Investigation of Caputo proportional fractional integro-di ﬀ erential equation with mixed nonlocal conditions with respect to another function

: In this manuscript, we analyze the existence, uniqueness and Ulam’s stability for Caputo proportional fractional integro-di ﬀ erential equation involving mixed nonlocal conditions with respect to another function. The uniqueness result is proved via Banach’s ﬁxed point theorem and the existence results are established by using the Leray-Schauder nonlinear alternative and Krasnoselskii’s ﬁxed point theorem. Furthermore, by using the nonlinear analysis techniques, we investigate appropriate conditions and results to study various di ﬀ erent types of Ulam’s stability. In addition, numerical examples are also constructed to demonstrate the application of the main results.


Introduction
Fractional calculus (FC), also known as non-integer calculus, has been widely studied in recent over the past decades (beginning 1695) in fields of applied sciences and engineering. FC deals with fractional-order integral and differential operators, which establishes phenomenon model as an increasingly realistic tool for real-world problems. In addition, it has been properly specified the term "memory" particularly in mathematics, physics, chemistry, biology, mechanics, electricity, finance, economics and control theory, we recommend these books to readers who require to learn more about the core ideas of fractional operators [1][2][3][4][5][6]. However, recently, several types of fractional operators have been employed in research education that mostly focus on the Riemann-Liouville (RL) [3], Caputo [3], Hadamard [3], Katugampola [7], conformable [8] and generalized conformable [9]. In 2017, Jarad et al. [10] introduced generalized RL and Caputo proportional fractional derivatives including exponential functions in their kernels. After that, in 2021, new fractional operators combining proportional and classical differintegrals have been introduced in [11]. Moreover, Akgül and Baleanu [12] studied the stability analysis and experiments of the proportional Caputo derivative.
Exclusive investigations in concepts of qualitative property in fractional-order differential equations (FDEs) have recently gotten a lot of interest from researchers as existence property (EP) and Ulam's stability (US). The EP of solutions for FDEs with initial or boundary value conditions has been investigated applying classical/modern fixed point theorems (FPTs). As we know, US is four types like Hyers-Ulam stability (HU), generalized Hyers-Ulam stability (GHU), Hyers-Ulam-Rassias stability (HUR) and generalized Hyers-Ulam-Rassias stability (GHUR). Because obtaining accurate solutions to fractional differential equations problems is extremely challenging, it is beneficial in various of optimization applications and numerical analysis. As a result, it is requisite to develop concepts of US for these issues, since studying the properties of US does not need us to have accurate solutions to the proposed problems. This qualitative theory encourages us to obtain an efficient and reliable technique for solving fractional differential equations because there exists a close exact solution when the purpose problem is Ulam stable. We suggest some interesting papers about qualitative results of fractional initial/boundary value problems (IVPs/BVPs) involving many types of non-integer order, see [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43] and references therein.
We are going to present some of the researches that inspired this manuscript. In recent years, pantograph equation (PE) is a type of proportional delay differential equation emerging in deterministic situations which first studied by Ockendon and Taylor [44]: (1.8) The problem (1.8) has been a broad area of applications in applied branchs such as science, medicine, engineering and economics that use the sake of PEs to model some phenomena of the problem at present which depend on the previous states. For more evidences of PEs ,see [45][46][47][48][49][50][51]. There are many researches of literature on nonlinear fractional differential equations involving a specific function with initial, boundary, or nonlocal conditions, for examples in 2013, Balachandran et al. [52] discussed the initial nonlinear PEs as follows: where u 0 ∈ R, C D α denotes the Caputo fractional derivative of order α and h ∈ C([0, 1] × R 2 , R). FC and FPTs were applied to discuss the existence properties of the solutions in their work. In 2018, Harikrishman and co-workers [53] examined the existence properties of ψ-Hilfer fractional derivative for nonlocal problem for PEs: , u(σt)), t ∈ (a, b), σ ∈ (0, 1), α ∈ (0, 1) where H D α,β;ψ a + represents the ψ-Hilfer fractional derivative of order α and type β ∈ [0, 1], I 1−γ;ψ a + is RL-fractional integral of order 1 − γ with respect to ψ so that ψ > 0 and h ∈ C([a, b] × R 2 , R). Asawasamrit and co-workers [54] used Schaefer's and Banach's FPTs to establish the existence properties of FDEs with mixed nonlocal conditions (MNCs) in 2019. In 2021, Boucenna et al. [55] investigated the existence and uniqueness theorem of solutions for a generalized proportional Caputo fractional Cauchy problem. They solved the proposed problem based on the decomposition formula. Amongst important fractional equations, one of the most interesting equations is the fractional integrodifferential equations, which provide massive freedom to explain processes involving memory and hereditary properties, see [56,57].
Recognizing the importance of all parts that we mentioned above, motivated us to generate this paper which deals with the qualitative results to the Caputo proportional fractional integro-differential equation (PFIDE) with MNCs: where Cρ D q,ψ a + is the Caputo-PFDO with respect to another increasing differentiable function ψ of order q = {α, β j } via 0 < β j < α ≤ 1, j = 1, 2, . . . , n, 0 < ρ ≤ 1, 0 < λ < 1, ρ I p,ψ a + is the PFIO with respect to another increasing differentiable function ψ of order p = {ω, δ r } > 0 for r = 1, 2, . . . , k, 0 < ρ ≤ 1, We use the help of the famous FPTs like Banach's, Leray-Schauder's nonlinear alternative and Krasnoselskii's to discuss the existence properties of the solutions for (1.11). Moreover, we employ the context of different kinds of US to discuss the stability analysis. The results are well demonstrated by numerical examples at last section.
The advantage of defining MNCs of the problem (1.11) is it covers many cases as follows: • If we set κ j = σ r = 0, then (1.11) is deducted to the proportional multi-point problem.
• If we set γ i = σ r = 0, then (1.11) is deducted to the PFD multi-point problem.
• If we set γ i = κ j = 0, then (1.11) is deducted to the PFI multi-point problem.
This work is collected as follows. Section 2 provides preliminary definitions. The existence results of solutions for (1.11) is studied in Section 3. In Section 4, stability analysis of solution for (1.11) in frame of HU, GHU, HUR and GHUR are given is established. Section 5 contains the example to illustrate the theoretical results. In addition, the summarize is provided in the last part.

