In this paper we continue our work on problems of the type Ex=Nx, Bx=0, where E:D(E)⊂X→Y is linear, closed, possibly nonselfadjoint, N:D(N)=X→Y is continuous, not necessarily linear, B is linear and homogeneous, X and Y are Banach spaces. Let P:X→X and Q:Y→Y be projections with X0=PX⊃kerE, Y0=QY⊃kerE∗, X1=kerP, Y1=kerQ. Let σ:X0→Y, S:Y→X0 be such that SQσ:X0→X0 is the identity on X0 and S−1(0)={0}. In the present paper, the authors consider the case that σ can be any linear map with some restrictions and indicate how S can be determined so as to satisfy the requirements above. We study in detail the problem (∗) Ex:=x′′=f(t)+g(t,x), t∈J:=[0,a], x′′(0)=x′′(a)=0, x(0)+x(a)=0 where f:J→R and g:J×R→R are continuous, with g bounded on J×R, and give sufficient conditions for (∗) to have at least one solution.
Further results in Nonlinear Analysis
PUCCI, Patrizia
1987
Abstract
In this paper we continue our work on problems of the type Ex=Nx, Bx=0, where E:D(E)⊂X→Y is linear, closed, possibly nonselfadjoint, N:D(N)=X→Y is continuous, not necessarily linear, B is linear and homogeneous, X and Y are Banach spaces. Let P:X→X and Q:Y→Y be projections with X0=PX⊃kerE, Y0=QY⊃kerE∗, X1=kerP, Y1=kerQ. Let σ:X0→Y, S:Y→X0 be such that SQσ:X0→X0 is the identity on X0 and S−1(0)={0}. In the present paper, the authors consider the case that σ can be any linear map with some restrictions and indicate how S can be determined so as to satisfy the requirements above. We study in detail the problem (∗) Ex:=x′′=f(t)+g(t,x), t∈J:=[0,a], x′′(0)=x′′(a)=0, x(0)+x(a)=0 where f:J→R and g:J×R→R are continuous, with g bounded on J×R, and give sufficient conditions for (∗) to have at least one solution.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
