On a fractional hybrid version of the Sturm–Liouville equation

It is well known that the Sturm–Liouville equation has many applications in different areas of science. Thus, it is important to review different versions of the well-known equation. The technique of α-admissible α-ψ-contractions was introduced by Samet et al. in (Nonlinear Anal. 75:2154–2165, 2012). Our aim in this work is to study a fractional hybrid version of the Sturm–Liouville equation by mixing the technique of Samet. In fact, by using the technique of α-admissible α-ψ-contractions, we investigate the existence of solutions for the fractional hybrid Sturm–Liouville equation by using the multi-point boundary value conditions. Also, we review the existence of solutions for a fractional hybrid version of the problem under the integral boundary value conditions. Finally, we provide two examples to illustrate our main results.


Introduction and preliminaries
What mathematics needs today is various applications to improve the standard of living of humanity. Although mathematics has had many uses in different fields so far, it can still have more beneficial effects in society. One of the most profitable ways to make mathematics more relevant in today's world is to produce modern software to reduce the consumption of minerals in chemical laboratories. Some chemistry experiments in software can be performed with high repeatability and by examining different pressure, temperature, and distinct conditions. It is a great advantage to do many experiments without the use of minerals. Computer software companies should pay particular attention to this issue.
It is logical that researchers concentrate on complicated fractional differential equations to increase their abilities for modeling of more real phenomena in the world. One of important methods in this way is working on different versions of well-known fractional differential equations. It is known that one of the famous ones is the Sturm-Liouville differential equation.
The Sturm-Liouville differential equation is an important differential equation in physics, applied mathematics, and other fields of engineering and science, and it has wide applications in quantum mechanics, classical mechanics, and wave phenomena (see, for example, [2] and [3] and the references therein). The existence of solutions and other properties for Sturm-Liouville boundary value problems have received considerable attention from many researchers during the last two decades (see, for example, [4][5][6][7][8][9][10][11][12][13][14][15][16][17]). Finally, a hybrid version of differential equations has a special appeal to everybody.
Nowadays, many researchers are currently studying various types of advanced mathematical modeling using fractional differential equations and its related inclusion version with more general boundary value conditions. Indeed, they try to model the processes so that it covers many general cases. In this situation, mathematicians would like to solve a wide range of these boundary value problems with advanced and complicate boundary conditions. Recently, many papers have been published on the existence of solutions for different fractional boundary value problems (see, for example, [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]). In the last few decades, fractional hybrid differential equations and inclusions with hybrid or non-hybrid boundary value conditions have received a great deal of interest and attention of many researchers (see, for example, [35][36][37][38][39][40][41]).

Lemma 1 ([1])
Let (X, d) be a complete metric space and T : X → X be an α-admissible α-ψ-contraction. Suppose that there exists x 0 ∈ X such that α(x 0 , Tx 0 ) ≥ 1 and α(x n , x) ≥ 1 for all n whenever {x n } is a sequence in X such that α(x n-1 , x n ) ≥ 1 for all n ≥ 1 and x n → x. Then T has a fixed point.

Main results
Now, we are ready to state and prove our main results. For study of problem (1), we consider the following hypotheses.
(D 1 ) The functions f ,f : R → R are differentiable on the interval [0, T] and ∂f ∂u and ∂f ∂u are bounded on [0, T] with ∂f ∂u ≤ K and ∂f ∂u ≤K, respectively. (D 2 ) The function p ∈ C 1 (I, R) has this property that p(t) = 0 for all t and inf t∈I |p(t)| = p.

p(t)
) t=0 , we get and so Thus, we obtain where = u(0) g(0,u(0)) . For simplicity, put A(t) = t 0p and By subtracting (6) from (7) and applying By substituting the value of in (5), we conclude that For the next part, by using (4) we have ). Hence, and so Thus, we obtain By using (4), we get ( (4) and (iii), we have Note that It is obvious that Finally, assume that ζ (t) = ( u(t) g(t,u(t)) ) . From (8) we know that ζ (t) ∈ C(I, R). Let (g(t, u(t))) ∈ C(I, R). Then . This completes the proof.
Now we are ready to state and prove our main result.

