On the solvability of some non-local boundary value problems for functional differential equations

Transcription

1 MASARYK UNIVERSITY Faculty of Science Department of Mathematics and Statistics On the solvability of some non-local boundary value problems for functional differential equations Ph.D. Dissertation Vita Pylypenko Supervisor: András Rontó, DrSc. Brno 2017

2 Bibliographic Entry Author: Title of dissertation: Degree Programme: Field of Study: Supervisor: Mgr. Vita Pylypenko Faculty of Science, Masaryk University Department of Mathematics and Statistics On the solvability of some non-local boundary value problems for functional differential equations Mathematics Mathematical Analysis Andras Ronto, DrSc. Institute of Mathematics Academy of Sciences of the Czech Republic Academic Year: 2016/2017 Number of Pages: 111 Keywords: boundary value problem; singular Cauchy problem; functional differential equation; non-local condition; unique solvability; singular solution

3 Bibliografický záznam Autor: Název práce: Studijní program: Studijní obor: Školitel: Mgr. Vita Pylypenko Přírodovědecká fakulta, Masarykova univerzita Ústav matematiky a statistiky O řešitelnosti některých nelokálních okrajových úlohách pro funkcionální diferenciální rovnice Matematika Matematická analýza András Rontó, DrSc. Akademický rok: 2016/2017 Počet stran: 111 Klíčová slova: okrajová úloha; singulární Cauchyova úloha; funkcionální diferenciální rovnice; nelokální podmínka; jednoznačná řešitelnost; singulární řešení

4 Abstract The dissertation is focused on several topics concerning boundary value problems for functional differential equations. We present general conditions sufficient for the unique solvability of a non-local boundary value problem for systems of linear functional differential equations. These theorems generalise several previously known results and are applicable, in particular, to neutral type functional differential equations. For linear functional differential equations with time singularities, we study the problem on singular solutions with a specified growth. Under a suitable positivity assumption on the underlying operator, the techniques based on two-sided monotone iterations allow us to prove new theorems on the existence and space localisation of such solutions. Corollaries for differential equations with argument deviations are presented. Sufficient conditions for the solvability of a singular Cauchy problem for functional differential equations with non-increasing non-linearities are established. For a system of linear functional differential equations with time singularities, we establish conditions under which the initial value problem is uniquely solvable in the class of functions with a specified growth rate. We show that the conditions are, in a sense, unimprovable.

5 Abstrakt Disertační práce je zaměřena na několik vybraných problémů teorie okrajových úloh pro funkcionální diferenciální rovnice. Uvádíme obecné věty o jednoznačné řešitelnosti nelokální okrajové úlohy pro soustavy lineárních funkcionálních diferenciálních rovnic. Nalezené podmínky zobecňují a doplňují řadu známých výsledků a lze je aplikovat mimo jiné i pro rovnice neutrálního typu. Pro lineární funkcionální diferenciální rovnice se singularitami je studovaná úloha o řešeních s předem danou rychlosti růstu. Za předpokladu splnění vhodné podmínky kladnosti příslušného operátoru nám techniky, využívající dvoustranných monotónních iterací, umožňují dokázat nové věty o existenci a prostorové lokalizaci takových řešení. Jsou také odvozeny důsledky pro diferenciální rovnice s odkloněným argumentem. Dále jsou dokázány věty o řešitelnosti singulární Cauchyovy úlohy pro funkcionální diferenciální rovnice s nerostoucí nelinearitou. Pro soustavu lineárních funkcionálních diferenciálních rovnic s a časovými singularitami jsou nalezeny podmínky zaručující jednoznačnou řešitelnost počáteční úlohy ve třídě funkcí s danou rychlosti růstu. Je ukázáno, že nalezené podmínky jsou v jistém smyslu nezlepšitelné.

6 Vita Pylypenko, Masaryk University, 2017

7 Hereby I declare that this dissertation has been written by me and all the external sources used in the course of the work on it are mentioned in the bibliography included Vita Pylypenko

8 Contents Preface 12 Notation 15 1 Introduction and bibliographic notes Methodological remarks Boundary value problems for singular equations Functional differential equations: different approaches to the notion of a solution Cauchy problem vs. boundary value problems General solvability conditions for a non-local problem Problem setting Main theorem More concrete solvability conditions Definitions The case where p 0 belongs to S a,r ([a, b], R n ) An interesting particular case Preliminaries from the cone theory: positively invertible operators Proof of the main theorem Linear functional differential equations with negative coefficients Problem setting Existence of solutions with restricted growth

9 3.3 A general theorem on the solvability Auxiliary statements Further notions and facts from the cone theory The space C loc; h ((a, b], R) Transformation to a fixed-point form. The operator T Proofs Proof of Theorem Proofs of Theorem 3.1 and corollaries Singular Cauchy problem for functional differential equations with monotone non-linearities Problem setting A general theorem Corollaries for linear equations An equation with minimum Auxiliary statements and proofs General notions Proof of Theorem Proofs of Corollaries 4.1 and Proofs of the results of Section Slowly growing solutions of linear functional differential systems Problem description and motivation Precise formulation of the problem Existence and uniqueness of a slowly growing solution Corollaries for equations with argument deviations Examples for equations with argument deviations Auxiliary statements: bounds for the spectrum of a positive operator Proofs Proof of Theorem Proof of Theorem

10 Conclusion 96 Bibliography 99 Author s publications References Index 108 Statement of contribution of co-authors

11 Acknowledgement I would like to thank sincerely to my supervisor András Rontó, DrSc, for his excellent mentoring, valuable advices, rich discussions, patience and understanding. I am also very grateful to doc. RNDr. Bedřich Půža, CSc. for his kind support during my study. 11

