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1 Introduction

We consider the nonlinear fractional differential equation

with the nonlocal boundary conditions

where . ∈ R, α ∈ (. − 1, n], n, m . N , n ≥ 3, β. ∈ . for all . = 0, . . . , m, 0 ≤ .. < β. < · · · < β. ≤ .. < α − 1, .. ≥ 1, . is a positive parameter, and denotes the Riemann–Liouville derivative of order . (for . = α, β., β., . . . , β.).

The integrals from the boundary conditions (2) are Riemann–Stieltjes integrals with H., i = 1, . . . , m, functions of bounded variation, the nonnegative function . (t, u) may have singularity at . = 0, and the nonnegative function .(.) may be singular at . = 0 and/or . = 1.

Under some assumptions for the functions . and . , we establish intervals for the parameter . such that problem (1), (2) has positive solutions (.(.) > 0 for all . (0, 1]). These intervals for . are expressed by using the principal characteristic value of an associated linear operator. In the proof of the main theorems, we use the fixed point index theory. In the case in which . 1 and . is a function which changes sign and has singularities at . = 0 and/or . = 1, we present two existence results for the positive solutions of this problem. In the proof of these results, we apply the Guo–Krasnosel’skii fixed point theorem. The boundary conditions (2) cover various cases, such as multi- point boundary conditions when the functions H. are step functions, or classical integral boundary conditions, or a combination of them.

We present below some papers, which investigate particular cases of our boundary value problem (1), (2) and other problems related to (1), (2). Equation (1) with .(.) 1 subject to the boundary conditions

where ξ. R, i = 1, . . . , m, 0 < ξ. < < ξ. < 1, p, q R, p [1, n 2], . [0, p], was investigated in [11]. In paper [11], the nonlinearity . changes sign, and it is singular only at . = 0 and/or . = 1. The authors of [11] apply the Guo–Krasnosel’skii fixed point theorem to prove the existence of positive solutions when the parameter belongs to various intervals. Equation (1) with . = 1 and .(.) 1 supplemented with the boundary conditions (2) with . = 1, where . may change sign and may be singular at the points . = 0, . = 1 and/or . = 0 has been studied in [20]. In the paper [20], the author presents some conditions for . , which contain height functions defined on special bounded sets under which he proves the existence and multiplicity of positive solutions. The existence of multiple positive solutions for equation (1) with . = 1 and .(.) 1 subject to the boundary conditions (2) was investigated in the recent paper [1]. The authors use in [1] various height functions of the nonlinearity defined on special bounded sets and two theorems from the fixed point index theory. In the paper [35], the authors prove the existence of at least three positive solutions for equation (1) with . = 1 and .(.) 1 with the boundary conditions

where . ≥ 1, . − . − 1 > 0, 0 < η ≤ 1, 0 ≤ . ∫ ˜.(.). ^{− −} dt < 1, ˜. ∈ . [0, 1] is nonnegative and may be singular at . = 0 and . = 1, and the function . is nonnegative and may be singular at the points . = 0, . = 1 and . = 0. Our boundary conditions (2) are more general than the above boundary conditions (3). Indeed, the last relation from (3)

can be written as D.u(.) d.(.) with .(.) = {. ∫ . ˜.(.) ds, t ∈ [0, η]; have a sum of Riemann–Stieltjes integrals from Riemann–Liouville derivatives of various orders. In the paper [35], the authors use different height functions of the nonlinear term on special bounded sets, the Krasnosel’skii theorem and the Leggett–Williams fixed point index theorem. We also mention the paper [33], where the authors prove the existence of positive solutions of fractional differential equation (1) supplemented with the boundary conditions

where . ∈ [1, n − 2], . ∈ [0, β], α. ≥ 0, . Σ− 1, 2, . . . , 0 < ξ. < ξ. < · · · < ξ.₋₁<our condition from (2). We mention that condition (I3) (see below, in Section 3) used in our results was first introduced in the paper [18], where the authors proved the existence of at least one positive solution for a fourth-order nonlinear singular Sturm–Liouville eigenvalue problem.

For some recent results on the existence, nonexistence and multiplicity of positive so- lutions for fractional differential equations and systems of fractional differential equations with various boundary conditions, we refer the reader to the monographs [10, 36] and the papers [2,3,8,12,13,17,19,25,28,30,31,34]. We also mention the books [14,15,24,26,27] and the papers [5–7, 21–23, 29] for applications of the fractional differential equations in various disciplines.

