Svatopluk Fučík

Ranges of nonlinear asymptotically linear operators

Abstract This paper deals with the solvability of the equation A ( u ) − S ( u ) = f , where A is a continuous self-adjoint operator defined on a real Hilbert space H with values in H , the null-space of A is nontrivial, and N is a nonlinear completely continuous perturbation. Sufficient, and necessary-sufficient conditions are given for the equation to be solvable. Abstract theorems are applied to solving boundary value problems for partial differential equations.

Generalized periodic solutions of nonlinear telegraph equations

paper completes the study of generalized periodic solutions of nonlinear telegraph equation of the form Bu, + u,, - Kr, - pa + +vu- ++(u)= h(t,x) (1.1) (where /I # 0, CL, v are real parameters,

Upper bound for the number of critical levels for nonlinear operators in Banach spaces of the type of second order nonlinear partial differential operators

is a continuous and bounded real function and h is square Lebesgue integrable over I2 with I = [0,2n]) initiated in [I]. Recall that a generalized periodic solution of (1.1) (shortly GPS) is a real function u E

Boundary value problems with bounded nonlinearity and general null-space of the linear part

Abstract Let f and g be two nonlinear functionals defined on a real Banach space X . Consider the eigenvalue problem λf ′( u ) = g ′( u ), u ϵ M r ( f ) = { x ϵ X ; f ( x ) = r } ( r > 0 is a prescribed number, f′ and g′ denote Frechet derivatives of f and g respectively). The value of the functional g at the critical point of the functional g with respect to the manifold M r ( f ) is called the critical level. Denote Γ the set of all critical levels. It is known that Γ is at least countable (see, for instance, E. S. Citlanadze , Trudy Mosk. Mat. Obsc . 2 (1953), 235–274). In this paper we give an abstract theory for upper bound for the number of points of Γ and an application to partial differential equations of the second order. The regularity properties of solutions of such equations are of great importance.