12 Preface The dissertation is devoted to several selected topics belonging to the theory of functional differential equations and is divided logically into five chapters. Chapter 1 is of introductory character, it contains a survey related to the topic of the work. The author s original results are presented in Chapters 2 5. In Chapter 2, non-local boundary value problems for systems of linear functional differential equations u (t) = (lu)(t) + f(t), t [a, b], (0.1) are considered. The boundary condition has the form u(a) = r(u) where r is a functional (generally speaking, non-linear) and includes, in particular, the two-point boundary conditions, multipoint conditions, and boundary conditions of an integral type. We prove theorems on the solvability of this problem on the assumption that l admits some suitable estimates by operators generating uniquely solvable problems. The idea here is to suggest a unified approach to obtain this general kind of conditions, which is based on suitable techniques related to monotone operators on partially ordered spaces. This allows one to use one and the same approach to obtain conditions that previously had to be proved directly in every particular case. If the functional r is constant and all the operators are well defined on the space of continuous functions, the theorems obtained in this way generally reduce to the results known for the Cauchy problem (with the exception for Theorem 2.3, which is new even in that case). The results of this chapter generalize, in particular, the corresponding statements from [31, 77]. In Chapter 3, we consider a particular class of linear functional differential equations of form (0.1) and study the question whether there exist a 12

13 solution which is slowly growing in the sense that sup h(t) u(t) < +, (0.2) t (a,b] where h : (a, b] R is continuous, non-decreasing and such that h(t) > 0 for t (a, b], lim t a+ h(t) = 0. Under the assumption on the non-positivity of the right-hand side, easily verifiable conditions sufficient for this are found. The setting here is close to the work [63] where the weighted Cauchy- Nicoletti problem is studied using the techniques of a priori bounds. The conditions obtained here, according to our knowledge, are new even for the particular case of equations with power functions as argument deviations. The proof of the main theorem is based on a suitable version of the technique of lower and upper functions. One of the key steps of the proof is to establish a certain regularity of the ordering in the resulting function space. Chapter 4 deals with the singular Cauchy problem for scalar non-linear functional differential equations. A theorem is proved providing a sufficient condition for the solvability in the case where the nonlinearity has a monotone character. The idea is to use a suitable fixed point theorem for a monotone mapping in a partially ordered space. An important point of the proof is to ensure that the mapping in question is limit monotone compact in the corresponding space of unbounded functions. The notion of a solution here requires only a local absolute continuity and involves a suitable weight function. As a consequence, solutions with a singularity at the given point are allowed (i. e., a solution may pass through the singularity). The theorem establishing the solvability also provides two-sided estimates for the solution and its derivative. In addition, the solution can be found as the limit of suitable monotone approximations. The results are illustrated on the case of a functional differential equation with a minimum. Finally, in Chapter 5, we consider a linear system of functional differential equation of the form u (t) = (lu)(t) + f(t), t [a, b), and study its solutions satisfying the conditions u(a) = λ 13

14 and sup h(t) u(t) < +, t [a,b) where f and λ are given and h : [a, b) [0, + ) is such that lim t b h(t) = 0. New efficient conditions sufficient for the existence of such a solution and its non-negativity are obtained. The proof uses suitable bounds for eigenvalues of operators leaving invariant a cone and requires appropriate positivity assumptions. Efficient corollaries for linear systems of equations with transformed argument are given. The dissertation is based on the author s publications [1 6]. 14

15 Notation The following notation is used in the text. 1. R := (, ), N := {1, 2, 3,... }. 2. C([a, b], R) is the Banach space of all the continuous functions u : [a, b] R with the standard norm C([a, b], R) u max u(t). t [a,b] 3. B([a, b], R) is the Banach space of all the bounded functions u : [a, b] R with the standard norm B([a, b], R) u sup u(t). t [a,b] 4. mes A is the Lebesgue measure of a set A R. 5. L 1 ([a, b], R) is the Banach space of all the Lebesgue integrable functions u : [a, b] R with the standard norm L 1 ([a, b], R) u a u(t) dt. 6. L 1; loc ((a, b], R) is the set of functions u : (a, b] R such that u [a+ε,b] L 1 ([a + ε, b], R) for any ε (0, b a). 7. C([a, b], R) is the Banach space of the absolutely continuous functions u : [a, b] R equipped with the norm C([a, b], R) u u(a) + a u (ξ) dξ. 8. Cloc ((a, b], R) is the set of all the locally absolutely continuous functions u : (a, b] R. 15

16 9. Cloc; h ((a, b], R) is the set of all the locally absolutely continuous functions u : (a, b] R such that hu L 1 ((a, b], R) and sup h(t) u(t) < +. t (a,b] 10. x := max i=1,2,...,n x i for any x = (x i ) n i=1 from Rn. 11. L 1 ([a, b], R n ) is the Banach space of all the Lebesgue integrable vector functions u : [a, b] R n with the standard norm L 1 ([a, b], R n ) u a u(ξ) dξ. 12. L 1; loc ((a, b], R n ) is the set of vector functions u = (u i ) n i=1 : (a, b] Rn such that u i L 1; loc ((a, b], R) for each i = 1, 2,..., n. 13. C([a, b], R n ) is the Banach space of the absolutely continuous functions u : [a, b] R n equipped with the norm C([a, b], R n ) u u(a) + a u (ξ) dξ. 14. Cloc ((a, b], R n ) is the set of all the locally absolutely continuous vector functions u = (u i ) n i=1 : (a, b] Rn. 15. If h = diag (h 1,..., h n ) : (a, b] R n is a continuous matrix-valued function, then C loc; h ((a, b], R n ) is the set of all the vector functions u = (u i ) n i=1 : (a, b] R such that u i C loc; hi ((a, b], R) for each i = 1, 2,..., n. 16. If r = (r k ) n k=1 : C([a, b], R n ) R n are certain operators, then the symbol C r ([a, b], R n ) denotes the set of all functions u = (u k ) n k=1 from C([a, b], R n ) for which u k (a) = r k (u), k = 1, 2,..., n. 17. The set C r,1 ([a, b], R n ) is defined by the formula C r,1 ([a, b], R n ) := { u = (u k ) n k=1 C r ([a, b], R n ) min ξ [a,b] u k(ξ) The set C r,2 ([a, b], R n ) is introduced by the formula for all k = 1, 2,..., n }. C r,2 ([a, b], R n ) := { u = (u k ) n k=1 C r ([a, b], R n ) min ξ [a,b] u k(ξ) 0 and vrai min u k(ξ) 0 for all k = 1, 2,..., n }. ξ [a,b] 16

