On two boundary value problems for mixed type equations with perpendicular lines of type change.
We construct fundamental solutions of some partial differential equations of order higher than two and examine properties of these solutions and of some related integrals. The results will be used in our next paper concerning boundary-value problems for these equations.
Realization theory for linear input-output operators and frequency-domain methods for the solvability of Riccati operator equations are used for the stability and instability investigation of a class of nonlinear Volterra integral equations in a Hilbert space. The key idea is to consider, similar to the Volterra equation, a time-invariant control system generated by an abstract ODE in a weighted Sobolev space, which has the same stability properties as the Volterra equation.
In this paper we study a linear integral equation , its resolvent equation , the variation of parameters formula , and a perturbed equation. The kernel, , satisfies classical smoothness and sign conditions assumed in many real-world problems. We study the effects of perturbations of and also the limit sets of the resolvent. These results lead us to the study of nonlinear perturbations.
Mathematics Subject Classification: 26A33, 34A25, 45D05, 45E10We consider ordinary fractional differential equations with Caputo-type differential operators with smooth right-hand sides. In various places in the literature one can find the statement that such equations cannot have smooth solutions. We prove that this is wrong, and we give a full characterization of the situations where smooth solutions exist. The results can be extended to a class of weakly singular Volterra integral equations.