Throughout the paper, we set E a Banach space and \mathrm{\Omega}\subset {R}^{n}. {L}^{p}(\mathrm{\Omega};E) denotes the space of all strongly measurable Evalued functions that are defined on Ω with the norm
{\parallel f\parallel}_{p}={\parallel f\parallel}_{{L}^{p}(\mathrm{\Omega};E)}={\left({\int}_{\mathrm{\Omega}}{\parallel f(x)\parallel}_{E}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{p}},\phantom{\rule{1em}{0ex}}1\le p<\mathrm{\infty}.
\mathit{BMO}(E) (bounded mean oscillation, see [25, 26]) is the space of all Evalued local integrable functions with the norm
\underset{B}{sup}{\int}_{B}{\parallel f(x){f}_{B}\parallel}_{E}\phantom{\rule{0.2em}{0ex}}dx={\parallel f\parallel}_{\ast ,E}<\mathrm{\infty},
where B ranges in the class of the balls in {R}^{n}, B is the Lebesgue measure of B and {f}_{B} is the average \frac{1}{B}{\int}_{B}f(x)\phantom{\rule{0.2em}{0ex}}dx.
For f\in \mathit{BMO}(E) and r>0, we set
\underset{\rho \le r}{sup}{\int}_{B}{\parallel f(x){f}_{B}\parallel}_{E}\phantom{\rule{0.2em}{0ex}}dx=\eta (r),
where B ranges in the class of balls with radius ρ.
We will say that a function f\in \mathit{BMO}(E) is in \mathit{VMO}(E) if {lim}_{r\to +0}\eta (r)=0. We will call \eta (r) the VMO modulus of f.
Note that if E=C, where C is the set of complex numbers, then \mathit{BMO}(E) and \mathit{VMO}(E) coincide with JohnNirenberg class BMO and Sarason class VMO, respectively.
The Banach space E is called a UMDspace if the Hilbert operator
(Hf)(x)=\underset{\epsilon \to 0}{lim}{\int}_{xy>\epsilon}\frac{f(y)}{xy}\phantom{\rule{0.2em}{0ex}}dy
is bounded in {L}_{p}(R,E), p\in (1,\mathrm{\infty}) (see, e.g., [27]). UMD spaces include, e.g., {L}_{p}, {l}_{p} spaces and Lorentz spaces {L}_{pq}, p,q\in (1,\mathrm{\infty}).
Let
{S}_{\phi}=\{\lambda \in C,arg\lambda \le \phi \}\cup \{0\},\phantom{\rule{1em}{0ex}}0\le \phi <\pi .
A linear operator A is said to be φpositive (or positive) in a Banach space E with bound M>0 if D(A) is dense on E and
{\parallel {(A+\lambda I)}^{1}\parallel}_{L(E)}\le M{(1+\lambda )}^{1}
for \lambda \in {S}_{\phi}, \phi \in (0,\pi ], I is an identity operator in E and L(E) is the space of bounded linear operators in E. Sometimes A+\lambda I will be written as A+\lambda and denoted by {A}_{\lambda}. It is known [[28], §1.15.1] that there exist fractional powers {A}^{\theta} of the positive operator A. Let E({A}^{\theta}) denote the space D({A}^{\theta}) with the graphical norm
{\parallel u\parallel}_{E({A}^{\theta})}={({\parallel u\parallel}^{p}+{\parallel {A}^{\theta}u\parallel}^{p})}^{\frac{1}{p}},\phantom{\rule{1em}{0ex}}1\le p<\mathrm{\infty},\mathrm{\infty}<\theta <\mathrm{\infty}.
Let {E}_{1} and {E}_{2} be two Banach spaces. A set W\subset L({E}_{1},{E}_{2}) is called Rbounded (see [14, 23]) if there is a positive constant C such that for all {T}_{1},{T}_{2},\dots ,{T}_{m}\in W and {u}_{1,}{u}_{2},\dots ,{u}_{m}\in {E}_{1}, m\in N,
{\int}_{0}^{1}{\parallel \sum _{j=1}^{m}{r}_{j}(y){T}_{j}{u}_{j}\parallel}_{{E}_{2}}\phantom{\rule{0.2em}{0ex}}dy\le C{\int}_{0}^{1}{\parallel \sum _{j=1}^{m}{r}_{j}(y){u}_{j}\parallel}_{{E}_{1}}\phantom{\rule{0.2em}{0ex}}dy,
where \{{r}_{j}\} is a sequence of independent symmetric \{1,1\}valued random variables on [0,1].
