$p(x)$-growth in nonlocal differential equations with convolution coefficients

The nonlocal differential equation \begin{equation} -A \big ( \big (b*u^{p(\cdot)} \big )(1) \big )u''(t) =\lambda f\big(t,u(t)\big)\text{, }t\in(0,1)\notag \end{equation} is considered, where $$ \displaystyle(a*b)(t):=\int_0^ta(t-s)b(s)\ ds , \ \ \ t\ge0 , $$ is a finite convolution. The existence of at least one positive solution is considered when the equation is subject to boundary conditions. A model case of the above problem occurs when $b(t)\equiv1$, as in this case the problem reduces to \begin{equation} -A \Big (\int_0^1\big(u(s)\big)^{p(s)}\ ds \Big )u''(t) =\lambda f\big(t,u(t)\big)\text{, }t\in(0,1).\notag \end{equation} As a consequence nonlocal coefficients possessing $p(x)$-growth are considered. By using a specialized order cone together with topological fixed point theory we are able to provide existence theorems with a minimum of assumptions on the coefficient function $A$. In particular, $A$ can vanish or change sign on intervals of infinite measure.