1.
Introduction
In the last years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives. These kind of equations have gained considerable importance due to their application in various sciences, such as physics, biology, economics, mechanics, chemistry, control theory, engineering, signal and image processing, etc [1,2,3,4,5,6,7].
Nonlinear coupled systems of fractional order differential equations appear often in investigations connected with disease models [8], anomalous diffusion [9] and ecological models [10]. Unlike the classical derivative operator, one can find a variety of its fractional counterparts such as Riemann-Liouville, Caputo, Hadamard, Erdelyi-Kober, Hilfer, Caputo-Hadamard, etc. Recently, a new class of fractional proportional derivative operators was introduced and discussed in [11,12,13]. Then, the concept of Hilfer type generalized proportional fractional derivative operators was proposed in [14]. For the detailed advantages of the Hilfer derivative, see [15] and a recent application in calcium diffusion in [16].
Many researchers studied initial and boundary value problems for differential equations and inclusions including different kinds of fractional derivative operators, for instance, see [17,18,19,20,21]. In [22], the authors studied a nonlocal-initial value problem of order in (0,1) involving a ¯ψ∗-Hilfer generalized proportional fractional derivative of a function with respect to another function. Recently, in [23], the authors investigated the existence and uniqueness of solutions for a nonlocal mixed boundary value problem for Hilfer fractional ¯ψ∗-proportional type differential equations and inclusions of order in (1,2] of the form
where Dχ,β,σ,ψc+, denotes the ψ-Hilfer generalized proportional fractional derivative operator of order χ∈{α,δk}, α,δk∈(1,2] and type β∈[0,1], respectively, σ∈(0,1], ηj,ζi,λk∈R are given constants, Υ: [w1,w2]×R→R is a given continuous function, Iϕi;σ,ψw1+ is the generalized proportional fractional integral operator of order ϕi>0 and ξj,θi,μk∈(w1,w2), j=1,2,⋯,m, i=1,2,⋯,n, k=1,2,⋯,r, are given points.
In [24], the authors discussed the existence of solutions for a nonlinear coupled system of (k,ψ)-Hilfer fractional differential equations of different orders in (1,2], complemented with coupled (k,ψ)-Riemann-Liouville fractional integral boundary conditions. In [25] a coupled system of Hilfer type generalized proportional fractional differential equations with nonlocal multi-point boundary conditions of the form
is investigated, in which Dδ1,η,σw+1 and Dδ2,η,σw+1 are the fractional derivatives of Hilfer generalized proportional type of order 1<δ1,δ2<2, the Hilfer parameter 0≤η≤1, σ∈(0,1], k,k1∈R, Υ1,Υ2: [w1,w2]×R→R are continuous functions, w1≥0, θj,εi∈R, ξj,λi∈(w1,w2) for i=1,2,3,⋯,n and j=1,2,⋯,m. Existence and uniqueness results are proved by applying classical Banach and Krasnosel'skii fixed-point theorems, and the Leray-Schauder alternative.
Very recently in [26] the authors established existence and uniqueness results for a class of coupled systems of nonlinear Hilfer-type fractional ¯ψ∗-proportional differential equations equipped with nonlocal multi-point and integro-multi-strip coupled boundary conditions of the form:
where Dρκ,φa,ϑ∗,¯ψ∗w1+ and κ=1,2 denote the Hilfer fractional ¯ψ∗-proportional derivative operator of the order ρκ∈(1,2] and type φi∈[0,1], ϑ∗∈(0,1], w1<ζj<ξi<ηi<w2, w1<δj<zi<ϵi<w2, κi,θj,ϕi,ϑj∈R, i=1,2,⋯,n, j=1,2,⋯,m, ¯ψ∗: [w1,w2]→R is an increasing function with ¯ψ′∗(z)≠0 for all z∈[w1,w2] and Ψ1,Ψ2: [w1,w2]×R×R→R are continuous functions.
