# Existence results for a class of high order differential equation associated with integral boundary conditions at resonance

## Le Cong Nhan, Do Huy Hoang, and Le Xuan Truong

Address:

Le Cong Nhan, An Giang University, 18, Ung Van Khiem Street, An Giang, Vietnam

Do Huy Hoang, Le Xuan Truong, Department of Mathematics and Statistics, University of Economics HoChiMinh City, 59C Nguyen Dinh Chieu Street, District 3, HoChiMinh City, Vietnam

E-mail:

lcnhanmathagu@gmail.com

lxuantruong@gmail.com

Abstract: By using Mawhinâ€™s continuation theorem, we provide some sufficient conditions for the existence of solution for a class of high order differential equations of the form
x^{(n)} =f(t,x,x^{\prime },\dots ,x^{(n-1)})\,, \quad t \in [0, 1]\,,
associated with the integral boundary conditions at resonance. The interesting point is that we shall deal with the case of nontrivial kernel of arbitrary dimension corresponding to high order differential operator which will cause some difficulties in constructing the generalized inverse operator.

AMSclassification: primary 34B10; secondary 34B15.

Keywords: coincidence degree, high order differential equation, resonance.

DOI: 10.5817/AM2017-2-111