Well-Posedness of Nonlocal Parabolic Differential Problems with Dependent Operators

Abstract

The nonlocal boundary value problem for the parabolic differential equation 𝑣^′(𝑡)+𝐴(𝑡)𝑣(𝑡)=𝑓(𝑡)(0≤𝑡≤𝑇),𝑣(0)=𝑣(𝜆)+𝜑,0<𝜆≤𝑇 in an arbitrary Banach space 𝐸 with the dependent linear positive operator 𝐴(𝑡) is investigated. The well-posedness of this problem is established in Banach spaces 𝐶₀^𝛽,𝛾(𝐸_𝛼−𝛽) of all 𝐸_𝛼−𝛽-valued continuous functions 𝜑(𝑡) on [0,𝑇] satisfying a Hölder condition with a weight (𝑡+𝜏)^𝛾. New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.

1. Introduction: A Cauchy Problem

It is known that (see, e.g., [1–5] and the references given therein) several applied problems in fluid mechanics, physics, and mathematical biology were formulated into nonlocal mathematical models. In general, such nonlocal problems were not well studied.

Before going to discuss the well-posedness of nonlocal boundary value problem, we will give the definition of positive operators in a Banach space and introduce the fractional spaces generated by positive operators that will be needed in the sequel.

Let 𝐸 be a Banach space and let 𝐴∶𝐷(𝐴)⊂𝐸→𝐸 be a linear unbounded operator densely defined in 𝐸. The operator 𝐴 is said to be positive in the Banach space 𝐸 if its spectrum 𝜎_𝐴 lies in the interior of the sector of angle 𝜙, 0<2𝜙<2𝜋, symmetric with respect to the real axis, and if, on the edges of this sector, 𝑆₁(𝜙)={𝜌𝑒^𝑖𝜙∶0≤𝜌≤∞} and 𝑆₂(𝜙)={𝜌𝑒^−𝑖𝜙∶0≤𝜌≤∞}, and outside of the sector the resolvent (𝜆−𝐴)⁻¹ is subject to the bound (𝐴−𝜆)⁻¹_𝐸→𝐸≤𝑀|𝜆|.1+ (1) The infimum of all such angles 𝜙 is called the spectral angle of the positive operator 𝐴 and is denoted by 𝜑(𝐴,𝐸). We call 𝐴 strongly positive in the Banach space 𝐸 if its spectral angle 𝜑(𝐴,𝐸)<𝜋/2.

For positive operator 𝐴 in the Banach space 𝐸, let us introduce the fractional spaces 𝐸_𝛽=𝐸_𝛽(𝐸,𝐴)(0<𝛽<1) consisting of those 𝜈∈𝐸 for which the norm ‖𝜈‖_𝐸𝛽=sup_𝜆>0𝜆^𝛽𝐴(𝜆+𝐴)⁻¹𝜈_𝐸+‖𝜈‖_𝐸 (2) is finite.

In the paper [6], the well-posedness in spaces of smooth functions of the nonlocal boundary value problem 𝑣^′(𝑡)(𝑡)(𝑡),𝑣(0)(𝜆)()+𝐴𝑣=𝑓)(0≤𝑡≤𝑇=𝑣+𝜇0<𝜆≤𝑇 (3) for the differential equation in an arbitrary Banach space 𝐸 with the strongly positive operator 𝐴 was established. The importance of coercive inequalities (well-posedness) is well known [7, 8].

Noted that theory and methods for approximate solutions of local and nonlocal boundary value problems for evolution differential equations have been studied extensively by many researchers (see [9–39] and the references therein).

Before going to establish theorems on the well-posedness of nonlocal boundary value problem for parabolic equations in an arbitrary Banach space with the strongly positive dependent space operators, let us consider the abstract Cauchy problem for the differential equation 𝑣^′(𝑡)+𝐴(𝑡)𝑣(𝑡)=𝑓(𝑡)(0≤𝑡≤𝑇),𝑣(0)=𝑣₀ (4) in an arbitrary Banach space 𝐸 with the strongly positive operators 𝐴(𝑡) in 𝐸 with domain 𝐷(𝐴(𝑡))=𝐷, independent of 𝑡 and dense in 𝐸.

A function 𝑣(𝑡) is called a solution of the problem (4) if the following conditions are satisfied.(i)𝑣(𝑡) is continuously differentiable on the segment??[0,𝑇]. The derivative at the endpoints of the segment is understood as the appropriate unilateral derivatives.(ii)The element 𝑣(𝑡) belongs to 𝐷=𝐷(𝐴(𝑡)) for all 𝑡∈[0,𝑇], and the function 𝐴(𝑡)𝑣(𝑡) is continuous on [0,𝑇].(iii)𝑣(𝑡) satisfies the equation and the initial condition (4).

A solution of problem (4) defined in this manner will from now on be referred to as a solution of problem (4) in the space 𝐶(𝐸)=𝐶([0,𝑇],𝐸). Here, 𝐶(𝐸) stands for the Banach space of all continuous functions 𝜑(𝑡) defined on [0,𝑇] with values in 𝐸 equipped with the norm 𝜑_𝐶(𝐸)=max_{0≤𝑡≤𝑇}𝜑(𝑡)_𝐸. (5) From the existence of the such solutions, it evidently follows that 𝑓(𝑡)∈𝐶(𝐸) and 𝑣₀∈𝐷.

We say that the problem (4) is well posed in 𝐶(𝐸) if the following conditions are satisfied.(1)Problem (4) is uniquely solvable for any 𝑓(𝑡)∈𝐶(𝐸) and any 𝑣₀∈𝐷. This means that an additive and homogeneous operator 𝑣(𝑡)=𝑣(𝑡;𝑓(𝑡),𝑣₀) is defined which acts from 𝐶(𝐸)×𝐷 to 𝐶(𝐸) and gives the solution of problem (4) in 𝐶(𝐸).(2)𝑣(𝑡;𝑓(𝑡),𝑣₀), regarded as an operator from 𝐶(𝐸)×𝐷 to 𝐶(𝐸), is continuous. Here, 𝐶(𝐸)×𝐷 is understood as the normed space of the pairs (𝑓(𝑡),𝑣₀), 𝑓(𝑡)∈𝐶(𝐸), and 𝑣₀∈𝐷, equipped with the norm 𝑓(𝑡),𝑣_{0𝐶(𝐸)×𝐷}=𝑓_𝐶(𝐸)+𝑣_0𝐷. (6)By Banach’s theorem, in 𝐶(𝐸) and these properties, one has coercive inequality 𝑣^′_𝐶(𝐸)+‖𝐴(⋅)𝑣‖_𝐶(𝐸)≤𝑀_𝐶𝑓_𝐶(𝐸)+𝑣_0𝐷, (7) where 𝑀_𝐶(1≤𝑀_𝐶<+∞) does not depend on??𝑣₀and 𝑓(𝑡).

Inequality (7) is called the coercivity inequality in 𝐶(𝐸) for (4). If 𝐴(𝑡)=𝐴, then the coercivity inequality implies the analyticity of the semigroup exp{−𝑠𝐴}(𝑠≥0), that is, the following estimates exp(−𝑠𝐴)_𝐸→𝐸,𝑠𝐴exp(−𝑠𝐴)_𝐸→𝐸≤𝑀(𝑠>0) (8) hold for some 𝑀∈[1,+∞). Thus, the analyticity of the semigroup exp{−𝑠𝐴}(𝑠≥0) is??necessary for the well-posedness of problem (4) in 𝐶(𝐸). Unfortunately, the analyticity of the semigroup exp{−𝑠𝐴}(𝑠≥0) is not sufficient for the well-posedness of problem (4) in 𝐶(𝐸).