17 Chapter 1 Introduction and bibliographic notes The purpose of this introductory chapter is to present a short review on the subject dealt with in what follows and to outline certain important points. The exposition here is based on [12, 79, 80, 86, 91]. 1.1 Methodological remarks The second part of the last century had been marked by a significant advance in the general theory of boundary value problems for systems of ordinary differential equations. In particular, the method of a priori estimates had been thoroughly developed, which gave a possibility to establish conditions for solvability and correctness of wide classes of non-linear boundary value problems with two-point conditions [37, 49, 90, 100], multipoint conditions [40, 68, 84, 88], and more general functional conditions (see, e. g., [41, 69, 85, 94, 95, 102]). For example, the work [86] presents a theory concerning the boundary value problem u = f(t, u), (1.1) h(u) = 0, (1.2) where f : [a, b] R n R n is a Carathéodory function and h is a continuous operator transforming the space of continuous functions to R n. The setting (1.1), (1.2) contains, in particular, multipoint boundary value problems, which are, in turn, a generalisation of the Cauchy Nicoletti problem. A characteristic feature of the corresponding results is the unilateral growth restrictions on the function f and a systematic use of the method of a priori estimates, which had later influenced many other works. 17

18 The last mentioned method is also commonly used in the works on boundary value problems of type (1.1), (1.2) where the topological degree theory is explicitly applied. This also concerns the method of upper and lower functions [24], which, in addition, provides a natural space localisation of solutions. A close idea may be seen in monotone-iterative techniques, which, together with the related studies of differential and integral inequalities, apart from the existence results, give an additional information on the character of dependence of a solution on external data. Order-theoretical methods are useful in this context. We refer, e. g., to the books [23, 24, 33, 38, 53, 99] for details on these very extensive subjects. 1.2 Boundary value problems for singular equations Boundary value problems for differential equations with non-integrable singularity are of particular interest. Such problems are studied, e. g., in [39, 40, 84, 88], mostly for second order equations. The systematic study of initial and boundary value problems for ordinary differential equations of the second order u = f(t, u, u ) (1.3) with singularities with respect to independent variable and one of the phase variable has about sixty years history and, in particular, is motivated by certain applied problems. For example, in the beginning of the last century, the singular Cauchy problem u = 2 t u u m, u(0) = c 0, u (0) = 0. where c 0 > 0, was studied in the work [30] concerning the equilibrium of a sphere of a polytropic gas. At present, singular Cauchy problems are studied non only for equations of type (1.3) but also for rather general classes of higher-order differential equations and systems (see, e. g., [21, 22, 39, 47, 70, 88, 89, 91, 101]). The fundamentals of the theory of singular boundary value problems for equation (1.3) are presented in the work [91], which is focused on twopoint problems and problems on bounded and monotone solutions. Twopoint problems for (1.3) are studied there provided that the function f : (a, b) R 2 R has non-integrable singularities with respect to the first argument at points a and b. Many statements of the theory concerning 18

19 equation (1.3) use essentialy the results of a through study of the linear differential equation under the boundary conditions u = p 1 (t)u + p 2 (t)u + p 0 u(a+) = c 1, u (i 1) ( b) = c 2, where i = 1, 2, < a < b < + and the functions p j : (a, b) R (j = 0, 1, 2,... ) are, generally speaking, non-integrable on [a, b] with singularities at the ends of the interval. The study of two-point problems for singular ordinary differential equations has led to significant advances (see, e. g., [8]). Differential equations with time singularities appear quite frequently in relation to various questions of either theoretical or applied character. For example, second order ordinary differential equations with time singularities often arise when studying radially symmetric solutions of certain partial differential equations (see, e. g., [62]). Solutions of initial value problems for time-singular ordinary differential equations with ϕ-laplacian, which are motivated, in particular, by applications in population dynamics and field theory, are studied in [21]. Numerical aspects of analysis of certain classes of time-singular problems for ordinary differential equations related to the application of shooting and collocation methods are treated in [22, 48, 50]. In contrast to the case of ordinary differential equations, significantly less is known at present for singular equations with argument deviations (see [45, 46, 51, 57 60, 75, 104]). We would like to especially mention the recent interesting works [18, 64 66] where a thourough study of certain linear scalar functional differential equations with time singularities is carried out based on the idea to study the solution space of a model equation (in the sense of [13]). The approach of the present work is based on other ideas and gives different results. 1.3 Functional differential equations: different approaches to the notion of a solution The interest to functional differential equations is constantly growing both due to their numerous applications and due to the needs of the general theory itself where a deeper knowledge of the objects involved is required so that they can be appropriately treated in a reasonably unified framework. 19