The paper is organized as follows. In Section 2, we present the solution of a linear fractional differential equation associated to equation (1) subject to the boundary condi- tions (2) and the properties of the corresponding Green functions. Some theorems from the fixed point index theory, the Guo–Krasnosel’skii fixed point theorem and an application of the Krein–Rutman theorem in the space .[0, 1] are recalled in Section 2, and they will be used in the next sections. In Section 3, we give and prove the main theorems for the existence of at least one positive solution for problem (1), (2). In Section 4, we present two existence results for the positive solutions of problem (1), (2) with . 1, where the nonlinearity changes sign, and it is singular at . = 0 and/or . = 1. Section 5 contains some examples, which illustrate the obtained results, and in Section 6, we give the conclusions for our fractional boundary value problems

2 Auxiliary results

In this section, we present some auxiliary results from [1] that we will use in the proof of the main theorems. We consider the fractional differential equation

with the boundary conditions (2), where . ∈ .(0, 1) ∩ ..(0, 1). We denote

Lemma 1

(See [1].) If ∆ = 0, then the unique solution u C[0, 1] of problem (5), (2) is given by

where

And

Based on some properties of functions .. and ._2., . = 1, . . . , m, given by (8) (see [11]), we have the following lemma.

Lemma 2

(See [1].) We suppose that ∆ > .. Then the Green function . given by (7) is a continuous function on [0, 1] × [0, 1] and satisfies the inequalities:

(i) G(t, s) ≤ J (.) for all t, s ∈ [0, 1], where

(ii) G(t, s) ≤ σt.⁻¹for all t, s ∈ [0, 1], where

Lemma 3

(See [1].) We suppose that ∆ > ., x ∈ .(0,1) ∩ ..(0, 1) and x(.) ≥ 0 for all t ∈ (0,1). Then the solution u of problem (5), (2) given by (6) satisfies the inequality u(.) ≥ t.⁻¹ǁ.ǁ for all t ∈ [0,1], where ǁ.ǁ = sup.∈[0,1] |.(.)|, and so u(.) ≥ 0 for all t ∈ [0, 1].

We recall now some theorems concerning the fixed point index theory and the Guo– Krasnosel’skii fixed point theorem. Let . be a real Banach space with the norm , C X a cone, “≤” the partial ordering defined by . and . the zero element in .. For . > 0, let B. . u X. u < Q be the open ball of radius . centered at ., its closure B . = u X: . ≤ . and its boundary ∂B. . u X: . = . . The proofs of our results are based on the following fixed point index theorems.

Theorem 1

(See [4].) Let A : B. ∩ . → C be a completely continuous operator. If there exists u. ∈ . \ {.} such that u − Au i(A, B. . C, C) = 0.λu. for all λ ≥ 0 and u . ∂B. . C, then

Theorem 2

(See [4].) Let A : B. ∩ . → C be a completely continuous operator. If A. /= µu for all u . ∂B. . C and µ ≥ 1, then i(A, B. . C, C) = 1.

Theorem 3

(See [9].) Let X be a Banach space, and let C . X be a cone in X. Assume Ω. and Ω. are bounded open subsets of X with θ ∈ .., Ω. ⊂ .., and let A : . ∩ (.. \ ..) → C be a completely continuous operator such that either

(i) ǁA.ǁ ≤ ǁ.ǁ, u ∈ . ∩ ∂Ω., and ǁA.ǁ ≥ ǁ.ǁ, u ∈ . ∩ ∂Ω., or

(ii) ǁA.ǁ ≥ ǁ.ǁ, u ∈ . ∩ ∂Ω., and ǁA.ǁ ≤ ǁ.ǁ, u ∈ . ∩ ∂Ω..

Then . has a fixed point in C ∩ (.. \ ..).

Let the space .[0, 1] and the cone . = u C[0, 1]: .(.) ≥ 0 . [0, 1] . We present next an application of the Krein–Rutman theorem in the space .[0, 1].

