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Study of fractional semipositone problems on RN

Biswas Nirjan

Opuscula Mathematica

2024

Vol. 44, no. 4

445--470

Let s ∈ (0, 1) and N > 2s. In this paper, we consider the following class of nonlocal semipositone problems: (−Δ)su = g(x)ƒa(u) in RN, u > 0 in RN, where the weight g ∈ L1(RN) ∩ L∞(RN) is positive, a > 0 is a parameter, and ƒa ∈ C(R) is strictly negative on (−∞, 0]. For ƒa having subcritical growth and weaker Ambrosetti–Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution ua, provided a is near zero. To obtain the positivity of ua, we establish a Brezis–Kato type uniform estimate of (ua) in Lr(RN) for every r ∈ [formula].

Fractional operators and their commutators on generalized Orlicz spaces

Karppinen Arttu

Opuscula Mathematica

2022

Vol. 42, no. 4

583--604

In this paper we examine boundedness of fractional maximal operator. The main focus is on commutators and maximal commutators on generalized Orlicz spaces (also known as Musielak-Orlicz spaces) for fractional maximal functions and Riesz potentials. We prove their boundedness between generalized Orlicz spaces and give a characterization for functions of bounded mean oscillation.

Comparative analysis of Riemann-Liouville, Caputo-Fabrizio, and Atangana-Baleanu integrals

Shaikh Asif Ali, Qureshi Sania

Journal of Applied Mathematics and Computational Mechanics

2022

Vol. 21, nr 1

91--101

This study analyzes the most commonly used operators of the Riemann-Liouville, the Caputo-Fabrizio, and the Atangana-Baleanu integral operators. Firstly, a numerical scheme for the Riemann-Liouville fractional integral has been discussed. Then, two numerical techniques have been suggested for the remaining two operators. The experimental order of convergence for the schemes is further supported by the computations of absolute relative error at the final nodal point over the integration interval [0, T ]. Comparative analysis of the integrals reveals that the Riemann-Liouville fractional integral yields the most minor errors and the most significant experimental order of convergence in the majority of functions, particularly when the fractional-order parameter α → 0. It is worth noting that the Atangana-Baleanu has proved to be an essential operator for solving many dynamical systems that a single RL operator cannot handle. All of the three integral operators coincide with each other for α = 1. Mathematica 11.3 for an Intel(R) Core(TM) i3-4500U procesor running on 1.70 GHz is used to carry out all the necessary computations.

Characterizations of compact operators on ℓp-type fractional sets of sequences

Özger Faruk

Demonstratio Mathematica

2019

Vol. 52, nr 1

105--115

Among the sets of sequences studied, difference sets of sequences are probably the most common type of sets. This paper considers some ℓp-type fractional difference sets via the gamma function. Although, we characterize compactness conditions on those sets using the main key of Hausdorff measure of noncompactness, we can only obtain sufficient conditions when the final space is ℓ∞. However, we use some recent results to exactly characterize the classes of compact matrix operators when the final space is the set of bounded sequences.