On One Interpolation Type Fractional Boundary-Value Problem

: We present some new results on the approximation of solutions of a special type of fractional boundary-value problem. The focus of our research is a system of three fractional differential equations of the mixed order, subjected to the so-called “interpolation” type boundary restrictions. Under certain conditions, the aforementioned problem is simpliﬁed via a proper parametrization technique, and with the help of the numerical-analytic method, the approximate solutions are constructed.


Introduction
Differential equations of the fractional order have a wide spectrum of applications, since they are often used to model problems in fluid dynamics, finance, biology, physics, engineering, etc. However, most of the exact methods of solution of the considered problems (see [1][2][3][4][5][6][7][8][9][10][11]) in the nonlinear case of differential equations and their systems are not applicable.
Recently, in the series of papers (see discussions [25][26][27][28]), a completely new approach for study of a class of fractional periodic and anti-periodic boundary-value problems was developed. To construct the approximate solutions of the studied problems, the numerical-analytic method, based on the successive iterations, was used. This technique was originally suggested and successfully applied to the boundary-value problems for ordinary differential systems with strong nonlinearities not only in the equations, but also in the boundary conditions (see [29][30][31][32]). This paper is a continuation of the obtained results in the aforementioned direction, and gives the important results for the study of a new class of the interpolation type fractional boundary-value problems. A suitable parametrization technique, sometimes called in literature a "freezing technique," transforms the given three-point boundary constraints into the two-point ones and allows us to construct the approximate solutions of the modified problem in an analytic form.
In Sections 2 and 3 the main terminology that will be used throughout the paper is given, and some important results already obtained in this direction are presented.
Sections 4-7 are devoted to the main result of the paper, particularly to the fractional boundary-value problems under the interpolation type boundary constraints. The primary interest of such boundary conditions is that we cannot apply the modification of the numerical-analytic scheme (see Section 3) straightforwardly. Therefore, we need to use a special "freezing," or parametrization technique to cope with these kinds of issues. Moreover, we reduce the order of the boundary conditions (from the initially three-point restrictions to the two-point ones) that plays an important role in the computational process. As a result: these techniques might be further studied in the more complicated cases of boundary conditions (e.g., two-and multipoint nonlinear boundary constraints), widely appearing in the mathematical models of applied sciences.

Main Notations and Supplementary Statements
which converges in the right half of the complex plane Re(z) > 0, is called the Gamma function.

Definition 2. [8]
The left and right Riemann-Liouville fractional integrals of order α ∈ R + are defined by respectively, provided the right-hand sides are pointwise defined on [a, b]. Definition 3. [8] The left and right Riemann-Liouville fractional derivatives of order α ∈ R + are defined by Let us also formulate some subsidiary lemmas, first proven by Fečkan, Marynets in [25] and reformulated here over an interval [a, b].
Lemma 1. Let f (t) be a continuous function for t ∈ [a, b]. Then, for all t ∈ [a, b], the following estimate is true: where where α 0 (t) = 1. Note α j 1 (t) is defined by Formula (2). Then, the following estimate holds: for m ∈ N 0 .

Resent Results in the Study of the Periodic and Anti-Periodic Fractional Boundary Value Problems
Since the approach of the numerical-analytic method [32] was appied to the fractional differential systems for the first time in [25][26][27][28], it is resonable to give an overview of the results that will allow the reader to follow and will open up possible perspectives for future research in this direction.

Integral Representation of the Solution of the PFBVP
To give an understanding of the numerical-analytic technique used for approximation of solutions of the fractional periodic and anti-periodic boundary-value problems (BVPs), let us first present the basic ideas of the aforementioned method.
Consider the simplest case of a periodic fractional BVP (PFBVP) (see discussion in [25]) of the form: where c 0 D p t is the generalized Caputo fractional derivative with the lower limit at 0 (see ([11], Definition 1. Together with the fractional differential system (FDS) (5) we consider an initial value problem for a perturbed system of the form: where ∆ is a term to be defined.

Remark 1.
Note, that the main advantage of the Caputo's approach is that the initial conditions for fractional differential equations with Caputo derivatives take on the same form as for the integer-order differential equations; i.e., contain the limit values of the integer-order derivatives of unknown functions at the lower limit t = a (see [8]).
Using an integral representation of the Cauchy problem for the FDS with the initial data (8) we get: where Γ(·) is the Gamma function. Substituting in (9) values t = 0, t = T and using the periodic boundary condition (6), we find an expression for the perturbation term: Note that in order to apply the numerical-analytic technique for approximation of solutions of the problem (5) and (6) we need the following hypothesis to be held (see also discussions in [29][30][31]): (H1) Function f in the system (5) is bounded by a constant vector M = col(M 1 , M 2 , . . . , M n ) ∈ R n and it satisfies the Lipschitz condition with a non-negative real matrix K = (k ij ) n i,j=1 ; i.e., the following inequalities Then, the approximate solution of the PFBVP may be found from the iterative formula: Note that the numerical values of the unknown vector x 0 are solutions of the so-called "bifurcation" system of the form: where ∆ : D → R n is defined by expression (10).
Let us now pass to the more general cases of FDS.

Study of the Lower Order Fractional Differential Systems
In [26] we considered the FDS where first n 1 equations had the fractional order p ∈ (0, 1] and the rest n 2 equations were of the order q ∈ (0, 1].
Here f : We were looking for a continuously differentiable solution of the system (11), subjected to the periodic boundary restrictions: Remark 2. The FDS (11) has a particular application to economic systems, since it may formulate a dynamical macroeconomic model of two national economies.
As mentioned in Section 3.1, particular requirements had to be met in order to apply the successive approximation's approach.
Indeed, in [26] we derived these conditions; i.e., we supposed that the BVP (11), (12) is such that the following conditions hold: for any t ∈ [0, T], x 1 , x 2 ∈ D f , y 1 , y 2 ∈ D g and some non-negative vectors M f ∈ R n 1 , M g ∈ R n 2 ; (B 1 ) Functions f , g satisfy the Lipschitz type conditions for any t ∈ [0, T], x 1 , x 2 ∈ D f , y 1 , y 2 ∈ D g and some non-negative matrixes K ij , i, j ∈ {1, 2}; The spectral radius r(Q) of the matrix Q := KΓ pq satisfies an inequality r(Q) < 1, where Then, we connected with the BVP (11), (12) the sequences of functions {x m , y m } m∈N , given by the iterative formulas: In addition, the following theorems were proved. Theorem 1. [26] Assume that conditions (A 1 )-(D 1 ) for the PFBPV (11), (12) hold. Then, for all fixed x 0 ∈ D β f , y 0 ∈ D β g it holds that: 1. Functions of the sequence (13) are continuous and satisfy periodic boundary conditions 2. The sequences of functions (13) for t ∈ [0, T] converge uniformly as m → ∞ to the appropriate limit functions 3. The limit functions x ∞ , y ∞ satisfy periodic boundary conditions and are the unique continuous solutions of the integral equations i.e., they are the unique solutions on [0, T] of the Cauchy problem: 4. The following error estimations hold: where I n is the n-dimensional identity matrix.

PFBVP with a Higher Order Caputo Type Fractional Derivative
As a generalization of the aforementioned in Sections 3.1 and 3.2 problems, in [27] we studied the generalized fractional differential system with periodic boundary conditions It was shown (see discussions in [27]) that if the FBVP (16), (17) satisfies the hypothesis: There is a non-negative vector M = col(M 1 , M 2 , . . . , M n ) ∈ R n , M i ≥ 0 and a non-negative real matrix L = (l ij ) n i,j=1 , l i,j ≥ 0 such that and B i and B i (z) are the Bernoulli numbers and polynomials; (C 2 ) The spectral radius r(Q) of matrix Q := κ m T p L satisfies the estimate r(Q) < 1, one can associate an iteration scheme where t ∈ [0, T], x 0 (t) = ξ 0 ∈ D β , with the solution of the given BVP (16), (17).
Similarly to Theorem 1 in Section 3.2 it was proved that a sequence of functions (18) is convergent to the limit function, which is a solution of the given BVP, if the corresponding determining equation has a root. Moreover in the case of the asymptotically constant scalar problems for the fractional differential equations of the form with boundary conditions (17) for n = 1, g ∈ C(R, R), h ∈ C([0, T], R) under hypothesis (A 2 )-(C 2 ) and if (D 2 ) There are g ± ∈ R such that lim x→±∞ g(x) = g ± , conditions of existence and nonexistence of solutions are obtained.

Anti-Periodic Fractional BVP
Recently we considered a boundary-value problem for a FDS with anti-periodic boundary conditions of the form:  (19), (20) was constructed in the form: with a modified form of zero approximation, given by the relation: (22) is optimal in the sense that it improves the estimate of difference between the m-th approximation x m (·, ξ 0 ) and its limit function x ∞ (·, ξ 0 ). In the case of selection x 0 (t, ξ 0 ) = x 0 , as it was for the PFBVPs (5), (6) and (11), (12), it would lead to additional terms in the estimates that would increase the error of computations.