Let S({R}^{n};E) denote the Schwartz class, i.e., the space of all Evalued rapidly decreasing smooth functions on {R}^{n}. Let F denote the Fourier transformation. A function \mathrm{\Psi}\in {L}^{\mathrm{\infty}}({R}^{n};B({E}_{1},{E}_{2})) is called a Fourier multiplier from {L}_{p}({R}^{n};{E}_{1}) to {L}_{p}({R}^{n};{E}_{2}) if the map u\to {\mathrm{\Lambda}}_{\mathrm{\Psi}}u={F}^{1}\mathrm{\Psi}(\xi )Fu, u\in S({R}^{n};{E}_{1}) is well defined and extends to a bounded linear operator
{\mathrm{\Lambda}}_{\mathrm{\Psi}}:{L}_{p}({R}^{n};{E}_{1})\to {L}_{p}({R}^{n};{E}_{2}).
The set of all multipliers from {L}_{p}({R}^{n};{E}_{1}) to {L}_{p}({R}^{n};{E}_{2}) will be denoted by {M}_{p}^{p}({E}_{1},{E}_{2}). For {E}_{1}={E}_{2}=E, it will be denoted by {M}_{p}^{p}(E).
Let
{U}_{n}=\{\beta =({\beta}_{1},{\beta}_{2},\dots ,{\beta}_{n})\in {N}^{n}:{\beta}_{k}\in \{0,1\}\}.
Definition 1 A Banach space E is said to be a space satisfying a multiplier condition if for any \mathrm{\Psi}\in {C}^{(n)}({R}^{n};L(E)), the Rboundedness of the set \{{\xi}^{\beta}{D}_{\xi}^{\beta}\mathrm{\Psi}(\xi ):\xi \in {R}^{n}\mathrm{\setminus}0,\beta \in {U}_{n}\} implies that Ψ is a Fourier multiplier in {L}_{p}({R}^{n};E), i.e., \mathrm{\Psi}\in {M}_{p}^{p}(E) for any p\in (1,\mathrm{\infty}).
Definition 2 The φpositive operator A is said to be an Rpositive in a Banach space E if there exists \phi \in [0,\pi ) such that the set
{L}_{A}=\{A{(A+\lambda )}^{1}:\lambda \in {S}_{\phi}\}
is Rbounded.
A linear operator A(x) is said to be positive in E uniformly in x if D(A(x)) is independent of x, D(A(x)) is dense in E and
\parallel {(A(x)+\lambda )}^{1}\parallel \le M{(1+\lambda )}^{1}
for all \lambda \in S(\phi ), \phi \in [0,\pi ).
Let {\sigma}_{\mathrm{\infty}}({E}_{1},{E}_{2}) denote the space of all compact operators from {E}_{1} to {E}_{2}. For {E}_{1}={E}_{2}=E, it is denoted by {\sigma}_{\mathrm{\infty}}(E). Assume {E}_{0} and E are two Banach spaces and {E}_{0} is continuously and densely embedded into E. Let m be a natural number. {W}^{m,p}(\mathrm{\Omega};{E}_{0},E) (the socalled SobolevLions type space) denotes a space of all functions u\in {L}^{p}(\mathrm{\Omega};{E}_{0}) possessing the generalized derivatives {D}_{k}^{m}u=\frac{{\partial}^{m}u}{\partial {x}_{k}^{m}} such that {D}_{k}^{m}u\in {L}^{p}(\mathrm{\Omega};E) is endowed with the norm
{\parallel u\parallel}_{{W}^{m,p}(\mathrm{\Omega};{E}_{0},E)}={\parallel u\parallel}_{{L}^{p}(\mathrm{\Omega};{E}_{0})}+\sum _{k=1}^{n}{\parallel {D}_{k}^{m}u\parallel}_{{L}^{p}(\mathrm{\Omega};E)}<\mathrm{\infty}.
For {E}_{0}=E the space {W}^{m,p}(\mathrm{\Omega};{E}_{0},E) will be denoted by {W}^{m,p}(\mathrm{\Omega};E). It is clear to see that
{W}^{m,p}(\mathrm{\Omega};{E}_{0},E)={W}^{m,p}(\mathrm{\Omega};E)\cap {L}^{p}(\mathrm{\Omega};{E}_{0}).
Let s be a positive parameter. We define in {W}^{m,p}(\mathrm{\Omega};{E}_{0},E) the following parameterized norm:
{\parallel u\parallel}_{{W}_{s}^{m,p}(\mathrm{\Omega};{E}_{0},E)}={\parallel u\parallel}_{{L}^{p}(\mathrm{\Omega};{E}_{0})}+\sum _{k=1}^{n}{\parallel s{D}_{k}^{m}u\parallel}_{{L}^{p}(\mathrm{\Omega};E)}.
Function u\in {W}^{2,p}(0,1;E(A),E,{L}_{k})=\{u\in {W}^{2,p}(0,1;E(A),E),{L}_{k}u=0\} satisfying equation (1) a.e. on (0,1) is said to be a solution of the problem (1) on (0,1).
From [21] we have the following theorem.
Theorem A1 Suppose the following conditions are satisfied:

(1)
E is a Banach space satisfying the multiplier condition with respect to p\in (1,\mathrm{\infty}) and A is an Rpositive operator in E;

(2)
\alpha =({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{n}) are ntuples of nonnegative integer numbers such that
\varkappa =\frac{\alpha }{m}\le 1\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}0<\mu \le 1\varkappa ;

(3)
\mathrm{\Omega}\in {R}^{n} is a region such that there exists a bounded linear extension operator from {W}^{m,p}(\mathrm{\Omega};E(A),E) to {W}^{m,p}({R}^{n};E(A),E).
Then the embedding
{D}^{\alpha}{W}^{m,p}(\mathrm{\Omega};E(A),E)\subset {L}^{p}(\mathrm{\Omega};E\left({A}^{1\varkappa \mu}\right))
is continuous and there exists a positive constant
{C}_{\mu}
such that
{s}^{\frac{\alpha }{m}}{\parallel {D}^{\alpha}u\parallel}_{{L}^{p}(\mathrm{\Omega};E({A}^{1\varkappa \mu}))}\le {C}_{\mu}[{h}^{\mu}{\parallel u\parallel}_{{W}_{s}^{m,p}(\mathrm{\Omega};E(A),E)}+{h}^{(1\mu )}{\parallel u\parallel}_{{L}^{p}(\mathrm{\Omega};E)}]
for all u\in {W}^{m,p}(\mathrm{\Omega};E(A),E) and 0<h\le {h}_{0}<\mathrm{\infty}.
Theorem A2 Suppose all conditions of Theorem A_{1} are satisfied. Assume Ω is a bounded region in {R}^{n} and {A}^{1}\in {\sigma}_{\mathrm{\infty}}(E). Then, for 0<\mu \le 1\varkappa, the embedding
{D}^{\alpha}{W}^{m,p}(\mathrm{\Omega};E(A),E)\subset {L}^{p}(\mathrm{\Omega};E\left({A}^{1\varkappa \mu}\right))
is compact.
In a similar way as in [[2], Theorem 2.1], we have the following result.
Lemma A1 Let E be a Banach space and f\in \mathit{VMO}(E). The following conditions are equivalent:

(1)
f\in \mathit{VMO}(E);

(2)
f is in the BMO closure of the set of uniformly continuous functions which belong to VMO;

