Existence and stability results for non-hybrid single-valued and fully hybrid multi-valued problems with multipoint-multistrip conditions

In this paper, we study a new class of non-hybrid single-valued fractional boundary value problems equipped with integro-non-hybrid-multiterm-multipoint-multistrip conditions and a fully hybrid integro-multi-valued fractional boundary value problem by some new methods including the Kuratowski measures based on Sadovskii’s theorem, Krasnoselskii–Zabreiko criterion, and Dhage’s technique. We generalize the Gronwall inequality in relation to a non-hybrid single-valued fractional boundary value problem, and then we investigate the stability notions in two versions. To examine the correctness of the results, we provide some examples. MSC: 34A08; 34A12

boundary value problems (FBVPs). A large number of mathematicians have constructive contributions in this regard, and in all of them, a trace of fixed point theory and the relevant techniques is observed. Some of their published works can be found in [8][9][10][11][12][13][14][15][16].
The main contribution of this work is that we combine some well-known fractional structures in the framework of two generalized FBVPs. In fact, a combination of the nonhybrid equations, fully hybrid equations, integro-differentials, multistrip conditions in multipoint positions, and a generalized inclusion is investigated in the supposed FBVPs (1.1) and (1.2) in this manuscript. Regarding the first novelty of this work, to establish results in relation to the existence criteria for this new abstract model, some pure methods arising in functional analysis will help us in this direction. In other words, with the help of some properties of the Kuratowski measure and by defining the condensing selfmaps on a convex and closed set, we prove our first result by Sadovskii's theorem on FBVP (1.1). We even have tried to derive the required conditions confirming the dependence of solutions via the generalized inequality of Gronwall type. The second novelty of this study is to apply the inclusion type of Dhage's method for generalized fully hybrid integro-multivalued FBVP with the integro-hybrid-multiterm-multipoint-multistrip boundary conditions (1.2).
We organize the paper as follows: some preliminaries in relation to our methods and techniques are recollected in Sect. 2. We consider a non-hybrid single-valued FBVP (1.1) in Sect. 3, and with the help of Sadovskii's fixed point, we prove our result, and by applying the generalized inequality of Gronwall type, the dependence of solutions is investigated. Also, the Krasnoselskii-Zabreiko criterion gives another existence result for the non-hybrid single-valued case. The stability property in some versions is proved in Sect. 4. For the fully hybrid-multi-valued FBVP (1.2), some results are established in Sect. 5 via Dhage's method. Section 6 is devoted to preparing some examples in the direction of our results. We end our study in Sect. 7 by giving conclusions.
The Kuratowski measure of noncompactness (denoted by ω(W )) is defined by In the following lemmas, we consider Y as a Banach space.

Lemma 2.5 ([32]) For a bounded and equi-continuous set W
We recall Sadovskii's fixed point theorem by assuming the same hypothesis on Y given above.
so that q > 0. Then Note that the above inequality is known as the generalized Gronwall inequality.
then there is u * ∈ Y such that Ku * = u * .

Results for non-hybrid FBVP (1.1)
In this section, a non-hybrid version of the given FBVP, defined by (1.1), is studied. In other words, in the first step, we aim to investigate the existence of solution for the given non-hybrid single-valued FBVP with integro-non-hybrid-multiterm-multipointmultistrip boundary conditions (1.1).
In the first step, the first given initial condition u * (t)| t=0 = 0 yields α 0 = 0. So Further, since thus the second given condition (u * ) (t)| t=0 = 0 gives α 1 = 0 immediately. In consequence, On the other hand, we have and for j = 1, . . . , k, By considering the nonzero constant μ defined in (3.3) and by the third boundary condition, we obtain the following coefficient: By substituting the above value α 2 in (3.6), we have Thus we see thatũ satisfies (3.2) and it is the solution of the mentioned integral equation, and so the proof is completed.
Before starting our theorems, we introduce the Banach space The Kuratowski measure of noncompactness will help us to continue our research on the non-hybrid FBVP (1.1).

