Mem. Differential Equations Math. Phys. 11(1997), 159–162

M. Ashordia

ON THE QUESTION OF SOLVABILITY OF THE PERIODIC BOUNDARY VALUE PROBLEM FOR A SYSTEM OF GENERALIZED

ORDINARY DIFFERENTIAL EQUATIONS

(Reported on February 10-17, 1997)

In the present note sufficient conditions are given for the solvability of the ω-periodic boundary value problem

dx(t) = dA(t) · f(t, x(t)),  (1)
x(0) = x(ω),  (2)

where ω is a positive number, A(t) = (a_{ik}(t))_{i,k=1}^{n}, a_{ik}(t) ≡ a_{ik}^{(1)}(t) − a_{ik}^{(2)}(t), a_{ik}^{(σ)} : R →

R (σ = 1, 2) are functions nondecreasing on [0, ω], a_{ik}^{(σ)}(t + ω) = a_{ik}^{(σ)}(t) + a_{ik}^{(σ)}(ω) for t ∈ R; f = (f_{k})_{k=1}^{n} : R × R^{n} → R^{n} is an ω-periodic with respect to the first variable vector-function such that its restriction on [0, ω]×R^{n} belongs to the Carathéodory classes corresponding to the matrix-functions A^{(1)} and A^{(2)}, A^{(σ)}(t) ≡ (a_{ik}^{(σ)}(t))_{i,k=1}^{n} (σ = 1, 2).

The following notation and definitions will be used: R =] − ∞, +∞[, R_{+} = [0, +∞[, [a, b] (a, b ∈ R) is a closed segment, R^{n×m} is the space of all real n × m-matrices X =

(x_{ik})_{i,k=1}^{n,m} with the norm ∥X∥ = max_{k=1,...,m} ∑_{i=1}^{n} |x_{ik}|; |X| = (|x_{ik}|)_{i,k=1}^{n,m}; if X ∈ R^{n×n},

then det(X) is the determinant of X, I_{n} is the identity n × n-matrix; R^{n} = R^{n×1} is the space of all real column n-vectors x = (x_{i})_{i=1}^{n}.

BV([a, b], R^{n×m}) is the set of all matrix-functions X = (x_{ik})_{i,k=1}^{n,m} : [a, b] → R^{n×m} such that every its component x_{ik} has a bounded total variation on [a, b].

s_{k} : BV([a, b], R) → BV([a, b], R) (k = 0, 1, 2) are the operators defined by s_{1}(x)(a) = s_{2}(x)(a) = 0,

s_{1}(x)(t) = ∑_{a<τ≤t} d_{1}x(τ) and s_{2}(x)(t) = ∑_{a≤τ<t} d_{2}x(τ) for t ∈]a, b],

s_{0}(x)(t) ≡ x(t) − s_{1}(x)(t) − s_{2}(x)(t).

BV_{ω}^{n×m} is the set of all matrix-functions X : R → R^{n×m} such that X(t+ω) = X(t)+ X(ω) for t ∈ R, and its restriction on [0, ω] belongs to BV([0, ω], R^{n×m}); X(t−) and X(t+) are the left and the right limits of X at the point t ∈ R; d_{1}X(t) = X(t) − X(t−), d_{2}X(t) = X(t+) − X(t).

If g : R → R is nondecreasing on the interval I ⊂ R, x : R → R and s < t (s, t ∈ I), then

∫_{s}^{t} x(τ) dg(τ) = ∫_{]s,t[} x(τ) dg(τ) + x(t)d_{1}g(t) + x(s)d_{2}g(s),

1991 Mathematics Subject Classification. 34B05.

Key words and phrases. Periodic problem, generalized ordinary differential equations, Lebesgue–Stieltjes integral, solvability. where ∫_{]s,t[} x(τ) dg(τ) is the Lebesgue–Stieltjes integral over the open interval ]s, t[ with

respect to the measure μ_{g} corresponding to the function g (if s = t, then ∫_{s}^{t} x(τ)dg(τ) =

0); L([a, b], R; g) is the set of all μ_{g}-measurable functions x : [0, ω] → R such that

∫_{a}^{b} |x(t)|dg(t) < +∞.

