MAXIMAL REGULAR BOUNDARY VALUE PROBLEMS IN BANACH-VALUED FUNCTION SPACES AND APPLICATIONS

The nonlocal boundary value problems for differential operator equations of second order with dependent coefficients are studied. The principal parts of the differential operators generated by these problems are non-selfadjoint. Several conditions for the maximal regularity and the Fredholmness in Banach-valued Lp-spaces of these problems are given. By using these results, the maximal regularity of parabolic nonlocal initial boundary value problems is shown. In applications, the nonlocal boundary value problems for quasi elliptic partial differential equations, nonlocal initial boundary value problems for parabolic equations, and their systems on cylindrical domain are studied.

Boundary value problems (BVPs) for differential operator equations (DOE) in H-valued (Hilbert space-valued) function spaces have been studied extensively by many researchers (see [4-7, 12, 15, 16, 18, 20, 22, 28-33, 37-39] and the references therein).In these works Hilbert-valued function spaces essentially were considered.The main objective of the present paper is to discuss the nonlocal BVP for DOE with variable coefficients in Banach-valued function spaces.In this work, (1) at first, nonhomogenous BVP for ordinary DOE is considered; (2) partial DOE with dependent coefficients in principal part is considered; (3) boundary conditions are, generally, nonlocal; (4) operators containing equations and boundary conditions are, in general, unbounded; (5) nonlocal initial boundary value problems (IBVP) for parabolic DOE are considered.The maximal regularity, positivity and, Fredholmness of these problems in Banach-valued L p -spaces are proved.These results are also applied to the nonlocal BVP for quasi elliptic partial differential equations, infinite systems of nonlocal BVP for elliptic equations with variable coefficients, and INBVP for parabolic equations on cylindrical domains.
Let E be a Banach space.L p (Ω;E) denotes a space all of strongly measurable E-valued functions that are defined on a domain Ω ⊂ R n with the norm (1.1) By L p,q (Ω) and W l p,q (Ω), we will denote a scalar-valued (p, q)-integrable function space and Sobolev space with mixed norms, respectively [8].Let B s pq denote the Besov space (see, e.g., [35,Section 2.3]).
A Banach space E is said to be ζ-convex space (see [9-11, 15, 23]) if there exists on E × E a symmetric real-valued function ζ(u,v) which is convex with respect to each of the variables, and satisfies the conditions (1.2) In literature the ζ-convex Banach spaces E are often called UMD-spaces and written as E ∈ UMD.It is shown in [10] that the Hilbert operator is bounded in L p (R;E), p ∈ (1,∞), for those and only those spaces E which possess the property of UMD spaces.UMD spaces include, for example, L p , l p spaces and Lorentz spaces L pq , p, q ∈ (1,∞).Let C be a set of complex numbers.S ϕ denotes an open sector with vertex 0, opening angle 2ϕ, which is symmetric with respect to the positive half-axis R + , that is, ( Let (1.5) A linear operator A is said to be positive in a Banach space E, with bound M if D(A) is dense on E and with ξ ∈ K ϕ , ϕ ∈ (0,π], where M is a positive constant and I an identity operator in E, where L(E) is a space of bounded linear operators acting in E. Sometimes instead of A + ξI, will be written A + ξ and denoted by A ξ .The operator A(t) is said to be positive in a Banach space E uniformly with respect to t, if D(A(t)) is independent of t, D(A(t)) is dense in E, and for all λ ∈ K(ϕ) , ϕ ∈ (0,π]. It is known [35,Section 1.15.1] that there exist fractional powers A θ of the positive operator A. Let E(A θ ) denote the space D(A θ ) with graphical norm defined as u E(A θ ) = u p + A θ u p 1/p , 1≤ p < ∞, −∞ < θ < ∞. (1.8)Let E 1 and E 2 be two Banach spaces.By (E 1 ,E 2 ) θ,p , 0 < θ < 1,1 ≤ p ≤ ∞, will be denoted an interpolation space for {E 1 ,E 2 } by the K-method [35,Section 1.3.1].By C(Ω;E) Veli B. Shakhmurov 3 and C (m) (Ω;E) will be denoted spaces of E-valued bounded continuous and m-times continuously differentiable function on Ω, respectively.Let S(R n ;E) denote a Schwarz class, that is, the space of all E-valued rapidly decreasing smooth functions ϕ on R n .The function , is well defined and extends to a bounded linear operator (1.9) We denote the set of all multipliers from L p (R n ;E 1 ) to L q (R n ;E 2 ) by be a collection of multipliers in M q p (E 1 ,E 2 ).We say that Φ h is a uniformly bounded multiplier with respect to h if there exists a constant C > 0, independent on h ∈ B(h), such that for all h ∈ K and u ∈ S(R n ;E 1 ).The exposition of the theory of L p -multipliers of the Fourier transformation, and some related references, can be found in [35, Sections 2.2.1-2.2.4].On the other hand, in vector-valued function spaces, Fourier multipliers have been studied by [11-13, 18, 26, 27, 36].
It is well known (see, e.g., [26]) that any Hilbert space satisfies the multiplier condition.There are, however, Banach spaces which are not Hilbert spaces but satisfy the multiplier condition, for example, UMD spaces (see [11,12,18,36]).

