Search Results (40)

In this article, a novel type of equation, namely the $\mathsf{\Phi }$-Hilfer fractional-order integro-differential delay system ($\mathsf{\Phi }$-HFOIDDS), is proposed. Here, we study the existence and Hyers–Ulam–Mittag–Leffler (H-U-M-L) stability of the aforementioned equation which are obtained by using the multivariate Mittag–Leffler function, Banach contraction principle, and Picard operator method as well as generalized Gronwall inequality. Finally, we conclude this paper by constructing a suitable example to illustrate the applicability of the principal outcomes. Full article

This study investigates the stability of fractional-order systems with infinite delay, which are prevalent in many fields due to their effectiveness in modeling complex dynamic behaviors. Recent advancements concerning the existence and various categories of stability for solutions to the given problem are also highlighted. This investigation utilizes tools such as the Picard operator approach, the Banach fixed-point theorem, an extended form of Gronwall’s inequality, and several well-known special functions. We establish key stability criteria for fractional differential equations using Hadamard fractional derivatives and illustrate these concepts using a numerical example. Specifically, graphical representations of the system’s responses demonstrate how fractional-order control enhances stability compared to traditional integer-order approaches. Our results emphasize the value of fractional systems in improving system performance and robustness. Full article

In this work, we introduce a new definition for the fractional differential operator that generalizes several well-known fractional differential operators. In fact, we introduce the notion of the $p$-proportional $\omega$-weighted $\kappa$-Hilfer derivative includes an exponential function, $D_{a,\lambda }^{\sigma ,\rho ,p,\kappa ,\omega }$, and then we consider a non-instantaneous impulse differential inclusion containing $D_{a,\lambda }^{\sigma ,\rho ,p,\kappa ,\omega }$ with order $\sigma \in (1,2)$ and of kind $\rho \in [0,1]$ in Banach spaces. We deduce the relevant relationship between any solution to the studied problem and the integral equation that corresponds to it, and then, by using an appropriate fixed-point theorem for multi-valued functions, we give two results for the existence of these solutions. In the first result, we show the compactness of the solution set. Next, we introduce the concept of the $(p,\omega ,\kappa )$-generalized Ulam-Hyeres stability of solutions, and, using the properties of the multi-valued weakly Picard operator, we present a result regarding the $(p,\omega ,\kappa )$-generalized Ulam-Rassias stability of the objective problem. Since many fractional differential operators are particular cases of the operator $D_{a,\lambda }^{\sigma ,\rho ,p,\kappa ,\omega }$, our work generalizes a number of recent findings. In addition, there are no past works on this kind of fractional differential inclusion, so this work is original and enjoyable. In the last section, we present examples to support our findings. Full article

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard’s iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two. Full article

Picard iteration is on the basis of a great number of numerical methods and applications of mathematics. However, it has been known since the 1950s that this method of fixed-point approximation may not converge in the case of nonexpansive mappings. In this paper, an extension of the concept of nonexpansiveness is presented in the first place. Unlike the classical case, the new maps may be discontinuous, adding an element of generality to the model. Some properties of the set of fixed points of the new maps are studied. Afterwards, two iterative methods of fixed-point approximation are analyzed, in the frameworks of b-metric and Hilbert spaces. In the latter case, it is proved that the symmetrically averaged iterative procedures perform well in the sense of convergence with the least number of operations at each step. As an application, the second part of the article is devoted to the study of fractal mappings on Hilbert spaces defined by means of nonexpansive operators. The paper considers fractal mappings coming from $\phi$-contractions as well. In particular, the new operators are useful for the definition of an extension of the concept of $\alpha$-fractal function, enlarging its scope to more abstract spaces and procedures. The fractal maps studied here have quasi-symmetry, in the sense that their graphs are composed of transformed copies of itself. Full article

