Nonlocal Initial Condition Research Articles

APPROXIMATIONS OF SOLUTIONS FOR A NONLOCAL FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION WITH DEVIATED ARGUMENT

This paper investigates the existence of mild solution for a fractional integro-differential equations with a deviating argument and nonlocal initial condition in an arbitrary separable Hilbert space H via technique of approximations. We obtain an associated integral equation and then consider a sequence of approximate integral equations obtained by the projection of considered associated nonlocal fractional integral equation onto finite dimensional space. The existence and uniqueness of solutions to each approximate integral equation is obtained by virtue of the analytic semigroup theory via Banach fixed point theorem. Next we demonstrate the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. We consider the Faedo-Galerkin approximation of the solution and demonstrate some convergenceresults. An example is also given to illustrate the abstract theory.

New qualitative properties of solutions to nonlinear nonlocal

We introduce new concepts of asymptotically anti-periodic function and semi-Lipschitz continuity. The former is a natural generalization of the well-known anti-periodic function. Then, sufficient conditions, ensuring the existence of asymptotically anti-periodic mild solutions to a Cauchy problem of nonlinear evolution equation with nonlocal initial condition, are established. It is mentioned that one of our main results is proved in the absence of the compactness and Lipschitz continuity of nonlocal item and of the Lipschitz continuity of nonlinearity. Finally, an example is presented as an application.

Approximate controllability of second order semilinear stochastic system with nonlocal conditions

In this paper, the approximate controllability of second order semilinear stochastic system involving nonlocal conditions is studied. By using Sadovskii’s Fixed Point theorem with stochastic analysis theory, we derive a new set of sufficient conditions for the approximate controllability of second order semilinear stochastic system with nonlocal conditions under the assumption that the corresponding linear system is approximately controllable. Finally, an application to a second order semilinear stochastic system with nonlocal initial condition is provided to illustrate the obtained theory.

Viability for delay evolution equations with nonlocal initial conditions

We prove a sufficient condition for a time-dependent closed set to be viable with respect to a delay evolution inclusion subjected to an implicit nonlocal initial condition. An application to a comparison problem is also included.

Non-autonomous impulsive Cauchy problems of parabolic type involving nonlocal initial conditions

We consider a nonautonomous impulsive Cau\-chy problem of parabolic type involving a nonlocal initial condition in a Banach space $X$, where the operators in linear part (possibly unbounded) depend on time $t$ and generate an evolution family. New existence theorems of mild solutions to such a problem, in the absence of compactness and Lipschitz continuity of the impulsive item and nonlocal item, are established. The non-autonomous impulsive Cauchy problem of neutral type with nonlocal initial condition is also considered. Comparisons with available literature are also given. Finally, as a sample of application, these results are applied to a system of partial differential equations with impulsive condition and nonlocal initial condition. Our results essentially extend some existing results in this area.

Approximate Controllability of Fractional Neutral Stochastic Evolution Equations with Nonlocal Conditions

In this paper, the approximate controllability of neutral stochastic fractional differential equations involving nonlocal initial conditions is studied. By using Sadovskii’s fixed point theorem with stochastic analysis theory, we derive a new set of sufficient conditions for the approximate controllability of semilinear fractional stochastic differential equations with nonlocal conditions under the assumption that the corresponding linear system is approximately controllable. Finally, an application to a fractional partial stochastic differential equation with nonlocal initial condition is provided to illustrate the obtained theory.

Results of local and global mild solution for impulsive fractional differential equation with state dependent delay

In this paper, we establish the existence of local and global mild solution for an impul- sive fractional integro-differential equation with state dependent delay subject to nonlocal initial condition. The existence results for local mild solution are proved by applying the Schauder, nonlinear Larey Schauder alternative and Banach fixed point theorems. Then, we prove global existence result. An example is presented to demonstrate the application of the established re- sults.

Fractional neutral evolution equations with nonlocal conditions

In the present paper, we deal with the fractional neutral differential equations involving nonlocal initial conditions. The existence of mild solutions are established. The results are obtained by using the fractional power of operators and the Sadovskii’s fixed point theorem. An application to a fractional partial differential equation with nonlocal initial condition is also considered. MSC:26A33, 34K30, 34K37, 34K40.

Global Continua of Positive Solutions for some Quasilinear Parabolic Equation with a Nonlocal Initial Condition

This paper is concerned with a quasilinear parabolic equation including a nonlinear nonlocal initial condition. The problem arises as equilibrium equation in population dynamics with nonlinear diffusion. We make use of global bifurcation theory to prove existence of an unbounded continuum of positive solutions.