Definition 1.3. A positive operator
A is said to be R-positive in the Banach space E if there exists ϕ ∈ (0,π] such that the set Note that in Hilbert spaces every norm bounded set is R-bounded.Therefore, in Hilbert spaces all positive operators are R-positive.If A is a generator of a contraction semigroup on L q , 1 ≤ q ≤ ∞ [23], A has bounded imaginary powers with (−A it ) B(E) ≤ Ce ν|t| ,ν < π/2 [14], or if A is a generator of a semigroup with Gaussian bound [19] in E ∈ UMD, then this operator is R-positive.
Let Ω ∈ R n and l = (l 1 ,l 2 ,...,l n ).Let E 0 and E be two Banach spaces and E 0 continuously and densely embedded into E. Let us consider a Banach-valued function space W l p (Ω;E 0 ,E) that consists of functions u ∈ L p (Ω;E 0 ) such that has the generalized derivatives (1.18) For E 0 = E the space W l p (Ω;E 0 ,E) will be denoted by W l p (Ω;E).For Ω = (a,b) ∈ R and l 1 = l 2 = ••• = l n = m the space W l p (Ω;E 0 ,E) will be denoted by W m p (a,b;E 0 ,E).By σ ∞ (E) will be denoted a space of all compact operators in E.

Background materials
Embedding theorems of vector-valued Sobolev spaces played important role in the present investigation.Embedding theorems in Banach-valued function spaces have been studied, for example, in [6,25,29,31,33].This section concentrates on anisotropic Banach-valued Sobolev spaces W l p (Ω;E 0 ,E) associated with Banach spaces E 0 , E. Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal regular in these spaces in terms of interpolations of E 0 and E. In particular, the most regular class of interpolation spaces E α between E 0 , E, depending on α and order of spaces are found that mixed derivatives D α are bounded and compact from this space to E α -valued L p spaces.This results are generalized and improve the result of Lions and Peetre [25] for Banach-valued spaces and the embedding theorems for scalar Sobolev spaces [8,Section 9].Multiplier theorems in the operator-valued L p spaces, are important tools in the theory of embedding of function spaces and PDE.Since the problems under consideration established the uniformly parameterized estimates, so we have to generalize multiplier theorems [18] for the case of L p multipliers depending on parameters.So, firstly by using a similar technique as [18] we show the following multiplier theorem.
Theorem 2.1.Let E be a UMD space with property (α) and let Ψ h ∈ C n (R n /{0};B(E)) and there is some C > 0 such that uniformly with respect to h, then Ψ h (ξ) is a uniform collection of multipliers in L p (R n ;E).
If n = 1, then the result remains true for E without having the property (α).
Note 2.2.It is clear that Theorem 2.1 is valid for the case of multipliers without parameter and without assumption of the uniformly boundedness condition.By virtue of [33] we obtain the following.
Theorem 2.3.Suppose the following conditions are satisfied: (1) E is a Banach space that satisfies the multiplier condition with respect to p and A is an R-positive operator in E for ϕ with 0 < ϕ ≤ π; (3) Ω ∈ R n is a region such that there exists a bounded linear extension operator acting from L p (Ω;E) to L p (R n ;E) and also from Then an embedding is continuous and there exists a positive constant C μ such that for all u ∈ W l p (Ω;E(A),E) and h with 0 < h < h 0 < ∞.Theorem 2.4.Suppose all conditions of Theorem 2.3 are satisfied and suppose is compact.
12), we obtain a multiplicative inequality ( By virtue of [29] the embedding is compact.Then from the estimate (2.7) we obtain assertion of Theorem 2.4.By a similar manner as Theorem 2.3 we have the following.
Theorem 2.5.Suppose all conditions of Theorem 2.3 are satisfied.Then for 0 < μ < 1 − κ an embedding is continuous and there exists a positive constant C μ such that for all u ∈ W l p (Ω;E(A),E) and h with 0 < h < h 0 < ∞ .By a similar manner as Theorem 2.4 we have the following.
Theorem 2.6.Suppose all conditions of Theorem 2.3 are satisfied and suppose ) Theorem 2.7 [32].Let E be a Banach space and A a positive operator in E. Let m be a positive integer, 1 ≤ p < ∞, and 1/2p < α < m + 1/2p.Let 0 ≤ γ < 1.Then for λ ∈ S(ϕ) the operator −A 1/2 λ generates a semigroup e −A 1/2 λ x which is holomorphic for x > 0 and strongly continuous for x ≥ 0.Moreover, there exists a constant C > 0 such that for every u ∈ (E, (2.12) By using a similar techniques as [25] (or [35, Section 1.8.1]) we obtain the following.
Theorem 2.8.Let the following conditions be satisfied: (1) l and s are integer numbers, and , the following inequality holds: (2.13)

Statement of problems
Consider a nonlocal BVP for elliptic DOE and nonlocal IBVP parabolic problem where

