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

Authors: Sh. Rezapour; B. Azzaoui; B. Tellab; S. Etemad; H. P. Masiha
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

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.


---

## Body

## 1. 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 [3–10].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
(1)Dvwz=Ξz,wz,1<v<2,0≤z≤1w0+w′0=0,w1+w′1=0,where Dν 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
(2)DRvwz=Ξz,wz,0≤z≤,1<v≤2,w0=0,w1=0,where Dν denotes the Riemann-Liouville fractional derivative and Ξ:0,1×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(3)Drwz=Ξz,wz,Dpwz,z∈J¯=0,1,w0=0,Dr−1w1=0,where 1<r<2, 0<p<1 and Ξ is a continuous positive function on 0,1×ℝ×ℝ and D· 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.


## 2. 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]).
Letν>0 and ϕ:0,+∞⟶ℝ be continuous. The integral
(4)Ivφz=1Γv∫0zz−sv−1φsds,is called the fractional integral in the Riemann-Liouville settings of order ν such that it has finite values.Definition 2 (see [20]).
Letk−1<ν<k with k=ν+1 and ϕ:0,+∞⟶ℝ belongs to ACk0,∞,ℝ. Then
(5)Dγφz=1Γk−v∫0zz−sk−v−1φksds,is called the Caputo fractional derivative of order ν such that it exists.Remark 3.
We have the following:
(RE1) For0≤ν<γ, the equality DνIγφz=Iγ−νφz holds
(RE2) Forγ>−1 with γ≠ν−jj=1,2,⋯,n and for each z≥0, we have
(6)Dvzγ=Γ1+γΓγ−v+1zγ−vandDvzv−1=0,j=1,2,⋯,n.Proposition 4 (see [21]).
Suppose thatϕ is contained in the space L0,1∩C0,1 and k=ν+1. Then
(7)IγDγwz=wz+c0+c1z+c2z2+⋯+ck−1zk−1,such that c0,c2,⋯,ck−1∈ℝ.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).Proposition 5.
Considerϱ∈C0,1,ℝ and 1<r<2. Then, the solution of the linear problem
(8)Drwz=ϱz,z∈J¯w0=0,Dr−1w1=0,is given by the following integral equation
(9)wz=∫01Hz,sϱsds,where
(10)Hz,s=z−sr−1Γr−Γ3−rz,0≤s≤z≤1−Γ3−rz,0≤z≤s≤1.Proof.
Ifw is a solution of the linear boundary value problem (8), then from Proposition (7), it is followed that
(11)wz=c0+c1z+Irϱz=c0+c1z+1Γr∫0zz−sr−1ϱsds.
Then, the first boundary condition givesc0=0. By applying the operator Dr−1 to both sides of (11) and using (6), we find that
(12)Dr−1wz=c1Γ3−rz2−r+I0+1ϱz,which in view of the second boundary condition, gives
(13)c1=−Γ3−r∫01ϱsds.
By substitutingc0 and c1 in (11), we get
(14)wz=1Γr∫0zz−sr−1ϱsds−Γ3−1z∫01ϱsds=∫01Hz,sϱsds,where Hz,s 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 functionH satisfies
(15)∫01Hz,sds≤1Γr−1+Γ3−r.Lemma 7.
The function∂Hz,s/∂z is integrable for each z∈0,1.Proof.
We have(16)∂Hz,s∂z=z−sr−2Γr−1−Γ3−r,0≤s≤z≤1,−Γ3−r,0≤z≤s≤1.
Then,(17)∫01∂Hz,s∂zds≤∫0zz−sr−2Γr−1ds+∫0zΓ3−rds+∫z1Γ3−rds=zr−1Γr+Γ3−rz+Γ3−r1−z≤1Γr+Γ3−r<∞.
This completes the proof.☐Remark 8.
Consider the spaceX=C10,1,ℝ. For 0<p<1 and w∈X, define the norm of w by
(18)wX=maxz∈0,1wz+maxz∈0,1w′z+maxz∈0,1Dpwz.Then, clearly,X,.X is a Banach space.


## 3. 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 α1,α3∈ℝ+ and α2,α4∈ℝ with α1<α3 and α2<α4. The upper control function
(19)∆^:0,1×α1,+∞×α2,+∞⟶ℝ+,and the lower control function δ^:0,1×−∞,α3×−∞,α4⟶ℝ+ is defined by
(20)∆^z,u,v=supα1≤θ≤uα2≤μ≤vΞz,θ,μandδ^z,u,v=infu≤θ≤α3v≤μ≤α4Ξz,θ,μ,respectively. We clearly have
(21)δ^z,u,v≤Ξz,u,v≤∆^z,u,v,for0≤z≤1,α1≤u≤α3,α2≤v≤α4.In addition to these, define the set(22)Λ~=w∈X:wz≥0,0≤z≤1,which is used in the sequel. Here, we mean by a positive solution, each function w satisfies w∈X,w0=0 and wz>0 for each 0<z≤1; in other words, w∈Λ~.


