Nonlocal Conformable-Fractional Differential Equations with a Measure of Noncompactness in Banach Spaces

This paper deals with the existence of mild solutions for the following Cauchy problem: dαxt/dtα=Axt+ft,xt,x0=x0+gx,t∈0,τ, where dα./dtα is the so-called conformable fractional derivative. The linear part A is the infinitesimal generator of a uniformly continuous semigroup Ttt≥0 on a Banach space X, f and g are given functions. The main result is proved by using the Darbo–Sadovskii fixed point theorem without assuming the compactness of the family Ttt>0 and the Lipshitz condition on the nonlocal part g.

Impulsive Fractional Differential Equations with p-Laplacian Operator in Banach Spaces

In this paper, we study a class of boundary value problem (BVP) with multiple point boundary conditions of impulsive p-Laplacian operator fractional differential equations. We establish the sufficient conditions for the existence of solutions in Banach spaces. Our analysis relies on the Kuratowski noncompactness measure and the Sadovskii fixed point theorem. An example is given to demonstrate the main results.

Existence of Mild Solutions for Sobolev-Type Hilfer Fractional Nonautonomous Evolution Equations with Delay

This paper treats the existence of mild solutions for Sobolev-type Hilfer fractional nonautonomous evolution equations with delay in Banach spaces. We first characterize the definition of mild solutions for the studied problem which was given based on an operator family generated by the operator pair (A,B) and probability density function. And then via Hilfer fractional derivative and combining the techniques of fractional calculus, measure of noncompactness and Sadovskii fixed-point theorem, we obtain new existence result of mild solutions for Sobolev-type Hilfer fractional nonautonomous evolution equations. Particularly, the existence or compactness of an operator $B^{-1} $ is not necessarily needed in our results. Furthermore, our results obtained improve and extend some related conclusions on this topic. At last, an example is given to illustrate our main results.

Solutions to Dirichlet-Type Boundary Value Problems of Fractional Order in Banach Spaces

We consider the boundary value problems with Dirichlet-type boundary conditions of nonlinear fractional differential equation in Banach space. The existence of the solution to the boundary value problems is established. Our analysis relies on the Sadovskii fixed point theorem. As an application, we give an example to demonstrate our results.

Complete Controllability of Fractional Neutral Differential Systems in Abstract Space

By using fractional power of operators and Sadovskii fixed point theorem, we study the complete controllability of fractional neutral differential systems in abstract space without involving the compactness of characteristic solution operators introduced by us.

Controllability of Second-Order Semilinear Impulsive Stochastic Neutral Functional Evolution Equations

We consider a class of impulsive neutral second-order stochastic functional evolution equations. The Sadovskii fixed point theorem and the theory of strongly continuous cosine families of operators are used to investigate the sufficient conditions for the controllability of the system considered. An example is provided to illustrate our results.

Dynamic equations x(Δm)(t) = f(t, x(t)) on time scales

In this paper we prove the existence of solutions and Carathéodory’s type solutions of the dynamic Cauchy problemThe Sadovskii fixed point theorem and Ambrosetti’s lemma are used to prove the main result.As dynamic equations are an unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions.

Null controllability of neutral evolution integrodifferential systems with infinite delay

We establish a set of sufficient conditions for the controllability of nonlinear neutral evolution integrodifferential systems with infinite delay in Banach spaces. The results are established by using the Sadovskiĭ fixed point theorem and generalize the previous results.