An Analysis on the Positive Solutions for a Fractional Configuration of the Caputo Multiterm Semilinear Differential Equation

In this paper, we consider a multiterm semilinear fractional boundary value problem involving Caputo fractional derivatives and investigate the existence of positive solutions by terms of different given conditions. To do this, we first study the properties of Green’s function, and then by defining two lower and upper control functions and using the wellknown Schauder’s fixed-point theorem, we obtain the desired existence criteria. At the end of the paper, we provide a numerical example based on the given boundary value problem and obtain its upper and lower solutions, and finally, we compare these positive solutions with exact solution graphically.


Introduction
At a vast level, it is understood that the hereditary properties and the memory of most processes, phenomena, and materials are predictable with the help of different modelings under the fractional operators. In this regard, differential equations involving fractional derivatives have recently been confirmed to be a useful tool in modeling of a considerable variety of structures in miscellaneous branches of sciences. For the sake of the increasing acceleration and advancements of studies and researches in the field of fractional calculus, several works have been done; see [1,2]. Since theoretical findings are used to achieve a deep understanding for the fractional models, a large number of mathematicians have also assigned their focus on studying the existence aspects of solutions for several structures of fractional equations by means of different techniques and methods. For instance, see [3][4][5][6][7][8][9][10].
In the next periods, a large number of researchers studied the notion of positive solutions for nonlinear fractional differential equations, and accordingly, many papers have been published in this direction. In 2003, Zhang [11] investigated the multiple and infinitely solvability of positive solutions for a nonlinear generalized fractional differential equation by relying on fixed point methods on cones. In 2007, El-Shahed [12] investigated the existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem in the Riemann-Liouville settings. The author used Krasnoselskii's fixed point theorem on cone preserving operators for deriving some required criteria. In [13], Guezane-Lakoud et al. presented a fourth-order mathematical model of elastic beam in three separate points of domain and studied the existence of positive solutions with the help of fixed point techniques.
In [14], Tian et al. turned to investigate positive solutions for a new class of fourpoint boundary value problem of fractional differential equations with p-laplacian operator and used the Leggett-Williams fixed point theorem on a cone to prove the multiplicity results of such solutions. More recently, Seemab et al. [15] established the existence results of positive solutions for a boundary value problem defined within generalized Riemann-Liouville and Caputo fractional operators by studying the properties of Green functions in three different types. Along with these, some other researchers investigated numerical methods and nonsingular fractional operators for obtaining numerical solutions of different fractional differential equations such as [16,17].
More specifically, in [18], Zhang studied the multiplicity and existence of positive solutions for the fractional nonlinear boundary value problem given by where D ν stands for the Caputo fractional derivative. To obtain the existence conditions, Zhang applied a method based on cones. Bai and Lu [19] also employed some nonlinear methods to establish the multiplicity and existence of positive solutions of the given problem as where D ν denotes the Riemann-Liouville fractional derivative and Ξ : ½0, 1 × ℝ >0 ⟶ ℝ >0 is a continuous function. Their method is based upon the reduction of the given boundary value problem to the equivalent Fredholm integral equation of the second kind. Inspired by the above works, in this paper, we derive some sufficient conditions to establish our main results on the existence of positive solutions to multiterm semilinear fractional boundary value problem given by where 1 < r < 2, 0 < p < 1 and Ξ is a continuous positive function on ½0, 1 × ℝ × ℝ and D ð·Þ denotes the Caputo fractional derivative. Note that the main method of this paper is to convert our multiterm semilinear boundary value problem (3) to an equivalent integral equation which allows us to convert it to a fixed point problem. In the following path, the criteria establishing the existence of positive solutions are guaranteed by imposing several sufficient conditions. In fact, we introduce two upper and lower control functions to achieve such aims, and after surveying some properties of the Green's function, the fundamental theorems in relation to the existence results are derived. To prove the existence results, by applying lower and upper control functions, we use the standard Schauder's fixed point theorem to obtain lower and upper solutions. In contrast to other works, we generalize and consider two terminal boundary conditions in the context of the Caputo fractional derivative of the unknown function and show our findings graphically. In other words, by plotting the graphs of lower and upper solutions, one can compare our results with the exact positive solution. It is notable that in the suggested problem (3), we have considered a multiterm semilinear boundary value problem, and in the future researches, one can implement this technique for the complicated boundary conditions and one can also cover other generalized nonlinear fractional boundary problems arising in real-world phenomena. The structure of the article is presented as follows: Sect. 2 recall the auxiliary and preliminary notions in relation to the fractional calculus and also some basic lemmas are provided. Sect. 3 is assigned to derive sufficient conditions to obtain positive solutions of the multiterm semilinear boundary value problem (3). Finally, a special example is provided to validate our theoretical findings according to the method implemented in theorems.