Ordinary DOE with constant coefficients
Let us first consider a nonlocal and nonhomogenous boundary value problem for ordinary DOE where ; a, α k , β k , δ k j , are complex numbers and x k j ∈ (0,b); A is a possible unbounded operator in E. Let ω j , j = 1,2, be roots of the equation Let the following conditions be satisfied: (1) A is a positive operator in a Banach space E for ϕ ∈ (0,π/2); (2) holds with respect to parameter λ .
Proof.From conditions (1.12) and (1.13), by virtue of [39, Lemma 5.3.2/1],for λ ∈ S(ϕ 0 ), there exists the holomorphic for x > 0 and strongly continuous for x ≥ 0 semigroups e xω1A 1/2 λ , e −(b−x)ω2A 1/2 λ , and the arbitrary solution of (4.3), belonging to space where By taking into account boundary conditions (4.4) we obtain algebraic linear equations with respect to g 1 , g 2 ; A system (4.8) is matrix-operator equations.Let D(λ) be a main operator determinant of (4.8).By virtue of the properties of positive operators and holomorphic semigroups [35,Section 1.14] it is clear to see that D(λ) B(E 2 ) → 0 for |λ| → ∞.Then by conditions η = 0 and λ ∈ S(ϕ), λ → ∞, the operator-matrix Q(λ) = [θ + D(λ)] −1 is invertible and bounded uniformly with respect to the parameters λ.Consequently, the system (4.8) has a unique solution for λ ∈ S(ϕ) and sufficiently large |λ|.From the expressions of operators D(λ) and Q(λ) it follows that these operators are bounded, and operators containing the expression D(λ) are commuting with any powers of operators A 1/2 λ .Consequently, substituting the values of g 1 , g 2 into (4.8),we obtain a representation of the solution of the problem (4.3)-(4.4): where B k j (λ) are bounded operators in E uniformly with respect to λ and By virtue of Theorem 2.7, the properties of holomorphic semigroups, in view of uniformly boundedness of operator Q(λ), and the representation of the solution (4.9), we obtain the estimate (4.5).
Proof.We have proved the uniqueness of the solution of the problem (4.1) in Lemma 4.2.Let (4.12) We now show that a solution of the problem (4.1) which belongs to space W 2 p (0,b; E(A)E) can be represented as a sum υ(x) = u 1 (x) + u 2 (x), where u 1 is a restriction on [0,b] of a solution u of an equation and u 2 is a solution of a problem The solution of (4.13) is given by formula where F f is a Fourier transform of a function f , and Due to R-positivity of operator A and by virtue of Kahane's contraction principle, we obtain (4.17) Veli B. Shakhmurov 11 Then in view of Definition 1.1 it follows from (4.17) that the operator-valued functions AL −1 (λ,ξ), |λ| 1− j/2 ξ j L −1 (λ,ξ), j = 0,1,2, are uniformly bounded Fourier multipliers in L p (R;E).Therefore, we obtain that the problem (4.13) has a solution u 0 ∈ W 2 p (R; E(A),E) and So, we obtain that u 1 ∈ W 2 p (0,b;E(A),E) is the solution of (4.13) on (0,b).By virtue of [25] we get that From (4.18) for λ ∈ S(ϕ) we obtain Therefore, by virtue of [25] and by the estimate (4.21) for x 0 ∈ [0,b], we have 12 Maximal regular problems Consider a BVP  Proof.From (1.12)-(1.13),part of Condition 4.1, we have By using the above representation and by using a similar technique as [39, Lemma 5.3.2/1]we obtain that ω k A λ , k = 1,2, for λ ∈ S(ϕ) are generators of the bounded analytic λ in E and a solution of (4.11) is represented as where where B k j (λ) are bounded operators in E uniformly with respect to λ. Due to holomorphic semigroups of U kλ , we have (see, e.g., [35,Section 1.14])

Partial DOE with constant coefficients
) α k , β k , δ k j are complex numbers, a k are complex numbers, and A is, generally speaking, an unbounded operator in E and 2) for f ∈ L p (G;E), λ ∈ S(ϕ), and for sufficiently large |λ| has a unique solution that belongs to the space W 2 p (G;E(A),E) and the coercive uniform estimate holds with respect to parameter λ; (b) the operator L 0 , generated by BVP (5.1)-( 5.2), is positive in L p (G;E).(5.5) in L p (0,b 1 ;E), where A is a positive operator in E and α 1 j , β 1 j , δ 1 ji are complex numbers.By virtue of Theorem 4.3 we obtain that for all f ∈ L p (0,b 1 ;E), λ ∈ S(ϕ), and sufficiently large |λ| the problem (5.5) has a unique solution that belongs to the space W 2 p (0,b 1 ;E(A),E), and coercive uniformity is defined by

