NONLOCAL STURM-LIOUVILLE PROBLEMS WITH INTEGRAL TERMS IN THE BOUNDARY CONDITIONS

Abstract

We consider a new type Sturm-Liouville problems whose main feature is the nature of boundary conditions. Namely, we study the nonhomogeneous Sturm-Liouville equation p(x)u '' (x) + (q(x) - lambda)u = f(x) on two disjoint intervals [-1, 0) and (0, 1], subject to the nonlocal boundary transmission conditions alpha(k)u((mk)) (-1) + beta(k)u((mk)) (-0) + eta ku((mk)) (+0) +gamma ku((mk)) (1) +Sigma(nk)(j=1) delta(kj)u((mk)) (xkj) + Sigma(2)(upsilon=1)Sigma(mk)(j=0)integral(Omega upsilon) k(kvj)(t)u((j)) (t)dt = f(k), k = 1,2,3,4. where Omega(1) := [-1,0), Omega(2) := (0,1] and xkj is an element of (-1,0) boolean OR (0,1) are internal points. By using our own approaches we establish such important properties as Fredholmness, coercive solvability and isomorphism with respect to the spectral parameter lambda.