Electron. J. Diff. Equ., Vol. 2011(2011), No. 02, pp. 1-13.

Initial-value problems for first-order differential recurrence equations with auto-convolution

Mircea Cirnu

A differential recurrence equation consists of a sequence of differential equations, from which must be determined by recurrence a sequence of unknown functions. In this article, we solve two initial-value problems for some new types of nonlinear (quadratic) first order homogeneous differential recurrence equations, namely with discrete auto-convolution and with combinatorial auto-convolution of the unknown functions. In both problems, all initial values form a geometric progression, but in the second problem the first initial value is exempted and has a prescribed form. Some preliminary results showing the importance of the initial conditions are obtained by reducing the differential recurrence equations to algebraic type. Final results about solving the considered initial value problems, are shown by mathematical induction. However, they can also be shown by changing the unknown functions, or by the generating function method. So in a remark, we give a proof of the first theorem by the generating function method.

Submitted September 10, 2010. Published January 4, 2011.
Math Subject Classifications: 11B37, 34A12.
Key Words: Differential recurrence equations; discrete auto-convolution; algebraic recurrence equations.

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Mircea Cîrnu
Department of Mathematics III, Faculty of Applied Sciences
University Politehnica of Bucharest, Romania
email: cirnumircea@yahoo.com

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