### 3.1. Required Assumptions

Now, for our main results, we need some assumptions given as follows:(A1) There arew⋆,w⋆∈Λ~ which satisfy α1≤w⋆z≤w⋆z≤α3 and α2≤Dpw⋆z≤Dpw⋆z≤α4 along with
(23)w⋆z≥∫01Hz,s∆^s,w⋆s,Dpw⋆sds,w⋆z≤∫01Hz,sδ^s,w⋆s,Dpw⋆sds,Dpw⋆z≥−1Γr−p∫0zz−sr−p−1Δ^s,w⋆s,Dpw⋆sds+Γ3−rΓ2−pz1−p∫01δ^s,w⋆s,Dpw⋆sds,Dpw⋆z≤−1Γr−p∫0zz−sr−p−1δ^s,w⋆s,Dpw⋆sds+Γ3−rΓ2−pz1−p∫01∆^s,w⋆s,Dpw⋆sds.(A2) There existξ>0 and nonnegative function θ∈L10,1 such that
(24)Ξz,w,v≤θz+ξw+v,0≤z≤1,w,v∈ℝ.(A3) There existsζ>0 such that
(25)A+B+BΓ2−p+ξζ1Γr+1Γr+1+2Γ3−r+1Γ2−pΓr+Γ3−rΓ2−p≤ζ.with
(26)A=maxz∈0,1∫01Hz,sθsdsandB=maxz∈0,1∫01∂Hz,s∂zθsds.At this moment, we are ready to present the first existence theorem.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⋆z≤wz≤w⋆z and Dpw⋆z≤Dpwz≤Dpw⋆z hold for each 0≤z≤1.Proof.
For eachζ>0, define the set Γζ as
(27)Γζ=w∈Λ~:wX≤ζ,w⋆z≤wz≤w⋆z,Dpw⋆z≤Dpwz≤Dpw⋆z,0,≤z≤1.
Obviously,Γζ is a convex, closed, and bounded set in X. Consider the operator P:Γζ⟶X under the following rule
(28)Pwz=−1Γr∫0zz−sr−1Ξs,ws,Dpwsds+Γ3−rz∫01Ξs,ws,Dpwsds=∫01Hz,sΞs,ws,Dpwsds.
To prove Theorem9, 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 wn which converges to w in X. We have
(29)Pwnz−Pwz=∫01Hz,sΞs,wns,Dpwns−Ξs,ws,D0+pwsds≤maxz∈0,1Ξz,wnz,Dpwnz−Ξz,wt,Dpwnz∫01Hz,sds≤1Γr+1+Γ3−rmaxz∈0,1Ξz,wnz,Dpwnz−Ξz,wz,Dpwz,(30)DpPwnz−DpPwz=1Γ1−p∫0zz−s−pPwn′s−Pw′sds≤Γ1−p∫0zz−s−p∫01∂Hs,λ∂sΞλ,wnλ,Dpwnλ−Ξλ,wλ,Dpwλdλds≤maxz∈0,1Ξz,wnz,Dpwnz−Ξz,wz,Dpwz1Γ1−p∫0zz−s−p∫01∂Hs,λ∂sdλds≤maxz∈0,1Ξz,wnz,Dpwnz−Ξz,wz,Dpwz1Γ1−p1Γr+Γ3−rΓ2∫0zz−s−pds≤1Γ2−p1Γr+Γ3−rmaxz∈0,1Ξz,wnz,Dpwnz−Ξz,wz,Dpwz,(31)Pwn′z−Pw′z=∫01∂Hz,s∂zΞs,wns,Dpwns−Ξs,ws,Dpwsds≤maxz∈0,1Ξz,wnz,Dpwnz−Ξz,wz,Dpwz∫01∂Hz,s∂zds≤1Γr+Γ3−rwz∈0,1Ξz,wnz,Dpwnz−Ξz,wz,Dpwz.
By tendingn⟶∞ and from the inequalities (29), (30), and (31), we follow that P is continuous in X.Step 2.
