On the single partial Caputo derivatives for functions of two variables

In this work we propose definitions (distinguishable from the standard ones) of single partial derivatives in a Caputo sense of functions of two variables on the rectangle P=[0,a]×[0,b]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P=[0,a]\times [0,b]$$\end{document}. Next, we give an integral representation of functions possessing such derivatives. Finally, we apply these derivatives to the study of the existence and uniqueness of solutions and the continuous dependence of solutions on controls to a fractional counterpart of a nonlinear continuous Roesser type model.


Introduction
Fractional calculus is a field of mathematics which is a generalization of classical differential calculus.Over the last decades it has been used successfully in various fields of science and engineering, for example in physics (classical and quantum mechanics, thermodynamics, optics, etc., cf.[12,13,30]), mechanics (non-conservative systems, mechanical systems including fractional oscillators, viscoelastic plane bodies and plates, cf.[7,22,27]), viscoelasticity (fractional models describing the behaviour of viscoelastic materials: polymers, gelatin phantoms, etc, cf.[4,11,25]), electrochemistry (ultracapacitors modelling, heat transfer models, cf.[8,9,31]), medicine (fractional epidemic models, cf.[1][2][3]10]), fractional calculus of variations (the Euler-Lagrange equations of fractional order, a fractional version of the Du Bois-Reymond Lemma, fractional Noether-type theorems, etc., cf.[5,6,15,24] and references therein) and optimal control (necessary and sufficient optimality conditions, cf.[14,[18][19][20]).Many models are formulated in terms of different types of fractional derivatives.One of the most popular fractional differential operator is the (left-sided) derivative in the Caputo sense of the form where f ∈ AC ([c, d], R n ) and I 1−α c+ is a fractional integral operator in the Riemann-Liouville sense of order 1 − α.This derivative is very popular among scientists because, as opposed to the Riemann-Liouville fractional operator, the Caputo derivative of a constant is zero and this operator is well adapted to deal with a fractional initial value problem (in which the standard initial condition is of the form f (c) = f 0 ).On the other hand, in definition (1.1) differentiability of the function f is required.This is very restrictive assumption, but this disadvantage can be eliminated by using the following alternative definition (cf.[23]): provided that f ∈ C([c, d], R n ) and the right side of (1.2) exists (here D α c+ denotes a fractional differential operator in the Riemann-Liouville sense of order α).It is easy to check that if f ∈ AC ([c, d], R n ), then the derivatives (1.1) and (1.2) coincide.
It is well known (cf.[28]) that the standard single partial Caputo derivatives (left-sided) of fractional order of a function z of two variables are given by (3.1) and (3.2).Similarly, as in the case of the Caputo derivative of a function of one variable, formulas (3.1) and (3.2) have disadvantages (the existence of the partial derivatives ∂z ∂ x and ∂z ∂ y a.e. on P, respectively, are required).In our paper, we propose an alternative definition (1.2) (then the assumption of the existence of partial derivatives is not required) for a function of two variables on the rectangle P.More precisely, for a function z := z(x, y) : P → R n we define the single partial Caputo derivatives (left and right-sided) with respect to x and y of order α ∈ (0, 1).Furthermore, we formulate and prove theorems on the integral representations of functions possessing such derivatives (all the definitions and representations mentioned above are contained in the main part of the paper, Sect.3).Results of such a type for single partial fractional derivatives in the Riemann-Liouville sense have been obtained in [16].Finally, in Sect.4, we consider a fractional Roesser type model with zero and nonzero boundary conditions involving new Caputo operators and study the existence and uniqueness of a solution to such a system as well as the continuous dependence of solutions on controls.We shall see that, using the operators we introduced, the problems of the existence and continuous dependence will be solved very easily.More precisely, the problem with zero boundary conditions is equivalent to the Roesser type problem with single partial Riemann-Liouville derivatives.Consequently, the existence and continuous dependence results can be immediately obtained by using the results of such a type for the Roesser problem involving Riemann-Liouville derivatives.The problem with nonzero boundary conditions can be easily solved by applying an appriopriate substitution (cf.Remark 4.6).
The rest of the paper is organized as follows.In Sect.2, some facts needed for the further discussion are given.The last section contains conclusions.

Preliminaries
This section contains the necessary definitions and some results obtained in [16] that will be useful later.They concern the left-sided fractional operators in the Riemann-Liouville sense for functions of two variables defined on the rectangle P.
Let ϕ ∈ L 1 (P, R n ) and α > 0. The functions I α x+ ϕ, I α y+ ϕ : P → R n given by are called the left-sided Riemann-Liouville integrals of order α on P of the function ϕ with respect to the variable x and y, respectively.
With the aid of the Fubini theorem one can show that these functions belong to L 1 (P, R n ).