Theorem 4

(See [16, 32].) Suppose that A : .[0, 1] → .[0, 1] is a completely contin- uous linear operator and A(. ) ⊂ P. If there exist v ∈ .[0, 1] \ (−. ) and a constantc > . such that cAv . v, then the spectral radius r(.) 0 and A has an eigenvector u. P θ corresponding to its principal characteristic value λ. = (.(.))⁻¹, that is λ.Au. = .. or Au. = .(.).., and so r(.) > 0.

3 Main results

In this section, we present intervals for the parameter . such that our problem (1), (2) has at least one positive solution. We consider the Banach space . = .[0, 1] with the supremum norm ǁ.ǁ = sup.∈[0,1] |.(.)|, and we define the cones

We define the operator A : . → . and the linear operator T : . → . by

We see that . is a solution of problem (1), (2) if and only if . is a fixed point of operator . For r > 0, we denote Q. . B. Q and Qr . B. Q

We introduce now the assumptions that we will use in what follows.

(I2) The function . ∈ .((0, 1), [0, ∞)), and . J (.).(.) ds < ∞.

(I3) The function . ∈ .([0, 1] × (0, ∞), [0, ∞)), and for any 0 < r < R, we have

where A. = [0, 1/n] ∪ [(. − 1)/n, 1].

Lemma 4

Assume that assumptions (I1)–(I3) hold. Then, for any . < r < R, the operator A : QR . Q. Q is completely continuous.

Proof. By (I3) we deduce that there exists a natural number .. ≥ 3 such that

For . ∈ QR . Q., there exists .. ∈ [r, R] such that ǁ.ǁ = .., and then

Let .. = max{. (t, x., t ∈ [1/n., (.. −1)/n.], x ∈ [r/n.⁻¹, R]}. By Lemma 2, (I2) and (I3) we find

where .. = max._∈[0,1] J (.). This implies that the operator A is well defined.

We show next that : QR Q. Q. Indeed, for any u QR Q. and . [0, 1],

we have

And then

On the other hand, by Lemma 2 we obtain

so u Q. Therefore (QR Q.) ..

W. prove now that : QR Q. Q is completely continuous. We assume that E QR Q. is an arbitrary bounded set. From the first part of the proof we know that (.) is uniformly bounded. Then we show that (.) is equicontinuous. Indeed, for ε > 0, there exists a natural number .. ≥ 3 such that

Since G(t, s) is uniformly continuous on [0, 1] × [0, 1], for the above ε > 0, there exists δ > 0 such that, for any .., t. ∈ [0, 1] with |.. − ..| < δ and . ∈ [1/n., (..−1)/n.], we have

Where

And

Then, for any . ∈ E, t., t. ∈ [0, 1] with |.. − ..| < δ, we deduce

This gives us that (.) is equicontinuous. By the Arzelà–Ascoli theorem we conclude that A: QR Qr > Q is comp.ct.

Finally, we prove that : QR Q. > Q is continuous. We suppose thatu.,u.QR/ Qr for all . ≥ 1 and un - u0. 0 as n> . Then . ≤ u. ≤ . for all . ≥ 0. By (I3), for ε > 0, there exists a natural number .. ≥ 3 such that

Because . (t, x) is uniformly continuous in [1/n., (.. − 1)/n.] × [r/n.⁻¹, R], we obtain

uniformly for . [1/n., (.. 1)/n.]. Then the Lebesgue dominated convergence theo- rem gives us

Thus, for the above ε > 0, there exists a natural number . such that, for n > N , we have

By (9) and (10) we conclude that

This implies that : QR Q. Q is continuous. Hence : QR Q. Q is completely continuous

Under assumptions (I1)–(I3), by the extension theorem the operator A has a com- pletely continuous extension (also denoted by A) from . to .

Lemma 5

Assume that assumptions (I1), (I2) hold. Then the spectral radius r(T ) /= 0, and . has an eigenfunction ψ. ∈ . \ {.} corresponding to the principal eigenvalue r(T ), that is T .. = .(T )... So r(T ) > 0.