A matrix-function is said to be nondecreasing if every of its components is such.

If G = (g_{ik})_{i,k=1}^{l,n} : R → R^{l×n} is a matrix-function nondeacreasing on the interval I ⊂ R and X = (x_{kj})_{k,j=1}^{n,m} : R → R^{n×m}, then

∫_{s}^{t} dG(τ) · X(τ) = ( ∑_{k=1}^{n} ∫_{s}^{t} x_{kj}(τ) dg_{ik}(τ))_{i,j=1}^{l,m} for s ≤ t (s, t ∈ I);

L([a, b], R^{n×m}; G) is the set of all matrix-functions X = (x_{kj})_{k,j=1}^{n,m} : [a, b] → R^{n×m} such that x_{kj} ∈ L([a, b], R; g_{ik}) (i = 1, …, l; k = 1, …, n; j = 1, …, m);

K([a, b]×D_{1}, D_{2}; G) (D_{1} ⊂ R^{n}, D_{2} ⊂ R^{n×m}) is the Carathéodory class corresponding to G, i.e., the set of all mappings F = (f_{kj})_{k,j=1}^{n,m} : [a, b] × D_{1} → D_{2} such that for each i ∈ {1, …, l}, j ∈ {1, …, m} and k ∈ {1, …, n}: a) the function f_{kj}(·, x) : [a, b] → R is μ_{g_{ik}} -measurable for every x ∈ D_{1}; b) the function f_{kj}(t, ·) : D_{1} → R is continuous for μ_{g_{ik}} -almost every t ∈ [a, b], and sup{|f_{kj}(·, x)| : x ∈ D_{0}} ∈ L([a, b], R; g_{ik}) for every compact D_{0} ⊂ D_{1}.

If G^{(σ)} : R → R^{l×n} (σ = 1, 2) are matrix-functions nondecreasing on the interval I ⊂ R, G = G^{(1)} − G^{(2)} and X : R → R^{n×m}, then

∫_{s}^{t} dG(τ) · X(τ) = ∫_{s}^{t} dG^{(1)}(τ) · X(τ) − ∫_{s}^{t} dG^{(2)}(τ) · X(τ) for s ≤ t (s, t ∈ I).

An ω-periodic vector-function x : R → R^{n} is said to be a solution of the problem (1),(2) if its restriction on [s, t] belongs to BV([s, t], R^{n}) and

x(t) = x(s) + ∫_{s}^{t} dA(τ) · f(τ, x(τ)) for s < t (s, t ∈ R).

Let B = (b_{ik})_{i,k=1}^{n} ∈ BV_{ω}^{n×n} and let natural numbers m, r_{1}, …, r_{m} and n_{1}, …, n_{m} (0 = n_{0} < n_{1} < ⋯ < n_{m} = n), nondecreasing functions g_{lj} : [0, ω] → R (l = 1, …, r_{j}; j = 1, …, m), functions α_{lj} ∈ L([0, ω], R; g_{lj}) (l = 1, …, r_{j}; j = 1, …, m) and matrixfunctions P_{lj} = (p_{ljik})_{i,k=1}^{n} (l = 1, …, r_{j}; j = 1, …, m), p_{ljik} ∈ L([0, ω], R; g_{lj}) (i, k = n_{j−1} + 1, …, n_{j}) be such that b_{ik}(t) ≡ 0 (i = n_{j−1} + 1, …, n_{j}; k = n_{j} + 1, …, n; j = 1, …, m − 1),

σ_{j} ∑_{i,k=n_{j−1}+1}^{n_{j}} p_{ljik}(t)x_{i}x_{k} ≥ α_{lj}(t) ∑_{i=n_{j−1}+1}^{n_{j}} x_{i}^{2}

for μ_{g_{lj}} -almost everywhere t ∈ [0, ω], (x_{i})_{i=1}^{n} ∈ R^{n} (l = 1, …, r_{j}; j = 1, …, m),