In this work, motivated by the above mentioned papers, we study a coupled system of ψ-Hilfer generalized proportional sequential fractional differential equations with mixed nonlocal and integro-multi-point boundary conditions, of the form
where HDX,βι,ρ,ψw1 denotes the ψ-Hilfer generalized proportional fractional derivative operator of order X∈{α1,α2} with the parameters βι,ι∈{1,2}, 1<X≤2, 0≤βι≤1, pIY,ρ,ψw1 is a generalized proportional fractional integral operator of order Y>0, Y∈{Φj,υl}, λ1,λ2,ηi,ζj,ℵk,Θl∈R∖{0}, ξi,θj,ϱk,ϑl∈(a,b), i=1,2,⋯,n, j=1,2,⋯,m, k=1,2,⋯,r, l=1,2,⋯,q and Υ1,Υ2: [w1,w2]×R×R→R are nonlinear continuous functions.
Here, we emphasize that problem (1.4) is novel, and its investigation will enhance the scope of the literature on nonlocal Hilfer-type fractional ψ-proportional systems. Note that, when ψ(t)=t, problem (1.4) reduces to a coupled system of Hilfer generalized proportional fractional differential equations with mixed nonlocal multi-point and integro-multi-point boundary conditions; while if ρ=1, reduces to a coupled system of ψ-Hilfer fractional differential equations with mixed nonlocal multi-point and integro-multi-point boundary conditions. If ψ(t)=t, ζj=0, Θl=0, problem (1.4) is reduced to problem (1.2).
In solving (1.4), we first convert it into an equivalent fixed point problem, with the help of an auxiliary result based on a linear variant (1.4). Afterward, under different assumptions, we apply different fixed point theorems to establish our results on existence and uniqueness of solutions. For the first result (Theorem 3.1), we apply the Leray-Schauder's alternative to show that there exists at least one solution for the problem (1.4). The second result (Theorem 3.2), relying on Krasnosel'skii's fixed point theorem, shows that the problem (1.4) has at least one solution under different assumptions, and the last result (Theorem 3.3), shows the existence of a unique solution to the problem (1.4) by means of Banach's contraction mapping principle. In Section 4, we illustrate all the obtained theoretical results with the aid of constructed numerical examples. We emphasize that the problem (1.4) is novel and its investigation will enhance the scope of the literature on coupled systems of ψ-Hilfer generalized proportional fractional differential equations with mixed nonlocal and integro-multi-point boundary conditions. The used method is standard, but its configuration in the problem (1.4) is new.
The structure of the rest of the paper is organized as follows: In Section 2, some necessary definitions and preliminary results related to our problem are presented. Section 3 contains the main results for the problem (1.4), while numerical examples illustrating these results are constructed in Section 4. A brief conclusion closes the paper.
2.
Preliminaries
In this section, we introduce some necessary definitions and preliminary results needed in main results later.
Definition 2.1. [11,12] Let the functions ϑ0,ϑ1: [0,1]×R→[0,∞) be continuous such that for all t∈R and for ρ∈[0,1], we get
and
Let also ψ(t) be a strictly positive increasing continuous function. So, the proportional differential operator of order ρ of function Υ(t) with respect to function ψ(t) is defined by
Moreover, if ϑ0(ρ,t)=ρ and ϑ1(ρ,t)=1−ρ, then operator pDρ,ψ becomes
The integral corresponding to the above proportional derivative is defined as
where
The generalized proportional integral of order n corresponding to proportional derivative pDn,ρ,ψΥ1(t), is given by
Based on the generalized proportional integral of order n, we can obtain the following general proportional fractional integral and derivative.
Definition 2.2. [11,12] Let ρ∈(0,1] and α>0. The fractional proportional integral of order α of the function f with respect to function ψ is defined by
Definition 2.3. [11,12] Let ρ∈(0,1] α>0 and ψ(t) is a continuous function on [w1,w2], ψ′(t)>0. The generalized proportional fractional derivative of order α of the function Υ with respect to function ψ is defined by
where
Below we present the generalized proportional fractional derivatives of order α of function Υ with respect to another function ψ in Hilfer sense.
