Existence of solutions for a higher order Riemann–Liouville fractional differential equation by Mawhin's coincidence degree theory

In this paper, we investigate the existence of at least one solution to the following higher order Riemann–Liouville fractional differential equation with Riemann–Stieltjes integral boundary condition at resonance: −(D0+αx)(t)=f(t,x(t),D0+α−1x(t)),n−1

To study problem (1.1)-(1.2), we apply Mawhin's coincidence theorem. We point out that Mawhin's coincidence theory is one of the powerful tool to study the existence theory of solutions of differential equations. This method have successfully been applied to differential equations and difference equations to study their positive solutions. It is recent that the method is applied to fractional differential equations to study the existence theory. There are relatively few papers on the use of Mawhin's coincidence theorem, as compared with the number dealing with boundary value problems 1-3 using different fixed point theorem. One may refer to previous studies 1,2,4-16 on the use of different fixed point theorems on Riemann-Liouville and Caputo-type fractional differential equations.
We now provide the list of articles dealing with the fractional differential equations, on the existence of solutions, using Mawhin's coincidence degree theory. Zou and He 17 used Mawhin's coincidence degree theory to obtain the existence of positive solutions of the problem x(t)), t ∈ (0, 1), where 2 < < 3, D 0+ x is the standard Riemann-Liouville fractional derivative x of order , ∶ [0, 1] × R 3 → R satisfies the Caratheodory conditions, A(t) is right continuous on [0, 1) and left continuous at t = 1, and ∫ 1 0 x(t)dA(t) is the Riemann-Stieltjes integral of x with respect to A. In an another work, Ma et al 11 used coincidence degree theory to study the problem x is the standard Caputo fractional derivative of x of the order , ∶ [0, 1] × R 3 → R satisfies the Caratheodory conditions, and A is as described in this article above. Jiang 18 studied the existence of solutions of the following fractional differential equations of Riemann-Liouville type at resonance using the coincidence degree theory due to Mawhin: The motivation for this present work has come from the above articles, [4][5][6][7][8][9][10][11][12][14][15][16] where the authors assumed that the integer k, defined in (1.2), is fixed, and used other fixed point theorems. In view of the above discussions, it seems that no work has been done on the existence of solutions for problem (1.1)-(1.2). Motivated by the papers, 11,17,18 we shall apply Mawhin's coincidence theory 19 to study problem (1.1)-(1.2) for ∈ (n − 1, n], n ≥ 2, and k is any integer between 0 and n − 1.
This work has been divided into five sections. Section 1 is introduction. All preliminary results and definitions are given in Section 2. The Mawhin's coincidence degree theorem is given in Section 2. The main results of this article is given in Section 3. Section 4 deals with the examples to apply our theorems obtained in Section 3. We have provided five examples for the the cases n = 2 and n = 3. Conclusion of this present work is given in Section 5.

PRELIMINARIES
In this section, we provide some definitions and lemmas that will be used to obtain the main results of this article. We start with classical definitions and properties of Riemann-Liouville fractional derivative and Riemann-Liouville fractional integrations of order > 0 with ∈ (n − 1, n]. Definition 1 (Previous studies 1-3 ). The (left-sided) fractional integral of the order > 0 of a function x ∶ (0, ∞) → R is given by provided the right-hand side integral is pointwise defined on (0, ∞) and Γ( ) is Euler gamma function defined by Definition 2 (Previous studies 1-3 ). The Riemann-Liouville fractional derivative of the order > 0 of a function x ∶ (0, 1] → R is given by ([ ] is the integer part of ) provided right-hand side derivative, which is pointwise defined on (0, ∞). [1][2][3]. The general solution to D 0+ x(t) = 0 with ∈ (n − 1, n] and n > 1 is of the form
Using the technique of Padhi et al, 12 we can prove the following lemma.
where k is an integer number between 0 and n − 1, in the form where G k (t, s), defined by is Green's function for problem (2.1).
Proof. Consider the triangle 0 < s < t < 1. In this triangle, Thus, G k (t, s) > 0 for 0 < s < t < 1. Obviously, G k (t, s) > 0 in the triangle 0 < t < s < 1. The lemma is proved. □ We now provide the essentials of the coincidence degree theory. Let X and Y be the real Banach spaces, and let L ∶ dom(L) ⊂ X → Y be Fredholm operator of index zero. If P ∶ X → X and Q ∶ Y → Y are two continuous projectors such that Im(P) = Ker(L), Ker(Q) = Im(L), X = Ker(L) ⊕ Ker(P), and Y = Im(L) ⊕ Im(Q), then the inverse operator of L| dom(L)∩Ker(P) ∶ dom(L) ∩ Ker(P) → Im(L) exists and is denoted by K p (generalized inverse operator of L). If Ω is an open bounded subset of X such that dom(L) ∩ Ω ≠ 0, the mapping N ∶ X → Y will be called L-compact onΩ, if QN(Ω) is bounded and K p (I − Q)N ∶Ω → X is compact. The abstract equation Lx = Nx is shown to be solvable in view of Theorem IV. 13. 19 Theorem 1 (Mawhin 19 ). Let L be a Fredholm operator of index zero and let N be the L-compact onΩ. Assume the following conditions are satisfied: Then, the equation Lx = Nx has at least one solution in dom(L) ∩Ω.
In this article, we use the classical Banach space Then, boundary value problem (1.1)-(1.2) becomes Finally, we define a generalized inverse operator

