The goal of this section is to derive the main equation in a Banach space, which plays a crucial role in the proofs of the main results. Our approach is based on the method of spectral mappings (see [14]). Since a part of the proofs repeat the standard technique of [8, Sect. 1.6] and [13], we omit the details and focus on the differences of our methods from the classical ones.
Let us consider two BVPs \(L = L(q(x), h, H)\) and \(\tilde{L} = (\tilde{q}(x), \tilde{h}, \tilde{H})\) with different coefficients. Fix \(N = N(L)\) and the contour \(\gamma _{N}\). Assume that the eigenvalues \(\{ \tilde{\lambda }_{n} \}_{n = 0}^{N}\) lie inside \(\gamma _{N}\) and \(\{ \tilde{\lambda }_{n} \}_{n = N+1}^{\infty }\) lie outside \(\gamma _{N}\).
Define
$$ D(x, \lambda , \xi ) := \frac{\varphi (x, \lambda ) \varphi '(x, \xi ) - \varphi '(x, \lambda ) \varphi (x, \xi )}{\lambda - \xi } = \int _{0}^{x} \varphi (t, \lambda ) \varphi (t, \xi ) \,dt. $$
For \(K \in \mathbb{N}\), consider the region \(\varUpsilon _{K} := \{ \lambda \in \mathbb{C} \colon -p < \operatorname{Re} \lambda < (K + \frac{1}{2})^{2}, |\operatorname{Im} \lambda | < p \}\) and its boundary \(\upsilon _{K} := \partial \varUpsilon _{K}\) with the counter-clockwise circuit. The constant p is chosen so that \(\operatorname{Re} \lambda _{n} > -p\), \(\operatorname{Re} \tilde{\lambda }_{n} > -p\), \(|\operatorname{Im} \lambda _{n}| < p\), \(|\operatorname{Im} \tilde{\lambda }_{n}| < p\) for all \(n \ge 0\). Using the contour integration (see [8, p. 53] for details), we obtain the relation
$$ \varphi (x, \lambda ) = \tilde{\varphi }(x, \lambda ) + \lim _{K \to \infty } \frac{1}{2 \pi i} \oint _{\upsilon _{K}} D(x, \lambda , \xi ) \hat{M}(\xi ) \tilde{\varphi }(x, \xi ) \,d\xi , $$
where \(\hat{M} := \tilde{M} - M\). Applying the residue theorem and observing that the function \((\hat{M}(\lambda ) - \hat{\mathcal{M}}_{N}(\lambda ))\) is analytic inside \(\gamma _{N}\), we obtain the relation
$$\begin{aligned} \varphi (x, \lambda ) & = \tilde{\varphi }(x, \lambda ) + \frac{1}{2 \pi i} \oint _{\gamma _{N}} D(x, \lambda , \xi ) \hat{\mathcal{M}}_{N}( \xi ) \tilde{\varphi }(x, \xi ) \,d\xi \\ &\quad {} + \sum_{n = N + 1}^{\infty } \bigl(\tilde{M}_{n} D(x, \lambda , \tilde{\lambda }_{n}) \tilde{\varphi }(x, \tilde{\lambda }_{n}) - M_{n} D(x, \lambda , \lambda _{n}) \tilde{\varphi }(x, \lambda _{n})\bigr). \end{aligned}$$
(3.1)
We use relation (3.1) for deriving the main equation of Inverse problem 2.1 in a special Banach space. Denote by \(B_{C}\) the Banach space of functions continuous on \(\gamma _{N}\) with the norm
$$ \Vert f_{C} \Vert _{B_{C}} = \max_{\lambda \in \gamma _{N}} \bigl\vert f_{C}( \lambda ) \bigr\vert ,\quad f_{C} \in B_{C}. $$
Denote by \(B_{D}\) the Banach space of bounded infinite sequences \(f_{D} = [ f_{n} ]_{n = 1}^{\infty }\) with the norm
$$ \Vert f_{D} \Vert _{B_{D}} = \sup_{n \ge 1} \vert f_{n} \vert ,\quad f_{D} \in B_{D}. $$
Define the Banach space
$$ B := \bigl\{ f = (f_{C}, f_{D}) \colon f_{C} \in B_{C}, f_{D} \in B_{D} \bigr\} ,\qquad \Vert f \Vert _{B} := \Vert f_{C} \Vert _{C} + \Vert f_{D} \Vert _{D}. $$
Here and below the lower indices C and D mean a “continuous” and a “discrete” part, respectively.
