We discuss the use of inequalities to obtain the solution of certain variational problems on time scales.
Viewpoint Left ventricular torsion: feeling the heat (pages 71–72) Excitatory amino acid receptors in the dorsomedial hypothalamus are involved in the cardiovascular and behavioural chemoreflex responses (pages 73–84) Maturation-related changes in the pattern of renal sympathetic nerve activity from fetal life to adulthood (pages 85–93) Angiotensin AT1… (More)
In this paper, we investigate the existence of nontrivial solutions to the nonlinear q-fractional boundary value problem (D α q y)(x) = −f (x, y(x)), 0 < x < 1, y(0) = 0 = y(1), by applying a fixed point theorem in cones.
We prove a version of the Euler-Lagrange equations for certain problems of the calculus of variations on time scales with higher-order delta derivatives.
We introduce a discrete-time fractional calculus of variations on the time scale hZ, h > 0. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They show that solutions to the considered fractional problems become the classical discrete-time… (More)
We prove some new retarded integral inequalities. The results generalize those in [
The calculus of variations is a classical subject which has gain throughout the last three hundred years a level of rigor and elegance that only time can give. In this note we show that, contrary to the classical field, available formulations and results on the recent calculus of variations on time scales are still at the heuristic level.
We prove a necessary optimality condition for isoperimetric problems on time scales in the space of delta-differentiable functions with rd-continuous derivatives. The results are then applied to Sturm-Liouville eigen-value problems on time scales.
We establish some nonlinear integral inequalities for functions defined on a time scale. The results extend some previous Gronwall and Bihari type inequalities on time scales. Some examples of time scales for which our results can be applied are provided. An application to the qualitative analysis of a nonlinear dynamic equation is discussed.