Resolvent Operators for Some Classes of Integro-Differential Equations

Abstract

Explicit representations are constructed for the resolvents of the operators of the form $B =\widehat{ A} + Q_{1}$ and $\mathbf{B} =\widehat{ A}^{2} + Q_{2}$, where $\widehat{A}$ and $\widehat{A}^{2}$ are linear closed operators with known resolvents and Q ₁ and Q ₂ are perturbation operators embedding inner products of $\widehat{A}$ and $\widehat{A}^{2}$ as they appear in integro-differential equations and other applications.

Appendix

Problem 1.

Let the operator $\widehat{A}: L_{2}(0,1) \rightarrow L_{2}(0,1)$ be defined by

$$\displaystyle{ \widehat{A}u = iu' = f,\quad D(\widehat{A}) =\{ u(t) \in H^{1}(0,1): u(0) + u(1) = 0\} }$$

(96)

Then $\hat{A}$ is closed and:

• (i) $\lambda \in \rho (\widehat{A})$ if and only if $\lambda \neq (2k + 1)\pi,\quad k \in \mathbf{Z}$, i.e.

  $$\displaystyle{ \rho (\widehat{A}) =\{\lambda \in \mathbf{C}:\lambda \neq (2k + 1)\pi,\quad k \in \mathbf{Z}\}. }$$

  (97)

• (ii) For $\lambda \in \rho (\widehat{A})$ the resolvent operator $R_{\lambda }(\widehat{A})$ is bounded and defined on the whole space L ₂(0, 1) by the formula

  $$\displaystyle{ R_{\lambda }(\widehat{A})f(t) = i\int _{0}^{1}e^{i\lambda (x-t)}\left [(e^{i\lambda } + 1)^{-1} -\eta (t - x)\right ]f(x)dx, }$$

  (98)

  where

  $$\displaystyle{\eta (t-x) = \left \{\begin{array}{lll} 1,&x \leq t& \\ & &\mbox{ is the Heaviside's function}.\\ 0, &x> t & \end{array} \right.}$$

Problem 2.

Let the operator $\widehat{A}: L_{2}(0,1) \rightarrow L_{2}(0,1)$ be defined by

$$\displaystyle{ \widehat{A}u = u' = f,\quad D(\widehat{A}) =\{ u(t) \in H^{1}(0,1):\, u(0) = u(1)\}. }$$

(99)

Then the quadratic operator $\widehat{A}^{2}: L_{2}(0,1) \rightarrow L_{2}(0,1)$ is closed and defined by

$$\displaystyle\begin{array}{rcl} \widehat{A}^{2}u = u'' = f,\quad D(\widehat{A}^{2}) =\{ u(t) \in H^{2}(0,1): u(0) = u(1),\,u'(0) = u'(1)\},& &{}\end{array}$$

(100)

the resolvent sets of $\widehat{A}$ and $\widehat{A}^{2}$ are

$$\displaystyle\begin{array}{rcl} \rho (\widehat{A}) =\{\lambda \in \mathbf{C}:\lambda \neq 2k\pi i,\,\,k \in \mathbf{Z}\},& &{}\end{array}$$

(101)

$$\displaystyle\begin{array}{rcl} \rho (\widehat{A}^{2}) =\{\lambda \in \mathbf{C}:\lambda \neq - 4k^{2}\pi ^{2},\,k \in \mathbf{Z}\}& &{}\end{array}$$

(102)

and the resolvent operators $R_{\pm \lambda }(\widehat{A}),\,R_{\lambda }(\widehat{A}^{2})$ are bounded and defined on the whole space L ₂(0, 1) by

$$\displaystyle\begin{array}{rcl} & & R_{\lambda }(\widehat{A})\,f(t)\, = \frac{1} {1 - e^{\lambda }}\int _{0}^{1}e^{\lambda (t-x+1)}f(x)dx +\int _{ 0}^{t}e^{\lambda (t-x)}f(x)dx{}\end{array}$$

(103)

$$\displaystyle\begin{array}{rcl} & & R_{-\lambda }(\widehat{A})f(t)\, = \frac{1} {1 - e^{-\lambda }}\int _{0}^{1}e^{-\lambda (t-x+1)}f(x)dx +\int _{ 0}^{t}e^{-\lambda (t-x)}f(x)dx{}\end{array}$$

(104)

$$\displaystyle\begin{array}{rcl} & & R_{\lambda }(\widehat{A}^{2})f(t) = \frac{1} {2\sqrt{\lambda }(1 - e^{\sqrt{\lambda }})}\int _{0}^{1}\left [e^{\sqrt{\lambda }(t-x+1)} + e^{-\sqrt{\lambda }(t-x)}\right ]f(x)dx \\ & & \phantom{mlmlmlmlmlm} + \frac{1} {2\sqrt{\lambda }}\int _{0}^{t}\left [e^{\sqrt{\lambda }(t-x)} - e^{-\sqrt{\lambda }(t-x)}\right ]f(x)dx. {}\end{array}$$

(105)