We provide sufficient conditions for solvability of a singular Dirichlet boundary value problem with $$-Laplacian$$ . ((u)) = f(t, u, u), u(0) = A, u(T) = B, . $$where$$ is an increasing homeomorphism, $\left(ℝ\right)=ℝ$, $\left(0\right)=0$, $f$ satisfies the Carathéodory conditions on each set $[a,b]\times {ℝ}^2$ with $[a,b]\subset (0,T)$ and $f$ is not integrable on $[0,T]$ for some fixed values of its phase variables. We prove the existence of a solution which has continuous first derivative on $[0,T]$.

The paper deals with the singular nonlinear problem $$u^{\text{'}\text{'}}\left(t\right)+f(t,u\left(t\right),u^{\text{'}}\left(t\right))=0,u\left(0\right)=0,u^{\text{'}}\left(T\right)=\psi \left(u\left(T\right)\right),$$ where $f\in \mathrm{𝐶𝑎𝑟}\left(\right(0,T)\times D)$, $D=(0,\infty )\times$. We prove the existence of a solution to this problem which is positive on $(0,T]$ under the assumption that the function $f(t,x,y)$ is nonnegative and can have time singularities at $t=0$, $t=T$ and space singularity at $x=0$. The proof is based on the Schauder fixed point theorem and on the method of a priori estimates.

The paper investigates singular nonlinear problems arising in hydrodynamics. In particular, it deals with the problem on the half-line of the form $${\left(p\left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}=p\left(t\right)f\left(u\left(t\right)\right),$$$$u^{\text{'}}(...$$

We study the existence of a solution to the nonlinear fourth-order elastic beam equation with nonhomogeneous boundary conditions $$\left\{\begin{array}{c}{u}^{\left(4\right)}\left(t\right)=f\left(t,u\left(t\right),u^{\text{'}}\left(t\right),u^{\text{'}\text{'}}\left(t\right),u^{\text{'}\text{'}\text{'}}\left(t\right)\right),\text{a.e.}t\in [0,1],\hfill \\ u\left(0\right)=a,u^{\text{'}}\left(0\right)=b,u\left(1\right)=c,u^{\text{'}\text{'}}\left(1\right)=d,\hfill \end{array}\right.$$ where the nonlinear term $f(t,u_0,u_1,u_2,u_3)$ is a strong Carathéodory function. By constructing suitable height functions of the nonlinear term $f(t,u_0,u_1,u_2,u_3)$ on bounded sets and applying the Leray-Schauder fixed point theorem, we prove that the equation has a solution provided that the integration of some height function has an appropriate value.

We study singular boundary value problems with mixed boundary conditions of the form $${\left(p\left(t\right)u^{\text{'}}\right)}^{\text{'}}+p\left(t\right)f(t,u,p\left(t\right)u^{\text{'}})=0,\underset{t\to 0+}{lim}p\left(t\right)u^{\text{'}}\left(t\right)=0,u\left(T\right)=0,$$ where $[0,T]\subset ℝ.$ We assume that ${ℝ}^2,$$f$ satisfies the Carathéodory conditions on $(0,T)\times$$p\in C...$

In this article, we present a new method for converting the boundary value problems for impulsive fractional differential systems involved with the Riemann-Liouville type derivatives to integral systems, some existence results for solutions of a class of boundary value problems for nonlinear impulsive fractional differential systems at resonance case and non-resonance case are established respectively. Our analysis relies on the well known Schauder’s fixed point theorem and coincidence degree theory....

This paper gives necessary and sufficient conditions for subgroups with trivial core to be of odd depth. We show that a subgroup with trivial core is an odd depth subgroup if and only if certain induced modules from it are faithful. Algebraically this gives a combinatorial condition that has to be satisfied by the subgroups with trivial core in order to be subgroups of a given odd depth. The condition can be expressed as a certain matrix with $\{0,1\}$-entries to have maximal rank. The entries of the matrix...

We discuss the existence and properties of solutions for systems of singular second-order ODEs in both sublinear and superlinear cases. Our approach is based on the variational method enriched by some topological ideas. We also investigate the continuous dependence of solutions on functional parameters.