, 2.15 be fulfilled, then the Cauchy function Ct, s of 2.14 and its derivative C  t t, s satisfy the inequalities Ct, s > 0 and C  t t, s ≥ 0 for 0 ≤ s ≤ t<∞. Consider the delay equation x   t  −  Sx    t  − m  i1 a i  t  x  g i  t    m  i1 b i  t  x  h i  t   f  t  ,t∈  0, ∞  , 2.16 8 Journal of Inequalities and Applications where x  ξ   0forξ<0, 2.17 with a i ,b i ∈ L ∞ 0,∞ and measurable functions g i and h i i  1, ,m. This equation is a particular case of 2.14. Theorem 2.3. Let S : L ∞ 0,∞ → L ∞ 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 Ct, s of 2.16 and its derivative C  t t, s satisfy the inequalities Ct, s > 0 and C  t t, s ≥ 0 for 0 ≤ s ≤ t<∞. Example 2.4. The inequality on deviating argument h i t ≤ g i t is essential as the following equation x   t  − x  0   b 1  t  x  h 1  t   0,t∈  0, ∞  , 2.18 demonstrates. This is a particular case of 2.16, where S is the zero operator, m  1,g 1 t ≡ 0,a 1 t ≡ 1, h 1  t   ⎧ ⎨ ⎩ 0, 0 ≤ t<2, 2, 2 ≤ t, b 1  t   ⎧ ⎪ ⎨ ⎪ ⎩ 0, 0 ≤ t<2, 1 2 , 2 ≤ t. 2.19 The function x  t  ≡ C  t, 0   ⎧ ⎪ ⎨ ⎪ ⎩ t  1, 0 ≤ t<2, 4 − 1 2 t, 2 ≤ t, 2.20 is a nontrivial solution of 2.18 and its Cauchy function Ct, s satisfies the equality C  t, s   ⎧ ⎪ ⎨ ⎪ ⎩ 1, 0 <s≤ 2, 0 ≤ t<2, 1 − 1 2  t − 2  , 0 <s≤ 2, 2 ≤ t, 2.21 that is, Ct, 0 > 0for0 ≤ t<8,Ct, 0 < 0fort>8,Ct, s > 0for0 <s≤ 2, 0 ≤ t<4,Ct, s < 0for0<s≤ 2,t>4. We see that each interval 0,ω, where ω<8, is a nonoscillation one for this equation, but Ct, s changes its sign for 0 <s≤ 2, 4 <t. Journal of Inequalities and Applications 9 Consider the integrodifferential equation x   t  −  Sx    t  − m  i1  g 2i  t  g 1i  t  m i  t, ξ  x  ξ  dξ  m  i1  h 2i  t  h 1i  t  k i  t, ξ  x  ξ  dξ  f  t  ,t∈  0, ∞  , x  ξ   0forξ<0, 2.22 as a particular case of 2.14. Let us define the functions h 0 ji tmax{0,h ji t} and g 0 ji tmax{0,g ji t}, where j  1, 2. Theorem 2.5. Let S : L ∞ 0,∞ → L ∞ 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 inequalities be fulfilled:  h 0 2i  t  h 0 1i  t  k i  t, ξ  dξ ≤  g 0 2i  t  g 0 1i  t  m i  t, ξ  dξ, t ∈  0, ∞  ,i 1, ,m, 2.23 then the Cauchy function of 2.22 and its derivative C  t t, s satisfy the inequalities Ct, s > 0 and C  t t, s ≥ 0 for 0 ≤ s ≤ t<∞. Consider the equation x   t  −  Sx    t  −  g 2  t  g 1  t  m  t, ξ  x  ξ  dξ  b  t  x  h  t   f  t  ,t∈  0, ∞  . x  ξ   0forξ<0. 2.24 Let us define χ  t, s   ⎧ ⎨ ⎩ 0,t<s, 1,t≥ s. 2.25 In the following assertion the integral term is dominant. Theorem 2.6. Let S : L ∞ 0,∞ → L ∞ 0,∞ be a positive Volterra operator, ρS < 1,ht ≤ g 1 t ≤ g 2 t ≤ t, mt, ξ ≥ 0,bt ≥ 0 for t, ξ ∈ 0, ∞ and the following inequalities be fulfilled: b  t  χ  h  t  , 0  ≤  g 0 2  t  g 0 1  t  m  t, ξ  dξ, t ∈  0, ∞  , 2.26 then the Cauchy function of 2.24 and its derivative C  t t, s satisfy the inequalities Ct, s > 0 and C  t t, s ≥ 0 for 0 ≤ s ≤ t<∞. 