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We discuss the use of inequalities to obtain the solution of certain variational problems on time scales.

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.

We study more general variational problems on time scales. Previous results are generalized by proving necessary optimality conditions for (i) variational problems involving delta derivatives of more than the first order, and (ii) problems of the calculus of variations with delta-differential side conditions (Lagrange problem of the calculus of variations… (More)