Lasota–Opial type conditions for periodic problem for systems of higher-order functional differential equations

In the paper we study the question of solvability and unique solvability of systems of the higher-order functional differential equations ui(mi)(t)=ℓi(ui+1)(t)+qi(t)(i=1,n‾) for t∈I:=[a,b]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u_{i}^{(m_{i})}(t)=\ell _{i}(u_{i+1}) (t)+ q_{i}(t) \quad (i= \overline{1, n}) \text{ for } t\in I:=[a, b] $$\end{document} and ui(mi)(t)=Fi(u)(t)+q0i(t)(i=1,n‾) for t∈I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u_{i}^{(m_{i})} (t)=F_{i}(u) (t)+q_{0i}(t) \quad (i = \overline{1, n}) \text{ for } t\in I $$\end{document} under the periodic boundary conditions ui(j)(b)−ui(j)(a)=cij(i=1,n‾,j=0,mi−1‾),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u_{i}^{(j)}(b)-u_{i}^{(j)}(a)=c_{ij} \quad (i=\overline{1, n},j= \overline{0, m_{i}-1}), $$\end{document} where un+1=u1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u_{n+1}=u_{1} $\end{document}, mi≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m_{i}\geq 1$\end{document}, n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\geq 2 $\end{document}, cij∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c_{ij}\in R$\end{document}, qi,q0i∈L(I;R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q_{i},q_{0i}\in L(I; R)$\end{document}, ℓi:C10(I;R)→L(I;R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell _{i}:C^{0}_{1}(I; R)\to L(I; R)$\end{document} are monotone operators and Fi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F_{i}$\end{document} are the local Caratheodory’s class operators. In the paper in some sense optimal conditions that guarantee the unique solvability of the linear problem are obtained, and on the basis of these results the optimal conditions of the solvability and unique solvability for the nonlinear problem are proved.


Introduction
Consider on the interval I = [a, b] the system of higher-order linear functional differential where u n+1 := u 1 , and the system of higher-order nonlinear functional differential equations where m i ≥ 1, q i , q 0i ∈ L(I; R), i : C 0 1 (I; R) → L(I; R) are linear bounded operators, and For arbitrary x ∈ R, we assume that Definition 1. 1 We will say that the operator F : C m 1 ,...,m n (I; R n ) → L(I; R) belongs to Caratheodory's local class K(C m 1 ,...,m n , L) if F is a continuous operator, and for arbitrary r > 0, the inclusion sup F(x)(·) : x C m 1 ,...,mn ≤ r, x ∈ C m 1 ,...,m n I; R n ∈ L(I; R + ) holds.
We will say that an operator is monotone if it is nonnegative or nonpositive.
By a solution of problem (2), (3) we understand a vector-function u := (u i ) n i=1 where u i ∈ C m i -1 (I; R) (i = 1, n), which satisfies equation (2) almost everywhere on I and satisfies conditions (3).
First of all, we would like to give a historical review which begins with Lasota and Opial's article [1] from the year 1964, which became the basis for a lot of interesting studies. In this article (see Theorems 5 and 6) authors proved that the problem where L 2 = 16 and the general term L n can be found by formula (40) from the paper [1], in which it is also shown that for n = 2 condition (5) is optimal, while for n ≥ 3 it is far from being optimal. In the article [2], we announced the Lasota-Opial type optimal results of unique solvability of the periodic problem for second-order linear functional differential equations, which in a more general form are considered in the paper [3] from 2006. In particular, in Theorem 1.1 of the paper [3] (see also [4]) it is proved that the functional differential equation under boundary conditions (4) for n = 2 is uniquely solvable if the optimal conditions b a 0 (1)(s) ds = 0 hold, where 0 : C 0 1 (I; R) → L(I; R) is a monotone operator, i = + i -i (i = 1, n -1) and ± i : C 0 1 (I; R) → L(I; R) are nonnegative operators (for n = 3 see [5]). For the general case, analogous conditions of the unique solvability of problem (6), (4) are proved in [6], for i = + i -i (i = 0, n -1), which if 0 is monotone operator and i ≡ 0 (i = 1, n -1) transforms where T 1 = 4, T 2 = 32, T 3 = 192, and the general term T n of this sequence can be found by formulas (9). In [6] (see also [7]) it is proved that the constants T n are sharp when n ≤ 7; for n > 7, the problem of sharpness of the numbers T n is still opened. In this brief historical review Bravyi's original studies of problem (6), (4) cannot be omitted. The method developed in these studies turned out to be very fruitful. Particularly in [8] the author proved that the condition where Y = N n (ba) n-1 min(P -, P + ), X = N n (ba) n-1 max(P -, P + ), the numbers N n are defined by the certain recurrent formula, and + 0 (1) L = P + , -0 (1) L = P -, is necessary and sufficient for solvability of problems (6), (4) if i ≡ 0 (i = 1, n -1) (N n = T -1 n-1 for n = 2, 7, but for n > 7 the validity of the last identity is unknown). For the case i ≡ 0 (i = 1, n -1), these results are generalized in [9]. It is interesting that the numbers N n are in some connection with Favard's, Bernoulli's, and Euler's numbers (see [10] and [11]) and if the operator 0 is monotone, then condition (8) Other interesting results about problems (6), (4) can be found also in the papers [10][11][12][13][14][15].
The next stage was the generalization of Lasota-Opial's results for the systems of linear functional differential equations. In particular in [16] it is proved that problem (1), (3) with m i = 1 (i = 1, n) is uniquely solvable if i are linear monotone operators, i = 0 (i = 1, n), and the condition n i=1 i (1) L < 4 n (which is optimal) holds. In this connection see also the papers [17] and [18].
The aims of this article are to establish Lasota-Opial type sufficient efficient optimal conditions of the solvability of problem (1), (3) and on the basis of these results to find the optimal efficient sufficient conditions of solvability and unique solvability of the nonlinear problem (2), (3).
Remark 2.1 In Remarks 1.2 and 1.3 of [7] it was shown that and T n < (2π) n (n ∈ N). Now we can formulate the first of our main theorems.

