A New Approach to Gene Network Modeling

This paper discusses mathematical modeling of artificial gene networks. We consider a phenomenological model of the simplest gene network, called a repressilator. This network contains three elements unidirectionally coupled into a ring. More specifically, the first element inhibits the synthesis of the second, the second inhibits the synthesis of the third, and the third, which closes the cycle, inhibits the synthesis of the first one. The interaction of protein concentrations and mRNA (messenger RNA) concentration is surprisingly similar to the interaction of six ecological populations, three predators and three preys. This makes it possible to propose a new phenomenological model, which is represented by a system of unidirectionally coupled ordinary differential equations. We investigate the problem of existence and stability of a relaxation periodic solution that is invariant with respect to cyclic permutations of coordinates. To find the asymptotics of this solution, a special relay system is constructed. We prove that the periodic solution of the relay system gives the asymptotic approximation of the orbitally asymptotically stable relaxation cycle of the problem under consideration.


STATEMENT OF THE PROBLEM AND DESCRIPTION OF RESULTS
Artificial genetic oscillators, combined in networks of various architectures, are of considerable interest, due to the fact that they can simulate certain key biological processes, including cell cycles and circadian rhythms. In [1], the simplest genetic oscillator called a repressilator was proposed, which consists of three elements Each of these elements unidirectionally inhibits the neighboring one. Namely, element inhibits the synthesis of element inhibits the synthesis of and the third element closing the cycle, inhibits the synthesis of The mathematical model of the mentioned gene network has the form (1) where Following [1], we believe that each element of the oscillator is a set of mRNA (messenger ribonucleic acid) with concentration and a protein with concentration We further assume that the temporal change in concentrations is characterized by synthesis and degradation processes. The first of these processes is described by the function , where is the concentration of the repressor protein for the th mRNA, is the coefficient of cooperativity, and is the transcription rate when no repressor is present. The second process is described by the linear term " ." Finally, additive in the equation for is responsible for the leakage of the promoter.
The situation is simpler in the case of the protein concentration . Namely, we assume that protein dynamics is characterized by linear processes of synthesis (the term " " in the equation for from system (1)) and degradation (the term " " in the same equation). Here is the ratio of the rate of protein degradation to the rate of mRNA degradation.  As a rule, model (1) is studied assuming the smallness of and In this situation, after replacing and discarding the additive , a singularly perturbed system is obtained, to which the well-known regularization principle of A.N. Tikhonov is then applied [2]. The result of this regularization is the system (2) The question of self-oscillations of system (2) has been studied by multiple authors (see, for example, [3][4][5][6][7][8][9][10]). These papers considered a case when its stable cycle arises as a result of the Andronov-Hopf bifurcation, as well as a case . In addition to model (2), a whole series of more general systems describing artificial gene networks has been proposed. For example, in [6,7], four classes of such systems were introduced, and not necessarily ring ones. If we consider only the simplest ring systems, the mentioned classes will be typically represented by models (3) (4) (5) (6) In all cases (3)- (6), it is assumed that , for , , , . Traveling waves of systems (3)- (6) are special periodic solutions that allow representations of the form (7) It should be added that traveling waves (7) are the most natural attractors of ring chains of unidirectionally coupled equations. However, more complex oscillations are also possible in such chains. As an example, we refer to paper [11], in which numerical analysis was used to show the feasibility of a chaotic self-oscillating regime in a ring of three unidirectionally coupled generators. It is also worth noting a whole series of works (see, for example, [12][13][14]) dedicated to the problem of transition to chaos in unidirectionally coupled chains of oscillators with an appropriate increase in the number of elements.
In [15,16], a new approach to modeling artificial gene networks was proposed. Namely, it was noted that the nature of the interaction between concentrations and described above resembles the interaction of six ecological populations, three predators and three preys. Indeed, suppose that and are population densities of predators and preys, respectively. Then, by virtue of (1), each predator feeds on only one prey (for , the population size of decays exponentially) and at the same time exerts pressure only on prey The latter is expressed in the fact that as increases, the population growth rate p j+1 decreases. In addition, in the absence of a predator-repressor ( ), the population size of tends as to the threshold value For mathematical modeling of our favorite gene network, the given ecological interpretation allows using the method of Yu.S. Kolesov [18]. In the framework of the interaction of predators and prey described above, the mentioned method leads to a system (8) where all constants are positive. Let us emphasize that in the equation for , violating its Voltaire structure, we, on purpose, included the term " ," similar to the addition of from (1). As will be shown below, in contrast to system (1), where we can put , in our case the requirement is mandatory.   (1), the new mathematical model of the repressilator (8) allows some simplification. Namely, we assume first that Then, according to the reduction principle [2] as we have As for components we obtain a system (9) which represents an alternative to the known model (2). The invariance of system (9) with respect to cyclic permutations of coordinates naturally raises questions about the existence and stability of a traveling-wave-type cycle of the form (10) where is the phase shift with period Earlier in [15], this problem was studied for and , in turn, in [16,17], the case was considered when , , and parameter is of the order of unity. More precisely, provided that (11) In this article, these questions are discussed under the additional assumptions (12) The main attention below is given to the case that is most interesting from a mathematical point of view: (13) when the shape of oscillations is as complex as possible (the case will be discussed separately in the final part of the paper). Inequality (13) allows us to introduce on the plane a curve (14) which is shown in Fig. 1. It turns out that curve (14), extended from the interval to the entire axis according to the -periodicity law, as well as the curves obtained as a result of shifting it along the axis by and , respectively, are zero approximations as for components of some periodic solution of system (9). Namely, the following statements are true. Theorem 1. There is a sufficiently large such that under conditions (12), (13) and for all , system (9) permits a self-symmetric cycle (15) of period For , the following asymptotic representations hold true for this cycle: ) ( 2 ( ) )) (0 ) 1 u u u u t r u t h r r u t h r r u r * * * * , , = , , + , , where is curve (14), (17) and denotes the Hausdorff distance between compact sets. Theorem 2. Cycle (15) referred to in the previous theorem is exponentially orbitally stable.
The proof of Theorem 1 is based on the fact that function appearing in (15) satisfies the auxiliary equation with delay (18) Thus, the problem of existence of the periodic regime (15) is reduced to finding a periodic solution to this equation with the period of and characterized by properties (16). As for the stability of cycle (15), it is determined separately by means of asymptotic analysis of the corresponding linear system in variations.
The relaxation properties of cycle (15) are graphically represented by the plotting of curve (17), constructed numerically for r = 10 (see Fig. 2).

