Nonlocal Initial Value Problem Research Articles

Enhancing mathematical simulation: a novel integro spline quasi-interpolation for nonlocal dynamical systems

ABSTRACT In this research, we present a novel and computationally streamlined method to tackle nonlocal initial value problems through a combination of modeling and simulation. Our approach utilizes the Integro spline quasi-interpolation technique, renowned for its effectiveness in approximating nonlocal integrals. To validate the efficacy of our proposed method, we conduct comprehensive comparisons with alternative approaches found in the existing literature for integrating nonlocal functions. Additionally, we successfully apply our suggested approach to address nonlocal Riccati, resistor-inductor circuit, and Airy oscillator dynamical systems. The numerical results obtained from these applications convincingly demonstrate the advantages and effectiveness of our developed approach in enhancing mathematical simulation.

Nonlocal semilinear second-order differential inclusions in abstract spaces without compactness

We study the existence of a mild solution to the nonlocal initial value problem for semilinear second-order differential inclusions in abstract spaces. The result is obtained by combining the Kakutani fixed point theorem with the approximation solvability method and the weak topology. This combination enables getting the result without any requirements for compactness of the right-hand side or of the cosine family generated by the linear operator.

Nonlocal Initial Value Problems for Ambartsumian Equation with Hilfer Generalized Proportional Fractional Derivative

Nonlocal Initial Value Problems for Ambartsumian Equation with Hilfer Generalized Proportional Fractional Derivative

Existence and stability theory of pantograph conformable fractional differential problem

The purpose of this paper is to investigate the existence and uniqueness (EU) of solutions to a class of conformable fractional differential equations (DE) with delay term using Krasnoselskii's fixed point theorem. The proposed problem is devoted to non-local initial value problems. Such problems are increasingly occurred in applications like in the filed of quantum mechanics and electrodynamics. The theoretical analysis is further enriched by establishing stability theory due to Ulam and its different kinds including ?Ulam-Hyers (UH), generalized Ulam-Hyers (GUH), Ulam-Hyers-Rassias (UHR), and generalized Ulam-Hyers-Rassias (GUHR)? stability for the considered class. The obtain analysis is then testified by an example.

Existence and Stability of Nonlocal Initial Value Problems Involving Generalized Katugampola Derivative

In this paper, the existence results for the solutions to nonlocal initial value problems involving generalized Katugampola derivative are established. Some fixed point theorem techniques are used to derive the existence results. In the sequel, we investigate the generalized Ulam-Hyers-Rassias stability corresponding to our problem. Some examples are given to illustrate our main results.

Exact Solvability Conditions for the Non-Local Initial Value Problem for Systems of Linear Fractional Functional Differential Equations

The exact conditions sufficient for the unique solvability of the initial value problem for a system of linear fractional functional differential equations determined by isotone operators are established. In a sense, the conditions obtained are optimal. The method of the test elements intended for the estimation of the spectral radius of a linear operator is used. The unique solution is presented by the Neumann’s series. All theoretical investigations are shown in the examples. A pantograph-type model from electrodynamics is studied.

On Fractional Semilinear Nonlocal Initial Value Problem with State Dependent Delay

On Fractional Semilinear Nonlocal Initial Value Problem with State Dependent Delay

Study of implicit-impulsive differential equations involving Caputo-Fabrizio fractional derivative

<abstract><p>This article is devoted to investigate a class of non-local initial value problem of implicit-impulsive fractional differential equations (IFDEs) with the participation of the Caputo-Fabrizio fractional derivative (CFFD). By means of Krasnoselskii's fixed-point theorem and Banach's contraction principle, the results of existence and uniqueness are obtained. Furthermore, we establish some results of Hyers-Ulam (H-U) and generalized Hyers-Ulam (g-H-U) stability. Finally, an example is provided to demonstrate our results.</p></abstract>

Dynamic Local and Nonlocal Initial Value Problems in Banach Spaces

Dynamic Local and Nonlocal Initial Value Problems in Banach Spaces

Global solvability for nonlinear nonautonomous evolution inclusions of Volterra-type and its applications

We study a nonlinear nonautonomous evolution inclusion of Volterra-type with time delay involving a multivalued nonlinearity. The family of operators (possibly unbounded) in the principal part of the evolution inclusion generates a noncompact evolution family. Global solvability for local and nonlocal initial value problems corresponding to the evolution inclusion is considered. The investigation in the present paper is based on two topological fixed point theorems. We impose a condition of Hausdorff-measure of noncompactness on the nonlinearity, which allows us to overcome the lack of compactness of the evolution family. Finally, we present an example to illustrate the abstract results.

