On the number of positive solutions to a class of integral equations

• Autorzy: Wang L. , Yu W. , Zhang L.
• Języki publikacji: EN

Abstrakty

EN

By using the complete discrimination system for polynomials, we study the number of positive solutions in C[0,1] to the integral equation phi(x) = integral[...] k(x,y)phi^n(y)dy, where k(x,y) = phi1(x)phi1(y)+phi2(x)phi2(y),[phi]i(x) > 0,[phi]i(y) > 0,0 < x,y < 1,i = 1,2, are continuous functions on [0,1], n is a positive integer. We prove the following results: when n = 1, either there does not exist, or there exist infinitely many positive solutions in C[0,1]; when n [is greater than or equal] 2, there exist at least 1, at most n + 1 positive solutions in C[0,1]. Necessary and sufficient conditions are derived for the cases: 1) n = 1, there exist positive solutions; 2) n [is greater than or equal to] 2, there exist exactly m (m belongs to {1,2,..., n + 1}) positive solutions. Our results generalize the ones existing in the literature, and their usefulness is shown by examples.

Słowa kluczowe

EN

integral equations; positive solutions; complete discrimination system for polynomials; number of solutions

PL

równanie całkowe; rozwiązanie dodatnie

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2003
• Tom: Vol. 32, no 2
• Strony: 383--395
• Opis fizyczny: Bibliogr. 21 poz.

Twórcy

autor

Wang L.

• Center for Systems and Control Department of Mechanics and Engineering Science Peking University, Beijing 100871, P.R. China

autor

Yu W.

• Laboratory for Complex Systems and Intelligent Control Institute of Automation, Chinese Academy of Sciences Beijing 100080, P.R. China

autor

Zhang L.

• Department of Automation, Tsinghua University Beijing 100084, P.R. China

Bibliografia

• ABRAHAM, R. A. and MARSDEN, J. E. (1978) Foundations of Mechanics. Benjamin and Cummings, Reading, MA.
• BASAR, T. and BERNHARD, P. (1990) Hoo Optimal Control and Related Minimax Des·ign Problems. Birkhiiuser, Boston.
• CORDUNEANU, C. (1973) Integral Equat·ions and Stab·ildy of Feedback Systems. Springer-Verlag, London and New York.
• COURANT, R. and HILBERT, D. (1953) Methods of Mathematical Physics. Interscience Publishing Company, New York.
• FEUER, A. and GOODWIN, G. C. (1996) Sarnpling in Digital Signal Pmcessing and Control. Birkhauser, Boston.
• FRANCIS, B. A. (1987) A Course in Hoo Control Theory. Springer-Verlag, Berlin, 1987.
• GANTMACHER, F. R. (1960) The Theor·y of Matr·ices. Chelsea, New York.
• GREUB, W. H. (1967) L-inear· Algebm. Springer-Verlag, Berlin.
• ISHII, H. and FRANCIS, B. A. (2002) Limited Data Rate ·in Control Systems wdh Networ-ks. Springer-Verlag, Berlin.
• LIN, C. C. and SEGEL, C. A. (2002) Mathematics Applied to Deterrninist·ic Problems in the Natural Sciences. Classics in Applied Mathematics, SIAM, Philadelphia.
• POPOV, V. M. (1973) Hype1·stab-ildy of Control Systems. Springer-Verlag, Berlin.
• SANDBERG, I. W. (1964) On the L2 -boundedness of solutions of nonlinear functional equations. Bell Systems Technical Journal, 42, 1981-1599.
• VAN DER SCHAFT, A. (1999) L2 -Gain and Passivity Techniques ·in Nonlinear· Control. Springer-Verlag, Berlin.
• VIDYASAGAR, M. (2002) Nonlinear· Systems Analysis, 2nd Edition. Classics in Applied Mathematics, SIAM, Philadelphia.
• WANG, 1. (2000) Composite Interval Control Systems: Some Kharitonov-like Properties. Reliable Computing - Special Issue on ContTol, S·ignals and Systems, 6, 3, 231- 246.
• WILLEMS, J. C. (1969) Some results on the LP-stability of linear time-varying systems. IEEE Tmns. on Automatic Contr·ol, 14, 660-665.
• YANG, 1., Hou, X. R. and ZENG Z. B. (1996) A complete discrimination system for polynomials. Science in China, E-39, 628-646.
• YANG, 1., ZHANG, J. Z. and Hou, X. R. (1996) Nonlinear Algebraic Equations and Machine Proving. Nonlinear Science Series, Shanghai Science and Education Press, Shanghai.
• YANG, 1. and XrA, B. C. (1997) Explicit criterion to determine the number of positive roots of a polynomial. Mathernatics-Mechan·ization Researdt ?reprints, 15, 134-145.
• YAO, P. (1991) On the number of positive solutions to an integral equation with rank 2 kernel. Journal of Mathernatical Physics, 11, 274-279.
• Yu, W. and WANG, 1. (2001) Anderson's Claim on Fourth-Order SPR Synthesis is True. IEEE Trans. on Circuits and Systems, 48, 4, 506-509.