Suppose that, for each 𝑡∈[0,𝑇], the operator −𝐴(𝑡) generates an analytic semigroup exp{−𝑠𝐴(𝑡)}(𝑠≥0) with exponentially decreasing norm, when 𝑠→+∞, that is, the following estimates exp(−𝑠𝐴(𝑡))_𝐸→𝐸,𝑠𝐴(𝑡)exp(−𝑠𝐴(𝑡))_𝐸→𝐸≤𝑀𝑒^−𝛿𝑠(𝑠>0) (9) hold for some 𝑀∈[1,+∞),??𝛿∈(0,+∞). From this inequality, it follows that the operator 𝐴⁻¹(𝑡) exists and is bounded, and hence 𝐴(𝑡) is closed in 𝐶(𝐸).

Suppose that the operator 𝐴(𝑡)𝐴⁻¹(𝑠) is Hölder continuous in 𝑡 in the uniform operator topology for each fixed 𝑠, that is, [𝐴(𝑡)−𝐴(𝜏)]𝐴⁻¹(𝑠)_{𝐸𝜇→𝐸𝜇}≤𝑀|𝑡−𝜏|^𝜀,0<𝜀≤1, (10) where 𝑀 and 𝜀 are positive constants independent of 𝑡,𝑠, and 𝜏 for 0≤𝑡,𝑠,𝜏≤𝑇,0≤𝜇≤𝜀.

From (9) and (10), it follows that 𝐸_𝛼(𝐸,𝐴(𝑡))=𝐸_𝛼(𝐸,𝐴(0)) for all 0<𝛼<1??and 0≤𝑡≤𝑇 (see, [14]).

An operator-valued function 𝑣(𝑡,𝑠), defined and strongly continuous jointly in 𝑡 and 𝑠 for 0≤𝑠<𝑡≤𝑇, is called a fundamental solution of (4) if(1)the operator 𝑣(𝑡,𝑠) is strongly continuous in 𝑡 and 𝑠 for 0≤𝑠<𝑡≤𝑇,(2)the following identity holds: 𝑣(𝑡,𝑠)=𝑣(𝑡,𝜏)𝑣(𝜏,𝑠),𝑣(𝑡,𝑡)=𝐼for0≤𝑠≤𝜏≤𝑡≤𝑇, (11)(3)the operator 𝑣(𝑡,𝑠) maps the region 𝐷 into itself; the operator 𝑢(𝑡,𝑠)=𝐴(𝑡)𝑣(𝑡,𝑠)𝐴⁻¹(𝑠) is bounded and strongly continuous in 𝑡 and 𝑠 for 0≤𝑠<𝑡≤𝑇,(4)on the region 𝐷, the operator 𝑣(𝑡,𝑠) is strongly differentiable relative to 𝑡 and 𝑠, while 𝜕𝑣(𝑡,𝑠)(𝑡)𝑣(),()𝜕𝑡=−𝐴𝑡,𝑠𝜕𝑣𝑡,𝑠()𝐴(𝑠).𝜕𝑠=𝑣𝑡,𝑠 (12)𝑣(𝑡,𝑠) is also called development family, evolution operator, Green’s function, and so forth [14].

If the function 𝑓(𝑡) is not only continuous, but also continuously differentiable on [0,𝑇], and 𝑣₀∈𝐷, it is easy to show that the formula 𝑣(𝑡)=𝑣(𝑡,0)𝑣₀+^𝑡₀𝑣(𝑡,𝑠)𝑓(𝑠)𝑑𝑠 (13) gives a solution of problem (4).

Now, we will give lemmas and estimates from [14] concerning the semigroup exp{−𝑠𝐴(𝑡)}(𝑠≥0) and the fundamental solution 𝑣(𝑡,𝑠) of (4) and theorem on well-posedness of (4) which will be useful in the sequel.

Lemma 1. For any 0<𝑠<𝑠+𝜏<𝑇,? 0≤𝑡≤𝑇, and 0≤𝛼≤1, one has the inequality exp(−𝑠𝐴(𝑡))−exp(−(𝑠+𝜏)𝐴(𝑡))_𝐸→𝐸𝜏≤𝑀^𝛼(𝑠+𝜏)^𝛼, (14) where 𝑀 does not depend on 𝛼,𝑡,𝑠, and 𝜏.

Lemma 2. For any 0≤𝑠, ? 𝜏, ?𝑡≤𝑇 and 0≤𝜀≤1, the following estimates hold: exp(−𝑡𝐴(𝜏))−exp(−𝑠𝐴(𝜏))𝐴⁻¹(𝜏)_𝐸→𝐸≤𝑀|𝑡−𝑠|^𝜀𝑒^{−𝛿min{𝑡,𝑠}},𝐴(𝑡)exp(−𝑡𝐴(𝜏))−exp(−𝑠𝐴(𝜏))𝐴⁻²(𝜏)_𝐸→𝐸≤𝑀|𝑡−𝑠|^𝜀𝑒^{−𝛿min{𝑡,𝑠}}, (15) where 𝑀≥0 and 𝛿>0 do not depend on 𝜀,𝑡,𝑠, and 𝜏.

Lemma 3. For any 0≤𝑠<𝑡≤𝑇 and 𝑢∈𝐷, the following identities hold:𝑣(){−()𝐴(𝑠𝑢+𝑡,𝑠𝑢=exp𝑡−𝑠)}_𝑡𝑠𝑣(𝑡,𝑧)[𝐴(𝑠)−𝐴(𝑧)]𝐴⁻¹(𝑠){−()𝐴(𝑠𝐴(𝑠)𝑣(){−()𝐴(𝑡𝑢+×exp𝑧−𝑠)}𝑢𝑑𝑧,𝑡,𝑠𝑢=exp𝑡−𝑠)}_𝑡𝑠{−()𝐴(𝑡×[𝐴(𝑧)(𝑡𝑣()exp𝑡−𝑧)}−𝐴)]𝑧,𝑠𝑢𝑑𝑧. (16)(17)

Lemma 4. For any 0≤𝑠<𝑡≤𝑡+𝑟≤𝑇, the following estimates hold: ‖𝑣(𝑡,𝑠)‖_𝐸→𝐸𝐴(𝑡)𝑣()𝐴≤𝑀,𝑡,𝑠⁻¹(𝑠)_𝐸→𝐸‖‖≤𝑀,𝐴(𝑡)𝑣(𝑡,𝑠)_𝐸→𝐸≤𝑀,𝑡−𝑠 (18)(19)(20) where 𝑀≥0 does not depend on 𝑡 and 𝑠.

A function 𝑣(𝑡) is said to be a solution of problem (4) in 𝐹(𝐸) if it is a solution of this problem in 𝐶(𝐸) and the functions??𝑣^′(𝑡) and 𝐴(𝑡)𝑣(𝑡) belong to 𝐹(𝐸).