Now, we show thatP:Γζ⟶Γζ is a selfmap on Γζ. Let w∈Γζ. By inequalities (15) and (17) along with the assumptions (A2) and (A3), we get
(32)Pwz=∫01Hz,sΞs,ws,Dpwsds≤∫01Hz,sΞs,ws,Dpwsds≤∫01Hz,sθs+ξws+Dpwsds≤∫01Hz,sθsds+ξζ∫01Hz,sds≤B+ξζ1Γr+Γ3−r,(33)Pw′z=∫01∂Hz,s∂zΞs,ws,Dpwsds≤∫01∂Hz,s∂zθs+ξws+Dpwsds≤∫01∂Hz,s∂zθsds+ξζ∫01∂Hz,s∂zds≤B+ξζ1Γr+Γ3−r,(34)DpPwz=1Γ1−p∫0zz−s−pPw′sds≤1Γ1−p∫0zz−s−p∫01∂Hs,λ∂sΞλ,wλ,Dpwλdλds≤1Γ1−p∫01z−s−p∫01∂Hs,λ∂sθλdλ+∫01∂Hs,λ∂sξxλ+Dpwλdλds≤BΓ1−p∫0zz−s−pds+ξζΓ1−p∫0zz−s−p∫01∂Hs,λ∂sdλds≤BΓ2−pz1−p+ξζΓ2−p1Γr+Γ3−rz1−p≤BΓ2−p+ξζΓ2−p1Γr+Γ3−r.
By virtue of inequalities (32), (33), (34), and the assumption (A3), we get PxX≤ζ.
In the sequel, we investigate the inequalitiesw⋆z≤Pwz≤w⋆z and also Dpw⋆z≤DpPwz≤Dpw⋆z for each 0≤z≤1. Since w belongs to Γζ, we obviously have w⋆z≤wz≤w⋆z. By using definitions of upper and lower control functions together with the assumption (A1), we get
(35)Pwz≤∫01Hz,s∆^s,ws,Dpwsds≤∫01Hz,s∆^s,w⋆s,Dpw⋆sds≤w⋆z,Pwz≥∫01Hz,sδ^s,ws,Dpwsds≥∫01Hz,sδ^s,w⋆s,Dpw⋆sds≥w⋆z,
Hence, we obtainw⋆z≤Pwz≤w⋆z. Now, we need to show that Dpw⋆z≤DpPwz≤Dpw⋆z. We have
(36)D0+pPwz=−1Γr−p∫0zz−sr−p−1Ξs,ws,Dpwsds+Γ3−rz1−pΓ2−p∫01Ξs,ws,Dpwsds≤−1Γr−p∫0zz−sr−p−1∆^s,w⋆s,Dpw⋆sds+Γ3−rz1−pΓ2−p∫01δ^s,w⋆s,Dpw⋆sds≤Dpw⋆z.
Similarly, we showed thatDpPwz≥Dpw⋆z. Therefore, PΓζ⊆Γζ.Step 3.
At the final step, we aim to prove thatP has the complete continuity property.
To see this, letw∈Γζ and take M=maxz∈0,1Ξz,wz,Dpwz. We have
(37)Pwz=1Γr∫0zz−sr−1Ξs,ws,Dpwsds−Γ3−rz∫01Ξs,ws,Dpwsds≤1Γr∫0zz−sr−1Ξs,ws,Dpwsds+Γ3−r∫01Ξs,ws,Dpwsds≤zrΓr+1+Γ3−rzM≤1Γr+1+Γ3−rM,Pw′z=∫01∂Hz,s∂zΞs,ws,Dpwsds≤M∫01∂Hz,s∂zds≤1Γr+Γ3−rM,DpPwz=1Γr−p∫0zz−sr−p−1Ξs,ws,Dpwsds−Γ3−rz1−pΓ2−p∫01Ξs,ws,Dpwsds≤1Γr−p∫01z−sr−p−1Ξs,ws,Dpwsds+Γ3−rz1−pΓ2−p∫01Ξs,ws,Dpwsds≤zr−pΓr−p+1+Γ3−rz1−pΓ2−pM≤zr−pΓr−p+1+Γ3−rΓ2−pM.
Thus,(38)∥Pw∥n≤1Γr+1Γr+1+1Γr−p+1+2F3−r+Γ3−rΓ2−pM.
Hence,PΓζ has the property of the uniform boundedness. Next, we show that Pw is equicontinuous. To do this, for each w∈Γζ and z1,z2∈0,1 with z1<z2, we have
(39)Pwz2−Pwz1=∣1Γr∫0z1z2−sr−1Ξs,ws,Dpwsds+1Γr∫z1z2z2−sr−1Ξs,ws,Dpwsds−1Γr∫0z1z1−sr−1Ξs,ws,Dpwsds−Γ3−rz2−z1∫01Ξs,ws,Dpwsds≤1Γr∫0z1z2−sr−1−z1−sr−1Ξs,ws,Dpwsds+1Γr∫0z2z2−sr−1Ξs,ws,Dpwsds−Γ3−rz2−z1∫01Ξs,ws,Dpwsds=MΓr∫0z1z2−sr−1−z1−sr−1ds+MΓr∫z1z2z2−sr−1ds+MΓ3−rz2−z1=Mz2r−z1rΓr+1+MΓ3−rz2−z1.