Definition 2.2
We say that a function z = z(x, y) : P → R n is absolutely continuous in x if there exists a function z ∈ L 1 (P, R n ) with the following properties: z = z a.e. on P and z satisfies conditions (a 1 )-(c 1 ).
The set of all absolutely continuous in x functions z : P → R n will be denoted by AC x (P, R n ) (AC x for short).We shall also identify any in x absolutely continuous function z with its representant z described above.
The following characterization of the set AC x [16,Theorem 6] holds: We put in such a case The set of all functions z possessing the partial derivatives D α x+ z, D α y+ z will be denoted by AC α x+ , AC α y+ , respectively.In [16,Theorem 11], the following characterization of the space AC α x+ has been obtained.
In such a case Analogously, the result of such a type for the set AC α y+ can be proved: In such a case

Single partial derivatives in the Caputo sense of functions of two variables
In this part of the paper we propose new definitions of single partial derivatives in the Caputo sense defined on the rectangle P. Next, we give their integral representations.We start with the definitions of some useful sets of functions.
Let us consider a class of functions z ∈ L 1 (P, R n ) such that for short) denotes the set of all functions z : P → R n such that there exists a function z ∈ L 1 (P, R n ) with z = z a.e. on P and z satisfies conditions (a 2 ) and(b 2 ).We shall identify any function z ∈ C x,σ with its representant z described above.
Similarly, we define the set of functions C y,ρ , where ρ ∈ {0, b} (condition (b 2 ) is replaced with the following one:

Left-sided partial derivatives
First, we recall a well-known definition of the single partial fractional Caputo derivative of a function of two variables.
) of order α ∈ (0, 1) on the rectangle P with respect to x (y) is defined as follows: Of course the assumption z ∈ AC x (z ∈ AC y ) is restrictive, therefore, we propose a different definition.
We put in such a case The set of all functions z possessing the partial derivatives x −α , (x, y) ∈ P a.e.

.5)
In such a case ) In this case it is possible that equalities (3.6), (3.7) do not hold.
x+ and thereby Consequently, from Theorem 2.6 it follows that there exist functions ϕ whereby ( C D α x+ z)(x, y) = ϕ(x, y), (x, y) ∈ P a.e., and and define Using the subaddivity of the one-dimensional measure we assert that Y is a nonempty set of full measure.
This fact and the assumption z ∈ C x,0 imply z ∈ C AC α x+ .The proof is completed.
Using similar arguments we can prove the following result of the same type for functions z possessing the derivative C D α y+ z:

.11)
In such a case ) ) such that z is given by (3.11), then z ∈ C AC α y+ .In this case it is possible that equalities (3.12) and (3.13) do not hold.Remark 3. 8 In order to obtain equalities (3.6) and (3.7) ((3.12) and (3.13)) at the point (b) of the above theorems, some additional assumptions are required.One of such assumptions will appear in Sect. 4 (Remark 4.2).

Right-sided partial derivatives
As a reminder, we give notions of the right-sided fractional partial operators in the Riemann-Liouville sense.
Let ϕ ∈ L 1 (P, R n ) and α > 0. The functions I α x− ϕ, I α y− ϕ : P → R n , given by are called the right-sided Riemann-Liouville integrals of order α on P of the function ϕ with respect to the variable x and y, respectively.
Similarly, as in the case of the left-sided partial integrals, using the Fubini theorem, one can show that these functions belong to L 1 (P, R n ).
We have the following semigroup property for the operator I α x+ (see [17, formula (6)]): We put in such a case The set of all functions z possessing the partial derivatives D α x− z, D α y− z will be denoted by AC α x− , AC α y− , respectively.Now, we formulate and prove a theorem on the integral representation of functions belonging to AC α x− .Theorem 3.11 Let α ∈ (0, 1), z ∈ L 1 (P, R n ).Then z ∈ AC α x− if and only if there exist functions ϕ ∈ L 1
The proof is completed.
Using similar arguments, the result of the same type for functions belonging to AC α y− can be proved: In such a case We put in such a case The set of all functions z possessing the partial derivatives C D α x− z, C D α y− z will be denoted by C AC α x− , C AC α y− , respectively.All results obtained in Sect.3.1 have a right-sided counterpart.For instance, based on Theorems 3.11 and 3.12, we can obtain the following characterizations of functions belonging to C AC α x− and C AC α y− :

.20)
In such a case ) ) such that z is given by (3.20), then z ∈ C AC α x− .In this case it is possible that equalities (3.21) and (3.22) do not hold.In such a case ) such that z is given by (3.23), then z ∈ C AC α y− .In this case it is possible that equalities (3.24) and (3.25) do not hold.

