This is a preview of subscription content, log in via an institution to check access.
References
R. Bellman,Stability theory of differential equations (New York, 1953).
O. Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen,Math. Zeitschrift,29 (1929), pp. 129–160.
See the notations in paragraph 1.
R. Bellman, The stability of solutions of linear differential equations,Duke Math. Journal,10 (1943), pp. 643–647.
I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations,Acta Math. Acad. Sci. Hung.,7 (1956), pp. 83–94.
R. Bellman, loc. cit.,Stability theory of differential equations (New York, 1953) p. 643.
If ω(u) vanishes in some region ofu=0, then were quire\(\int\limits_0^\infty {|| B (t)||} dt< \propto \). Assuming\(\omega (u) \equiv \mathop {\max }\limits_{|| \zeta || \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} u} ||\varphi (\zeta )||\), the function ω(u) is non-decreasing and satisfies the condition of the lemma. Taking\(\omega (u) \mathbin{\lower.3ex\hbox{$\buildrel>\over{\smash{\scriptstyle=}\vphantom{_x}}$}} \mathop {\max }\limits_{|| \zeta || \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} u} ||\varphi (\zeta )||\), we must suppose this explicitly.
If ω(u)=0 for 0≦u≦σ, then assuming ||c||<δ, c1|| c||<δ and supposing that |z| attains the value σ fort=t 0>0 we get from (6′) δ≦c1||c|+0<δ what is a contradiction. Therefore |z|<σ fort≧0.
The function ω(u)=u α is playing a part e.g. in thescalar equationz′+z=b(t)z α.
Originally more was assumed.J. Czipszer simplified the theorem and the proof too.
Compare toBellman's book, p. 36.
As we shall see immediately the boundedness of |Y −1(t)| ought not to be supposed. It is a consequence of the boundedness of |Y(t)| and the condition 1. The functionY −1(t) satisfies the adjoint equationd Z/dt=−ZA(t).
Here is\(Y = \left( {\begin{array}{*{20}c} {y_1 } & {y_2 } \\ {y_1^\prime } & {y_2^\prime } \\\end{array}} \right)\), wherey 1 andy 2 are the solutions of (14) withy 1(0)=1,y′ 1(0)=0,y′ 2(0)=0,y′ 2(0)=1.
L. Cesari, Sulla stabilità delle soluzioni delle equazioni differenziali lineari,Annali R. Scuola Norm. Sup. Pisa, Ser. 2,8 (1939), pp. 131–148.
SeeBellman's book, pp. 112–115. At these equations the degree of non-linearity is so great that a discussion by matrices is impossible.
In connection with the same equation (27) and analogous types oscillation and monotonity problems will be discussed in a forthcoming paper of the author.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bihari, I. Researches of the boundedness and stability of the solutions of non-linear differential equations. Acta Mathematica Academiae Scientiarum Hungaricae 8, 261–278 (1957). https://doi.org/10.1007/BF02020315
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02020315