Preliminaries
Before proving, assume that E = C(J, R) is the Banach space of all continuous functions from J into R provided with u = sup t∈J {|u(t)|}. The symbol ρ I q,ψ a + [F u (s)(c)] means that In order to convert the considered problem into a fixed point problem, (1.11) must be transformed to corresponding an integral equation. We discuss the following key lemma.
Applying the nonlocal conditions in (2.2), we have Solving the above equation, we get the value where Ω is given as in (2.4). Taking c 1 in (2.5), we obtain (2.3).
On the other hand, it is easy to show by direct computing that u(t) is provided as in (2.3) verifies (2.2) via the given MNCs. The proof is done.

Existence results
By using Lemma 2.1, we will set the operator K : It should be noted that (1.11) has solutions if and only if K has fixed points. Next, we are going to examine the existence properties of solutions for (1.11), which is discussed by employing Banach's FPT, Leray-Schauder's nonlinear alternative and Krasnoselskii's FPT. For the benefit of calculation in this work, we will provide the constants: (3.5)

Uniqueness property
Firstly, the uniqueness result for (1.11) will be stidied by applying Banach's FPT.
Lemma 3.1. (Banach contraction principle [59]) Assume that B is a non-empty closed subset of a Banach space E. Then any contraction mapping K from B into itself has a unique fixed point.
Proof. First, we will convert (1.11) into u = Ku, where K is given as in (3.1). Clearly, the fixed points of K are solutions to (1.11). By using the Banach's FPT, we are going to prove that K has a FP which is a unique solution of (1.11).
For each u ∈ B r 1 , we obtain From the assumption (H 1 ), it follows that Then This implies that By using the fact of 0 < e ρ−1 Step II. We prove that K : From (H 1 ) again, we can compute that Then By inserting (3.9) into (3.8), one has that K is contraction. Then, (from Lemma 3.1), we conclude that K has the unique fixed point, that is the unique solution to (1.11) in E.

