Сompactness of one class of fractional integrating operator with variable upper limit

The paper establishes a characterization of the compactness for fractional operators of a general class, including the Riemann-Liouville, Hadamard and Erdelyi-Kober operators. The paper considers an integral fractional integration operator of Hardy type with nonnegative kernels and a variable limit of integration (a function as the upper limit of integration) and under certain conditions on the kernel, a criterion of the compactness in weighted Lebesgue spaces is obtained for this operator, when the parameters of the spaces satisfy the conditions Moreover, more general results are obtained for the weighted differential inequality of Hardy type on the set of locally absolutely continuous functions that vanish and infinity at the ends of the interval, covering the previously known results, and more precise estimates for the best constant are given. The localization method, Schauder’s theorem, the Kantorovich test, and the theorem on the uniform limit of compact operators were used in the proof of the main theorem. The obtained results of the study the compactness of fractional integration operators can be used in the estimation of solutions of differential equations that model various processes in mathematics. In particular, these results yield new results in the theory of Hardy-type inequalities.

A note on convolution operators on Riesz Bounded variation spaces

We show some estimates and approximation results of operators of convolution type defined on Riesz Bounded variation spaces in Rn. We also state some embedding results that involve the collection of generalized absolutely continuous functions.

A qualitative study on generalized Caputo fractional integro-differential equations

The aim of this article is to discuss the uniqueness and Ulam–Hyers stability of solutions for a nonlinear fractional integro-differential equation involving a generalized Caputo fractional operator. The used fractional operator is generated by iterating a local integral of the form $(I_{a}^{\rho }f)(t)=\int _{a}^{t}f(s)s^{\rho -1}\,ds$ ( I a ρ f ) ( t ) = ∫ a t f ( s ) s ρ − 1 d s . Our reported results are obtained in the Banach space of absolutely continuous functions that rely on Babenko’s technique and Banach’s fixed point theorem. Besides, our main findings are illustrated by some examples.

Explicit and Implicit Non-convex Sweeping Processes in the Space of Absolutely Continuous Functions

We show that sweeping processes with possibly non-convex prox-regular constraints generate a strongly continuous input-output mapping in the space of absolutely continuous functions. Under additional smoothness assumptions on the constraint we prove the local Lipschitz continuity of the input-output mapping. Using the Banach contraction principle, we subsequently prove that also the solution mapping associated with the state-dependent problem is locally Lipschitz continuous.

On the nonlinear Hadamard-type integro-differential equation

This paper studies uniqueness of solutions for a nonlinear Hadamard-type integro-differential equation in the Banach space of absolutely continuous functions based on Babenko’s approach and Banach’s contraction principle. We also include two illustrative examples to demonstrate the use of main theorems.

VARIATION PROBLEM CONTAINING SECOND DERIVATIVES OF UNKNOWN FUNCTIONS

The property subdifferential of an integral and terminal functional in a space of the type of absolutely continuous functions is studied. Necessary and sufficient conditions for an extremum for a variational problem containing the second derivatives of unknown functions are obtained. With the help of the subdifferential introduced by the author, a nonconvex generalized variational problem containing the second derivatives of unknown functions is considered, and the necessary condition for an extremum is obtained.

ON THE EXACT PENALTY OF HIGH ORDER FOR EXTREME PROBLEMS OF DIFFERENTIAL INCLUSIONS

In this paper, using theorems on the continuous dependence of the solution of differential inclusions on the perturbation, we obtain high-order exact penalty theorems for nonconvex extremal problems of differential inclusions in the space of Banach-valued absolutely continuous functions. Using the type of the distance function in the classes of 𝜙 − (𝜙, 𝜙, 𝜙, 𝜙, 𝜙) locally Lipschitz functions, the nonconvex extremal problem for differential inclusions is reduced to a variational problem and the necessary condition for the extremum of a high-order is obtained. The paper also shows that the used functions type of distance function satisfy the 𝜙 − (𝜙, 𝜙, 𝜙, 𝜙, 𝜙) locally Lipschitz conditions.

Approximation by sampling-type nonlinear discrete operators in φ-variation

In the present paper, our purpose is to obtain a nonlinear approximation by using convergence in ?-variation. Angeloni and Vinti prove some approximation results considering linear sampling-type discrete operators. These types of operators have close relationships with generalized sampling series. By improving Angeloni and Vinti?s one, we aim to get a nonlinear approximation which is also generalized by means of summability process. We also evaluate the rate of approximation under appropriate Lipschitz classes of ?-absolutely continuous functions. Finally, we give some examples of kernels, which fulfill our kernel assumptions.