The author considers a nonlinear Caputo fractional-order delay differential equation (CFrDDE) with multiple variable delays. First, we study the existence and uniqueness of the solutions of the CFrDDE with multiple variable delays. Second, we obtain two new results on the Ulam–Hyers–Mittag–Leffler (UHML) stability of the same equation in a closed interval using the Picard operator, Chebyshev norm, Bielecki norm and the Banach contraction principle. Finally, we present three examples to show the applications of our results. Although there is an extensive literature on the Lyapunov, Ulam and Mittag–Leffler stability of fractional differential equations (FrDEs) with and without delays, to the best of our knowledge, there are very few works on the UHML stability of FrDEs containing a delay. Thereby, considering a CFrDDE containing multiple variable delays and obtaining new results on the existence and uniqueness of the solutions and UHML stability of this kind of CFrDDE are the important aims of this work. Full article

Spectral curves are algebraic curves associated to commutative subalgebras of rings of ordinary differential operators (ODOs). Their origin is linked to the Korteweg–de Vries equation and to seminal works on commuting ODOs by I. Schur and Burchnall and Chaundy. They allow the solvability of the spectral problem $Ly=\lambda y$, for an algebraic parameter $\lambda$ and an algebro-geometric ODO L, whose centralizer is known to be the affine ring of an abstract spectral curve $\Gamma$. In this work, we use differential resultants to effectively compute the defining ideal of the spectral curve $\Gamma$, defined by the centralizer of a third-order differential operator L, with coefficients in an arbitrary differential field of zero characteristic. For this purpose, defining ideals of planar spectral curves associated to commuting pairs are described as radicals of differential elimination ideals. In general, $\Gamma$ is a non-planar space curve and we provide the first explicit example. As a consequence, the computation of a first-order right factor of $L-\lambda$ becomes explicit over a new coefficient field containing $\Gamma$. Our results establish a new framework appropriate to develop a Picard–Vessiot theory for spectral problems. Full article

This paper is dedicated to the memory of the esteemed Serbian mathematician Slaviša B. Prešić (1933–2008). The primary aim of this survey paper is to compile articles on Prešić-type mappings published since 1965. Additionally, it introduces a novel class of symmetric contractions known as Prešić–Menger and Prešić–Ćirić–Menger contractions, thereby enriching the literature on Prešić-type mappings. The paper endeavors to furnish young researchers with a comprehensive resource in functional and nonlinear analysis. The relevance of Prešić’s method, which generalizes Banach’s theorem from 1922, remains significant in metric fixed point theory, as evidenced by recent publications. The overview article addresses the growing importance of Prešić’s approach, coupled with new ideas, reflecting the ongoing advancements in the field. Additionally, the paper establishes the existence and uniqueness of fixed points in Menger spaces, contributing to the filling of gaps in the existing literature on Prešić’s works while providing valuable insights into this specialized domain. Full article

We define and study enriched Suzuki mappings in Hadamard spaces. The results obtained here are extending fundamental findings previously established in related research. The extension is realized with respect to at least two different aspects: the setting and the class of involved operators. More accurately, Hilbert spaces are particular Hadamard spaces, while enriched Suzuki nonexpansive mappings are natural generalizations of enriched nonexpansive mappings. Next, enriched Suzuki nonexpansive mappings naturally contain Suzuki nonexpansive mappings in Hadamard spaces. Besides technical lemmas, the results of this paper deal with (1) the existence of fixed points for enriched Suzuki nonexpansive mappings and (2) $\mathsf{\Delta }$ and strong (metric) convergence of Picard iterates of the $\alpha$-averaged mapping, which are exactly Krasnoselskij iterates for the original mapping. Full article

The main objective of this study is to determine the existence and uniqueness of solutions to the fractional Black–Scholes equation. The solution to the fractional Black–Scholes equation is expressed as an infinite series of converging Mittag-Leffler functions. The method used to discover the new solution to the fractional Black–Scholes equation was the Daftardar-Geiji method. Additionally, the Picard–Lindelöf theorem was utilized for the existence and uniqueness of its solution. The fractional derivative employed was the Caputo operator. The search for a solution to the fractional Black–Scholes equation was essential due to the Black–Scholes equation’s assumptions, which imposed relatively tight constraints. These included assumptions of a perfect market, a constant value of the risk-free interest rate and volatility, the absence of dividends, and a normal log distribution of stock price dynamics. However, these assumptions did not accurately reflect market realities. Therefore, it was necessary to formulate a model, particularly regarding the fractional Black–Scholes equation, which represented more market realities. The results obtained in this paper guaranteed the existence and uniqueness of solutions to the fractional Black–Scholes equation, approximate solutions to the fractional Black–Scholes equation, and very small solution errors when compared to the Black–Scholes equation. The novelty of this article is the use of the Daftardar-Geiji method to solve the fractional Black–Scholes equation, guaranteeing the existence and uniqueness of the solution to the fractional Black–Scholes equation, which has not been discussed by other researchers. So, based on this novelty, the Daftardar-Geiji method is a simple and effective method for solving the fractional Black–Scholes equation. This article presents some examples to demonstrate the application of the Daftardar-Gejji method in solving specific problems. Full article

