On the existence of solutions for nonlocal sequential boundary fractional differential equations via ψ -Riemann–Liouville derivative

The purpose of this paper is to study a generalized Riemann–Liouville fractional diﬀerential equation and system with nonlocal boundary conditions. Firstly, some properties of the Green function are presented and then Lyapunov-type inequalities for a sequential ψ -Riemann–Liouville fractional boundary value problem are established. Also, the existence and uniqueness of solutions are proved by using Banach and Schauder ﬁxed-point theorems. Furthermore, the existence and uniqueness of solutions to a sequential nonlinear diﬀerential system is established by means of Schauder’s and Perov’s ﬁxed-point theorems. Examples are given to validate the theoretical results.

Inspired and motivated by [26,30] and the aforementioned works, in the first part of this paper, we study Lyapunov-type inequalities for a sequential ψ-RL FBVP under nonlocal boundary condition where Then, we present some existence results to problem (10) by applying Banach's and Schauder's fixed point theorems.It should be noted that our results extend and complete those obtained in [16].
Next, we investigate the existence and uniqueness of solutions for the following sequential ψ-RL nonlinear system for ω ∈ (a, b) and In [34], the authors introduced a most generalized variant of the Hilfer derivative, namely (k, ψ)-Hilfer fractional derivative.Clearly, the ψ-Riemann-Liouville derivatives are a special case of (k, ψ)-Hilfer fractional derivative, see also [35].Our main tools here are Schauder's and Perov's fixed point theorems.At the end of this paper, we give some examples to show the applicability of our results.

Lemma 2.1 ([1]
) Let λ 1 , λ 2 > 0 and : → R be an integrable function.Then we have Theorem 2.3 (Banach's fixed-point theorem [37]) Consider a Banach space X B and let A : X B → X B be an operator for which A n is a contraction, where n is a positive integer, then A has a unique fixed point.
Theorem 2.4 (Schauder's fixed-point theorem [38]) Consider a Banach space X B , a closed bounded convex subset D = ∅ of it and A : D → D a completely continuous operator.Then A has at least one fixed point.
A square matrix M of real numbers is said to be convergent to zero if M k → 0, as k → ∞.In other words, this means that its spectral radius ρ(M) is strictly less than 1.
Theorem 2.5 ([39, 40]) For any nonnegative square matrix M, the following assertions are equivalent (i) M is convergent to zero; (ii) The matrix I -M is nonsingular and (I - The eigenvalues of M are located inside the open unit disc of the complex plane; (iv) The matrix I -M is nonsingular and (I -M) -1 has nonnegative elements.

Lemma 2.6 ([39]
) If M is a square matrix that converges to 0 and the elements of another square matrix M are small enough, then M + M also converges to 0.
Let X be a nonempty set.A vector-valued metric on X is a map d : X × X → R n s.t has non-negative, symmetry and zero properties along with triangle inequality.The pair (X , d) is called a generalized metric space.Furthermore, the convergence and completeness are similar to those in usual metric spaces.If ṙ, ṡ ∈ R n , ṙ = (r 1 , r 2 , . . ., r n ), ṡ = (s 1 , s 2 , . . ., s n ), by ṙ ≤ ṡ we mean r ι ≤ s ι for ι = 1, 2, . . . ,n.An operator A : (X , d) → (X , d) is said to be contractive if there exists a convergent to 0 matrix M s.t Theorem 2.7 (Perov's fixed-point theorem [39,41]) Let (X , d) be a generalized metric space and A : X → X a contractive operator with Lipschitz matrix M. Then A has a unique fixed point and for each ∈ X , we have has an integral solution given by where κ (ω) := ψ a (ω) ψ a (κ) and Proof Firstly, we apply R L I λ,ψ a + to both sides of (17), we get From Lemma 2.2, we may reduce (21) to an equivalent integral equation where c 1 , c 2 ∈ R are arbitrary constants.By using the boundary conditions (a) = 0, we get c 2 = 0.Then, Eq. ( 22) takes the following form From (b) = ( (η)), we have Thus, the solution of FBVP ( 17)-( 18) is Now we derive some properties of Green's functions G(ω, ξ ).