Proof. The operator T : . → . is a linear completely continuous operator. By Lemma 2 we know that G(t, s. > 0 for all t, s ∈ (0, 1). By (I2) we deduce that there exists an interval [c, d] ⊂ (0, 1) (0 < c < d < 1) such that .(.) > 0 for all . ∈ [c, d]. We consider a function . ∈ .[0, 1] satisfying the conditions .(.) > 0 for . ∈ (c, d) and .(.) = 0 for . ∈. (c, d). Then, for all . ∈ [c, d], we have

Hence there exists a constant a > 0 (. = max._∈[c,d] .(.). min._∈[c,d](T .)(.)), which satisfies the inequality .(T .)(.) ≥ .(.) for all . ∈ [0, 1]. By Theorem 4 we conclude that the spectral radius .(T ) /= 0 and T has an eigenfunction .. ∈ . \ {.} corresponding to its principal characteristic value .. = (.(T ))⁻¹ such that T .. = .(T ).., and so .(T ) > 0.

Using a similar argument as that used in the proof of Lemma 4 for operator A, we obtain that T (.) ⊂ ..

Theorem 5

Assume that assumptions (I1)–(I3) hold. If

then, for any λ ∈ (1.(f.r(T )), 1.(f. r(T ))), problem (1), (2) has at least one positive

Proof. We consider . ∈ (1.(f.r(T )), 1.(f. r(T ))). For f., we have the cases: f. ∈ (0, ∞) with .0 > 1.(λr(T )) and .0 = ∞. In the first case, .0 ∈ (0, ∞) with f. > 1.(λr(T )), we obtain

By taking . = f. − 1.(λr(T )) we deduce that there exists .. > 0 such that . (t, u./u ≥ 1.(λr(T )) for all . ∈ [0, 1] and . ∈ (0, r.. ], and so . (t, u) ≥ u/.λr(T )) for all . ∈ [0, 1] and . ∈ [0, r.. ].

In the case f. = ∞, we have

So for . = 1.(λr( )), we deduce that there exists ..^′′> 0 such that . (t, u) ≥ u/.λr(T)) for all . [0, 1] and . [0, r1′′].

Hence, in the above both cases, we conclude that there exists .. > 0 such that . (t, u) ≥ u/(λr(T )) for all . ∈ [0, 1] and . ∈ [0, r.].

Then, for any . ∈ ∂Q.1 , we find

We assume that has no fixed point on ∂Q.1 , (otherwise the proof is finished). We will prove that

where .. is given in Lemma 5. We suppose that there exist .. ∈ ∂Q.1 and .. ≥ 0 such that .. − A.. = ..... Then .. > 0 and .. = A.. + .... ≥ ..... We denote .. = sup{.: .. ≥ µψ.}. Then .. ≥ .., .. ≥ .... and

Hence .. = ..+.... ≥ ....+.... = (..+..).., which contradicts the definition of ... So relation (11) holds, and by Theorem 1 we deduce that

For f. , we have also two cases:

By taking . = 1.(2λr(T )) − f. /2 we deduce that there exists ..

In the case f.

So for . = 1.(2λr(T )), we deduce that there exists .2′′ 1.(2λr(T )). for all . ∈ [0, 1] and . ∈ [.2′′, ∞). > r. such that . (t, u) ≤

Therefore, in the above both cases, we conclude that there exist . ∈ (0, 1) and .. > r. such that . (t, u) ≤ .1.(λr(T )). for all . ∈ [0, 1] and . ∈ [.., ∞).

We define now the operator T. : . → . by

The operator T. is linear and bounded, and T.(.) ⊂ .. Because . ∈ (0, 1), we obtain

.(T.) = θ < 1. We consider the set

By (13) and the definition of operator , for any u Z, µ ≥ 1 and . [0, 1], we deduce

where .(.) = min{.(.), r.} for all . ∈ [0, 1] (which satisfies ..t.⁻¹ ≤ .˜(.) ≤ .. for all t 2 [0; 1]), J0 = sups2[0;1] J (s), and M1 = supu2Qr2nQr1 R 1 0 h(s)f(s; u(s)) ds (as in the proof of Lemma 4, we obtain that M1 < 1). By the Gelfand formula we know that (I 􀀀 T1)��1 exists and (I 􀀀 T1)􀀀1 =

P1 i=1 T i 1 , which implies (I 􀀀 T1)􀀀1(Q) _ Q. This, together with (14), gives us u(t) 6 (I 􀀀 T1)􀀀1(_J0M1), and so u(t) 6 _J0M1 _ k(I 􀀀 T1)􀀀1k for all t 2 [0; 1], which means that Z is bounded. Now we choose R > maxfr2; supfkuk; u 2 Zgg. Then we obtain that _u 6= Au for all u 2 @QR and _ > 1. By Theorem 2 we conclude that

By (12), (15) and the additivity property of the fixed point index we deduce that

So operator has at least one fixed point on Q. Qr1 , which is a positive solution of problem (1), (2).

