Let (L_{0}) be the set of all Lebesgue measurable functions on I.
Let be the set of Lebesgue measurable functions on such that (3.7).
; is the space of real Lebesgue measurable functions on which is a Banach space for the -norm, ;, where,.
Let X be the set of Lebesgue measurable functions on ([0,1]) such that (int_{0}^{1} x t),mathrm{d}t<1).
; (ii) is the space of real Lebesgue measurable functions on which is a Banach space for the -norm, ; (iii), where,.
Let X be the set of Lebesgue measurable functions on [ 0, 1 ] such that ∫ 0 1 | f ( x ) | 2 d x < ∞.
Let X be the set of Lebesgue measurable functions on ([0, 1]) such that (int_{0}^{1}|x(t)|, dt<infty).
Let X be the set of Lebesgue measurable functions on ([0,1]) such that (int_{0}^{1}vert f(x vert ^{2}, dx<infty).
If there exists x 0 ∈ X such that x 0 ≤ T x 0, then T has a fixed point in X. Example 2.1 Let X be the set of Lebesgue measurable functions on [ 0, 1 ] such that ∫ 0 1 | x ( t ) | d t < ∞.
Notations: Throughout the paper, L 2 is the space of scalar value Lebesgue measurable functions on Ω and is a Banach space for the L 2 -norm ∥ v ∥ 2 = ( ∫ Ω | v ( x ) | d x ) 1 2, v ∈ L 2.
We consider the set of Lebesgue measurable functions on ([0,1] ) such that (int_{0}^{1} {|alpha(x)|}^{q},dx < infty), where (q>0 ) is a real number denoted by (mathfrak{M} ).