σ_{i}(b_{jii}(t) − b_{jii}(s) − ∑_{l=1}^{r_{j}} ∫_{s}^{t} p_{ljii}(τ) dg_{lj}(τ)) ≥ 0

for 0 ≤ s ≤ t ≤ ω (i = n_{j−1} + 1, …, n_{j}; j = 1, …, m)

and

b_{jik}(t) = ∑_{l=1}^{r_{j}} ∫_{0}^{t} p_{ljik}(τ) dg_{lj}(τ) for t ∈ [0, ω]

(i ≠ k; i, k = n_{j−1} + 1, …, n_{j}; j = 1, …, m),

where σ_{j} ∈ {−1, 1} (j = 1, …, m) and

b_{jik}(t) ≡ b_{ik}(t) − \frac{1}{2}( ∑_{0<τ≤t} ∑_{σ=n_{j−1}+1}^{n_{j}} d_{1}b_{σi}(τ) · d_{1}b_{σk}(τ) −

− ∑_{0≤τ<t} ∑_{σ=n_{j−1}+1}^{n_{j}} d_{2}b_{σi}(τ) · d_{2}b_{σk}(τ)) (i, k = n_{j−1} + 1, …, n_{j}; j = 1, …, m).

Then we will say that

B ∈ Q_{ω}^{n×n}(m, (r_{j}, n_{j}, (g_{lj}, α_{lj}, P_{lj})_{l=1}^{r_{j}})_{j=1}^{m}).  (3)

Theorem 1. Let the conditions (3),

|f(t, x) − P(t)x| ≤ q(t, ∥x∥),
det (I_{n} + (−1)^{σ}d_{σ}A(t) · P(t)) ≠ 0 (σ = 1, 2),

lim_{ρ→+∞} \frac{1}{ρ} ∫_{0}^{ω} d(A^{(1)}(t) + A^{(2)}(t)) · q(t, ρ) = 0,  (4)

(1 + σ_{j})d_{1}g_{j}(t) + (1 − σ_{j})d_{2}g_{j}(t) < 2,  (5)
(1 − σ_{j})d_{1}g_{j}(t) + (1 + σ_{j})d_{2}g_{j}(t) ≠ −2  (6)

and

exp (s_{0}(g_{j})(ω)) > \frac{1}{2}[(1 + σ_{j}) ∏_{0<τ≤ω}(1 − d_{1}g_{j}(τ)) ∏_{0≤τ<ω}(1 + d_{2}g_{j}(τ))^{−1} +

+ (1 − σ_{j}) ∏_{0<τ≤ω}(1 + d_{1}g_{j}(τ))^{−1} ∏_{0≤τ<ω}(1 − d_{2}g_{j}(τ))]  (7)

be fulfilled on [0, ω]×R^{n} for every j ∈ {1, …, m}, where σ_{j} ∈ {−1, 1}; m, r_{1}, …, r_{m} and n_{1}, …, n_{m} (0 = n_{0} < n_{1} < ⋯ < n_{m} = n) are natural numbers; α_{lj} ∈ L([0, ω], R; g_{lj}), P_{lj} = (p_{ljik})_{i,k=1}^{n}, p_{ljik} ∈ L([0, ω], R; g_{lj}), g_{lj} : [0, ω] → R is nondecreasing (i, k =

n_{j−1} + 1, …, n_{j}); P ∈ ∩^{2}_{σ=1}L([0, ω], R^{n×n}; A^{(σ)}); B(t) = ∫_{0}^{t} dA(τ) · P(τ) for t ∈ [0, ω],

B(t + ω) = B(t) + B(ω); g_{j}(t) ≡ 2 ∑_{l=1}^{r_{j}} ∫_{0}^{t} α_{lj}(τ)dg_{lj}(τ); q ∈ ∩^{2}_{σ=1}K([0, ω] × R_{+}, R_{+}^{n}; A^{(σ)})

is a vector-function nondecreasing in the second variable. Then the problem (1),(2) is solvable.