Nonlinear problem
be nonnegative linear operators, then we will say that and for arbitrary monotone operators i : where u n+1 := u 1 , has no nontrivial solution.
Also note that in all our propositions below the functions η i : I × R + → R + (i = 1, n) are summable in the first argument, nondecreasing in the second one, and admit to the conditions ..,m n (I; R n ), x n+1 = x 1 , and inclusion (13) hold. Moreover, let the function g ∈ L(I; R n + ) be such that, for all i ∈ {1, . . . , n} on I, the conditions Then problem (2), (3) has at least one solution.
On the other hand, in the example below we construct the operator F for which conditions (21) hold and (22) do not hold, and therefore condition (20) cannot be omitted because, as it follows from Remark 2.4, condition (21) is improvable.
For the case when (2) is the system of higher-order differential equations with the argument deviation of the form where u n+1 := u 1 and f i : I × R → R (i = 1, n) are the functions from Caratheodory's class, the following corollary is true.

Corollary 2.3 Let the function h ∈ L(I; R n + ) be such that the conditions
hold. Moreover, let the function g ∈ L(I; R n + ) and the numbers σ i ∈ {-1, 1} (i = 1, n), r 0 > 0 be such that conditions (21) and hold. Then, for arbitrary τ ∈ M n (I), problem (24), (3) has at least one solution.
Remark 2.4 Theorem 2.2 is optimal in the sense that there does not exist such i 0 ∈ {1, . . . , n}, for which i 0 th inequality of condition (21) can be replaced by the inequality no matter how small ε > 0 would be. Indeed, let I = [0, 1], and Then, for the arbitrary functions p i ∈ L(I; R) such that a |p i (s)| ds ≤ 4 n , and due to Corollary 2.1 inclusion (13) with h i (x)(t) = h i (t)x(τ i (t)) holds. Also it is not difficult to verify that instead of the i 0 th inequality of condition (21) inequality (27) is satisfied, but all the other assumptions of Corollary 2.3 hold with r 0 = 1. Nevertheless, in that case problem (24), (3) is not solvable because Example 2.2 Consider the system of the differential equations where u n+1 = u 1 , σ i ∈ {-1, 1}, the functions τ ∈ M n (I), h ∈ L(I; R n + ) are such that inclusion (13) holds with h i (x)(t) = h i (t)x(τ i (t)), and b a q 0i (s) ds = 0 (i = 1, n). Then, from Theorem 2.2 with g 0i ≡ g i ≡ 0, the solvability of problems (28), (3) with c i m i -1 = 0 (i = 1, n) follows.

