Application of Fixed-Point Index Theory for a Nonlinear Fractional Boundary Value Problem with an Advanced Argument

In this paper, we investigate the existence of positive solutions of a class of fractional three-point boundary value problem with an advanced argument by using fixed-point index theory. Our results improve and extend some known results in the literature. Two examples are given to demonstrate the effectiveness of our results.

A fixed-point approach for decaying solutions of difference equations

A boundary value problem associated with the difference equation with advanced argument * Δ ( a n Φ ( Δ x n ) ) + b n Φ ( x n + p ) = 0 , n ≥ 1 is presented, where Φ ( u ) = | u | α sgn u , α > 0, p is a positive integer and the sequences a , b , are positive. We deal with a particular type of decaying solution of (*), that is the so-called intermediate solution (see below for the definition). In particular, we prove the existence of this type of solution for (*) by reducing it to a suitable boundary value problem associated with a difference equation without deviating argument. Our approach is based on a fixed-point result for difference equations, which originates from existing ones stated in the continuous case. Some examples and suggestions for future research complete the paper. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

Nonlinear Boundary Value Problem of Fractional Differential Equations with Arguments under Integral Boundary Condition

In this paper, we develop the existence and uniqueness theory of fractional differential equation involving Riemann-Liouville di¤erential operator of order 0 < < 1, with advanced argument under integral boundary condition. We show the uniqueness of solution by using Banach xed point theorem with a weighted norm. We apply the comparison result and obtain the existence and uniqueness of solution by monotone iterative technique also by use weakly coupled extremal solution of (1.1).

Initial Value Problems for Second Order Neutral Impulsive Integro-differential Equations with Advanced Argument

This paper discusses initial value problems for second order neutral impulsive integro-differential equations with advanced argument. By using the fixed point theorem of either Leray-Schaude or Banach, two existence results are obtained. By comparison, each of them has his own strong and weak points. If appropriate changes are made to some conditions for two results, the same results can be got. Two examples to illustrate our main results are given, which are compared with the existence results for impulsive differential equations from existing literature.