Proof. Let us first consider a nonlocal boundary value problem for ordinary DOE
with respect to λ; the estimate holds for the solution of the problem (5.5).Consider in where The problem (5.7) can be expressed in the following view: where B is a differential operator in L p (0,b 1 ;E) generated by problem (5.5).It is known (see, e.g., [10,11]) due to E ∈ UMD, p ∈ (1,∞), that the space L p (0,b 1 ;E) is UMD space.Moreover, by virtue of Theorem 4.4 operator B is R-positive in L p (0,b 1 ;E).Then again applying Theorem 4.3 we obtain that for all f ∈ L p (G 2 ;E), λ ∈ S(ϕ), and sufficiently large |λ| the problem (5.9), that is, the problem (5.7), has a unique solution that belongs to the space W 2 p (G 2 ;E(A),E), and the coercive uniform estimate holds with respect to λ.Moreover, the estimate (5.6) implies (2) A(x)A −1 (x 0 ) ∈ C(G;B(E)) and for any ε > 0, for a.e.x ∈ G, and for u (5.12) Then (a) the problem (3.1) for f ∈ L p (G;E), λ ∈ S(ϕ), and for sufficiently large |λ| has a unique solution that belongs to the space W 2 p (G;E(A),E).And the coercive uniform estimate Proof.Let G 1, G 2 ,...,G N be regions in R and ϕ 1 ,ϕ 2 ,...,ϕ N be corresponding a partition of unique that, functions ϕ j are smooth on R, σ j = suppϕ j ⊂ G j and N j=1 ϕ j (x) = 1.Then for all u ∈ W 2 p (G;E(A),E) we have u(x) = N j=1 u j (x), where u j (x) = u(x)ϕ j (x).Moreover, due to the nonlocalness of boundary conditions, functions ϕ j are chosen such that σ j = suppϕ j adjoin with boundary G k0 , G kb and consist of the sets; σ jk0 and σ jkb that is, σ jk = σ jk0 ∪ σ jkb , where σ jk0 are parts of σ jk adjoin with G k0 and σ jkb is the part of σ jk adjoin with G kb .Let us consider the case when the regions G j adjoin with the boundary points and contain σ jm .Let u ∈ W 2 p (G;E(A),E).Then from (3.1) we obtain where Let suppϕ j partially belong to G. Freezing coefficients in (5.14) obtain that where (5.17) Suppose functions ϕ j (x) such that L mk u j = 0, m = 1,2, for all u ∈ W 2 p (G;E(A),E).Since functions u j (x) have the compact supports, then extending u j (x) on outsides of suppϕ j from (5.16) we obtain boundary value problems with constant coefficients: L mk u j = 0, m = 1,2, j = 1,2,...,N.