(3)
{lim}_{y\to 0}{\parallel f(xy)f(x)\parallel}_{\ast ,E}=0.
For f\in {L}^{p}(\mathrm{\Omega};E), p\in (1,\mathrm{\infty}), a\in {L}^{\mathrm{\infty}}({R}^{n}), consider the commutator operator
H[a,f](x)=a(x)Hf(x)H(af)(x)=\underset{\epsilon \to 0}{lim}{\int}_{xy>\epsilon ,}\frac{[a(x)a(y)]}{xy}f(y)\phantom{\rule{0.2em}{0ex}}dy.
Proof Indeed, we observe that if f\in \mathit{VMO}(E) with VMO modulus η, there exists a constant C such that {\parallel f(xy)f(x)\parallel}_{\ast ,E}\le C\eta (r) for \parallel y\parallel \le r so that the Evalued usual mollifiers converge to f in the BMO norm. More precisely, given f\in \mathit{VMO}(E) with VMO modulus \eta (r), we can find a sequence of Evalued {C}^{\mathrm{\infty}} functions \{{f}_{h}\} converging to f in Evalued BMO spaces as h\to 0 with VMO moduli {\eta}_{h} such that {\eta}_{h}\le \eta (r). In a similar way, other cases are derived. □
From [[26], Theorem 1] and [[29], Corollary 2.7], we have the following.
Theorem A3 Let E be a UMD space and a\in \mathit{VMO}\cap {L}^{\mathrm{\infty}}({R}^{n}). Then H[a,f] is a bounded operator in {L}^{p}(R;E), p\in (1,\mathrm{\infty}).
From Theorem A_{3} and the property (2) of Lemma A_{1}, we obtain, respectively:
Theorem A4 Assume all conditions of Theorem A_{3} are satisfied. Also, let a\in \mathit{VMO}\cap {L}^{\mathrm{\infty}}({R}^{n}) and let η be the VMO modulus of a. Then, for any \epsilon >0, there exists a positive number \delta =\delta (\epsilon ,\eta ) such that
{\parallel H[a,f]\parallel}_{{L}^{p}(0,r;E)}\le M\epsilon {\parallel f\parallel}_{{L}^{p}(0,r;E)},\phantom{\rule{1em}{0ex}}r\in (0,\delta ).
Theorem A5 Let E be a UMD space, p\in (1,\mathrm{\infty}) and A(\cdot ) uniformly Rpositive in E. Moreover, let A(\cdot ){A}^{1}({x}_{0})\in {L}_{\mathrm{\infty}}(R;L(E))\cap \mathit{BMO}(L(E)), {x}_{0}\in R. Then the following commutator operator is bounded in {L}^{p}(R;E):
\begin{array}{rcl}H[A,f](x)& =& A(x){A}^{1}({x}_{0})Hf(x)H(A(x){A}^{1}({x}_{0})f)(x)\\ =& \underset{\epsilon \to 0}{lim}{\int}_{xy>\epsilon ,}\frac{[A(x){A}^{1}({x}_{0})A(y){A}^{1}({x}_{0})]}{xy}f(y)\phantom{\rule{0.2em}{0ex}}dy.\end{array}
Note that singular integral operators in Evalued {L}^{p} spaces were studied, e.g., in [30].
Theorem A6 Assume all conditions of Theorem A_{5} are satisfied and η is a VMO modulus of A(\cdot ){A}^{1}({x}_{0}).
Then, for any \epsilon >0, there exists a positive number \delta =\delta (\epsilon ,\eta ) such that
{\parallel H[A,f]\parallel}_{{L}^{p}({\mathrm{\Omega}}_{r};E)}\le M\epsilon {\parallel f\parallel}_{{L}^{p}({\mathrm{\Omega}}_{r};E)},\phantom{\rule{1em}{0ex}}r\in (0,\delta ).
Consider the nonlocal BVP for parameterdependent DOE with constant coefficients
where {\nu}_{k}\in \{0,1,\dots ,2m1\}, a, {\alpha}_{ki}, {\beta}_{ki} are complex numbers, {\mu}_{i}=\frac{i}{2m}+\frac{1}{2mp}, {\theta}_{k}=\frac{{\nu}_{k}}{2m}+\frac{1}{2mp}, s is a positive and λ is a complex parameter; {A}_{\lambda}=A+\lambda and A is a linear operator in E. Let {\omega}_{1},{\omega}_{2},\dots ,{\omega}_{2m} be roots of the equation a{\omega}^{2m}+1=0, [{\upsilon}_{ij}] be a 2mdimensional matrix and \eta =[{\upsilon}_{ij}] be a determinant of the matrix [{\upsilon}_{ij}], where
It is known that (see, e.g., [[24], §1.15]) if the operator A is φpositive in E, then operators {\omega}_{k}{s}^{\frac{1}{2m}}{A}_{\lambda}^{\frac{1}{2m}}, k=1,2,\dots ,2m generate the following analytic semigroups:
Let
{E}_{k}={(E(A),E)}_{{\theta}_{k},p}.
From [[19], Theorem 1] and [[22], Theorem 3.2], we obtain the following.
Theorem A7 Assume the following conditions are satisfied:

(1)
E is a Banach space satisfying the multiplier condition with respect to p\in (1,\mathrm{\infty});

(2)
A is an Rpositive operator in E for 0\le \phi <\pi and \eta \ne 0;

(3)
arg{\omega}_{j}\pi \le \frac{\pi}{2}\phi, j=1,2,\dots ,m, arg{\omega}_{j}\le \frac{\pi}{2}\phi, j=m+1,\dots ,2m and \frac{\lambda}{{\omega}_{j}}\in S(\phi ).
Then

(1)
for f\in {L}_{p}(0,1;E), {f}_{k}\in {E}_{k}, \lambda \in S(\phi ) and for sufficiently large \lambda , the problem (2) has a unique solution u\in {W}^{2m,p}(0,1;E(A),E). Moreover, the following coercive uniform estimate holds:
\sum _{i=0}^{2m}{s}^{\frac{i}{2m}}\lambda {}^{1\frac{i}{2m}}{\parallel {u}^{(i)}\parallel}_{{L}^{p}(0,1;E)}+{\parallel Au\parallel}_{{L}^{p}(0,1;E)}\le C[{\parallel f\parallel}_{{L}^{p}(0,1;E)}+\sum _{k=1}^{2m}{\parallel {f}_{k}\parallel}_{{E}_{k}}].