As in the case of the space 𝐶(𝐸), we say that the problem (4) is well posed in 𝐹(𝐸) if the following two conditions are satisfied.(1)For any 𝑓∈𝐹(𝐸) and 𝑣₀∈𝐷, there exists the unique solution 𝑣(𝑡)=𝑣(𝑡;𝑓(𝑡),𝑣₀) in 𝐹(𝐸) of problem (4). This means that an additive and homogeneous operator 𝑣(𝑡;𝑓(𝑡),𝑣₀) is defined which acts from 𝐹(𝐸)×𝐷 to 𝐹(𝐸) and gives the solution of (4) in 𝐹(𝐸).(2)𝑣(𝑡;𝑓(𝑡),𝑣₀), regarded as an operator from 𝐹(𝐸)×𝐷 to 𝐹(𝐸), is continuous. Here, 𝐹(𝐸)×𝐷 is understood as the normed space of the pairs (𝑓(𝑡),𝑣₀),𝑓(𝑡)∈𝐹(𝐸), and 𝑣₀∈𝐷, equipped with the norm 𝑓(𝑡),𝑣_{0𝐹(𝐸)×𝐷}=𝑓_𝐹(𝐸)+𝑣_0𝐷. (21)

We set 𝐹(𝐸) equal to 𝐶₀^𝛽,𝛾(𝐸),(0≤𝛾≤𝛽,0<𝛽<1) space, obtained by completion of the set of all smooth 𝐸-valued functions 𝜑(𝑡) on [0,𝑇] with respect to the norm 𝜑_𝐶0^𝛽,𝛾_(𝐸)=max_{0≤𝑡≤𝑇}𝜑(𝑡)_𝐸+sup_{0≤𝑡<𝑡+𝜏≤𝑇}(𝑡+𝜏)^𝛾𝜑(𝑡+𝜏)−𝜑(𝑡)_𝐸𝜏^𝛽. (22) Let us give the following theorems on well-posedness of (4) in 𝐶₀^𝛽,𝛾(𝐸_𝛼−𝛽),0≤𝛾≤𝛽≤𝛼,0<𝛼<1 from [14]. To these, there correspond the spaces of traces 𝐸^𝛽,𝛾_𝛼−𝛽, which consist of elements 𝑤∈𝐸 for which the norm |𝑤|^𝛽,𝛾_𝛼−𝛽=max_{0≤𝑧≤𝑇}𝑒^{−𝑧𝐴(𝑡)}𝑤_{𝐸𝛼−𝛽}+sup_{0≤𝑧<𝑧+𝜏≤𝑇}𝜏^−𝛽(𝑧+𝜏)^𝛾𝑒^{−(𝑧+𝜏)𝐴(𝑡)}−𝑒^{−𝑧𝐴(𝑡)}𝑤_{𝐸𝛼−𝛽} (23) is finite.

Theorem 5. Suppose 𝑣^′₀∈𝐸_𝛼−𝛾,??𝑓(𝑡)∈𝐶₀^𝛽,𝛾(𝐸_𝛼−𝛽)(0≤𝛾≤𝛽≤𝛼,0<𝛼<1). Suppose that the assumptions (9) and (10) hold and 0<𝛼≤𝜀<1. Then, for the solution 𝑣(𝑡) in 𝐶₀^𝛽,𝛾(𝐸_𝛼−𝛽) of the Cauchy problem (4), the coercive inequality 𝑣^′_{𝐶0𝛽,𝛾(𝐸𝛼−𝛽)}+‖𝐴(⋅)𝑣‖_𝐶0^𝛽,𝛾_{(𝐸𝛼−𝛽)}+𝑣^′_{𝐶(𝐸𝛼−𝛾)}≤𝑀𝛼⁻¹(1−𝛼)⁻¹𝑣^′_{0𝐸𝛼−𝛾}+𝑓_𝐶0^𝛽,𝛾_{(𝐸𝛼−𝛽)} (24) holds, where 𝑀 does not depend on 𝛼,𝛽,𝛾,𝑣^′₀, and 𝑓(𝑡).

Theorem 6. Suppose 𝑣^′₀∈𝐸^𝛽,𝛾_𝛼−𝛽,??𝑓(𝑡)∈𝐶₀^𝛽,𝛾(𝐸_𝛼−𝛽)(0≤𝛾≤𝛽≤𝛼,0<𝛼<1). Suppose that the assumptions (9) and (10) hold and 0<𝛼≤𝜀<1. Then, for the solution 𝑣(𝑡) in 𝐶₀^𝛽,𝛾(𝐸_𝛼−𝛽) of the Cauchy problem (4), the coercive inequality 𝑣^′_{𝐶0𝛽,𝛾(𝐸𝛼−𝛽)}+‖𝐴(⋅)𝑣‖_𝐶0^𝛽,𝛾_{(𝐸𝛼−𝛽)}+𝑣^′_{𝐶(𝐸𝛽,𝛾𝛼−𝛽)}≤𝑀𝑣^′₀^𝛽,𝛾_𝛼−𝛽+𝛼⁻¹(1−𝛼)⁻¹𝑓_𝐶0^𝛽,𝛾_{(𝐸𝛼−𝛽)} (25) holds, where 𝑀 does not depend on 𝛼,𝛽,𝛾,𝑣^′₀, and 𝑓(𝑡).

Note that the spaces of smooth functions 𝐶₀^𝛽,𝛾(𝐸_𝛼−𝛽)(0≤𝛾≤𝛽≤𝛼,0<𝛼<1), in which coercive solvability has been established, depend on the parameters 𝛼,𝛽, and 𝛾. However, the constants in the coercive inequalities depend only on 𝛼. Hence, we can choose the parameters 𝛽 and 𝛾 freely, which increases the number of functional spaces in which problem (4)??is well posed. In particular, Theorems 5 and 6 imply theorems on well-posedness of the nonlocal boundary value problem (4) in 𝐶(𝐸_𝛼)(0<𝛼<1).

Finally, in the paper [40], the initial-value problem 𝑑𝑣(𝑡)𝑑𝑡+𝐷_𝑡^1/2𝑣(𝑡)+𝐴(𝑡)𝑣(𝑡)=𝑓(𝑡),0<𝑡<1,𝑣(0)=0 (26) for the fractional parabolic equation in an arbitrary Banach space 𝐸 with the strongly positive operators 𝐴(𝑡) was investigated. Here, 𝐷_𝑡^1/2=𝐷^1/2₀₊ is standard Riemann-Liouville’s derivative of order 1/2. The well-posedness of problem (26) in 𝐶(𝐸_𝛼) spaces was established. New exact estimates in Hölder norms for the solution of initial-boundary value problems for fractional parabolic equations were obtained.

In the paper [41], the nonlocal boundary value problem for the parabolic differential equation 𝑣^′(𝑡)(𝑡)𝑣(𝑡)(𝑡),𝑣(0)(𝜆)+𝐴=𝑓)(0≤𝑡≤𝑇=𝑣+𝜑,0<𝜆≤𝑇 (27) in an arbitrary Banach space 𝐸 with the strongly positive operators 𝐴(𝑡) in 𝐸 with domain 𝐷(𝐴(𝑡))=𝐷, independent of 𝑡 and dense in 𝐸, was investigated. The well-posedness of problem (27) in 𝐶₀^𝛽,𝛾(𝐸) spaces was established. New exact estimates in Hölder norms for the solution of three nonlocal boundary value problems for parabolic equations were obtained.

In the present paper, the well-posedness of problem (27) in 𝐶₀^𝛽,𝛾(𝐸_𝛼−𝛽) spaces is established. New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.

The paper is organized as follows. Section 1 is introduction. In Section 2, new theorems on well-posedness of problem (27) in 𝐶₀^𝛽,𝛾(𝐸_𝛼−𝛽) spaces are established. In Section 3, theorems on the coercive stability estimates for the solution of two nonlocal boundary value parabolic problems are obtained. Finally, Section 4 is conclusion.

2. Well-Posedness of Nonlocal Boundary Value Problem (27)

Now, we will give lemmas on the fundamental solution 𝑣(𝑡,𝑠) of (4) from paper [41].