This paper is dedicated to exploring the existence, uniqueness and Ulam stability analysis applied to a specific class of mathematical equations known as nonlinear impulsive Volterra Fredholm integro-dynamic adjoint equations within finite time scale intervals. The primary aim is to establish sufficient conditions that demonstrate Ulam stability for this particular class of equations on the considered time scales. The research methodology relies on the Banach contraction principle, Picard operator and extended integral inequality applicable to piecewise continuous functions on time scales. To illustrate the applicability of the findings, an example is provided. Full article

Recent years have seen an increase in scientific interest in the El Nio/La Nia Southern Oscillation (ENSO), a quasiperiodic climate phenomenon that takes place throughout the tropical Pacific Ocean over five years and causes significant harm. It is associated with the warm oceanic stage known as El Nio and the cold oceanic stage known as La Nia. In this research, the ENSO model is considered under a fractional operator, which is defined via a nonsingular and nonlocal kernel. Some theoretical features, such as equilibrium points and their stability, bifurcation maps, the existence of a unique solution via the Picard–Lindelof approach, and the stability of the solution via the Ulam–Hyres stability approach, are deliberated for the proposed ENSO model. The Adams–Bashforth numerical method, associated with Lagrangian interpolation, is used to obtain a numerical solution for the considered ENSO model. The complex dynamics of the ENSO model are displayed for a few fractional orders via MATLAB-18. Full article

In this paper, we introduce a new type of contractions on a metric space $(X,d)$ in which the distance $d(x,y)$ is replaced with a function, depending on a parameter $\lambda$, that is not symmetric in general. This function generalizes the usual case when $\lambda =1/2$ and can take bigger values than $m_{1/2}.$ We call these new types of contractions λ-weak contractions and we provide some of their properties. Moreover, we investigate cases when these contractions are Picard operators. Full article

Artificial neural networks with different structures are used for identification of complex dynamic plant with distributed parameters. The plant is a high-temperature plasma in the spherical Globus-M2 tokamak. Experimental data from it were processed by plasma reconstruction code based on Picard iterations, namely, the Flux-Current Distribution Identification (FCDI) code. This represents smart technology employed to obtain distributed plasma parameters by minimizing the difference between measured and reconstructed signals. An artificial neural network was then applied to identify the data obtained by the FCDI code on the hardware as a real-time testbed realized on a Speedgoat computer. The aim of this repeated identification is to increase the operational response speed in real time in the closed-loop control system of the plasma shape. Full article

The Delta and Omicron variants’ system was used in this research study to replicate the complex process of the SARS-CoV-2 outbreak. The generalised fractional system was designed and rigorously analysed in order to gain a comprehensive understanding of the transmission dynamics of both variants. The proposed dynamical system has heredity and memory effects, which greatly improved our ability to perceive the disease propagation dynamics. The non-singular Atangana–Baleanu fractional operator was used to forecast the current pandemic in order to meet this challenge. The Picard recursions approach can be used to ensure that the designed fractional system has at least one solution occupying the growth condition and memory function regardless of the initial conditions. The Hyers–Ulam–Rassias stability criteria were used to carry out the stability analysis of the fractional governing system of equations, and the fixed-point theory ensured the uniqueness of the solution. Additionally, the model exhibited global asymptotically stable behaviour in some conditions. The approximate behaviour of the fatal virus was investigated using an efficient and reliable fractional numerical Adams–Bashforth approach. The outcome demonstrated that there will be a significant decline in the population of those infected with the Omicron and Delta SARS-CoV-2 variants if the vaccination rate is increased (in both the symptomatic and symptomatic stages). Full article