20 This class of equations includes, in particular, equations with argument deviations and integro-differential equations; the role of some classes of such equations in modern physics, biology and economics is becoming increasingly significant. It should also be noted that certain problems arising in physics, economics, and immunology are described by functional differential equations with discontinuous coefficients which are often non-integrable on the given time interval. The theory of functional differential equations is by no means a straightforward extension of the classical theory of ordinary differential equations. The principal difference between the two theories is that the basic notions characterizing ordinary differential equations are of a local character, which is not the case for the differential equations that may contain argument deviations. In fact, many methods and statements of the theory of the equation u (t) = f(t, u(t)), t [a, b], (1.4) which is a central object of the theory of ordinary differential equations, are based on the properties of the corresponding Nemytskii operator u N f u := f(, u( )), for which the following holds: for any instant of time t 0, the value of N f u in any neighbourhood U 0 of t 0 is completely determined by the values of u on U 0. The operators possessing the property mentioned are called local operators [67, 105]. In this sense, equation (1.4) is an object of a local character (the operator of differentiation u u obviously has the same property), and the theory of boundary value problems for (1.4) is very rich (see, e.g., [37, 86 88] and the references therein). Due to the locality of the operator N f, equation (1.4) has a number of properties which are absent in other functional differential equations. In particular, every segment of a solution curve of equation (1.4) can be treated as the graph of a solution of an independent initial or boundary value problem posed for the restriction of the equation to the appropriate time interval, i.e., roughly speaking, a partial segment of the solution is a solution as well. This property fails to be satisfied for functional differential equations. For example, in the simplest differential equation with a relection u (t) = u( t), t [ 1, 1], (1.5) the restriction u [0,1] makes no sense since a function cannot be substituted into (1.5) unless its values on [ 1, 0) are specified. Every solution of (1.5) should therefore be defined as a function mapping [ 1, 1] to R, and no restriction of it to a narrower set is possible. This situation is generic and 20

21 should be taken into account when introducing the notion of a solution of a general functional differential equation of the form u (t) = (Fu)(t), t [a, b], (1.6) where operator F is defined on a suitable set of differentiable functions. Various equations of type (1.6) arise in applications when the limitations imposed by the local character of the Nemytskii operator do not allow one to remain in the class of ordinary differential equations (e.g., problems of economy and immunology [35, 93, 96]). The class of equations (1.6) includes, in particular, ordinary differential equations, differential equations with aftereffect, integro-differential equations, differential equations with argument deviation, and equations with distributed deviating argument. A correct definition of a solution of equation (1.6) requires a systematic approach taking into account the generality of the situation. Such an approach, which is very natural and simulteneously provides a considerable degree of generality, was developed by the Perm school on differential equations [79, 80]. A new definition of solution had been introduced, according to which the fulfilment of (1.6) was required almost everywhere on [a, b], and by a solution itself one understood a function absolutely continuous on the given interval. The derivative of u is assumed to be only integrable. In this way, as turns out, one gains a very general setting and looses surprisingly little (e.g., results on the increasing, with time, smoothness of the solution of a delay equation proved by the method of steps). The new definition has numerous advantages over the original one, which can be seen even in the case of a delay equation. For example, consider the delay differential equation u (t) = g(t, u(t δ)), t [a, b], (1.7) where δ (0, b a). To ensure that (1.7) determines a rigorously described mathematical object, it is necessary to specify an initial function, i.e., consider (1.7) together with the condition u(t) = h(t) for t [a δ, a), (1.8) where the function h is given. Without (1.8), relation (1.7) makes no sense and, thus, both (1.7) and (1.8) should be parts of the definition of the delay equation. Thus, rigourously speaking, the delay differential equation we have in mind is the object obtained as the union of (1.7) and (1.8). The question is which function space should be chosen to contain its solutions. In the older works on this kind of equations, a solution of (1.7), (1.8) was understood as a continuously differentiable function u : [a, b] R which is 21

22 continuously glued to the initial function, i.e., such that the equality u(a) = h(a) (1.9) holds (see the pioneering monograph of Myshkis [97] on which this approach is based). The motivation for this clearly lies in the ordinary case: for smooth enough right-hand sides, one should like to have continuous solutions, which is impossible without (1.9). Condition (1.9) is, however, superfluous from the point of view of (1.8), since (1.8) alone is sufficient for all the operations at the right-hand side of (1.7) to make sense. In this way, instead of the delay equation formed by relations (1.7) and (1.8), the older approach suggests to study the initial value problem (1.7), (1.8), (1.9). This logical inconsistency, when the equation and problems posed for it were somewhat mixed together, had been long an obstacle for the creation of a general theory. The new definition, where u is only integrable, allows one to avoid incorporating (1.9) into the definition of a solution and makes the situation look natural. It also makes unnecessary the old construction where the initial function was often considered as a part of the solution (i.e., u : [a δ, b] R in the case of (1.7), (1.8), (1.9); see, e.g., [16, 34, 103, 106]). Furthermore, the general setting based on the ideas of [79] is fruitful also in the sense that it allows one to consider various mixed kinds of argument deviations in a unified way (e.g., neutral type equations and equations with both retarted and advanced argument), which is not possible at all with the old definition. In particular, all the equations studied by means of the corresponding evolution families (see, e.g., [35]) have, in fact, a retarded character because the corresponding notions are otherwise undefined and, as a result, the methods used do not work. The well-known method of steps, which is traditionally used to construct solutions of equations of type (1.7), (1.8), makes sense for delay differential equations only and is not applicable to any other kinds of equations. It should be noted that differential equations with mixed type argument deviation, besides their importance from the theoretical point of view, arise also in purely applied problems (e.g., of economy [93] and immunology [96] ) where models may involve both past and future states of a process. Earlier works on equations with argument deviations (see, for example, [82, 98] and references therein) are devoted to variuous problems for special types equations of form (1.6) (in fact, mostly of form (1.7), (1.8)). However, as noted in [79], one observes that particular equations, as a rule, were studied separately and without essential connection with one another in a common theoretical framework. Furthermore, the absolute majority of 22

