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 constant**C**does not depend on**x*, *δ**and on the choice of**G̃**satisfying 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*.