Theorem 3 Assume that hypotheses (D 1 )-(D 4 ) hold. Then the fractional hybrid Sturm-
Proof By using Lemma (6), problem (1) is equivalent to the integral equation (3). Define the map Θ : where By using (D 4 ), there exists r > 0 such that Consider the closed ball B r , where B r = {u ∈ X : u ≤ r}. Clearly, B r is a closed and bounded subset of X. Define the map α : |g(s, u(s))| ≤ φ u + g 0 , and |f (u(s))| ≤ K u + f 0 . We prove that the operator Θ satisfies the conditions of Lemma 1. We prove it in some steps.
Step 1: In this step, we prove Θu ≤ r whenever u ∈ B r . Let u ∈ B r . Then we have and Since and |E||ν| n j=1 Moreover, we have Since by using (10)-(18), we find |Hu(t)| ≤ A 1 r + A 2 , where Hence, Θu ≤ r and so ΘB r ⊆ B r .
Step 2: Let u, v ∈ B r . By using a similar method to that in step 1, we get Hence, |Hu(t) -Hv(t)| ≤ A 1 uv . This implies that By using the first step, Θu, Θv ∈ B r and so α(Θu, Θv) ≥ 1. Assume that {u n } is a sequence in C(I, R) such that α(u n-1 , u n ) ≥ 1 for all n ≥ 1 and u n → u ∈ C(I, R). Then {u n } is a sequence in B r . Since B r is closed, u ∈ B r and so α(u n , u) ≥ 1 for all n. Let u 0 ∈ B r ⊂ X.
Since ΘB r ⊂ B r , Θu 0 ∈ B r and so α(u 0 , Θu 0 ) ≥ 1. Now, by using Lemma 1, Θ has a fixed point in C(I, R) which is a solution for problem (1).
Example 1 Consider the fractional hybrid Sturm-Liouville differential equation
Also,p(t), q(t), and h(t) are absolutely continuous functions on I.

Corollary 1 Assume that hypotheses (M 1 )-(M 2 ) hold and there exists a number r
|η j | + 1 < 1, Then the fractional hybrid Sturm-Liouville differential equation with hybrid multi-point boundary condition has a solution u ∈ C 1 (I, R) if and only if u solves the integral equation Proof Note that problem (21)-(22) is a special case of problem (1) with g(t, x) = 1 for all t ∈ I and x ∈ R. Now, by using Theorem 3, we can conclude that problem (21)-(22) has a solution u ∈ C 1 (I, R).
In (L 2 ) The function p ∈ C 1 (I, R) with p(t) = 0 for all t ∈ I, inf t∈I |p(t)| = p. Also, q(t) and h(t) are absolutely continuous functions on I. (L 3 ) The function g : I × R → R {0} is continuous in its two variables and there exists a function φ(t) ≥ 0 (∀t ∈ I) such that |g(t, x)g(t, y)| ≤ φ(t)|x -y| for all (t, x, y) ∈ I × R × R. In this case, we obtain the next result.
Proof By a method similar to that in the proof of Corollary 1, we can conclude that problem 23 has a solution u ∈ C(I, R) (also, u ∈ C 1 (I, R) whenever (g(t, u(t))) ∈ C(I, R)).

Continuous dependence
In this section we are going to investigate continuous dependence(on the coefficient ξ i and η j of the hybrid multi-point condition) of the solution of the fractional hybrid Sturm-Liouville differential equation (21) with the hybrid multi-point boundary condition (22). Note that the main theorem of this section is a hybrid version of Theorem 3.2 in [42].

Fractional hybrid Sturm-Liouville equation with integral boundary value conditions
In this section, we investigate the fractional hybrid Sturm-Liouville equation with integral boundary value conditions.
Step 2: Let u, v ∈ B r . By using a method similar to that in step 1, we get