Lemma 7. Assume that 𝐴(𝑡)𝐴(𝑝)⁻¹=𝐴(𝑡+𝜆)𝐴(𝑝)⁻¹,𝑝∈[0,𝑇] for any 0≤𝑡≤𝑡+𝜆. Then, for any 0≤𝑠<𝑡≤𝑡+𝜆 and 𝑢∈𝐷, the following identity holds 𝑣(𝑡,𝑠)𝑢=𝑣(𝑡+𝜆,𝑠+𝜆)𝑢. (28)

Lemma 8. Under the assumption of Lemma 7 there exists the inverse of the operator 𝐼−𝑣(𝜆,0) in 𝐸 and the following estimate holds: (𝐼−𝑣(𝜆,0))⁻¹_𝐸→𝐸(𝜆),((≤𝑀𝐴(0)𝐼−𝑣𝜆,0))⁻¹𝐴(𝜆)⁻¹_𝐸→𝐸≤𝑀(𝜆). (29)(30)

A function 𝑣(𝑡) is called a solution of the problem (27) if the following conditions are satisfied.(i)𝑣(𝑡) is continuously differentiable on the segment [0,𝑇].(ii)The element 𝑣(𝑡) belongs to 𝐷 for all 𝑡∈[0,𝑇], and the function 𝐴(𝑡)𝑣(𝑡) is continuous on [0,𝑇].(iii)𝑣(𝑡) satisfies the equation and the nonlocal boundary condition (27).

We say that the problem (27) is well posed in 𝐶(𝐸) if the following conditions are satisfied.(1)Problem (27) is uniquely solvable for any 𝑓(𝑡)∈𝐶(𝐸) and any 𝜑∈𝐷. This means that an additive and homogeneous operator 𝑣(𝑡)=𝑣(𝑡;𝑓(𝑡),𝜑) is does not depend on defined which acts from 𝐶(𝐸)×𝐷 to 𝐶(𝐸) and gives the solution of problem (4) in 𝐶(𝐸).(2)𝑣(𝑡;𝑓(𝑡),𝜑), regarded as an operator from 𝐶(𝐸)×𝐷 to 𝐶(𝐸), is continuous. Here, 𝐶(𝐸)×𝐷 is understood as the normed space of the pairs (𝑓(𝑡),𝜑),? 𝑓(𝑡)∈𝐶(𝐸), and 𝜑∈𝐷, equipped with the norm (𝑓(𝑡),𝜑)_{𝐶(𝐸)×𝐷}=𝑓_𝐶(𝐸)+𝜑_𝐷. (31)By Banach’s theorem in 𝐶(𝐸) and these properties, one has coercive inequality 𝑣^′_𝐶(𝐸)+‖𝐴(⋅)𝑣‖_𝐶(𝐸)≤𝑀_𝐶𝑓_𝐶(𝐸)+𝜑_𝐷, (32) where 𝑀_𝐶(1≤𝑀_𝐶<+∞) does not depend on 𝜑??and 𝑓(𝑡).

Inequality (32) is called the coercivity inequality in 𝐶(𝐸) for (27). ?If 𝐴(𝑡)=𝐴, then the coercivity inequality implies the analyticity of the semigroup exp{−𝑠𝐴}(𝑠≥0). Thus, the analyticity of the semigroup exp{−𝑠𝐴}(𝑠≥0) is necessary for the well-posedness of problem (27) in 𝐶(𝐸). Unfortunately, the analyticity of the semigroup exp{−𝑠𝐴}(𝑠≥0) is not sufficient for the well-posedness of problem (27) in 𝐶(𝐸).

A function 𝑣(𝑡) is said to be a solution of problem (27) in 𝐹(𝐸) if it is a solution of this problem in 𝐶(𝐸) and the functions??𝑣^′(𝑡) and 𝐴(𝑡)𝑣(𝑡) belong to 𝐹(𝐸).

As in the case of the space 𝐶(𝐸), we say that the problem (27) is well posed in 𝐹(𝐸) if the following two conditions are satisfied.(1)For any 𝑓∈𝐹(𝐸) and 𝜑∈𝐷, there exists the unique solution 𝑣(𝑡)=𝑣(𝑡;𝑓(𝑡),𝜑) in 𝐹(𝐸) of problem (27). This means that an additive and homogeneous operator 𝑣(𝑡;𝑓(𝑡),𝜑) is defined which acts from 𝐹(𝐸)×𝐷 to 𝐹(𝐸) and gives the solution of (27) in 𝐹(𝐸).(2)𝑣(𝑡;𝑓(𝑡),𝜑), regarded as an operator from 𝐹(𝐸)×𝐷 to 𝐹(𝐸), is continuous. Here, 𝐹(𝐸)×𝐷 is understood as the normed space of the pairs (𝑓(𝑡),𝜑),𝑓(𝑡)∈𝐹(𝐸), and 𝜑∈𝐷, equipped with the norm (𝑓(𝑡),𝜑)_{𝐹(𝐸)×𝐷}=𝑓_𝐹(𝐸)+𝜑_𝐷. (33)If 𝑣(𝑡) is a solution in 𝐶₀^𝛽,𝛾(𝐸) of problem (27), then it is a solution in 𝐶(𝐸) of this problem. Hence, by (13), we get the following representation for the solution of problem (27): 𝑣(𝑡)=𝑣(𝑡,0)𝑣(0)+^𝑡₀𝑣()𝑓(𝑠)𝑣(0)=((𝑡,𝑠𝑑𝑠,𝐼−𝑣𝜆,0))^−1𝜆₀𝑣(𝜆,𝑠)𝑓(𝑠)𝑑𝑠+𝜑. (34) Using (27) and formula (34), we get 𝑣^′₀=𝑣^′(0)(0)𝑣(0)(0)(0(=−𝐴+𝑓=𝐴)(𝐼−𝑣𝜆,0))^−1𝜆₀𝑣(𝑓(𝜆)(𝑠(0(𝜆,𝑠)−𝑓)𝑑𝑠=𝐴)(𝐼−𝑣𝜆,0))^−1𝜆₀𝑣()𝑓(𝜆)(𝑠)(0(𝜆,𝑠−𝑓𝑑𝑠+𝐴)(𝐼−𝑣𝜆,0))⁻¹𝐴⁻¹(0(0)(𝜆)(0(0()−𝐴𝜑−𝑓+𝑓)−𝐴)(𝐼−𝑣𝜆,0))⁻¹×^𝜆₀𝑣(𝜆,𝑠)[𝐴(𝜆)−𝐴(𝑠)]𝐴⁻¹(𝜆)𝑓(𝜆)(0(𝑑𝑠+𝐴)(𝐼−𝑣𝜆,0))⁻¹𝑣()×𝐴𝜆,0₋₁(𝜆)𝑓(𝜆)−𝐴⁻¹(0)𝑓(0)+𝐴(0)(𝐼−𝑣(𝜆,0))⁻¹𝑣(𝜆,0)𝐴⁻¹(𝜆)×(𝐴(𝜆)−𝐴(0))𝐴⁻¹(0)𝑓(𝜆)=𝐾₁+𝐾₂+𝐾₃+𝐾₄, (35) where 𝐾₁=𝐴(0)(𝐼−𝑣(𝜆,0))⁻¹×^𝜆₀𝑣()𝑓(𝜆)(𝑠)𝐾𝜆,𝑠−𝑓𝑑𝑠,₂=−𝐴(0)(𝐼−𝑣(𝜆,0))⁻¹×^𝜆₀𝑣(𝜆,𝑠)[𝐴(𝜆)−𝐴(𝑠)]𝐴⁻¹(𝜆)𝑓(𝜆)𝐾𝑑𝑠,₃=𝐴(0)(𝐼−𝑣(𝜆,0))⁻¹𝐴⁻¹(0)×(0)(𝜆)(0),𝐾−𝐴𝜑−𝑓+𝑓₄=𝐴(0)(𝐼−𝑣(𝜆,0))⁻¹𝑣()×𝐴𝜆,0₋₁(𝜆)𝑓(𝜆)−𝐴⁻¹(0)𝑓(0)+𝐴(0)(𝐼−𝑣(𝜆,0))⁻¹𝑣(𝜆,0)𝐴⁻¹(𝜆)×(𝐴(𝜆)−𝐴(0))𝐴⁻¹(0)𝑓(𝜆). (36) The main result of the present paper is the following theorem on well-posedness of (27) in the spaces 𝐶₀^𝛽,𝛾(𝐸_𝛼−𝛽),0≤𝛾≤𝛽≤𝛼,0<𝛼<1.

