We consider a nonlinear integral transform and show that the transform acts as a homeomorphism between certain metric spaces of positive functions. We apply the result to the inverse bifurcation problem of determining the nonlinear term of a certain nonlinear Sturm-Liouville problem from its first bifurcating branch, and we establish the well-posedness of the inverse problem. An application to an inverse problem of determining a restoring force from a time-map is also given.
@article{ASNSP_2011_5_10_4_863_0, author = {Kamimura, Yutaka}, title = {A nonlinear integral transform and a global inverse bifurcation theory}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {863--911}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 10}, number = {4}, year = {2011}, mrnumber = {2932896}, zbl = {1253.44003}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2011_5_10_4_863_0/} }
TY - JOUR AU - Kamimura, Yutaka TI - A nonlinear integral transform and a global inverse bifurcation theory JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2011 SP - 863 EP - 911 VL - 10 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2011_5_10_4_863_0/ LA - en ID - ASNSP_2011_5_10_4_863_0 ER -
%0 Journal Article %A Kamimura, Yutaka %T A nonlinear integral transform and a global inverse bifurcation theory %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2011 %P 863-911 %V 10 %N 4 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2011_5_10_4_863_0/ %G en %F ASNSP_2011_5_10_4_863_0
Kamimura, Yutaka. A nonlinear integral transform and a global inverse bifurcation theory. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 4, pp. 863-911. http://www.numdam.org/item/ASNSP_2011_5_10_4_863_0/
[1] B. Alfawicka, Inverse problems connected with periods of oscillations described by , Ann. Polon. Math. 44 (1984), 297–308. | EuDML | MR | Zbl
[2] B. Alfawicka, Inverse problem connected with half-period function analytic at the origin, Bull. Pol. Acad. Sci. Math. 32 (1984), 267–274. | MR | Zbl
[3] A. Cima, A. Gasull and F. Mañosas, Period function for a class of Hamiltonian systems, J. Differential Equations 168 (2000), 180–199. | MR | Zbl
[4] A. Cima, F. Mañosas and J. Villadelprat, Isochronicity for several classes of Hamiltonian systems, J. Differential Equations 157 (1999), 373–413. | MR | Zbl
[5] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340. | MR | Zbl
[6] A. Denisov, Inverse problems for nonlinear differential equations, Dokl. Akad. Nauk. 307 (1989), 1040-1042. | MR | Zbl
[7] A. M. Denisov and A. Lorenzi, Identification of nonlinear terms in boundary value problems related to ordinary differential equations, Differential Integral Equations 5 (1992), 567–579. | MR | Zbl
[8] A. M. Denisov and A. Lorenzi, Recovering nonlinear terms with a priori unknown domains of definition in second order nonlinear differential equations, J. Inverse Ill-Posed Probl. 11 (2003), 137–159. | MR | Zbl
[9] R. Gorenflo and S. Vessella, “Abel Integral Equations”, Lecture Notes in Mathematics, Vol. 1461, Springer, Berlin, 1991. | MR | Zbl
[10] M. Henrard and F. Zanolin, Bifurcation from a periodic orbit in perturbed planar Hamiltonian systems, J. Math. Anal. Appl. 277 (2003), 79–103. | MR | Zbl
[11] K. Iwasaki and Y. Kamimura, An inverse bifurcation problem and an integral equation of the Abel type, Inverse Problems 13 (1997), 1015–1031. | MR | Zbl
[12] K. Iwasaki and Y. Kamimura, Convolution calculus for a class of singular Volterra integral equations, J. Integral Equations Appl. 11 (1999), 461–499. | MR | Zbl
[13] K. Iwasaki and Y. Kamimura, Inverse bifurcation problem, singular Wiener-Hopf equations, and mathematical models in ecology, J. Math. Biol. 43 (2001), 101–143. | MR | Zbl
[14] Y. Kamimura, An inverse problem in bifurcation theory, J. Differential Equations 106 (1993), 10–26. | MR | Zbl
[15] Y. Kamimura, An inverse problem in bifurcation theory, II, J. Math. Soc. Japan 46 (1994), 89–110. | MR | Zbl
[16] Y. Kamimura, An inverse problem of determining a nonlinear term in an ordinary differential equation, Differential Integral Equations 11 (1998), 341-359. | MR | Zbl
[17] Y. Kamimura, Conductivity identification in the heat equation by the heat flux, J. Math. Anal. Appl. 235 (1999), 192–216. | MR | Zbl
[18] Y. Kamimura, Global existence of a restoring force realizing a prescribed half-period, J. Differential Equations 248 (2010), 2562–2584. | MR | Zbl
[19] J. B. Keller, Inverse problems, Amer. Math. Monthly 83 (1976), 107–118. | MR
[20] L. D. Landau and E. M. Lifshitz, “Mechanics”, 3rd ed., Course of Mathematical Physics, Vol. 1, Pergamon Press, Oxford, 1976. | MR
[21] A. Lorenzi, An inverse spectral problem for a nonlinear ordinary differential equation, Applicable Analysis 46 (1992), 129–143. | MR | Zbl
[22] F. Mañosas and P. J. Torres, Two inverse problems for analytic potential systems, J. Differential Equations 245 (2008), 3664–3673. | MR | Zbl
[23] Z. Opial, Sur les périodes des solutions de l’équation differentielle , Ann. Polon. Math. 10 (1961), 49–72. | EuDML | MR | Zbl
[24] P. H. Rabinowitz, Nonlinear Sturm-Liouville problem for second order ordinary differential equations, Comm. Pure Appl. Math. 23 (1970), 939–961. | MR | Zbl
[25] P. H. Rabinowitz, A global theorem for nonlinear eigenvalue problems and applications, In: “Contributions to Nonlinear Functional Analysis”, E. H. Zarantonello (ed.), Academic Press, New York-London, 1971, 11–36. | MR | Zbl
[26] S. G. Samko, A. A. Kilbas and O. I. Marichev, “Fractional Integrals and Derivatives”, Gordon and Breach, Switzerland, 1993. | MR | Zbl
[27] R. Schaaf, “Global Solution Branches of Two Point Boundary Value Problems”, Lecture Notes in Mathematics, Vol. 1458, Springer, Berlin, 1990. | MR | Zbl
[28] J. T. Schwartz, “Nonlinear Functional Analysis”, Gorgon and Breach, New York, 1969. | MR | Zbl
[29] T. Shibata -inverse spectral problems for semilinear Sturm-Liouville problems, Nonlinear Anal. 69 (2008), 3601–3609. | MR | Zbl
[30] M. Urabe, The potential force yielding a periodic motion whose period is an arbitrary continuous function of the amplitude of the velocity, Arch. Ration. Mech. Anal. 11 (1962), 27–33. | MR | Zbl
[31] M. Urabe, Relation between periods and amplitudes of periodic solutions of , Funkc. Ekvacioj 6 (1964), 63–88. | MR | Zbl
[32] M. Urabe, “Nonlinear Autonomous Oscillations”, Academic Press, New York, 1967. | MR | Zbl
[33] P. Zhidkov, On an inverse eigenvalue problem for a semilinear Sturm-Liouville operator, Nonlinear Anal. 68 (2008), 639–644. | MR | Zbl