Mem. Differential Equations Math. Phys. 15(1998), 145–149

Z. Sokhadze

ON THE SOLVABILITY OF THE WEIGHTED INITIAL VALUE

PROBLEM FOR HIGH ORDER EVOLUTION SINGULAR FUNCTIONAL DIFFERENTIAL EQUATIONS

(Reported on April 20, 1998)

In the present paper on the basis of the results obtained in [1, 2] optimal, in a certain sense, sufficient conditions for solvability of the weighted initial value problem

u^{(n)}(t) = f(u)(t),  (1)
lim_{t→a} \frac{u^{(k)}(t)}{h^{(k)}(t)} = 0 (k = 0, …, n − 1)  (2)

are established, where f ∈ C^{n−1}([a, b]; ℝ^{m}) → L_{loc}(]a, b]; ℝ^{m}) is a continuous Volterra operator and h : [a, b] → [0, +∞[ is an (n − 1)-times continuously differentiable function such that

h^{(k)}(a) = 0 (k = 0, …, n − 2), h^{(n−1)}(t) > 0 for a < t ≤ b.  (3)

The problem (1), (2) for the case n = 1 has been investigated in [1, 2]. Therefore below we will assume that n ≥ 2.

Throughout the paper the use will be made of the following notation. ℝ^{m} is the space of m-dimensional column vectors x = (x_{i})_{i=1}^{m} with real components

x_{i} (i = 1, …, m) and the norm ∥x∥ = ∑_{i=1}^{m} |x_{i}|.

ℝ_{ρ}^{m} = {x ∈ ℝ^{n} : ∥x∥ ≤ ρ}. If x = (x_{i})_{i=1}^{m} ∈ ℝ^{m}, then sgn(x) = (sgn x_{i})_{i=1}^{m}. x · y is the scalar product of the vectors x and y ∈ ℝ^{m}. C^{n−1}([a, b]; ℝ^{m}) is the space of (n − 1)-times continuously differentiable vector func-

tions x : [a, b] → ℝ^{m} with the norm

∥x∥_{Cn−1} = max { ∑_{k=1}^{n−1} ∥x^{(k−1)}(t)∥ : a ≤ t ≤ b}.

C_{h}^{n−1}([a, b]; ℝ^{m}) is the set of u ∈ C^{n−1}([a, b]; ℝ^{m}) such that

sup \{\frac{∥u^{(k)}(t)∥}{h^{(k)}(t)} : a < t ≤ b\} < +∞ (k = 0, …, n − 1).

C_{h,ρ}^{n−1}([a, b]; ℝ^{m}) is the set of u ∈ C^{n−1}([a, b]; ℝ^{m}) satisfying the inequalities

|u^{(k)}(t)| ≤ ρh^{(k)}(t) for a < t ≤ b (k = 0, …, n − 1).

1991 Mathematics Subject Classification. 34K10. Key words and phrases. Evolution singular functional differential equation, initial

value problem, local solvability, global solvability.

If x : ]a, b] → ℝ^{m} is a bounded function and a ≤ s < t ≤ b, then

ν(x)(s, t) = sup {∥x(ξ)∥ : s < ξ < t}.

L_{loc}(]a, b]; ℝ^{m}) is the space of vector functions x : ]a, b] → ℝ^{m} which are summable on each segment from ]a, b] with the topology of convergence in the mean on each segment from ]a, b].

Definition 1. f : C^{n−1}([a, b]; ℝ^{m}) → L_{loc}(]a, b]; ℝ^{m}) is called a Volterra operator if the equality f(x)(t) = f(y)(t) holds almost everywhere on ]a, t_{0}[ for any t_{0} ∈]a, b] and any vector functions x and y ∈ C^{n−1}([a, b]; ℝ^{m}) satisfying the condition x(t) = y(t) for a ≤ t ≤ t_{0}.

Definition 2. We will say that the operator f : C^{n−1}([a, b]; ℝ^{m}) → L_{loc}(]a, b]; ℝ^{m}) satisfies the local Carathéodory conditions if it is continuous and there exists a nondecreasing with respect to the second argument function γ : ]a, b] × [0, +∞[ → [0, +∞[ such that γ(·, ρ) ∈ L_{loc}(]a, b]; ℝ) for any ρ ∈]0, +∞[ , and the inequality

∥f(x)(t)∥ ≤ γ(t, ∥x∥_{Cn−1} )

is fulfilled for any x ∈ C^{n−1}([a, b]; ℝ^{m}) almost everywhere on ]a, b[ .