23 research concerned only retarded differential equations (alternatively called equations with delay, delay differential equations, or equations with aftereffect), i.e., those equations (1.6) where the operator F has the Volterra property [79]: for any pair of functions (u 1, u 2 ) and any t [a, b], the equality u 1 (s) = u 2 (s) for all s [a, t] implies that (Fu 1 )(s) = (Fu 2 )(s) for a.e. s [a, t]. This property means that the current state of the process under consideration may depend on the past only and, in particular, guarantees the unique solvability of the corresponding initial value problem under quite general conditions. The general theory of functional differential equations had appeared only by 1980ies [11, 79, 81]. A crucial advance was the construction of a theory of the equation (Lu)(t) = f(t), t [a, b] (1.10) with a linear operator L defined on the Banach space C([a, b], R n ) of absolutely continuous functions u : [a, b] R n and having values in the Banach space L 1 ([a, b], R n ) of Lebesgue itegrable function f : [a, b] R n. A general theory of such equations is presented in the books [79, 80] and is essentially based on a special representation of equation which uses isomorphism of the space C([a, b], R n ) and the direct product L 1 ([a, b], R n ) R n, as a result of which the notion of a principle part of an equation arises [79]. The new formalism [79], together with the basic theory available for the linear equations (1.10), now allows one to deal, in a unified systematic way, with numerous equations and problems involving argument deviations, which had been earlier studied separately (see, e.g., [16, 34, 97, 103, 106]) but, with the new definition of a solution, in fact, could be treated as natural representatives of the class of linear equations (1.10). One should also outline an essential practical advantage of the new definition consisting in the possibility to formulate problems with argument deviations without use of initial functions. Indeed, e.g., the setting of the delay differential equation (1.7), (1.8) contains relation (1.8) which introduces the initial function h : [ δ, 0) R. As follows from the above-said, it is an integral part of the object description and cannot be dropped. However, the new definition allows one [79] to easily rewrite (1.7), (1.8) in the form u (t) = g(t, u(τ(t))), t [a, b], (1.11) where no extra conditions of type (1.8) are needed. Indeed, assume that g satisfies the Carathéodory conditions, put τ(t) := t δ, t [a, b], (1.12) 23

24 and introduce the function g : [a, b] R R by setting { g(t, z) if τ(t) [a, b], g(t, z) := g(t, h(τ(t))) if τ(t) [a, b]. (1.13) Then an absolutely continuous function u : [a, b] R satisfying (1.7), (1.8) also satisfies (1.11) almost everywhere on [a, b] and vice versa, i.e., (1.11) is an equivalent form of (1.7), (1.8). Note that, in (1.11), there is no need to specify any initial functions because all the external data are already incorporated into the equation and all the operations involved in (1.11) are well defined. Clearly, this approach works for other argument deviations, one should only modify (1.12) appropriately to deal with another τ( ). An important technical consequence of this argument is that the initial value problem for equation (1.11) is posed in the same way as for an ordinary differential equation at a single point: u(t 0 ) = α, which is indeed sufficient because the history of the process is included into equation itself. In this way, the rather common opinion on the necessity of an initial function and the technical inconveniences caused by its presence are overcome simply by passing to a more relevant definition. 1.4 Cauchy problem vs. boundary value problems The role played by the Cauchy problem in the theory of functional differential equations differs from that in the ordinary case. Indeed, in the class of ordinary differential equations, the initial value problem is the basic and most wide spread problem that is ever posed. In particular, it is well-known that the Cauchy problem u (t) + A(t)u(t) = f(t), t [a, b], (1.14) u(a) = α, (1.15) where A is an n n-matrix function with columns belonging to L 1 ([a, b], R n ), has a unique solution for any f L 1 ([a, b], R n ) and α R n. In other words, for an arbitrary fixed f L 1 ([a, b], R n ), every solution u of equation (1.14) is uniquely determined by its initial value u(a). In the theory of ordinary differential equations, this fact had led to the tradition to reduce various problems to the Cauchy problem. The extension of this tradition to functional differential equations (1.10) is, however, both impossible and, in a sense, unnatural since, e.g., the properties of the equation u (t) + A(t)u(τ(t)) = f(t), t [a, b], (1.16) 24

25 where τ : [a, b] [a, b], may be completely different from those of (1.14) (clearly, both (1.14) and (1.16) are particular cases of (1.10)). Consider a simple example [79] of the scalar equation u (t) u(1) = f(t), t [0, 1], (1.17) which, obviously, has form (1.16) with a = 0, b = 1, and A(t) = 1, τ(t) = 1 for t [0, 1]. The general solution of this equation is u(t) = 1 t f(s)ds + ct, t [0, 1], (1.18) where c is arbitrary, and it is easy to see from (1.18) that the Cauchy problem u(0) = 0 (1.19) for equation (1.17) has no solutions, e.g., if f(t) = 1, t [0, 1], and has infinitely many solutions, e.g., if f(t) = 0, t [0, 1] (the function u(t) = λt, t [0, 1], is a solution of (1.17), (1.19) for any λ in the latter case). On the other hand, replacing the initial condition (1.19) by the boundary condition φ(u) = α, (1.20) where φ : C([a, b], R) R is a linear functional, we find that the boundary value problem (1.17), (1.20) has a unique solution for every pair of (f, α) C([a, b], R) R if φ satisfies the condition φ(u 0 ) 0 (1.21) for u 0 (t) := t, t [0, 1]. Assumption (1.21) holds, in particular, if φ is given by one of the following formulas: φ(u) = u(1), (1.22) φ(u) = u(0) u(1), (1.23) φ(u) = u(0) + u(1), φ(u) = 1 0 u(s)ds (1.24) for any u. Condition (1.21) serves here as a test for the absence of nontrivial solutions of the corresponding homogeneous problem. From this point of view, e.g., the Cauchy problem at the right boundary ((1.20) with φ of form (1.22) and α = 0), the periodic problem ((1.20) with φ of form (1.23) and α = 0) and the zero mean value problem ((1.20) with 25

