Nonlocal Initial Condition Research Articles

A Nonlinear Fractional Problem with Mixed Volterra-Fredholm Integro-Differential Equation: Existence, Uniqueness, H-U-R Stability, and Regularity of Solutions

This paper considers nonlinear fractional mixed Volterra-Fredholm integro-differential equation with a nonlocal initial condition. We propose a fixed-point approach to investigate the existence, uniqueness, and Hyers-Ulam-Rassias stability of solutions. Results of this paper are based on nonstandard assumptions and hypothesis and provide a supplementary result concerning the regularity of solutions. We show and illustrate the wide validity field of our findings by an example of problem with nonlocal neutral pantograph equation, involving functional derivative and ψ -Caputo fractional derivative.

On Hilfer generalized proportional fractional derivative

Motivated by the Hilfer and the Hilfer–Katugampola fractional derivative, we introduce in this paper a new Hilfer generalized proportional fractional derivative, which unifies the Riemann–Liouville and Caputo generalized proportional fractional derivative. Some important properties of the proposed derivative are presented. Based on the proposed derivative, we consider a nonlinear fractional differential equation with nonlocal initial condition and show that this equation is equivalent to the Volterra integral equation. In addition, the existence and uniqueness of solutions are proven using fixed point theorems. Furthermore, we offer two examples to clarify the results.

Approximate controllability of non-autonomous Sobolev type integro-differential equations having nonlocal and non-instantaneous impulsive conditions

The aim of this article is to study approximate controllability of a class of non-autonomous Sobolev type integro-differential equations having non-instantaneous impulses with nonlocal initial condition. The results will be proved with the help of evolution system and Krasnoselskii fixed point theorem. An example is presented to show how our abstract results can be applied.

Approximate controllability of a semilinear impulsive stochastic system with nonlocal conditions and Poisson jumps

The objective of this paper is to investigate the approximate controllability of a semilinear impulsive stochastic system with nonlocal conditions and Poisson jumps in a Hilbert space. Nonlocal initial condition is a generalization of the classical initial condition and is motivated by physical phenomena. The results are obtained by using Sadovskii’s fixed point theorem. Finally, an example is provided to illustrate the effectiveness of the obtained result.

Time-varying integro-differential inclusions with Clarke sub-differential and non-local initial conditions: existence and approximate controllability

In this paper, we mainly consider a time-varying semi-linear integro-differential inclusion with Clarke sub-differential and a non-local initial condition. By a suitable Green function combined with a resolvent operator, we firstly formulate its mild solutions and show that it admits at least one mild solution which can exist in a well-defined ball with a radius big enough. Through constructing a proper functional, we then derive a useful characterization of the approximate controllability for its related linear system in Green function terms, and establish a sufficient condition for the approximate controllability of the time-varying semi-linear integro-differential inclusion. Lastly, we also consider the finite approximate controllability of the time-varying semi-linear integro-differential inclusion via variational method.

Observability of Fractional-Order Impulsive Control Integro-Differential System with Nonlocal Initial Condition

The article describes a new concept for initial and exactly observability of nonlocal fractional-order impulsive control integro-differential system. This is based on the concepts of the abstract Cauchy problem, which depended on some necessary and sufficient conditions. These conditions established on the semigroup theory of bounded operators as a dynamical operator system, which generated by bounded linear operators. Moreover, invertible operators play a primary role, and we presented a necessary condition for some nonlinear multi variables functions. Thus, all these operators were treated in nonlinear functional analysis to guaranty the initial observable and exactly observability. Therefore, from the mild solution of the system and exactly homogenous part, we proved the equivalent concepts between the initial observability and exactly the observability. Thus, our approach in this article is to prove the uniqueness of initial nonlocal values with admissible control, which belongs to the second-order Lebesgue integrable. The interest of observability results in this article lies by proving a unique fixed point, which is nonlocal initial values that are described in the proposal formula by using Banach’s fixed point theory. The processing observability for complexly systems (such as this system) with all components and properties was established and can be used for many control system applications.

Monotone Iterative Technique for Nonlocal Impulsive Finite Delay Differential Equations of Fractional Order

The paper is concerned with the extension of a monotone iterative technique to impulsive finite delay differential equations of fractional order with a nonlocal initial condition in an ordered Banach space. We study the existence of extremal mild solutions with or without assuming the compactness of a semigroup and also prove the uniqueness of the mild solution of the system. The results are obtained with the help of fractional calculus, a measure of non-compactness, the semigroup theory and monotone iterative technique. Finally, an example is provided to show the application of our main.

Approximate controllability of nonlocal and impulsive neutral integro-differential equations using the resolvent operator theory and an approximating technique

Our purpose in this paper is to study the approximate controllability of nonlocal and impulsive neutral integro-differential equations using the resolvent operator theory, the approximating technique, semigroup theory and the theory of fixed point. We also derive a variation of constants formula for representing a solution of the given system by resolvent operators and then study the existence of a mild solution of the system. We prove the main results by taking the impulsive functions to be continuous only and taking the nonlocal initial condition function to be Lipschitz continuous in the first case and continuous only in the second case. Finally, the main results are illustrated using examples.

Hilfer fractional evolution hemivariational inequalities with nonlocal initial conditions and optimal controls

In this paper, we mainly consider a control system governed by a Hilfer fractional evolution hemivariational inequality with a nonlocal initial condition. We first establish sufficient conditions for the existence of mild solutions to the addressed control system via properties of generalized Clarke subdifferential and a fixed point theorem for condensing multivalued maps. Then we present the existence of optimal state-control pairs of the limited Lagrange optimal systems governed by a Hilfer fractional evolution hemivariational inequality with a nonlocal initial condition. The optimal control results are derived without uniqueness of solutions for the control system.