Theorem 9. Suppose 𝑓(𝑡)∈𝐶₀^𝛽,𝛾(𝐸_𝛼−𝛽)(0≤𝛾≤𝛽≤𝛼,0<𝛼<1). Suppose that the assumptions (9) and (10) hold and 0<𝛼≤𝜀<1. If 𝐴(0)𝜑+𝑓(𝜆)−𝑓(0)∈𝐸_𝛼−𝛾, then for the solution 𝑣(𝑡) in 𝐶₀^𝛽,𝛾(𝐸_𝛼−𝛽) of the nonlocal boundary value problem (27) the coercive inequality 𝑣^′_{𝐶0𝛽,𝛾(𝐸𝛼−𝛽)}+‖𝐴(⋅)𝑣‖_𝐶0^𝛽,𝛾_{(𝐸𝛼−𝛽)}+𝑣^′_{𝐶(𝐸𝛼−𝛾)}≤𝑀(𝜆)𝛼⁻¹(1−𝛼)⁻¹×𝐴(0)𝜑+𝑓(𝜆)−𝑓(0)_{𝐸𝛼−𝛾}+𝑓_𝐶0^𝛽,𝛾_{(𝐸𝛼−𝛽)} (37) holds, where 𝑀(𝜆) does not depend on 𝛼,𝛽,𝛾,𝜑, and 𝑓(𝑡).

