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Donal O'Regan

Publications and source records attributed to Donal O'Regan.

At least 19 recordsLinked to original sources
arXiv2025

Asymptotic behavior for a finitely degenerate semilinear pseudo-parabolic equation

This paper investigates the initial boundary value problem of a finitely degenerate semilinear pseudo-parabolic equation associated with Hörmander's operator. Based on the global existence of solutions in previous literature, the exponential decay estimate of the energy functional is obtained. Moreover, by developing some novel estimates about solutions and using the energy method, the upper bounds of both blow-up time and blow-up rate and the exponential growth estimate of blow-up solutions are determined. In addition, the lower bound of blow-up rate is estimated when a finite time blow-up occurs. Finally, it is established that as time approaches infinity, the global solutions strongly converge to the solution of the corresponding stationary problem. These results complement and improve the ones obtained in the previous literature.

math-ph↗
arXiv2025

The Fractal Lie Derivative: Theory and Applications

This paper presents a new Lie theoretic approach to fractal calculus, which in turn yields such new results as a Fractal Noether's Theorem, a setting for fractal differential forms, for vector fields, and Lie derivatives, as well as k-fractal jet space, and algorithms for k-th fractal prolongation. The symmetries of the fractal nonlinear \(n\)-th \(α\)-order differential equation are examined, followed by a discussion of the symmetries of the fractal linear \(n\)-th \(α\)-order differential equation. Additionally, the symmetries of the fractal linear first \(α\)-order differential equation are derived. Several examples are provided to illustrate and highlight the details of these concepts.

math.GM↗
arXiv2023

Best Ulam constants for two-dimensional non-autonomous linear differential systems

This study deals with the Ulam stability of non-autonomous linear differential systems without assuming the condition that they admit an exponential dichotomy. In particular, the best (minimal) Ulam constants for two-dimensional non-autonomous linear differential systems with generalized Jordan normal forms are derived. The obtained results are applicable not only to systems with solutions that exist globally on $(-\infty,\infty)$, but also to systems with solutions that blow up in finite time. New results are included even for constant coefficients. A wealth of examples are presented, and approximations of node, saddle, and focus are proposed. In addition, this is the first study to derive the best Ulam constants for non-autonomous systems other than periodic systems.

math.CA↗
arXiv2023

Best Ulam constants for damped linear oscillators with variable coefficients

This study uses an associated Riccati equation to study the Ulam stability of non-autonomous linear differential vector equations that model the damped linear oscillator. In particular, the best (minimal) Ulam constants for these non-autonomous linear differential vector equations are derived. These robust results apply to vector equations with solutions that blow up in finite time, as well as to vector equations with solutions that exist globally on $(-\infty,\infty)$. Illustrative, non-trivial examples are presented, highlighting the main results.

math.CA↗
arXiv2023

A New Shrinking projection Algorithm for an infinite family of Bregman weak relatively nonexpansive mappings in a Banach Space

In this paper, using a new shrinking projection method and generalized resolvents of maximal monotone operators and generalized projections, we consider the strong convergence for finding a common point of the fixed points of a Bregman quasi-nonexpansive mapping, and common fixed points of a infinite family of Bregman weak relatively nonexpansive mappings, and common zero points of a finite family of maximal monotone mappings, and common solutions of an equilibrium problem in a reflexive Banach space.

math.FA↗
arXiv2021

A strong convergence theorem for solving an equilibrium problem and a fixed point problem using the Bregman distance

In this paper, using the Bregman distance, we introduce a new projection-type algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points. Then the strong convergence of the sequence generated by the algorithm will be established under suitable conditions. Finally, using MATLAB software, we present a numerical example to illustrate the convergence performance of our algorithm.

math.OC↗
arXiv2021

A generalized strong convergence algorithm in the presence of the errors for the variational inequality problems in Hilbert spaces

In this paper, we study the strong convergence of an algorithm to solve the variational inequality problem which extends(Thong et al, Numerical Algorithms. 78, 1045-1060 (2018)). We have reduced and refined some of their algorithm's conditions and we have proved the convergence of the algorithm in the presence of some computational errors. Then using MATLAB software, the result will by illustrated in some numerical examples. Finally, we compare our algorithm with some other well known algorithms.

math.NA↗
arXiv2021

Existence of group nonexpansive retractions and ergodic theorems in topological groups

