Hybrid System of Proportional Hilfer-Type Fractional Differential Equations and Nonlocal Conditions with Respect to Another Function

: In this paper, a new class of coupled hybrid systems of proportional sequential ψ -Hilfer fractional differential equations, subjected to nonlocal boundary conditions were investigated. Based on a generalization of the Krasnosel’ski˘i’s fixed point theorem due to Burton, sufficient conditions were established for the existence of solutions. A numerical example was constructed illustrating the main theoretical result. For special cases of the parameters involved in the system many new results were covered. The obtained result is new and significantly contributes to existing results in the literature on coupled systems of proportional sequential ψ -Hilfer fractional differential equations.


Introduction
Fractional calculus (differentiation and integration of arbitrary order) has proved to be an important tool in describing many mathematical models in science and engineering.In fact, this branch of calculus has found its application in physics, mechanics, control theory, economics, biology, signal and image precessing, etc. Fractional differential equations describe many real world processes more accurately than classical differential equations and have been addressed by many researchers.For theoretical and application details of fractional differential equations, we refer the reader to the books [1][2][3][4][5][6], while an extensive study of fractional boundary value problems can be found in the monograph [7].Usually, fractional derivative operators are defined via fractional integral operators and in the literature one can find a variety of such operators, such as Riemann-Liouville, Caputo, Hadamard, Erdélyi-Kober, Hilfer fractional derivatives, etc., to name some of them.In [8], with the help of Euler's k-gamma function, the k-Riemann-Liouville fractional integral operator was introduced, generalizing the concept of Riemann-Liouville fractional integral operator, which was used to define the k-Riemann-Liouville fractional derivative in [9].The Hilfer fractional derivative [10] extends both Riemann-Liouville and Caputo fractional derivatives.For applications of Hilfer fractional derivatives in mathematics, physics, etc., see [11][12][13][14][15][16].For recent results on boundary value problems for fractional differential equations and inclusions with Hilfer fractional derivative, see the survey paper by Ntouyas [17].The ψ-Riemann-Liouville fractional integral and derivative operators, which are fractional calculus with respect to a function ψ, are discussed in [1], while the ψ-Hilfer fractional derivative is discussed in [18].
Fractional coupled systems are also important, as such systems appear in the mathematical models associated with fractional dynamics [26], bio-engineering [27], financial economics [28], etc.In [29], the authors studied the existence and Ulam-Hyers stability results of a coupled system of ψ-Hilfer sequential fractional differential equations.In [30], by using Krasnosel'ski ȋ's fixed point theorem, the existence of solutions are established for the following nonlinear system involving generalized Hilfer fractional operators where H D A,B,ψ is the ψ-Hilfer fractional derivative of order A ∈ {δ, ν} with δ, ν ∈ (0, 1) and types B ∈ {ϑ, κ}, ϑ, κ ∈ [0, 1], I 1−γ,ψ , I q i ,ψ are the ψ-Riemann-Liouville fractional integrals of order 1 − γ > 0, In [31], the existence and uniqueness results are derived for a coupled system of Hilfer-Hadamard fractional differential equations with fractional integral boundary conditions.Recently, in [32] a coupled system of nonlinear fractional differential equations involving the (k, ψ)-Hilfer fractional derivative operators complemented with multi-point nonlocal boundary conditions was discussed.Moreover, Samadi et al. [33] have considered a coupled system of Hilfer-type generalized proportional fractional differential equations.
In this article, motivated by the above works, we study a hybrid system of proportional Hilfer-type fractional differential equations of the form: subject to coupled nonlocal boundary conditions where H D δ,ϑ 1 ,ρ,ψ denotes the ψ-Hilfer generalized proportional derivatives of order To establish our main existence result, we first transform the problem (3) and ( 4) into a fixed-point problem by using a linear variant of the problem (3) and ( 4), and then apply a generalization of the Krasnosel'ski ȋ's fixed-point theorem due to Burton.
Our problem enriches the literature on hybrid sequential coupled systems of proportional ψ-Hilfer differential equations of fractional order with nonlocal boundary conditions.The nonlocal boundary conditions can be applied in physics, thermodynamics, wave propagation, etc., and are more general than classical boundary conditions.For some applications see [34,35] and the references cited therein.For applications of Hilfer fractional derivative operators in applied sciences (such as physics, filtration processes, cobweb economics model, stochastic equations etc.), we refer the reader to [36][37][38][39][40][41] and their references.
Comparing our problem with the problem studied in [30], we note that: • We study a system involving ψ-Hilfer proportional fractional derivatives.

•
Our equations are more general as the contained fractional derivatives have different orders.

•
Our system contains nonlocal coupled boundary conditions.