Proof. The proof of Theorem 9 is based on Theorem 5 and the following estimate: 𝑣^′_{0𝐸𝛼−𝛾}≤𝑀(𝜆)𝛼⁻¹(1−𝛼)⁻¹×𝐴(0)𝜑+𝑓(𝜆)−𝑓(0)_{𝐸𝛼−𝛾}+𝑓_𝐶0^𝛽,𝛾_{(𝐸𝛼−𝛽)}. (38) Let us estimate 𝐾_𝑚 for any 𝑚=1,2,3,4 in 𝐸_𝛼−𝛾, separately. We start with 𝐾₁. Applying the inequality (30), we get 𝐾_{1𝐸𝛼−𝛾}≤𝑀(𝜆)𝐴(𝜆)^𝜆₀𝑣(𝜆,𝑠)𝑓(𝜆)−𝑓(𝑠)𝑑𝑠_{𝐸𝛼−𝛾}. (39) Therefore, we will estimate 𝐴(𝜆)^𝜆₀𝑣(𝜆,𝑠)𝑓(𝜆)−𝑓(𝑠)𝑑𝑠_{𝐸𝛼−𝛾}. (40) We have that 𝐴(𝜆)^𝜆₀𝑣()𝑓(𝜆)(𝑠)(𝜆)𝜆,𝑠−𝑓𝑑𝑠=𝐴^𝜆₀{−()𝐴(𝑠𝑓(𝜆)(𝑠)(𝜆)exp𝜆−𝑠)}−𝑓𝑑𝑠+𝐴^𝜆₀𝑣(){−()𝐴(𝜆×𝑓(𝜆)(𝑠)𝜆,𝑠−exp𝜆−𝑠)}−𝑓𝑑𝑠=𝐾₁₁+𝐾₁₂, (41) where 𝐾₁₁=𝐴(𝜆)^𝜆₀{−()𝐴(𝜆𝑓(𝜆)(𝑠)𝐾exp𝜆−𝑠)}−𝑓𝑑𝑠,₁₂=𝐴(𝜆)^𝜆₀𝑣(){−()𝐴(𝜆×𝑓(𝜆)(𝑠)𝜆,𝑠−exp𝜆−𝑠)}−𝑓𝑑𝑠. (42) By the definition of spaces 𝐸_𝛼−𝛽 and using estimates (9), we obtain 𝑧^{1−(𝛼−𝛾)}𝐴(𝜆)exp{−𝑧𝐴(𝜆)}𝐾_11𝐸≤𝑧^{1−(𝛼−𝛾)}𝐴(𝜆){(𝜆×exp−𝑧𝐴)}_𝜆0𝐴(𝜆)exp{−(𝜆−𝑠)𝐴(𝜆)}𝑓(𝜆)−𝑓(𝑠)𝑑𝑠_𝐸≤𝑧^{1−𝛼+𝛾𝜆}₀𝐴²(𝜆)exp{−(𝜆−𝑠+𝑧)𝐴(𝜆)}𝑓(𝜆)−𝑓(𝑠)_𝐸𝑑𝑠≤𝑀𝑧^{1−𝛼+𝛾𝜆}₀(𝜆−𝑠)^𝛽𝑑𝑠(𝑧+𝜆−𝑠)^{2−𝛼+𝛽}𝜆^𝛾𝑓_𝐶0^𝛽,𝛾_{(𝐸𝛼−𝛽)} (43) for any 𝑧>0. We will consider separately the cases 𝑧≤𝜆 and 𝑧>𝜆. If 𝑧≤𝜆, then 𝑧^{1−𝛼+𝛾𝜆}₀(𝜆−𝑠)^𝛽𝑑𝑠(𝑧+𝜆−𝑠)^{2−𝛼+𝛽}𝜆^𝛾≤𝑧^{1−𝛼+𝛾}𝜆^𝛾𝜆₀𝑑𝑠(𝑧+𝜆−𝑠)^2−𝛼≤𝑧^{1−𝛼+𝛾}(1−𝛼)𝜆^𝛾𝑧^1−𝛼≤1𝑧1−𝛼𝜆^𝛾≤1.1−𝛼 (44) If 𝑧>𝜆, then 𝑧^{1−𝛼+𝛾𝜆}₀(𝜆−𝑠)^𝛽𝑑𝑠(𝑧+𝜆−𝑠)^{2−𝛼+𝛽}𝜆^𝛾≤1𝑧^𝛼−𝛾𝜆^𝛾𝜆₀𝑑𝑠(𝑧+𝜆−𝑠)^1−𝛼≤1𝜆^𝛼−𝛾𝜆^𝛾𝜆₀𝑑𝑠(𝜆−𝑠)^1−𝛼=1𝛼. (45) Combining these estimates, we get 𝑧^{1−(𝛼−𝛾)}𝐴(𝜆)exp{−𝑧𝐴(𝜆)}𝐾_11𝐸≤𝑀𝛼()𝑓1−𝛼^{𝐶0𝛽,𝛾(𝐸𝛼−𝛽)} (46) for any 𝑧>0. Therefore, 𝐾_{11𝐸𝛼−𝛾}≤𝑀𝛼()𝑓1−𝛼^{𝐶0𝛽,𝛾(𝐸𝛼−𝛽)}. (47) By the definition of spaces 𝐸_𝛼−𝛽 and using estimates (9), (18), (19), and (28), we obtain𝑧^{1−(𝛼−𝛾)}𝐴(𝜆)exp{−𝑧𝐴(𝜆)}𝐾_12𝐸≤𝑧^{1−(𝛼−𝛾)}𝐴(𝜆)exp{−𝑧𝐴(𝜆)}^𝜆₀𝐴(𝜆)×𝑣(){−()𝐴(𝜆×𝑓(𝜆)(𝑠𝜆,𝑠−exp𝜆−𝑠)}−𝑓)𝑑𝑠_𝐸≤𝑧^{1−𝛼+𝛾𝜆}₀𝐴²(𝜆){(𝜆×𝑣(){−()𝐴(𝜆exp−𝑧𝐴)}𝜆,𝑠−exp𝜆−𝑠)}_𝐸→𝐸×𝑓(𝜆)−𝑓(𝑠)_{𝐸𝛼−𝛽}𝑑𝑠≤𝑀𝑧^{1−𝛼+𝛾𝜆}₀1min𝑧²,1(𝜆−𝑠)²(𝜆−𝑠)^𝜀(𝜆−𝑠)^𝛽𝜆^−𝛾×𝑓𝑑𝑠_{𝐶0𝛽,𝛾(𝐸𝛼−𝛽)}≤𝑀₁𝑧^{1−𝛼+𝛾𝜆}₀(𝜆−𝑠)^𝜀(𝜆−𝑠)^𝛽𝑑𝑠(𝑧+𝜆−𝑠)²𝜆^𝛾𝑓_𝐶0^𝛽,𝛾_{(𝐸𝛼−𝛽)} (48) for any 𝑧>0. We will consider separately the cases 𝑧≤𝜆 and 𝑧>𝜆. If 𝑧≤𝜆, then 𝑧^{1−𝛼+𝛾𝜆}₀(𝜆−𝑠)^𝜀(𝜆−𝑠)^𝛽𝑑𝑠(𝑧+𝜆−𝑠)²𝜆^𝛾≤𝑧^1−𝛼𝜆₀(𝜆−𝑠)^𝛼𝑑𝑠(𝑧+𝜆−𝑠)²𝜆^{𝜀−𝛼+𝛽}≤𝑧^1−𝛼𝜆₀𝑑𝑠(𝑧+𝜆−𝑠)^2−𝛼𝑇^{𝜀−𝛼+𝛽}≤𝑇².1−𝛼 (49) If 𝑧>𝜆, then 𝑧^{1−𝛼+𝛾𝜆}₀(𝜆−𝑠)^𝜀(𝜆−𝑠)^𝛽𝑑𝑠(𝑧+𝜆−𝑠)²𝜆^𝛾≤𝑧^{1−𝛼+𝛾𝜆}₀(𝜆−𝑠)^𝛼𝑑𝑠(𝑧+𝜆−𝑠)²𝜆^𝛾𝜆^{𝜀−𝛼+𝛽}≤1𝑧^𝛼−𝛾𝜆^𝛾𝜆₀𝑑𝑠(𝜆−𝑠)^1−𝛼𝑇^{𝜀−𝛼+𝛽}≤𝜆^𝛼−𝛾𝛼𝑧^𝛼−𝛾𝑇²<𝑇²𝛼. (50) Combining these estimates, we get 𝑧^{1−(𝛼−𝛾)}𝐴(𝜆)exp{−𝑧𝐴(𝜆)}𝐾_12𝐸≤𝑀𝛼()𝑓1−𝛼^{𝐶0𝛽,𝛾(𝐸𝛼−𝛽)} (51) for any 𝑧>0. Therefore, 𝐾_{12𝐸𝛼−𝛾}≤𝑀𝛼()𝑓1−𝛼^{𝐶0𝛽,𝛾(𝐸𝛼−𝛽)}. (52) Applying the triangle inequality and estimates (47) and (52), we obtain 𝐴(𝜆)^𝜆₀𝑣(𝜆,𝑠)𝑓(𝜆)−𝑓(𝑠)𝑑𝑠_{𝐸𝛼−𝛾}≤𝑀𝛼()𝑓1−𝛼^{𝐶0𝛽,𝛾(𝐸𝛼−𝛽)}. (53) From this estimate and estimate (39), it follows the following estimate 𝐾_{1𝐸𝛼−𝛾}≤𝑀𝛼()𝑓1−𝛼^{𝐶0𝛽,𝛾(𝐸𝛼−𝛽)}. (54) Now, we estimate 𝐾₂. Applying inequality (30), we get 𝐾_{2𝐸𝛼−𝛾}≤𝑀(𝜆)𝐴(𝜆)^𝜆₀𝑣()×[𝐴(𝜆)(𝑠𝐴𝜆,𝑠−𝐴)]⁻¹(𝜆)𝑓(𝜆)𝑑𝑠_{𝐸𝛼−𝛾}. (55) Using estimates (9), (18), (19), and (28), we obtain 𝑧^{1−(𝛼−𝛾)}𝐴(𝜆)exp{−𝑧𝐴(𝜆)}^𝜆₀𝐴(𝜆)𝑣()×[𝐴(𝜆)(𝑠𝐴𝜆,𝑠−𝐴)]⁻¹(𝜆)𝑓(𝜆)𝑑𝑠_𝐸≤𝑧^{1−𝛼+𝛾𝜆}₀𝐴²(𝜆)exp{−𝑧𝐴(𝜆)}𝑣(𝜆,𝑠)_𝐸→𝐸×[𝐴(𝜆)−𝐴(𝑠)]𝐴⁻¹(𝜆)_𝐸→𝐸𝑓(𝜆)_𝐸𝑑𝑠≤𝑧^{1−𝛼+𝛾𝜆}₀1min𝑧²,1(𝜆−𝑠)²(𝜆−𝑠)^𝜀𝑑𝑠𝑓_𝐶(𝐸)≤𝑀₁𝑧^{1−𝛼+𝛾𝜆}₀(𝜆−𝑠)^𝜀𝑑𝑠(𝑧+𝜆−𝑠)²𝑓_𝐶0^𝛽,𝛾_{(𝐸𝛼−𝛽)} (56) for all 𝑧>0. We will prove that 𝑧^{1−𝛼+𝛾𝜆}₀(𝜆−𝑠)^𝜀𝑑𝑠(𝑧+𝜆−𝑠)²≤𝑀𝛼(1−𝛼) (57) for any 𝑧>0. If 𝑧≤𝜆, then 𝑧^{1−𝛼+𝛾𝜆}₀(𝜆−𝑠)^𝜀𝑑𝑠(𝑧+𝜆−𝑠)²≤𝑧^1−𝛼𝜆^{𝛾+𝜀−𝛼𝜆}₀𝑑𝑠(𝑧+𝜆−𝑠)^2−𝛼≤𝜆^{𝛾+𝜀−𝛼}≤𝑇1−𝛼².1−𝛼 (58) If 𝜆<𝑧, then 𝑧^{1−𝛼+𝛾𝜆}₀(𝜆−𝑠)^𝜀𝑑𝑠(𝑧+𝜆−𝑠)²≤1𝑧^{𝛼−𝛾𝜆}₀𝑑𝑠(𝜆−𝑠)^1−𝜀=𝜆^𝜀𝜀𝑧^𝛼−𝛾<𝜆^𝜀𝜀𝜆^𝛼−𝛾≤𝑇²𝛼. (59) Combining these estimates, we get (57). Applying estimates (55), (56), and (57), we obtain 𝐾_{2𝐸𝛼−𝛾}≤𝑀𝛼()𝑓1−𝛼^{𝐶0𝛽,𝛾(𝐸𝛼−𝛽)}. (60) Applying inequality (30), we get 𝐾_{3𝐸𝛼−𝛾}≤𝑀(𝜆)𝐴(0)𝜑+𝑓(𝜆)−𝑓(0)_{𝐸𝛼−𝛾}. (61) Finally, we estimate 𝐾₄. Applying inequality (30), we get 𝐾_{4𝐸𝛼−𝛾}≤𝑀₁(𝜆)𝐴(𝜆)𝑣(𝜆,0)𝐴⁻¹(𝜆)𝑓(𝜆)−𝐴⁻¹(0)𝑓(0)+𝐴⁻¹(𝜆)[𝐴(𝜆)−𝐴(0)]𝐴⁻¹(0)𝑓(𝜆)_{𝐸𝛼−𝛾}. (62) Using estimates (9), (18), (19), and (28), we obtain 𝑧^{1−(𝛼−𝛾)}𝐴²(𝜆){(𝜆𝑣()×𝐴exp−𝑧𝐴)}𝜆,0₋₁(𝜆)𝑓(𝜆)−𝐴⁻¹(0)𝑓(0)+𝐴⁻¹(𝜆)(𝐴(𝜆)−𝐴(0))𝐴⁻¹(0)𝑓(𝜆)_𝐸≤𝑧^{1−𝛼+𝛾}𝐴²(𝜆)exp{−𝑧𝐴(𝜆)}𝑣(𝜆,0)𝐴⁻¹(𝜆)_𝐸→𝐸×𝑓(𝜆)_𝐸+𝐴(𝜆)𝐴⁻¹(0)_𝐸→𝐸𝑓(0)_𝐸+[𝐴(𝜆)−𝐴(0)]𝐴⁻¹(0)_𝐸→𝐸𝑓(𝜆)_𝐸≤𝑀𝑧^{1−𝛼+𝛾}1min𝑧,1𝜆𝑓_𝐶0^𝛽,𝛾_(𝐸)≤𝑀₁𝑧^{1−𝛼+𝛾}𝑓𝑧+𝜆^{𝐶0𝛽,𝛾(𝐸)}≤𝑀₂𝜆^−𝛼+𝛾𝑓_𝐶0^𝛽,𝛾_{(𝐸𝛼−𝛽)} (63) for all 𝑧>0. Applying estimates (63) and (62), we get 𝐾_{4𝐸𝛼−𝛾}≤𝑀₂(𝜆)𝑓_𝐶0^𝛽,𝛾_{(𝐸𝛼−𝛽)}. (64) Combining the estimates (54), (60), (61), and (64), we get estimate (38). Theorem 9 is proved.