A Unique Solution of Stochastic Partial Differential Equations with Non-Local Initial condition

In this paper, we shall discuss the uniqueness ”pathwise uniqueness” of the solutions of stochastic partial differential equations (SPDEs) with non-local initial condition,We shall use the Yamada-Watanabe condition for ”pathwise uniqueness” of the solutions of the stochastic differential equation; this condition is weaker than the usual Lipschitz condition. The proof is based on Bihari’sinequality.

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.

Approximate Controllability of Semilinear Stochastic Integrodifferential System with Nonlocal Conditions

The objective of this paper is to analyze the approximate controllability of a semilinear stochastic integrodifferential system with nonlocal conditions in Hilbert spaces. The nonlocal initial condition is a generalization of the classical initial condition and is motivated by physical phenomena. The results are obtained by using Sadovskii’s fixed point theorem. At the end, an example is given to show the effectiveness of the result.

The two-level finite difference schemes for the heat equation with nonlocal initial condition

In this paper, the two-level finite difference schemes for the one-dimensional heat equation with a nonlocal initial condition are analyzed. As the main result, we obtain conditions for the numerical stability of the schemes. In addition, we revise the stability conditions obtained in [21] for the Crank–Nicolson scheme. We present several numerical examples that confirm the theoretical results within linear, as well as nonlinear problems. In some particular cases, it is shown that for small regions of the time step size values, the explicit FTCS scheme is stable while certain implicit methods, such as Crank–Nicolson scheme, are not.

Nonlocal Integro-Differential Equations Without the Assumption of Equicontinuity on the Resolvent Operator in Banach Spaces

In this work, we study the existence of mild solutions for the nonlocal integro-differential equation $$\begin{aligned} \left\{ \begin{array}{l} x'(t)=Ax(t)+\displaystyle \int _{0}^{t}B(t-s)x(s)\text {d}s+f(t,x_{t})\quad \text {for}\;\; t\in [0,b]\\ x_{0}=\phi +g(x)\in C([-r,0];X), \end{array} \right. \end{aligned}$$ without the assumption of equicontinuity on the resolvent operator and without the assumption of separability on the Banach space X. The nonlocal initial condition is assumed to be compact. Our main result is new and its proof is based on a measure of noncompactness developed in Kamenskii et al. (Condensing multivalued maps and semilinear differential inclusions in Banach spaces. Walter De Gruyter, Berlin, 2001) together with the well-known Monch fixed point Theorem. To illustrate our result, we provide an example in which the resolvent operator is not equicontinuous.

Fractional parabolic problems with a nonlocal initial condition

Abstract In this work we will consider a class of non local parabolic problems with nonlocal initial condition, more precisely we deal with the problem where is a bounded domain, the function a can be a singular potential, g is a suitable function which will be specified later and p, θ Abstract. In this work we will consider a class of non local parabolic problems with nonlocal initial condition, more precisely we deal with the problem under natural conditions on the data. The question of global existence is also analyzed.

The Generalized Taylor Expansion Method for Solving Some Types of Fractional Non-local Problems

The aim of this paper is to prove the existence and the uniqueness of the solution for some types of fractional non-local problems, namely the non-linear non-local initial value problems for fractional Fredholm-Volterra integro-differential equations. Also, the generalized Taylor expansion method is used to solve the non-local initial value problem that consists of the linear fractional Fredholm-Volterraintegro-differential equation together with the linear non-local initial condition with some illustrative examples.

Nonlocal m-Dissipative Evolution Inclusions in General Banach Spaces

In this paper, we study a class of a multivalued perturbations of m-dissipative evolution inclusions with nonlocal initial condition in arbitrary Banach spaces. We prove the existence of solutions when the multivalued right hand side is Lipschitz and admits nonempty closed bounded, but in general case, neither convex nor compact values. Illustrative example is provided.

Abstract functional second-order stochastic evolution equations with applications

We investigate a class of abstract second-order damped functional stochastic evolution equations driven by a fractional Brownian motion in a separable Hilbert space. The global existence of mild solutions is established under various growth and compactness conditions. The case of a nonlocal initial condition is addressed. A related convergence result is discussed, and the theory is applied to stochastic wave and beam equations, as well as a spring-mass system, for illustrative purposes.

A study on existence of solutions for fractional functional differential equations

In this research article, we establish the existence results of mild solutions for semi-linear impulsive neutral fractional order integro-differential equations with state dependent delay subject to nonlocal initial condition by applying well known classical fixed point theorems. At last, we present an example of partial derivative to illuminate the results.

Nonclassical Problem for Ultraparabolic Equation in Abstract Spaces

Nonclassical problem for ultraparabolic equation with nonlocal initial condition with respect to one time variable is studied in abstract Hilbert spaces. We define the space of square integrable vector-functions with values in Hilbert spaces corresponding to the variational formulation of the nonlocal problem for ultraparabolic equation and prove trace theorem, which allows one to interpret initial conditions of the nonlocal problem. We obtain suitable a priori estimates and prove the existence and uniqueness of solution of the nonclassical problem and continuous dependence upon the data of the solution to the nonlocal problem. We consider an application of the obtained abstract results to nonlocal problem for ultraparabolic partial differential equation with second-order elliptic operator and obtain well-posedness result in Sobolev spaces.