Nonlinear second-order equations with functional implicit impulses and nonlinear functional boundary conditions

In this paper we deal with nonlinear second-order boundary value problems with impulses. The impulsive functions depend implicitly on the different considered variables and the boundary value conditions are nonlinear. In both cases functional dependence on the solution is allowed. The existence results follow from the existence of a pair of well-ordered lower and upper solutions.     

Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations

By establishing a comparison result and using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the existence of solutions for systems of nonlinear fractional differential equations.     

Extremal solutions of second order nonlinear periodic boundary value problems

We study a periodic boundary value problem for a nonlinear ordinary differential equation of second order when the nonlinearity is given by a Carathéodory function. We generalize the monotone iterative method to cover the fully nonlinear case.     

The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions

In this paper, first, we consider the existence of a positive solution for the nonlinear fractional differential equation boundary value problem where 0≤λ < 1,^CD^α is the Caputo's differential operator of order α, and f:[0,1] × [0,∞)→[0,∞) is a continuous function. Using some cone theoretic techniques, we deduce a general existence theorem for this problem. Then, we consider two following more general problems for arbitrary α, 1≤n < α≤n + 1: Problem 1: where , 0≤λ < k + 1; Problem 2: where 0≤λ≤α and D^α is the Riemann–Liouville fractional derivative of order α. For these problems, we give existence results, which improve recent results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Third-order right-focal multi-point problems on time scales

We are concerned with the existence and form of positive solutions to a third-order multi-point boundary-value problem on time scales with mixed derivatives. We find and utilize the Green function for the corresponding homogeneous right-focal problem as the kernel of an integral equation of Hammerstein-type. Two examples are included to illustrate the results.     

Positive solutions of nonlinear fractional differential equations with integral boundary value conditions

In this paper, we consider the existence of positive solutions for a class of nonlinear boundary-value problem of fractional differential equations with integral boundary conditions. Our analysis relies on known Guo–Krasnoselskii fixed point theorem.     

Extremal solutions and Green's functions of higher order periodic boundary value problems in time scales

In this paper we develop the monotone method in the presence of lower and upper solutions for the problem

     

First order differential equations with piecewise constant arguments and nonlinear boundary value conditions

This paper is devoted to the study of differential equations with piecewise constant arguments coupled with nonlinear boundary value conditions. Under suitable assumptions on the data of the equation, by means of the method of (weakly coupled) lower and upper solutions, we derive the existence of extremal solutions and extremal quasi-solutions. Moreover some results are given concerning the uniqueness of solutions. Furthermore, we deduce some maximum principles related to the linear equation which allow us to develop the monotone iterative method. Some illustrative examples are also presented.