Definition 2.4. [27] For ρ∈(0,1]. Let functions Υ,ψ∈Cm([w1,w2],R) and ψ be positive and strictly increasing with ψ′(t)≠0, for all t∈[w1,w2]. The ψ-Hilfer generalized propotional fractional derivative of order α and type β for Υ with respect to another function ψ is defined by
where order n−1<α<n and 0≤β≤1.
If γ=α+β(n−α), then the ψ-Hilfer generalized proportional derivative HDα,β,ρ,ψw1 is equivalent to
Lemma 2.1. [27] Let n−1<α<n∈N, 0<ρ≤1, 0≤β≤1 and n−1<γ<n such that γ=α+nβ−αβ. If Υ∈C([w1,w2],R) and I(n−γ,ρ,ψ)w1∈Cn([w1,w2],R), then
Lemma 2.2. Let 1<α1,α2<2, 0≤β1,β2≤1, γi=αi+βi(2−αi), i=1,2, ¨Λ≠0 and z,w∈C([w1,w2],R). Then the pair (k1,k2) is the solution of the coupled system
if and only if
and
where ¨Λ=X1Y1−X2Y2,
Proof. Let the pair (k1,k2) be the solution of the system (2.1). We take the Riemann-Liouville integrals to Eq (2.1),
Then, applying Lemma 2.1 with n=2 to Eq (2.5), we get
and
where
From the conditions k1(w1)=0 and k2(w1)=0 we get c1=0 and c3=0, since γ1∈[α1,2] and γ2∈[α2,2] (see [27]), and Eqs (2.6) and (2.7) are reduced to
From the boundary conditions
and
we get
and
From Eqs (2.10) and (2.11), by using the notations (2.4) we get the system
where
By solving the above system, we obtain the constants c0 and c2 as
Now substitute the values of c0 and c2 into Eqs (2.8) and (2.9) and yield Eqs (2.2) and (2.3), as desired. We can prove the converse of the lemma by direct computation. □
3.
Existence results
In this section, we prove the existence and uniqueness results for the problem (1.4) by using three fixed ponit theorems.
First, we defined the spaces
with the norm
and
with the norm
Then it is well known that (X,‖⋅‖) and (Y,‖⋅‖) are Banach apaces. Obviously, the product space of X×Y endowed with norm ‖(k1,k2)‖=‖k1‖+‖k2‖ for (k1,k2)∈X×Y is a Banach space.
In view of the Lemma 2.2, we define an operator T: X×Y→X×Y by
where
and
Then, we introduce the following notation for computational convenience.
Notation 3.1. Let Ai,Bi,Ci for i=1,2 be the constants:
Now we prove our first existence result via Leray-Schauder alternative [28].
Theorem 3.1. Let Υ1,Υ2: [w1,w2]×R2→R be continuous functions. Suppose that:
(H1) There exist ui,vi≥0 for i=1,2 and u0,v0>0 such that for each k1,k2∈R, t∈[w1,w2],
If
and
where the constants Ai,Bi,Ci for i=1,2 are defined in the Notation 3.1, then the problem (1.4) has at lest one solution on [w1,w2].
Proof. Since f and g are continuous functions, then T is a continuous operator. We prove that TBr is uniformly bounded, where Br is the closed ball
For all (k1,k2)∈Br, by (H1) we have
and similarly
So, for any k1,k2∈Br, we have
and hence
In the same way, we can obtain that
Consequently,
Therefore, the set TBr is uniformly bounded.
Next, it is proven that TBr is equicontinuous. Let (k1,k2)∈Br and t1,t2∈[w1,w2] with t1<t2. Then, we have
Then, we obtain that
when t2→t1, independently of k1 and k2. Similarly,
as t2→t1. Therefore TBr is equicontinuous on [w1,w2]. From the above three steps and Arzelaˊ-Ascoli theorem, we conclude that T is completely continuous.
Let
We prove that U is bounded. Let
be any solution of (k_1, k_2) = \mu T(k_1, k_2). For each t\in [\mathtt{w}_1, \mathtt{w}_2] , we have
Then
and thus
Similarly, we have
Thus we obtain
This imply that,
where
Then, the set U is bounded. Therefore, by Leray-Schauder alternative the problem (1.4) has at least one solution on [\mathtt{w}_1, \mathtt{w}_2]. □
Now, we prove the second existence of results by applying Krasnosel'skii point theorem [29].