General Study Scheme
As already mentioned above, the justification of Theorem 1 is associated with the analysis of auxiliary equation (18), and more precisely, with the search for its nonconstant -periodic solution.
Under condition (13), we make the substitution in this equation and put As a result, for the new variable , we obtain the equation Everywhere below, we assume that its delay runs through the set (20) where the value of constant will be specified later. For now, we assume satisfaction of the inequality (21) ( ) 10 r = Now we describe a class of initial conditions for equation (19). In this connection, we take a sufficiently small constant which satisfies the conditions (22) (these inequalities are possible, because by (21) we have for all ). Next, we introduce a set of initial functions continuous for , which is given by the equality (23) where are some universal (independent of ) constants, whose values will be discussed later.
Let us consider the solution of equation (19) with an arbitrary initial condition for as we denote the second positive root of the equation (if it exists) and define operator which acts from to space of continuous functions for by the rule (25) As will be shown in the following, with an appropriate choice of parameters , operator (25) is defined on set (23) and, moreover, for all Further, since set is closed, bounded, and convex, and operator by virtue of inequality is compact, then according to the Schauder principle, this operator has in at least one fixed point It is also clear that solution of equation (19) is periodic with period As for the available parameter from set (20), it is determined from the equation (26) It turns out that equation (26) permits solution which is bounded in and such that as And this, in turn, implies that the auxiliary equation (19) for has our favorite -periodic solution

Asymptotic Integration of an Auxiliary Scalar Equation
In order to implement the program of actions described in the previous paragraph, it is necessary to know the asymptotic behavior as of solution uniform for on the time interval (27) In the process of constructing this asymptotic, interval (27) is divided into 11 parts. To clarify the matter, going ahead, we note that on the indicated interval the graph of function is asymptotically close to the curve similar to (14): (for , it coincides with and has the form shown in Fig. 1). It is clear that in justifying this proximity, we need to separately discuss the intervals of variation in which are adjacent to the breaks of curve and the intervals at which the graph of is asymptotically close to one of the vertical sections of Examination of each of the resulting intervals is a separate stage with its own corresponding lemma.
Stage 1 is related to the time interval (28) The following statement holds true.

