# Approximate solutions of operator equations by Mingjun Chen; Zhongying Chen; G Chen

By Mingjun Chen; Zhongying Chen; G Chen

Those chosen papers of S.S. Chern talk about subject matters akin to critical geometry in Klein areas, a theorem on orientable surfaces in 4-dimensional house, and transgression in linked bundles Ch. 1. advent -- Ch. 2. Operator Equations and Their Approximate ideas (I): Compact Linear Operators -- Ch. three. Operator Equations and Their Approximate strategies (II): different Linear Operators -- Ch. four. Topological levels and glued aspect Equations -- Ch. five. Nonlinear Monotone Operator Equations and Their Approximate recommendations -- Ch. 6. Operator Evolution Equations and Their Projective Approximate suggestions -- App. A. primary practical research -- App. B. advent to Sobolev areas

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**Example text**

I) Suppose f(x) e L2{Sl). Show that for any fixed u € HQ(Q), a(-,u) is a bounded linear functional on HQ(£1). Show also that if q{x) > 0 22 Chapter 1 C/iapier for all x e xeQ, € H, il, then the bilinear functional a(-, •) defines an equivalent inner product. u) o(t;,«), V . 6VveHfcn), fi0'(O), (v,AlU) = a(v,u), where, as in the text, (», {v, t) I) represents the evaluation of the functional £ I at v. ,. i, .. := ll^l^llfflil* • = Ssup P Kv,iMl^dlMltfi, I fvt-ofilAu) i t O l S^ cc ii ll FMI II *W!

Let X ^ 0 be a regular value of operator TK. 12) has a unique solution un€Vn. 7) in the space X = L2J-1,1). 14) where aj(t), f(t) € C[a, 6], j = 0,1, • • •• m - 11 and 4 0 > A ; =, 1 , •, m, m,a continuous linear functional denned on C7m-1[a,6]. , . 7). ,m; (iii) for any x(t) € Cm-l[a,b] n Hm(a,b), Sobolev space, where ffm(a,&) is the mth-order fbH(t,s){Dmx)(s)ds, x(t)=x0(t)+ (Dtx)(t) = (Dixo){t)+ (D*x)(t) (D*x0)(t)+ /bH^°\t,s){Dmx)(s)ds, / JO Ja j = 1, • • •• m - 11 where x0(t) is the unique polynomial of degree m - 1 that satisfies 4 ( x 0 ) = 4 ( # ) f o r a l l f c = 1,•-••m.

In the beginning of the century, I. Fredholm studied the integral equation