Nonlocal Initial Conditions Research Articles

Controllability of some nonlocal‐impulsive Volterra evolution systems via measures of noncompactness

This study investigates the controllability of a Volterra evolution equation with impulsive terms and nonlocal initial conditions. With the aid of the resolvent operator generated by the linear part of the equation, mild solutions can be defined. Notably, the resolvent operator lacks compactness and equicontinuity. Additionally, the compactness of the impulsive and nonlocal functions is not required. Sufficient conditions for controllability are obtained through measures of noncompactness in Banach spaces. Functional differential equations and hyperbolic partial differential equations can be solved with these results. An example is given to illustrate the validity of the presented results.

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.

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.

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.

Theoretical and computational results for implicit non-singular hybrid fractional differential equation subject to multi-terms non-local initial conditions

Theoretical and computational results for implicit non-singular hybrid fractional differential equation subject to multi-terms non-local initial conditions

Efficient Computational Methods for Solving the One-Dimensional Parabolic Equation with Nonlocal Initial and Boundary Conditions

Efficient Computational Methods for Solving the One-Dimensional Parabolic Equation with Nonlocal Initial and Boundary Conditions

Non-autonomous fractional nonlocal evolution equations with superlinear growth nonlinearities

We carry out an analysis of the existence of solutions for a class of nonlinear fractional partial differential equations of parabolic type with nonlocal initial conditions. Sufficient conditions for the solvability of the desired problem are presented by transforming it into an abstract non-autonomous fractional evolution equation, and constructing two families of solution operators based on the Mittag-Leffler function, the Mainardi Wright-type function and the analytic semigroup generated by the closed densely defined operator −A(⋅). The discussions are based on the fractional power theory as well as the Banach fixed point theorem in the interpolation space Xpυ (0⩽υ<1, 1<p<∞).

Fractional Neutral Integro-Differential Equations with Nonlocal Initial Conditions

We primarily investigate the existence of solutions for fractional neutral integro-differential equations with nonlocal initial conditions, which are crucial for understanding natural phenomena. Taking into account factors such as neutral type, fractional-order integrals, and fractional-order derivatives, we employ probability density functions, Laplace transforms, and resolvent operators to formulate a well-defined concept of a mild solution for the specified equation. Following this, by using fixed-point theorems, we establish the existence of mild solutions under more relaxed conditions.

DETERMINATION OF A KERNEL IN A NONLOCAL PROBLEM FOR THE TIME-FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION

In this work, we consider an inverse problem of determining the kernel of a fractional diffusion equation. The backward problem is the initial-boundary/value problem for this equation with non-local initial and homogeneous Dirichlet conditions. To determine the kernel, an overdetermination condition of the integral form is specified for the solution of the backward problem. Using the Fourier method and an ordinary fractional differential equation with a non-local boundary condition, the inverse problem is reduced to an equivalent problem. Further by using the fixed point argument in suitable Sobolev spaces, the global theorems of existence and uniqueness for the solution of the inverse problem are obtained.

Fractional Evolution Equations with Nonlocal Initial Conditions and Superlinear Growth Nonlinear Terms

Fractional Evolution Equations with Nonlocal Initial Conditions and Superlinear Growth Nonlinear Terms

Analysis of nonlinear implicit fractional differential equations with the Atangana-Baleanu derivative via measure of non-compactness

&lt;p&gt;In this study, we proved existence results for nonlinear implicit fractional differential equations with the Caputo version of the Atangana-Baleanu derivative, subject to the boundary and nonlocal initial conditions. The Kuratowski's measure of non-compactness and its associated fixed point theorems–Darbo's fixed point theorem and Mönchh's fixed point theorem, are the foundation for the analysis in this paper. We support our results with examples of nonlinear implicit fractional differential equations involving the Caputo version of the Atangana-Baleanu derivative subject to both boundary and nonlocal initial conditions. In addition, we provide solutions to the problems we considered.&lt;/p&gt;

Ulam-Hyers and generalized Ulam-Hyers stability of fractional functional integro-differential equations

Unique solvability conditions for general fractional functional integro-differential equations with nonlocal initial conditions are established using fixed-point theory. Ulam-Hyers and generalized Ulam-Hyers stability conditions for a solution of these problems are formulated. An example of the nonlinear fractional functional integro-differential equation related to wheel shimmy models is considered as well.

