On the asymptotic integration of nonlinear differential equations

The aim of the present paper is twofold. Firstly, the paper surveys the literature concerning a specific topic in asymptotic integration theory of ordinary differential equations: the class of second order equations with Bihari-like nonlinearity. Secondly, some general existence results are established with regard to a condition that has been found recently to be of significant use in the theory of elliptic partial differential equations.

[1]  Erik Wahlén,et al.  Positive solutions of second-order differential equations , 2004 .

[2]  Angelo B. Mingarelli,et al.  Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions , 1983 .

[3]  O. Haupt.  Über Lösungen linearer Differentialgleichungen mit Asymptoten , 1942 .

[4]  Z. Opial.  Sur un système d'inégalités intégrales , 1957 .

[5]  F. Brauer,et al.  On the asymptotic relationships between solutions of two systems of ordinary differential equations , 1969 .

[6]  Man Kam Kwong,et al.  An application of integral inequality to second order nonlinear oscillation , 1982 .

[7]  P. Hartman.  Ordinary Differential Equations , 1965 .

[8]  Upper and lower solutions for singular problems with nonlinear boundary data , 2002 .

[9]  P. Waltman.  On the asymptotic behavior of solutions of a nonlinear equation , 1964 .

[10]  Richard Bellman,et al.  The boundedness of solutions of linear differential equations , 1947 .

[11]  On the asymptotic integration of ordinary differential equations , 1963 .

[12]  Yuri V. Rogovchenko,et al.  On asymptotic integration of a nonlinear second-order differential equation , 2006 .

[13]  Yuri V. Rogovchenko,et al.  Global existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations , 2002 .

[14]  H. Usami.  Global existence and asymptotic behavior of solutions of second-order nonlinear differential equations , 1987 .

[15]  Philip Hartman,et al.  On Non-Oscillatory Linear Differential Equations of Second Order , 1952 .

[16]  Ravi P. Agarwal,et al.  Existence theory for single and multiple solutions to singular positone boundary value problems , 2001 .

[17]  A Note on Tong Paper: The Asymptotic Behavior of a Class of Nonlinear Differential Equations of Second Order , 1990 .

[18]  M. S. P. Eastham,et al.  The asymptotic solution of linear differential systems : applications of the Levinson theorem , 1989 .

[19]  Manabu Naito.  Integral averages and the asymptotic behavior of solutions of second order ordinary differential equations , 1992 .

[20]  William F. Trench,et al.  Existence of global solutions with prescribed asymptotic behavior for nonlinear ordinary differential equations , 1985 .

[21]  Yuri V. Rogovchenko.  On the asymptotic behavior of solutions for a class of second order nonlinear differential equations , 1998 .

[22]  Ravi P. Agarwal,et al.  Infinite Interval Problems for Differential, Difference and Integral Equations , 2001 .

[23]  J. Tong,et al.  The asymptotic behavior of a class of nonlinear differential equations of second order , 1982 .

[24]  B. G Pachpatte,et al.  On some integral inequalities similar to Bellman-Bihari inequalities , 1975 .

[25]  F. V. Vleck,et al.  Stability and Asymptotic Behavior of Differential Equations , 1965 .

[26]  A. Constantin.  On the existence of positive solutions of second order differential equations , 2005 .

[27]  Z. Yin.  Monotone positive solutions of second-order nonlinear differential equations , 2003 .

[28]  W. F. Trench.  Systems of differential equations subject to mild integral conditions , 1983 .

[29]  W. F. Trench,et al.  Global Existence Theorems for Solutions of Nonlinear Differential Equations with Prescribed Asymptotic Behaviour , 1985 .

[30]  H. Weyl.  Comment on the Preceding Paper , 1946 .

[31]  R. Bellman.  Stability theory of differential equations , 1953 .

[32]  C. Philos.  Asymptotic behaviour of a class of nonoscillatory solutions of differential equations with deviating arguments , 1983 .

[33]  A multidimensional nonlinear Gronwall inequality , 1974 .

[34]  A. Orpel.  On the existence of positive radial solutions for a certain class of elliptic BVPs , 2004 .

[35]  A. Constantin.  Existence of positive solutions of quasilinear elliptic equations , 1996, Bulletin of the Australian Mathematical Society.

[36]  GLOBAL SOLUTIONS OF NONLINEAR PERTURBATIONS OF LINEAR DIFFERENTIAL EQUATIONS , 1992 .

[37]  Mary L. Boas,et al.  The growth of solutions of a differential equation , 1942 .

[38]  T. H. Gronwall.  Note on the Derivatives with Respect to a Parameter of the Solutions of a System of Differential Equations , 1919 .

[39]  A. Constantin.  Global existence of solutions for perturbed differential equations , 1995 .

[40]  Octavian G. Mustafa,et al.  Initial value problem with infinitely many linear-like solutions for a second-order differential equation , 2005, Appl. Math. Lett..

[41]  I. K. Purnaras,et al.  Asymptotic to polynomials solutions for nonlinear differential equations , 2004 .

[42]  P. Hartman.  Unrestricted solution fields of almost-separable differential equations , 1948 .

[43]  Ravi P. Agarwal,et al.  Existence and asymptotic behavior of solutions of a boundary value problem on an infinite interval , 2005, Math. Comput. Model..

[44]  Mats Ehrnström.  On radial solutions of certain semi-linear elliptic equations , 2006 .

[45]  C. Avramescu.  Sur l’existence des solutions convergentes des systèmes d’équations différentielles non linéaires , 1969 .

[46]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[47]  W. F. Trench.  Asymptotic behavior of solutions of Lu = g(t, u,..., u(k−1)) , 1972 .

[48]  I. T. Kiguradze,et al.  Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations , 1992 .

[49]  G. Seifert.  Global asymptotic behavior of solutions of positively damped Liénard equations , 1990 .

[50]  Hiroyuki Usami,et al.  Asymptotic behavior of solutions of a class of second order nonlinear differential equations , 1986 .

[51]  R. Bellman.  The stability of solutions of linear differential equations , 1943 .

[52]  B. Philip.  ON THE ASSIGNMENT OF ASYMPTOTIC VALUES FOR THE SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER. , 1955 .

[53]  D. O’Regan.  Upper and lower solutions for singular problems arising in the theory of membrane response of a spherical cap , 2001 .

[54]  Olivia Lipovan.  ON THE ASYMPTOTIC BEHAVIOUR OF THE SOLUTIONS TO A CLASS OF SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS , 2003, Glasgow Mathematical Journal.

[55]  Yuri V. Rogovchenko,et al.  ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS , 2000 .

[56]  Yuri V. Rogovchenko,et al.  Global Existence and Asymptotic Behavior of Solutions of Nonlinear Differential Equations , 2004 .

[57]  Yuri V. Rogovchenko,et al.  Asymptotic integration of a class of nonlinear differential equations , 2006, Appl. Math. Lett..

[58]  Z. Zhao.  Positive solutions of nonlinear second order ordinary differential equations , 1994 .

[59]  M. Grammatikópoulos.  Oscillatory and asymptotic behavior of differential equations with deviating arguments , 1976 .

[60]  I. Bihari.  A generalization of a lemma of bellman and its application to uniqueness problems of differential equations , 1956 .

[61]  J. Wong.  On Two Theorems of Waltman , 1966 .

[62]  O. Mustafa.  ON THE EXISTENCE OF SOLUTIONS WITH PRESCRIBED ASYMPTOTIC BEHAVIOUR FOR PERTURBED NONLINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER , 2005, Glasgow Mathematical Journal.

[63]  Yuri V. Rogovchenko,et al.  Asymptotic behaviour of solutions for second order nonlinear autonomous differential equations , 1997 .

[64]  Ravi P. Agarwal,et al.  Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations , 2002 .

[65]  A. Constantin.  Positive Solutions of Quasilinear Elliptic Equations , 1997 .

[66]  I. Bihari.  Researches of the boundedness and stability of the solutions of non-linear differential equations , 1957 .

[67]  Global Solutions Approaching Lines at Infinity to Second Order Nonlinear Delay Differential Equations , 2007 .

[68]  O. Haupt.  Über das asymptotische Verhalten der Lösungen gewisser linearer gewöhnlicher Differentialgleichungen , 1942 .

[69]  Asymptotic Integration of Ordinary Differential Equations , 1983 .

[70]  Mats Ehrnström.  Positive solutions for second-order nonlinear differential equations , 2006 .

[71]  F. Dannan.  Integral Inequalities of Gronwall-Bellman-Bihari Type and Asymptotic Behavior of Certain Second Order Nonlinear Differential Equations , 1985 .

[72]  P. Hartman,et al.  On the Asymptotic Behavior of the Solutions of a Non-Linear Differential Equation , 1946 .

[73]  Bounds for solutions of nonlinear differential systems , 1957 .

[74]  Aurel Wintner.  Comments on “flat” oscillations of low frequency , 1957 .

[75]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .