Problem with Integral Conditions for Differential-Operator Equation

Abstract

We propose a method for the solution of the problem with inhomogeneous integral conditions for homogeneous differential-operator equations with abstract operator in a linear space H . For the righthand sides of the integral conditions from a special subspace L ⊆ H in which the vectors are represented in the form of Stieltjes integrals with respect to certain measures, the solution of the problem is represented in the form of Stieltjes integrals with respect to the same measures. We give an example of application of the method to the solution of the ill-posed problem for the second-order partial differential equation in the time variable (in which the integral conditions are given) and, in general, an infiniteorder partial differential equation in the space variable.

Closed-Form Solutions of Linear Ordinary Differential Equations with General Boundary Conditions

“…An overview of non-local boundary value problems and their historical evolution can also be found in the survey papers [13][14][15][16]. Boundary value problems with integral constraints have been considered in [17][18][19][20][21][22][23][24][25], to mention but a few. Boundary value problems with multipoint and integral conditions have been studied in [26][27][28][29][30][31][32], and others.…”

Regularity of complexified hyperbolic equations with integral conditions

“…For hyperbolic wave equations, related results and the references can be found, e.g., in [5,4,7,8,12,13,15,17].…”

“…In [12,13,17], hyperbolic wave equations under integral conditions with respect to time have been considered. In [12,13], the eigenfunction expansion method have been used, and the regularity result have been affected by the so-called "small denominators" ("small divisors") problem) that often causes instability of solutions for hyperbolic wave equations with non-local with respect to time conditions. The solvability was obtained in [12,13] was obtained for the case where the spectrum for the inputs and solutions does not contain resonance points.…”

“…In [12,13], the eigenfunction expansion method have been used, and the regularity result have been affected by the so-called "small denominators" ("small divisors") problem) that often causes instability of solutions for hyperbolic wave equations with non-local with respect to time conditions. The solvability was obtained in [12,13] was obtained for the case where the spectrum for the inputs and solutions does not contain resonance points. In [17], a regularity condition without these restrictions on the spectrum have been obtained for the hyperbolic wave equation with a Laplacian.…”