Theorem 3.2 Let h be a continuous function with the real values defined on I × Y. Moreover, assume that there is a continuous function
holds for any t ∈ I and u ∈ Y. Furthermore, assume that there is a function n h : , (3.10) and n * h = sup t∈I |n h (t)|.
Proof In relation to the non-hybrid-FBVP with integro-non-hybrid-multitermmultipoint-multistrip boundary conditions (1.1), by Lemma 3.1, we define P : To hold Theorem 2.8, we prove the continuity of P on B ε . Let {u n } n≥1 ⊂ B ε such that u n → u for all u ∈ B ε . For the sake of the continuity of the function h on I × Y, we obtain lim n→∞ h(t, u n (t)) = h(t, u(t)). By the dominated convergence theorem attributed to Lebesgue, it gives Hence, we get lim n→∞ (Pu n )(t) = (Pu)(t). Now, consider the member u ∈ B ε . By (3.7), we whereˆ is introduced in (3.10). In consequence, the above estimate becomes as t 1 tends to t 2 (independent of u ∈ B ε ). Hence P is equi-continuous. It follows that P is completely continuous by the Arzela-Ascoli theorem, and it is compact on B ε . We show that P is condensing on B ε . Lemma 2.4 gives this fact that there is a countable set W 0 = {u n } n≥1 ⊂ W for each bounded set W ⊂ B ε such that ω(P(W )) ≤ 2ω(P(W 0 )). Hence, by Lemmas 2.3, 2.5, and 2.6, we get the following inequalities: Hence, whereˆ is defined in (3.10). Accordingly, by (3.9), we get ω(P(W )) < ω(W ). So P is a condensing map on B ε . By Theorem 2.8, P has at least one fixed point in B ε , and accordingly, there is a solution for the given non-hybrid single-valued-FBVP with integro-non-hybridmultiterm-multipoint-multistrip boundary conditions (1.1).
In the present step, we aim to investigate the dependence of solutions for the non-hybrid single-valued FBVP with integro-non-hybrid-multiterm-multipoint-multistrip boundary conditions (1.1). Indeed, this part of the paper states that the solution of the non-hybrid single-valued FBVP (1.1) depends on some parameters so that the nonlinear map h satisfies Theorem 3.2, which ensures the existence of solutions, and the continuous dependence of solutions on the coefficients and orders gives the stability in relation to the solutions of (1.1). We act on the solutions of the non-hybrid single-valued FBVP with integronon-hybrid-multiterm-multipoint-multistrip boundary conditions (1.1) by changing the order of the non-hybrid single-valued FBVP (1.1) to a small value. The generalized Gronwall inequality will be useful for our purpose.
for all u, u ∈ Y and t ∈ I. Moreover, let u be the solution of the non-hybrid singlevalued FBVP with integro-non-hybrid-multiterm-multipoint-multistrip boundary conditions (1.1) and v be the solution of (3.13) Then the following inequality is valid: (3.14)
Proof Prior to proceeding to derive inequality (3.14), we know that the existence of solution for two non-hybrid single-valued-FBVPs (1.1) and (3.13) is guaranteed by the same proof done above, and so the solutions of these two non-hybrid-single-valued-FBVPs are obtained by (3.2) and respectively. Then, the following estimate for uv is calculated as follows: so that F and B are introduced by (3.15) and (3.16). Thus, by the generalized Gronwall inequality presented in Theorem 2.9 and by assumingȗ( , and the latter inequality completes the proof. The next fixed point theorem is due to Krasnoselskii and Zabreiko, and we prove our existence result with the help of it for the non-hybrid single-valued FBVP (1.1).

Theorem 3.4 Let
(J5) h : I × R → R be continuous and for some t ∈ I, h(t, 0) = 0 and Then there exists at least one solution for the non-hybrid single-valued FBVP (1.1) on I such that whereˆ is given by (3.10).
Proof Assume that {u n } n∈N tends to u. We know that h is continuous. As n → ∞, we get Thus, for t ∈ I, and by defining P : Y → Y given by (3.11), we write Thus (3.20) tends to zero. This yields the continuity of P. For r > 0, we set N = u ∈ C(J, R); u ≤ r and h = sup (t,u)∈I×N |h(s, u(s))|. So which yields Pu ≤ Aˆ r, whereˆ is given by (3.10). This gives the uniform boundedness of P.
The equicontinuity of P is established similar to the proof of Theorem 3.2. Immediately, the Arzelà-Ascoli theorem confirms the compactness of P on N . Now, by considering the non-hybrid single-valued FBVP (1.1), and by taking h t, u(t) = ρ(t)u(t), the operator L, by Lemma 3.1, is defined by We further claim that 1 is not an eigenvalue of L. If it is so, then by (3.19) we compute This is invalid. Hence our claim is correct. To conclude the proof, we claim that P(u) -L(u) / u vanishes when u → ∞. For t ∈ I, one may write This means that By (3.18) and letting u → ∞, it is concluded that | h(·,u(·)) u ρ(·)| → 0. Thus we obtain