(5.18)By using Theorem 5.2 we obtain that the problem (5.18) has a unique solution u j and for λ ∈ S(ϕ) and sufficiently large |λ| the following coercive estimate: holds.Whence, using properties of the smoothness of coefficients of (5.15), (5.17) and choosing diameters of σ j sufficiently small, we get that where ε is a sufficiently small and C(ε) is a continuous function.Consequently, from , we get ( Choosing ε < 1 from the above inequality, we have By a similar manner we also obtain estimates (5.22) for regions G j entirely belonging to G. Then using the equality u(x) = N j=1 u j (x) and by virtue of the estimate (5.22) for u ∈ W 2 p (G;E(A),E), we have Let u ∈ W 2 p (G;E(A),E) be a solution of the problem (3.1).Then for λ ∈ S(ϕ), we have Then by embedding Theorems 2.3, 2.5 and by virtue of (5.23), (5.24) for sufficiently large |λ| we have Veli B. Shakhmurov 17 Let us consider an operator O λ , acting in L p (G;E), that is generated by the problem (3.1), that is, The estimate (5.25) implies that the problem (3.1) has only a unique solution and the operator O λ has an invertible operator in its rank space.We need to show that this rank space coincide with the space L p (0,b;E).We consider the smooth functions g j = g j (x) with respect to the partition of the unique ϕ j = ϕ j (y) on the region G that equals one on suppϕ j , where suppg j ⊂ G j and |g j (x)| < 1.Let us construct for all j the function u j , that is defined on the regions Ω j = G ∩ G j and satisfying the problem (3.1).Consider at first when G j adjoin to the boundary points.The problem (3.1) can be expressed in the form L km u j = 0, j = 1,2,...,N. ( Consider operators O jλ , acting in L p (G j ;E) that is generated by boundary value problems (5.18).By virtue of Theorem 4.4 for all f ∈ L p (G j ;E), for λ ∈ S(ϕ), and sufficiently large |λ| we have (5.28) Extending u j zero on the outside of suppϕ j in equalities (5.27) and passing substitutions u j = O −1 jλ υ j obtained from (5.1), operator equations with respect to υ j , υ j = K jλ υ j + g j f , j = 1,2,...,N. (5.29) By virtue of Theorem 2.3 and the estimate (5.28), in view of the smoothness of the coefficients of the expression K jλ , and in view of condition (2.7) for λ ∈ S(ϕ) and sufficiently large |λ|, we have K jλ < ε, where ε is sufficiently small.Consequently, (5.29) has unique solutions υ j = [I − K jλ ] −1 g j f .Moreover, (5.30) Whence, [I − K jλ ] −1 g j are bounded linear operators from L p (G;E) to L p (G j ;E).Thus, we obtain that the functions perturbation theory [12] we obtain that the operator O is Fredholm from W 2 p (G;E(A),E) into L p (G;E).Proof.Really, problem (3.2) can be expressed in space L p (R + ;F) in the following form: where with nonlocal boundary conditions in L p (G + ;E) space.

