Theory of Third-Order Differential Equations

This book discusses the theory of third-order differential equations. Most of the results are derived from the results obtained for third-order linear homogeneous differential equations with constant coefficients. M. Gregus, in his book written in 1987, only deals with third-order linear differential equations. These findings are old, and new techniques have since been developed and new results obtained.

Chapter 1 introduces the results for oscillation and non-oscillation of solutions of third-order linear differential equations with constant coefficients, and a brief introduction to delay differential equations is given. The oscillation and asymptotic behavior of non-oscillatory solutions of homogeneous third-order linear differential equations with variable coefficients are discussed in Ch. 2. The results are extended to third-order linear non-homogeneous equations in Ch. 3, while Ch. 4 explains the oscillation and non-oscillation results for homogeneous third-order nonlinear differential equations. Chapter 5 deals with the z-type oscillation and non-oscillation of third-order nonlinear and non-homogeneous differential equations. Chapter 6 is devoted to the study of third-order delay differential equations. Chapter 7 explains the stability of solutions of third-order equations. Some knowledge of differential equations, analysis and algebra is desirable, but not essential, in order to study the topic.

“This monograph is devoted to the qualitative behavior of solutions (oscillation, non-oscillation, stability, asymptotic behaviors, etc.) of various ordinary differential equations of third order with and without delay. It is suitable for those mathematicians and of other sciences dealing with mathematics and engineering. … In summary, this monography is useful for researches investigating the qualitative behavior of solutions of ordinary differential equations of third order.” (Cemil Tunç, zbMATH 1308.34002, 2015)

“This is a comprehensive monograph on third-order differential equations, spanning more than 500 pages and collecting recent results on qualitative behavior of solutions of these equations. … the book may serve as a basis for understanding the oscillatory and asymptotic theory of third-order differential equations, offering a comprehensive account of today’s knowledge in the field and a rich source of references for specialists.” (Zuzana Došlá, Mathematical Reviews, November, 2014)

SESHADEV PADHI is associate professor of mathematics at Birla Institute of Technology, Mesra, Ranchi, Jharkhand, India. He received his PhD on the topic “oscillation theory of third-order differential equations”. He was awarded the Better Opportunities for Young Scientists in Chosen Areas of Science and Technology (BOYSCAST) fellowship by the Department of Science and Technology (DST), Government of India in 2004, to visit Mississippi State University, USA, where he subsequently did his postdoctoral research. In addition, Dr. Padhi has visited several institutes of international repute: Florida Institute of Technology, Melbourna, Florida, USA, in 2006 to work in collaboration with Prof. T. Gnanabhaskar; Texas State University, San Marcos, Texas, USA, in 2009 to work in collaboration with Prof. Julio G.Dix; University of Tennessee, Chattanooga, Tennessee, USA, in 2011, 2012 and 2013 to work in collaboration with Prof. John R. Graef; University of Szeged, Szeged, Hungary, in 2007 and 2011 to work in collaboration with Prof. Tibor Krisztin. Besides, he also visited Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland, under Borel Set Theory Programme in 2005, and many other countries to deliver lectures in international conferences and workshops. Dr. Padhi has published more than 60 research papers in international journals of repute and has been working as referee for Mathematical Review for more than 30 international journals since 2006.

SMITA PATI is a postdoctoral fellow in the Department of Applied Mathematics, Birla Institute of Technology, Mesra, Ranchi, India, working under the postdoctoral grant from the National Board for Higher Mathematics (NBHM), Department of Atomic Energy (DAE), Government of India. Besides, Dr. Pati has been working as a referee for seven international mathematics journals and for Mathematical Review. She was awarded with Marqui's Whos Who in the year 2012 and has published 15 research papers in international journals of repute, in addition to publishing several research papers in collaboration with Prof. Julio G. Dix, Texas State University, USA, and John R. Graef, University of Tennessee, Chattanooga, Tennessee, USA. She has visited several institutes in Hungary and Italy to deliver lectures in different conferences on Differential Equations.