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Some topological properties preserved by nearness between operators and applications to P.D.E.

1996; Springer Nature; Volume: 46; Issue: 4 Linguagem: Inglês

10.21136/cmj.1996.127323

1572-9141

Antonio Tarsia,

Approximation Theory and Sequence Spaces

ІNTRODUCTION In tliis papеr somе topological propеrtiеs which arе prеsеrvеd by nearness bеtлvееn opеratoľs and which liavе signiñcant applications in thе еxistеncе tliеory of solutions for diffеľеntial еquations arе studiеd.Bеforе еxplaining shortly thе contеnt of еach part of tliе papег, wе shall givе thе dеfìnition of near opcrators and thе basic tliеoríтn on isoniorpliism (Cf.[Ci]).Lеt Л' bе a sеt and B a Banach spaco noпnеd with || ||.Lеt Л, D bе oporators 1)(Twе(тi Л' and B. Dеfìnition 0.1.Wе say that Л is ncar D if tliеro еxist tлvo positivе constants o and k. witli k Є (0, 1).sucli that wе liavе (0.1) \\B(.rt ) -B(x 2 ) -a[A(.Гl )-A(.r 2 )}\\ íC к\\B(.Гl ) -B(.r 2 )l V.T,,.Г 2 Є Д'.Thеorеm 0.1.Let Л be nonr D. Iť D is a Ыjective operнtoг betweeu Л' aud B. then Л is bijectivc bctwecn Л' нnd B, too.Wе rocall that thе thеory of ncar opеrators was introducеd in [Ci] for s» udying thе oxistonco and rеgularity of solutions for nonlinеar non-variational еllipüo systíтns.Subsoquеntly.it was also appliеd to noiilinеar paral)olic systешs ([C 2 ]. [TJ).Tho tlкч)ry of n.cxir opеrators is connеctеd witli tlк-nюnotonе opеraroг tlкч)rv (SÍM 1 foг oxamplе [C;,]).In §1 wе sliall provе somе piч^hшinary rеsults about topological propеrtios of op-(T -ators Ф, bеtлvееn B and B, tliat arе ncar Iц (idеntity шap on B).Tliеoiчтn 1.1 in §l ìччpiiľеs somе furtliеľ rеmarks.It еstablishеs that if Ф: íì { -^ B ìs ncar Iß and if íži is opеn in B thеn ӣ-2 = Ф(í-i) is oj)(тi in B. Conso(iuontly.Г Пl(Ч)ľЄlll 1.1, whеn B has a íinitе doinеnsion, is a. particular caso of tlк^ following topology thеorеin.