Theorem 2. Let the conditions (3)–(7) be fulfilled. Let, moreover,

|f(t, x) − P(t, x)x| ≤ q(t, ∥x∥),

σ_{j} ∑_{i,k=n_{j−1}+1}^{n_{j}} ( ∑_{ν=1}^{n} a_{liν}(τ)p_{νk}(τ, y)x_{i}x_{k} ≥ α_{lj}(τ) ∑_{i=n_{j−1}+1}^{n_{j}} x_{i}^{2}

for μ_{g_{lj}} -almost every τ ∈ [0, ω] (l = 1, …, r_{j}),

(−1)^{σ}σ_{j} ∑_{i,k,l=n_{j−1}+1}^{n_{j}} ( ∑_{ν,μ=1}^{n} p_{νi}(t, y)p_{μk}(t, y)d_{σ}a_{lν})t) · d_{σ}a_{lμ}(t))x_{i}x_{k} ≥

≥ d_{σ}β_{j}(t) ∑_{i=n_{j−1}+1}^{n_{j}} x_{i}^{2} (σ = 1, 2),

|p_{ik}(t, y)| ≤ φ_{ik}(t) (i, k = 1, …, n)

and

∑_{j=1}^{m} ∑_{i,k=n_{j−1}+1}^{n_{j}} ∑_{l=1}^{n} φ_{lk}(t)d_{σ}(a_{il}^{(1)}(t) + a_{il}^{(2)}(t)) < 1 (σ = 1, 2)

be fulfilled for (t, x, y) ∈ [0, ω] × R^{2n} (j = 1, …, m) where σ_{j} ∈ {−1, 1}; m, r_{1}, …, r_{m} and n_{1}, …, n_{m} (0 = n_{0} < n_{1} < ⋯ < n_{m} = n) are natural numbers; β_{j} ∈ BV([0, ω], R); a_{lik} and α_{lj} belong to L([0, ω], R; g_{lj}), g_{lj} : [0, ω] → R are nondecreasing (i, k = n_{j−1} +

1, …, n_{j}); P = (p_{ik})_{i,k=1}^{n} ∈ ∩^{2}_{σ=1}K([0, ω] × R^{n}, R^{n×n}; A^{(σ)}), (φ_{ik})_{i,k=1}^{n} ∈ ∩^{2}_{σ=1}L([0, ω],

R_{+}^{n×n}; A^{(σ)}); a_{ik}(t) = ∑_{l=1}^{n} ∫_{0}^{t} a_{lik}(τ)dg_{lj}(τ) and ∑_{l=1}^{n} ∫_{0}^{t} p_{lk}(τ, y_{1}, …, y_{n})da_{il}(τ) = 0 for

t ∈ [0, ω] (i = n_{j−1} + 1, …, n_{j}; k = n_{j} + 1, …, n; j = 1, …, m − 1); g_{j}(t) ≡

2 ∑_{l=1}^{r_{j}} ∫_{0}^{t} α_{lj}(τ)dg_{lj}(τ) + ∑_{σ=1}^{2} s_{σ}(β_{j})(t); and q ∈ ∩^{2}_{σ=1}K([0, ω] × R_{+}, R_{+}^{n}; A^{(σ)}) is a vector-

function nondecreasing in the second variable. Then the problem (1),(2) is solvable.

The analogous question has been considered in [1] for a system of ordinary differential equations.

References

1. I. T. Kiguradze, Boundary value problems for systems of ordinary differential equations. (Russian) Itogi Nauki i Tekhniki. VINITI, 30(1987), 3–103.

Author’s address:

Sukhumi Branch of Tbilisi State University

19, Al. Chavchavadze St., Tbilisi 380049

Georgia