On a boundary value problem for nonlinear functional differential equations
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We consider the problem u (t) = H(u)(t) + Q(u)(t), u(a) = h(u), where H,Q : C([a,b];R) αβ → L([a,b];R) are, in general, nonlinear continuous operators, H ∈ Ᏼab (g0 ,g1 , p0 , p1 ), and h : C([a,b];R) → R is a continuous functional. Efficient conditions sufficient for the solvability and unique solvability of the problem considered are established. 1. Notation The following notation is used throughout the paper: N is the set of all natural numbers. R is the set of all real numbers, R+ = [0,+∞[,[x]+ = (1/2)(|x| + x), [x]− = (1/2)(|x| − x). C([a,b];R) is the Banach space of continuous functions u : [a,b] → R with the norm uC = max{|u(t)| : t ∈ [a,b]}. C([a,b];R) is the set of absolutely continuous functions u : [a,b] → R. L([a,b];R) is the Banach space of Lebesgue integrable functions p : [a,b] → R with the b norm pL = a | p(s)|ds. L([a,b];R+ ) = { p ∈ L([a,b];R) : p(t) ≥ 0 for t ∈ [a,b]}. ᏹab is the set of measurable functions τ : [a,b] → [a,b]. ab is the set of continuous operators F : C([a,b];R) → L([a,b];R) satisfying the Carath`eodory condition, that is, for each r > 0 there exists qr ∈ L([a,b];R+ ) such that F(v)(t) ≤ qr (t)
for t ∈ [a,b], v ∈ C [a,b];R , vC ≤ r.
(1.1)
K([a,b] × A;B), where A ⊆ R2 , B ⊆ R, is the set of functions f : [a,b] × A → B satisfying the Carath`eodory conditions, that is, f (·,x) : [a,b] → B is a measurable function for all x ∈ A, f (t, ·) : A → B is a continuous function for almost all t ∈ [a,b], and for each r > 0 there exists qr ∈ L([a,b];R+ ) such that f (t,x) ≤ qr (t)
for t ∈ [a,b], x ∈ A, x ≤ r.
Copyright © 2006 Hindawi Publishing Corporation Boundary Value Problems 2005:3 (2005) 263–288 DOI: 10.1155/BVP.2005.263
(1.2)
264
On a BVP for nonlinear FDE
2. Statement of the problem We consider the equation u (t) = H(u)(t) + Q(u)(t),
(2.1)
αβ
where H ∈ Ᏼab (g0 ,g1 , p0 , p1 ) (see Definition 2.1) and Q ∈ ab . By a solution of (2.1) we understand a function u ∈ C([a,b];R) satisfying the equality (2.1) almost everywhere in [a,b]. αβ
Definition 2.1. We will say that an operator H belongs to the set Ᏼab (g0 ,g1 , p0 , p1 ), where g0 ,g1 , p0 , p1 ∈ L([a,b];R+ ) and α,β ∈ [0,1[, if H ∈ ab is such that, on the set C([a,b];R), the inequalities −mg0 (t) − µ(m,α)M 1−α g1 (t) ≤ H(v)(t) ≤ M p0 (t) + µ(M,β)m1−β p1 (t)
for t ∈ [a,b] (2.2)
are fulfilled, where
M = max v(t)
+
m = max v(t)
: t ∈ [a,b] ,
−
: t ∈ [a,b] ,
(2.3)
and the function µ : R+ × [0,1[→ R+ is defined by 1
µ(x, y) =
xy
if x = 0, y = 0, otherwise.
(2.4)
αβ
The class Ᏼab (g0 ,g1 , p0 , p1 ) contains all the positively homogeneous operators H and, in particular, those defined by the formula
H(u)(t) = p0 (t) u τ1 (t)
++
p1 (t) u τ2 (t)
β
u τ3 (t)
+
1−β −
α 1−α − g0 (t) u ν1 (t) − + g1 (t) u ν2 (t) − u ν3 (t) + ,
(2.5)
where τi ,νi ∈ ᏹab (i = 1,2,3), α = 0, β = 0. The class of equations (2.1) contains various equations with “maxima” studied, for example, in [3, 4, 33, 35, 36, 38, 41]. For example, the eq
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