Vol. 56, No. 1, 1975

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ISSN: 0030-8730
A product integral solution of a Riccati equation

Burrell Washington Helton

Vol. 56 (1975), No. 1, 113–130
Abstract

Product integrals are used to show that, if dw, G, H and K are functions from number pairs to a normed complete ring N which are integrable and have bounded variation on [a,b] and v1 exists and is bounded on [a,b], then the integral equation

                    ∫
x
β (x) = w(x)+ (LRLR ) a (βH + G β + βK β)

has a solution β(x) = v1(x)u(x) on [a,b], where u and v are defined by the matrix equation

                   x     [        ]
[u(x),v(x)] = [w(a),1]∏ (I + H   − K  )
a      dw   − G

The above results are used to show that if p,q,h and r are quasicontinuous functions from the numbers to N such that h is left continuous and has bounded variation and p,q and h commute, then the solution on [a,b] of the differential-type equation f∗∗ + fp + fq = r is

           x               ∫ x   x
f(x) = f (a)∏ (1 − βdh) +(R )   dz∏ (1− β dh),
a                a    t

where f(x) f(a) = (L) axfdh, β is the solution of

         ∫ x          ∫ x              ∫ x
β(x) = (L)  qdh + (LL )   β(− p dh)+ (LR)  β dhβ,
a            a                a

and z is defined in terms of p,q,r,h and β.

Mathematical Subject Classification 2000
Primary: 34A99
Secondary: 26A39, 45A05
Milestones
Received: 12 September 1973
Revised: 11 June 1974
Published: 1 January 1975
Authors
Burrell Washington Helton