Theorem 3.2. Suppose \Upsilon_1, \Upsilon_2 : [\mathtt{w}_1, \mathtt{w}_2] \times \mathbb{R}^{2} \rightarrow \mathbb{R} are continuous functions. In addition we assume that:
(H_2) There exist positive functions \varphi_1, \varphi_2 \in C([\mathtt{w}_1, \mathtt{w}_2], \mathbb{R}^{+}), such that
If
then, the problem (1.4) has at least one solution on [\mathtt{w}_1, \mathtt{w}_2].
Proof. First, we separate the operator T as
with
We claim that TB_r \subset B_{r} where
We set
for i = 1, 2 and choose
Let (k_1, k_2), (\bar k_1, \bar k_2)\, \in B_{r}. Acconding the proof of Theorem 3.1, the following inequalities are obtained
and therefore
Hence,
and
Consider the operators T_{11} and T_{21}. By continuity of f and g, T_{11} and T_{21} are continuous operators. For any (k_1, k_2) \in B_r, we have
In a similar way, we can get
Hence, we obtain that
which yields that (T_{11}, T_{12})B_r is uniformly bounded. For any t_{1}, \, t_{2} \in [\mathtt{w}_1, \mathtt{w}_2], t_2 > t_1 and for all (k_1, k_2) \in B_r the operators (T_{11}, T_{12})B_r are equicontinuous by the proof of Theorem 3.1.
Lastly, it is proven that the operators T_{12} and T_{22} are contraction mappings. For all (k_1, k_2), (\bar k_1, \bar k_2) \in B_{r}, we have:
Additionally, we also obtain that
Combining the above inequalities, we have
We have (T_{11}, T_{21}) is a contraction. Consequently, by Krasnosel'skii's fixed point theorem, the problem (1.4) has at least one solution on [\mathtt{w}_1, \mathtt{w}_2]. □
Banach's fixed point theorem [30] is applied to obtain our uniqueness and existence result.
Theorem 3.3. Let \Upsilon_1, \Upsilon_2 : [\mathtt{w}_1, \mathtt{w}_2] \times \mathbb{R}^{2} \rightarrow \mathbb{R} such that:
(H_3) There exist positive constants L_1, L_2, such that, for all t\in [\mathtt{w}_1, \mathtt{w}_2] and o_i, \bar{o}_i \in \mathbb{R}, i = 1, 2, we have
Then, the problem (1.4) has a unique solution, provided that
Proof. Let
and
with
For all (k_1, k_2) \in B_r and t\in [\mathtt{w}_1, \mathtt{w}_2]. By applying (H_3), we obtain the following inequalities
We have
Thus,
We also have
Therefore
Hence, T(B_r) \subset B_r. Now, it is shown that (T_1, T_2) is a contraction mapping. For all (k_1, k_2), (\bar k_1, \bar k_2)\in B_r, we have
By a similar argument, we have
Hence, we obtain that
By assumption (3.4), the operator T is a contraction mapping. By Banach's fixed point theorem the problem (1.4) has a unique solution. □
4.
Examples
Example 4.1. Consider the following coupled system of \psi -Hilfer genneralized proportional fractional differential equation,
Here, we take
From the above data, we obtain
{\rm{(i)}} In order to illustrate Theorem 3.1, consider the functions f and g, defined by
Then, we have
Thus, (H_1) is satisfied with
Then
and
Thus, all assumptions of Theorem 3.1 are satisfied. Hence, the proplem (4.1), with f and g, given by (4.2), has at least one solution on [1/10, 3].
{\rm{(ii)}} Consider now the following functions f and g ,
It is obvious to check that the above functions satisfy
Then we find that \mathcal{C}_1+\mathcal{C}_2 \approx0.4610 < 1. Hence, by Theorem 3.2, the coupled system (4.1), with f and g, given by (4.3), has at lest one solution on an interval [1/10, 3] .
