Solvability and Volterra property of nonlocal problems for mixed fractional-order diffusion-wave equation

The paper is devoted to the study of one class of problems with nonlocal conditions for a mixed diffusion-wave equation with two independent variables. The main results of the work are the proof of regular and strong solvability, as well as the Volterra property of three problems with conditions pointwise connecting the values of the tangent derivative of the desired solution on one of the characteristics with derivatives in various directions of the solution on an arbitrary curve lying inside the characteristic triangle for a fractional-order diffusion-hyperbolic equation.


Introduction
In recent years, there has been an increased interest in the study of fractional differential equations, in which an unknown function is contained under the sign of a fractional derivative. This is due to the development of fractional integration theory and differentiation itself, as well as applications in various fields of science: physics, mechanics, chemistry, engineering, anomalous diffusion processes, and other areas of natural science.
Since the fractional-order equations generalize the integer-order equations, and there are a relatively small number of systematized analytical and numerical methods for such equations, this direction is the priority of the general theory of differential equations.
As far as we know, the spectral properties, including the Volterra property of the mixed fractional equations, are almost not studied.
Note that the solvability issues and spectral properties of local and nonlocal problems for a mixed parabolic-hyperbolic equation of the second and third orders are studied in [5-8, 12, 13, 19].
This work is devoted to one of the most important problems, the study of the solvability and spectral properties (Volterra property) of three nonlocal problems for the diffusionhyperbolic equation (of fractional order).
In the case α = 1, problem M 1 B coincides with a nonlocal problem for a mixed parabolic-hyperbolic equation with noncharacteristic line of changing type. In this case the issues of regular and strong solvability, as well as the Volterra property of problem M 1 B, are investigated in [7,8].
In the domain 0 we consider the following auxiliary problem.
Then for any function f (x, y) ∈ C 1 (¯ 0 ), the solution of problem C 1 admits the a priori estimate Hereafter symbol C will denote a positive constant that does not depend on z(x, y), not necessarily the same.
Proof of Lemma 2.1 Multiplying equation (1.1) for y > 0 by z(x, y), integrating from 0 to 1 over y, and taking into account conditions (2.1) and (2.2), after some transformations, we have (2.9) Integrating (2.9) over t from 0 to x and taking into account and τ (0) = 0, we have In the latter, on the right side, applying the known inequalities, we obtain (2.10) In the left part of (2.10), omitting the first two terms and applying the Gronwall-Bellman inequality, we have Taking into account the last term of (2.10), we get Similarly, omitting the second term of the left part in (2.11) and applying Lemma 2 in [3], we have from which, taking into account Due to the conditions imposed on the function γ (x), equation of the curve AD in characteristic variables ξ , η allows the representation (2.14) In (2.13) satisfying condition (2.3), after some simple transformations, we have The ratio (2.15) is the main functional relationship between τ (x) and ν(x) brought to the segment AB from the hyperbolic domain 1 .
Substituting the obtained expression of ν(x) into (2.13) and taking into account (2.16), after some transformations, we get the following presentation of the solution z(ξ , η) in the domain 1 : Taking into account (2.14) and (2.16), after some calculations, it is not difficult to establish the following estimate: Now in (2.5), assuming that δ(x) = (x) and taking into account (2.18), it is not difficult to verify the validity of the following lemma.
Then for any function f (x, y) ∈ C 1 (¯ ), f (0, 0) = 0, the solution to problem M 1 B admits the a priori estimate Lemma 2.2 implies the following estimate: where L 2 ( ) is the space of square-summable functions in . Consider the following auxiliary problem C 2 : In the domain 0 , find a solution of equation ( [25]. Differentiating (2.21) over y, we have Using the known formulas [19,25] d n dt t μ-1 e μ,δ α,β ct α = t μ-n-1 e μ-n,δ after some calculations, from (2.22) it is not difficult to establish that Further, from (2.24), taking into account τ (0) = 0 and applying formulas [25] -βte (2.25) Now taking into account (2.25), from (2.23), as y → 0, we have Then for functions E(x, y, y 1 ) and E y (x, y, y 1 ), we have the following estimates: The proof is carried out using the inequality Thus by (2.27) problem M 1 B is equivalently (in the sense of unambiguous solvability) reduced to a Volterra-type integral equation of the second kind with weak singularity (2.28). Therefore by (2.32) there is a unique solution of equation (2.28) from the class C 1 [0, 1], representable as where R(x) is the resolvent of the integral equation (2.28), From (2.33), taking into account τ (0) = 0, we have Similarly, substituting (2.34) into (2.17) and taking into account (2.16) and (2.29), after some calculations, we get From (2.35) and (2.36) we have Taking into account explicit types of functions it is not difficult to establish that in (2.38) all terms are bounded, with the exception of the first, M 01 (x, y, x 1 , y 1 ), in which by Lemma 2.3 the summand E(xx 1 , y, y 1 ) may be not limited. Therefore it is sufficient to show that