It is seen that the right-hand side of (39) does not depend on w and tends to zero whenever z1⟶z2 which leads to Bwz2−Bwz1⟶0. Further, we have
(40)Pw′z2−Pw′z1=∫01∂Hz2,s∂z2Ξs,ws,Dpwsds−∫01∂Hz1,s∂z1Ξs,ws,Dpwsds≤∫01∂Hz2,s∂z2−∂Hz1,s∂z1Ξs,ws,Dpwsds≤MΓr−1∫0z1z2‐sr−2−z1‐sr−2ds+∫zzz2z2‐sr−2ds=MΓrz2r−1−z1r−1,which tends to zero whenever z1⟶z2. In addition,
(41)DpBwz2−DpBwz1=1Γ1−p∫0z2z2−s−pBw′sds−∫0z1z1−s−pBw′sds=1Γ1−p∫0z1z1−s−p−z2−s−p∫01∂Hs,λ∂sΞλ,wλ,Dpwλdλds+∫z1t2z2−s−p∫01∂Hs,λ∂sΞλ,wλ,Dpwλdλds≤1Γ1−p∫0z1z1−s−p−z2−s−p∫01∂Hs,λ∂sΞλ,wλ,Dpwλdλds+1Γ1−p∫z1z2z2−s−p∫01∂Hs,λ∂sΞλ,wλ,Dpwλdλds≤MΓ1−p∫0z1z1−s−p−z2−s−p∫01∂Hs,λ∂sdλds+MΓ1−p∫z1z2z2−s−p∫01∂Hs,λ∂sdλds≤MΓ2−p1Γr+3−r2z2−z11−p+z11−p−z21−p,which tends to zero as z1⟶z2. Therefore, inequalities (39), (40), and (41) imply that Pw is equicontinuous. Knowing that it is uniformly bounded, we find that P is completely continuous. Schauder’s fixed point theorem implies that P has a fixed point w∈Γζ which is a solution for the multiterm semilinear boundary value problem (3) and the proof is completed.☐Corollary 10.
LetΞ be continuous on 0,1×ℝ×ℝ and there exists ξ>0 such that
(42)ξ≤11/Γr+1/Γr+1+2Γ3−r+1/Γ2−pΓr+Γ3−r/Γ2−p.Then, a solution exists for the multiterm semilinear boundary value problem (3).Proof.
We chooseθz=0 and maxz∈0,1Ξz,w,v=ξw+v. So, the assumption (42) allows us to apply Theorem 9 which affirms the existence of a solution for the mentioned multiterm semilinear problem (3).☐Corollary 11.
Assume that there exist two real numbersη,ν>0 such that
(43)η≥sup0<z≤1w∈ℝ+,v∈ℝΞz,w,vandν≤inf0<z≤1w∈ℝ+,v∈ℝΞz,w,v.
Then, the multiterm semilinear boundary value problem (3) has at least a positive solution on 0,Z∗, where
(44)Z∗=vΓr+1Γ3−rη1/r−1.☐Proof.
From definitions of the functionsδ^z,u,v and Δ^z,u,v, it is followed that
(45)v≤δ^z,u,v≤Δ^z,u,v≤η,0≤z≤Z∗,w∈ℝ+,v∈ℝ.
Define(46)w∗z=Γ3−rηz−vzrΓr+1,0≤z≤Z∗,w∗z=Γ3−rv−ηzrΓr+1,0≤z≤Z∗,
So, we get0≤w∗z≤w∗zfor0≤z≤Z∗ and also
(47)w∗z≥η∫01Hz,sds≥∫01Hz,sΔ^s,w∗s,Dpw∗sds,0≤z≤Z∗,w∗z≤v∫01Hz,sds≤∫01Hz,sδ^s,w∗s,Dpw∗sds,0≤z≤Z∗.
Moreover, by some direct computations, we get(48)Dpw∗z=Γ3−rηz1−pΓ2−p−vzr−pΓr−p+1,0≤z≤Z∗,Dpw∗z=Γ3−rvz1−pΓ2−p−ηzr−pΓr−p+1,0≤z≤Z∗.