Lyapunov-type inequality and existence results for problem (10)
In this section, we investigate Lyapunov-type inequality and present existence results for our problem.Consider To state our main results on Lyapunov-type inequality, we assume that the following condition holds: (H 1 ) There exists a function q ∈ C( , R), and δ > 0 s.t Theorem 3.1 Assume that (H 1 ) holds and 2|γ | ψ a (b) + δ < 1.If the sequential ψ-RL FBVP (10) has a nontrivial solution on , then Proof Assume that (ω) is a nontrivial solution of problem (10).Since ψ is an increasing function on , from (H 1 ) and ( 30), we obtain that is Thus, inequality (32) holds.
Remark 3.1 (i) For ψ(ω) = ω, inequality (32) can be rewritten as (iii) If v e is an eigenvalue of the problem (10), i.e., q(ξ ) = v e for each ξ ∈ , then we have Next, to present existence results, we make the following assumptions.
for any ω ∈ , 1 , 2 ∈ R; for any ω ∈ , ∈ R. In the following we give a result on the existence and uniqueness of solutions via the generalization of Banach contraction principle.Theorem 3.2 Assume that (H 2 ) holds.Then the sequential ψ-RL FBVP (10) has a unique solution on , whenever * 1 Proof Clearly, the existence of a solution for ψ-RL FBVP (10) is equivalent to the existence of a fixed point for operator A. For all 1 , 2 ∈ X B and ω ∈ , using assumption (H 2 ), we have Then Similarly, we obtain Using an induction method, we get A n 1 -A n 2 ≤ κn 1 -2 , for any n ∈ N. According to (40), since κ < 1, then we can choose enough large n, s.t κn < 1.By means of Theorem 2.3, the operator A has a unique fixed point, giving a unique solution to problem (10).Theorem 3.3 Assume that (H 3 ) and the following assertions hold: where , then the sequential ψ-RL FBVP (10) has at least one solution on .
≤ M}.Also, define the map A : B M → X B as follows: Clearly, B M is a closed convex subset of X B .For ∈ B M and ω ∈ , we have Then, we have A ≤ M, for each ∈ B M .Thus, A maps B M into itself.Now, we show that the operator A is completely continuous.We divide the proof into three steps.
Step 1: Let n be a sequence s.t n → , as n → ∞ in B M .From the continuity of the functions , and Lemma 2.9, for each ω ∈ , we have which implies that Therefore, the operator A is continuous.
Step 2: From (45), it follows immediately that A maps bounded sets into bounded sets in X B .
Step 3: Next, we will show that A(B M ) is uniformly bounded and equicontinuous.For a ≤ ω 1 ≤ ω 2 ≤ b and each ∈ B M , using Lemma 2.9, assumption (H 3 ), and the fact that Now, we will consider three cases.
In this case, we have Case 2: a ≤ ξ ≤ ω 1 ≤ ω 2 ≤ b.In this case, we have Using a similar argument to that have used to prove the second case, we have Thus, |(Au)(ω 2 ) -(Au)(ω 1 )| → 0 uniformly as ω 2 → ω 1 .Then, A is a completely continuous operator.Schauder's fixed point theorem 2.4 guarantees that A has a fixed point, which is a solution of the sequential ψ-RL fractional boundary value problem (10).

Existence results for system (11)
In this section, we present some existence results for the nonlocal sequential system of fractional differential equations via ψ-RL derivative (11).Let A : X 2 B → X 2 B be the operator defined as A = (A 1 , A 2 ), where A 1 , A 2 are given by It is worth noting that a solution of the system (11) can be considered as a fixed point in X 2 B for the completely continuous operator A.
Firstly, an existence result can be obtained for system (11) by applying Schauder's fixedpoint theorem in X 2 B endowed with vector-valued norm For this, we assume the following assumptions (H 4 ) Theorem 4.1 Assume that (H 4 ) and the following assertion hold: (H 5 ) The matrix M defined as converges to zero, where τ ij = max ω∈ τ ij (ω) and i = 1, 2, j = 1, 3. Then system (11) Proof According to the proof of Theorem 3.3, we can easily check that A is completely continuous.Next, we have to show that A has a fixed point in a bounded, closed and convex subset of X 2 B of the form where For each 1 , 2 ∈ X B and ω ∈ , we have that Then, we obtain Similarly, we get If ( 1 , 2 ) ∈ B, then Eqs. ( 60) and ( 61) can be rewritten as or equivalently Therefore, from (58), we obtain Next, we use Perov's fixed-point theorem to prove the existence of unique solution of the sequential system (11).Before we state our result, we list the following assumptions on ι and ι, ι = 1, 2 for convenience: Theorem 4.2 Assume that (H 5 ) and the following assertion hold: (H 6 ) The matrix M defined as converges to zero, where Proof We shall prove that A is a generalized contraction.Let ( 1 , 2 ), ( 1 , 2 ) ∈ X 2 B and ω ∈ .Using (H 5 ), we have Therefore Similarly, one has Equations ( 68)-(69) can be expressed as This shows that A is a generalized contraction.Thus, by applying Perov's fixed-point theorem, the operator A has a unique fixed point, which is a solution to system (11).
The next example shows the correctness of the results based on the changes of function and .
In the next example, we check the provided Theorem 4.1 for the sequential ψ-RL nonlinear system with the changes of the ψ function.

Conclusion
In this paper, we first establish the Lyapunov-type inequalities of sequential ψ-RL FBVPs, and then using the Banach contraction principle and Schauder's fixed point, we present two results of the existence and uniqueness of solutions for our problem.Also, two results on the existence of at least one solution, and uniqueness for a sequential nonlinear system of ψ-RL fractional derivative are obtained by applying Schauder's and Perov's fixed-point theorems.Examples illustrating the theoretical main results are given.
) So, we have shown that A(B) ⊂ B. Hence, A has a fixed point by Schauder's fixed-point Theorem 2.4.