Solvability of boundary value problems with Riemann-Stieltjes 忖-integral conditions for second-order dynamic equations on time scales at resonance

作者：Li Yongkun; Shu Jiangye

来源：Advances in Difference Equations, 2011.

• boundary value problem with Riemann-Stieltjes &Delta
• -integral conditions
• resonance
• time scales

In this paper, by making use of the coincidence degree theory of Mawhin, the existence of the nontrivial solution for the boundary value problem with Riemann-Stieltjes 忖-integral conditions on time scales at resonance x 忖 忖 ( t ) = f ( t , x ( t ) , x 忖 ( t ) ) + e ( t ) , a . e . t ﹋ [ 0 , T ] T , x 忖 ( 0 ) = 0 , x ( T ) = ÷ 0 T x 考 ( s ) 忖 g ( s ) is established, where f : [ 0 , T ] T ℅ ℅ ↙ satisfies the Carath谷odory conditions and e : [ 0 , T ] T ↙ is a continuous function and g : [ 0 , T ] T ↙ is an increasing function with ÷ 0 T 忖 g ( s ) = 1 . An example is given to illustrate the main results.