By Lemma 2.3 from estimate (2.30) by direct calculation we have
From (2.17) by (2.40) and direct calculation it is not difficult to establish that (2.42) We call a function z(x, y) ∈ V a regular solution of problem M 1 B in the domain , where Thus, summarizing the above statements, we have proved the following theorem.
The function z(x, y) ∈ L 2 ( ) is called a strong solution to problem M 1 B if there exists the sequence of functions {z n (x, y)}, z n (x, y) ∈ V , satisfying conditions (2.1)-(2.3) such that Proof of Theorem 2.2 Now let us show that for f (x, y) ∈ L 2 ( ), the solution to problem M 1 B is strong. Due to the density in L 2 ( ), For Then equation (2.28) can be considered as an integral equation of the second kind in the space . Therefore the function z n (x, y) defined by formulas (2.13) and (2.21) (here it is necessary to replace the functions τ (x), ν(x), f (x, y) by τ n (x), ν n (x), f n (x, y), respectively) belongs to class V . However, on the other hand, by Lemma 2.4, Q(x) ∈ L 2 (0, 1) when f (x, y) ∈ L 2 ( ). Therefore equation (2.28) can be considered as a Volterra integral equation of the second kind in the space L 2 (0, 1). Equation (2.28) in the space L 2 (0, 1) is unambiguously solvable, τ (x) ∈ L 2 (0, 1), and τ (x) L 2 (0,1) ≤ C Q(x) L 2 (0,1) . As before, by (2.15) we have ν(x) ∈ L 2 (0, 1).
In this case the function z(x, y) defined by formulas (2.13) and (2.21) at least belongs to class C(¯ ) ∩ W 0,1 2 ( 0 ) ∩ W 1,1 2 ( 1 ). By (2.39) it is also not difficult to verify estimate (2.43). Now, due to the completeness of the space L 2 ( ), the sequence {f n (x, y)} we constructed above is fundamental. From the linearity of equation (1.1) and estimate (2.43) we obtain that z n (x, y)z m (x, y) L 2 ( ) ≤ C f n (x, y)f m (x, y) L 2 ( ) , i.e., the sequence {z n (x, y)} is fundamental in L 2 ( ). Taking into account the completeness of the space L 2 ( ), we get that there exists a limit z(x, y) ∈ L 2 ( ) of the sequence z n (x, y), which will be the desired strong solution to problem M 1 B with the right-hand part f (x, y) ∈ L 2 ( ).
Analyzing the above facts, it is also not difficult to establish that a strong solution z(x, y) to problem M 1 B is representable as (2.37). Theorem 2.2 is proved. Now let us establish the Volterra property of problem M 1 B. By B 1 we denote the closure in space L 2 ( ) of the fractional differential operator satisfying conditions (2. From Theorem 2.2 it follows that the operator B 1 is closed and its domain is dense in L 2 ( ); the inverse operator B -1 1 exists, is defined on the whole L 2 ( ), and is completely continuous. In this regard, a natural question arises: is there an eigenvalue of the operator B -1 1 and hence of problem M 1 B? The main result is the theorem on the absence of eigenvalues of the operator B -1 1 .

Lemma 2.5
For the iterated kernels M n (x, y, x 1 , y 1 ), we have the following estimate: Proof of Lemma 2.5 We use mathematical induction over n. For n = 1, the inequality follows from representation (2.38) taking into account (2.30).
Let (2.47) be valid for n = k -1. Let us prove the validity of this formula for n = k. Using inequality (2.47) for n = 1 and n = k -1, we have which proves the lemma.
Using the well-known Schwarz inequality and Lemma 2.5, from representation (2.46) we have from which we obtain From the latter it is not difficult to establish equality (2.45). Theorem 2.3 is proved.

