Keywords
- Periodic Solution
- Differential System
- Nonlinear Differential Equation
- Nonlinear Wave Equation
- Michigan Math
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References
References on the Cesari-Hale method
H.R. Bailey and L. Cesari, Boundedness of solutions of linear differential systems with periodic coefficients, Arch. Rational Mech. Anal. 1 (1958), 246–271.
H.R. Bailey and R.A. Gambill, On stability of periodic solutions of weakly nonlinear differential equations, J. Math. Mech. 6 (1957), 655–668.
C. Banfi, Sulla determinazione delle soluzioni periodiche di equazioni non lineari periodiche, Boll. Un. Mat. Ital. (4) 1 (1968), 608–619.
—, Su un metodo di successive approssimazioni per lo studio delle soluzioni periodiche di sistemi debolmente non lineari, Atti Accad. Sci. Torino 100 (1968), 1065–1066.
C. Banfi and G. Casadei, Calcolo di soluzioni periodiche di equazioni differenziali non lineari periodiche, Congresso AICA, Napoli, Sept. 1968.
L. Cesari, Sulla stabilitá delle soluzioni dei sistemi di equazioni differenziali lineari a coefficienti periodici, Mem. Accad. Italia (6) 11 (1941), 633–695.
—, Existence theorems for periodic solutions of nonlinear Lipschitzian differential equations and fixed-point theorems, Contributions to the Theory of Nonlinear Oscillations 5, Princeton, 1960, pp. 115–172.
— Existence theorems for periodic solutions of nonlinear differential systems, Bol. Soc. Mat. Mexicana 5 (1960), 24–41.
Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, 2nd ed., Springer, 1963.
— Functional analysis and Galerkin's method, Michigan Math. J. 11 (1964), 385–418.
— Functional analysis and periodic solutions of nonlinear differential equations, Contributions to Differential Equations 1 (1963), 149–187.
Periodic solutions of hyperbolic partial differential equations, Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, 1963, 33–57.
— A criterion for the existence in a strip of periodic solutions of hyperbolic partial differential equations, Rend. Circ. Mat. Palermo (2) 14 (1965), 95–118.
— Existence in the large of periodic solutions of hyperbolic partial differential equations, Arch. Rational Mech. Anal. 20 (1965), 170–190.
— Smoothness properties of periodic solutions in the large of nonlinear hyperbolic differential systems, Funkcial. Ekvac. 9 (1966), 325–338.
— A nonlinear problem in potential theory, Michigan Math. J. 16 (1969), 3–20.
— Functional analysis and differential equations, SIAM Studies in Applied Mathematics 5 (1969), 143–155.
L. Cesari and J.K. Hale, Second order linear differential systems with periodic L-integrable coefficients, Riv. Mat. Univ. Parma 2 (1954), 55–61; 6 (1955), 159.
— — A new sufficient condition for periodic solutions of weakly nonlinear differential systems, Proc. Amer. Math. Soc. 8 (1957), 757–764.
P.A.T. Christopher, A new class of subharmonic solutions to Duffing's equation, Co A.Rep. 195, The College of Aeronautics, Cranfield, Bedford, England, 1967.
— An extended class of subharmonic solutions to Duffing's equations, Co A. Rep. 99, ibid. 1967.
— The response of a second order nonlinear system to a step-function disturbance, Co A. Report 205, ibid. 1969.
J. Cronin, Fixed Points and Topological Degree in Nonlinear Analysis, Amer. Math. Soc., Providence, 1964.
R.A. Gambill, Stability criteria for linear differential systems with periodic coefficients, Riv. Mat. Univ. Parma 5 (1954), 169–181.
— Criteria for parametric instability for linear differential systems with periodic coefficients, Riv. Mat. Univ. Parma 6 (1955), 37–43.
— A fundamental system of real solutions for linear differential systems with periodic coefficients, Riv. Mat. Univ. Parma 7 (1956), 311–319.
R.A. Gambill and J.K. Hale, Subharmonic and ultraharmonic solutions for weakly nonlinear systems, J. Rational Mech. Anal. 5 (1956), 353–398.
A Halanay, Differential Equations, Academic Press, 1966 (particularly pp. 308–317).
J.K. Hale, Evaluations concerning products of exponential and periodic functions, Riv. Mat. Univ. Parma 5 (1954), 63–81.
— On boundedness of the solutions of linear differential systems with periodic coefficients, Riv. Mat. Univ. Parma 5 (1954), 137–167.
— Periodic solutions of nonlinear systems of differential equations, Riv. Mat. Univ. Parma 5 (1954), 281–311.
— On a class of linear differential equations with periodic coefficients, Illinois J. Math. 1 (1957), 98–104.
— Linear systems of first and second order differential equations with periodic coefficients, Illinois J. Math. 2 (1958), 586–591.
— Sufficient conditions for the existence of periodic solutions of systems of weakly nonlinear first and second order differential equations, J. Math. Mech. 7 (1958), 163–172.
— A short proof of a boundedness theorem for linear differential systems with periodic coefficients, Arch. Rational Mech. Anal. 2 (1959), 429–434.