Lemma 1.
On interval (28) as , the following asymptotic representation uniform for holds true: Here and below, the same letter denotes some universal (independent of ) positive constants whose exact values are irrelevant.
Proof. Since the length of interval (28) is then automatically and therefore, by virtue of (23), as we have Further, suppose that on the considered interval an a priori estimate of the following form is satisfied: where, as in the case of letter here and in the following, symbol denotes various positive constants independent of . Substituting in (19) (33) and taking into account relations (31), (32), to find , we arrive at the Cauchy problem of the form where Further, combining relations (35), (37) to solve problem (34), we obtain the following asymptotic representation uniform for variable from set and for : And from here and from (33) the required equality (29) follows automatically. To complete the justification of the lemma, it remains to verify the validity of condition (32). Combining the explicit formula for (see (30)) with the asymptotic representation (29) (so far a priori), we conclude that this condition actually holds for any fixed constant Thus, with the specified choice of , all our previous constructions, which were conventional in nature, become validated, and, in particular, the required asymptotic equality (29) holds true.
Let us draw attention to one characteristic moment of this stage, which will be repeated in subsequent stages. Namely, we were able to find solution in explicit form with accuracy up to an exponentially small additive (that is, quantity of the order where ). As for the justification of the specified order of smallness of the aforementioned additive, both at this stage and in the future, it is carried out similarly, according to the above scheme. In this connection, the corresponding fragments of proofs in the following lemmas will be omitted.
It is interesting to note that on intervals (the length of the second is positive by virtue of conditions (22)), formula (29) can be simplified. Indeed, from the obvious asymptotic properties it follows that as uniformly for Stage 2 consists in considering the interval (40) where is an arbitrarily fixed constant from interval Under the specified , by virtue of the condition (see (22)), there are inclusions And this and (38) imply the asymptotic equality uniform : As for function the following holds true: Lemma 2. For , the following asymptotic representation holds true uniformly for from interval (40) and for : where (43) Proof. As in the case of Lemma 1, we justify formulas (42), (43) first under the a priori assumption (32) and then verify the validity of this assumption itself.
In the right-hand side of equation (19), we consider relations (32), (41). As a result, it is converted According to the previous stage (see (39)), this equation should be supplemented with the initial condition (45) Further, it is easy to show that the function (46) satisfies the Cauchy problem (44), (45) with accuracy up to values of the order with respect to the residual. Therefore, reasoning as in the justification of Lemma 1, we are convinced that under the a priori condition (32), the asymptotic representation (42) does indeed hold true. This information allows us to make the following statement.