Existence property via Leray-Schauder's type
Next, Leray-Schauder's nonlinear alternative is employed to analyze in the second property.
(Leray-Schauder's nonlinear alternative [59]) Assume that E is a Banach space, C is a closed and convex subset of M, X is an open subset of C and 0 ∈ X. Assume that F : X → C is continuous, compact (that is, F(X) is a relatively compact subset of C) map. Then either (i) F has a fixed point in X, or (ii) there is x ∈ ∂X (the boundary of X in C) and ∈ (0, 1) with z = F(z).
Then the Caputo-PFIDE with MNCs (1.11) has at least one solution.
Proof. Assume that K is given as in (3.1). Next, we are going to prove that K maps bounded sets (balls) into bounded sets in E. For any r 2 > 0, assume that B r 2 := {u ∈ E : u ≤ r 2 } ∈ E, we have, for each t ∈ J,
Next, we will prove that there is B ⊆ E where B is an open set, u K(u) for ∈ (0, 1) and u ∈ ∂B. Assume that u ∈ E is a solution of u = Ku, ∈ (0, 1). Hence, it follows that Note that K : Q → E is continuous and completely continuous. By the option of Q, there is no u ∈ ∂Q so that u = Ku, ∃ ∈ (0, 1). Thus, (by Lemma 3.3), we conclude that K has fixed point u ∈ Q which verifies that (1.11) has at least one solution.

Existence property via Krasnoselskii's fixed point theorem
By applying Krasnoselskii's FPT, the existence property will be achieved.  [60]) Let M be a closed, bounded, convex and nonempty subset of a Banach space. Let K 1 , K 2 be the operators such that (i) K 1 x + K 2 y ∈ M whenever x, y ∈ M; (ii) K 1 is compact and continuous; (iii) K 2 is contraction mapping. Then there exists z ∈ M such that z = K 1 z + K 2 z. Theorem 3.6. Suppose that (H 1 ) holds and f ∈ C(J × R 4 , R) so that: then the Caputo-PFIDE with MNCs (1.11) has at least one solution.
Proof. Define sup t∈J |g(t)| = g and picking we consider B r 3 = {u ∈ E : u ≤ r 3 }. Define K 1 and K 2 on B r 3 as (3.2) and (3.3). For any u, v ∈ B r 3 , we obtain α+δ r ρ α+δ r Γ(α + δ r + 1) This implies that K 1 u + K 2 v ∈ B r 3 , which verifies Lemma 3.5 (i). Next, we are going to show that Lemma 3.5 (ii) is verified. Assume that u n is a sequence so that u n → u ∈ E as n → ∞. Hence, we get Since f is continuous, verifies that F u is also continuous. By the Lebesgue dominated convergent theorem, we have |(K 1 u n )(t) − (K 1 u)(t)| → 0 as n → ∞.
Thus, implies that K 1 u is continuous. Also, the set K 1 B r 3 is uniformly bounded as Next step, we will show the compactness of K 1 . Define sup{| f (t, u, v, w, z)|; (t, u, v, w, z) ∈ J × R 4 } = f * < ∞, thus, for each τ 1 , τ 2 ∈ J with τ 1 ≤ τ 2 , it follows that Clearly, the right-hand side of the above inequality is independent of u and |(K 1 u)(τ 2 ) − (K 1 u)(τ 1 )| → 0, as τ 2 → τ 1 . Hence, the set K 1 B r 3 is equicontinuous, also K 1 maps bounded subsets into relatively compact subsets, which implies that K 1 B r 3 is relatively compact. By the Arzelá-Ascoli theorem, then K 1 is compact on B r 3 .
Finally, we are going to show that K 2 is contraction. For each u, v ∈ B r 3 and t ∈ J, we get Since (3.10) holds, implies that K 2 is contraction and also Lemma 3.5 (iii) verifies. Therefore, the assumptions of Lemma 3.5 are verified. Then, (by Lemma 3.5) which verifies that (1.11) has at least one solution.

