Nonlocal Conditions Research Articles

Intuitionistic fuzzy evolution problem with nonlocal conditions

In this manuscript, we investigate the existence and uniqueness of solutions for the intuitionistic fuzzy evolution problem with non-local conditions, employing a generalized Caputo derivative <math>^C_{gH}D_{0^+}^{\gamma}</math> of order <math>0 < \gamma < 1</math>. Our methodology involves utilizing the intuitionistic fuzzy semigroup and the concept of contraction mapping.

Local and nonlocal conditions for electron runaway in a gas gap with a conical cathode with a variable opening angle

The conditions for the generation of runaway electrons in an air gap are compared at different degrees of inhomogeneity of the electric field distribution provided by varying the opening angle of the conical cathode: in the range 40°–120° in experiments and 0°–180° in calculations. It is demonstrated that, in a weakly inhomogeneous electric field (according to the proposed classification, this corresponds to cones with angles greater than the Taylor angle of 98.6°), the runaway condition has a local character. The transition of free electrons into the runaway mode is determined by the local distribution of the electric field near their starting point—the tip of the cone. The local electric field strength must exceed a threshold value comparable to the strength critical for the runaway of electrons in a uniform field. In a strongly inhomogeneous field (cones with angles less than 98.6°), this condition is not sufficient for electrons to run away throughout the gap. Electrons accelerating in the near-cathode region may begin to slow down in a weak field at a distance from the cathode. In this case, the runaway condition becomes nonlocal. It is determined by the dynamics of electrons in the entire gap, primarily in the near-anode region, and reduces to the requirement that the potential difference applied to the gap exceeds a certain threshold value.

Local and nonlocal problems for fractional impulsive evolution systems with order $\nu\in(1,2)$

In this paper, we study the existence and uniqueness of $PC$-mild solutions for fractional impulsive evolution systems with Caputo derivative. First, we give a compact result of the solution operators of linear fractional evolution system while the cosine family is compact. Second, the representation of $PC$-mild solutions of linear fractional impulsive systems with initial value conditions and nonlocal initial value conditions are given by using the Laplace transform method. Third, several sufficient conditions for judging the existence and uniqueness of solutions of our problem are established via some fixed point theorems and operator semigroup theory. Finally, an example is given to illustrate the results of our paper.

Existence results for some fractional order coupled systems with impulses and nonlocal conditions on the half line

Abstract. In this paper, we deal with initial value problems for coupled systems of nonlinear fractional differential equations, subject to coupled nonlocal initial and impulsive conditions on the half line. Global existence-uniqueness results are obtained under weak conditions allowing the reaction part of the problem to increase indefinitely with time. Our approach relies mainly to some fixed point theorem of Perov’s type in generalized gauge spaces. The obtained results improve, generalize and complement many existing results in the literature. An example illustrating our main finding is also given. Mathematics Subject Classification (2010): 26A33, 93C23, 35E15, 47H10. Keywords: Fractional differential equation, coupled systems, generalized spaces in Perov’s sens, nonlocal initial conditions, impulses.

Sticky Brownian motions on star graphs

AbstractThis paper is concerned with the construction of Brownian motions and related stochastic processes in a star graph, which is a non-Euclidean structure where some features of the classical modeling fail. We propose a probabilistic construction of the Sticky Brownian motion by slowing down the Brownian motion when in the vertex of the star graph. Later, we apply a random change of time to the previous construction, which leads to a trapping phenomenon in the vertex of the star graph, with characterization of the trap in terms of a singular measure $$\varPhi $$ Φ . The process associated to this time change is described here and, moreover, we show that it defines a probabilistic representation of the solution to a heat equation type problem on the star graph with non-local dynamic conditions in the vertex that can be written in terms of a Caputo-Džrbašjan fractional derivative defined by the singular measure $$\varPhi $$ Φ . Extensions to general graph structures can be given by applying to our results a localisation technique.

Effects of fractional derivative and Wiener process on approximate boundary controllability of differential inclusion

One of the core concepts of contemporary control theory is the idea that a dynamical system can be controlled. Several abstract settings have been developed to describe the distributed control systems in a domain in which the control is acted through the boundary. In this manuscript, we investigate the boundary approximate controllability (ABC) of stochastic differential inclusion (SDI) with Hilfer fractional derivative (HFD) and nonlocal condition by implementing the principle of stochastic analysis, the fixed-point theorem, fractional calculus and multi-valued map. Moreover, an example is offered to define the primary results.

A study on approximate controllability of evolution systems with nonlocal conditions

The purpose of this paper is to present the existence of mild solutions and approximate controllability for a class of non-autonomous measure driven evolution systems with nonlocal conditions in Banach spaces. Firstly, we obtain the existence of mild solutions for the concerned problem by the semigroup theory and Schauder's theorem. Secondly, some sufficient conditions of approximate controllability are proved. At the end, an example is also given to illustrate the feasibility of our theoretical results.

A Study of Positive Solutions for Semilinear Fractional Measure Driven Functional Differential Equations in Banach Spaces

In this paper, we deal with the delayed measure differential equations with nonlocal conditions via measure of noncompactness in ordered Banach spaces. Combining (β,γk)-resolvent family, regulated functions and fixed point theorem with respect to convex-power condensing operator and measure of noncompactness, we investigate the existence of positive mild solutions for the mentioned system under the situation that the nonlinear function satisfies measure conditions and order conditions. In addition, we provide an example to verify the rationality of our conclusion.

