Mem. Differential Equations Math. Phys. 16(1999), 150–153

M. Ashordia

ON EXISTENCE OF SOLUTIONS OF THE PERIODIC BOUNDARY VALUE PROBLEM FOR NONLINEAR SYSTEM OF GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS

(Reported on October 5, 1998)

Consider the ω-periodic problem for the nonlinear system

dx(t) = dA(t) · f(t, x(t)),  (1)

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

(a_{ik}^{(m)})_{i,k=1}^{n} ∈BV_{ω}(ℝ, ℝ^{n×n}) (m=1, 2) are nondecreasing on [0, ω], f(t, x)=(f_{k}(t, x))_{k=1}^{n}, the functions f_{k} : ℝ×ℝ^{n} → ℝ (k = 1, …, n) are periodic with respect to the first variable by period ω, i.e., f_{k}(t + ω, x) ≡ f_{k}(t, x) (k = 1, …, n), and their restriction on [0, ω] belong to K([0, ω] × ℝ^{n}, ℝ; a_{ik}) for every i ∈ {1, …, n}.

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}|;

ℝ_{+}^{n×m} = {(x_{ik})_{i,k=1}^{n,m} : x_{ik} ≥ 0 (i = 1, …, n; k = 1, …, m)};

ℝ^{n} = ℝ^{n×1} is the space of all real column n-vectors x = (x_{i})_{i=1}^{n}; ℝ_{+}^{n} = ℝ_{+}^{n×1}; δ_{ij} is the Kroneker symbol, i.e., δ_{ij} = 1 if i = j, δ_{ij} = 0 if i ≠ j.

BV([a, b], D), where D ⊂ 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 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),

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}(ℝ, ℝ^{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).

1991 Mathematics Subject Classification. 34B05.

Key words and phrases. Periodic boundary value problem, system of nonlinear generalized ordinary differential equations, Lebesgue-Stieltjes integral, solvability.

If x ∈ BV([0, ω], ℝ), 1 + (−1)^{j}d_{j}x(t) ≠ 0 for t ∈ [0, ω] (j = 1, 2) and λ(x)(ω) ≠ 1, where

λ(x)(t) = exp (s_{0}(x)(t)) ∏_{0≤τ<t} (1 + d_{2}x(τ))/ ∏_{0<τ≤t} (1 − d_{1}x(τ)),  (2)

then

g_{0}(x)(t, τ) = \begin{cases}(1 − λ(x)(ω))^{−1}λ(x)(ω) · λ(x)(t)[λ(x)(τ)]^{−1} & for 0 ≤ t ≤ τ ≤ ω, \\ (1 − λ(x)(ω))^{−1} · λ(x)(t)[λ(x)(τ)]^{−1} & for 0 ≤ τ < t ≤ ω; \end{cases} (3)
g_{j}(x)(t, τ) = (1 + (−1)^{j}d_{j}x(τ))^{−1}g_{0}(x)(t, τ) for t ≠ τ, t, τ ∈ [0, ω] (j = 1, 2) (4)

and

g_{j}(x)(t, t) = (1 + (−1)^{j}d_{j}x(t))^{−1} · λ^{j−2}(x)(ω)g_{0}(x)(t, t) for t ∈ [0, ω] (j = 1, 2). (5)

If g : [a, b] → ℝ is a nondecreasing function, x : [a, b] → ℝ and a ≤ s < t ≤ b, then

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

where g_{k} : [a, b] → ℝ (k = 1, 2) are nondecreasing functions such that

g_{1}(t) − g_{2}(t) ≡ s_{0}(g)(t)

and ∫_{]s,t[} x(τ) dg_{k}(τ) is the Lebesgue–Stieltjes integral over the open interval ]s, t[ with re-

spect to the measure μ(g_{k}), corresponding to the function g_{k} (if s=t, then ∫_{s}^{t} x(τ)dg(τ)= 0); L([a, b], ℝ; g) is the set of all μ(g_{k})-measurable (i.e., measurable with respect to the measures μ(g_{1}) and μ(g_{2})) functions x : [a, b] → ℝ such that ∫_{a}^{b} |x(t)|dg(t) < +∞.