Thus,(49)Γ3−rz1−pΓ2−p∫01δ^s,w∗s,Dpw∗sds−1Γr−p∫0zz−sr−p−1Δ^s,w∗s,Dpw∗sds≤Γ3−rηz1−pΓ2−p−vzr−pΓr−p+1=D0+pw∗z,0≤z≤Z∗,Γ3−rz1−pΓ2−p∫01Δ^s,w∗s,Dpw∗sds−1Γr−p∫0zz−sr−p−1δ^s,w∗s,Dpw∗sds≥Γ3−rvz1−pΓ2−p−ηzr−pΓr−p+1=D0+Pw∗z,0≤z≤Z∗.
This means that the assumption (A1) is satisfied. Finally, if (A2) holds, then we can chooseζ such that
(50)ζ≥A+B+BΓ2−p+η1Γr+1Γr+1+2Γ3−r+1Γ2−pΓr+Γ3−rΓ2−p.
Now, all hypotheses of Theorem9 hold. Consequently, the multiterm semilinear boundary value problem (3) has at least a positive solution w∈Γζ, where 0≤w∗z≤wz≤w∗z and Dpw∗z≤Dpwz≤Dpw∗z for each z∈0,Z∗ and the corollary is proved.☐To validate the theoretical findings, we provide a special example corresponding to the suggested multiterm semilinear boundary value problem (3).Example 12.
According to the multiterm semilinear boundary value problem (3), in the present example, we take r=1.5,p=0.5,η=1,ν=0.5, Z∗=0.3 and
(51)Ξz,w,y=ν+η−νz=0.5+0.5z.
By taking into account the definition of the functionΞ, we clearly have ν≤Ξz,w,v≤η. Now, we choose upper and lower control functions Δ^z,u,v=η and δ^z,u,v=ν, respectively, and then, we get
(52)W⋆z=Γ1.5z−12Γ2.5z1.5=0.8862z−0.3761z1.5,W⋆z=0.5×Γ1.5z−1Γ2.5z1.5=0.4431z−0.7522z1.5,wz=3Γ1.54z−13Γ1.5z1.5−215Γ1.5z2.5=0.6647z−0.3761z1.5−0.1505z2.5.Therefore, by some simple calculations, we obtain(53)DPw∗z=z0.5−0.5z,DPw∗z=0.5z−z,Dpwz=0.75z0.5−0.5z−0.25z2.Note that in this example, we have taken the interval0,Z∗⊂0,1 to hold the essential condition 0≤w∗z≤wz≤w∗z. The graphs of positive solutions and their derivatives are illustrated in Figures 1 and 2.Figure 1
Graphs ofw,w∗, and w∗.Figure 2
Graphs ofA=Dpw∗,B=Dpw∗,andC=Dpw.


## 3.1. Required Assumptions

Now, for our main results, we need some assumptions given as follows:(A1) There arew⋆,w⋆∈Λ~ which satisfy α1≤w⋆z≤w⋆z≤α3 and α2≤Dpw⋆z≤Dpw⋆z≤α4 along with
(23)w⋆z≥∫01Hz,s∆^s,w⋆s,Dpw⋆sds,w⋆z≤∫01Hz,sδ^s,w⋆s,Dpw⋆sds,Dpw⋆z≥−1Γr−p∫0zz−sr−p−1Δ^s,w⋆s,Dpw⋆sds+Γ3−rΓ2−pz1−p∫01δ^s,w⋆s,Dpw⋆sds,Dpw⋆z≤−1Γr−p∫0zz−sr−p−1δ^s,w⋆s,Dpw⋆sds+Γ3−rΓ2−pz1−p∫01∆^s,w⋆s,Dpw⋆sds.(A2) There existξ>0 and nonnegative function θ∈L10,1 such that
(24)Ξz,w,v≤θz+ξw+v,0≤z≤1,w,v∈ℝ.(A3) There existsζ>0 such that
(25)A+B+BΓ2−p+ξζ1Γr+1Γr+1+2Γ3−r+1Γ2−pΓr+Γ3−rΓ2−p≤ζ.with
(26)A=maxz∈0,1∫01Hz,sθsdsandB=maxz∈0,1∫01∂Hz,s∂zθsds.At this moment, we are ready to present the first existence theorem.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⋆z≤wz≤w⋆z and Dpw⋆z≤Dpwz≤Dpw⋆z hold for each 0≤z≤1.Proof.
For eachζ>0, define the set Γζ as
(27)Γζ=w∈Λ~:wX≤ζ,w⋆z≤wz≤w⋆z,Dpw⋆z≤Dpwz≤Dpw⋆z,0,≤z≤1.
Obviously,Γζ is a convex, closed, and bounded set in X. Consider the operator P:Γζ⟶X under the following rule
(28)Pwz=−1Γr∫0zz−sr−1Ξs,ws,Dpwsds+Γ3−rz∫01Ξs,ws,Dpwsds=∫01Hz,sΞs,ws,Dpwsds.