For every \(x \in [0, \pi ]\), define the element \(\psi (x) = (\psi _{C}(x), \psi _{D}(x))\), where
$$\begin{aligned}& \psi _{C}(x, \lambda ) = \varphi (x, \lambda ), \quad \lambda \in \gamma _{N},\qquad \psi _{D}(x) = \bigl[\psi _{n}(x) \bigr]_{n = 1}^{\infty }, \\& \psi _{2j-1}(x) = \varphi (x, \tilde{\lambda }_{N + j}), \qquad \psi _{2j}(x) = \chi _{N + j} \bigl(\varphi (x, \lambda _{N + j}) - \varphi (x, \tilde{\lambda }_{N + j})\bigr),\quad j \ge 1, \\& \chi _{n} := \textstyle\begin{cases} \xi _{n}^{-1}, & \text{if } \xi _{n} \ne 0, \\ 0, & \text{if } \xi _{n} = 0. \end{cases}\displaystyle \end{aligned}$$
The element \(\tilde{\psi }(x)\) is defined analogously by using φ̃ instead of φ.
For the solution \(\varphi (x, \lambda )\), the following standard asymptotics is valid:
$$ \varphi (x, \lambda ) = \cos \rho x + O \bigl( \rho ^{-1} \exp \bigl( \vert \operatorname{Im} \rho \vert x\bigr) \bigr), \quad \vert \rho \vert \to \infty , $$
(3.2)
where \(\rho = \sqrt{\lambda }\), \(\operatorname{Re} \rho \ge 0\). Using (1.4) and (3.2), we obtain the estimates
$$ \bigl\vert \varphi (x, \lambda _{n}) \bigr\vert \le C, \bigl\vert \varphi (x, \lambda _{n}) - \varphi (x, \tilde{\lambda }_{n}) \bigr\vert \le C \xi _{n},\quad x \in [0, \pi ], n \ge 0, $$
where the constant C does not depend on x and n. Analogous relations are valid for \(\tilde{\varphi }(x, \lambda )\). Consequently, for each fixed \(x \in [0, \pi ]\), we have \(\psi (x) \in B\) and \(\tilde{\psi }(x) \in B\).
For each fixed \(x \in [0, \pi ]\), we define the linear bounded operator \(R(x) \colon B \to B\) as follows:
$$\begin{aligned}& R(x) = \begin{pmatrix} R_{{CC}}(x) & R_{{CD}}(x) \\ R_{{DC}}(x) & R_{{DD}}(x) \end{pmatrix}, \\& R_{{CC}}(x) \colon B_{C} \to B_{C},\qquad R_{{CD}}(x) \colon B_{D} \to B_{C}, \\& R_{{DC}}(x) \colon B_{C} \to B_{D},\qquad R_{{DD}}(x) \colon B_{D} \to B_{D}, \\& R(x) f = \bigl(R_{{CC}}(x) f_{C} + R_{{CD}}(x) f_{D}, R_{{DC}}(x) f_{C} + R_{{DD}}(x) f_{D}\bigr),\quad f = (f_{C}, f_{D}) \in B, \\& \bigl(R_{{CC}}(x) f_{C}\bigr) (\lambda ) = \frac{1}{2 \pi i} \oint _{\gamma _{N}} D(x, \lambda , \xi ) \hat{\mathcal{M}}_{N}( \xi ) f_{C}(\xi ) \,d\xi , \end{aligned}$$
(3.3)
