A descriptive definition of some multidimensional gauge integrals
A direct approach to one variable noncommutative calculus
A direct extension of Meller's calculus.
A family of singular functions and its relation to harmonic fractal analysis and fuzzy logic
We study a parameterized family of singular functions which appears in a paper by H. Okamoto and M. Wunsch (2007). Various properties are revisited from the viewpoint of fractal geometry and probabilistic techniques. Hausdorff dimensions are calculated for several sets related to these functions, and new properties close to fractal analysis and strong negations are explored.
A fixed point approach to the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation
In this paper, we have presented and studied two types of the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation. We prove that the fractional order delay integral equation is Mittag-Leffler-Hyers-Ulam stable on a compact interval with respect to the Chebyshev and Bielecki norms by two notions.
A formula for calculation of metric dimension of converging sequences
Converging sequences in metric space have Hausdorff dimension zero, but their metric dimension (limit capacity, entropy dimension, box-counting dimension, Hausdorff dimension, Kolmogorov dimension, Minkowski dimension, Bouligand dimension, respectively) can be positive. Dimensions of such sequences are calculated using a different approach for each type. In this paper, a rather simple formula for (lower, upper) metric dimension of any sequence given by a differentiable convex function, is derived....
A Fractional Analog of the Duhamel Principle
Mathematics Subject Classification: 35CXX, 26A33, 35S10The well known Duhamel principle allows to reduce the Cauchy problem for linear inhomogeneous partial differential equations to the Cauchy problem for corresponding homogeneous equations. In the paper one of the possible generalizations of the classical Duhamel principle to the time-fractional pseudo-differential equations is established.* This work partially supported by NIH grant P20 GMO67594.
A fractional calculus approach to the mechanics of fractal media.
A Fractional LC − RC Circuit
Mathematics Subject Classification: 26A33, 30B10, 33B15, 44A10, 47N70, 94C05We suggest a fractional differential equation that combines the simple harmonic oscillations of an LC circuit with the discharging of an RC circuit. A series solution is obtained for the suggested fractional differential equation. When the fractional order α = 0, we get the solution for the RC circuit, and when α = 1, we get the solution for the LC circuit. For arbitrary α we get a general solution which shows how the...
A full characterization of multipliers for the strong -integral in the euclidean space
We study a generalization of the classical Henstock-Kurzweil integral, known as the strong -integral, introduced by Jarník and Kurzweil. Let be the space of all strongly -integrable functions on a multidimensional compact interval , equipped with the Alexiewicz norm . We show that each element in the dual space of can be represented as a strong -integral. Consequently, we prove that is strongly -integrable on for each strongly -integrable function if and only if is almost everywhere...
A full descriptive definition of the BV-integral
We present a Cauchy test for the almost derivability of additive functions of bounded BV sets. The test yields a full descriptive definition of a coordinate free Riemann type integral.
A functioanl identity characterizing polynomials.
A gauge approach to an ordinal index of Baire one functions
We develop a calculus for the oscillation index of Baire one functions using gauges analogous to the modulus of continuity.
A general form of the product integral and linear ordinary differential equations
A general integral [Book]
A general method to solve fractional differential equations on
A general result on the equivalence between derivation of integrals and weak inequalities for the Hardy-Littlewood maximal operator
A Generalisation Of The Concept Of Convexity
A generalization of Andersson's inequality.
A generalization of the Lagrange mean value theorem to the case of vector-valued mappings.