Results on generalized neutral fractional impulsive dynamic equation over time scales using nonlocal initial condition

&lt;abstract&gt;&lt;p&gt;This paper explored the existence and uniqueness of a neutral fractional impulsive dynamic equation over time scales that included nonlocal initial conditions and employed the Caputo-nabla derivative (C$ \nabla $D). The establishment of existence and uniqueness relies on the fine fixed point theorem. Furthermore, a comparison was conducted between the fractional order C$ \nabla $D and the Riemann-Liouville nabla derivative (RL$ \nabla $D) over time scales. Theoretical findings were substantiated through a numerical methodology, and an illustrative graph using MATLAB was presented for the provided example.&lt;/p&gt;&lt;/abstract&gt;

Stability of nonlinear impulsive higher order differential – fractional integral delay equations with nonlocal initial conditions

The aim of this paper is to investigate some types of stability such as generalized Hyers-Ulam- Rassias stability(G-H-U-R-stabile) and the relation with Hyers-Ulam(H-U-stabile) stable and Hyers-Ulam-Rassias stable (H-U-R-stabile) and generalized Hyers-Ulam stable (G-H-U- stable) to obtain which one guarantee to satisfy stability of equations included a nonlinear function some of them contains a delay time of solution and the other contain a vector of different order of derivatives for the solution to n-time and vector of fractional order of integrals with different fractional orders and that was the for using a claculse of fractional calculus to satisfies the issue of this techniques. Moreover, the nonlocal initial values for the proposal equation of nonlinear impulsive higher order differential – fractional integral delay time equations which are adding more interesting for nonlinear analytic object of nonlinear higher order integro – fractional order impulsive classes, and the impulsive difference of the equation has some necessary conditions to prove the results of solution to be stable with certain type has related with other types. The necessary and sufficient conditions which assumed on this nonlinear higher order integro-differential impulsive equation have been achieved the stability with interesting certain estimates obtain through the proving technique. Also the uniqueness of solution has been studied with same conditions was presented for stability and used for that issue a contraction fixed point theorem.

Optimal control problem of fractional evolution inclusions with Clarke subdifferential driven by quasi-hemivariational inequalities

In this paper, we introduce and study an optimal control problem for a new class of fractional differential quasi-hemivariational inequalities with nonlocal initial conditions and Clarke subdifferentials. The main tools in our study are the theory of fractional calculus, the fixed point argument for condensing multimaps and Balder’s theorem. We prove the solvability of abstract nonlocal fractional differential quasi-hemivariational inequalities and then show the existence of solutions to the associated optimal control problems.

Mild Solutions of Fractional Integrodifferential Diffusion Equations with Nonlocal Initial Conditions via the Resolvent Family

Fractional integrodifferential diffusion equations play a significant role in describing anomalous diffusion phenomena. In this paper, we study the existence and uniqueness of mild solutions to these equations. Firstly, we construct an appropriate resolvent family, through which the related equicontinuity, strong continuity, and compactness properties are studied using the convolution theorem of Laplace transform, the probability density function, the Cauchy integral formula, and the Fubini theorem. Then, we construct a reasonable mild solution for the considered equations. Finally, we obtain some sufficient conditions for the existence and uniqueness of mild solutions to the considered equations by some fixed point theorems.

A class of stochastic nonlocal evolution equations with nonlocal initial conditions

In this paper, we consider the existence and uniqueness of mild solutions for stochastic nonlocal evolution equations with Lévy diffusion operator and nonlocal initial conditions. Based on the continuity of the Lévy semigroup and the technique of the measure of noncompactness, we establish the local existence of mild solutions in [Formula: see text] under some weaker growth conditions. Moreover, we obtain the existence of mild solutions on any finite interval by using the general growth conditions on the nonlinear. Finally, the global existence and uniqueness of mild solutions follow from the additional Lipschitz conditions on nonlinear terms.

On Existence and Attractivity of Ψ-Hilfer Hybrid Fractional-order Langevin Differential Equations

The work reported in this article studies the equivalence relationship between fractional integral equation and Ψ-Hilfer Hybrid Langevin Differential Equations of fractional order with nonlocal initial conditions, and then we use this relationship to establish the existence of the results by means of Banach algebra and Schauder’s fixed point theorem. We then demonstrate the uniform local attractiveness of all the solutions.

A nonlinear fractional partial integro‐differential equation with nonlocal initial value conditions

In this work, we study a new nonlinear partial integro‐differential equation with nonlocal initial value conditions and investigate the solutions of this equation. By considering an equivalent implicit integral equation via series, we prove the uniqueness of solutions of the equation by Babenko's approach, Banach's contraction principle, and the multivariable Mittag–Leffler function. We also demonstrate the application of our key theorem with an illustrative example.