Existence of Solutions for a Class of Damped Vibration Problems on Time Scales

• Main Authors: Li Yongkun, Zhou Jianwen
• Format: Article
• Language: English
• Published: SpringerOpen 2010-01-01
• Series: Advances in Difference Equations
• Online Access: http://www.advancesindifferenceequations.com/content/2010/727486
• ISSN: 1687-1839; 1687-1847

<p/> <p>We present a recent approach via variational methods and critical point theory to obtain the existence of solutions for a class of damped vibration problems on time scale <inline-formula><graphic file="1687-1847-2010-727486-i1.gif"/></inline-formula>, <inline-formula><graphic file="1687-1847-2010-727486-i2.gif"/></inline-formula><inline-formula><graphic file="1687-1847-2010-727486-i3.gif"/></inline-formula>-a.e. <inline-formula><graphic file="1687-1847-2010-727486-i4.gif"/></inline-formula>, <inline-formula><graphic file="1687-1847-2010-727486-i5.gif"/></inline-formula> where <inline-formula><graphic file="1687-1847-2010-727486-i6.gif"/></inline-formula> denotes the delta (or Hilger) derivative of <inline-formula><graphic file="1687-1847-2010-727486-i7.gif"/></inline-formula> at <inline-formula><graphic file="1687-1847-2010-727486-i8.gif"/></inline-formula>, <inline-formula><graphic file="1687-1847-2010-727486-i9.gif"/></inline-formula>, <inline-formula><graphic file="1687-1847-2010-727486-i10.gif"/></inline-formula> is the forward jump operator, <inline-formula><graphic file="1687-1847-2010-727486-i11.gif"/></inline-formula> is a positive constant, <inline-formula><graphic file="1687-1847-2010-727486-i12.gif"/></inline-formula>, <inline-formula><graphic file="1687-1847-2010-727486-i13.gif"/></inline-formula>, and <inline-formula><graphic file="1687-1847-2010-727486-i14.gif"/></inline-formula>. By establishing a proper variational setting, three existence results are obtained. Finally, three examples are presented to illustrate the feasibility and effectiveness of our results.</p>