To prove Theorem9, 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 wn which converges to w in X. We have
(29)Pwnz−Pwz=∫01Hz,sΞs,wns,Dpwns−Ξs,ws,D0+pwsds≤maxz∈0,1Ξz,wnz,Dpwnz−Ξz,wt,Dpwnz∫01Hz,sds≤1Γr+1+Γ3−rmaxz∈0,1Ξz,wnz,Dpwnz−Ξz,wz,Dpwz,(30)DpPwnz−DpPwz=1Γ1−p∫0zz−s−pPwn′s−Pw′sds≤Γ1−p∫0zz−s−p∫01∂Hs,λ∂sΞλ,wnλ,Dpwnλ−Ξλ,wλ,Dpwλdλds≤maxz∈0,1Ξz,wnz,Dpwnz−Ξz,wz,Dpwz1Γ1−p∫0zz−s−p∫01∂Hs,λ∂sdλds≤maxz∈0,1Ξz,wnz,Dpwnz−Ξz,wz,Dpwz1Γ1−p1Γr+Γ3−rΓ2∫0zz−s−pds≤1Γ2−p1Γr+Γ3−rmaxz∈0,1Ξz,wnz,Dpwnz−Ξz,wz,Dpwz,(31)Pwn′z−Pw′z=∫01∂Hz,s∂zΞs,wns,Dpwns−Ξs,ws,Dpwsds≤maxz∈0,1Ξz,wnz,Dpwnz−Ξz,wz,Dpwz∫01∂Hz,s∂zds≤1Γr+Γ3−rwz∈0,1Ξz,wnz,Dpwnz−Ξz,wz,Dpwz.
By tendingn⟶∞ and from the inequalities (29), (30), and (31), we follow that P is continuous in X.Step 2.
Now, we show thatP:Γζ⟶Γζ is a selfmap on Γζ. Let w∈Γζ. By inequalities (15) and (17) along with the assumptions (A2) and (A3), we get
(32)Pwz=∫01Hz,sΞs,ws,Dpwsds≤∫01Hz,sΞs,ws,Dpwsds≤∫01Hz,sθs+ξws+Dpwsds≤∫01Hz,sθsds+ξζ∫01Hz,sds≤B+ξζ1Γr+Γ3−r,(33)Pw′z=∫01∂Hz,s∂zΞs,ws,Dpwsds≤∫01∂Hz,s∂zθs+ξws+Dpwsds≤∫01∂Hz,s∂zθsds+ξζ∫01∂Hz,s∂zds≤B+ξζ1Γr+Γ3−r,(34)DpPwz=1Γ1−p∫0zz−s−pPw′sds≤1Γ1−p∫0zz−s−p∫01∂Hs,λ∂sΞλ,wλ,Dpwλdλds≤1Γ1−p∫01z−s−p∫01∂Hs,λ∂sθλdλ+∫01∂Hs,λ∂sξxλ+Dpwλdλds≤BΓ1−p∫0zz−s−pds+ξζΓ1−p∫0zz−s−p∫01∂Hs,λ∂sdλds≤BΓ2−pz1−p+ξζΓ2−p1Γr+Γ3−rz1−p≤BΓ2−p+ξζΓ2−p1Γr+Γ3−r.
By virtue of inequalities (32), (33), (34), and the assumption (A3), we get PxX≤ζ.
In the sequel, we investigate the inequalitiesw⋆z≤Pwz≤w⋆z and also Dpw⋆z≤DpPwz≤Dpw⋆z for each 0≤z≤1. Since w belongs to Γζ, we obviously have w⋆z≤wz≤w⋆z. By using definitions of upper and lower control functions together with the assumption (A1), we get
(35)Pwz≤∫01Hz,s∆^s,ws,Dpwsds≤∫01Hz,s∆^s,w⋆s,Dpw⋆sds≤w⋆z,Pwz≥∫01Hz,sδ^s,ws,Dpwsds≥∫01Hz,sδ^s,w⋆s,Dpw⋆sds≥w⋆z,
Hence, we obtainw⋆z≤Pwz≤w⋆z. Now, we need to show that Dpw⋆z≤DpPwz≤Dpw⋆z. We have
(36)D0+pPwz=−1Γr−p∫0zz−sr−p−1Ξs,ws,Dpwsds+Γ3−rz1−pΓ2−p∫01Ξs,ws,Dpwsds≤−1Γr−p∫0zz−sr−p−1∆^s,w⋆s,Dpw⋆sds+Γ3−rz1−pΓ2−p∫01δ^s,w⋆s,Dpw⋆sds≤Dpw⋆z.
Similarly, we showed thatDpPwz≥Dpw⋆z. Therefore, PΓζ⊆Γζ.Step 3.
