Saddlepoint Characterizations of Solutions to Semilinear Operator Equations |
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Page 16
... Suppose now that H can be decomposed into the product Ax B , where A and B are closed subspaces . A saddle point for a functional is a critical point that is neither a local minimum nor a local maximum . With respect to the product set ...
... Suppose now that H can be decomposed into the product Ax B , where A and B are closed subspaces . A saddle point for a functional is a critical point that is neither a local minimum nor a local maximum . With respect to the product set ...
Page 18
... suppose that for con- stants , C1 and C2 > 0 , 1. ) ( V1F ( x1 , y ; w ) – V1 F ( x2 , y ; w ) , x1 − x2 ) ≥ c1 || 1 - x2 || 2 for all y , w , - - 2. ) ( V2F ( x . y1 ; w ) - V2 F ( x , y2 ; w ) , Y1 — Y2 ) ≥ c2 || y1 y2 || 2 for all ...
... suppose that for con- stants , C1 and C2 > 0 , 1. ) ( V1F ( x1 , y ; w ) – V1 F ( x2 , y ; w ) , x1 − x2 ) ≥ c1 || 1 - x2 || 2 for all y , w , - - 2. ) ( V2F ( x . y1 ; w ) - V2 F ( x , y2 ; w ) , Y1 — Y2 ) ≥ c2 || y1 y2 || 2 for all ...
Page 33
... Suppose L satisfies ( L1 ) and ( L2 ) , and N satisfies ( N1 ) , ( N2 ) , and ( N4 ) . Suppose , in addition , the solvability condition ( LL1 ) is met . Then there exists û in H1 with the following properties : i 33 The One-Sided ...
... Suppose L satisfies ( L1 ) and ( L2 ) , and N satisfies ( N1 ) , ( N2 ) , and ( N4 ) . Suppose , in addition , the solvability condition ( LL1 ) is met . Then there exists û in H1 with the following properties : i 33 The One-Sided ...
Contents
Preliminary Functional Analysis and Operator Theory | 6 |
Preliminary Convex Analysis | 13 |
The Saddlepoint Reduction | 18 |
6 other sections not shown
Common terms and phrases
Amann anti-coercive boundary value problems bounded operator Chapter Conditions L1 convex analysis convex functional critical point defined Definition differentiable double-sided resonance eigenspace eigenvalues elliptic boundary value exists F[xk(wk finite-dimensional Fréchet derivative Gâteaux derivatives Gâteaux differentiable H₁ Hilbert space hypotheses L²(N Landesman Lazer Lemma linear growth restriction linear operator Lu(w max inf F(x max max inf max min sup Mitchell 14 N₁ nonlinear term normed vector space obtain a solution one-sided resonance orthogonal point of F properties Proposition 3.4 require saddle saddlepoint reduction method saddlepoint solution self-adjoint operator Semilinear Operator Equations spectral gap spectrum strictly convex strongly monotone subspaces sup F(x Theorem Uk wk uk(wk um(wm variational characterization VF(u weak solution x(wk x(wn X₁ xk wk xk(wk xm(wm y(wk y(wm y(wn y(wo Y₁ yk wk yk(wk Ym Wm Ym(Wm