Stability results
This part is proving different kinds of US like HU stable, GHU stable, HUR stable and GHUR stable of the Caputo-PFIDE with MNCs (1.11).
there exists the solution u ∈ E of (1.11) so that there is the solution u ∈ E of (1.11) so that Definition 4.3. The Caputo-PFIDE with MNCs (1.11) is called HUR stable with respect to Φ ∈ C(J, R + ) if there is a constant ∆ f,Φ > 0 such that for every > 0 and for any the solution z ∈ E of (4.3) there is the solution u ∈ E of (1.11) so that there is the solution u ∈ E of (1.11) so that Remark 4.6. z ∈ E is the solution of (4.1) if and only if there is the function w ∈ E (which depends on z) so that: (i) |w(t)| ≤ , ∀t ∈ J; (ii) Cρ D α,ψ a + [z(t)] = F z (t) + w(t), t ∈ J. Remark 4.7. z ∈ E is the solution of (4.3) if and only if there is the function v ∈ E (which depends on z) so that:

HU stability and GHU stability
From Remark 4.6, the solution of , t ∈ J, can be rewritten as Firstly, the key lemma that will be applied in the presents of HU stable and GHU stable.
Next, we will show the HU and GHU stability results.
Proof. Assume that z ∈ E is the solution of (4.1) and assume that u is the unique solution of By using |x − y| ≤ |x| + |y| with Lemma 4.8, one has where Λ is given as in (3.5). This offers |z(t) − u(t)| ≤ ∆ f , where Then, the Caputo-PFIDE with MNCs (1.11) is HU stable. In addition, if we input Φ( ) = ∆ f via Φ(0) = 0, hence (1.11) is GHU stable.

The HUR stability and GHUR stability
Thanks of Remark 4.7, the solution can be rewritten as For the next proving, we state the following assumption: (H 5 ) there is an increasing function Φ ∈ C(J, R + ) and there is a constant n Φ > 0, so that, for each t ∈ J, Lemma 4.10. Assume that z ∈ E is the solution of (4.3). Hence, z verifies where Λ is given as in (3.5).
Finally, we are going to show HUR and GHUR stability results. Proof. Assume that > 0. Suppose that z ∈ E is the solution of (4.6) and u is the unique solution of (1.11). By using the triangle inequality, Lemma 4.8 and (4.11), we estamate that where Λ is given as in (3.5), verifies that Then, the Caputo-PFIDE with MNCs (1.11) is HUR stable. In addition, if we input Φ(t) = Φ(t) with Φ(0) = 0, then (1.11) is GHUR stable.

Numerical examples
This part shows numerical instances that demonstrate the exactness and applicability of our main results.
(I) If we set the nonlinear function For any u i , v i , w i , z i ∈ R, i = 1, 2 and t ∈ [0, 1], one has The assumption (H 1 ) is satisfied with L 1 = L 2 = L 3 = 1 27 . Hence All conditions of Theorem 3.2 are verified. Hence, the nonlinear Caputo-PFIDE with MNCs (5.1) has a unique solution on [0, 1]. Moreover, we obtain Hence, from Theorem 4.9, the nonlinear Caputo-PFIDE with MNCs (5.1) is HU stable and also GHU stable on [0, 1]. In addition, by taking Thus, (4.12) is satisfied with n Φ = 2 Hence, from Theorem 4.11, the nonlinear Caputo-PFIDE with MNCs (5.1) is HUR stable and also GHUR stable.
Example 5.2. Discussion the following linear Caputo-PFIDE with MNCs of the form: where We consider several cases of the following function ψ(t): (I) If ψ(t) = t α then the solution of linear Caputo-PFIDE with MNCs (5.2) is given as in where

Conclusions
The qualitative analysis is accomplished in this work. The authors proved the existence, uniqueness and stability of solutions for Caputo-PFIDE with MNCs which consist of multi-point and fractional multi-order boundary conditions. Some famous theorems are employed to obtain the main results such as the Banach's FPT is the important theorem to prove the uniqueness of the solution, while Leray-Schauder's nonlinear alternative and Krasnoselskii's FPT are used to investigate the existence results. Furthermore, we established the various kinds of Ulam's stability like HU, GHU, HUR and GHUR stables. Finally, by using Python, numerical instances allowed to guarantee the accuracy of the theoretical results.
This research would be a great work to enrich the qualitative theory literature on the problem of nonlinear fractional mixed nonlocal conditions involving a particular function. For the future works, we shall focus on studying the different types of existence results and stability analysis for impulsive fractional boundary value problems.