10 Journal of Inequalities and Applications Consider the equation x   t  −  Sx    t  − a  t  x  g  t     h 2  t  h 1  t  k  t, ξ  x  ξ  dξ  f  t  ,t∈  0, ∞  , x  ξ   0forξ<0. 2.27 In the following assertion the term atxgt is a dominant one. Theorem 2.7. Let S : L ∞ 0,∞ → L ∞ 0,∞ be a positive Volterra operator, ρS < 1,kt, ξ ≥ 0,h 1 t ≤ h 2 t ≤ gt ≤ t, at ≥ 0 for t, ξ ∈ 0, ∞ and the following inequalities be fulfilled:  h 0 2  t  h 0 1  t  k  t, ξ  dξ ≤ a  t  ,t∈  0, ∞  , 2.28 then the Cauchy function of 2.27 and its derivative C  t t, s satisfy the inequalities Ct, s > 0 and C  t t, s ≥ 0 for 0 ≤ s ≤ t<∞. Consider now the equation x   t  −  Sx    t  − a  t  x  g  t    b  t  x  h  t  −  g 2  t  g 1  t  m  t, ξ  x  ξ  dξ   h 2  t  h 1  t  k  t, ξ  x  ξ  dξ  f  t  ,t∈  0, ∞  , x  ξ   0forξ<0. 2.29 In the following assertion we do not assume inequalities kt, ξ ≤ mt, ξ or bt ≤ at. Here the sum atxgt   g 2 t g 1 t mt, ξxξdξ is a dominant term. Theorem 2.8. Let S : L ∞ 0,∞ → L ∞ 0,∞ be a positive Volterra operator, ρS < 1,ht ≤ g 1 t ≤ g 2 t ≤ t, h 1 t ≤ h 2 t ≤ gt ≤ t, kt, ξ ≥ 0,mt, ξ ≥ 0,at ≥ 0,bt ≥ 0 for t, ξ ∈ 0, ∞ and let the following inequalities be fulfilled: b  t  χ  h  t  , 0  ≤  g 0 2  t  g 0 1  t  m  t, ξ  dξ,  h 0 2  t  h 0 1  t  k  t, ξ  dξ ≤ a  t  , 2.30 for t ∈ 0, ∞, then the Cauchy function Ct, s of 2.29 and its derivative C  t t, s satisfy the inequalities Ct, s > 0 and C  t t, s ≥ 0 for 0 ≤ s ≤ t<∞. [...]... t for t ∈ 0, ∞ 4 Maximum Boundaries Principles in Existence and Uniqueness of Boundary Value Problems Consider the boundary value problems of the following type Mx t ≡ x t − Sx t − Ax t lx c, Bx t f t , t ∈ 0, ω , 4.1 4.2 where l : D 0,ω → R1 is a linear bounded functional and c ∈ R1 It was explained in Remark 2.14 that Lemma 2.13 can be considered as the maximum boundaries principle for 4.1 , and. .. References 1 M H Protter and H F Weinberger, Maximum Principles in Differential Equations, Springer, New York, NY, USA, 1984 2 A Domoshnitsky, Maximum principles and nonoscillation intervals for first order Volterra functional differential equations,” Dynamics of Continuous, Discrete & Impulsive Systems, vol 15, no 6, pp 769–814, 2008 3 A Domoshnitsky, Maximum principles and boundary value problems, mathematical... one for every s > 0 and consequently As 1 t 1/2 for t ∈ 2, ∞ and condition 2.31 is not fulfilled for s 2 and S : L∞ → L∞ be Lemma 2.13 Let A : C 0,∞ → L∞ , B : C 0,∞ → L∞ 0,∞ 0,∞ 0,∞ 0,∞ positive Volterra operators, ρ S < 1 and inequality 2.31 be fulfilled for every s ∈ 0, ∞ Then C t, s > 0, ∂/∂t C0 t, s ≥ 0 and (∂/∂t C t, s ≥ 0 for 0 ≤ s ≤ t < ∞ Proof According to Lemma 2.11, we have C0 t, s > 0 for. .. Volterra operator and its spectral radius satisfy Theorem 4.2 Let S : L∞ 0,∞ 0,∞ the inequality ρ S < 1, and for each s ∈ 0, ω the inequality A s 1 t ≥ Bs 1 t for t ∈ s, ω , 4.4 be fulfilled Then the following assertions are true: 1 If l : C 0,ω → R1 is a linear nonzero positive functional, then boundary value problem 4.1 , 4.2 is uniquely solvable for each f ∈ L 0,ω , c ∈ R1 2 The boundary value problem... 