Parameter identification in anomalous diffusion equations with nonlocal conditions and weak-valued nonlinearities

Parameter identification in anomalous diffusion equations with nonlocal conditions and weak-valued nonlinearities

Analysis of a parametric delay functional differential equation with nonlocal integral condition

Analysis of a parametric delay functional differential equation with nonlocal integral condition

Vector valued piecewise continuous almost automorphic functions and some consequences

In the present work, for X a Banach space, the notion of piecewise continuous Z-almost automorphic functions with values in finite dimensional spaces is extended to piecewise continuous Z-almost automorphic functions with values in X. Several properties of this class of functions are provided, in particular it is shown that if X is a Banach algebra, then this class of functions constitute also a Banach algebra; furthermore, using the theory of Z-almost automorphic functions, a new characterization of compact almost automorphic functions is given. As consequences, with the help of Z-almost automorphic functions, it is presented a simple proof of the characterization of almost automorphic sequences by compact almost automorphic functions; the method permits us to give explicit examples of compact almost automorphic functions which are not almost periodic. Also, using the theory developed here, it is shown that almost automorphic solutions of differential equations with piecewise constant argument are in fact compact almost automorphic. Finally, it is proved that the classical solution of the 1D heat equation with continuous Z-almost automorphic source is also continuous Z-almost automorphic; furthermore, we comment applications to the existence and uniqueness of the asymptotically continuous Z-almost automorphic mild solution to abstract integro-differential equations with nonlocal initial conditions.

TOPOLOGICAL DEGREE METHOD FOR A COUPLED SYSTEM OF $\psi$-FRACTIONAL SEMILINEAR DIFFERENTIAL EQUATIONS WITH NON LOCAL CONDITIONS

This paper's primary goal is to build the existence of solutions for non-local coupled semi-linear differential equations involving $\psi$-Caputo differential derivatives of an arbitrary $l\in (0,1).$ We establish it with non-local conditions under which the examined coupled system has at least one solution by utilizing the notion of topological degree. Semilinear equations are an extremely unusual use of the corresponding approach.

Strict Solutions to Abstract Differential Equations with State-Dependent Nonlocal Conditions

Strict Solutions to Abstract Differential Equations with State-Dependent Nonlocal Conditions

Fractional boundary value problems and elastic sticky brownian motions

AbstractWe extend the results obtained in [14] by introducing a new class of boundary value problems involving non-local dynamic boundary conditions. We focus on the problem to find a solution to a local problem on a domain $$\varOmega $$ Ω with non-local dynamic conditions on the boundary $$\partial \varOmega $$ ∂ Ω . Due to the pioneering nature of the present research, we propose here the apparently simple case of $$\varOmega =(0, \infty )$$ Ω = ( 0 , ∞ ) with boundary $$\{0\}$$ { 0 } of zero Lebesgue measure. Our results turn out to be instructive for the general case of boundary with positive (finite) Borel measures. Moreover, in our view, we bring new light to dynamic boundary value problems and the probabilistic description of the associated models.

Analytical and Computational Results for Neutral Impulsive Fractional System on Time Scales

This paper delves into the existence and uniqueness of neutral fractional integro-differential impulsive dynamic equations across various time scales, enriched by nonlocal initial conditions using the Caputo-Nabla derivative. By leveraging the refined fixed point theorem, the study provides a robust framework for establishing existence. The theoretical findings are elegantly illustrated through detailed graphical representations, enhancing the comprehension and appeal of the results.

Properties of a Partial Fredholm Integro-differential Equations with Nonlocal Condition and Algorithms

Properties of a Partial Fredholm Integro-differential Equations with Nonlocal Condition and Algorithms

Impulsive integro‐differential inclusions with nonlocal conditions: Existence and Ulam's type stability

This article focuses on the existence and Ulam–Hyers–Rassias stability outcomes pertaining to a specific category of impulsive integro‐differential inclusions (with instantaneous and non‐instantaneous impulses). These problems are examined using resolvent operators, drawing from the Grimmer perspective. Our analysis is based on Bohnenblust–Karlin's and Darbo's fixed point theorems for multivalued mappings in Banach spaces. Additionally, we provide an illustrative example to reinforce and demonstrate the validity of our findings.

Fixed point theorems for xi-alpha-eta-Gamma F-fuzzy contraction with an application to neutral fractional integro-differential equation with nonlocal conditions

Fixed point theorems for xi-alpha-eta-Gamma F-fuzzy contraction with an application to neutral fractional integro-differential equation with nonlocal conditions

T-controllability of higher-order fractional stochastic delay system via integral contractor

The existence, uniqueness and trajectory (T)-controllability results for mild solutions to the fractional neutral stochastic integro-delay differential system are presented in this article. To demonstrate the results, the concept of bounded integral contractors is combined with the stochastic result and sequencing technique. In contrast to previous publications, we do not need to specify the induced inverse of the controllability operator to prove the controllability results, and the relevant non-linear function does not have to meet the Lipschitz condition. Furthermore, T-controllability results for neutral integro-delay differential systems with non-local conditions have been established. Finally, an application to demonstrate the acquired results is discussed.

Existence analysis on multi-derivative nonlinear fractional neutral impulsive integro-differential equations

This article utilizes the Atangana–Baleanu (AB) fractional derivative to examine the behavior of multi-derivative fractional neutral impulsive integro-differential equations under non-local conditions. To demonstrate the existence, uniqueness, and controllability of the solutions, we utilize fixed point theory as our primary analytical tool. Furthermore, we include a detailed example to illustrate and validate the theoretical results obtained from our study.