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Abstract The purpose of this paper is to study a generalized Riemann–Liouville fractional differential equation and system with nonlocal boundary conditions. Firstly, some properties of the Green function are presented and then Lyapunov-type inequalities for a sequential ψ-Riemann–Liouville fractional boundary value problem are established. Also, the existence and uniqueness of solutions are proved by using Banach and Schauder fixed-point theorems. Furthermore, the existence and uniqueness of solutions to a sequential nonlinear differential system is established by means of Schauder’s and Perov’s fixed-point theorems. Examples are given to validate the theoretical results.

The purpose of this paper is to study a generalized Riemann–Liouville fractional differential equation and system with nonlocal boundary conditions. Firstly, some properties of the Green function are presented and then Lyapunov-type inequalities for a sequential ψ-Riemann–Liouville fractional boundary value problem are established. Also, the existence and uniqueness of solutions are proved by using Banach and Schauder fixed-point theorems. Furthermore, the existence and uniqueness of solutions to a sequential nonlinear differential system is established by means of Schauder's and Perov's fixed-point theorems. Examples are given to validate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the existence of mild solutions as well as optimal controls for non-autonomous impulsive evolution equations with nonlocal conditions. Using the Schauder’s fixed-point theorem as well as the theory of evolution family, we prove the existence of mild solutions for the concerned problem. Furthermore, without the Lipschitz continuity of the nonlinear term, the optimal control result is derived by setting up minimizing sequences twice. An example is given of the application of the results.

In this paper, we investigate the existence of mild solutions as well as optimal controls for non-autonomous impulsive evolution equations with nonlocal conditions. Using the Schauder's fixed-point theorem as well as the theory of evolution family, we prove the existence of mild solutions for the concerned problem. Furthermore, without the Lipschitz continuity of the nonlinear term, the optimal control result is derived by setting up minimizing sequences twice. An example is given of the application of the results. [ABSTRACT FROM AUTHOR]

Abstract In this article, we study the existence of positive periodic solutions of second order damped boundary value problem u ″ + p ( t ) u ′ + q ( t ) u = g ( t , u , u ′ ) $u'' + p(t)u'+q(t)u = g(t,u,u')$ , u ( 0 ) = u ( T ) $u(0) = u(T )$ , u ′ ( 0 ) = u ′ ( T ) $u'(0) = u'(T )$ . The main tools are the nonlinear alternative principle of Leray–Schauder and Schauder’s fixed point theorem. We emphasize that the damped term and nonnegative Green’s functions are the key points. We also apply the results to examples for testing. Some recent results in the literature are improved and generalized.

Abstract This paper is mainly devoted to studying one kind of the second order differential equation. Under periodic-integrable boundary value condition, the existence of the solutions of this equation is discussed by the method of the operator theory and the Schauder fixed point theorem.

In this article, we study the existence of positive periodic solutions of second order damped boundary value problem u ″ + p (t) u ′ + q (t) u = g (t , u , u ′) , u (0) = u (T) , u ′ (0) = u ′ (T) . The main tools are the nonlinear alternative principle of Leray–Schauder and Schauder's fixed point theorem. We emphasize that the damped term and nonnegative Green's functions are the key points. We also apply the results to examples for testing. Some recent results in the literature are improved and generalized. [ABSTRACT FROM AUTHOR]

This paper is mainly devoted to studying one kind of the second order differential equation. Under periodic-integrable boundary value condition, the existence of the solutions of this equation is discussed by the method of the operator theory and the Schauder fixed point theorem. [ABSTRACT FROM AUTHOR]

In this article, we first of all convert the boundary value problems for impulsive fractional differential equations to integral equations. Second, we construct a weighted function space and prove the completely continuous property of a nonlinear operator. Finally, we establish existence results for solutions of a boundary value problem for a nonlinear impulsive fractional differential system. Our analysis relies on the well-known Schauder fixed point theorem. An example is given to illustrate main results. [ABSTRACT FROM AUTHOR]

In this article, we first of all convert the boundary value problems for impulsive fractional differential equations to integral equations. Second, we construct a weighted function space and prove the completely continuous property of a nonlinear operator. Finally, we establish existence results for solutions of a boundary value problem for a nonlinear impulsive fractional differential system. Our analysis relies on the well-known Schauder fixed point theorem. An example is given to illustrate main results. [ABSTRACT FROM AUTHOR]

By application of Green's function and some fixed point theorems, i.e., Leray-Schauder alternative principle and Schauder's fixed point theorem, we establish two new existence results of positive periodic solutions for a nonlinear fourth-order singular differential equation, which extend and improve significantly existing results in the literature. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the existence of solutions to second order nonlinear boundary value problems (BVPs) involving the distributional Henstock-Kurzweil integral. The present results in this article are generalizations of previous results in the literature. [ABSTRACT FROM AUTHOR]

We establish the existence of positive periodic solutions of the second-order differential equation x" + a(t)x = f (t, x) + c(t) via Schauder’s fixed point theorem, where a ∈ L¹(R/TZ;R+), c ∈ L¹(R/TZ;R), f is a Caratheodory function and it is singular at x = 0. Our main results generalize some recent results by Torres (J. Differ. Equ. 232:277-284, 2007). [ABSTRACT FROM AUTHOR]

In this paper we deal with one kind of second order periodic-integrable boundary value problem. Using the lemma on bilinear form and Schauder's fixed point theorem, we give the existence and uniqueness of solutions for the problem under Lazer type nonresonant condition. MSC: 34B15, 34B16, 37J40. [ABSTRACT FROM AUTHOR]