Definition 4.1
The non-hybrid single-valued FBVP with integro-non-hybrid-multitermmultipoint-multistrip conditions (1.1) is Ulam-Hyers stable if there is 0 < σ h ∈ R such that, for each > 0 and for each u * (t) ∈ Q satisfying there is u(t) ∈ Q satisfying the non-hybrid single-valued FBVP (1.1) with

Definition 4.2
The non-hybrid single-valued FBVP via integro-non-hybrid-multitermmultipoint-multistrip conditions (1.1) is generalized Ulam-Hyers stable if there is σ h ∈ C(R + , R + ) with σ h (0) = 0 such that, for each > 0 and for each u * (t) ∈ Q satisfying the inequality Remark 4.4 Notice that u * (t) ∈ Q is a solution for (4.1) if and only if there is z ∈ Q depending on u * such that, for each t ∈ I, The Ulam-Hyers stability is discussed here for the non-hybrid single-valued FBVP (1.1).

Theorem 4.5 Suppose that there is a constant β > 0 such that
for each u, u ∈ Y and t ∈ I. Then the non-hybrid single-valued FBVP with integro-nonhybrid-multiterm-multipoint-multistrip conditions (1.1) is Ulam-Hyers stable on I and is the generalized Ulam-Hyers stable provided that βˆ < 1, whereˆ is given by (3.10).
Proof For every > 0 and for each u * (t) ∈ C(I, R) satisfying we can find a function z(t) satisfying If u is the unique solution of the non-hybrid single-valued FBVP (1.1), then u(t) is given by Then Consequently, whereˆ is defined in (3.10). In consequence, it follows that If we let σ h =ˆ 1-βˆ , then its Ulam-Hyers stability is proved. Further, for with σ h (0) = 0, the generalized Ulam-Hyers stability will be proved.

Lemma 5.1 Let
which is given as where μ is a constant given by (3.3).
Proof Let the functionũ be the solution of the linear hybrid-FDE (5.1). Then By utilizing I q 0 on both sides of the hybrid differential equation (5.3), we get for some α 0 , α 1 , α 2 ∈ R.
In the first step, the first given initial condition (ũ (t) y(t,ũ(t)) )| t=0 = 0 yields α 0 = 0. Sõ Further, since thus the second given condition (ũ (t) y(t,ũ(t)) ) | t=0 = 0 gives α 1 = 0. In consequence, and for i = 2, . . . , p, By assuming the nonzero constant μ given by (3.3) and by the third condition, we obtain the following coefficient: By substituting the above value α 2 in (5.5), from which we see thatũ satisfies (5.2) and it is the solution of the mentioned integral equation, and so the proof is completed.
Based on the above lemma, we aim to define the solution of supposed fully hybrid integro-multi-valued FBVP (1.2).

Definition 5.2
The absolutely continuous map u : I → R is called a solution to the fully hybrid integro-multi-valued FBVP (1.2) if an integrable mapping κ ∈ L 1 (I, R) with κ(t) ∈ G t, u(t), 1 0 u(s) ds for almost all t ∈ I satisfies integro-hybrid-multiterm-multipoint-multistrip boundary conditions and The first theorem in relation to the inclusion problem (1.2) is proved here.

Theorem 5.3
Let G : I × R 2 → P cp,cv (R) and y : I × R → R\{0} be continuous and: (J1) There is M : I → R + (it is bounded) so that for each u, v ∈ R and for all t ∈ I, we have for all u ∈ R and for almost all t ∈ I; (J4) There isã ∈ R + such that a > y * ˆ

then the fully hybrid integro-multi-valued FBVP with integro-hybrid-multitermmultipoint-multistrip boundary conditions (1.2) has a solution.
Proof For each u ∈ Y, we define as the selections of G for almost all t ∈ I, and define E : Y → P(Y) by for all t ∈ I. By this structure, h 0 satisfies the fully hybrid integro-multi-valued FBVP (1.