Remark 3. A choice of the zero-approximation in the form
In addition to the aforementioned convergence conditions and corollaries about the connection of the limit function of the sequence (21) to the exact solution of the AFBVP (19), (20), a proper solvability analysis of the original problem was provided.
In particular, the following statements have been proven.

Remark 4.
On the basis of Theorem 4, we established an algorithm of approximate localization of the point ξ * 0 , that defines solution x(·) of the original BVP (19), (20). The set Ω was rewritten as an association of the finite value of subsets : In every subset Ω i from (23) we picked a pointξ 0 i ∈ Ω i , and for some fixed m calculated the m-th approximation x m ·,ξ 0 i , by using the recurrent Formula (21).
Then, we obtained the value of the determining function and excluded from the set (23) subsets Ω i , for which the inequality is not true. When N and m tend to ∞, Ω m,N "follows" to the set Ω * that may contain value ξ * 0 and defines solution of the anti-periodic BVP (19), (20).
Every pointξ 0 ∈ Ω m,N one can observe as an approximation to the value ξ * 0 that determines solution of the given BVP (19), (20). In this case it is clear that and the value of function x m t,ξ 0 that is calculated by iterative Formula (21), may be taken as an approximate solution of the BVP (19), (20).

Consider a system of FDEs
for some p, q, r ∈ (0, 1], where f : G f → R, g : G g → R and h : G h → R are continuous functions; , and D f ⊂ R, D g ⊂ R and D h ⊂ R are closed and bounded domains; and the system is subjected to the interpolation type boundary constraints x(a) = d 1 , where d i ∈ R, i = 1, 3 are some given values.
The problem is to find a triple (x(·), y(·), z(·)) that satisfies the mixed-order FDS (24) and the three-point boundary conditions of the interpolation type (25) in the class of continuously differentiable functions x, y, z : [a, b] → R.
Assume that the FBVP (24), (27) satisfies the following conditions: (i) Functions f , g, h in the system (24) are bounded by some non-negative constants M f , M g , M h ∈ R.
(ii) Functions f , g, h in the system (24) satisfy the Lipschitz type condition for any t ∈ [a, b], x 1 , x 2 ∈ D f , y 1 , y 2 ∈ D g , z 1 , z 2 ∈ D h and some non- are non-empty, where The following convergence theorem holds.
Theorem 5. Let functions f : G f → R, g : G g → R and h : G h → R satisfy conditions (i)-(iii) in the domains D l , l = { f , g, h} with a matrix K for which where λ Q is the spectral radius of matrix Q = Γ pqr K (34) . Then, for all fixed ξ ∈ Ξ and λ ∈ Λ the statements to be hold: 1. All functions of the sequences (28)- (30) are absolutely continuous on the interval [a, b] , have values in the domains D l , l = { f , g, h} respectively, and satisfy the two-point boundary conditions 2. The sequences of functions (28)- (30) for t ∈ [a, b] converge uniformly as m → ∞ to the limit functions: 3. The limit functions satisfy the initial conditions and the two-point boundary conditions 4. The functions x ∞ (·, ξ, λ), y ∞ (·, ξ, λ), z ∞ (·, ξ, λ) are the unique continuously differentiable solutions of the integral equations In other words, x ∞ (·, ξ, λ), y ∞ (·, ξ, λ), z ∞ (·, ξ, λ) satisfy the Cauchy problem for the modified system of FDEs: where ∆ i (ξ, λ) : Ξ × Λ → R, i = 1, 3 are the mappings given by formulas: 5. The following error estimate holds: where I 3 is the identity three-dimensional matrix and M = col(M f , M g , M h ) is a vector, defined in the condition (i).
Proof. As already mentioned, the sequences of functions (28)- (30) are constructed in such a form that they satisfy the parametrized boundary restrictions in advance. Due to this fact, the statement 1 of the theorem holds. Let us prove that all functions x m (·, ξ, λ), y m (·, ξ, λ), z m (·, ξ, λ) of the sequences (28)-(30) remain in the domains D l , l = { f , g, h}, where each of them is defined; i.e., the iteration process can be continued until infinity. For this purpose we estimate the differences: The uniqueness of the solutions can be proven in a similar way as in [26] by generalization of the interval to [a, b] and consideration of three functions x(·), y(·), z(·), instead of two, as it was done in the aforementioned paper.

Some Solvability and Applicability Remarks
In addition to the results above, we can prove that the system of determining Equations (61) and (62) determines all possible solutions of the interpolation type FBVP (24), (25).
We have thus specified in (66) the values of a pair (ξ, λ) that satisfy the system of determining Equations (61), (62), which proves the theorem.

Remark 6.
In practice it is more suitable to consider the approximate determining equations