A FUNCTIONAL EQUATION V. S. Kozyakin UDC 513.88 Existence theorems for and the determination of continuous solutions, defined on the real axis R, of the functional equationf (t) = A[t,f (at-b),f (a~t-c)], where a, b, and c are real param- eters, A : R • E • E --~ E is a continuous operator, and E is a Banach space. Let E be a Banach space and let R be the real line. Consider the functional equation l(t) =A [t,/(at-- b), ](at--c)], (1) for a > 1 and e > b, where A : R x E • E --~ E. We seek continuous solutions of (1) defined on R. A similar equation was investigated in [1], where G. E. Zhuravlev obtained the following functional equation for the distribution of states of a forgetting automaton: = ,)i(-:)+,i where 0 <p < 1, and 0 < vt< 1, are parameters determining the state of the automaton. Since (1) has in general several solutions, conditions similar to the initial conditions for a differential equation must be imposed 9 Here we are mainly interested in obtaining existence theorems under various assumptions concerning the operator A and the parameter a; these theorems will be proved by using the contraction-mapping principle. The existence theorems derived can be used in constructing solutions of (1) 9 1 We use the following notation: b b 9 a--~ -c-~b=tl' ~--i= t2' G e a_l----Q, ~'~-~ -~c--b=te. in E, and employ the norm b C r b a a (a -- i) ta, We write C(E, [d, el) for the space of continuous functions on [d, el with values I[ x (t)l] ~-- max I] x (t)ll~. reid, el Definition 1. We call a function iv(t) E C(E, it l, t6]) a left initial value if it satisfies (1) at t2 9 We call a function ~(t) E C(E, it 5, t6]) a right initial value if it satisfies (1) at ts. Definition 2. A functionf : R --~ E is said to satisfy the initial conditions { iv, ~}, iff (t) is defined on it l, t 6] andf(t) = iv(t) on [tl, t2], andf(t) =~(t) on it5, t6]. THEOREM 1 9 Let the continuous operator A : R • E • E --* E satisfy the following conditions: 1) llh (t, u, v)-A (t, u', v')tl -< K Ilu-u' ] + Nllv-v'tl (K, N -> 0) for t E it 2, ts], and u, u', v, v' E E; 2) the equation A(t, u, v) =w has a unique solution v for t ~ t 2 and u, w E E, and the operator v = v(t, u, w) is continuous in its domain of definition; 3) the equation A(t, u, v) =w has a unique solution u for t ->t 5 and u, w E E, and the operator u =u(t, v, w) is continuous in its domain of definition. Let {iv, ~} be initial values and let any one of the following conditions hold: Voronezh State University 9 Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 161-170, February, 1971. Original article submitted November 26, 1969. 91971 Consultants Bureau,. a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00. 95