Furthermore, the method of proof of Theorem 6 and scheme of proof of Theorem 9 enable us to establish the following theorem on well-posedness of (27) in spaces 𝐶₀^𝛽,𝛾(𝐸_𝛼−𝛽),0≤𝛾≤𝛽≤𝛼,0<𝛼<1.

Theorem 10. Suppose 𝑓(𝑡)∈𝐶₀^𝛽,𝛾(𝐸_𝛼−𝛽)(0≤𝛾≤𝛽≤𝛼,0<𝛼<1). Suppose that the assumptions (9) and (10) hold and 0<𝛼≤𝜀<1. If 𝐴(0)𝜑+𝑓(𝜆)−𝑓(0)∈𝐸^𝛽,𝛾_𝛼−𝛽, then for the solution 𝑣(𝑡) in 𝐶₀^𝛽,𝛾(𝐸_𝛼−𝛽) of the nonlocal boundary value problem (27) the coercive inequality 𝑣^′_{𝐶0𝛽,𝛾(𝐸𝛼−𝛽)}+‖𝐴(⋅)𝑣‖_𝐶0^𝛽,𝛾_{(𝐸𝛼−𝛽)}+𝑣^′_{𝐶(𝐸𝛽,𝛾𝛼−𝛽)}≤𝑀(𝜆)𝐴(0)𝜑+𝑓(𝜆)−𝑓(0)^𝛽,𝛾_𝛼−𝛽+𝛼⁻¹(1−𝛼)⁻¹𝑓_𝐶0^𝛽,𝛾_{(𝐸𝛼−𝛽)} (65) holds, where 𝑀(𝜆) does not depend on 𝛼,𝛽,𝛾,𝜑, and 𝑓(𝑡).

Note that the spaces of smooth functions 𝐶₀^𝛽,𝛾(𝐸_𝛼−𝛽)(0≤𝛾≤𝛽≤𝛼,0<𝛼<1), in which coercive solvability has been established, depend on the parameters 𝛼,𝛽, and 𝛾. However, the constants in the coercive inequalities depend only on 𝛼. Hence, we can choose the parameters 𝛽 and 𝛾 freely, which increases the number of functional spaces in which problem (27) is well posed. In particular, Theorems 9 and 10 imply theorems on well-posedness of the nonlocal boundary value problem (27) in 𝐶₀^𝛽,𝛾(𝐸)(0≤𝛾≤𝛽,0<𝛽<1) which is established in the paper [41].

3. Applications

First, we consider the nonlocal boundary value problem for the parabolic equation 𝜕𝑢()𝜕𝜕𝑡−𝑎𝑡,𝑥²𝑢𝜕𝑥²(),𝑢()()(𝑥)𝑢()()+𝛿𝑢=𝑓𝑡,𝑥0<𝑡<𝑇,0<𝑥<1,0,𝑥=𝑢𝜆,𝑥+𝜑,0≤𝑥≤1,𝑡,0=𝑢𝑡,1,𝑢_𝑥(𝑡,0)=𝑢_𝑥(𝑡,1),0≤𝑡≤𝑇, (66) where 𝜑(𝑥)(𝑥∈[0,1]),??𝑎(𝑡,𝑥), and 𝑓(𝑡,𝑥)??(𝑡∈(0,𝑇),𝑥∈(0,1)) are given sufficiently smooth functions and 𝑎(𝑡,𝑥)=𝑎(𝑡+𝜆,𝑥)≥𝑎>0 and they satisfy every compatibility conditions which guarantee that the problem (66) has a smooth solution 𝑢(𝑡,𝑥). Here,??𝛿>0 is a sufficiently large number.

We introduce the Banach spaces 𝐶^𝛽[0,1](0<𝛽<1) of all continuous functions 𝜑(𝑥) satisfying a Hölder condition for which the following norms are finite: 𝜑_{𝐶𝛽[0,1]}=𝜑_𝐶[0,1]+sup_{0≤𝑥<𝑥+𝜏≤1}𝜑(𝑥+𝜏)−𝜑(𝑥)𝜏^𝛽, (67) where 𝐶[0,1] is the space of the all continuous functions 𝜑(𝑥) defined on [0,1] with the usual norm 𝜑_𝐶[0,1]=max_0≤𝑥≤1𝜑(𝑥). (68) It is known that the differential expression 𝐴^𝑡,𝑥𝑣(𝑡,𝑥)=−𝑎(𝑡,𝑥)𝑣_𝑥𝑥(𝑡,𝑥)+𝛿𝑣(𝑡,𝑥) (69) defines a strongly positive operator 𝐴^𝑡,𝑥 acting in 𝐶^𝛽[0,1] with the domain 𝐶^𝛽+2[0,1] and satisfying nonlocal boundary conditions 𝑣(𝑡,0)=𝑣(𝑡,1),𝑣_𝑥(𝑡,0)=𝑣_𝑥(𝑡,1) for any 0≤𝑡≤𝑇.

Therefore, we can replace the nonlocal boundary value problem (66) by the abstract nonlocal boundary value problem (27) in a Banach space 𝐸=𝐶[0,1] with a strongly positive operator 𝐴^𝑡,𝑥 defined by formula (69).

Theorem 11. For the solution of nonlocal boundary value problem (66), the following coercive inequality: ‖𝑢‖_𝐶0^1+𝛽,𝛾_{(𝐶2(𝛼−𝛽)+𝜇[0,1])}+‖𝑢‖_𝐶0^𝛽,𝛾_{(𝐶2(𝛼−𝛽)+𝜇[0,1])}+‖𝑢‖_𝐶(𝐶^{2(𝛼−𝛾)+𝜇}_[0,1])≤𝑀𝜆,𝜇,𝛼𝑓_𝐶0^𝛽,𝛾_{(𝐶2(𝛼−𝛽)+𝜇[0,1])}+()𝜕−𝑎0,⋅²𝜑(⋅)𝜕𝑥²(⋅)(+𝛿𝜑+𝑓(𝜆,⋅)−𝑓0,⋅)_{𝐶2(𝛼−𝛾)+𝜇[0,1]},0<2𝛼−𝛾+𝜇<1,0≤𝛾≤𝛽,0≤𝜇≤1,0<𝛼<1 (70) holds, where 𝑀(𝜆,𝜇,𝛼) does not depend on 𝛾, 𝛽,??𝑓(𝑡,𝑥),??𝜑(𝑥).