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

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

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

If D_{1} ⊂ ℝ^{n} and D_{2} ⊂ ℝ^{n×m}, then K([a, b] × D_{1}, D_{2}; G) is the Caratheodory class, 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] → D_{2} is μ(g_{ik})-measurable for every x ∈ D_{1}; (b) the function f_{kj}(t, ·) : D_{1} → D_{2} is continuous for μ(g_{ik})-almost all 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_{j} : [a, b] → ℝ^{l×n} (j = 1, 2) are nondecreasing matrix-functions, G = G_{1} − G_{2} and X : [a, b] → ℝ^{n×m}, then

∫_{s}^{t} dG(τ) · X(τ) = ∫_{s}^{t} d G_{1}(τ) · X(τ) − ∫_{s}^{t} dG_{2}(τ) · X(τ) for a ≤ s ≤ t ≤ b.

Inequalities between both vectors and matrices are understood componentwise.

A vector-function x : ℝ → ℝ^{n} is said to be a solution of the system (1) (of the system os generalized ordinary differential inequalities dx(t) ≤ dA(t) · f(t, x(t)) 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(t)) = 0 (≤ 0) for s < t (s, t ∈ ℝ).

If σ_{i} ∈ \{−1, 1\}, then t_{i} = \frac{1−σ_{i}}{2} ω, j_{i} = \frac{3+σ_{i}}{2} , and I_{ω}^{σ_{i}} = [0, ω]\\{t_{i}\}.

Definition 1. Let σ_{i} ∈ {−1, 1} (i = 1, …, n). We shall say that a matrix-function C = (c_{il})_{i,l=1}^{n} : ℝ → ℝ^{n×n} belongs to the set U_{ω}^{σ_{1},...,σ_{n}} if c_{il} ∈ BV_{ω}(ℝ, ℝ) (i, l = 1, …, n), the functions c_{il} (i ≠ l; i, l = 1, …, n) are nondecreasing on [0, ω] and continuous at the point t_{i},

d_{j}c_{ii}(t_{i}) ≥ 0 (j = 1, 2; i = 1, …, n),  (6)
∥d_{j}C(t)∥ < 1 for t ∈ [0, ω] (j = 1, 2)  (7)

and the system

σ_{i}dy_{i}(t) ≤ ∑_{l=1}^{n} y_{l}(t)dc_{il}(t) (i = 1, …, n)

has no nontrivial nonnegative ω-periodic solution.

Theorem 1. Let the conditions

(−1)^{m+1}σ_{i}f_{k}(t, x_{1}, …, x_{n}) sign x_{i} ≤ ∑_{l=1}^{n} p_{mikl}(t)|x_{l}| + q_{k}(t)

for μ(a_{ik}^{(m)})- almost all t ∈ I_{ω}^{σ_{i}} (i, k = 1, …, n)  (8)

and

[(−1)^{m+1}σ_{i}f_{k}(t_{i}, x_{1}, …, x_{n}) sign x_{i} − ∑_{l=1}^{n} α_{mikjl}|x_{l}| − q_{k}(t_{i})]d_{j_{i}}a_{ik}^{(m)}(t_{i}) ≤ 0

(j = 1, 2; i, k = 1, …, n)  (9)

be fulfilled on ℝ^{n} for every m ∈ {1, 2}, where σ_{i} ∈ {−1, 1}, α_{mikl} ∈ ℝ, (p_{mikl})_{k,l=1}^{n} ∈ L([0, ω], ℝ^{n×n}; A^{(m)}), q = (q_{k})_{k=1}^{n} ∈ L([0, ω], ℝ_{+}^{n}; A). Let, moreover,

∑_{m=1}^{2} ∑_{k=1}^{n} ∫_{s}^{t} p_{mikl}(τ)da_{ik}^{(m)}(τ)≤c_{il}(t) − c_{il}(s) for s < t; s, t ∈ I_{ω}^{σ_{i}} (i, l = 1, …, n) (10)

and

∑_{m=1}^{2} ∑_{k=1}^{n} α_{mikl}d_{j_{i}}a_{ik}^{(m)}(t_{i}) ≤ δ_{il}d_{j_{i}}c_{ii}(t_{i}) (i, l = 1, …, n),  (11)

where C = (c_{il})_{i,l=1}^{n} ∈ U_{ω}^{σ_{1},...,σ_{n}}. Then the system (1) has at least one ω-periodic solution.

