Document Type: Research Paper

**Authors**

School of Mathematics and Physics, Changzhou University, Changzhou 213164, China.

**Abstract**

This paper gives an abstract order-type existence theorem for second order nonlocal boundary value problems at resonance and obtain existence criteria for at least two positive solutions, where $f$ is a continuous function. Our results generalize or extend related results in the literature and give a positive answer to the question raised in the literature. An example is given to illustrate the new results.

**Keywords**

**Main Subjects**

C. Bai and J. Fang, Existence of positive solutions for three-point boundary value problems at resonance, *J. Math. Anal. Appl. ***291 **(2004), no. 2, 538--549.

Y. Chen and X. Tang, Positive solutions of fractional differential equations at resonance on the half-line, *Bound. Value Probl. ***2012 **(2012), Article no. 64, 13 pages.

J. Chu and F. Wang, An order-type existence theorem and applications to periodic problems, *Bound. Value Probl. ***2013 **(2013), Article no. 37, 12 pages.

C.T. Cremins, A fixed-point index and existence theorems for semilinear equations in cones, *Nonlinear Anal. ***46 **(2001), no. 6, 789--806.

Y. Cui and J. Sun, Fixed point theorems for a class of nonlinear operators in Hilbert spaces with lattice structure and application, *Fixed Point Theory Appl. ***2013 **(2013), Article no. 345, 9 pages.

Y. Cui and X. Zhang, Fixed points for discontinuous monotone operators, *Fixed Point Theory Appl. ***2010 **(2010), Article ID 926209, 11 pages.

D. Franco, G. Infante and M. Zima, Second order nonlocal boundary value problems at resonance, *Math. Nachr. ***284 **(2011), no. 7, 875--884.

X. Han, Positive solutions for a three-point boundary value problem at resonance, *J. Math. Anal. Appl. ***336 **(2007), no. 1, 556--568.

G. Infante and M. Zima, Positive solutions of multi-point boundary value problems at resonance, *Nonlinear Anal. ***69 **(2008), no. 8, 2458--2465.

S. Liang and L. Mu, Multiplicity of positive solutions for singular three-point boundary value problems at resonance, *Nonlinear Anal. ***71 **(2009), no. 7-8, 2497--2505.

Y. Liu and C. Shen, On the existence of positive solution for a kind of multi-point boundary value problem at resonance, *Nonlinear Anal. ***72 **(2010), no. 11, 4211--4220.

Y. Liu and C. Shen, Positive solutions for second order four-point boundary value problems at resonance, *Topol. Methods Nonlinear Anal. ***38 **(2011), no. 1, 1--15.

J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, *J. Differential Equations ***12 **(1972) 610--636.

D. O'Regan and M. Zima, Leggett-Williams norm-type theorems for coincidences, *Arch. Math. (Basel) ***87 **(2006), no. 3, 233--244.

W.V. Petryshyn, Using degree theory for densely defined *A*-proper maps in the solvability of semilinear equations with unbounded and noninvertible linear part, *Nonlinear Anal. ***4 **(1980), no. 2, 259--281.

A. Yang and W. Ge, Existence of positive solutions for self-adjoint boundary value problems with integral boundary conditions at resonance, *Electron. J. Differential Equations ***2011 **(2011), no. 11, 8 pages.

A. Yang and H. Wang, Positive solutions of two-point boundary value problems of nonlinear fractional differential equation at resonance, *Electron. J. Qual. Theory Differ. Equ. ***2011 **(2011), no. 71, 15 pages.

H. Zhang and J. Sun, Positive solutions of third-order nonlocal boundary value problems at resonance, *Bound. Value Probl. ***2012 **(2012), Article ID 102, 10 pages.

M. Zima, Existence of positive solutions for a second-order three-point boundary value problem at resonance, in: Dynamic Systems and Applications, Vol. 5, pp. 527--532, Dynamic, Atlanta, GA, 2008.

M. Zima, Positive solutions for first-order boundary value problems at resonance, *Commun. Appl. Anal. ***13 **(2009), no. 4, 671--680.

Volume 43, Issue 5

September and October 2017

Pages 1045-1055

**Receive Date:**15 September 2015**Revise Date:**12 April 2016**Accept Date:**14 April 2016