Displaying similar documents to “Existence Principles for Singular Vector Nonlocal Boundary Value Problems with φ -Laplacian and their Applications”

On a two-point boundary value problem for second order singular equations

Alexander Lomtatidze, P. Torres (2003)

Czechoslovak Mathematical Journal

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The problem on the existence of a positive in the interval ] a , b [ solution of the boundary value problem u ' ' = f ( t , u ) + g ( t , u ) u ' ; u ( a + ) = 0 , u ( b - ) = 0 is considered, where the functions f and g ] a , b [ × ] 0 , + ∞ [ → ℝ satisfy the local Carathéodory conditions. The possibility for the functions f and g to have singularities in the first argument (for t = a and t = b ) and in the phase variable (for u = 0 ) is not excluded. Sufficient and, in some cases, necessary and sufficient conditions for the solvability of that problem are established.

Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem

Qingliu Yao (2013)

Applications of Mathematics

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We consider the classical nonlinear fourth-order two-point boundary value problem u ( 4 ) ( t ) = λ h ( t ) f ( t , u ( t ) , u ' ( t ) , u ' ' ( t ) ) , 0 < t < 1 , u ( 0 ) = u ' ( 1 ) = u ' ' ( 0 ) = u ' ' ' ( 1 ) = 0 . In this problem, the nonlinear term h ( t ) f ( t , u ( t ) , u ' ( t ) , u ' ' ( t ) ) contains the first and second derivatives of the unknown function, and the function h ( t ) f ( t , x , y , z ) may be singular at t = 0 , t = 1 and at x = 0 , y = 0 , z = 0 . By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.