Lemma 3. On interval (48), for
, the following asymptotic representation uniform for ϕ holds true: Here and in the following formulas, exponent is defined by equality Proof. We justify formula (51) under a priori assumptions (52) and then make sure the conditions (52) themselves are true. We consider relations (49), (52) in the right-hand side of equation (19), switch to variable (see (48)), and supplement it with the initial condition (50). As a result, we arrive at a Cauchy problem of the form Further, a simple check shows that the function (54) satisfies the Cauchy problem (53) with accuracy up to quantities of the order with respect to the residual. And this, in turn, implies that under conditions (52) the required asymptotic equality (51) is indeed satisfied.
Let us separately discuss the verification of a priori assumptions (52). In this regard, we note that for function (54) the corresponding estimates hold for constants Thus, initially choosing constants in (52) in the manner described above, we verify that these estimates are also valid for solution That means that the asymptotic representation (51), which was conventional previously, becomes valid.
Stage 4 consists in considering the time interval where Based on formula (38) again, we conclude that in this case Next, we put (57) The following statement holds true. Lemma 4. As , for function (57), the following asymptotic representation holds true uniformly for : Proof. Let us replace the time (55) in equation (19), substitute relation (56) in its right-hand side, and put This fact allows us to use the scheme described in the proof of Lemma 1 to obtain the required asymptotic representation (58).
Let us separately discuss information about function which can be extracted from formula (59). To this end, we note that function (60) is expanded in a convergent series of the form where are algebraic polynomials of degree and Substituting then the series (65) in (59) and performing the reexpansion for for function we obtain a similar series (67) which, unlike series (65), is only uniformly asymptotic for variable . A simple calculation using formulas (66) leads to equalities (68) where In the general case, based on formulas (68) and the method of mathematical induction, we obtain asymptotic representations: where and are some algebraic polynomials of degree at most and , respectively. Expansions (67), (69), and (70) give the full asymptotic behavior of function (59). It follows from the above expansions that, in addition to formula (58), for , the following asymptotic representation holds true: , the following asymptotic equality holds true for : Proof. The justification of this lemma is similar to the proof of Lemma 4. Namely, we substitute time (72) in equation (19), take equality (74) into its right-hand side, and switch to a new variable v by formula (61). As a result, to find function (75), we obtain an equation similar to (62): By virtue of the previous asymptotic representation (58), it should be supplemented with the initial condition (80) To analyze the Cauchy problem (79), (80), we require some information about function (77). A direct check shows that it is a solution to the simplified Cauchy problem (81) Further, taking into account that function (78) monotonically increases for from (77) we have And this and the explicit form of (see (73)) implies that This information suggests that and, therefore, after substituting in (79), (80) the equality and transferring all terms from the left-hand sides of the resulting expressions to the right-hand sides, we receive residuals of the order of smallness and , respectively. And this means (see the similar part in the proof of Lemma 1) that on interval (72) the asymptotic representation (76) indeed holds true.
As in the previous stage, we dwell separately on revealing the asymptotic properties of function (77). In this regard, we note that in the case of the Cauchy problem (81) we are within the framework of applicability of the known results of A.N. Tikhonov [2]. The above results suggest that for as , the following asymptotic representation uniform for holds true: As is known [2], the terms of series (82) are calculated by substituting it in the equation from (81) and equating the coefficients for powers As a result, for functions we get a recurrence sequence of algebraic equations, the first of which is nonlinear, and the rest are linear. From the above equations, these functions are uniquely determined. In this way, in particular, we obtain formulas (83) In what follows, we need information about the behavior of function as In order to deal with this issue, we put in the equation from (81) As a result, for we deal with the equation Further, a simple analysis of equation (84)  for parameter (which we assume to be satisfied everywhere below) and as we have an asymptotic representation uniform for ϕ: where The following statement holds true.
where is function (30). Proof. In equation (19), we take a new independent variable take into account equality (90) in its right-hand side, and make the substitution (93) As a result, to find function (91) we arrive at an equation of the form (94) where According to equalities (88), (93), we supplement it with the initial condition We first consider the simplified Cauchy problem It is easy to verify that for its solution the following formulas hold true: Further, taking into account in (97) and (98) the asymptotic representation for (see (95)), the equality and estimates we conclude that uniformly for (99) The inverse transition from problem (96) to (94), (95) is standard: it follows from the found equality (99) that  and by virtue of this the function satisfies equation (94) with the accuracy up to with respect to the residual. And this, in turn, implies the validity of the required asymptotic representation (92).
Concluding the consideration of stage 6, we add that on an interval narrower than (89), formulas (91), (92) can be simplified. Relying on the obvious property for the indicated values of time, we obtain the following asymptotic equality uniform for : At stage 7, we turn to the values of from the interval (101) where the value of is the same as in (40). Since, in this case, the argument changes on interval (40), by virtue of (42) we have (102) where is function (43). Next, we introduce a function similar to (91): where now The following holds true.

Lemma 7. As
, for function (103), the following asymptotic equality holds true uniformly for from interval and for : Proof. The justification of this lemma is similar to the proof of Lemma 6. Namely, in equation (19) we substitute time take into account relation (102) in its right-hand side, and switch to a new variable v according to the rule similar to (93), As a result, we obtain an equation similar to (94): where, as before, According to the previous stage, it should be supplemented with the initial condition (106) where, by virtue of (100), As above, we first consider the simplified Cauchy problem It is easy to see that its solution is given by explicit formulas similar to (97), (98): The final stage of the substantiation of the lemma, associated with the transition from problem (108) to (105), (106) does not cause difficulties. Indeed, taking into account equality (110), we find that function satisfies equation (105) with accuracy up to values of the order of smallness with respect to the residual. Then, relying on the standard scheme described in the proof of Lemma 1, we obtain the required asymptotic representation (104 As usual, we first consider the simplified Cauchy problem corresponding to problem (116), (117): (118) and note that its solution can be written out explicitly. Namely, the following equalities hold true: where Further, based on the formula it is easy to show that our favorite solution (119) of the simplified problem (118) permits for the following asymptotics uniform for (120) To complete the proof of the lemma, we note that by virtue of (120) for , the term from the right-hand side of equation (116)  Its asymptotic properties are described by the following lemma. Lemma 9. As , for function (123), the following asymptotic equality uniform for from interval (121) and for holds true: (1 ) Proof. The justification scheme of this lemma is standard: first, we replace time (121) in equation (19), then substitute equality (122) into its right-hand side, and switch to the new variable by the formula (127) As a result, for we obtain the equation Combining these estimates with condition we conclude that And from here the required fact follows automatically.
We show now that function satisfies the initial condition (129) with exponential accuracy. To verify this property, it is necessary to know the asymptotic behavior of the integral for : Performing double integration by parts, we successively derive: where the remainder is uniform for Then, taking into account relations (130)-(132) for in formula (125), we see that when we substitute function in (129), we obtain the  (125) permits an asymptotic representation Therefore, for , the following equalities hold true: In the case when , first, relying on the explicit form of function which appears in (135), we obtain an asymptotic representation Concluding stage 9, we provide the simplest version of the asymptotic equality for which can be obtained from expansion (134). Namely, relying on the explicit formulas (135) and properties (136), (140) shown above, we conclude that as (141)  In analyzing the Cauchy problem (145), (146), we need the asymptotic equality for more detailed than that given by formula (141). In this regard, we note that where We also note that function can be expanded into an asymptotic series in integer powers and for , it coincides with constant (138). And finally, taking into account the above relations in formula (125), we conclude that (147) In order to formulate a rigorous result corresponding to this stage, we substitute the resulting equality (147) into (146) and proceed from (145), (146) to the simplified Cauchy problem Note further that the solution to this problem is written explicitly by means of the formula    A simple analysis of the resulting Cauchy problem (155), (156) leads to the conclusion that, first, its solution is given by the equality (157) second, for function , the required estimates (154) are satisfied with constants And this means that formula (157) becomes valid and the final lemma holds true.

