Existence principle for higher-order nonlinear differential equations with state-dependent impulses via fixed point theorem
Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, 77146, Czech Republic
Boundary Value Problems 2014, 2014:118 doi:10.1186/1687-2770-2014-118
Dedicated to Professor Ivan Kiguradze for his merits in mathematical sciencesPublished: 14 May 2014
The paper provides an existence principle for a general boundary value problem of the form , a.e. , , , with the state-dependent impulses , where the impulse points t are determined as solutions of the equations , , . Here, , , the functions , , are Lebesgue integrable on and satisfies the Carathéodory conditions on . The impulse functions , , , and the barrier functions , , are continuous on and , respectively. The functionals , , are linear and bounded on the space of left-continuous regulated (i.e. having finite one-sided limits at each point) on vector functions. Provided the data functions h and are bounded, transversality conditions which guarantee that each possible solution of the problem in a given region crosses each barrier at the unique impulse point are presented, and consequently the existence of a solution to the problem is proved.
MSC: 34B37, 34B10, 34B15.