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Symmetric Quantum Calculus
Artur M. C. Brito da Cruz
We generalize the Hahn variational calculus by studying problems of the calculus of variations with higher-order derivatives. The symmetric quantum calculus is studied, namely the…
The Diamond Integral on Time Scales
Artur M. C. Brito da Cruz, Natalia Martins, Delfim F. M. Torres
We define a more general type of integral on time scales. The new diamond integral is a refined version of the diamond-alpha integral introduced in 2006 by Sheng et al. A mean valu…
Symmetric Differentiation on Time Scales
Artur M. C. Brito da Cruz, Natalia Martins, Delfim F. M. Torres
We define a symmetric derivative on an arbitrary nonempty closed subset of the real numbers and derive some of its properties. It is shown that real-valued functions defined on tim…
A symmetric Norlund sum with application to inequalities
Artur M. C. Brito da Cruz, Natalia Martins, Delfim F. M. Torres
Properties of an -symmetric Norlund sum are studied. Inspired in the work by Agarwal et al., -symmetric quantum versions of Holder, Cauchy-Schwarz and Minkowski inequalit…
A symmetric quantum calculus
Artur M. C. Brito da Cruz, Natalia Martins, Delfim F. M. Torres
We introduce the -symmetric difference derivative and the -symmetric Nörlund sum. The associated symmetric quantum calculus is developed, which can be seen as a generaliz…
L'Hopital-Type Rules for Monotonicity with Application to Quantum Calculus
Natalia Martins, Delfim F. M. Torres
We prove new l'Hopital rules for monotonicity valid on an arbitrary time scale. Both delta and nabla monotonic l'Hopital rules are obtained. As an example of application, we give s…