Abstract
Abstract In this paper, we consider the existence of positive solutions for a second-order discrete boundary value problem Δ ( g ( k − 1 ) Δ u ( k − 1 ) ) + w ( k ) f ( k , u ( k ) ) = 0 subject to the boundary conditions: a u ( 0 ) − b g ( 0 ) Δ u ( 0 ) = ∑ i = 1 n − 1 h ( i ) u ( i ) , a u ( n ) + b g ( n − 1 ) Δ u ( n − 1 ) = ∑ i = 1 n − 1 h ( i ) u ( i ) , where a , b > 0 , Δ u ( k ) = u ( k + 1 ) − u ( k ) for k ∈ { 0 , 1 , … , n − 1 } , g ( k ) > 0 is symmetric on { 0 , 1 , … , n − 1 } , w ( k ) is symmetric on { 0 , 1 , … , n } , f : { 0 , 1 , … , n } × [ 0 , + ∞ ) is continuous, f ( k , u ) = f ( n − k , u ) for all ( k , u ) ∈ { 0 , 1 , … , n } × [ 0 , + ∞ ) , and h ( i ) is nonnegative and symmetric on { 0 , 1 , … , n } . By the fixed point theorem and the Hölder inequality, we study the existence of symmetric positive solutions for the above difference equation with sum form boundary conditions.
Highlights
A class of boundary value problems (BVPs) with integral boundary conditions arise in thermal conduction problems, semiconductor problems, and hydrodynamic problems [ – ]
By the fixed point theorem and the Hölder inequality, we study the existence of symmetric positive solutions for the above difference equation with sum form boundary conditions
1 Introduction A class of boundary value problems (BVPs) with integral boundary conditions arise in thermal conduction problems, semiconductor problems, and hydrodynamic problems [ – ]
Summary
A class of boundary value problems (BVPs) with integral boundary conditions arise in thermal conduction problems, semiconductor problems, and hydrodynamic problems [ – ]. We consider the existence of positive solutions for a second-order discrete boundary value problem (g(k – 1) u(k – 1)) + w(k)f (k, u(k)) = 0 subject to the boundary conditions: au(0) – bg(0) u(0) = By the fixed point theorem and the Hölder inequality, we study the existence of symmetric positive solutions for the above difference equation with sum form boundary conditions. In [ ], Feng considered the following differential equation BVP with integral boundary conditions: g(t)u (t) + w(t)f t, u(t) = , < t < ,
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