Auxiliary propositions
First we formulate a result from [7] (see Theorem 1.1, and Remark 1.1) in a suitable for us form.
where the constants T m are defined by equalities (9) and Proof Validity of (35) follows from the inequalities max t∈ [a,b] {w(t)}w(t) ≥ 0, w(t)min t∈ [a,b] {w(t)} ≥ 0 and the nonnegativity of the linear operator σ .
Then there exist a subsequence ( k r ) ∞ r=1 of the sequence ( k ) ∞ k=1 and the linear operator 0 : σ 0 is a nonnegative linear operator, and Proof Assume that P := {p 1 , p 2 , . . .} is a set of all polynomials with rational coefficients, then from (36) it is clear that, for the monotone linear operators k (k ∈ N ) and arbitrary p j ∈ P, the inequalities hold. Then by the Arzela-Ascoli lemma, from the sequence ( k ) ∞ k=1 we can choose the subsequence ( k m 1 ) ∞ m 1 =1 convergent on the polynomial p 1 ∈ P, and if the subsequence of the operators ( k mr ) ∞ m r =1 convergent on the polynomial p r ∈ P is already chosen, then we can choose its subsequence ( k m r+1 ) ∞ m r+1 =1 of the operators convergent on the polynomial p r+1 ∈ P. Therefore it is clear that the sequence ( k mr ) ∞ r=1 is convergent for the all polynomials from P. Consequently, without loss of generality, we can assume that the first subsequence ( k r ) ∞ r=1 is convergent for arbitrary p j ∈ P, and then in view of (39) and the fact that P is dense in C 0 1 (I; R), from the Banach-Steinhaus theorem (see [19], Theorem 3, p. 203), there follows the existence of linear operator 0 : C 0 1 (I; R) → C 0 (I; R) such that condition (37) holds, where 0 (x)(t) : From the monotonicity of the operators k and conditions (36) it follows that σ k are the nonnegative operators, and for arbitrary x ∈ C 0 1 (I; R + ) we have From the last inequalities by (37) it is clear that σ 0 is a nonnegative linear operator and inequality (38) holds.

Now consider the problem
where u := (u i ) n i=1 and u n+1 = u 1 . It is not difficult to verify that the following proposition is true.  = 1, n). Then, for arbitrary λ ∈ (0, 1), problems (40), (41) and (16), (17) are equivalent respectively to the problems For all i ∈ 1, n, the following identities hold: where I : C 0 1 (I, R) → L(I, R) is the identical operator. Moreover, it is clear that w C 0 m = u C m 1 ,...,mn .
Taking into account Proposition 3.1 we get the following modification of Corollary 2 of paper [20], which is formulated for the system of first-order equations.
Also due to (44) all the assumptions of Lemma 3.3 hold, and then without loss of generality we can assume that the sequence of operators ( ik ) ∞ k=1 is convergent for arbitrary fixed i ∈ {1, . . . , n}, and there exist the monotone operators i : C 0 1 (I; R) → L(I; R) such that if the operators ik , i : C 0 1 (I; R) → C 0 (I; R) are defined as σ i i is a nonnegative linear operator, and Thus from the inequalities