— On the behavior of the solutions of linear periodic differential systems near resonance points, Contributions to the Theory of Nonlinear Oscillations, Vol. V, pp. 55–89, Princeton Univ. Press, Princeton, 1960.
— On the stability of periodic solutions of weakly nonlinear periodic and autonomous differential systems, Contributions to the Theory of Nonlinear Oscillations, Vol. V, pp. 91–113, Princeton Univ. Press, Princeton, 1960.
— On the characteristic exponents of linear periodic differential systems, Bol. Soc. Mat. Mexicana (2) 5 (1960), 58–66.
— Oscillations in Nonlinear Systems, McGraw-Hill, New York, 1963.
— Periodic solutions of a class of hyperbolic equations, Arch. Rational Mech. Anal. 23 (1967), 380–398.
Ordinary Differential Equations, Wiley-Interscience, 1969.
J.K. Hale, S. Bancroft, and D. Sweet, Alternative problems for nonlinear functional equations, J. Differential Equations 4 (1968), 40–56.
W.S. Hall, Periodic solutions of a class of weakly nonlinear evolution equations, Arch. Rational Mech. Anal. (to appear).
W.A. Harris, Y. Sibuya, and J. Weinberg, Holomorphic solutions of linear differential systems at singular points, Arch. Rational Mech. Anal. 35 (1969), 245–248.
W.A. Harris, Holomorphic solutions of nonlinear differential equations at singular points, SIAM Studies in Applied Mathematics 5 (1969), 184–187.
C. Imaz, Sobre ecuaciones differenciales lineales periodicas con un parametro pequeno, Bol. Soc. Mat. Mexicana (2) 6 (1961), 19–51.
H.W. Knobloch, Remarks on a paper of Cesari on functional analysis and nonlinear differential equations, Michigan Math. J. 10 (1963), 417–430.
— Eine neue Methode zur Approximation von periodischen Loesungen nicht linear Differentialgleichungen zweiter Ordnung, Math. Z. 82 (1963), 177–197.
— Comparison theorems for nonlinear second order differential equations, J. Differential Equations 1 (1965), 1–25.
E.M. Landesman and A.C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970), 609–623.
J. Locker An existence analysis for nonlinear equations in Hilbert space, Trans. Amer. Math. Soc. 128 (1967), 403–413.
— An existence analysis for nonlinear boundary value problems, SIAM J. Appl. Math. 19 (1970), 199–207.
J. Mawhin, Application directe de la méthode de Cesari a l'étude des solutions périodiques de systèmes différentiels faiblement non linéaires, Bull. Soc. Roy. Sci. Liège 36 (1967), 193–210.
— Solutions périodiques de systèmes différentiels faiblement non linéaires, ibid. 36 (1967), 491–499.
— Familles de solutions périodiques dans les systèmes différentiels faiblement non linéaries, ibid. 36 (1967), 500–509.
— Degré topologique et solutions périodiques des les systèmes différentiels non linéaires, ibid. 38 (1969), 308–398.
A. Naparstek, (a) Composition of functions in certain Sobolev spaces (to appear). — (b) Periodic solutions of weakly nonlinear wave equations in Sobolev spaces (to appear). — (c) On the Cesari method and periodic perturbation problem for certain hyperbolic equations (to appear).
C. Perello, A note on periodic solutions of nonlinear differential equations with time lags, Differential Equations and Dynamical Systems (J.P. LaSalle, ed.) Academic Press, 1967, 185–188.
D. Petrovanu, (a) Solutions périodiques pour certaines équations hyperboliques, An. Sti. Univ. Iasi 14 (1968), 327–357. — (b) Periodic solutions of the Tricomi problem, Michigan Math. J. 16 (1969), 331–348.
A.M. Rodionov, Periodic solutions of nonlinear differential equations with time lag, Trudy Sem. Differential Equations Lumumba University 2 (1963), 200–207 (Russian).
C.D. Stocking, Nonlinear boundary value problems in a circle and related questions on Bessel functions, Thesis, University of Michigan, 1971.
S.A. Williams, A connection between the Cesari and Leray-Schauder methods, Michigan Math. J. 15 (1968), 441–448.
Other references
L. Hormander, Linear Partial Differential Operators, 3rd Ed., Springer, 1969.
J. Leray and J. Schauder, Topologie et équations fonctionelles, Ann. Sci. École Norm. Sup. 51 (1934), 45–78.
L.A. Liusternik and V.J. Sobolev, Elements of Functional Analysis, Ungar, 1961.
L.V. Ovcjannikov, A singular operator in a scale of Banach spaces, Soviet Math. Dokl. 163 (1965), 1025–1028.
I.G. Petrovsky, Lectures on Partial Differential Equations, Interscience, 1954.
F. Treves, Ovcjannikov theorem and hyperdifferential operators, Inst. Mat. Pura Appl., Rio de Janeiro, 1968.
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Cesari, L. (1971). Functional analysis and boundary value problems. In: Hsieh, P.F., Stoddart, A.W.J. (eds) Analytic Theory of Differential Equations. Lecture Notes in Mathematics, vol 183. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0060418
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DOI: https://doi.org/10.1007/BFb0060418
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