Applications
In this section, we apply the fractional partial operators we introduced to investigate the following fractional Roesser type system: for a.e.(x, y) ∈ P with boundary conditions where α, β ∈ (0, 1), We shall study the existence and uniqueness of solutions and the continuous dependence of solutions on controls u ∈ U M to problem (4.1)-(4.2),where (x, y) ∈ P a.e.}, for 1 < p < ∞.We shall see that, using the operators introduced, the existence and continuous dependence can be obained more easily than the results of the same type for problem (4.1)-(4.2) with partial Caputo derivatives from Definition 3.1.

Proof
We shall prove only the first part of this lemma.The proof of the second one is analogous.
First, let us note that for a.e.
The proof is completed.
x+ will denote the set of all functions that have integral representation (3.5) y+ is the set of all functions that have integral representation (3.11) with We introduce in C AC α, p x+ and C AC β, p y+ norms in the following way: .   Consequently, [21, Theorems 3.1 and 3.2] immediately imply the following results: ) is measurable on P for all z 1 , z 2 ∈ R n , u ∈ M and the function f i (x, y, z 1 , z 2 , •) is continuous on M for a.e.(x, y) ∈ P and all z 1 , z 2 ∈ R n ; 2. There exists a constant N > 0 such that for a.e.(x, y) ∈ P and all z 1 , z 2 , w 1 , w 2 ∈ R n , u ∈ M; and 3.There exist a function r ∈ L p (P, R + 0 ) and a constant γ > 0 such that p < α, β < 1, then the sets of solutions of such problems are the same.Therefore, we can use the existence and continuous dependence results proved in [21] to obtain analogous results for problem (4.1)-(4.2).Furthermore, we see that all assumptions of Theorem 4.3 are easy to verify.Conditions more complicated to verify appear if we use the Caputo derivatives from Definition 3.1.This is a consequence of the fact that then the existence of partial derivatives (almost everywhere on P) of a solution is required ((z 1 , z 2 ) ∈ AC x × AC y ).Remark 4. 6 Using an appriopriate substitution, we can easily obtain the above results for the problem for a.e.(x, y) ∈ P with nonzero boundary conditions where α, β ∈ (0, 1), then the pair is a solution to problem (4.1)-(4.2).
From the above observation and Theorems 4.3 and 4.4 we obtain the following

Conclusions
In this paper we studied single partial Caputo derivatives of order α ∈ (0, 1) for functions of two variables defined on the rectangle P. We introduced alternative definitions of such operators (with the aid of single partial derivatives in the Riemann-Liouville sense).Integral representations of functions possessing such derivatives (left and right-sided) were presented.
The most important advantages of the fractional operators introduced are: -less restrictive assumptions on functions possessing the Caputo derivative: the existence of classical partial derivatives of functions is not required (in contrast to Definition 3.1), -facility of investigation of problems using the new operators (cf.Remark 4.5).
Motivated by this paper, the aim of a forthcoming work is to study mixed partial fractional derivatives in the Caputo sense of functions of two variables.
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Theorem 4 . 4 Remark 4 . 5
for a.e.(x, y) ∈ P and all u ∈ M, then problem (4.1)-(4.2) has a unique solution (z 1 , z 2 ) ∈ C AC α, p x+ × C AC β, p y+ corresponding to any control u ∈ U M .If all assumptions of Theorem 4.3 are satisfied and the sequence of controls (u l ) l∈N tends to ũ in the space L p (P, R m ), then the sequence of corresponding solutions (z l ) l∈N = (z l 1 , z l 2 ) l∈N to problem (4.1)-(4.2) tends to ẑ = (ẑ 1 , ẑ2 ) in the space C AC α, p x+ × C AC β, p y+ .Let us note that due to Definition 3.2 and Lemma 4.1 we can replace problem (4.1)-(4.2) with an equivalent problem involving the partial Riemann-Liouville derivatives.If 1
In a similar way, we define and characterize a function absolutely continuous in y.The set of all such functions will be denoted by AC y (P, R n ) (AC y for short).
The proof of the second part is analogous.The proof is completed.