26 φ of form (1.24) and α = 0) for equation (1.17) are more natural to be considered than the initial value problem (1.19). The same logic is applicable to functional differential equations in general. Thus, for functional differential equations, the role of the Cauchy problem as a basic problem, which is uniquely solvable under the mildest assumptions, should be played by other boundary value problems, possibly with non-local boundary condition. The last mentioned type of boundary conditions is described by (1.20) where φ is a non-local functional, i.e., the values of φ depend on the function over an interval. A typical example is φ given by (1.24). Efficient conditions sufficient for the solvability as well as unique solvability of the Cauchy problem and other types of boundary value problem for systems of functional differential equations are established in [19, 20, 25 27, 32, 43 45, 71 74, 87]. Conditions for the existence of solutions of linear functional differential equations with a given growth rate are established in [9]. The Cauchy problem for singular functional differential equations with aftereffect is studied in [104]. Problems with non-local boundary conditions for delay differential equations with impulses are treated in [28, 29]. The up-to-date state of the theory of boundary value problems for functional differential equations in their general form can be seen, in particular, from the recent monographs [7, 33, 43, 80, 83], where one can also find an extensive bibliography on the subject. Non-local boundary value problems and boundary value problems for functional differential equations with singular coefficients are currently studied to a less extent. In this work, new statements on the solvability of such problems are presented, which generalize and complete known results and, due to certain techniques of the proofs, may also be of interest from the methodological point of view. 26

27 Chapter 2 General solvability conditions for a non-local problem This chapter deals with the question on the existence and uniqueness of a solution of a non-local boundary-value problem for linear functional differential equations. Equations of a rather general form are studied. 2.1 Problem setting Consider the system of functional differential equations u k(t) = (l k u)(t) + f k (t), t [a, b], k = 1, 2,..., n, (2.1) subjected to the non-local boundary conditions u k (a) = r k (u), k = 1, 2,..., n, (2.2) where < a < b < +, n N, l k : C([a, b], R n ) L 1 ([a, b], R), k = 1, 2,..., n, are linear operators, {f k k = 1, 2,..., n} L 1 ([a, b], R) are given functions, and r k : C([a, b], R n ) R, k = 1, 2,..., n, are continuous linear functionals. By a solution of problem (2.1), (2.2), as usual (see, e. g., [13]), we mean a vector function u = (u k ) n k=1 : [a, b] Rn whose components are absolutely continuous, satisfy system (2.1) almost everywhere on the interval [a, b], and possess property (2.2). It should be noted that equations (2.1) may contain terms with derivatives and, thus, the statements presented in what follows are applicable, in particular, to neutral type linear functional differential equations. 27

28 Our aim here is to prove the unique solvability of problem (2.1), (2.2) on the assumption that the linear operators l k, k = 1, 2,..., n, appearing in (2.1) can be estimated by certain other linear operators generating problems with conditions (2.2) for which the statement on the integration of a differential inequality holds. The precise formulation of the property mentioned is given by the following definition. Definition 2.1. A linear operator l = (l k ) n k=1 : C([a, b], R n ) L 1 ([a, b], R n ) is said to belong to the set S a,r ([a, b], R n ) if the boundary value problem (2.1), (2.2) has a unique solution u = (u k ) n k=1 for any {f k k = 1, 2,..., n} L 1 ([a, b], R) and, moreover, the solution of (2.1), (2.2) possesses the property min u k(t) 0, k = 1, 2,..., n, (2.3) t [a,b] whenever the functions f k, k = 1, 2,..., n, appearing in (2.1) are nonnegative almost everywhere on [a, b]. Note that S a,r ([a, b], R) contains the set Ṽ+ ab (h) defined in [55], where certain efficient conditions sufficient for the inclusion l Ṽ+ ab (h) are established in the case where the linear operator l admits a continuous extension to the space of continuous functions. 2.2 Main theorem The following theorem provides general conditions sufficient for the existence and uniqueness of a solution of problem (2.1), (2.2). Theorem 2.1. Let there exist linear operators p i = (p ik ) n k=1 : C([a, b], R n ) L 1 ([a, b], R n ), i = 0, 1 satisfying the inclusions p 1 S a,r ([a, b], R n ), p 0 + p 1 S a,r ([a, b], R n ), (2.4) and such that the inequalities (l k u)(t) (p 1k u)(t) (p 0k u)(t), t [a, b], k = 1, 2,..., n, (2.5) hold for an arbitrary non-negative absolutely continuous vector function u : [a, b] R n with property (2.2). Then the boundary value problem (2.1), (2.2) has a unique solution for arbitrary {f k k = 1, 2,..., n} L 1 ([a, b], R). 28

29 Theorem 2.1 generalises [77, Theorem 3.3]. Note that assumption (2.4) in Theorem 2.1 can be replaced neither by the condition (1 ε) p 1 S a,r ([a, b], R n ), p 0 + p 1 S a,r ([a, b], R n ), (2.6) nor by the condition p 1 S a,r ([a, b], R n ), (1 ε) (p 0 + p 1 ) S a,r ([a, b], R n ), (2.7) where ε is an arbitrarily small positive number. Indeed, let us fix some ε [0, 1) and consider the homogeneous Cauchy problem u 1 (a) = 0, u 2 (a) = 0, (2.8) for the system u 1 1(t) = 2 (b a) (u 1(b) u 2 (b)), t [a, b], (2.9) u 1 2(t) = 2 (b a) (u 1(b) u 2 (b)), t [a, b]. (2.10) It is clear that (2.8), (2.9), (2.10) is a particular case of the problem (2.1), (2.2), where n = 2, f 1 = f 2 = 0, (l k u)(t) = ( 1)k+1 2(b a) (u 1(b) u 2 (b)), t [a, b], k = 1, 2, and r 1 = r 2 = 0. Problem (2.8), (2.9), (2.10) has the family of solutions u k (t) = λ ( 1) k (t a), t [a, b], k = 1, 2, where λ R is arbitrary. However, condition (2.7) in this case is satisfied for all ε (0, 1) with p 1 := 0 and ( ) 1 u1 (b) + u p 0 u := 2 (b) 2 (b a) u 1 (b) + u 2 (b) because problem (2.8) for the system u 1(t) = 1 ε 2 (b a) (u 1(b) + u 2 (b)) + f 1 (t), t [a, b], u 2(t) = 1 ε 2 (b a) (u 1(b) + u 2 (b)) + f 2 (t), t [a, b], as is easy to see, has a unique solution for any f k L 1 ([a, b], R), k = 1, 2, and this solution is non-negative for non-negative f k, k = 1, 2. In a similar way, one can specify an example showing the optimality of condition (2.6). 29

