Local attractivity for integro-differential equations with noncompact semigroups

In this paper, we are devoted to study the existence and local attractivity of solutions for a class of integro-differential equations.Under the situation that the nonlinear term satisfy Carathéodory conditions and a noncompactness measure condition, we establish some existence and local attractivity of mild solutions by utilizing Mönch fixed point theorem, Kuratowski measure of noncompactness and resolvent operator theory in the sense of Grimmer.Our investigations will be situated in the Banach space of real functions which are defined, continuous, and bounded on the right-hand real half axis 𝕉+. Moreover an example is given to illustrate our outcomes.

Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups

The aim of this paper is to discuss the existence of mild solutions for a class of semilinear stochastic partial differential equation with nonlocal initial conditions and noncompact semigroups in a real separable Hilbert space. Combined with the theory of stochastic analysis and operator semigroups, a generalized Darbo’s fixed point theorem and a new estimation technique of the measure of noncompactness, we obtained the existence of mild solutions under the situation that the nonlinear term and nonlocal function satisfy some appropriate local growth conditions and a noncompactness measure condition. In addition, the condition of uniformly continuity of the nonlinearity is not required and also the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example to illustrate the feasibility of the main results is also given.

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 Results for Generalized Bagley-Torvik Type Fractional Differential Inclusions with Nonlocal Initial Conditions

In this article, we prove the existence of solutions for the generalized Bagley-Torvik type fractional order differential inclusions with nonlocal conditions. It allows applying the noncompactness measure of Hausdorff, fractional calculus theory, and the nonlinear alternative for Kakutani maps fixed point theorem to obtain the existence results under the assumptions that the nonlocal item is compact continuous and Lipschitz continuous and multifunction is compact and Lipschitz, respectively. Our results extend the existence theorems for the classical Bagley-Torvik inclusion and some related models.

Integrable Solutions of a Nonlinear Integral Equation via Noncompactness Measure and Krasnoselskii's Fixed Point Theorem

We study the existence of solutions of a nonlinear Volterra integral equation in the space L1[0,+∞). With the help of Krasnoselskii’s fixed point theorem and the theory of measure of weak noncompactness, we prove an existence result for a functional integral equation which includes several classes on nonlinear integral equations. Our results extend and generalize some previous works. An example is given to support our results.

Monotone Iterative Technique for Fractional Evolution Equations in Banach Spaces

We investigate the initial value problem for a class of fractional evolution equations in a Banach space. Under some monotone conditions and noncompactness measure conditions of the nonlinearity, the well-known monotone iterative technique is then extended for fractional evolution equations which provides computable monotone sequences that converge to the extremal solutions in a sector generated by upper and lower solutions. An example to illustrate the applications of the main results is given.

Initial value problems for first order impulsive integro-differential equations of Volterra type in Banach spaces

In this paper, we use a new method and combining the partial ordering method to study the existence of the solutions for the first order nonlinear impulsive integro-differential equations of Volterra type on finite interval in Banach spaces and for the first order nonlinear impulsive integro-differential equations of Volterra type on infinite interval with infinite number impulsive times in Banach spaces. By introducing an interim space and using progressive estimation method, some restrictive conditions on impulsive terms, used before, such as, prior estimation, noncompactness measure estimations are deleted.