Suppose that $G$ is a topological group and $ C $ a compact subset of $G$. In this paper we define group nonexpansive mappings and then we consider $\sc = \{T_{i} : i \in I \}$ as a family of the group nonexpansive mappings on $C$. Also we study the existence of group nonexpansive retractions $P_{i}$ from $C$ onto $\text{Fix}(\sc)$ such that $P_{i}T_{i} = T_{i}P_{i} = P_{i}$.

math.FA↗
arXiv2021

A Simple proof for the algorithms of relaxed $(u, v)$-cocoercive mappings and $α$-inverse strongly monotone mappings

In this paper, a simple proof is presented for the convergence of the algorithms for the class of relaxed $(u, v)$-cocoercive mappings and $α$-inverse strongly monotone mappings. Based on $α$-expansive maps, for example, a simple proof of the convergence of the recent iterative algorithms by relaxed $(u, v)$-cocoercive mappings due to Kumam-Jaiboon is provided. Also a simple proof for the convergence of the iterative algorithms by inverse-strongly monotone mappings due to Iiduka-Takahashi in a special case is provided. These results are an improvement as well as a refinement of previously known results.

math.FA↗
arXiv2020

A new class of fractional impulsive differential hemivariational inequalities with an application

We consider a new fractional impulsive differential hemivariational inequality which captures the required characteristics of both the hemivariational inequality and the fractional impulsive differential equation within the same framework. By utilizing a surjectivity theorem and a fixed point theorem, we establish an existence and uniqueness theorem for such a problem. Moreover, we investigate the perturbation problem of the fractional impulsive differential hemivariational inequality to prove a convergence result which describes the stability of the solution in relation to perturbation data. Finally, our main results are applied to obtain some new results for a frictional contact problem with the surface traction driven by the fractional impulsive differential equation.

math.OC↗
arXiv2020

Backward problem for time fractional reaction-diffusion equation with nonlinear source and discrete data

In this paper, we study the problem of finding the solution of a multi-dimensional time fractional reactiondiffusion equation with nonlinear source from the final value data. We prove that the present problem is not well-posed. Then regularized problems are constructed using the truncated expansion method (in the case of two-dimensional) and the quasi-boundary value method (in the case of multi-dimensional). Finally, convergence rates of the regularized solutions are given and investigated numerically.

math.AP↗
arXiv2019

On existence and regularity of a terminal value problem for the time fractional diffusion equation

In this paper we consider a final value problem for a diffusion equation with time-space fractional differentiation on a bounded domain $D$ of $ \mathbb{R}^{k}$, $k\ge 1$, which includes the fractional power $\mathcal L^β$, $0<β\le 1$, of a symmetric uniformly elliptic operator $\mathcal L$ defined on $L^2(D)$. A representation of solutions is given by using the Laplace transform and the spectrum of $\mathcal L^β$. We establish some existence and regularity results for our problem in both the linear and nonlinear case.

math.AP↗
arXiv2017

On the solutions of certain fractional kinetic equations involving $E^{γ,q}_{k,α,β}(.)$

We develop a new generalized form of the fractional kinetic equation involving a generalized k-Bessel function. The generalized $k$-Mittag-leffler function $E^{γ,q}_{k,α,β}(.)$ is discussed in terms of the solution of the fractional kinetic equation in the present paper. The results obtained here are quite general in nature and capable of yielding known and as well new results.

math.AP↗
arXiv2016

An inequality concerning the growth bound of a discrete evolution family on a complex Banach space

We prove that the uniform growth bound $ω_0(\mathcal{U})$ of a discrete evolution family $\mathcal{U}$ of bounded linear operators acting on a complex Banach space $X$ satisfies the inequality $$ω_0(\mathcal{U})c_{\mathcal{U}}(\mathcal{X})\le -1;$$ here $c_{\mathcal{U}}(\mathcal{X})$ is the operator norm of a convolution operator which acts on a certain Banach space $\mathcal{X}$ of $X$-valued sequences.

math.FA↗
arXiv2013

A Kamenev-type oscillation result for a linear $(1+α)$--order fractional differential equation

We investigate the eventual sign changing for the solutions of the linear equation $\left(x^{(α)}\right)^{\prime}+q(t)x=0$, $t\geq0$, when the functional coefficient $q$ satisfies the Kamenev-type restriction $\limsup\limits_{t\rightarrow+\infty}\frac{1}{t^{\varepsilon}}\int_{t_0}^{t}(t-s)^{\varepsilon}q(s)ds=+\infty$ for some $\varepsilon>2$, $t_{0}>0$. The operator $x^{(α)}$ is the Caputo differential operator and $α\in(0,1)$.

math.CA↗