The proof of Theorem 11 is based on abstract Theorem 9 and on the following theorem on the structure of fractional spaces 𝐸_𝛼(𝐶^𝜇[0,1],𝐴^𝑡,𝑥).

Theorem 12. 𝐸_𝛼(𝐶^𝜇[0,1],𝐴^𝑡,𝑥)=𝐶^2𝛼+𝜇[0,1] for all 0<2𝛼+𝜇<1,0≤𝜇≤1 and 0≤𝑥≤1,0≤𝑡≤𝑇 [42].

Second, on the range {0≤𝑡≤𝑇,𝑥∈ℝ^𝑛}, we consider the nonlocal boundary value problem for the 2𝑚th order multidimensional parabolic equation 𝜕𝑢𝜕𝑡+_|𝑟|=2𝑚𝑎_𝑟()𝜕𝑡,𝑥^|𝜏|𝑢𝜕𝑥^𝑟1₁,…,𝜕𝑥^𝑟𝑛_𝑛+𝛿𝑢(𝑡,𝑥)=𝑓(𝑡,𝑥),0<𝑡<𝑇,𝑥,𝑟∈ℝ^𝑛,|𝑟|=𝑟₁+⋯+𝑟_𝑛,𝑢(0,𝑥)=𝑢(𝜆,𝑥)+𝜑(𝑥),𝑥∈ℝ^𝑛, (71) where 𝑎_𝑟(𝑡,𝑥)=𝑎_𝑟(𝑡+𝜆,𝑥)??and 𝜑(𝑥)(𝑥∈ℝ^𝑛),?𝑎_𝑟(𝑡,𝑥), and 𝑓(𝑡,𝑥)??(𝑡∈(0,𝑇),𝑥∈ℝ^𝑛) are given sufficiently smooth functions. Here,??𝛿>0 is a sufficiently large number.

Let us consider a differential operator with constant coefficients of the form 𝐵=_|𝑟|=2𝑚𝑏_𝑟𝜕^{𝑟1+⋯+𝑟𝑛}𝜕𝑥^𝑟1₁,…,𝜕𝑥^𝑟𝑛_𝑛, (72) acting on functions defined on the entire space??𝑅^𝑛. Here, 𝑟∈𝑅^𝑛 is a vector with nonnegative integer components,??|𝑟|=𝑟₁+⋯+𝑟_𝑛. If 𝜑(𝑦)(𝑦=(𝑦₁,…,𝑦_𝑛)∈𝑅^𝑛) is an infinitely differentiable function that decays at infinity together with all its derivatives, then, by means of the Fourier transformation one establishes the equality𝐹𝐵_𝜑(𝜉)=𝐵(𝜉)𝐹𝜑(𝜉). (73) Here, the Fourier transform operator is defined by the rule 𝐹𝜑(𝜉)=(2𝜋)^−𝑛/2_𝑅𝑛exp−𝑖𝑦,𝜉𝜑𝑦𝑑𝑦,𝑦,𝜉=𝑦₁𝜉₁+⋯+𝑦_𝑛𝜉_𝑛. (74) The function 𝐵(𝜉) is called the symbol of the operator 𝐵 and is given by𝐵(𝜉)=_|𝑟|=2𝑚𝑏_𝑟𝑖𝜉₁^𝑟1,…,𝑖𝜉_𝑛^𝑟𝑛. (75) We will assume that the symbol 𝐵^𝑡,𝑥(𝜉)=_|𝑟|=2𝑚𝑎_𝑟(𝑡,𝑥)𝑖𝜉₁^𝑟1,…,𝑖𝜉_𝑛^𝑟𝑛,𝜉=𝜉₁,…,𝜉_𝑛∈𝑅^𝑛 (76) of the differential operator of the form 𝐵^𝑡,𝑥=_|𝑟|=2𝑚𝑎_𝑟()𝜕𝑡,𝑥^|𝑟|𝜕𝑥^𝑟1₁,…,𝜕𝑥^𝑟𝑛_𝑛, (77) acting on functions defined on the space ℝ^𝑛, satisfies the inequalities 0<𝑀₁𝜉^2𝑚≤(−1)^𝑚𝐵^𝑡,𝑥(𝜉)≤𝑀₂𝜉^2𝑚<∞ (78) for 𝜉≠0 and for any 0≤𝑡≤𝑇. Assume that the all compatibility conditions hold which guarantees that the problem (71) has a unique smooth solution 𝑢(𝑡,𝑥). This allows us to reduce the nonlocal boundary value problem (71) by the abstract nonlocal boundary value problem (27) in a Banach space 𝐸=𝐶^𝜇(𝑅^𝑛) of all continuous bounded functions defined on ℝ^𝑛 satisfying a Hölder condition with the indicator 𝜇∈(0,1) with a strongly positive operator 𝐴^𝑡,𝑥=𝐵^𝑡,𝑥+𝛿𝐼 defined by (77).

Theorem 13. For the solution of the nonlocal boundary value problem (71), the following coercivity inequality: ‖𝑢‖_𝐶0^1+𝛽,𝛾_{(𝐶2𝑚(𝛼−𝛽)+𝜇(𝑅𝑛))}+_|𝜏|=2𝑚𝜕^|𝑟|𝑢𝜕𝑥^𝑟1₁,…,𝜕𝑥^𝑟𝑛_𝑛𝐶0^𝛽,𝛾_{(𝐶2𝑚(𝛼−𝛽)+𝜇(𝑅𝑛))}+‖𝑢‖_𝐶(𝐶^{2𝑚(𝛼−𝛾)+𝜇}_(𝑅^𝑛₎₎≤𝑀𝜆,𝜇,𝛼𝑓_𝐶0^𝛽,𝛾_{(𝐶2𝑚(𝛼−𝛽)+𝜇(𝑅𝑛))}+_|𝑟|=2𝑚𝑎_𝑟()𝜕0,⋅^|𝜏|𝜑(⋅)𝜕𝑥^𝑟1₁,…,𝜕𝑥^𝑟𝑛_𝑛(⋅)()()+𝛿𝜑+𝑓𝜆,⋅−𝑓0,⋅_{𝐶2𝑚(𝛼−𝛾)+𝜇(𝑅𝑛)},0<2𝛼−𝛾+𝜇<1,0≤𝛾≤𝛽,0≤𝜇≤1 (79) holds, where 𝑀(𝜆,𝜇,𝛼) does not depend on??𝛾,??𝛽,??𝑓(𝑡,𝑥), ??𝜑(𝑥).

The proof of Theorem 13 is based on abstract Theorem 9, on the coercivity inequality for an elliptic operator 𝐴^𝑡,𝑥 in 𝐶^𝜇(𝑅^𝑛), and on the following theorem on the structure of fractional spaces 𝐸_𝛼(𝐶^𝜇(𝑅^𝑛),𝐴^𝑡,𝑥).

Theorem 14. 𝐸_𝛼(𝐶^𝜇(𝑅^𝑛),𝐴^𝑡,𝑥)=𝐶^{2𝑚𝛼+𝜇}(𝑅^𝑛) for all 0<2𝑚𝛼+𝜇<1,0≤𝜇≤1??and 𝑥∈ℝ^𝑛, ?0≤𝑡≤𝑇 [7].

4. Conclusion

In the present study, the well-posedness of the nonlocal boundary value parabolic problem with dependent coefficients in Hölder spaces with a weight is established. In practice, new Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are obtained. Moreover, applying the result of the monograph [14], the high order of accuracy single-step difference schemes for the numerical solution of the nonlocal boundary value parabolic problem with dependent coefficients can be presented. Of course, the coercive stability estimates for the solution of these difference schemes have been established without any assumptions about the grid steps.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

The authors would like to thank Professor Pavel E. Sobolevskii, for his helpful suggestions. They??would like also to thank the referees for their efforts to improve the paper.