Lemma 11. As
, the solution of permits the asymptotic representation (157) uniform for from interval (151) and for arguments Let us separately discuss the geometric interpretation of all eleven stages of constructing the asymptotics of function To do this, we need the root of of equation (24), which was mentioned in Section 2.1. Lemmas 1-11 and conditions (22) imply that time belongs to interval (151). By virtue of representation (157) and equality (155), the mentioned root is simple and, as , has the following asymptotics uniform for : In turn, the presence of root (158) allows us to determine on the plane the curve The geometric meaning of Lemmas 1-11 is that as uniformly as where is the curve introduced at the beginning of this section and is the Hausdorff distance. Indeed, the above proximity between curves and is successively shown in eleven different intervals, and from the asymptotic representations obtained for these sections for the required equality (159) follows automatically.

Completing the Proof of Theorem 1
We proceed to the implementation of the scheme described in Section 2.1. In this regard, we turn to operator (25). Our constructions suggest that this operator is well defined on set (23). Let us show now that for an appropriate choice of constants in (23) we have the inclusion Indeed, relying on the asymptotic formula (159), it is easy to see that the inequalities hold true for any fixed constants , it is true that the required inclusion Further, by virtue of inequality following from (158), operator is compact. Thus, according to the Schauder principle, in it has at least one fixed point As already mentioned in Section 2.1, the corresponding solution of equation (19) turns out to be periodic with period Let us also add that at for from equality (158) suggests the following asymptotic representation uniform for : Further, we turn to equation (26) to find the free parameter . Using the asymptotic formula (161), we conclude that the above equation permits solution with the asymptotics The constructions carried out in this paragraph, together with the asymptotic analysis in Section 2.2, make it possible to deal with our main problem, the justification of Theorem 1. To do this, we introduce the function (163) which, by construction, is periodic with period Next, we consider root of equation which is asymptotically close to zero. It follows from representation (38) and formula on interval , that this root is simple and has asymptotics Assuming then we obtain the desired cycle (15) of system (9), (12).
In conclusion, we note that the found periodic solution permits asymptotics (16). Indeed, the desired formula for obviously follows from (162), the asymptotic representation for follows from (159), (162), (164), and the formulas for the maximum and minimum of function are obtained from (165) given the asymptotic behavior of the solution of Theorem 1 is completely proven.