30 2.3 More concrete solvability conditions The principle described by Theorem 2.1 allows one to obtain various solvability conditions by using the freedom available in the choice of p 0 and p 1 involved in estimate (2.5) Definitions Definition 2.2. A vector function u = (u k ) n k=1 C([a, b], R n ) is said to belong to the set C r ([a, b], R n ) if u k (a) = r k (u), k = 1, 2,..., n, where r k : C([a, b], R n ) R, k = 1, 2,..., n, are certain operators. Definition 2.3. A vector function u = (u k ) n k=1 belong to the set C r,1 ([a, b], R n ) if C r ([a, b], R n ) is said to min u k(ξ) 0, k = 1, 2,..., n. (2.11) ξ [a,b] Definition 2.4. A linear operator l = (l k ) n k=1 : C([a, b], R n ) L 1 ([a, b], R n ) is said to have positive restriction on C r ([a, b], R n ) if the inequality vrai min (l k u)(t) 0, k = 1, 2,..., n, (2.12) t [a,b] is true for any vector function u = (u k ) n k=1 from C r,1 ([a, b], R n ). Definition 2.5. A linear operator l = (l k ) n k=1 : C([a, b], R n ) L 1 ([a, b], R n ) is said to be positive if (2.12) holds for any non-negative vector function u = (u k ) n k=1 from C([a, b], R n ). Remark 2.1. Note that an operator l : C([a, b], R n ) L 1 ([a, b], R n ) having positive restriction on C r ([a, b], R n ) need not be positive. This is the case, in particular, for the operator l = (l k ) n k=1, (l k u)(t) = (p k u)(t) + r k (u) u k (a), t [a, b], k = 1, 2,..., n, where p = (p k ) n k=1 : C([a, b], R n ) L 1 ([a, b], R n ) is positive and mes {t [a, b] (p k v)(t) < v k (a) r k (v)} = 0 for some v = (v k ) n k=1 C([a, b], R n ) and k {1, 2,..., n}. The latter property is present, e. g., if v k (t) = (t a)(b a) 1, r k (u) = u k (b), 30

31 (p k u)(t) = α kj (t)u j (ω kj (t)), t [a, b], k = 1, 2,..., n, j=1 for all u = (u k ) n k=1 from C([a, b], R n ) where the integrable functions α kj : [a, b] R and the measurable functions ω kj : [a, b] [a, b], k = 1, 2,..., n, j = 1, 2,..., n, satisfy the condition { min mes t [a, b] k=1,2,...,n j=1 } α kj (t)(ω kj (t) a) < b a The case where p 0 belongs to S a,r ([a, b], R n ) Theorem 2.2. Let there exist linear operators l i = (l ik ) n k=1 : C([a, b], R n ) L 1 ([a, b], R n ), i = 0, 1, which have positive restriction on C r ([a, b], R n ), satisfy the inclusions l 0 S a,r ([a, b], R n ), 1 2 l 1 S a,r ([a, b], R n ), (2.13) and are such that the inequalities (l k u)(t) + (l 1k u)(t) (l 0k u)(t), t [a, b], k = 1, 2,..., n, (2.14) hold for an arbitrary function u : [a, b] R n from the set C r,1 ([a, b], R n ). Then the boundary value problem (2.1), (2.2) has a unique solution for arbitrary {f k k = 1, 2,..., n} L 1 ([a, b], R). Proof. It follows from assumption (2.14) and the positivity property of the operator l 1 Cr([a,b],R n ) that, for any vector function u from C r,1 ([a, b], R n ), the relations (l k u)(t) (l 1ku)(t) = (l k u)(t) + (l 1k u)(t) 1 2 (l 1ku)(t) (l 0k u)(t) (l 1ku)(t) = (l 0k u)(t) (l 1ku)(t), t [a, b], k = 1, 2,..., n, are true. This means that l admits estimate (2.5) with the operators p 0 and p 1 defined by the equalities p 0 := l l 1, p 1 := 1 2 l 1. (2.15) 31

32 It now remains to note that assumption (2.13) ensures the validity of inclusion (2.4) for operators (2.15). Thus, under conditions (2.13) and (2.14), the operators p i : C([a, b], R n ) L 1 ([a, b], R n ), i = 0, 1, defined by formulae (2.15) satisfy conditions (2.4) and (2.5) of Theorem 2.1. Applying Theorem 2.1, we arrive at the required assertion. Remark 2.2. Arguing similarly, one can show that the assertion of Theorem 2.2 is preserved if its condition (2.13) is replaced by the assumption that l 0 + (1 2θ) l 1 S a,r ([a, b], R n ), θl 1 S a,r ([a, b], R n ) (2.16) for a certain θ (0, 1). Remark 2.3. Theorem 2.2 generalises, in particular, [31, Theorem 2.2] An interesting particular case Condition (2.14) is satisfied, in particular, if l can be represented in the form l = l 0 l 1, (2.17) where l 0 and l 1 are certain linear operators whose restrictions to the set C r ([a, b], R n ) are positive operators. In the case where the operator l admits decomposition (2.17), the following statement is also true. Theorem 2.3. Let us assume that the operator l admits representation (2.17) where l i : C([a, b], R n ) L 1 ([a, b], R n ), i = 0, 1, are certain linear operators which have positive restriction on C r ([a, b], R n ) and such that the inclusions are satisfied. l 0 S a,r ([a, b], R n ), 1 2 (l 0 l 1 ) S a,r ([a, b], R n ) (2.18) Then problem (2.1), (2.2) has a unique solution for arbitrary {f k k = 1, 2,..., n} L 1 ([a, b], R). Proof. The positivity of the operators l i Cr([a,b],R n ), i = 0, 1, and equality (2.17) imply that, for any vector function u from C r,1 ([a, b], R n ), the inequalities (l 1k u)(t) (l k u)(t) (l 0k u)(t), k = 1, 2,..., n, (2.19) are true for almost every t from [a, b]. 32

