scholarly journals Existence and multiplicity of solutions for fourth-order boundary value problems with parameters

2007 ◽  
Vol 327 (1) ◽  
pp. 362-375 ◽  
Author(s):  
Xi-Lan Liu ◽  
Wan-Tong Li
2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Shoucheng Yu ◽  
Zhilin Yang

We study the existence and multiplicity of positive solutions for the system of fourth-order boundary value problems x(4)=ft,x,x′,-x′′,-x′′′,y,y′,-y′′,-y′′′,  y(4)=gt,x,x′,-x′′,-x′′′,y,y′,-y′′,-y′′′,  x(0)=x′(1)=x′′(0)=x′′′(1)=0, and y(0)=y′(1)=y′′(0)=y′′′(1)=0, where f,g∈C([0,1]×R+8,R+)  (R+:=[0,∞)). We use fixed point index theory to establish our main results based on a priori estimates achieved by utilizing some integral identities and inequalities and R+2-monotone matrices.


2010 ◽  
Vol 2010 ◽  
pp. 1-26
Author(s):  
Bo Zheng

We consider the existence and multiplicity of solutions to discrete conjugate boundary value problems. A generalized asymptotically linear condition on the nonlinearity is proposed, which includes the asymptotically linear as a special case. By classifying the linear systems, we define index functions and obtain some properties and the concrete computation formulae of index functions. Then, some new conditions on the existence and multiplicity of solutions are obtained by combining some nonlinear analysis methods, such as Leray-Schauder principle and Morse theory. Our results are new even for the case of asymptotically linear.


2012 ◽  
Vol 45 (3) ◽  
Author(s):  
Marek Galewski ◽  
Joanna Smejda

AbstractUsing critical point theory and some monotonicity results we consider the existence and multiplicity of solutions to nonlinear discrete boundary value problems represented as a nonlinear system


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