PROOF OF THEOREM 2 3.1. General Study Plan
To analyze the stability properties of cycle (15), in system (9), under conditions (12), (13), we substitute variables and put As a result, we obtain a system of the form (166) Further, we note that in the framework of system (166), the self-symmetric cycle (15) has a corresponding periodic solution (167) where are functions (162), (163). In turn, the stability of cycle (167) depends on multipliers of the linear system in variations (168) with coefficients (169) Note that system (166) is a special case of a ring system of unidirectionally coupled equations, and cycle (167) is a periodic mode of the traveling wave type. For such periodic solutions, a special method for analyzing stability properties was developed in [14][15][16]; we describe it below.
Along with (168), we consider an auxiliary scalar linear equation with delay (170) where is the phase shift from (167), is a complex-valued function, and is an arbitrary parameter. More precisely, we are interested in its multipliers , numbered in descending order of modules.
Let us explain the meaning of the term "multiplier" as applied to equation (170). In this regard, we consider the Banach space of complex-valued functions continuous for with the norm where denotes the solution of equation (170) on the time interval with the initial function Note that, due to the compactness of operator , its spectrum is discrete. By analogy with the case of ordinary differential equations, as the multipliers of equation (170) we consider eigenvalues of operator (172).
Let us discuss the association between the multipliers of system (168) and equation (170). The following statement holds true (see [14][15][16] In the next two sections, we carry out an asymptotic calculation of multipliers and analyze equations (174). In this way, for multipliers of system (168), we will obtain relations of the form exp( ) meaning that cycle (15) is exponentially orbitally stable.

Analysis of the Auxiliary Linear Equation
We take an arbitrary constant and consider the set (176) where ||*|| is norm (171). We are interested in solution of equation (170) (178), where these properties are uniform for from interval (177) and for the initial condition from set (176). The corresponding constructions are divided into the same 11 stages as in Section 2.2. Therefore, omitting the technical details, here we only summarize the final results.
At stage 1, when changes on interval (28), by virtue of (23), (29), (30), (169), as , we have Taking these circumstances and equalities (179) into account, we arrive at the following asymptotic formulas uniform for : At stage 2, for the coefficients of equation (170) as , the following asymptotic equalities hold true: where is function (43). Combining these equalities with the already known information (180), we see that for uniformly for from interval (40) and for At stage 3, for as , we obtain the equalities and the delayed components are given by the equalities following from (180): g t g g t g g t g g t g , , , ε = , , ε + , , ε + , , ε + , , ε .
, ,ε ≡ ∈ −σ , − − σ , = , . (1 )   At stage 9, we consider the time interval (121). In this case, for , coefficients (169) permit asymptotic representations (192) where and functions are given by equalities of the form (183) after replacing by and substituting Based on these facts and taking into account the previous formulas (191), we see that as uniformly for (193) where is function (125), and is the variable from (192).
Stage 10 concerns the time interval (142), on which for we have (194) where And from here, in turn, taking into account the equalities already known from the previous stages for and formulas (193), it is easy to deduce that for we have the following asymptotic representations uniform for : (195) where variable is the same as in (194 (κ ) ( ) κ ( ) κ ( ) κ ( ) where are bounded linear operators. Moreover, assuming constant in set (176) to be equal to 1 and relying on the asymptotic analysis of the solution of which was carried out in Section 3.2, we see that estimates of the following form hold true: (198) where We turn further to equations (174) and show that for a fixed sufficiently large they have no roots in the set Indeed, from (197) As for multiplier it is simple for , analytically depends on , and as , it permits the following asymptotic representations uniform for : Proof. We arbitrarily take and assume that parameter from (172) belongs to set (200). Next, we represent operator (172) in the form (203) where The asymptotic analysis of solution from Section 3.2 and equality suggest that operators that appear in (203) We first study the spectral properties of operator It follows from formulas (203) that it is finitedimensional, and its spectrum consists of two points, a simple eigenvalue where and an eigenvalue of infinite multiplicity. For eigenvalue from representations (196) (regarding the case ), we obtain the following asymptotic equations uniform for : We now turn to the original operator and note that, by virtue of the relation where is the unit operator in space , any value for which (206) belongs to the resolvent set of this operator. Recall further that operator permits an estimate from (204). In the case of operator relying on its explicit form and the second estimate from (204), we obtain the inequality (207) where constant does not depend on At the final stage of the proof of Lemma 13, we combine estimates (204), (207) with asymptotic representations (205). As a result, we see that condition (206) is satisfied for any where (208) and constant is fixed and sufficiently small. Thus, the spectrum of operator (172) certainly belongs to the pairs (208). And hence inequality (201) follows in an obvious way.
To justify relations (202), we note that, when perturbing the operator by analytic for additive of the order of smallness , the eigenvalue becomes a simple and analytically dependent on eigenvalue where (209) (in the -metric for variable ). Combining then relations (205), (209), we conclude that for the multiplier has all the required properties. The final stage of the proof of Theorem 2 is related to the analysis of equations (174). We take constant satisfying the condition (199). Then, when considering these equations, we can only consider values of Further, we arbitrarily take and consider set (200). By virtue of relations