ON A PSEUDO FUCIK SPECTRUM FOR STRONGLY NONLINEAR 2ND-ORDER ODES AND AN EXISTENCE RESULT
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Date
1994
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ELSEVIER SCIENCE BV
Abstract
We study here the existence of solutions to the nonlinear Dirichlet problem (P): (phi(u'))' + f(t, u) = q(t), a.e. for t is an element of [a, b], u(a)=u(b)=0, where phi is an increasing odd homeomorphism of R, f:[a, b]XR --> R satisfies the Caratheodory assumptions and q is an element of L(1)([a, b], R).
With a view towards defining a pseudo Fucik spectrum (PFS) we study, using a time-mapping approach, the eigenvalue-like problem (phi(u'))' + A phi(u(+)) - B phi(u(-)) = 0, u(a) = 0 = u(b), where A > 0, B > 0.
We show here that this PFS, which we denote by P, consists of a set of curves in (R(+))(2) resembling the classical Fucik spectrum, i.e., when phi(s) = s.
Our main existence result, which deals with nonresonance with respect to the PFS, can be roughly stated as follows: if for s > 0 sufficiently large and for almost every t is an element of [a, b], the pair (f(t, - s)/phi(- s>, f(t, s)/phi(s)) lies in a compact rectangle contained in an open component of (R(+))(2)\P which intersects the diagonal, then problem (P) has at least a solution.
With a view towards defining a pseudo Fucik spectrum (PFS) we study, using a time-mapping approach, the eigenvalue-like problem (phi(u'))' + A phi(u(+)) - B phi(u(-)) = 0, u(a) = 0 = u(b), where A > 0, B > 0.
We show here that this PFS, which we denote by P, consists of a set of curves in (R(+))(2) resembling the classical Fucik spectrum, i.e., when phi(s) = s.
Our main existence result, which deals with nonresonance with respect to the PFS, can be roughly stated as follows: if for s > 0 sufficiently large and for almost every t is an element of [a, b], the pair (f(t, - s)/phi(- s>, f(t, s)/phi(s)) lies in a compact rectangle contained in an open component of (R(+))(2)\P which intersects the diagonal, then problem (P) has at least a solution.
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Keywords
NONRESONANCE, TIME MAPPING, PSEUDO-EIGENVALUE, PSEUDO FUCIK SPECTRUM, CONTINUATION LEMMA