At the final step, we aim to prove thatP has the complete continuity property.
To see this, letw∈Γζ and take M=maxz∈0,1Ξz,wz,Dpwz. We have
(37)Pwz=1Γr∫0zz−sr−1Ξs,ws,Dpwsds−Γ3−rz∫01Ξs,ws,Dpwsds≤1Γr∫0zz−sr−1Ξs,ws,Dpwsds+Γ3−r∫01Ξs,ws,Dpwsds≤zrΓr+1+Γ3−rzM≤1Γr+1+Γ3−rM,Pw′z=∫01∂Hz,s∂zΞs,ws,Dpwsds≤M∫01∂Hz,s∂zds≤1Γr+Γ3−rM,DpPwz=1Γr−p∫0zz−sr−p−1Ξs,ws,Dpwsds−Γ3−rz1−pΓ2−p∫01Ξs,ws,Dpwsds≤1Γr−p∫01z−sr−p−1Ξs,ws,Dpwsds+Γ3−rz1−pΓ2−p∫01Ξs,ws,Dpwsds≤zr−pΓr−p+1+Γ3−rz1−pΓ2−pM≤zr−pΓr−p+1+Γ3−rΓ2−pM.
Thus,(38)∥Pw∥n≤1Γr+1Γr+1+1Γr−p+1+2F3−r+Γ3−rΓ2−pM.
Hence,PΓζ has the property of the uniform boundedness. Next, we show that Pw is equicontinuous. To do this, for each w∈Γζ and z1,z2∈0,1 with z1<z2, we have
(39)Pwz2−Pwz1=∣1Γr∫0z1z2−sr−1Ξs,ws,Dpwsds+1Γr∫z1z2z2−sr−1Ξs,ws,Dpwsds−1Γr∫0z1z1−sr−1Ξs,ws,Dpwsds−Γ3−rz2−z1∫01Ξs,ws,Dpwsds≤1Γr∫0z1z2−sr−1−z1−sr−1Ξs,ws,Dpwsds+1Γr∫0z2z2−sr−1Ξs,ws,Dpwsds−Γ3−rz2−z1∫01Ξs,ws,Dpwsds=MΓr∫0z1z2−sr−1−z1−sr−1ds+MΓr∫z1z2z2−sr−1ds+MΓ3−rz2−z1=Mz2r−z1rΓr+1+MΓ3−rz2−z1.
It is seen that the right-hand side of (39) does not depend on w and tends to zero whenever z1⟶z2 which leads to Bwz2−Bwz1⟶0. Further, we have
(40)Pw′z2−Pw′z1=∫01∂Hz2,s∂z2Ξs,ws,Dpwsds−∫01∂Hz1,s∂z1Ξs,ws,Dpwsds≤∫01∂Hz2,s∂z2−∂Hz1,s∂z1Ξs,ws,Dpwsds≤MΓr−1∫0z1z2‐sr−2−z1‐sr−2ds+∫zzz2z2‐sr−2ds=MΓrz2r−1−z1r−1,which tends to zero whenever z1⟶z2. In addition,
(41)DpBwz2−DpBwz1=1Γ1−p∫0z2z2−s−pBw′sds−∫0z1z1−s−pBw′sds=1Γ1−p∫0z1z1−s−p−z2−s−p∫01∂Hs,λ∂sΞλ,wλ,Dpwλdλds+∫z1t2z2−s−p∫01∂Hs,λ∂sΞλ,wλ,Dpwλdλds≤1Γ1−p∫0z1z1−s−p−z2−s−p∫01∂Hs,λ∂sΞλ,wλ,Dpwλdλds+1Γ1−p∫z1z2z2−s−p∫01∂Hs,λ∂sΞλ,wλ,Dpwλdλds≤MΓ1−p∫0z1z1−s−p−z2−s−p∫01∂Hs,λ∂sdλds+MΓ1−p∫z1z2z2−s−p∫01∂Hs,λ∂sdλds≤MΓ2−p1Γr+3−r2z2−z11−p+z11−p−z21−p,which tends to zero as z1⟶z2. Therefore, inequalities (39), (40), and (41) imply that Pw is equicontinuous. Knowing that it is uniformly bounded, we find that P is completely continuous. Schauder’s fixed point theorem implies that P has a fixed point w∈Γζ which is a solution for the multiterm semilinear boundary value problem (3) and the proof is completed.☐Corollary 10.
LetΞ be continuous on 0,1×ℝ×ℝ and there exists ξ>0 such that
(42)ξ≤11/Γr+1/Γr+1+2Γ3−r+1/Γ2−pΓr+Γ3−r/Γ2−p.Then, a solution exists for the multiterm semilinear boundary value problem (3).Proof.
