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A class of singular fourth-order boundary value problems with nonhomogeneous nonlinearity

Qingliu Yao (2013)

Annales Polonici Mathematici

We study the existence of positive solutions to a class of singular nonlinear fourth-order boundary value problems in which the nonlinearity may lack homogeneity. By introducing suitable control functions and applying cone expansion and cone compression, we prove three existence theorems. Our main results improve the existence result in [Z. L. Wei, Appl. Math. Comput. 153 (2004), 865-884] where the nonlinearity has a certain homogeneity.

A unified approach to singular problems arising in the membrane theory

Irena Rachůnková, Gernot Pulverer, Ewa B. Weinmüller (2010)

Applications of Mathematics

We consider the singular boundary value problem ( t n u ' ( t ) ) ' + t n f ( t , u ( t ) ) = 0 , lim t 0 + t n u ' ( t ) = 0 , a 0 u ( 1 ) + a 1 u ' ( 1 - ) = A , where f ( t , x ) is a given continuous function defined on the set ( 0 , 1 ] × ( 0 , ) which can have a time singularity at t = 0 and a space singularity at x = 0 . Moreover, n , n 2 , and a 0 , a 1 , A are real constants such that a 0 ( 0 , ) , whereas a 1 , A [ 0 , ) . The main aim of this paper is to discuss the existence of solutions to the above problem and apply the general results to cover certain classes of singular problems arising in the theory of shallow membrane caps, where we are especially interested in...

Dead cores of singular Dirichlet boundary value problems with φ -Laplacian

Ravi P. Agarwal, Donal O'Regan, Staněk, Svatoslav (2008)

Applications of Mathematics

The paper discusses the existence of positive solutions, dead core solutions and pseudodead core solutions of the singular Dirichlet problem ( φ ( u ' ) ) ' = λ f ( t , u , u ' ) , u ( 0 ) = u ( T ) = A . Here λ is the positive parameter, A > 0 , f is singular at the value 0 of its first phase variable and may be singular at the value A of its first and at the value 0 of its second phase variable.

Error analysis of splitting methods for semilinear evolution equations

Masahito Ohta, Takiko Sasaki (2017)

Applications of Mathematics

We consider a Strang-type splitting method for an abstract semilinear evolution equation t u = A u + F ( u ) . Roughly speaking, the splitting method is a time-discretization approximation based on the decomposition of the operators A and F . Particularly, the Strang method is a popular splitting method and is known to be convergent at a second order rate for some particular ODEs and PDEs. Moreover, such estimates usually address the case of splitting the operator into two parts. In this paper, we consider the splitting...

Existence and iteration of positive solutions for a singular two-point boundary value problem with a p -Laplacian operator

De-xiang Ma, Weigao Ge, Zhan-Ji Gui (2007)

Czechoslovak Mathematical Journal

In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative scheme for the equation ( φ p ( u ' ) ) ' + q ( t ) f ( u ) = 0 , 0 < t < 1 , where φ p ( s ) : = | s | p - 2 s , p > 1 , subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient q ( t ) may be singular at t = 0 , 1 .

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