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We consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral...

We consider the singular boundary value problem $${\left(t^n{u}^{\text{'}}\left(t\right)\right)}^{\text{'}}+t^{n}f(t,u\left(t\right))=0,\underset{t\to 0+}{lim}{t}^n{u}^{\text{'}}\left(t\right)=0,a_{0}u\left(1\right)+a_1{u}^{\text{'}}(1-)=A,$$ where $f(t,x)$ is a given continuous function defined on the set $(0,1]\times (0,\infty )$ which can have a time singularity at $t=0$ and a space singularity at $x=0$. Moreover, $n\in \mathbb{N}$, $n\ge 2$, and $a_0$, $a_1$, $A$ are real constants such that $a_0\in (0,\infty )$, whereas $a_1,A\in [0,\infty )$. The main aim of this paper is to discuss the existence of solutions to the above problem and apply the general results to cover certain classes of singular problems arising in the theory of shallow membrane caps, where we are especially interested in...