MAIN RESULTS
In this section, we prove our main results of this paper. We shall apply Theorem 1 to prove our theorem. For the rest of this section, we denote To study problem (1.1)-(1.2), we use the following assumptions to provide our results: (A5) There exists a constant B > 0 such that either of the following holds: for |c| > B and c ∈ R.
We start with the following lemma.

Lemma 7. K p is the inverse of L| dom(L)∩Ker(P) .
Proof. If ∈ Im(L), then For x ∈ dom(L) ∩ Ker(P) and Lx = , we have that is, K p = (L| dom(L)∩Ker(P) ) −1 . This completes the proof of the lemma. □ Next, using (2.3) and (2.4), we can write Moreover, Proof. We can use K p (t) from (2.4) and applying Lemma 3, we get D −1 for n ≥ 3, k = 0; and hold. Then, where Δ(n, k) is defined in (3.1). The proof is complete.

CONCLUSION
In this article, we have applied Mawhin's coincidence theorem to study the existence of a nontrivial solution for problem (1.1)-(1.2). An advantage of studying problem (1.1)-(1.2) is that we consider the parameter k to be any integer in between 0 and n − 1. Fractional calculus, which is an extension of ordinary calculus, in which the derivatives and integrals are defined for arbitrary real orders. [20][21][22] Due to the generalization nature of fractional calculus, it found its vital presence not only in pure mathematics but also in many real-world phenomenon. It is well known that many natural/physical processes such as fluid mechanics, electromagnetism, electrochemistry, viscoelasticity, population dynamics, and dynamical systems exhibit nonlocal effects. These nonlocal behavior can effectively be handled by fractional-order differential and integral operators. 23,24 Thus, the fractional calculus finds its way to modeling many physical, chemical, and natural systems, such as anomalous convection, ecological effects, infectious disease spreading, blood flow issues, and control phenomenon. 2,3,20,[25][26][27] Tenreiro Machado et al 28 present few other applications, namely, fractional control of hexapod robot, redundant manipulators, robot trajectory control, circuit synthesis, and heat diffusion. Further, fractional derivatives are useful in investigating various properties of materials and processes such as memory and hereditary. Recent studies by Abu Arqub et al [29][30][31][32] discuss the application of integrodifferenatial equations of fractional order and their numerical solutions. The applications of fractional differential calculus in studying the wave patterns are discussed in Goncalves and Zeidan 33 and Zeidan and Sekhar. 34 As the presence of fractional differential equations in various fields is increasing, it is essential to investigate the existence and stability of solutions of different fractional-order differential systems. One observes from the above articles, especially, on the application point of view, positive solutions are important. Since Mawhin's coincidence theorem is one of the powerful tool in the theory of differential equations to study the existence of solutions, it would be interesting to study the existence of positive solutions of (1.1)-(1.2) using the Mawhin's coincidence degree theory. This is an open problem.
Although we provide five examples for the cases n = 2 and n = 3 with various values of k according to the choices of n, it is always interesting to provide a numerical solution to the examples considered in Section 4. This is still an open problem.