On solutions of nonlinear BVPs with general boundary conditions by using a generalized Riesz–Caputo operator

In this work, we study the existence, uniqueness, and continuous dependence of solutions for a class of fractional differential equations by using a generalized Riesz fractional operator. One can view the results of this work as a refinement for the existence theory of fractional differential equations with Riemann–Liouville, Caputo, and classical Riesz derivative. Some special cases can be derived to obtain corresponding existence results for fractional differential equations. We provide an illustrated example for the unique solution of our main result.


Introduction
Fractional differential equations are considered as prolongation of the concept of derivative operator from integer order to any real or complex order. Fractional differential equations usually describe the nonlocal effects. Over the last two decades, there has been a blistering growth in the field of fractional calculus. Owing to the vast amount of applications, many mathematicians focused their engrossment on fractional calculus.
There exist several definitions for fractional derivatives and fractional integrals in the literature like Riemann-Liouville, Caputo, Hadamard, Riesz, Grunwald-Letnikov, Marchaud, Erdelyi-Kober, etc. The process of developing these operators began with a series of stages ranging from exponential functions to different classes of functions. Having lately come into holocene Udita N. Katugampola [1] generalized the above mentioned integral and differential operators. Meanwhile the well-developed theory and many more applications of the said operators are still a spotlight area of research in applied sciences.
As most fractional derivatives are computed using the corresponding integrals, researchers describe the nonlocal effects in terms of left and the right derivative. Thus, many mathematicians are in a hunt to generalize the notions further. In this context, Riesz [19] demonstrated the two-sided fractional operators using both left and right Riemann-Liouville's fractional differential and integral operators.
Due to the two-sided nature of Riesz's differential operator, the interesting differential is specifically used for fractional modeling on a finite domain. Some optimality conditions are discussed by Almeida for fractional variational problems with Riesz-Caputo derivative [20]. Frederico et al. derived Noether's theorem for variational problems having Riesz-Caputo derivatives. In [21], Mandelbrot demonstrated that there is a close connection between Brownian motion and fractional calculus.
In [22], the authors solved the fractional Poisson equation having Riesz derivative using Fourier transform. Due to the validity of Riesz derivative operator on the whole domain, it appears in the fractional turbulent diffusion model. In [23], the authors numerically solved the advection-diffusion equation having Riesz derivative. For further applications of Riesz derivative on the anomalous diffusion, see [24][25][26][27][28][29].
In this work, we define the generalized Riesz-Caputo type derivative operator by using the generalized operators. We present basic perspectives on the existence and uniqueness of solutions of fractional differential equations. Motivated by [30,31], we provide the analysis on existence of solutions for the following nonlinear fractional differential equation involving generalized Riesz-Caputo type derivative operator with general boundary conditions: where φ 0 and φ T are constants, while g : [0, T] × R 2 → R is continuous with 1 < α ≤ 2, 0 < α * ≤ 1, and 1 < ρ < ∞.
The rest of the paper is organized as follows: Sect. 2 presents some basic definitions and lemmas from literature. In Sect. 3 we introduce the generalized Riesz-Caputo's fractional operators and derived some useful results, while in Sect. 4 we establish some equivalence results for boundary value problem (1) and establish the results for the existence and uniqueness of solutions for BVP (1). The last section of this paper presents the stability of solutions for BVP (1) by means of continuous dependence on parameters.

Preliminaries
In this section we demonstrate some useful results including definitions and lemmas related to Riesz-Caputo derivatives and integrals that will help us in our later discussions. Following the same traditional definitions of Riesz-Caputo derivative and integral [19,30,32], we can generalize these definitions using a generalized Caputo type derivative operator. Some preliminary structural properties, which we will frequently use in our later discussion, are also introduced in this section. In 2010, Om Prakash Agrawal defined the generalized fractional in the following way. is defined as where the kernel function K α (μ, η) may depend on α and a < μ < T and r, s ∈ R.
This is the generalized fractional integral operator which, by using the specific kernel function, leads to the specific operator. For example, if K α (μ, η) = (μ-η) α-1 (α) and by taking T = 0 leads to the left sided R-L integral operator and by taking with T = 0 gives the left generalized integral defined below. Furthermore, the limits of integration a and T can be extended to -∞ and ∞ respectively.

Theorem 2.3 ([35])
Let α, ρ ∈ R and ρ, a > 0. Then, for φ ∈ X p c (a, b), the following relation holds: Similarly, the inverse property holds for a right-hand-sided integral and a derivative operator as well.