Decay of solutions of discontinuous parabolic equations with nonlocal initial conditions

In this paper we investigate the asymptotic behavior of solutions of a class of discontinuous parabolic partial differential equations with nonlocal initial condition of integral type. Some sufficient conditions are discussed. Our technique is based on the maximum principle and the Green’s function for linear parabolic partial differential equations.

Abstract fractional integro-differential equations involving nonlocal initial conditions in α-norm

In the present paper, we deal with the Cauchy problems of abstract fractional integro-differential equations involving nonlocal initial conditions in α-norm, where the operator A in the linear part is the generator of a compact analytic semigroup. New criterions, ensuring the existence of mild solutions, are established. The results are obtained by using the theory of operator families associated with the function of Wright type and the semigroup generated by A, Krasnoselkii's fixed point theorem and Schauder's fixed point theorem. An application to a fractional partial integro-differential equation with nonlocal initial condition is also considered. Mathematics subject classification (2000) 26A33, 34G10, 34G20

Nonlocal conditions for differential inclusions in the space of functions of bounded variations

We discuss the existence of solutions of an abstract differential inclusion, with a right-hand side of bounded variation and subject to a nonlocal initial condition of integral type.

A note on the fractional Cauchy problems with nonlocal initial conditions

Of concern is the Cauchy problems for fractional integro-differential equations with nonlocal initial conditions. Using a new strategy in terms of the compactness of the semigroup generated by the operator in the linear part and approximating technique, a new existence theorem for mild solutions is established. An application to a fractional partial integro-differential equation with a nonlocal initial condition is also considered.

Existence of Solutions to Fractional Mixed Integrodifferential Equations with Nonlocal Initial Condition

We study the existence and uniqueness theorem for the nonlinear fractional mixed Volterra-Fredholm integrodifferential equation with nonlocal initial condition , where , , and is a given function. We point out that such a kind of initial conditions or nonlocal restrictions could play an interesting role in the applications of the mentioned model. The results obtainded are applied to an example.

Nonlocal Conditions for Lower Semicontinuous Parabolic Inclusions

We discuss conditions for the existence of at least one solution of a discontinuous parabolic equation with lower semicontinuous right hand side and a nonlocal initial condition of integral type. Our technique is based on fixed point theorems for multivalued maps.

Discontinuous Parabolic Problems with a Nonlocal Initial Condition

We study parabolic differential equations with a discontinuous nonlinearity and subjected to a nonlocal initial condition. We are concerned with the existence of solutions in the weak sense. Our technique is based on the Green's function, integral representation of solutions, the method of upper and lower solutions, and fixed point theorems for multivalued operators.

On a class of retarded integro-differential equations with nonlocal initial conditions

Of concern is the following Cauchy problem for fractional integro-differential equations with time delay and nonlocal initial condition { u ′ ( t ) − ∫ 0 t ( t − s ) μ − 2 Γ ( μ − 1 ) A u ( s ) d s = F ( t , u ( t ) , u ( κ ( t ) ) ) , t ≥ 0 , u ( t ) + H t ( u ) = ϕ ( t ) , − τ ≤ t ≤ 0 , where 1 < μ < 2 , τ > 0 , A : D ( A ) ⊂ X → X is a generator of a solution operator on a complex Banach space X , the convolution integral in the equation is known as the Riemann–Liouville fractional integral, κ ( t ) : [ 0 , + ∞ ) → [ − τ , + ∞ ) representing the delay property, is a function, and H t is an operator defined from [ − τ , 0 ] × C ( [ − τ , 0 ] , X ) into X for some T > 0 which constitutes a nonlocal condition. The local existence and uniqueness of mild solutions for the Cauchy problem, under various criteria, are proved. Moreover, we present an existence result of the global solution. Also an example is given to illustrate the applications of the abstract results.

Solutions to Fractional Differential Equations with Nonlocal Initial Condition in Banach Spaces

A new existence and uniqueness theorem is given for solutions to differential equations involving the Caputo fractional derivative with nonlocal initial condition in Banach spaces. An application is also given.

Solutions to Fractional Differential Equations with Nonlocal Initial Condition in Banach Spaces

Solutions to Fractional Differential Equations with Nonlocal Initial Condition in Banach Spaces

On the stability of a fractional-order differential equation with nonlocal initial condition

The topic of fractional calculus (integration and differentiation of fractional-order), which concerns singular integral and integro-differential operators, is enjoying interest among mathematicians, physicists and engineers (see [1]-[2] and [5]-[14] and the references therein). In this work, we investigate initial value problem of fractional-order differential equation with nonlocal condition. The stability (and some other properties concerning the existence and uniqueness) of the solution will be proved.