A condition for solvability of an integral equation which is connected with the first boundary value problem for the heat equation is investigated. It is shown that if this condition is fulfilled then the boundary considered is 1⁄2-Hölder. Further, some simple concrete examples are examined.
We consider an elliptic pseudodifferential equation in a multi-dimensional cone, and using the wave factorization concept for an elliptic symbol we describe a general solution of such equation in Sobolev-Slobodetskii spaces. This general solution depends on some arbitrary functions, their quantity being determined by an index of the wave factorization. For identifying these arbitrary functions one needs some additional conditions, for example, boundary conditions. Simple boundary value problems, related to Dirichlet and Neumann boundary conditions, are considered. A certain integral representation for this case is given.
We give sufficient conditions for the existence of at least one integrable solution of equation x(t) = f(t) + ∫ t 0 K(t, s)g(s, x(s)) ds. Our assumptions and proofs are expressed in terms of measures of noncompactness.
In this paper we study a linear integral equation x(t) = a(t)− ∫ t 0 C(t, s)x(s) ds, its resolvent equation R(t, s) = C(t, s) − ∫ t s C(t, u)R(u, s) du, the variation of parameters formula x(t) = a(t) − ∫ t 0 R(t, s)a(s) ds, and a perturbed equation. The kernel, C(t, s), satisfies classical smoothness and sign conditions assumed in many real-world problems. We study the effects of perturbations of C and also the limit sets of the resolvent. These results lead us to the study of nonlinear perturbations.