•
Our system covers many special cases by fixing the parameters involved in the problem.For example, by taking f, g = 1 in the problem (3), we have the following new nonlocal coupled system of sequential Hilfer-type proportional fractional differential equations , λ, µ constants, then we have a nonlocal coupled system of sequential Hilfer-type proportional fractional differential equations of Langevin-type which is a generalization of the well-known classical results in [42].
The structure of this article has been organized as follows: In Section 2, some necessary concepts and basic results concerning our problem are presented.The main result for the problem (3) and ( 4) is proved in Section 3, while Section 4 contains an example illustrating the obtained result.
which ψ is positive and strictly increasing with ψ ′ (t) ̸ = 0 for all t ∈ [a 0 , b 0 ].The ψ-Hilfer generalized proportional fractional derivative of order δ and type ϑ for F with respect to another function ψ is defined by where n − 1 < δ < n and 0 ≤ ϑ ≤ 1.

Remark 1 ([43]
).It is assumed that the parameters δ, ϑ and γ (involved in the above definitions) satisfy the relations: To prove the main result we need the following lemma, which concerns a linear variant of the ψ-Hilfer proportional coupled system (3) and ( 4), and is used to convert the nonlinear problem in system ( 3) and ( 4) into a fixed-point problem.
Then the pair (u, s) is a solution of the system if and only if and where Proof.Due to Lemma 1 with n = 1, we obtain where c 0 , d 0 ∈ R. Now applying the boundary conditions Now, by taking the operators p I δ 2 ,ρ,ψ and p I δ 4 ,ρ,ψ into both sides of (10) and using Lemma 1, we obtain Applying in (11) the conditions u(a 0 ) = s(a 0 ) = 0, we obtain and γ 4 ∈ [δ 4 , 2] (Remark 1).Thus we have In view of (12) and the conditions u(b 0 ) = θ 1 s(ξ 1 ) and s(b and Due to ( 8) and ( 13), ( 14), we have where By solving the above system, we conclude that Replacing the values c 1 and d 1 in the Equation ( 12) we obtain the solutions ( 6) and (7).The converse follows by direct computation.The proof is completed.
The following version of Krasnosel'ski ȋ's fixed point theorem due to Burton is the basic tool in proving our main existence result.

Lemma 3 ([44]
).Let S be a nonempty, convex, closed, and bounded set such that S ⊂ X, and let A : X → X and B : S → X be two operators which satisfy the following: Then there exists a solution of the operator equation x = Ax + Bx.
where for convenience we have put ) Then the ψ-Hilfer proportional coupled system (3) and ( 4) has at least one solution on [a 0 , b 0 ].
Proof.Firstly, we consider a subset S of Y × Y defined by S = {(u, s) ∈ Y × Y : ∥(u, s)∥ ≤ r}, where r is given by where +Ψ(b 0 , δ 4 )∥τ∥∥L 2 ∥Ψ(b 0 , δ 3 ) , and Let us define the operators and Then we have and Also, we obtain and Moreover, we have and Finally, we obtain and Now we split the operator U as In the following steps, we will prove that the operators U 1 , U 2 fulfill the assumptions of Lemma 3.
Step 1.In the first step we will prove that the operators U 1,2 and U 2,2 are contraction mappings.For all (u, s), (u Similarly we can find Consequently, we obtain ) is a contraction.
Step 2. In the second step we will prove that the operator (U 1,1 , U 2,1 ) is completely continuous on S. For continuity of U 1,1 , take any sequence of points (u n , s n ) in S converging to a point (u, s) ∈ S.Then, by Lebesgue dominated convergence theorem, we have Next, we show that the operator (U 1,1 , U 2,1 ) is uniformly bounded on S. For any (u, s) ∈ S we have Similarly we can prove that Therefore ∥U 1,1 ∥ + ∥U 2,1 ∥ ≤ Λ 1 + Λ 2 , (u, s) ∈ S, which shows that the operator (U 1,1 , U 2,1 ) is uniformly bounded on S. Finally we show that the operator (U 1,1 , U 2,1 ) is equicontinuous.Let τ 1 < τ 2 and (u, s) ∈ S.Then, we have As τ 2 − τ 1 → 0, the right-hand side of the above inequality tends to zero, independently of (u, s).
Step 3. In the third step we will prove that condition (iii) of Lemma 3 is fulfilled.Let Then, we have In the given system (21), Hence the nonlocal sequential Hilfer-type coupled system of nonlinear proportional fractional differential quations ( 21) satisfies all conditions in Theorem 1, and thus it has at least one solution (u, s) on [1/8, 13/8].

Conclusions
In this research, we have presented the existence result for a new class of coupled systems of sequential ψ-Hilfer proportional fractional differential equations with nonlocal boundary conditions.The main existence result was proved via a Burton's generalization of Krasnosel'ski ȋ's fixed point theorem.The main result was illustrated by a numerical constructed example.Our results are new and enrich the existing literature on coupled systems of ψ-Hilfer proportional fractional differential equations.For special values of the parameters involved in the system at hand, we cover many new problems.Thus, by taking ψ(t) = t, our problem reduces to a coupled system of Hilfer generalized proportional fractional differential equations with boundary conditions, while if ρ = 1, reduces to a coupled system of ψ-Hilfer fractional differential equations.In future work, we can implement these techniques on different boundary value problems equipped with complicated integral multi-point boundary conditions.