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... < ε for m ≥ N1 ε , and there exists N2 ε such that |xm t2 − x t2 | < ε for m ≥ N2 ε It is clear that xm t1 > xm t2 for m ≥ max{N1 ε , N2 ε } This contradicts to the fact that xm t nondecreases We have proven that for every positive ω, the solution x of 2.33 is nondecreasing for t ∈ s, ω It means that the solution x of 2.11 is nondecreasing for every t ∈ s, ∞ and consequently ∂/∂t C0 t, s ≥ 0 for. .. − mx c, 4.5 and the norm of the linear functional m : C 0,ω → R1 is less than one is uniquely solvable for each f ∈ L 0,ω , c ∈ R1 ; 3 The boundary value problem 4.1 , 4.6 , where 2k αj x tj c, 0 ≤ t1 < t2 < · · · < t2k ≤ ω, 4.6 j 1 with 0 ≤ −α2j−1 ≤ α2j , j 1, , k, and there exists an index i such that −α2i−1 < α2i , is uniquely solvable for each f ∈ L 0,ω , c ∈ R1 4 The boundary value problem... norms of the operators A and B Journal of Inequalities and Applications 21 The assertions about existence and uniqueness are based on the known Fredholm alternative for functional differential equations → L∞ be less than one, then Lemma 4.1 4 Let the spectral radius of the operator S : L∞ 0,∞ 0,∞ boundary value problem 4.1 , 4.2 is uniquely solvable for each f ∈ L 0,ω , c ∈ R1 if and only if the homogeneous... Uravneniya, vol 19, no 9, pp 1475–1482, 1983 15 S A Gusarenko and A I Domoshnitski˘, “Asymptotic and oscillation properties of first-order linear ı scalar functional- differential equations,” Differentsial’nye Uravneniya, vol 25, no 12, pp 2090–2103, 1989 ˇ 16 R Hakl, A Lomtatidze, and J Sremr, Some Boundary Value Problems for First Order Scalar Functional Differential Equations, FOLIA, Masaryk University,... case of the neutral equation x t x t−1 −x g t x h t 0, t ∈ 0, ∞ , 3.32 20 Journal of Inequalities and Applications where h t g t ⎧ ⎨0, 0 ≤ t < 3, ⎩2, 3 ≤ t, ⎧ ⎨0, 0 ≤ t < 2, ⎩2, 2 ≤ t, 3.33 the inequality h t ≤ g t for t ∈ 0, ∞ does not avoid the changes of signs of C0 t, s and its derivative: C0 t, 2 ⎧ ⎨t − 1, 2 ≤ t < 3, 3.34 ⎩5 − t, 3 ≤ t, and ∂C0 t, 2 /∂t −1 < 0 for t > 3 and C0 t, 2 < 0 for t > 5 . Inequalities and Applications 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. in the case when ht ≤ gt for t ∈ 0, ∞. 4. Maximum Boundaries Principles in Existence and Uniqueness of Boundary Value Problems Consider the boundary value problems of the following type  Mx  t  ≡. 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