## Asymptotic equivalence of second order differential equations.(English)Zbl 0555.34048

The asymptotic properties of the equation (1) $$[p(t)x']'+q(t)x=0$$ and the perturbed equation (2) $$[p(t)y']'+q(t)y=f(t,y,y')$$ are compared. It is supposed that $$p,q\in C^ 0(j)$$, $$p>0$$ on j, $$f\in C^ 0(j\times R^ 2)$$, $$j=[t_ 0,+\infty)$$. Let $$\mu_ 0$$, $$\mu_ 1\in C^ 0(j)$$. The equations (1) and (2) are called $$\mu_ 0$$-asymptotically equivalent $$(\mu_ 0$$-a.e.) if for each solution x(t) of (1) there exists a solution y(t) of (2) defined for large t such that $$x(t)$$-y(t)$$=o[\mu_ 0(t)]$$, $$t\to +\infty$$, and conversely. Analogously the weak $$\mu_ 1$$- a.e. $$(x'$$-y’$$=o(\mu_ 1))$$ and the strong $$(\mu_ 0,\mu_ 1)$$-a.e. $$(x$$-y$$=o(\mu_ 0)$$, $$x'$$-y’$$=o(\mu_ 1))$$ is defined. It is assumed that there exists a nonnegative function F(t,r,s), $$F\in C^ 0(j\times R^ 2_+)$$, nondecreasing in r and s for each fixed $$t\in j$$ such that $$| f(t,r,s)| \leq F(t,| r|,| s|)$$. Then general theorems on a.e. are proved using Tikhonov’s fixed point theorem. Also some special perturbations are considered as $$F=h(t)| r|$$, $$F=k(t)| s|$$ and $$F=a(t)g(| r|)$$, $$F=b(t)h(| s|)$$. In the last part the more detailed results on a.e. are obtained for nonoscillatory equation (1). Here the concept of the principal solution is used. A large list of references is contained.

### MSC:

 3.4e+11 Perturbations, asymptotics of solutions to ordinary differential equations

### Keywords:

asymptotic equivalence; nonoscillatory equation
Full Text:

### References:

  Wintner A.: Small perturbations. Am. J. Math. 67 (1945), 417-430. · Zbl 0063.08285  Levinson N.: The asymptotic behavior of a system of linear differential equations. Am. J. Math. 68 (1946), 1-6. · Zbl 0061.19706  Weyl H.: Comment on the preceding paper. Am. J. Math. 68 (1946), 7-12. · Zbl 0061.19707  Wintner A.: Linear variation of constants. Am. J. Math. 68 (1946), 185-213. · Zbl 0063.08291  Jakubovič V. A.: On asymptotic behaviour of solutions of a system of differential equations. (Russian), Mat. Sbornik, T 28 (70), 1951, 217-240.  Brauer F.: Asymptotic equivalence and asymptotic behavior of linear systems. Michigan J. Math. P (1962), 33-43. · Zbl 0111.08603  Onuchic N.: Relationship among the solutions of two systems of ordinary differential equations. Michigan J. Math. 10 (1963), 129-139. · Zbl 0115.30301  Onuchic N.: Nonlinear perturbation of a linear system of ordinary differential equations. Michigan J. Math. 11 (1964), 237-242. · Zbl 0126.30003  Kato J.: The asymptotic relations of two systems of ordinary differential equations. Contr. Diff. Eqs. i(1964), 141-161. · Zbl 0137.28101  Brauer F.: Nonlinear differential equations with forcing terms. Proc. Am. Math. Soc. 15 (1964), 758-765. · Zbl 0126.30004  Onuchic N.: Asymptotic relationships at infinity between the solutions of two systems of ordinary differential equations. J. Diff. Eq. 3 (1967), 47-58. · Zbl 0153.11901  Brauer F., Wong: On the asymptotic relationship between solutions of two systems of ordinary differential equations. J. Diff. Eq. 6 (1969), 527-543. · Zbl 0185.16601  Brauer F., Wong: On asymptotic behavior of perturbed linear systems. J. Diff. Eq. 6 (1969), 142-153. · Zbl 0201.11703  Hallam T. G.: On asymptotic equivalence of the bounded solutions off two systems of differential equations. Michigan J. Math. 16 (1969), 353-363. · Zbl 0191.10401  Hallam T. G., Onuchic N.: Asymptotic relations between perturbed linear systems of ordinary differential equations. Pac. J. Math. 45 (1973), 187-199. · Zbl 0261.34039  Hallam T. G.: Asymptotic relationships between the solutions of two second order differential equations. Ann. Polon. Math. 24 (1971), 295-300. · Zbl 0217.40103  Švec M.: Asymptototic relationship between solutions of two systems of differential equations. Czech. Math. J. 24 (99), Nr. 1 (1974), 44-58.  Ráb M.: Asymptotic relationships between the solutions of two systems of differential equations. Ann. Polon. Math. 30 (1974), 119-124.  Kitamura Y.: Remarks on the asymptotic relationships between solutions of two systems of ordinary differential equations. Hiroshima Math. J. 6, No. 2 (1976), 403 - 420. · Zbl 0337.34045  Marlin Struble: Asymptotic equivalence of nonlinear systems. J. Diff. Eq. 6 (1969), 578- 596. · Zbl 0218.34036  Hallam T. G.: Asymptotic expansions in certain second order non-homogeneous differential equations. Mathematika 15 (1968), 30-38. · Zbl 0162.11704  Charlamov P. V.: On estimates of solutions of a system of differential equations. (Russian), Ukrainskij mat. zurnal 7 (1955), 471-473.  Bihari I.: A generalisation of a lemma of Bellman and its applications to uniqueness problems of differential equations. Acta Math. Acad. Sci. Hungar 7 (1956), 71 - 94. · Zbl 0070.08201  Lakshmikantham V., Leela S.: Differential and Integral Inequalities. Volume I. New York and London, Academic Press 1969. · Zbl 0177.12403  Hartman P.: Ordinary Differential Equations. New York-London-Sydney, John Wiley & Sons, Inc. 1964. · Zbl 0125.32102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.