Sturm-Liouville oscillation theory for impulsive problems

Dissipative extensions of a symmetric relation generated by a system of integral equations with operator measures

“…Here for constructing boundary values we adhere to the scheme described in [15], because the boundary values obtained in papers cited above cannot be used in the case under consideration. Note that the class of integral equations with operator measures includes equations with the Stieltjes integrals [16] and differential equations whose coefficients are generalized functions. The intensive study of such differential equations was started in [17].…”

Invertibility of linear relations generated by integral equation with operator measures

“…Integral equations with operator measures are rather general. For instance, they cover integral-differential equations with Stiltjes integral [1], differential equations with the coefficients being generalized functions [2] (the way for reducing an integral equation to the equation in [2] was provided in [3]).…”

Stieltjes differential in nonlinear momentum problems

“…(2) that, at any discontinuity point ξ of the function σ(x), we have On the coefficients p(x) and Q(x) and the functions f(x, u) we impose the following conditions: (i) p(x) and Q(x) are σ absolutely continuous on ; (ii) the function p(x) is positive and bounded away from zero; (iii) the function Q(x) does not decrease on ; (iv) f(x, u) satisfies the Carathéodory condition, i.e., (a) for each fixed u, the function f(x, u) is σ mea surable; (b) f(x, u) is continuous in u for all x ∈ [0; ᐉ]; (c) there exists a function m(x) such that some power p ∈ [1, ∞) of this function is σ integrable and |f(x, u)| ≤ m(x) for almost all x (in the sense of the σ measure) and u. The latter requirement ensures that the superposition operator [Fu](x) = f(x, u(x)) is a con tinuous operator from the space C[0; ᐉ] of continuous functions in [0; ᐉ] to the space L p, σ ([0; ᐉ]) of functions σ integrable to some power p in [0; ᐉ].In[9,10], it was proved that, under conditions (i)-(iii), the linear problem (with f(x, u) = f(x), where f(x) is a σ integrable function) is solvable in the class E and the linear boundary value problem Lu = f, u(0) = u(ᐉ) = 0 has a unique influence function…”

“…Krein (see comments in [8]), he replaced the equation with generalized coefficients and right hand side by the pointwise equation In [9,10], Eq. (1), which is of pointwise character similar to that of ordinary differential equations, was used to develop qualitative methods for analyzing clas sical oscillatory properties for the Sturm-Liouville problem…”