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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