(2)
For {f}_{k}=0, the solution is represented as
\begin{array}{rcl}u(x)& =& {\int}_{0}^{1}{G}_{\lambda s}(x,y)f(y)\phantom{\rule{0.2em}{0ex}}dy,{G}_{\lambda s}(x,y)\\ =& \sum _{k=1}^{2m}\sum _{j=1}^{2m}\sum _{i=0}^{{\nu}_{k}}[{B}_{kij}(\lambda ){\left({s}^{1}{A}_{\lambda}\right)}^{\frac{1}{2m}(2m+{\nu}_{k}i1)}{U}_{j\lambda s}(x){U}_{k\lambda s}(1y)]+{U}_{0\lambda s}(xy),\end{array}
(3)
where {B}_{kij}(\lambda ) are uniformly bounded operators in E and
{U}_{0\lambda s}(xy)=\{\begin{array}{c}{a}^{1}\{{s}^{1\frac{1}{2m}}{A}_{\lambda}^{(1\frac{1}{2m})}{\sum}_{i=1}^{m}{(1)}^{2m+i}{P}_{i}^{1}{U}_{i\lambda s}(xy),x\ge y\},\hfill \\ {a}^{1}\{{s}^{1\frac{1}{2m}}{A}_{\lambda}^{(1\frac{1}{2m})}{\sum}_{i=m+1}^{2m}{(1)}^{2m+i}{P}_{i}^{1}{U}_{i\lambda s}(xy),x\le y\},\hfill \end{array}
where
{P}_{i}=({\omega}_{i}{\omega}_{1})\cdots ({\omega}_{i}{\omega}_{i1})({\omega}_{i+1}{\omega}_{i})\cdots ({\omega}_{2m}{\omega}_{i}),\phantom{\rule{1em}{0ex}}i=1,2,\dots ,2m.
Consider the BVP for DOE with variable coefficients
where a=a(x) is a complexvalued function, {m}_{k}\in \{0,1,\dots ,2m1\}, {\alpha}_{ki}, {\beta}_{ki} are complex numbers, s is a positive and λ is a complex parameter, {\theta}_{k}=\frac{{\nu}_{k}}{2m}+\frac{1}{2mp} and A(x) is a linear operator in E.
Let {\omega}_{1}={\omega}_{1}(x), {\omega}_{2}={\omega}_{2}(x),\dots ,{\omega}_{2m}={\omega}_{2m}(x) be roots of the equation a(x){\omega}^{2m}+1=0, [{\upsilon}_{ij}] be a 2mdimensional matrix and \eta (x)=[{\upsilon}_{ij}] be a determinant of the function matrix [{\upsilon}_{ij}], where
In the next theorem, we consider the case when principal coefficients are continuous. The wellposedness of this problem occurs in studying of equations with VMO coefficients. From [[19], Theorem 3] and [[22], Theorem 3.2], we get the following.
Theorem A8 Suppose the following conditions are satisfied:

(1)
E is a Banach space satisfying the multiplier condition with respect to p\in (1,\mathrm{\infty});

(2)
arg{\omega}_{j}\pi \le \frac{\pi}{2}\phi, j=1,2,\dots ,m, arg{\omega}_{j}\le \frac{\pi}{2}\phi, j=m+1,\dots ,2m and \frac{\lambda}{{\omega}_{j}}\in S(\phi ) a.e. x\in (0,1);

(3)
a\in C[0,1], a(0)=a(1) and \eta (x)\ne 0 for a.e. x\in [0,1];

(4)
A(x) is a uniformly Rpositive operator in E and
A(\cdot ){A}^{1}({x}_{0})\in C([0,1];L(E)),\phantom{\rule{1em}{0ex}}{x}_{0}\in (0,1),\phantom{\rule{2em}{0ex}}A(0)=A(1).
Then, for f\in {L}^{p}(0,1;E), \lambda \in S(\phi ) and for sufficiently large \lambda , there is a unique solution u\in {W}^{2,p}(0,1;E(A),E) of the problem (4). Moreover, the following coercive uniform estimate holds:
\sum _{i=0}^{2m}{s}^{\frac{i}{2m}}\lambda {}^{1\frac{i}{2m}}{\parallel {u}^{(i)}\parallel}_{{L}^{p}(0,1;E)}+{\parallel Au\parallel}_{{L}^{p}(0,1;E)}\le C{\parallel f\parallel}_{{L}^{p}(0,1;E)}.