Basic and Auxiliary Concepts
Before proving some preliminary lemmas, we need to present some definitions and properties on fractional calculus which are useful throughout our research work.
Definition 1 (see [20]). Let ν > 0 and ϕ : ð0,+∞Þ ⟶ ℝ be continuous. The integral is called the fractional integral in the Riemann-Liouville settings of order ν such that it has finite values.
The following proposition is important and specifies the structure of the equivalent solution of the integral equation arising in the multiterm semilinear boundary value problem (3).

2
Journal of Function Spaces Proposition 5. Consider ϱ ∈ Cð½0, 1, ℝÞ and 1 < r < 2. Then, the solution of the linear problem is given by the following integral equation where Proof. If w is a solution of the linear boundary value problem (8), then from Proposition (7), it is followed that Then, the first boundary condition gives c 0 = 0. By applying the operator D r−1 to both sides of (11) and using (6), we find that which in view of the second boundary condition, gives By substituting c 0 and c 1 in (11), we get where Hðz, sÞ is given by (10). In this case, we follow that w will be a solution of (9). This completes the proof. ☐ Remark 6. It is easy to show by a simple computation that the function H satisfies Lemma 7. The function j∂Hðz, sÞ/∂zj is integrable for each z ∈ ½0, 1.
Proof. We have Then, This completes the proof. ☐ Remark 8. Consider the space X = C 1 ð½0, 1, ℝÞ. For 0 < p < 1 and w ∈ X, define the norm of w by Then, clearly, ðX, k:k X Þ is a Banach space.

Existence Criteria for Positive Solutions
In this section, several conditions are derived for which the existence of positive solutions to the multiterm semilinear boundary value problem (3) is guaranteed. Let α 1 , α 3 ∈ ℝ + and α 2 , α 4 ∈ ℝ with α 1 < α 3 and α 2 < α 4 . The upper control function and the lower control function b δ : respectively. We clearly have In addition to these, define the set 3 Journal of Function Spaces which is used in the sequel. Here, we mean by a positive solution, each function w satisfies w ∈ X, wð0Þ = 0 and wðzÞ > 0 for each 0 < z ≤ 1; in other words, w ∈Λ.
Theorem 9. Suppose that the assumptions (A1)−(A3) hold. Then, the multiterm semilinear boundary value problem (3) has at least a positive solution w in X such that all inequalities w ⋆ ðzÞ ≤ wðzÞ ≤ w ⋆ ðzÞ and D p w ⋆ ðzÞ ≤ D p wðzÞ ≤ D p w ⋆ ðzÞ hold for each 0 ≤ z ≤ 1.
Proof. For each ζ > 0, define the set Γ ζ as Obviously, Γ ζ is a convex, closed, and bounded set in X. Consider the operator P : Γ ζ ⟶ X under the following rule To prove Theorem 9, we will show that the hypotheses of Schauder's fixed point theorem hold. So, the process of proof will be done in several steps.
Step 1. P is continuous in X. To prove such a claim, we consider a sequence fw n g which converges to w in X. We have Journal of Function Spaces By tending n ⟶ ∞ and from the inequalities (29), (30), and (31), we follow that P is continuous in X.
In the sequel, we investigate the inequalities w ⋆ ðzÞ ≤ PwðzÞ ≤ w ⋆ ðzÞ and also D p w ⋆ ðzÞ ≤ D p PwðzÞ ≤ D p w ⋆ ðzÞ for each 0 ≤ z ≤ 1. Since w belongs to Γ ζ , we obviously have w ⋆ ðzÞ ≤ wðzÞ ≤ w ⋆ ðzÞ. By using definitions of upper and lower control functions together with the assumption (A1), we get Hence, we obtain w ⋆ ðzÞ ≤ PwðzÞ ≤ w ⋆ ðzÞ. Now, we need to show that D p w ⋆ ðzÞ ≤ D p PwðzÞ ≤ D p w ⋆ ðzÞ. We have Similarly, we showed that D p PwðzÞ ≥ D p w ⋆ ðzÞ. Therefore, PðΓ ζ Þ ⊆ Γ ζ .

Journal of Function Spaces
Step 3. At the final step, we aim to prove that P has the complete continuity property.

Conclusion
In this paper, we considered a new fractional class of the multiterm semilinear differential equation in the context of the standard Caputo differentiation operator. The main purpose here is to derive several criteria of the existence of positive solutions for mentioned multiterm boundary value problem. To achieve such an aim, we first obtained the relevant Green function of the equivalent integral equation and showed that the absolute value of the first-order partial derivative of this function is integrable. After introducing two lower and upper control functions, we defined an operator on the given Banach space and used Schauder's fixed point theorem for establishing the existence of positive solutions. With the help of a special numerical example at the end of the study, we validated our theoretical findings according to the method implemented here. By plotting upper and lower positive solutions and their derivatives, we compared our results with the exact solution graphically. The results and methods presented in this research can be widely applied in different complicated classes of boundary value problems involving modern integral conditions in the future works. Also, one can design such models by using new generalized fractional operators involving singular or nonsingular kernels for describing their dynamical behaviors in better settings.

Data Availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.