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Yuji Liu,Xingyuan Liu 충청수학회 2012 충청수학회지 Vol.25 No.3
Su±cient conditions for the existence of at least one solution of a class of multi-point boundary value problems of the fractional di??erential equations at resonance are established. The main theorem generalizes and improves those ones in [Liu, B., Solv-ability of multi-point boundary value problems at resonance(II), Appl. Math. Comput., 136(2003)353-377], see Remark 2.3. An example is presented to illustrate the main results.
Yuji Liu 충청수학회 2011 충청수학회지 Vol.25 No.4
Motivated by Agarwal and O Regan ( Boundary value problems for general discrete systems on in¯nite intervals, Comput. Math. Appl. 33(1997)85-99), this article deals with the discrete type BVP of the in¯nite di??erence equations. The su±cient condi-tions to guarantee the existence of at least three positive solutions are established. An example is presented to illustrate the main re- sults. It is the purpose of this paper to show that the approach to get positive solutions of BVPs by using multi-¯xed-point theorems can be extended to treat BVPs for in¯nite di??erence equations. The strong Caratheodory (S-Caratheodory) function is de¯ned in this paper.
Liu, Yuji,Shi, Haiping,Liu, Xingyuan Department of Mathematics 2013 Kyungpook mathematical journal Vol.53 No.2
In this article, we establish the existence of at least three unbounded positive solutions to a boundary-value problem of the nonlinear singular fractional differential equation. Our analysis relies on the well known fixed point theorems in the cones.
Existence and Non-Existence of Positive Solutions of BVPs for Singular ODEs on Whole Lines
LIU, YUJI,YANG, PINGHUA Department of Mathematics 2015 Kyungpook mathematical journal Vol.55 No.4
This paper is concerned with integral type boundary value problems of second order singular differential equations with quasi-Laplacian on whole lines. Sufficient conditions to guarantee the existence and non-existence of positive solutions are established. The emphasis is put on the non-linear term $[{\Phi}({\rho}(t)x^{\prime}(t))]^{\prime}$ involved with the nonnegative singular function and the singular nonlinearity term f in differential equations. Two examples are given to illustrate the main results.
Liu, Yuji,Xia, Jianye The Korean Society for Computational and Applied M 2010 Journal of applied mathematics & informatics Vol.28 No.1
This paper is concerned with the neutral impulsive functional differential equations $$\{{x'(t)\;+\;a(t)x(t)\;=\;f(t,\;x(t\;-\;\tau(t),\;x'(t\;-\;\delta(t))),\;a.e.\;t\;{\in}\;R, \atop {\Delta}x(t_k)\;=\;b_kx(t_k),\;k\;{\in}\;Z.$$ Sufficient conditions for the existence of at least three positive T-periodic solution are established. Our results generalize and improve the known ones. Some examples are presented to illustrate the main results.
Liu, Yuji 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.23 No.1
The existence of solutions of the following multi-point boundary value problem $${x^{(n)}(t)=f(t,\;x(t),\;x'(t),{\cdots}, x^{(n-2)}(t))+r(t),\;0<t<1,\;x^{(i)}({\xi}_i)=0\;for\;i=0,\;1,{\cdots},\;n-3,\;{\alpha}x^{(n-2)}(0)-{\beta}x^{(n-1)}(0)={\gamma}x^{(n-1)}(1)+{\tau}x^{(n-2)}(1)=0$$ is studied. Sufficient conditions for the existence of at least one solution of BVP(*) are established. It is of interest that the growth conditions imposed on f are allowed to be super-linear (the degrees of phases variables are allowed to be greater than 1 if it is a polynomial). The results are different from known ones since we don't apply the Green's functions of the corresponding problem and the method to obtain a priori bounds of solutions are different enough from known ones. Examples that can not be solved by known results are given to illustrate our theorems.
Yuji Liu,Patricia J.Y. Wong 한국전산응용수학회 2015 Journal of applied mathematics & informatics Vol.33 No.3
By employing a fixed point theorem in a weighted Banach space, we establish the existence of a solution for a system of impulsive singular fractional differential equations. Some examples are presented to illustrate the efficiency of the results obtained.
Liu, Yuji Department of Mathematics 2008 Kyungpook mathematical journal Vol.48 No.4
New sufficient conditions for the existence of at least one solution of Neumann type boundary value problems for second order nonlinear differential equations $$\array{\{{p(t)\phi(x'(t)))'=f(t,x(t),\;x(\tau_1(t)),\;{\cdots},\;x(\tau_m(t))),\;t\in[0,T],\\x'(0)=0,\;x'(T)=0,}\,}$$, are established.
YUJI LIU 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.23 No.1
The existence of solutions of the following multi-point boundary value problem ( x(n)(t) = f(t, x(t), x0(t), · · · , x(n−2)(t)) + r(t), 0 < t < 1, x(i)(i) = 0 for i = 0, 1, · · · , n − 3, x(n−2)(0) − x(n−1)(0) = x(n−1)(1) + x(n−2)(1) = 0 () is studied. Sufficient conditions for the existence of at least one solution of BVP() are established. It is of interest that the growth conditions imposed on f are allowed to be super-linear (the degrees of phases variables are allowed to be greater than 1 if it is a polynomial). The results are different from known ones since we don’t apply the Green’s functions of the corresponding problem and the method to obtain a priori bounds of solutions are different enough from known ones. Examples that can not be solved by known results are given to illustrate our theorems.