An exact functional-analytic representation of solutions to a Hamilton-Jacobi equation of Riccati type

N.K.Prykarpatska; E.Wachnicki

A generalized characteristic method [5],[7] preliminaries are described and used for studying functional-analytic solutions to the Cauchy problem of noncanonical Hamilton-Jacobi equations. If the equation is of the Riccati type solutions are obtained and investigated making use of the classical Leray-Schauder fixed point theory.

1. Arnold V.I. Introduction into partial differential equations. Interfactor, Moscow, 2002.

2. Abraham R., Marsden J. Foundations of Mechanics. Cummings Co, USA, 1978.

3. Cartan E. Lecons sur invariants integraux. Hermann, Paris, 1971.

4. Arnold V.I. Mathematical Methods of Classical Mechanics., Springer, NY, 1978.

5. Prykarpatska N.K., Pytel-Kudela M. On the structure of characteristics surfaces related with partial differentail equations of first and higher orders, Part 1. Opuscula Mathematica, 2005, N25/2, p. 299-306

6. Prykarpatska N.K., Blackmore D.L., Prykarpatsky A.K. and M. Pytel-Kudela. On the inf-type extremality solutions to Hamilton-Jacobi equations, their regularity properties and some generalizations$.$ Miskolc Mathematical Notes, 2003, v. 4, p. 157-180

7. Prykarpatska N.K., Wachnicki E. On the geometric structure of characteristic vector fields related nonlinear equations of the Hamilton-Jacobi type. \ Opuscula Mathematica, 2006, N26 (in print).

8. Evans L.C. Partial Differential equations, AMS, U.S.A., 1998

9. Schwartz J.T. Nonlinear functional analysis, Gordon and Breach Science Publishers, New York, London, Paris, 1969

10. Prykarpatska N.K. On the structure of characteristic surfaces related with nonlinear partial differential equations of first and higher orders, Part 2. Nonlinear Oscillations, 2005, v. 8, N4

11. Crandall M.G., Ishii H., Lions P.L. User's guide to viscosity solutions of second order partial differential equations. Bulletin of AMS, 1992, N1, p.1-67

12. Prykarpatsky A.K., Mykytiuk I.V. Algebraic integrability of nonlinear dynamical systems on manifolds: classical and quantum aspects. -- Dordrecht-Boston-London: Kluwer Academic Publishers, 1998. -- 553~p.

13. Weinberg B.R. Asymptotical methods in equations of mathematical physics, Moscow State University Publisher, 1982

14. Bliss G.A. Lectures on the calculus of variations. Chikago, Illinois, USA, 1946.

	An exact functional-analytic representation of solutions to a Hamilton-Jacobi equation of Riccati type
Author	N.K.Prykarpatska, E.Wachnicki prykanat@cybergal.com, natpats@yahoo.com The AGH University of Science and Technology, Department,of Applied Mathematics, Krakow 30059 Poland, , , The Pedagogical Academy, Institute of Mathematics, Krakow 30059 Poland
Abstract	A generalized characteristic method [5],[7] preliminaries are described and used for studying functional-analytic solutions to the Cauchy problem of noncanonical Hamilton-Jacobi equations. If the equation is of the Riccati type solutions are obtained and investigated making use of the classical Leray-Schauder fixed point theory.
Keywords	functional-analytic solution, noncanonical Hamilton-Jacobi equation, Riccati type solution
DOI	doi:10.30970/ms.26.2.154-160
Reference	1. Arnold V.I. Introduction into partial differential equations. Interfactor, Moscow, 2002. 2. Abraham R., Marsden J. Foundations of Mechanics. Cummings Co, USA, 1978. 3. Cartan E. Lecons sur invariants integraux. Hermann, Paris, 1971. 4. Arnold V.I. Mathematical Methods of Classical Mechanics., Springer, NY, 1978. 5. Prykarpatska N.K., Pytel-Kudela M. On the structure of characteristics surfaces related with partial differentail equations of first and higher orders, Part 1. Opuscula Mathematica, 2005, N25/2, p. 299-306 6. Prykarpatska N.K., Blackmore D.L., Prykarpatsky A.K. and M. Pytel-Kudela. On the inf-type extremality solutions to Hamilton-Jacobi equations, their regularity properties and some generalizations$.$ Miskolc Mathematical Notes, 2003, v. 4, p. 157-180 7. Prykarpatska N.K., Wachnicki E. On the geometric structure of characteristic vector fields related nonlinear equations of the Hamilton-Jacobi type. \ Opuscula Mathematica, 2006, N26 (in print). 8. Evans L.C. Partial Differential equations, AMS, U.S.A., 1998 9. Schwartz J.T. Nonlinear functional analysis, Gordon and Breach Science Publishers, New York, London, Paris, 1969 10. Prykarpatska N.K. On the structure of characteristic surfaces related with nonlinear partial differential equations of first and higher orders, Part 2. Nonlinear Oscillations, 2005, v. 8, N4 11. Crandall M.G., Ishii H., Lions P.L. User's guide to viscosity solutions of second order partial differential equations. Bulletin of AMS, 1992, N1, p.1-67 12. Prykarpatsky A.K., Mykytiuk I.V. Algebraic integrability of nonlinear dynamical systems on manifolds: classical and quantum aspects. -- Dordrecht-Boston-London: Kluwer Academic Publishers, 1998. -- 553~p. 13. Weinberg B.R. Asymptotical methods in equations of mathematical physics, Moscow State University Publisher, 1982 14. Bliss G.A. Lectures on the calculus of variations. Chikago, Illinois, USA, 1946.
Pages	154-160
Volume	26
Issue	2
Year	2006
Journal	Matematychni Studii
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