It is well-known that the pioneer of mixed type boundary value problems is F. G. Tricomi (1923) with his Tricomi equation: $yu_{xx}+u_{yy}=0$. In this paper we consider the more general case of above equation so that \[ Lu\equiv K_{1}(y)u_{xx}+(K_{2}(y)u_{y})^{\prime}+ru=f \] is hyperbolic-elliptic and parabolic, and then prove the existence of weak solutions for the corresponding Tricomi problem by employing the well-known a-b-c energy integral method to establish an a-priori estimate. This result is interesting in fluid mechanics.

In this paper by using Ros’s method we prove the following result which has been obtained by Tani M. [1]: An $n$-dimensional $(n\geq 3)$ compact conformally flat manifold with positive Ricci curvature and constant scalar curvature must be of constant sectional curvature.

We investigate the solvability of the extension problem for complete preimages from the given class $\mathscr{F}$ of surjective, perfect mappings of metric spaces, which consists of representing an arbitrary mapping $f_{0}$: $X_{0}\rightarrow Y_{0}\in \mathscr{F}$ as the restriction of another mapping $f$ : $X\rightarrow Y\in \mathscr{F}$, onto the complete preimage $f_{0}^{-1}(Y_{0})=X_{0}$, where $Y$ is an arbitrary metric space, containing $Y_{0}$ as a closed subset. We prove that this problem can be solved for the class of open $UV^{n}$-mappings. Along the way, we also establish that the subset $\exp_{UV^{n}}(\ell_{2}(\tau))$ of the exponent $exp(\ell_{2}(\tau))$ of the Hilbert space $\ell_{2}(\tau)$ of density $\tau$, consisting of $UV^{n}$-compacta, belongs to the class of absolute retracts.

The formulas of the Bessel functions are applied to obtain the estimates of the limiting absorption principle. As an application we prove a result on smoothing effect for the Schrödinger equation.

We treat partially conformal geodesic transformations with respect to submanifolds in almost Hermitian manifolds. Non-isometric ones only exist when the submanifold is a real hypersurface or reduces to a point. In these two cases, we derive necessary and sufficient conditions for the existence in terms of the Jacobi operator and show how this existence influences the geometry of the hypersurface and that of the ambient space. As an application, we use these transformations to obtain a new characterization of complex space forms.