Generalized Riesz-Caputo fractional operators
In this section we introduce the generalized Riesz-Caputo fractional integrals and derivative operators.
Following the same mechanism, we generalize the Riesz fractional integral by means of Definition 2.2 as follows. ) and α, ρ > 0. Then, for 0 ≤ μ ≤ T, the generalized Riesz type integral is defined as Accordingly, the Riesz-Caputo derivative [19] can be generalized by means of generalized Caputo type derivative operators [1] as follows. Definition 3.3 Let α, ρ ∈ C with Re(α), Re(ρ) > 0 and g(μ) ∈ X ρ c (a, b) for 0 ≤ μ ≤ T. Then the generalized Riesz-Caputo type derivative operator is defined as where ρ * D α 0,μ and ρ * D α μ,T are left and right generalized Caputo type derivatives [37] as follows: where n = α .
Since for α = 1 the right generalized derivative is the negative of the left generalized derivative, so for integer values of α, the generalized Riesz-Caputo type derivative defined above comes to term with the conventional definitions of derivative.
Then the following relation is true: Proof Using the above definitions, we can write and the proof is finished.
Proof The proof simply follows by using n = 1 in Lemma 3.4 and Lemma 2.5, which yields the required result.
Theorem 3.6 Let α > 0 and {φ j } ∞ j=1 be a uniformly convergent sequence of continuous functions on [a, b]. Then we can interchange the generalized fractional integral operator and the limit, i.e., Proof Let φ be the limit of the sequence {φ j }. Since {φ j } is the convergent sequence of continuous functions, so φ is also continuous. To prove that under the given conditions we can interchange fractional integral and limit, it is enough to show that the sequence Now, we first shall evaluate the integral Now, using the result Consequently, from equation (2), we arrive at Since (φ j ) is a uniformly convergent sequence, thus Therefore, the sequence { ρ I α a+ φ j } ∞ j=1 is also uniformly convergent, and hence the result follows.
The similar result holds true for the right-sided generalized fractional integral as well.
In particular, ρ I α a 0 φ is also analytic.
Proof Since φ is an analytic function, thus it can be written in the form of convergent power series, i.e.,

Using Definition 2.2, we get
Using Theorem 3.6, the summation and integral sign are interchanged as follows: be a convergent sequence of nonnegative real numbers with limit λ. Then Proof Let the sequence {λ j } ∞ j=1 converge to the limit λ. Then, by definition, and by taking limit on both sides and by using Theorem 3.6, we have and this ends the proof.
Proof The result follows taking into account Definition 3.2, Theorem 3.6, and the fact that sum of two convergent sequence is convergent.
Likewise, the Gronwall inequality for generalized right-sided generalized fractional operator is expressed as follows.
then the following inequality holds true: Lemma 3.13 Let α > 0 and assume that g(μ), u 1 (μ), and v 1 (μ) are defined in the same way as in Lemma 3.10. Furthermore, if Proof From Lemma 3.12, Since u 2 is a nondecreasing function, therefore u 2 (μ) ≤ u 2 (η) for all η ∈ [0, T], and hence and the proof is ended.

Existence and stability
For the upcoming existence results and discussion for boundary value (1), we use the following conditions. Let J = [0, T] and C(J) be the space of all continuous functions defined on J. We define the space and taking into account Theorem 3.6 and Theorem 2.3, we get v(μ) = ρ * D α * u(μ). This completes the proof. α ∈ (1, 2), α * ∈ (0, 1), and g ∈ C(J). Then problem (1) is equivalent to the following integral equation:

Lemma 4.2 Let
where, for μ > η, and for η > μ, Proof Let φ(μ) ∈ X be a solution of boundary value problem (1). Then, by applying the generalized Riesz-type integral operator on both sides of equation (1) and using Definition 3.2, Lemma 2.5, and Lemma 3.4, we obtain Using the boundary conditions φ(0) = φ 0 and φ(T) = φ T into the above equation, we get Now again substituting these values of constants into the above equation, we get where, for μ > η, and for η > μ, Conversely, let φ(μ) ∈ X be a solution of the fractional integral operator (3), and we denote the right-hand side of equation (3) by (μ), i.e., Now taking the left and the right generalized Caputo derivative on both sides of the above equation, we get and Here, we have used Theorem 2.3, and some simple calculation leads to the facts that ρ * D α 0,μ (μ ρ ) = 0 and ρ * D α μ,T (μ ρ -T ρ ) = 0. Consequently, from equations (4), (5) and Definition 3.3, the required result follows, i.e., and the proof is completed.
Furthermore, let By means of local integrability of U(μ), K * exists certainly. Define a set . Then manifestly the set A r is a closed, bounded, and convex subset of the above defined Banach space (X, · X ).
Proof The result follows from Theorem 4.3.

Dependence of solutions on the parameters
The stability analysis of fractional differential equations has been carried out by many mathematicians. For details, one can see [36,[39][40][41][42] and the references therein. The solutions satisfy various types of stability, and continuous dependence on the initial data is one of them. This section demonstrates that the solution of problem (1) depends on the parameters α, φ 0 , φ T , and g provided that the function g satisfies conditions (H * 2 ) and (H * 2 ). Continuous dependence of solutions on the parameters indicates the stability of solutions. Theorem 4.8 Assume that φ 1 (η) is the solution of BVP (1) and φ 2 (η) is the solution of the following problem: where 1 < α -< α ≤ 2, 0 < α * ≤ 1, and g is continuous. Then φ 1φ 2 = O(ε).
This completes the proof.
This completes the proof.

Concluding remarks
We presented a generalization of the Riesz fractional operator in this work. We provided some results and inequalities for the new generalized Riesz fractional operators. Furthermore, we proved some equivalence results for the nonlinear fractional differential equation involving the generalized Riesz derivative operator. By using suitable fixed point theorems, we provided the uniqueness of solution of the problem and some several mathematical techniques. Also, we discussed the stability of solutions and showed continuous dependence onto given parameters. An instructive comparison with literature shows that these results present the generalization of various old theorems in the related areas.