$$\begin{aligned}& \begin{aligned}[b] \bigl(R_{{CD}}(x) f_{D}\bigr) (\lambda ) = {}&\sum_{k = 1}^{\infty } \bigl(\bigl( \tilde{M}_{N + k} D(x, \lambda , \tilde{\lambda }_{N + k}) - M_{N + k} D(x, \lambda , \lambda _{N + k})\bigr) f_{2k-1} \\ &{}- \xi _{N + k} M_{N + k}D(x, \lambda , \lambda _{N + k}) f_{2k}\bigr), \end{aligned} \\& \bigl(R_{{DC}}(x) f_{C}\bigr)_{2j-1} = \frac{1}{2 \pi i} \oint _{\gamma _{N}} D(x, \tilde{\lambda }_{N + j}, \xi ) \hat{\mathcal{M}}_{N}(\xi ) f_{C}( \xi ) \,d\xi , \\& \bigl(R_{{DC}}(x) f_{C}\bigr)_{2j} = \frac{1}{2\pi i} \oint _{\gamma _{N}} \bigl(D(x, \lambda _{N + j}, \xi ) - D(x, \tilde{\lambda }_{N + j}, \xi )\bigr) \chi _{N + j} \hat{ \mathcal{M}}_{N}(\xi ) f_{C}(\xi ) \,d\xi , \\& \begin{aligned} \bigl(R_{{DD}}(x) f_{D}\bigr)_{2j-1} ={}& \sum _{k = 1}^{\infty } \bigl(\bigl( \tilde{M}_{N + k}D(x, \tilde{\lambda }_{N + j}, \tilde{\lambda }_{N + k}) - M_{N + k} D(x, \tilde{\lambda }_{N + j}, \lambda _{N + k})\bigr) f_{2k-1} \\ &{}- \xi _{N + k} M_{N + k} D(x, \tilde{\lambda }_{N + j}, \lambda _{N + k}) f_{2k}\bigr), \end{aligned} \\& \begin{aligned} \bigl(R_{{DD}}(x) f_{D}\bigr)_{2j} ={}& \chi _{N + j} \sum_{k = 1}^{\infty } \bigl( \bigl( \tilde{M}_{N + k} \bigl(D(x, \lambda _{N + j}, \tilde{ \lambda }_{N + k}) - D(x, \tilde{\lambda }_{N + j}, \tilde{ \lambda }_{N + k})\bigr) \\ &{}- M_{N + k} \bigl(D(x, \lambda _{N + j}, \lambda _{N + k}) - D(x, \tilde{\lambda }_{N + j}, \lambda _{N + k})\bigr)\bigr) f_{2k-1} \\ &{}- \xi _{N + k} M_{N + k} \bigl(D(x, \lambda _{N + j}, \lambda _{N + k}) - D(x, \tilde{\lambda }_{N + j}, \lambda _{N + k})\bigr) f_{2k}\bigr), \end{aligned} \end{aligned}$$
(3.4)
where \(\lambda \in \gamma _{N}\), \(j \ge 1\), \(f_{D} = [ f_{k} ]_{k = 1}^{\infty }\).
Taking \(\lambda \in \gamma _{N}\), \(\lambda = \tilde{\lambda }_{n}\) and \(\lambda = \lambda _{n}\), \(n > N\), in (3.1), we obtain the so-called main equation in the Banach space B:
$$ \psi (x) = \bigl(I + R(x)\bigr) \tilde{\psi }(x),\quad x \in [0, \pi ]. $$
(3.5)
Here, I is the identity operator in B.
Now suppose that the problem L and the data \(\tilde{G} = \{ \tilde{\lambda }_{n}, \tilde{M}_{n} \}_{n = 0}^{\infty }\) satisfy the conditions of Theorem 2.2. We choose \(\delta _{0}\) to be so small that the values \(\{ \tilde{\lambda }_{n} \}_{n = 0}^{N}\) definitely lie inside \(\gamma _{N}\) and the values \(\{ \tilde{\lambda }_{n} \}_{n > N}\) definitely lie outside \(\gamma _{N}\). It is not known whether the data G̃ correspond to any problem L̃ or not. Let \(\psi (x)\) and \(R(x)\) be constructed by L and G̃ via the formulas above. Then the following assertion holds.
