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Advanced Nonlinear Studies 13 (2013), 893–919 Positive Solutions for Quasilinear Systems with Critical Growth Yuxia Guo∗, Xiangqing Liu†, Fukun Zhao‡ ∗Department of Mathematics Tsinghua University, Beijing 100084, China e-mail: yguo@math.tsinghua.edu.cn †Department of Mathematics Yunnan Normal University, Kunming 650092, China e-mail: lxq8u8@163.com ‡Department of Mathematics Yunnan Normal University, Kunming 650092, China e-mail: fukunzhao@163.com Received 08 November 2012 Communicated by E.N. Dancer Abstract In this paper, we consider the following system

Georgian Mathematical Journal Volume 14 (2007), Number 4, 699–710 POSITIVE SOLUTIONS FOR NEUTRAL DIFFERENCE EQUATIONS WITH CONTINUOUS ARGUMENTS XIANYI LI Abstract. Some “sharp” conditions are established for a kind of linear neu- tral difference equations with continuous arguments not to possess eventually positive solutions. The existence and asymptotic behavior are obtained for positive solutions of the kind of equations. The results for linear cases are further extended to nonlinear ones. A comparison principle, which is a nec- essary and sufficient condition

of positive solutions of the following nonlinear perturbation of ( 1.5 ) u ( t ) = g ( t , u ( t ) ) + ∫ 0 1 k ( t , s ) f ( s , u ( s ) ) d s , t ∈ [ 0 , 1 ] . $$\begin{array}{} \displaystyle u(t)=g(t,u(t))+\int\limits_0^1k(t,s)f(s,u(s))\text{d} s, \qquad t\in[0,1]. \end{array} $$ (1.6) Similar problems have been considered for example in [ 4 ], [ 6 ], [ 21 ] and [ 25 ]. In [ 6 ], the authors studied the Hammerstein perturbed equation u ( t ) = g ( u ( t ) ) + λ ∫ Ω k ( t , s ) [ ∑ i = 0 m a i ( s ) u α i ( s ) ] − 1 d s , λ > 0 , $$\begin

) -superlinear in the x -variable, but need not satisfy the Ambrosetti-Rabinowitz condition which is common in problems with superlinear reactions. Also, λ > 0 \lambda \gt 0 is a parameter. We are looking for positive solutions of

References [1] I. Bachar, H. Mâagli, N. Zeddini, Estimates on the Green function and existence of positive solutions of nonlinear singular elliptic equations, Com- mun. Contemp. Math. 5(3) (2003) 401-434. [2] Ruipeng Chen, Existence of positive solutions for semilinear elliptic systems with indefinite weight, Electronic Journal of Differential Equations. 164 (2011) 1-8. [3] K.L. Chung, Z. Zhao, From Brownian motion to Schrodinger's equation, Springer Verlag 1995. [4] R. Dalmasso, Existence and uniqueness of positive solutions of semilinear elliptic systems

{\beta=0} , then we recover the Neumann problem. Our aim in this paper is to study the precise dependence of the set of positive solutions on the parameter λ > 0 {\lambda>0} . In this direction, we prove a bifurcation-type theorem for small values of the parameter, that is, we show that there exists a critical parameter value λ * ∈ ( 0 , + ∞ ) {\lambda^{*}\in(0,+\infty)} such that • for all λ ∈ ( 0 , λ * ) {\lambda\in(0,\lambda^{*})} , problem ( 1.1 ) admits at least two positive solutions; • for λ = λ * {\lambda=\lambda^{*}} , problem ( 1.1 ) has at least one

[1] C. Bai, Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative. J. Math. Anal. Appl. 384, No 2 (2011), 211–231. http://dx.doi.org/10.1016/j.jmaa.2011.05.082 [2] M. El-Shahed, Positive solutions for boundary value problems of nonlinear fractional differential equation. Abs. Appl. Anal. Volume 2007, Article ID 10368. [3] D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones. Academic Press, Orlando (1988). [4] D. Guo, Positive solutions of Hammerstein integral

Abstract

In this paper we consider a fractional difference inclusion of the form (α-1Δαy)(t)λy(t+α-1)+f(t+α-1,y(t+α-1)), y(α-1)=y(α-1+T)+j=1NFj(y(tj)), where 0<α<1. Consequently, we treat the situation in which the boundary condition is both nonlocal and nonlinear. Thus, the boundary conditions under study can be very general. We provide conditions under which this problem has at least one positive solution and then discuss the application of the results. Finally, since we allow Fj0, for each j, our results are also new for the case of periodic boundary conditions.

Advanced Nonlinear Studies 15 (2015), 789–803 Multiplicity of Positive Solutions for Second Order Nonlinear Dirichlet Problem with One-dimension Minkowski-Curvature Operator Ruyun Ma∗, Yanqiong Lu Department of Mathematics Northwest Normal University, Lanzhou, 730070, P R China e-mail:mary@nwnu.edu.cn (R. Ma), linmu8610@163.com (Y. Lu) Received 08 August 2014 Communicated by Jean Mawhin Abstract In this work, we study the existence and multiplicity of positive solutions for nonlinear Dirichlet problem with one-dimension Minkowski-curvature operator ( u′ √ 1 − κu

GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 4, 1999, 347-362 POSITIVE SOLUTIONS TO SECOND ORDER SINGULAR DIFFERENTIAL EQUATIONS INVOLVING THE ONE-DIMENSIONAL M-LAPLACE OPERATOR TOMOYUKI TANIGAWA Abstract. We consider a class of second order quasilinear differential equations with singular ninlinearities. Our main purpose is to investi- gate in detail the asymptotic behavior of their solutions defined on a positive half-line. The set of all possible positive solutions is classified into five types according to their asymptotic behavior near infinity, and sharp