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Existence of solutions for nonlinear boundary value problems. (English) Zbl 1145.34007

The authors study a one-dimensional Φ-Laplacian equation of the form

(Φ(u ' )) ' +f(t,u,u ' )=0,0<t<1(1)

together with the boundary conditions

u(0)=g(u(t 1 ),u ' (t 1 ),,u(t m ),u ' (t m )),u(1)=h(u(t 1 ),u ' (t 1 ),,u(t m ),u ' (t m ))·(2)

The numbers t i belong to [0,1]; f is Carath√©odory in ]0,1[× 2 ; g and h are continuous and nondecreasing in the u-variables; Φ is an odd homeomorphism of .

They introduce the notion of coupled lower and upper solutions as a pair of functions αβ of C 1 [0,1] satisfying, in addition to the standard differential inequalities, boundary inequalities that make sense by virtue of a Nagumo condition that f verifies with respect to the pair α, β. Under these conditions, they show that the problem (1)-(2) has at least one solution between α and β.

Several consequences follow. In particular, if

|f(t,x,y 1 )-f(t,x,y 2 )|χ(t)|Φ(y 1 )-Φ(y 2 )|

with χL 1 (0,1) and f is nondecreasing in x, there exists a unique solution. Also, an application is given to the existence of radial solutions in an annulus R 1 |z|R 2 of the equation



u(R i )= j=1 m (a ij u(R i )+g ij (u ' (R i )))+λ i ,i=1,2·

Here A(|x|)x is an increasing homeomorphism of , the a ij are nonnegative, ij a ij <1 and sup y m |g 1j (y j )|+|g 2j (y j )|<.

34B15Nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
47H10Fixed point theorems for nonlinear operators on topological linear spaces