Lemma 3.1
For each fixed\(x \in [0, \pi ]\), the following estimate is valid:
$$ \bigl\Vert R(x) \bigr\Vert _{B \to B} \le C \delta , $$
(3.6)
where the constantCdoes not depend onx, δand on the choice ofG̃satisfying the conditions of Theorem 2.2.
Proof
In order to prove (3.6), it is sufficient to obtain similar estimates for \(\| R_{{CC}}(x) \|_{B_{C} \to B_{C}}\), \(\| R_{{CD}}(x) \|_{B_{D} \to B_{C}}\), \(\| R_{{DC}}(x) \|_{B_{C} \to B_{D}}\), and \(\| R_{{DD}}(x) \|_{B_{D} \to B_{D}}\). Using (2.2) and (3.3), we get
$$ \bigl\Vert R_{{CC}}(x) \bigr\Vert _{B_{C} \to B_{C}} \le \frac{1}{2\pi }\mbox{length}( \gamma _{N}) \cdot \max _{\lambda , \xi \in \gamma _{N}} \bigl\vert D(x, \lambda , \xi ) \bigr\vert \cdot \max_{\xi \in \gamma _{N}} \bigl\vert \hat{\mathcal{M}}_{N}(\xi ) \bigr\vert \le C \delta . $$
The standard estimates (see [8, Lemma 1.6.2]) imply
$$ \bigl\vert D(x, \lambda , \tilde{\lambda }_{n}) - D(x, \lambda , \lambda _{n}) \bigr\vert \le \frac{C \exp ( \vert \operatorname{Im} \rho \vert x) \xi _{n}}{ \vert \rho - n \vert + 1},\quad n \ge 0. $$
Combining the latter relation with (3.4), (2.3) and the obvious estimates
$$ \vert M_{n} \vert \le C, \qquad \vert \tilde{M}_{n} - M_{n} \vert \le \xi _{n},\quad n \ge 0, $$
(3.7)
we get
$$\begin{aligned} \bigl\Vert R_{{CD}}(x) \bigr\Vert _{B_{D} \to B_{C}} & \le \max _{\lambda \in \gamma _{N}} \sum_{k = 1}^{\infty } \bigl( \vert \tilde{M}_{N + k}- M_{N + k} \vert \bigl\vert D(x, \lambda , \tilde{\lambda }_{N + k}) \bigr\vert \\ &\quad {} + \vert M_{N + k} \vert \bigl\vert D(x, \lambda , \tilde{ \lambda }_{N + k}) - D(x, \lambda , \lambda _{N + k}) \bigr\vert \\ &\quad {}+ \xi _{N + k} \vert M_{N + k} \vert \bigl|D(x, \lambda , \lambda _{N + k})\bigr|\bigr) \\ & \le C \sum_{n = N + 1}^{\infty } \frac{\xi _{n}}{ \vert \rho - n \vert + 1} \le C \Biggl( \sum_{n = N + 1}^{\infty } (n \xi _{n})^{2} \Biggr)^{1/2} \le C \delta . \end{aligned}$$
One can similarly study the components \(R_{{DC}}(x)\) and \(R_{{DD}}(x)\) and finally arrive at the assertion of the lemma. □
Corollary 3.2
There exists\(\delta _{0} > 0\)such that, for every\(\delta \le \delta _{0}\)and\(x \in [0, \pi ]\), the estimate\(\| R(x) \|_{B \to B} \le \frac{1}{2}\)holds. In this case, for each fixed\(x \in [0, \pi ]\), the operator\((I + R(x))\)has a bounded inverse, and the main equation (3.5) has a unique solution.