By using a similar approach as that used in the proof of Theorem 5, we obtain the following result.

Theorem 6

Assume that assumptions (I1)–(I3) hold. If

then, for any λ ∈ (1.(f. r(T )), 1.(f.r(T ))), problem (1), (2) has at least one positive

4 Some remarks on a related semipositone problem

In this section, we present two existence results for a semipositone problem associated to problem (1), (2). More precisely, we consider the fractional differential equation

subject to the boundary conditions (2). We suppose that assumption (I1) holds and . sat- isfies the conditions

(I2.) The function . ∈ .((0, 1) × [0, ∞), R) may be singular at . = 0 and/or . = 1, and there exist the functions p, q ∈ .((0, 1), [0, ∞)), . ∈ .([0, 1] × [0, ), [0, )) such that .(.) ≤ . (t, u) ≤ .(.).(t, u) for all . (0, 1) and

(I3.) There exists . ∈ (0, 1.2) such that lim._→∞ min._∈[ζ,1−.. .˜(t, u./u =

By using the Guo–Krasnosel’skii fixed point theorem (Theorem 3) and similar argu- ments as those used in [11] (Theorems 3.1 and 3.2) we obtain the following results for problem (16), (2).

Theorem 7

Assume that (I1), (I2.) and (I3.) hold. Then there exists λ. > . such that, for any λ ∈ (0, λ.], the boundary value problem (16), (2) has at least one positive solution.

In the proof of Theorem 7, we consider .. > σ ∫ .(.) dt > 0, and we define

with .. = max{max._∈[0,1],._∈[0,R1] .(t, u), 1}. The solution .(.), . ∈ [0, 1], satisfies the condition .(.) ≥ ..t.⁻¹ for all . ∈ [0, 1], where ..

Theorem 8

Assume that (I1), (I2.) and

(I4) There exists ζ ∈ (0, 1.2) such that the following hold:

Then there exists λ. > . such that, for any λ ≥ .., the boundary value problem (16), (2) has at least one positive solution.

By (I4) we know that for . ∈ (0, 1.2) and for a fixed number .. > 0, there exists M > 0 such that . (t, u) ≥ . for all . [ζ, 1 .] and . ≥ . . In the proof of Theorem 8, we define .. = . (ζ.⁻¹. ∫ 1 .(.) d.)⁻¹. The solution .(.), . ∈ [0, 1], satisfies the condition .(.) ≥ ..t.⁻¹ for all . ∈ [0, 1], where .. = ../ζ.⁻¹

5 Examples

Let . = 10.3, . = 4, .. = 11.5, . = 2, .. = 1.2, .. = 5.4, ..(.) = . for all . [0, 1], ..(.) = 0 for . [0, 1.2); 1 for . [1.2, 1] .

We consider the fractional differential equations

subject to the boundary conditions

We have . 1.12792427 > 0 and . 0.94443688. So assumption (I1) is satisfied. In addition, we obtain

Example 1. We consider the functions

The cone . from Section 3 is here . = {. ∈ .[0, 1]: .(.) ≥ .^7.3ǁ.ǁ ∀. ∈ [0, 1]}. For 0 < r < R and . ∈ QR . Q., we deduce

Besides, we obtain ∫ 1 J (.).(.) d. ≤ . Γ(2.3)Γ(1.3) < ∞, . 0.781. Hence assumption (I2) is satisfied

For . ∈ QR . Q. and A. = [0, 1/n] ∪ [(.−1)/n, 1], we find

and then lim._→∞ sup C. = 0 because ..(.) = 1.(√3 .(1 − .)2) ∈ ..(0, 1) and ..(.) = 1.(.^11.12(1 − .)^2.3) ∈ ..(0, 1). Hence assumption (I3) is satisfied. We also have f.= 0 and f.= ∞. Then by using Theorem 5 we deduce that, for any λ ∈ (0, ∞), problem (17), (19) has at least one positive solution .(.), . ∈ [0, 1], which satisfies the condition .(.) ≥ .^7.3ǁ.ǁ for all . ∈ [0, 1]

