Luxemburg norm localisation for nonlocal differential equations in variable-exponent Lebesgue spaces

We investigate a class of variable growth nonlocal differential equations of Kirchhoff-type having the general form aligned -A\!( ∫ ₀¹ b(1-s) (u(s) )ᵖ⁽ˢ⁾\,ds) u''(t) = λ f (t,u(t) ), t∈ (0,1), aligned - A ∫ 0 1 b ( 1 - s ) ( u ( s ) ) p ( s ) d s u ′ ′ ( t ) = λ f ( t , u ( t ) ) , t ∈ ( 0 , 1 ) , where A is a possibly sign-changing function. Our analysis is carried out in the variable-exponent Lebesgue space Lp(· )([0,1]) L p ( · ) ( [ 0 , 1 ] ) under the standing hypothesis $$p(t)>1$$ p ( t ) > 1 . We demonstrate that using the Luxemburg norm allows for a sharper localisation of the solution to the nonlocal problem. Moreover, conditions imposed on both λ λ and f are appreciably weakened by analysing the problem within the Luxemburg norm framework. An example explicitly demonstrates both the qualitative and the quantitative advantages over earlier techniques.