\text{(iii)} To illustrate Theorem 3.3, we consider the functions f and g as
We have
and therefore the Lipschitz condition for f and g is satisfied with L_1 = 1/10 and L_2 = 1/9. In addition, we find that
Thus, by Theorem 3.3, problem (4.1), with f and g, given by (4.4), has a unique solution on [1/10, 3].
Example 4.2. We investigate the behavior of solutions by replacing the values of proportional constant \rho by 0.1, 0.2, \cdots, 0.9, in the following coupled linear system of \psi -Hilfer generalized proportional fractional differential equations of the form:
Here, we set
By using integrating factor technique, we can obtain
where \gamma_{1} = 7/4 , \gamma_{2} = 31/20 and
with the canstants M, N, O, P, Q, R, S and T are defined by
For finding the analytic solutions, we use two constants c_1 and c_3 from Table 1 and substitute them in Eqs (4.6) and (4.7), respectively.
Next, using the Matlab program, we can find the approximate analytical solutions of k_1(t) and k_2(t) with different values of \rho as 0.1, 0.2, 0.3, \cdots, 0.9 . Two graphs of k_1(t) and k_2(t) can be drawn.
From Figure 1, if the value \rho is increasing then the value of k_1(t) is decreasing for each point t\in [0, 1] . From the Figure 2, we see that if the value of \rho increases, then the value of k_2(t) decreases for each t\in [0, 1] . The lower and upper bounds for the above two curves correspond to \rho = 0.1 and \rho = 0.9 , respectively, when the value of t increases.
5.
Conclusions
In this paper, we investigated a coupled system of \psi -Hilfer fractional proportional differential equations supplemented with nonlocal integro-multipoint boundary conditions. We rely on standard fixed point theorems, Banach, Krasnosel'skii and Leray-Schauder alternative to establish the desired existence and uniqueness results. The obtained theoretical results are well illustrated by numerical examples. Our results are new and contribute significialy to the existing results in the literature concerning \psi -Hilfer fractional proportional nonlocal integro-multi-point coupled systems.
Our results are novel and contribute to the existing literature on nonlocal systems of nonlinear \psi -Hilfer generalized fractional proportional differential equations. Note that the results presented in this paper are wider in scope and produced a variety of new results as special cases. For instance, fixing the parameters in the nonlocal integro-multi-point \psi Hilfer generalized proportional fractional system in (1.4), we obtained some new results as special cases associated with the following:
● Nonlocal \psi -Hilfer generalized proportional fractional systems of order in (1, 2] if \zeta_j = 0, \; \Theta_{l} = 0, j = 1, 2, \cdots, m, l = 1, 2, \cdots, q.
● Integro-multi-point nonlocal \psi -Hilfer generalized proportional fractional systems of order in (1, 2] if \eta_i = 0, \; \aleph_{k} = 0, i = 1, 2, \cdots, n, k = 1, 2, \cdots, r.
● Nonlocal Integro-multi-point Hilfer generalized proportional fractional systems of order in (1, 2] if \psi (t) = t.
● Integro-multi-point nonlocal Hilfer generalized fractional systems of order in (1, 2] if \rho = 1.
Furthermore, additional new results can be recorded as special cases for different combinations of the parameters \zeta_j, \Theta_{l} , j = 1, 2, \cdots, m , l = 1, 2, \cdots, q , \eta_i = 0 , \aleph_{k} = 0 , i = 1, 2, \cdots, n , k = 1, 2, \cdots, r involved in the system (1.4). For example, by taking all values where \eta_j = 0 , j = 1, 2, \cdots, m, we obtain the results for a coupled system of nonlinear \psi -Hilfer generalized proportional fractional differential equations supplemented by the following nonlocal boundary conditions:
Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Acknowledgments
This research was funded by National Science, Research and Innovation Fund (NSRF) and King Mongkut's University of Technology North Bangkok with Contract No. KMUTNB-FF-66-11.
Conflict of interest
The authors declare that there are no conflicts of interest.