By Lucia M., Magtone R., Zhou H.-S.
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Extra resources for A dirichlet problem with asymptotically linear and changing sign nonlinearity
The notation a ≥ b (meaning b ≤ a) can also be regarded as a deﬁnition of a partial order relation. The relation ≥ satisﬁes also (i) – (iii), if ≤ does, and so < A, ≥> is a partially ordered set. Then < A, ≥> is called the dual of < A, ≤>. Let ϕ be a statement about < A, ≤>. If in ϕ we change all occurrences of ≤ by ≥ we get the dual of ϕ. The importance of duals follows from the Duality Principle which we formulate following Gr¨ atzer (1998). The Principle is often used in lattice theory to shorten proofs.
5, A− is a g-inverse since AA− A = A. Moreover, if A− 0 is a speciﬁc g-inverse, choose Z = A− − A− 0 . Then − − − A− 0 + Z − A0 A0 ZA0 A0 = A . 5 can also be used as a deﬁnition of a generalized inverse of a matrix. This way of deﬁning a g-inverse has been used in matrix theory quite often. Unfortunately a generalized inverse matrix is not uniquely deﬁned. Another disadvantage is that the operation of a generalized inverse is not transitive: when A− is a generalized inverse of A, A may not be a generalized inverse of A− .
Let us now prove suﬃciency. 13) has a solution for a speciﬁc b. Therefore, a vector w exists which satisﬁes Aw = b. 13). 1. All g-inverses A− of A are generated by − − A− = A− 0 + Z − A0 AZAA0 , where Z is an arbitrary matrix and A− 0 is a speciﬁc g-inverse. 5, A− is a g-inverse since AA− A = A. Moreover, if A− 0 is a speciﬁc g-inverse, choose Z = A− − A− 0 . Then − − − A− 0 + Z − A0 A0 ZA0 A0 = A . 5 can also be used as a deﬁnition of a generalized inverse of a matrix. This way of deﬁning a g-inverse has been used in matrix theory quite often.