We chooseθz=0 and maxz∈0,1Ξz,w,v=ξw+v. So, the assumption (42) allows us to apply Theorem 9 which affirms the existence of a solution for the mentioned multiterm semilinear problem (3).☐Corollary 11.
Assume that there exist two real numbersη,ν>0 such that
(43)η≥sup0<z≤1w∈ℝ+,v∈ℝΞz,w,vandν≤inf0<z≤1w∈ℝ+,v∈ℝΞz,w,v.
Then, the multiterm semilinear boundary value problem (3) has at least a positive solution on 0,Z∗, where
(44)Z∗=vΓr+1Γ3−rη1/r−1.☐Proof.
From definitions of the functionsδ^z,u,v and Δ^z,u,v, it is followed that
(45)v≤δ^z,u,v≤Δ^z,u,v≤η,0≤z≤Z∗,w∈ℝ+,v∈ℝ.
Define(46)w∗z=Γ3−rηz−vzrΓr+1,0≤z≤Z∗,w∗z=Γ3−rv−ηzrΓr+1,0≤z≤Z∗,
So, we get0≤w∗z≤w∗zfor0≤z≤Z∗ and also
(47)w∗z≥η∫01Hz,sds≥∫01Hz,sΔ^s,w∗s,Dpw∗sds,0≤z≤Z∗,w∗z≤v∫01Hz,sds≤∫01Hz,sδ^s,w∗s,Dpw∗sds,0≤z≤Z∗.
Moreover, by some direct computations, we get(48)Dpw∗z=Γ3−rηz1−pΓ2−p−vzr−pΓr−p+1,0≤z≤Z∗,Dpw∗z=Γ3−rvz1−pΓ2−p−ηzr−pΓr−p+1,0≤z≤Z∗.
Thus,(49)Γ3−rz1−pΓ2−p∫01δ^s,w∗s,Dpw∗sds−1Γr−p∫0zz−sr−p−1Δ^s,w∗s,Dpw∗sds≤Γ3−rηz1−pΓ2−p−vzr−pΓr−p+1=D0+pw∗z,0≤z≤Z∗,Γ3−rz1−pΓ2−p∫01Δ^s,w∗s,Dpw∗sds−1Γr−p∫0zz−sr−p−1δ^s,w∗s,Dpw∗sds≥Γ3−rvz1−pΓ2−p−ηzr−pΓr−p+1=D0+Pw∗z,0≤z≤Z∗.
This means that the assumption (A1) is satisfied. Finally, if (A2) holds, then we can chooseζ such that
(50)ζ≥A+B+BΓ2−p+η1Γr+1Γr+1+2Γ3−r+1Γ2−pΓr+Γ3−rΓ2−p.
Now, all hypotheses of Theorem9 hold. Consequently, the multiterm semilinear boundary value problem (3) has at least a positive solution w∈Γζ, where 0≤w∗z≤wz≤w∗z and Dpw∗z≤Dpwz≤Dpw∗z for each z∈0,Z∗ and the corollary is proved.☐To validate the theoretical findings, we provide a special example corresponding to the suggested multiterm semilinear boundary value problem (3).Example 12.
According to the multiterm semilinear boundary value problem (3), in the present example, we take r=1.5,p=0.5,η=1,ν=0.5, Z∗=0.3 and
(51)Ξz,w,y=ν+η−νz=0.5+0.5z.
By taking into account the definition of the functionΞ, we clearly have ν≤Ξz,w,v≤η. Now, we choose upper and lower control functions Δ^z,u,v=η and δ^z,u,v=ν, respectively, and then, we get
(52)W⋆z=Γ1.5z−12Γ2.5z1.5=0.8862z−0.3761z1.5,W⋆z=0.5×Γ1.5z−1Γ2.5z1.5=0.4431z−0.7522z1.5,wz=3Γ1.54z−13Γ1.5z1.5−215Γ1.5z2.5=0.6647z−0.3761z1.5−0.1505z2.5.Therefore, by some simple calculations, we obtain(53)DPw∗z=z0.5−0.5z,DPw∗z=0.5z−z,Dpwz=0.75z0.5−0.5z−0.25z2.Note that in this example, we have taken the interval0,Z∗⊂0,1 to hold the essential condition 0≤w∗z≤wz≤w∗z. The graphs of positive solutions and their derivatives are illustrated in Figures 1 and 2.Figure 1
Graphs ofw,w∗, and w∗.Figure 2
Graphs ofA=Dpw∗,B=Dpw∗,andC=Dpw.


## 4. 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.



---

*Source: 6022941-2021-06-23.xml*