The Eigenvalue Problem And Perturbation Theory Of Accretive And Monotone Operators

Accretive and monotone operators attract many scholars because oftheir wide application. As for the case about the perturbations, there are many results. Generally, their results are in the research into compact perturbations with the application of Leray-Schauder degree. This thesis studies the eigenvalue problem and perturbation theory of accretive and monotone operators. We establish some results for $k-$set contractions and condensing mappings perturbations with toplogy degree theory. And we also extend some results about eigenvalue problem for accretive operators with compact perturbations.In chapter two, for accretive operators we establish the following theoryTHEOREM 2.1.1 Let D∈Xbe open, bounded with θ∈D. Let T:D|-→X be bounded, accretive with T(θ)=θ, and let C:D|-→X be compact. Assume further, thatTx≠θ,x∈аD, and that T is φ-expansive onаD. Let the constant α>0be such that |Cx|≥α,x∈аD and let one of the following conditions be satisfied:(i) X⁺is uniformly convex and T is demicontinuous;(ii) Tis continuous.Then there exists (λ₀,x₀)∈(0,∞)×аD and (μ₀,y₀)∈(-∞,0)×аDsuch that Tx₀-λ+0Cx+0=θ Ty₀-μ₀Cy₀=θTHEOREM 2.2.2 Let D∈ X be open, bounded, and K be a quasinormal wedge with quasinormal constant σ of X, and D ∩ K≠φ. Let T:K→Xbe bounded, accretive with . Assume that implies that for all . Let be k- set contractions . Assume that there is a constant such that Then each one of the following statements is true:(i) For every there exists and such that (ii) In addition , ,and is compact, then there exists and such that .THEOREM 2.3.1 Let be a bounded, open and convex subset of . Let be accretive and be condensing for all . (a) is condensing. Assume that and that Then .Assume that one of the following conditions holds:(i) is uniformly convex and is completely continuous;(ii) instead of (a), let be compact and be continuousand bounded. Then .In chapter three, for monotone operators we establish the following theoryTHEOREM 3.1.1 Let be bounded, open and convex, and be a quasinormal wedge with quasinormal constant of . Let be bounded, monotone, demicontinuous, and for all . Let be set contractions, and be set contractions . Assume that there exists ,such that (i) For every there exists and such that (ii) If is completely continuous and , then there exists and such that (iii) If is compact, then there exists and such that .THEOREM 3.2.1 Let be maximal monotone and be condensing for all (a) is condensing. Assume that there exist and such that : for every with and every we have .Then , for every .Assume further, that , Then. Let one of the following conditionshold:(i) is completely continuous;(ii) instead of (a), let be compact and be continuous and bounded. Then...