We study the existence of positive solutions of the integral equation $$x\left(t\right)=\mu {\int }_0^{1}k(t,s)f(s,x\left(s\right),x^{\text{'}}\left(s\right),...,x^{(n-1)}\left(s\right))ds,n\ge 2$$ in both $C^{n-1}[0,1]$ and $W^{n-1,p}[0,1]$ spaces, where $p\ge 1$ and $\mu >0$. Throughout this paper $k$ is nonnegative but the nonlinearity $f$ may take negative values. The Krasnosielski fixed point theorem on cone is used.

In this paper we study the existence and uniqueness of positive and periodic solutions of nonlinear delay integral systems of the type $$x\left(t\right)={\int }_{t-{\tau }_1}^{t}f(s,x\left(s\right),y\left(s\right))ds$$$$y\left(t\right)=...$$

We consider a convolution-type integral equation u = k ⋆ g(u) on the half line (−∞; a), a ∈ ℝ, with kernel k(x) = x α−1, 0 < α, and function g(u), continuous and nondecreasing, such that g(0) = 0 and 0 < g(u) for 0 < u. We concentrate on the uniqueness problem for this equation, and we prove that if α ∈ (1, 4), then for any two nontrivial solutions u 1, u 2 there exists a constant c ∈ ℝ such that u 2(x) = u 1(x +c), −∞ < x. The results are obtained by applying Hilbert projective metrics....