Corollary 1. Let the conditions (6)–(11) and

σ_{i}λ(σ_{i}c_{ii})(ω) < 1 (i = 1, …, n)

be fulfilled on ℝ^{n} for every m ∈ {1, 2}, where σ_{i} ∈ {−1, 1}, α_{mikl} ∈ ℝ, (p_{mikl})_{k,l=1}^{n} ∈ L([0, ω], ℝ^{n×n}; A^{(m)}), q = (q_{k})_{k=1}^{n} ∈ L([0, ω], ℝ_{+}^{n}; A), C = (c_{il})_{i,l=1}^{n} ∈ BV_{ω}(ℝ, ℝ^{n×n}), the functions c_{il} (i ≠ l) are nondecreasing on [0, ω] and continuous at the point t_{i}, the function λ(σ_{i}c_{ii}) is defined by (2). Let, moreover, the module of every characteristic value of the matrix S = (s_{il})_{i,l=1}^{n} be less than 1, where

s_{ii} = 0, s_{il} = sup { ∑_{j=0}^{2} ∫_{0}^{ω} σ_{i}g_{j}(σ_{i}c_{ii})(t, τ)ds_{j}(c_{il})(τ) : t ∈ [0, ω]}

(i ≠ l; i, l = 1, …, n),

g_{j} (j = 0, 1, 2) are operators defined by (3)–(5). Then the conclusion of Theorem 1 is true.

Corollary 2. Let the conditions (8)–(11),

c_{il}(t) = η_{il}α_{i}(t) for t ∈ ℝ (i, l = 1, …, n)

and

(|η_{ii}| + ∑_{l≠i,l=1}^{n} η_{il})d_{j}α_{i}(t) < 1 for t ∈ [0, ω] (j = 1, 2; i = 1, …, n)

be fulfilled on ℝ^{n} for every m ∈ {1, 2}, where σ_{i} ∈ {−1, 1}, α_{mikl} ∈ ℝ, (p_{mikl})_{k,l=1}^{n} ∈ L([0, ω], ℝ^{n×n}; A^{(m)}), q = (q_{k})_{k=1}^{n} ∈ L([0, ω], ℝ_{+}^{n}; A), η_{ii} ∈ ℝ, η_{il} ∈ ℝ_{+} (i ≠ l), α_{i} ∈ BV_{ω}(ℝ, ℝ) is nondecreasing on [0, ω] and continuous at the point t_{i}, α_{i}(ω) ≠ 0. Let, moreover, the real part of every characteristic value of the matrix H = (η_{il})_{i,l=1}^{n} be negative. Then the conclusion of Theorem 1 is true.

Corollary 3. Let the conditions (6), (7), and (8)–(11) be fulfilled on ℝ^{n} for every m ∈ {1, 2}, where σ_{1} = ⋯ = σ_{n} = σ_{0}, σ_{0} ∈ {−1, 1}, α_{mikl} ∈ ℝ, (p_{mikl})_{k,l=1}^{n} ∈ L([0, ω], ℝ^{n×n}; A^{(m)}), q = (q_{k})_{k=1}^{n} ∈ L([0, ω], ℝ_{+}^{n}; A), C = (c_{il})_{i,l=1}^{n} ∈ BV_{ω}(ℝ, ℝ^{n×n}), the functions c_{il} (i ≠ l) are nondecreasing on [0, ω] and continuous at the point t_{i}. Let, moreover, the module of every multiplicator of the system dy(t) = dC_{σ_{0}}(t) · y(t), where C_{σ_{0}}(t) = σ_{0}C(σ_{0}t + \frac{1−σ_{0}}{2} ω), be less than 1. Then the conclusion of Theorem 1 is true.

Acknowledgement

This paper was supported by INTAS Grant 96-1060.

References

1. I. T. Kiguradze, Boundary value problems for systems of ordinary differential equations. (Russian) Current Problems in Mathematics. Newest results, vol. 30, 3–103; Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuzn. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987.

Author’s address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Aleksidze St., Tbilisi 380093

Georgia