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Periodic solutions of the third order parametric differential equations involving large nonlinearities. (English) Zbl 0753.34025

Summary: Sufficient conditions are obtained for the existence of periodic solutions to the triad of parametric third order equations \(x'''+L(t,x)=a(x'')\), \(x'''+L(t,x)=b(x')\), \(x'''+L(t,x)=c(x)\), where \(L(t,x):=f(x)x''+g(t)x'+h(x)+p(t)+q(t,x,x',x'')\). The main endeavour is that the involved nonlinearities \(a(x'')\) or \(b(x')\) or \(c(x)\) be not restricted too much. Uniqueness and instability criteria are established in special cases as well.

MSC:

34C25 Periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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[1] VILLARI G.: Criteri di esistenza di soluzioni periodiche per particolari classi di equazioni differenziali del terz’ ordine non lineari. Matematiche (Catania) 19 (1964), 96-113. · Zbl 0128.08702
[2] VILLARI G.: Contributi alio studio dell’ esistenza di soluzioni periodiche per i sistemi di equazioni differenziali ordinarie. Ann. Mat. Pura Appl. 69 (1965), 171-190. · Zbl 0132.31902
[3] REISSIG R., SANSONE G., CONTI R.: Nichtlineare Differentialgleichungen höherer Ordnung. Cremonese, Roma, 1969. · Zbl 0172.10801
[4] EZEILO J. O. C : A further result on the existence of periodic solutions on the equation x”’ + ax” + bx’ + h(x) = p(t, x, x’, x”). Math. Proc. Camb. Philos. Soc. 77 (1975), 547-551. · Zbl 0307.34038
[5] EZEILO J. O. C.: Further results on the existence of periodic solutions of a certain third order differential equation. Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 64 (1978), 48-58. · Zbl 0405.34042
[6] EZEILO J. O. C.: Periodic solutions of third-order differential equations in the past twenty five years or so. Invited paper presented at the 2nd Pan-African Congress of the African Mathematical Union, 23-29 March 1986, University of Jos, Nigeria.
[7] KONSTANTINOV M. S.: Existence of periodic solutions of certain differential equation. (Russian), Differ. Uravn., Sb. Tr. Mat. Kafedr Pedinst. RSFSR 9 (1977), 48-56.
[8] KONSTANTINOV M. S.: On a third order equation. (Russian), Differ. Uravn., Sb. Tr. Mat. Kafedr Pedinst. RSFSR 11 (1978), 93-99.
[9] KONSTANTINOV M. S.: Some sufficient conditions for existence of periodic and bounded solutions of the equation x”’ + ax’ + g(x’) + cx = \Psi (t, x, x’,x”). (Russian). Diff. Urav. Qualitative Theory, ed I. P. Makarov. Rjazan. Gos. Ped. Inst., Rjazan, 1984, pp. 92-100.
[10] ANDRES J., VORÁČEK J.: Periodic solutions to a nonlinear parametric differential equation of the third order. Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 77 (1984), 81-86. · Zbl 0607.34039
[11] MEHRI B.: On the existence of periodic solutions for certain nonlinear differential equation. Rev. Roum. Math. Pures Appl. 23 (1978), 1367-1373. · Zbl 0405.34040
[12] ANDRES J.: Periodic boundary value problems for certain nonlinear differential equations of the third order. Math. Slovaca 35 (1985), 305-309. · Zbl 0591.34037
[13] EZEILO J. O. C., ONYA J.: Non-resonant oscillations for third-order differential equations. J. Nig. Math. Soc. 3 (1984), 83-96. · Zbl 0599.34055
[14] EZEILO J. O. C., NKASHAMA M. N.: Non-uniform non-resonance and existence of periodic solutions of some third order non-linear ordinary differential equations. Nonlinear Anal., Theory Methods Appl. 12 (1988), 1029-1046. · Zbl 0676.34021
[15] EZEILO J. O. C., OMARI P.: Non-resonant oscillations for some third-order differential equations II. ICTP, Trieste IC/87/374 (1987).
[16] FONDA A., HABETS P.: Periodic solutions of asymptotically positively homogeneous differential equations. Rapp., Semin. Math. Appl. Mec, Univ. Cathol. Louvain No. 130 (1988). · Zbl 0692.34041
[17] AFUWAPE A. U., OMARI P., ZANOLIN F.: Nonlinear perturbations of differential operators with nontrivial kernel and applications to third order periodic boundary value problems. J. Math. Anal. Appl. 143 (1989), 35-56. · Zbl 0695.47044
[18] AFTABIZADEH A. R., XU J.-M., GUPTA C. P.: Periodic boundary value problems for third order ordinary differential equations. Nonlinear Anal., Theory Methods Appl. 14 (1990), 1-10. · Zbl 0706.34018
[19] BEBERNES J., GAINES R., SCHMITT K.: Existence of periodic solutions for third and fourth order ordinary differential equations via coincidence degree. Ann. Soc. Sci. Brux., Ser. I (T) 88 (1974), 25-36. · Zbl 0695.47044
[20] KONSTANTINOV M. S.: Some sufficient conditions for the existence of periodic solutions of third order nonlinear differential equations. (Russian), Differ. Uravn., Sb. Tr. Mat. Kafedr Pedinst. RSFSR 9 (1977), 57-61.
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