A problem for a diffusion-hyperbolic equation with a nonlocal condition with derivatives in different characteristic directions
This section is devoted to the study of a nonlocal problem with derivatives in different characteristic directions for equation (1.1).
The main goal is to show that for the correctness and Volterra property of problem M 2 B considered in this section, in contrast to problem M 1 B, it is essential to consider the ratio between the coefficient of "compression" μ(0) at the origin of the derivative in the direction of the characteristic BC and the polar angle ω formed by the curve AD and the abscissa axis.
Problem M 2 B Find a solution to equation (1.1) satisfying the conditions (2.1), (2.2), and is the equation of the curve AD in characteristic coordinates ξ = x + y, η = xy, and μ(t) is a given function.
Then for any function f (x, y) ∈ C 1 (¯ ), f (A) = 0, there is a unique regular solution to problem M 2 B, which satisfies inequality (2.43) and can be represented in the form where M 2 (x, y, x 1 , y 1 ) ∈ L 2 ( × ).
Proof As before, denoting z( where

Relation (3.4) is the main relation between τ (x) and ν(x)
, brought to the segment AB from the hyperbolic part 1 .
The main functional relation between τ (x) and ν(x), brought to the segment AB from the parabolic part of the domain, has the form (2.26). Now, excluding the function ν(x) from relations (2.26) and (3.4), for τ (x), we obtain the integro-differential equation Thus, problem M 2 B in the sense of unique solvability is equivalently reduced to integrofunctional equation (3.6). Note that similar integro-functional equations have been studied in [7,8].
Consider the equation First, we present the following lemma, which will be needed later.

Lemma 3.1 ([7]) Let
Then for any function F 2 (x) ∈ L 2 (0, 1), there is a unique solution ϕ(x) ∈ L 2 (0, 1) to equation (3.9), and it satisfies inequality Proof The proof of the lemma is given in [7]. For ease of reading, we briefly outline it here. Let us consider the operator acting by the formula Taking in to account (3.12), equation (3.9) can be presented in the form where E is the identity operator.
Proof Consider equation (3.9) in the class C 1 [0, 1]. It is obvious that if F 2 (x) ∈ C 1 [0, 1] and F 2 (0) = 0, then ϕ(0) = 0. Therefore differentiating (3.9), for ϕ (x), we obtain the equation where Since we consider equation (3.9) in a narrower class than L 2 (0, 1), the solution of equation (3.9) and therefore the solution of (3.14) is unique. Also, it is obvious that T 1 is completely continuous in C[0, 1]. Therefore solvability of equation (3.14) in C[0, 1] is equivalent to the existence of the operator It is easy to see that the operator A -1 2 exists, is bounded in C[0, 1], and A -1 Indeed, similarly to Lemma 3.1, we have Taking into account that λ n (1) → 0 as n → ∞, we get Therefore by (3.13) the number series ∞ n=1 b n converges, and which shows the continuity of operator A -1 2 in C[0, 1]. Lemma 3.2 is proved.
Proof of Lemma 3.3 Using the explicit form of the function E(x, y, y 1 ), by Lemma 2.3 it is not difficult to establish that the function Q 0 (x) defined by formula (3.8) belongs to class C 1 [0, 1] and Q 0 (0) = 0. From (3.5), taking into account the conditions imposed on the function μ(t), it is easy to establish that F 1 (x) ∈ C 1 [0, 1] and F 1 (0) = 0. Hence the proof of Lemma 3.3 follows by (3.7).

