Three symmetric positive solutions for second-order nonlocal boundary value problems

Using the Leggett-Williams fixed point theorem, we will obtain at least three symmetric positive solutions to the second-order nonlocal boundary value problem of the form $ begin{gathered} u'left( t right) + gleft( t right)fleft( {t,uleft( t right)} right) = 0, 0 < t < 1, hfill uleft( 0 right) = uleft( 1 right) = int_0^1 {mleft( s right)u} left( s right)ds, hfill end{gathered} $ begin{gathered} u'left( t right) + gleft( t right)fleft( {t,uleft( t right)} right) = 0, 0 < t < 1, hfill uleft( 0 right) = uleft( 1 right) = int_0^1 {mleft( s right)u} left( s right)ds, hfill end{gathered} where m ∈ L ¹[0, 1], g: (0, 1) → [0,∞) is continuous, symmetric on (0, 1) and maybe singular at t = 0 and t = 1, f: [0, 1] × [0,∞) → [0,∞) is continuous and f(·, x) is symmetric on [0, 1] for all x ∈ [0,∞).