Mem. Differential Equations Math. Phys. 5(1995), 117–118

M. Ashordia

ON THE STABILITY OF SOLUTIONS OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR SYSTEMS OF GENERALIZED ORDINARY

DIFFERENTIAL EQUATIONS

(Reported on April 10 and 17, 1995)

Let A = (a_{ij})_{i,j=1}^{n} ∈ BV_{n×n}(a, b), f = (f_{i})_{i=1}^{n} ∈ K_{n}(a, b; A), and let h : BV_{n}(a, b) → R^{n} be a continuous operator such that the nonlinear boundary value problem

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

has a solution x^{∘}.

Consider sequences A_{k} ∈ BV_{n×n}(a, b) (k = 1, 2, …) and f_{k} ∈ K_{n}(a, b; A_{k}) (k = 1, 2 …) and a sequence of continuous operators h_{k} : BV_{n}(a, b) → R^{n} (k = 1, 2, …).

In this paper, sufficient conditions are given guaranteeing both the solvability of the problem

dx(t) = dA_{k}(t) · f_{k}(t, x(t)), h_{k}(x) = 0  (1_{k})

for any sufficiently large k and the convergence of its solutions as k → ∞ to the solution x^{∘} of the problem (1).

We use the following notation and definitions: R =] − ∞, ∞[, R_{+} = [0, ∞[; R^{n×m} is

the space of all real n × m-matrices X = (x_{ij})_{i,j=1}^{n,m} with the norm ∥X∥ = max{∑_{i=1}^{n} |x_{ij}| :

j = 1, …, m}; |X| = (|x_{ij}|)_{i,j=1}^{n,m}, [X]_{+} = (|X| + X)/2; I is the identity n × nmatrix; R^{n} = R^{n×1} is the space of all matrix-functions X = (x_{ij})_{i,j=1}^{n,m} : [a, b] → R^{n×m} such that var_{a}^{b} x_{ij} < +∞ with the norm ∥X∥_{s} = sup{∥X(t)∥ : t ∈ [a, b]}; V (X)(t) ≡ (var_{a}^{t} x_{ij})_{i,j=1}^{n,m}, d_{1}X(t) = X(t) − X(t − 0), d_{2}X(t) = X(t + 0) − X(t). U(y, r) = {x ∈ BV_{n}(a, b) : ∥x − y∥_{s} < r}; D(y, r) is the set of all x ∈ R^{n} such that inf{∥x − y(τ)∥ : τ ∈ [a, b]} < r.

If g ∈ BV_{1}(a, b), x : [a, b] → R and a ≤ s < t ≤ b, 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), where ∫_{]s,t[} x(τ)dg(τ) is the Lebesque–Stieltjes integral over the

open interval ]s, t[. If G(t) = (g_{ij}(t))_{i,j=1}^{l,n} and X(t) = (x_{jk}(t))_{j,k=1}^{n,m}, then

∫_{s}^{t} dG(τ) · X(τ) = ( ∑_{j=1}^{n} ∫_{s}^{t} x_{jk}(τ)dg_{ij}(τ))_{i,k=1}^{l,m}.

L_{n×m}(a, b; G) is the set of all matrix-functions (x_{jk}(t))_{j,k=1}^{n,m} such that x_{jk} is integrable with respect to g_{ij} (i = 1, …, l). K_{n×m}(a, b; G) is the Carathéodory class, i.e., the set of all mappings F = (f_{jk})_{j,k=1}^{n,m} : [a, b] × R^{n} → R^{n×m} such that (a) f_{ik}(·, x) is measurable

1991 Mathematics Subject Classification. 34B15.

Key words and phrases. System of generalized ordinary differential equations, nonlinear boundary value problem, stability. with respect to the measures V (g_{ij}) and V (g_{ij}) − g_{ij} for x ∈ R^{n} (i = 1, …, l); (b) F (t, ·) is continuous for t ∈ [a, b] and sup{|F (·, x)| : x ∈ D} ∈ L_{n×m}(a, b; G) for every compact D ⊂ R^{n}.

Inequalities between the matrices are understood to be componentwise. If B_{1} and B_{2} are normed spaces, then an operator g : B_{1} → B_{2} is called positively homogeneous if g(λx) = λg(x) for every λ ∈ R_{+} and x ∈ B_{1}.

