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This paper intends to study the analytical properties of the Riemann–Liouville fractional integral and fractal dimensions of its graph on ℝ n . We show that the Riemann–Liouville fractional integral preserves some analytical properties such as boundedness, continuity and bounded variation in the Arzelá sense. We also deduce the upper bound of the box dimension and the Hausdorff dimension of the graph of the Riemann–Liouville fractional integral of Hölder continuous functions. Furthermore, we prove that the box dimension and the Hausdorff dimension of the graph of the Riemann–Liouville fractional integral of a function, which is continuous and of bounded variation in Arzelá sense, are n. [ABSTRACT FROM AUTHOR]

This paper intends to estimate the box dimension of the Weyl–Marchaud fractional derivative (Weyl–M derivative) for various choices of continuous functions on a compact subset of ℝ. We show that the Weyl–M derivative of order γ of a continuous function satisfying Hölder condition of order μ also satisfies Hölder condition of order μ − γ and the upper box dimension of the Weyl–M derivative increases at most linearly with the order γ. Moreover, the upper box dimension of the Weyl–M derivative of a continuous function satisfying the Lipschitz condition is not more than the sum of the box dimension of the function itself and order γ. Furthermore, we prove that the box dimension of the Weyl–M derivative of a certain continuous function which is of bounded variation is one. [ABSTRACT FROM AUTHOR]

We introduce a fractal operator on C [ 0 , 1 ] which sends a function f ∈ C (I) to fractal version of f where fractal version of f is a super fractal interpolation function corresponding to a countable data system. Furthermore, we study the continuous dependence of super fractal interpolation functions on the parameters used in the construction. We know that the invariant subspace problem and the existence of a Schauder basis gained lots of attention in the literature. Here, we also show the existence of non-trivial closed invariant subspace of the super fractal operator and the existence of fractal Schauder basis for C (I) . Moreover, we can see the effect of the composition of Riemann-Liouville integral operator and super fractal operator on the fractal dimension of continuous functions. We also mention some new problems for further investigation. [ABSTRACT FROM AUTHOR]

In this article, we provide a rigorous study on the fractal dimension of the graph of the mixed Riemann-Liouville fractional integral for various choices of continuous functions on a rectangular region. We estimate bounds for the box dimension and the Hausdorff dimension of the graph of the mixed Riemann-Liouville fractional integral of the functions which belong to the class of continuous functions and the class of Hölder continuous functions. We also show that the box dimension of the graph of the mixed Riemann-Liouville fractional integral of two-dimensional continuous functions is also two. Furthermore, we give the construction of unbounded variational continuous functions. Later, we prove that the box dimension and the Hausdorff dimension of the graph of the mixed Riemann-Liouville fractional integral of unbounded variational continuous functions are two. Moreover, we illustrate our results by using some examples. [ABSTRACT FROM AUTHOR]

The goal of this article is to study the box dimension of the mixed Katugampola fractional integral of two-dimensional continuous functions on [ 0 , 1 ] × [ 0 , 1 ] . We prove that the box dimension of the mixed Katugampola fractional integral having fractional order (α = (α 1 , α 2) ; α 1 > 0 , α 2 > 0) of two-dimensional continuous functions on [ 0 , 1 ] × [ 0 , 1 ] is still two. Moreover, the results are also established for the mixed Hadamard fractional integral. Our new results are to improve the existing studies. We pose also some open problems for further research. [ABSTRACT FROM AUTHOR]

The calculus of the mixed Weyl–Marchaud fractional derivative has been investigated in this paper. We prove that the mixed Weyl–Marchaud fractional derivative of bivariate fractal interpolation functions (FIFs) is still bivariate FIFs. It is proved that the upper box dimension of the mixed Weyl–Marchaud fractional derivative having fractional order γ = (p , q) of a continuous function which satisfies μ -Hölder condition is no more than 3 − μ + (p + q) when 0 < p , q < μ < 1 , p + q < μ , which reveals an important phenomenon about linearly increasing effect of dimension of the mixed Weyl–Marchaud fractional derivative. Furthermore, we deduce box dimension of the graph of the mixed Weyl–Marchaud fractional derivative of a continuous function which is defined on a rectangular region in ℝ 2 and also, we analyze that the mixed Weyl–Marchaud fractional derivative of a function preserves some basic properties such as continuity, bounded variation and boundedness. The results are new and compliment the existing ones. [ABSTRACT FROM AUTHOR]