Partial differential equations: Proc. symposia in pure by Charles Bradfield Morrey Jr.

By Charles Bradfield Morrey Jr.

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B) Assume that (H1 ), (H3 ), (H5 ), and (H6 ) hold. 5) converges to a finite limit. (b ) Assume that (H1 ), (H3 ), (H5 ), and (H6 ) hold. 5) bounded from above and from below by positive constants converges to a finite limit. PROOF We will prove (a ) and (b ) together. Let {xn } be a solution bounded from above and from below by positive constants. Clearly, there exists a full limiting sequence {L−q }∞ q=0 such that L0 = S. We divide the proof into the following two cases: Case 1: S, J = ∅ and all the indices is1 , .

12) f (M1 , . . , Mk ) − M = f (m1 , . . , mk ) − m = 0. 13) or We also define the following sets: S = {is ∈ {i1 , . . , ik } : f strictly increases in xn−is } = {is1 , . . , isr } and J = {ij ∈ {i1 , . . , ik } : f strictly decreases in xn−ij } = {ij1 , . . , ijt }. Clearly when H1 or H1 holds, S J = {i1 , . . , ik }. (H4 ) : The set S consists of even indices only and the set J consists of odd indices only. (H5 ) : Either the set S contains at least one odd index, or the set J contains at least one even index.

A + Bxn + Cxn−1 + Dxn−2 is bounded. Indeed, for n ≥ 0, xn+1 = α + βxn + γxn−1 max{α, β, γ}(1 + xn + xn−1 ) ≤ A + Bxn + Cxn−1 + Dxn−2 min{A, B, C, D}(1 + xn + xn−1 + xn−2 ) ≤ max{α, β, γ} . min{A, B, C, D} A minor extension of the above idea establishes the boundedness of every solution of any rational equation of the form xn+1 = α + βxn + γxn−1 + δxn−2 , Bxn + Cxn−1 + Dxn−2 n = 0, 1, . . 1) with α > 0 under the condition that when β > 0, then B > 0, when γ > 0, then C > 0, and when δ > 0, then D > 0.

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