by (48) and (50) it follows that
uniformly on I. Therefore if we integrate the equations of system (46) from a to t and pass to the limit as k → +∞, due to conditions (45), (46), (48), and (52), we find that v is a solution of problem (16), (17). On the other hand, from the inclusion h ∈ P(I) and conditions (51), it follows that problem (16), (17) has only the zero solution v ≡ 0 if i (1) ≡ 0 (i = 1, n). But if there exists i 0 ∈ {1, . . . , n} such that i 0 (1) ≡ 0, then from the i 0 th equation of system (16) it (17) we have v i 0 ≡ Const, and due to (49) it is clear that v i 0 ≡ 0, and from the (i 0 -1)th equation of system (16), it follows that v (m i 0 -1 ) i 0 -1 ≡ 0. After analogous n -1 steps we get that v i ≡ 0 (i = 1, n). Therefore we get the contradiction with the second equality of (48), i.e., our assumption is invalid and estimation (43) holds.

Proof of main results
Proof of Theorem 2.1 It is known from the general theory of boundary value problems for the functional differential equations that problem (1), (3) has Fredholm's property (see [14]), and therefore our problem is uniquely solvable iff the homogeneous problem (16), (17) has only the trivial solution. Assume to the contrary that problem (16), (17) has the nontrivial solution (v i ) n i=1 , and introduce the notations: Let there exist r ∈ {1, n} such that v r+1 ≡ Const. Then from (16) and (17) we get v r+1 (t) b a r (1)(s) ds ≡ 0. From the last equality and conditions (10), (16), it is obvious that v r+1 ≡ 0 and v (m r ) r ≡ 0. But from the identity v (m r ) r ≡ 0, due to conditions (17), we get that v r ≡ Const. After analogous n -1 steps we get that v i ≡ 0 for all i ∈ 1, n, which is the contradiction with our assumption, i.e., Now define for all i ∈ 1, n the numbers and notice that due to (17) and (54) the functions v (m i -1) i (i = 1, n) change the sign, and then in view of (34) we have Therefore in view of (16) and (55) we get Assume that σ i i is a nonnegative operator, then from equalities (56) by Lemma 3.2 with = σ i i , σ = 1, we have It is not difficult to verify that inequality (57) holds even when the operator σ i i is nonpositive. On the other hand, due to (17) and (54), all the assumptions of Lemma 3.1 hold for the functions v i with m = m i+1 -1; and consequently, (57) by (33) implies Finally, if we multiply inequalities (58) for all i ∈ 1, n and take into account notations (53), we get the contradiction to condition (11). Therefore our assumption is invalid and v i ≡ 0 (i = 1, n), which definitely proves our theorem.
Let ρ 0 be a number defined in Lemma 3.4, then due to condition (18) there exists such a constant ρ 1 > r 0 that where α(t) = 1+ n i=1 |g 0i (s)|. Now assume that u C m 1 ,...,mn ≥ ρ 1 and introduce the notation for t ∈ I.
Then, in view of conditions (19), we have On the other hand it is not difficult to verify that u is a solution also of the system (1), with where due to inequalities (62) the following estimations are valid: which due to the assumption u C m 1 ,...,mn ≥ ρ 1 contradicts (61). Therefore our assumption is invalid and estimation (42) holds, and then from Proposition 3.2 the solvability of problem (2), (3) follows.
Also, it is clear that due to (26), for arbitrary x ∈ C m 1 ,...,m n (I; R n ), the conditions for t ∈ s ∈ I : x i+1 τ i (s) ≥ r 0 (i = 1, n) are fulfilled. On the other hand, if g i (t) := max f i (t, y) : |y| ≤ r 0 , g 0i (t) := ming i (t), g i (t) (i = 1, n), then, in view of (26) and the fact that the functions η i are nondecreasing in the second argument, we obtain Consequently, if we introduce the notations F i (x)(t) = f i (t, x i+1 (τ i (t))), we get that all the assumptions of Theorem 2.2 are valid, from which the validity of our corollary immediately follows.
From the last two inequalities and (21) it is clear that all the assumptions of Corollary 2.3 hold when the functions g i are defined by (31) and η i (t, x) = h i (t)r + max{|f (t, r)|, |f (t, -r)|} for t ∈ I, |x| > r. Therefore it remains to prove that problem (24), (3) has no more than one solution.