A vector-function x ∈ BV_{n}(a, b) is said to be a solution of the problem (1) if h(x) = 0

and x(t) = x(s) + ∫_{s}^{t} dA(τ) · x(τ) for a ≤ s ≤ t ≤ b.

Let ℓ : BV_{n}(a, b) → R^{n} be a linear continuous operator and let ℓ_{0} : BV_{n}(a, b) → R_{+}^{n} be a positively homogeneous continuous operator. We say that a matrix-function P ∈ K_{n×n}(a, b; A) satisfies the Opial condition with respect to the triplet (ℓ, ℓ_{0}; A) if (a) there exists a matrix-function Φ ∈ L_{n×n}(a, b; A) such that |P (t, x)| ≤ Φ(t) on [a, b] × R^{n}; b) det(I + (−1)^{j}d_{j}B(t)) ≠ 0 for t ∈ [a, b] (j = 1, 2) and the problem dx(t) = dB(t) · x(t), |ℓ(x)| ≤ ℓ_{0}(x) has only trivial solution for every B ∈ BV_{n×n}(a, b) for which there

exists a sequence y_{k} ∈ BV_{n}(a, b) (k = 1, 2, …) such that lim_{k→∞} ∫_{a}^{t} dA(τ) · P (τ, y_{k}(τ)) =

B(t) uniformly on [a, b].

x^{∘} is said to be strongly isolated in the radius r if there exist P ∈ K_{n×n}(a, b; A), q ∈ K_{n}(a, b; A), a linear continuous operator ℓ : BV_{n}(a, b) → R^{n} and a positively homogeneous operator _{ℓ}: BV_{n}(a, b) → R^{n} such that (a) f(t, x) = P (t, x)x + q(t, x) for t ∈ [a, b], ∥x − x^{∘}(t)∥ < r and h(x) = ℓ(x) + _{ℓ}(x) is fulfilled on U(x^{∘}; r); (b) the vector-functions α(t, ρ) = max{|q(t, x)| : ∥x∥ ≤ ρ} and β(ρ) = sup{[|_{ℓ}(x)| − ℓ_{0}(x)]_{+} :

∥x∥_{s} ≤ ρ\} satisfy lim_{ρ→∞} \frac{1}{ρ} ∫_{a}^{b} dV (A)(t) · α(t, ρ) = 0, lim_{ρ→∞} β(ρ)/ρ = 0; (c) the problem

dx(t) = dA(t) · [P (t, x(t))x(t) + q(t, x(t))], ℓ(x) + _{ℓ}(x) = 0 has no solution different from x^{∘}; (d) P satisfies the Opial condition with respect to the triplet (ℓ, ℓ_{0}; A).

The notation ((A_{k}, f_{k}, h_{k}))_{k=1}^{∞} ∈ W_{r}(A, f, h; x^{∘}) means that (a) lim_{k→∞} ∫_{a}^{t} dA_{k}(τ) ×

f_{k}(τ, x) = ∫_{a}^{t} dA(τ) · f(τ, x) uniformly on [a, b] for every x ∈ D(x^{∘}; r); (b) lim_{k→∞} h_{k}(x) =

h(x) uniformly on U(x^{∘}; r); (c) lim_{s→0+} sup{∥ ∫_{a}^{b} dV (A_{k})(t) · ω_{k}(t, s)∥ : k = 1, 2, … } = 0 on

[a, b]×D(x^{∘}; r), where ω_{k}(t, s)=max{|f_{k}(t, x)−f_{k}(t, y)|:∥x∥, ∥y∥≤∥x^{∘}∥_{s}+r; ∥x−y∥≤s}.

The problem (1) is said to be (x^{∘}; r)-correct if for every ε ∈]0, r[ and ((A_{k}, f_{k}, h_{k}))_{k=1}^{∞} ∈ W_{r}(A, f, h; x^{∘}) there exists a natural number k_{0} such that the problem (1_{k}) has at least one solution contained in U(x^{∘}; r), and any such solution belongs to U(x^{∘}; r) for any k ≥ k_{0}.

Theorem. If the problem (1) has a solution x^{∘} which is strongly isolated in the radius r, then it is (x^{∘}; r)-correct.

Similar results for ordinary differential equations can be found in [1].

References

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

Author’s address:

A. Razmadze Mathematical Institute, Georgian Academy of Sciences

1, M. Aleksidze St., Tbilisi 380093, Georgia