Lemma 3.5 Let condition (3.10) be fulfilled. Then for any function F(x)
∈ L 2 ( ), there is a unique solution to equation (3.6). This solution belongs to the class L 2 (0, 1) and satisfies the inequality Proof We introduce the integral operator T acting in L 2 (0, 1) according to the formula Since x β m(x) is a continuous function, it is obvious that T is a completely continuous operator in L 2 (0, 1).
Taking into account (3.12) and (3.16), from (3.6), passing to the operator record, we obtain (3.17) By Lemma 3.1 the operator (E + A) -1 is bounded. Applying the operator (E + A) -1 to (3.17), we have For n = 1, it follows from Lemma 3.1 that (3.20) where k = ∞ n-1 a n < ∞. By direct calculation we can prove the following estimates: We claim that The proof follows from estimates (3.20)-(3.22) and from equations (3.23) and (3.24). The latter implies the convergence in L 2 (0, 1) of the series (3.25) which is majorized in L 2 (0, 1) by the convergent numerical series It is not difficult to make sure that the constructed function τ (x) satisfies equation (3.18). In fact, summing up the recurrent relations (3.23) and (3.24) over n from 1 to k, we obtain Passing to the limit as k → ∞ in this equality, taking advantage of the limitations of operators A and T, due to the convergence of the series (3.25), we obtain equation (3.18). Now let us show the uniqueness of the solution to equation (3.18). For this, as is known, it is sufficient to show that the corresponding homogeneous equation (3.18) has only a zero solution. Letτ (x) ∈ L 2 (0.1) be a solution to homogeneous equation (3.18): (3.26) We apply to (3.26) the method of successive approximations, takingτ 0 (x) =τ (x) and Since the functionτ (x) is a solution of equation (3.26), then by the unambiguous solvability of equation (3.9) in L 2 (0, 1) every next approximation will coincide with itτ n (x) = τ (x) . . . . Reasoning similarly, i.e., as in the derivation of the inequality for ψ n (x), we get Taking into account thatτ n (x) ≡τ (x) and passing here to the limit as n → ∞, we obtain thatτ (x) ≡ 0, as required.
Note that from the convergence of the series (3.25) in L 2 (0, 1) we get inequality (3.15) or, more precisely, Lemma 3.5 is proved. Proof It is clear that if τ (x) is a solution to equation (3.6), then τ (x) is a solution to equation (3.9), where Since T is an operator with weak singularity (see (3.16) and (2.27)) and is completely continuous as an operator from L 2 (0, 1) to C[0, 1], and the operator A is bounded operator in C[0, 1], by direct calculation we obtain F 2 (x) ∈ C 1 [0, 1] and F 2 (0) = 0. Next, applying Lemma 3.2, we get the statement of Lemma 3.6. To complete the proof of Theorem 3.1, we note that if conditions (3.10) (of Lemma 3.1) are met. Then conditions (3.13) (of Lemma 3.2) are also met, since 0 < λ (0) < 1.
Conditions (3.10) are equivalent to condition (3.2). Indeed, it easily follows from the equation of the curve AD: ξ = λ(η) in characteristic coordinates that Proof Note at once that by Lemmas 3.3-3.6 and representations (2.13) and (2.21) we get inequality (2.43) for all f (x, y) ∈ L 2 ( ).
Evaluation (2.43) also implies the uniqueness of a strong solution to problem M 2 B. Due to the density in L 2 ( ) of the set for any function f (x, y) ∈ L 2 ( ), there is a sequence f n (x, y) ∈ C 1 0 (¯ ) such that f n (x, y)f (x, y) 0 → 0 as n → ∞.
By z n (x, y) we denote a regular solution to problem M 2 B for equation (1.1) with righthand part f n (x, y) and initial conditions τ n (x) = z n (x, 0), ν n (x) = z ny (x, 0). By Lemma 3.7 we have τ n (x) ∈ C 2 [0, 1], τ n (0) = τ n (0) = 0, ν n (x) ∈ C 1 [0, 1], and therefore by formulas (2.13) and (2.21) we get z n (x, y) ∈ V for all f n (x, y) ∈ C 1 0 (¯ ). By the completeness of the space L 2 ( ) the sequence f n (x, y) is fundamental. From the linearity of equation (1.1) and estimate (2.43) we obtain that i.e., the sequence {z n (x, y)} is fundamental in L 2 ( ). Taking into account the completeness of the space L 2 ( ), we obtain that there is a unique limit z(x, y) ∈ L 2 ( ) of the sequence {z n (x, y)}, which will be the desired strong solution of problem M 2 B for equation (1.1) with the right part f (x, y) ∈ L 2 ( ).
As noted above, the operator corresponding to problem M 2 B is denoted by B 2 . The main result of this section is the following: Theorem 3.3 Let condition (3.2) be fulfilled. Then problem M 2 B is Volterra, that is, for any complex number λ, the solution to the equation exists and is unique for all f (x, y) ∈ L 2 ( ).