Ulam stability for nonlocal differential equations involving the Hilfer–Katugampola fractional derivative

In this paper, we establish the existence and uniqueness of solutions to nonlocal initial value problem for differential equations with Hilfer–Katugampola type fractional derivative, also, the stability of this class of problem. The arguments are based upon the Banach contraction principle, and Schaefer’s fixed point theorem. An example is included to show the applicability of our results.

Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type Fractional Implicit Differential Equations

Abstract In this paper, we prove some existence results of solutions for a class of nonlocal initial value problem for nonlinear fractional hybrid implicit differential equations under generalized Hilfer fractional derivative. The result is based on a fixed point theorem on Banach algebras. Further, examples are provided to illustrate our results.

A Study on Impulsive Hilfer Fractional Evolution Equations with Nonlocal Conditions

Abstract In this paper, we concern with the existence of mild solution to nonlocal initial value problem for nonlinear Sobolev-type impulsive evolution equations with Hilfer fractional derivative which generalized the Riemann–Liouville fractional derivative. At first, we establish an equivalent integral equation for our main problem. Second, by means of the properties of Hilfer fractional calculus, combining measure of noncompactness with the fixed-point methods, we obtain the existence results of mild solutions with two new characteristic solution operators. The results we obtained are new and more general to known results. At last, an example is provided to illustrate the results.

Solvability of the Nonlocal Initial Value Problem and Application to Design of Controller for Heat-equation with Delay

Solvability of the Nonlocal Initial Value Problem and Application to Design of Controller for Heat-equation with Delay

RETRACTED ARTICLE: Existence of solutions of nonlocal initial value problems for differential equations with Hilfer–Katugampola fractional derivative

RETRACTED ARTICLE: Existence of solutions of nonlocal initial value problems for differential equations with Hilfer–Katugampola fractional derivative

On a multipoint nonlocal initial value problem for a singularly-perturbed first-order ODE

Abstract A linear first order ordinary differential equation (ODE) with a positive parameter ɛ and a multipoint nonlocal initial value condition (NLIVC) is considered. The existence of a classical solution of the multipoint nonlocal initial value problem (NLIVP) is proved. A uniform on ɛ a priori estimate and asymptotic expansion of smooth solution is obtained. The differential problem with integral kind of NLIVC is considered and reduced to appropriate multipoint NLIVP.

Existence and stability results for nonlocal initial value problems for differential equations with Hilfer fractional derivative

In this paper, we establish sufficient conditions for the existence and stability of solutions for a class of nonlocal initial value problems for differential equations with Hilfer's fractional derivative, The arguments are based upon the Banach contraction principle. Two examples are included to show the applicability of our results.

English

This paper is devoted to studying the existence of solutions of a nonlocal initial value problem involving generalized Katugampola fractional derivative. By using fixed point theorems, the results are obtained in weighted space of continuous functions. Illustrative examples are also given.

Nonlocal initial value problems for Hadamard-type fractional differential equations and inclusions

In this paper, we study initial value problems of fractional differential equations and inclusions of Hadamard type, supplemented with nonlocal conditions. Some new existence and uniqueness results are obtained by using fixed point theorems for single valued and multi-valued maps. Examples illustrating the main results are also presented.

Nonlocal initial value problems for implicit differential equations with Hilfer–Hadamard fractional derivative

In this paper, the Schaefer's fixed-point theorem is used to investigate the existence of solutions to nonlocal initial value problems for implicit differential equations with Hilfer–Hadamard fractional derivative. Then the Ulam stability result is obtained by using Banach contraction principle. An example is given to illustrate the applications of the main result.