Let h ∈ L¹[0,1] ∩ C(0,1) be nonnegative and f(t,u,v) + h(t) ≥ 0. We study the existence and multiplicity of positive solutions for the nonlinear fourth-order two-point boundary value problem $u^{\left(4\right)}\left(t\right)=f(t,u\left(t\right),u^{\text{'}}\left(t\right))$, 0 < t < 1, u(0) = u’(0) = u’(1) =u”’(1) =0, where the nonlinear term f(t,u,v) may be singular at t=0 and t=1. By constructing a suitable cone and integrating certain height functions of f(t,u,v) on some bounded sets, several new results are obtained. In mechanics, the problem models the deflection of...

This paper studies the existence of multiple positive solutions to a nonlinear fourth-order two-point boundary value problem, where the nonlinear term may be singular with respect to both time and space variables. In order to estimate the growth of the nonlinear term, we introduce new control functions. By applying the Hammerstein integral equation and the Guo-Krasnosel'skii fixed point theorem of cone expansion-compression type, several local existence theorems are proved.

The paper investigates the structure and properties of the set S of all positive solutions to the singular Dirichlet boundary value problem u″(t) + au′(t)/t − au(t)/t 2 = f(t, u(t),u′(t)), u(0) = 0, u(T) = 0. Here a ∈ (−∞,−1) and f satisfies the local Carathéodory conditions on [0,T]×D, where D = [0,∞)×ℝ. It is shown that S c = {u ∈ S: u′(T) = −c} is nonempty and compact for each c ≥ 0 and S = ∪c≥0 S c. The uniqueness of the problem is discussed. Having a special case of the problem, we introduce...

Realization theory for linear input-output operators and frequency-domain methods for the solvability of Riccati operator equations are used for the stability and instability investigation of a class of nonlinear Volterra integral equations in a Hilbert space. The key idea is to consider, similar to the Volterra equation, a time-invariant control system generated by an abstract ODE in a weighted Sobolev space, which has the same stability properties as the Volterra equation.

This paper focuses on the automatic recognition of map projection, its inverse and re-projection. Our analysis leads to the unconstrained optimization solved by the hybrid BFGS nonlinear least squares technique. The objective function is represented by the squared sum of the residuals. For the map re-projection the partial differential equations of the inverse transformation are derived. They can be applied to any map projection. Illustrative examples of the stereographic and globular Nicolosi projections...

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.