Definition 3. If f : C^{n−1}([a, b]; ℝ^{m}) → L_{loc}(]a, b]; ℝ^{m}) is a Volterra operator and b_{0} ∈]a, b], then:

(i) for any u ∈ C^{n−1}([a, b_{0}]; ℝ^{m}) by f(u) is understood the vector function given by the equality f(u)(t) = f(\overline{u})(t) for a ≤ t ≤ b_{0}, where

\overline{u}(t) = \begin{cases}u(t) & for a ≤ t ≤ b_{0}, \\ ∑_{k=1}^{n} \frac{(t − b_{0})^{k−1}}{(k − 1)!} u^{(k−1)}(b_{0}) & for b_{0} < t ≤ b; \end{cases}

(ii) a function u ∈ C^{n−1}([a, b_{0}]; ℝ^{m}) is called a solution of the equation (1) on the segment [a, b_{0}] if u^{(n−1)} is absolutely continuous on each segment contained in ]a, b_{0}] and u^{(n)}(t) = f(u)(t) almost everywhere on ]a, b_{0}[ ;

(iii) a solution u of the equation (1) on the segment [a, b_{0}], satisfying the initial conditions (2) is called a solution of the problem (1), (2) on the segment [a, b_{0}].

Definition 4. The problem (1), (2) is said to be locally solvable (globally solvable) if it has at least one solution on a segment [a, b_{0}] ⊂ [a, b[ (on the segment [a, b]).

In what follows, we will assume that f : C^{n−1}([a, b]; ℝ^{m}) → L_{loc}(]a, b]; ℝ^{m}) is a continuous Volterra operator satisfying the local Carathéodory conditions.

Theorem 1. Let there exist a positive number ρ and summable functions p_{k} : [a, b] → [0, +∞[ (k = 0, …, n − 1) and q : [a, b] → [0, +∞[ such that

lim sup_{t→a} (\frac{1}{h^{(n−1)}(t)} ∑_{k=0}^{n−1} ∫_{a}^{t} p_{k}(s) ds) < 1, lim_{t→a} (\frac{1}{h^{(n−1)}(t)} ∫_{a}^{t} q(s) ds) = 0 (4)

and for any u ∈ C_{h,ρ}^{n−1}([a, b]; ℝ^{m}) the inequality

f(u)(t) · sgn(u^{(n−1)}(t)) ≤ ∑_{k=0}^{n−1} p_{k}(t)ν(\frac{u^{(k)}}{h^{(k)}} )(a, t) + q(t)  (5)

is fulfilled almost everywhere on ]a, b[ . Then the problem (1), (2) is locally solvable. Proof. For any x ∈ C([a, b]; ℝ^{m}) assume

w(x)(t) = \frac{1}{(n − 2)!} ∫_{a}^{t}(t − s)^{n−2}x(s) ds, _{f}(x)(t) = f(w(x))(t).  (6)

Then by (3)

h(t) = w(h^{(n−1)})(t) = \frac{1}{(n − 2)!} ∫_{a}^{t}(t − s)^{n−2}h^{(n−1)}(s) ds.  (7)

Obviously, _{f}: C([a, b]; ℝ^{m}) → L_{loc}(]a, b]; ℝ^{m}) is a continuous Volterra operator satisfying the local Carathéodory conditions.

Assume now that y ∈ C([a, b]; ℝ^{m}), ∥y∥_{C} ≤ ρ and

u(t) = w(h^{(h−1)}y)(t).  (8)

Then by virtue of (6) and (7)

∥u^{(k)}(t)∥ = \frac{1}{(n − 2 − k)!} ∥ ∫_{a}^{t}(t − s)^{n−2−k}h^{(n−1)}(s)y(s) ds∥ ≤

≤ \frac{1}{(n − 2 − k)!} ( ∫_{a}^{t}(t − s)^{n−2−k}h^{(n−1)}(s) ds)ν(y)(a, t) =

= h^{(k)}(t)ν(y)(a, t) for a < t ≤ b (k = 0, …, n − 2),

and

u^{(n−1)}(t) = h^{(n−1)}(t)y(t), ∥u^{(n−1)}(t)∥ ≤ h^{(n−1)}(t)ν(y)(a, t) for a < t ≤ b.