Nonlocal boundary value problems for elliptic equations.
The Fredholm property of boundary value problems for elliptic equations with parameters in smooth domains was studied in [1-3, 12, 24] also for nonsmooth domains was investigated in [17,21,28,34].
where D j = −i(∂/∂y j ), m k ∈ {0, 1}, α k , β k , δ k ji are complex numbers, y = (y 1 ,..., y l ), and (5.41) Let Ω ⊂ R l be an open connected set with compact C 2m -boundary ∂Ω.Recall that for all y 0 ∈ ∂Ω local coordinates corresponding to y 0 are defined as coordinates obtained from the original ones by a rotation and a shift which transfers y 0 to the origin and after which the positive y l -axis has the direction of the interior normal to ∂Ω at y 0 .
Proof.Let E = L q (G).Then by virtue of [18,Theorem 3.6] part (1) of Condition 5.1 is satisfied.Consider an operator A which is defined by the equalities (5.45) For x ∈ G also consider operators has a unique solution for f ∈ L q (G), and for arg λ ∈ S(ϕ), |λ| → ∞; and the differential operator A generating by (5.47) is R-positive in L q .Then by virtue of (3.1)-(3.2) Condition 5.1 is fulfilled.It is known that the embedding W 2m q (G) ⊂ L q (G) is compact (see, e.g., [35,Theorem 3.2.5]).Then by using interpolation properties of Sobolev spaces (see, e.g., [35,Section 4]) it is clear to see that conditions (1.12) and (1.13) of Theorem 5.3 are fulfilled too.

Mathematical Problems in Engineering
Special Issue on Time-Dependent Billiards

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
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. 11 ). 4 . 5 . 1 .
Then estimates (5.10) and(5.11)imply the assertion of Theorem 4.3 for n = 2.By expending this process we obtain the assertion (a).The assertion (b) is obtained by using Theorem 4Partial DOE with variable coefficients.Consider the boundary value problem (3.1).Theorem 5.3.Let Condition 5.1 be satisfied for all x ∈ G and (1) A(x) is an R positive in E uniformly with respect to x and A(G 0k ) = A(G bk ), a k (x) are continuous functions on Ḡ such that a k (G j0 ) = a k (G jb ), k, j = 1,2,...,n; Veli B. Shakhmurov 15

13 )
holds with respect to parameter λ; (b) the operator O generating by BVP (3.1) is positive in L p (G;E).
We say that the elliptic problem (3.1) is a maximal L p -regular, if for all f ∈ L p (G;E) there exists a unique solution u ∈ W 2 p (G;E(A),E) of the problem (3.1) satisfying this problem almost everywhere and there exists a positive constant C independent of f , such that has an estimate We say that the parabolic problem (3.2) is a maximal L p -regular, if for all f ∈ L p (G + ;E) there exists a unique solution u satisfying the (3.2) problem almost everywhere and there exists a positive constant C independent of f , such that has an estimate 2,...,n;(3.3)αjk , β jk , δ jki are complex numbers, a k is real-valued function on G, and A(x), A k (x) for x, y ∈ G are generally speaking, unbounded operators in E.
.19) Hence, L k u 1 ∈ E k .Thus by virtue of Theorem 4.3 the problem (4.14) has a unique solution u 2 (x) that belongs to space W 2 p (0,b;E(A),E), and for sufficiently large |λ| we have 2 j=0