COERCIVE SOLVABILITY OF THE NONLOCAL BOUNDARY VALUE PROBLEM FOR PARABOLIC DIFFERENTIAL EQUATIONS

The nonlocal boundary value problem, v′(t)+Av(t)= f (t) (0 ≤ t ≤ 1), v(0)= v(λ)+μ (0 < λ≤ 1), in an arbitrary Banach space E with the strongly positive operator A, is considered. The coercive stability estimates in Hölder norms for the solution of this problem are proved. The exact Schauder’s estimates in Hölder norms of solutions of the boundary value problem on the range {0 ≤ t ≤ 1, x ∈ Rn} for 2m-order multidimensional parabolic equations are obtained.


Introduction
We consider the following nonlocal boundary value problem for the differential equation v (t) + Av(t) = f (t) (0 ≤ t ≤ 1), v(0) = v(λ) + µ (0 < λ ≤ 1), (1.1) in an arbitrary Banach space with linear (unbounded) operator A. It is known (cf.[4]) that various nonlocal boundary value problems for the parabolic equations can be reduced to the boundary value problem (1.1).We obtain the coercive solvability of problem (1.1) in some function Banach space.The role played by coercive inequalities in the study of boundary value problems for elliptic and parabolic partial differential equations is well known (cf.[5,6,7]).Coercivity inequalities for the solutions of an abstract differential equation of parabolic type of the linear bounded operators with exponentially decreasing norm (see [3]).
In [1], the coercive solvability of a Cauchy problem (1.2) was established in C β,γ 0 (E) (0 ≤ γ ≤ β, 0 < β < 1)-the space obtained by completion of the space of all smooth E-valued functions ϕ(t) on [0, 1] in the norm that is established in the following theorem.
Theorem 1.1.Let A be a strongly positive operator in a Banach space E and holds, where M does not depend on β, γ , v 0 , and f (t).
Theorem 1.2.Let A be a strongly positive operator in a Banach space E and holds, where M does not depend on α, β, γ , v 0 , and f (t).
Here C(E) stands for the Banach space of all continuous functions ϕ(t) defined on [0, 1] with values in E equipped with the norm and the Banach space E α (0 < α < 1) consists of those v ∈ E for which the norm (see [3]) From the existence of such solutions evidently follows that f (t) ∈ C β,γ

(E) and µ ∈ D(A).
In the present paper, the coercive inequalities in the norms of the same spaces for the solutions of the boundary value problem (1.1) are obtained.The exact Schauder's estimates in Hölder norms of the solution of the boundary value problem on the range {0 ≤ t ≤ 1, x ∈ R n } for 2m-order multidimensional parabolic equations are obtained.

Coercive solvability in
It is known that an operator A is strongly positive in E if and only if −A is the generator of the analytic semigroup exp{−tA} (t ≥ 0) of the linear bounded operators in E with exponentially decreasing norm when t → +∞, that is, the following estimates hold. ( For a strongly positive operator A we have that (2.2) Theorem 2.1.Let A be a strongly positive operator in a Banach space E and

(E) of the boundary value problem (1.1), the coercive inequality
holds, where M does not depend on β, γ , µ, and f (t).
Remark 2.2.Note that the spaces of smooth functions C β,γ 0 (E) in which coercive solvability has been established depend on the parameters β and γ .However, the constants in the coercive inequality (2.3) depend only on β.Hence, γ can be chosen freely in [0, β], which increases the number of function spaces in which problem (1.1) is coercively solvable.For example, it is important that problem (1.1) is coercively solvable in the Hölder space without a weight (γ = 0).

Coercive solvability in C
β,γ 0 (E α−β ) Theorem 3.1.Let A be a strongly positive operator in a Banach space E and

of the boundary value problem (1.1), the coercive inequality
holds, where M does not depend on β, γ , α, and f (t).
Proof.The proof of Theorem 3.1 follows from the inequality (1.5) and the estimate Using formula (2.7) and the estimates (2.1), (2.2) for any z > 0, we have that we have that .5) for any z > 0, and it follows that Now we estimate I 1 in the norm E α−γ .Using formula (2.6) and the estimates (2.1), (2.2) for any z > 0, we obtain A. Ashyralyev et al. 59 If z ≤ λ, then (3.9) Therefore, for any z > 0 we have that From the last estimate and (3.7), it follows that (3.11) Using the estimates (3.6), (3.11), and the triangle inequality, we obtain the estimate (3.2).This completes the proof of Theorem 3.1.Remark 3.2.Using this approach we can obtain the same results for solutions of the general boundary value problem where 0 < t 1 < t 2 < • • • < t p ≤ 1, if the operator I − p i=1 c i e −t i A has a bounded inverse in E.

Applications
We consider the boundary value problem on the range {0 ≤ t ≤ 1, x ∈ R n } for 2m-order multidimensional differential equations of parabolic type ∂v (t, x) ∂t ) where a r (x), f (y, x) are given sufficiently smooth functions and δ > 0 is a sufficiently large number.We will assume that the symbol of the differential operator of the form acting on functions defined on the space R n , satisfies the inequalities A. Ashyralyev et al. 61 where M(ε) do not depend on β, γ , α, µ(x), and f (t,x) and M(α, β, γ ) do not depend on µ(x), and f (t,x).Here C ε (R n ) is the space of functions satisfying a Hölder condition with the indicator ε ∈ (0, 1).
The proof of Theorem 4.1 is based on Theorems 2.1 and 3.1, the strong positivity of the operator A in C ε (R n ), the coercive inequality for the solution of the resolvent equation of the elliptic operator A in C ε (R n ), and equivalent of the norms in the spaces E β = E β (A, C(R n )) and C 2mβ (R n ) when 0 < β < 1/2m (see [2,3]).

Theorem 4 . 1 .
) for ξ = 0. Problem (4.1) has a unique smooth solution.This allows us to reduce the boundary value problem (4.1) to the boundary value problem (1.1) in the Banach space E with a strongly positive operator A = B +δI defined by (4.3).We give in Theorem 4.1 a number of corollaries to Theorems 2.1 and 3.1.The solutions of the boundary value problem (4.1) satisfy the following coercive inequalities: