The general ordinary quasi-differential expression M of n-th order with complex coefficients and its formal adjoint M + are considered over a regoin (a, b) on the real line, −∞ ≤ a < b ≤ ∞, on which the operator may have a finite number of singular points. By considering M over various subintervals on which singularities occur only at the ends, restrictions of the maximal operator generated by M in L2|w (a, b) which are regularly solvable with respect to the minimal operators T0 (M ) and T0...

In this paper, we study a kind of fractional differential system with impulsive effect and find the formula of general solution for the impulsive fractional-order system by analysis of the limit case (as impulse tends to zero). The obtained result shows that the deviation caused by impulses for fractional-order system is undetermined. An example is also provided to illustrate the result.

We consider the initial-boundary-value problem for the one-dimensional fast diffusion equation $u_t=[sign\left(u_x\right)log|u_x{\left|\right]}_x$ on $Q_T=[0,T]\times [0,l]$. For monotone initial data the existence of classical solutions is known. The case of non-monotone initial data is delicate since the equation is singular at $u_x=0$. We ‘explicitly’ construct infinitely many weak Lipschitz solutions to non-monotone initial data following an approach to the Perona-Malik equation. For this construction we rephrase the problem as a differential inclusion which enables us...

In this paper, the general ordinary quasi-differential expression $M_p$ of $n$-th order with complex coefficients and its formal adjoint $M_p^{+}$ on any finite number of intervals $I_p=(a_p,b_p)$, $p=1,\cdots ,N$, are considered in the setting of the direct sums of $L_{w_p}^2(a_p,b_p)$-spaces of functions defined on each of the separate intervals, and a number of results concerning the location of the point spectra and the regularity fields of general differential operators generated by such expressions are obtained. Some of these are extensions or generalizations...