Therefore

u ∈ C_{h,ρ}^{n−1}([a, b]), sgn(u^{(n−1)}(t)) = sgn(y(t)),  (9)
ν(\frac{u^{(k)}}{h^{(k)}} )(a, t) ≤ ν(y)(a, t) for a < t ≤ b (k = 0, …, n − 1).  (10)

On the basis of the conditions (5), (6) and (8)–(10), almost everywhere on ]a, b[ the inequality

_{f}(h^{(n−1)}y)(t) · sgn(y(t)) = f(u)(t) · sgn(u^{(n−1)}(t)) ≤

≤ ∑_{k=0}^{n−1} p_{k}(t)ν(\frac{u^{(k)}}{h^{(k)}} )(a, t) + q(t) ≤ ∑_{k=0}^{n−1} p_{k}(t)ν(y)(a, t) + q(t),

is fulfilled, that is,

_{f}(h^{(n−1)}y)(t) sgn(y(t)) ≤ p(t)ν(y)(a, t) + q(t), where p(t) = ∑_{k=0}^{n−1} p_{k}(t).

On the other hand, as it follows from (4),

lim sup_{t→a} (\frac{1}{h^{(n−1)}(t)} ∫_{a}^{t} p(s) ds) < 1.

Hence all the conditions of Theorem 2.1 from [1] are fulfilled for the problem

\frac{dx(t)}{dt} = _{f}(x)(t), lim_{t→a} \frac{x(t)}{h^{(n−1)}(t)} = 0.  (11)

Therefore this problem is locally solvable.

Let x be a solution of the problem (11) on a segment [a, b_{0}], and u(t) = w(x)(t). Then, owing to (6), the function u is a solution of the problem (1), (2) on [a, b_{0}].

Applying Corollary 1 of [2] and repeating the arguments used in proving Theorem 1, we convince ourselves that the following theorem is valid.

Theorem 2. Let for any u ∈ C_{h}^{n−1}([a, b]; ℝ^{n}) the inequality

f(u)(t) · sgn(u^{(n−1)}(t)) ≤ ∑_{k=0}^{n−1} p_{k}(t, ρ_{0}(u)(t))ν(\frac{u^{(k)}}{h^{(k)}} )(a, t) + q(t, ρ_{0}(u)(t))

be fulfilled almost everywhere on ]a, b[ , where

ρ_{0}(u)(t) = ∑_{j=0}^{n−1} ν(\frac{u^{(j)}}{h^{(j)}} )(a, τ(t)),

τ : [a, b] → [a, b] is a continuous function, p_{k} (k = 0, …, n − 1) and q : [a, b] × [0 + ∞[ → [0, +∞[ are summable with respect to the first argument and continuous and nondecreasing with respect to the second argument. Let furthermore τ(t) < t for a < t ≤ b and

lim sup_{t→a} (\frac{1}{h^{(n−1)}(t)} ∑_{k=0}^{n−1} ∫_{a}^{t} p_{k}(s, ρ) ds) < 1, lim_{t→a} (\frac{1}{h^{(n−1)}(t)} ∫_{a}^{t} q(s, ρ) ds) = 0

for some positive constant ρ. Then the problem (1), (2) is globally solvable. A particular case of the equation (1) is the vector differential equation with delay

u^{(n)}(t) = f_{0}(t, u(τ_{10}(t)), ..., u^{(n−1)}(τ_{1 n−1}(t)), ..., u(τ_{l0}(t)), ..., u^{(n−1)}(τ_{l n−1}(t))), (12)

where f_{0} : ]a, b] × ℝ^{lmn} → ℝ^{m} satisfies the local Carathéodory conditions, and τ_{ik} : [a, b] → [a, b] are measurable functions such that τ_{ik}(t) ≤ t for a ≤ t ≤ b (i = 1, …, l; k = 0, …, n − 1).

Theorems 1 and 2 result in the following

Corollary 1. Let τ_{l n−1}(t) ≡ t and there exist a positive number ρ, summable functions p_{ik} : [a, b] → [0, +∞[ (i = 1, …, l; k = 0, …, n − 1) and q : [a, b] → [0, +∞[ such that

lim sup_{t→a} (\frac{1}{h^{(n−1)}(t)} ∑_{k=0}^{n−1} ∑_{i=1}^{l} ∫_{a}^{t} p_{ik}(s) ds) < 1, lim_{t→a} (\frac{1}{h^{(n−1)}(t)} ∫_{a}^{t} q(s) ds) = 0.