2) if and only if Eh
for all t ∈ I. Then, we obtain E(u) = (F 1 u)(F 2 u). We show that both operators F 1 and F 2 satisfy Theorem 2.2. We show that F 1 is Lipschitz. Let u 1 , u 2 ∈ Y. Thus assumption (J1) implies that for all t ∈ I. Hence, we get for all u 1 , u 2 ∈ Y. Thus F 1 is Lipschitz with the constant M * > 0. Further, we claim that F 2 is convex-valued. Let u 1 , u 2 ∈ F 2 u. Choose κ 1 , κ 2 ∈ S G,u such that for almost all t ∈ I. Let c ∈ (0, 1). Then As G is convex-valued, S G,u is too, and this gives cκ 1 (t) + (1c)κ 2 (t) ∈ S G,u , and so F 2 u is convex for each u ∈ Y.
We investigate the complete continuity of F 2 . For ε * ∈ R + , set For every u ∈ V ε * and ∈ F 2 u, there is κ ∈ S G,u such that whereˆ is given in (5.7). Thus, ≤ˆ L 1 and F 2 (Y) is uniformly bounded. Let u ∈ V ε * and ∈ F 2 u. Choose κ ∈ S G,u such that for all t ∈ I. Assume that t 1 , t 2 ∈ I with t 1 < t 2 . Then we have as t 2 → t 1 (independent of u ∈ V ε * ). The Arzela-Ascoli theorem gives the complete continuity of F 2 . Assume that u n ∈ V ε and n ∈ (F 2 u n ) with u n → u * and n → * . We claim that * ∈ (F 2 u * ). For every n ≥ 1 and n ∈ (F 2 u n ), choose κ n ∈ S G,u n such that We claim that there is κ * ∈ S G,u * such that * (t) = 1 (q) t 0 (ts) q-1 κ * (s) ds Theorem 2.1 implies that F • S G has a closed graph. On the other hand, n ∈ F(S G,u n ) and u n → u * . So there exists κ * ∈ S G,u * such that for all t ∈ I. Hence, * ∈ (F 2 u * ) and so F 2 has a closed graph, and it is upper semicontinuous. Therefore F 2 is compact and upper semi-continuous. By (J1), Thenˆ M * < 1 2 . By Theorem 2.2 in relation to F 2 , one of (a) or (b) will be held. By (J4), let u ∈ Q * be such that u =ã. Then ρ * u(t) ∈ (F 1 u)(t)(F 2 u)(t) for all ρ * > 1. Choose κ ∈ S G,u . Then, for each ρ * > 1,  Evidently, M * = sup t∈I |M(t)| 0.0001971. We define G : I × R 2 → P(R) by We selectã > 0 such thatã > 0.0020017. Also, by the above given values for the parameters, we findˆ 3.89692. Therefore The conclusion of Theorem 5.3 gives this fact that there exists a solution for the fully hybrid-FBVP inclusion with integro-hybrid-multiterm-multipoint-multistrip boundary conditions (6.3).

Conclusions
In the present manuscript, two novel generalized non-hybrid single-valued FBVP and fully hybrid integro-multi-valued FBVP with integro-hybrid-multiterm-multipoint-multistrip boundary conditions were considered and the qualitative results were proved in relation to its solutions. Precisely, on the non-hybrid-multi-valued FBVP (1.1), we established an existence theorem based on Sadovskii's method, and in the sequel, the Krasnoselskii-Zabreiko theorem was utilized for the second existence result. We got help from the Gronwall inequality in its generalized version to investigate the dependence of solutions of the nonhybrid multi-valued FBVP (1.1). Stability analysis was implemented in the sense of Ulam-Hyers. Further, on the fully hybrid multi-valued FBVP (1.2), we derived the corresponding multipoint integral equation and used Dhage's techniques to establish the third existence theorem. Two numerical examples have been designed to examine the correctness of theorems. Our boundary conditions are general and cover different simple forms defined in numerous FBVPs. We will continue our study in the context of newly-defined notions of q-calculus and (p, q)-calculus.