33 Let us put Then it is easy to verify that p i := 1 2 (l 0 + ( 1) i l 1 ), i = 0, 1. (2.20) p 0 + ( 1) i p 1 = l i, i = 0, 1. (2.21) Considering (2.21), we conclude that inequalities (2.19) guarantee the fulfilment of conditions (2.4) and (2.5) of Theorem 2.1 for the operators p i : C([a, b], R n ) L 1 ([a, b], R n ), i = 0, 1, given by formulae (2.20). Application of Theorem 2.1 completes the proof. Remark 2.4. Arguing similarly to Sec. 2.2, one can show that conditions (2.13), (2.16), and (2.18) are optimal in a certain sense. 2.4 Preliminaries from the cone theory: positively invertible operators The proof of Theorem 2.1 is based on rather general order-theoretical considerations. Let us consider the abstract operator equation Fx = z, (2.22) where F : E 1 E 2 is a mapping, E 1, E1 is a normed space, E 2, E2 is a Banach space over the field R, K i E i, i = 1, 2, are closed cones, and z is an arbitrary element from E 2. We now recall several definitions and facts that are used below. Let E, be a normed space over R and let K be a cone in E. Definition 2.6 (see [52] ). Cone K is a non-empty closed subset of E possessing the properties K ( K) = {0} and α 1 K+α 2 K K for all {α 1, α 2 } [0, + ). The cones K i, i = 1, 2, induce the natural partial orderings of the respective spaces. Thus, for each i = 1, 2, we write x Ki y and y Ki x if and only if {x, y} E i and y x K i. The following statement is due to Krasnoselskii, Lifshits, Pokornyi, and Stetsenko [92, Theorem 7] (see also [54, Theorem 49.4]). 33

34 Theorem 2.4. Let the cone K 2 be normal and generating. Furthermore, let B k : E 1 E 2, k = 1, 2, be additive and homogeneous operators such that B 1 1 and (B 1 + B 2 ) 1 exist and possess the properties B 1 1 (K 2 ) K 1, (B 1 + B 2 ) 1 (K 2 ) K 1, (2.23) and, furthermore, the relation {Fx Fy B 1 (x y), B 2 (x y) Fx + Fy} K 2 (2.24) is satisfied for any pair (x, y) E 2 1 such that x y K 1. Then equation (2.22) has a unique solution u E 1 for an arbitrary element z E 2. Let us recall that a cone K E in a Banach space E, E is normal if and only if the relation inf {γ (0, + ) x E γ y E {x, y} K : y x K} < + is true. By definition, the cone K is generating in E if and only if an arbitrary element x from E can be represented in the form x = u v, where u and v belong to K (see, e. g., [52, 54]). 2.5 Proof of the main theorem Definition 2.7. A vector function u = (u k ) n k=1 belong to the set C r,2 ([a, b], R n ) if C r ([a, b], R n ) is said to min u k(ξ) 0, k = 1, 2,..., n, ξ [a,b] and Lemma 2.1. vrai min u k(ξ) 0, k = 1, 2,..., n. ξ [a,b] 1. Cr ([a, b], R n ) is a normed space with the norm C r ([a, b], R n ) u a u (ξ) dξ + u(a). 2. The set C r,1 ([a, b], R n ) is a cone in the space C r ([a, b], R n ). 3. The set C 0,2 ([a, b], R n ) is a normal and generating cone in the space C 0 ([a, b], R n ). 34

35 Proof. The assertions of Lemma 2.1 follow immediately from the definitions of the sets C r ([a, b], R n ) and C r,1 ([a, b], R n ) (see Definitions 2.2 and 2.3). The next lemma establishes the relation between the property described by Definition 2.1 and the positive invertibility of a certain operator. Lemma 2.2. If l = (l k ) n k=1 : C([a, b], R n ) L 1 ([a, b], R n ) is a linear operator such that l S a,r ([a, b], R n ) (2.25) then the operator V l : C r ([a, b], R n ) C 0 ([a, b], R n ) given by the formula C r ([a, b], R n ) u V l u := u is invertible and, moreover, its inverse V 1 l a (lu)(ξ) dξ r(u) (2.26) satisfies the inclusion V 1 l ( C 0,2 ([a, b], R n )) C r,1 ([a, b], R n ). Proof. Let the mapping l belong to the set S a,r ([a, b], R n ). Given an arbitrary function y = (y k ) n k=1 C 0 ([a, b], R n ), consider the equation V l u = y. (2.27) In view of the definition of the space C r ([a, b], R n ), we have y k (a) = 0, k = 1, 2,..., n. (2.28) By virtue of assumption (2.25), there exists a unique absolutely continuous vector function u = (u k ) n k=1 such that u k(t) = (l k u)(t) + y k(t), t [a, b], k = 1, 2,..., n, (2.29) u k (a) = r k (u), k = 1, 2,..., n. (2.30) Moreover, if the functions y k, k = 1, 2,..., n, are non-negative and nondecreasing, the components of u possess property (2.3). Integrating both parts of (2.29) and taking (2.28) and (2.30) into account, we find that u = (u k ) n k=1 is the unique solution of equation (2.27). The following statement is an obvious consequence of formula (2.26). Lemma 2.3. For any linear operators p i : C([a, b], R n ) L 1 ([a, b], R n ), i = 1, 2, the identity V p1 + V p2 = 2V 1 2 (p 1+p 2 ) is true. 35