Let furthermore the inequality

f_{0}(t, h(τ_{10}(t))x_{10}, …, h^{(n−1)}(τ_{1n}(t))x_{1 n−1}, …,

h(τ_{l0}(t))x_{l0}, …, h^{(n−1)}(τ_{l n−1}(t))x_{l n−1}) · sgn(x_{l n−1}) ≤ ∑_{k=0}^{n−1} ∑_{i=1}^{l} p_{ik}(t)∥x_{ik}∥ + q(t)

be fulfilled on ]a, b] × ℝ_{ρ}^{lmn}. Then problem (12), (2) is locally solvable.

Corollary 2. Let there exist a number l_{0} ∈ {1, …, l − 1} and a continuous function τ : [a, b] → [a, b] such that τ_{l_{0} n−1}(t) ≡ t,

τ_{ik}(t) ≤ τ(t) < t for a < t ≤ b (i = l_{0} − 1, …, l; k = 0, …, n − 1)

and let the inequality

f_{0}(t, h(τ_{10}(t))x_{10}, …, h^{(n−1)}(τ_{1n}(t))x_{1 n−1}, …,
h(τ_{l0}(t))x_{l0}, …, h^{(n−1)}(τ_{l n−1}(t))x_{l n−1}) · sgn(x_{l_{0} n−1}) ≤

≤ ∑_{k=0}^{n−1} ∑_{i=1}^{l_{0}} p_{ik}(t, ∑_{j=0}^{n−1} ∑_{i=l_{0}+1}^{l} ∥x_{ij}∥)|x_{ik}| + q(t, ∑_{j=0}^{n−1} ∑_{i=l_{0}+1}^{l} ∥x_{ij}∥)

be fulfilled on ]a, b] × ℝ^{lmn}, where the functions p_{ik} : [a, b] × [0, +∞[ → [0, +∞[ (i = 1, …, l_{0}; k = 0, …, n − 1), q : [a, b] × [0, +∞[ → [0, +∞[ are summable with respect to the first argument and continuous and nondecreasing with respect to the second argument. Let furthermore

lim sup_{t→a} (\frac{1}{h^{(n−1)}(t)} ∑_{k=0}^{n−1} ∑_{i=1}^{l_{0}} ∫_{a}^{t} p_{ik}(s, ρ) ds) < 1, lim_{t→a} (\frac{1}{h^{(n−1)}(t)} ∫_{a}^{t} q(s, ρ) ds) = 0

for some positive constant ρ. Then problem (12), (2) is globally solvable.

Remark 1. Under the conditions of the above-mentioned propositions the right sides of differential equations may have singularities of arbitrary orders. Indeed, as an example let us consider on the interval [a, b] the scalar differential equation

u^{(n)}(t) = ∑_{k=0}^{n−1} [\frac{α_{k}}{t^{(λ−k)μ_{k}+n−λ}} u^{(k)}(t^{μ_{k}}) + \frac{β_{k}}{t^{(λ−k)μ_{k}γ_{k}+n−k}} |u^{(k)}(t^{μ_{k}})|^{γ_{k}}] −

− ∑_{k=1}^{k_{0}} g_{k}(t, u(t), …, u^{(n−1)}(t))u^{(n−1)}(t) + ct^{λ_{0}−n}  (13)

with the initial conditions

lim_{t→a} \frac{u^{(k)}(t)}{t^{λ−k}} = 0 (k = 0, …, n − 1)  (14)

where b ∈]0, 1[ , α_{k} and β_{k} ∈ ℝ, μ_{k} > 1, γ_{k} > 1, c ∈ ℝ, λ_{0} > λ, g_{k} : ]0, b] × ℝ^{n} → [0, +∞[ are continuous functions. By Corollary 2, for the global solvability of problem (13), (14) it is sufficient that

∑_{k=0}^{n−1} \frac{|α_{k}|}{(λ − k) ⋯ (λ − n + 1)} < 1.

Remark 2. There exists an example which shows that condition (4) in Theorem 1 is optimal and it cannot be replaced by the condition

lim sup_{t→a} (\frac{1}{h^{(n−1)}(t)} ∑_{k=0}^{n−1} ∫_{0}^{t} p_{k}(s) ds) ≤ 1.

References

1. I. Kiguradze and Z. Sokhadze, On the Cauchy problem for evolution singular functional differential equations. (Russian) Differentsial’nye Uravneniya 33(1997), No. 1, 48–59.

2. I. Kiguradze and Z. Sokhadze, On the global solvability of the Cauchy problem for singular functional differential equations. Georgian Math. J. 4(1997), No. 4, 355-372.

Author’s address:

A. Tsereteli Kutaisi State University

55, Queen Tamar St., Kutaisi 384000

Georgia