We prove Cauchys formula for repeated integration on time scales. The obtained relation gives rise to new notions of fractional integration and differentiation on arbitrary nonempty closed sets.
Motivated by the supersymmetric extension of Liouville theory in the recent physics literature, we couple the standard Liouville functional with a spinor field term. The resulting functional is conformally invariant. We study geometric and analytic aspects of the resulting Euler-Lagrange equations, culminating in a blow up analysis.
This note concerns the general formulation by Preiss and Uher of Kestelmans influential result pertaining the change of variable, or substitution, formula for the Riemann integral.
We introduce and present the general solution of three two-term fractional differential equations of mixed Caputo/Riemann Liouville type. We then solve a Dirichlet type Sturm-Liouville eigenvalue problem for a fractional differential equation derived from a special composition of a Caputo and a Riemann-Liouville operator on a finite interval where the boundary conditions are induced by evaluating Riemann-Liouville integrals at those end-points. For each $1/2<alpha<1$ it is shown that there is a finite number of real eigenvalues, an infinite number of non-real eigenvalues, that the number of such real eigenvalues grows without bound as $alpha to 1^-$, and that the fractional operator converges to an ordinary two term Sturm-Liouville operator as $alpha to 1^-$ with Dirichlet boundary conditions. Finally, two-sided estimates as to their location are provided as is their asymptotic behavior as a function of $alpha$.
We continue the study of a non self-adjoint fractional three-term Sturm-Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left-Riemann-Liouville fractional integral under {it Dirichlet type} boundary conditions. We study the existence and asymptotic behavior of the real eigenvalues and show that for certain values of the fractional differentiation parameter $alpha$, $0<alpha<1$, there is a finite set of real eigenvalues and that, for $alpha$ near $1/2$, there may be none at all. As $alpha to 1^-$ we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm-Liouville problem with the composition of the operators becoming the operator of second order differentiation.
Delfim F. M. Torres
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(2021)
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"Cauchys formula on nonempty closed sets and a new notion of Riemann--Liouville fractional integral on time scales"
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Delfim F. M. Torres
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