Example 2. We consider the function

For this example, we have .(.) = − ln . and .(.) = 1.(√4 .(1 − .)3) for all . ∈ (0, 1), Γ(3.4)Γ(1.4) 4.44288. Then assumption (I2.) is satisfied. In addition, for . (0, ½), Rfixed, assumption (I30) is also satisfied. By some computations we obtain that J (.)(.(.)+.(.)) d. ≈ 2.71742073. We choose . = 2, which satisfies the condition and then we deduce M2 = 11 and 0:0669084. By Theorem 7 we conclude that, for any _ 2 (0; __], problem (18), (19) has at least one positive solution u(t), t 2 [0; 1], which satisfies the condition u(t) > _1t7=3 for all t 2 [0; 1], where _1 _ 1:05556.

Example 3. We consider the function

Here we have .(.) = 1.√3 . and .(.) = 1. 5 .3(1 .)2 for all . (0, 1), .(t, u) . + 1.3 for all . ∈ [0, 1] and . ≥ 0. Because ∫ 1 .(.) d. = 3.2, ∫ 1 .(.) d. ≈ 3.30327, assumption (I2.) is satisfied. In addition, for . ∈ (0,1.2), we obtain that lim._→∞ min._∈[ζ,1−.. . (t, u) = and lim._→∞ max._∈[0,1] .(t, u./u = 0, and then assumption (I4) is also satisfied. We choose . = 1.4 and .. = 100, and then we find .. = 5805 and .. ≈ 104075. Then by Theorem 8 we deduce that, for any . ≥ .., problem (18), (19) has at least one positive solution .(.), . ∈ [0, 1], which satisfies the inequality .(.) ≥ .˜1.... for all . ∈ [0, 1], where .˜1 ≈ 147438.

6 Conclusion

In this paper, we study the existence of positive solutions for the nonlinear Riemann– Liouville fractional boundary value problem (1), (2), where . is a positive parameter. The function . is nonnegative, and it may be singular at the second variable, and the function . is also nonnegative, and it may have singularities at . = 0 and/or . = 1. We present conditions for . and . and intervals for ., which are expressed in term of the principal characteristic value of an associated linear operator. In the proof of the existence theo- rems, we use two results from the fixed point index theory. We also investigate a related semipositone problem, namely, equation (1) with . 1 and . a sign-changing function with singularities at . = 0 and/or . = 1 subject to the nonlocal boundary conditions (2). For this problem, we give two existence results for the positive solutions when . belongs to various intervals. Three examples, which illustrate the obtained existence theorems, are finally presented.

References

1. R.P. Agarwal, R. Luca, Positive solutions for a semipositone singular Riemann–Liouville fractional differential problem, Int. J. Nonlinear Sci. Numer. Simul.,20(7–8):823–832, 2019, https://doi.org/10.1515/ijnsns-2018-0376.

2. B. Ahmad, R. Luca, Existence of solutions for sequential fractional integro–differential equations and inclusions with nonlocal boundary conditions, Appl. Math. Comput.,339:516– 534, 2018, https://doi.org/10.1016/j.amc.2018.07.025.

3. B. Ahmad, S.K. Ntouyas, Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions, Appl. Math. Comput.,266:615–622, 2015, https://doi.org/10.1016/j.amc.2015.05.116.

4. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18:620–709, 1976, https://doi.org/10.1137/1018114.

5. A.A.M. Arafa, S.Z. Rida, M. Khalil, Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection, Nonlinear Biomed. Phys., .(1):1–7, 2012, https://doi.org/ 10.1186/1753-4631-6-1.

6. Y. Ding, H. Ye, A fractional-order differential equation model of HIV infection of CD4+ T- cells, Math. Comput. Model.,50:386–392, 2009, https://doi.org/10.1016/j.mcm. 2009.04.019.

7. V. Djordjevic, J. Jaric, B. Fabry, J. Fredberg, D. Stamenovic, Fractional derivatives embody essential features of cell rheological behavior, Ann. Biomed. Eng., 31:692–699, 2003, https://doi.org/10.1114/1.1574026.

8. S.R. Grace, J.R. Graef, E. Tunc, On the boundedness of nonoscillatory solutions of certain fractional differential equations with positive and negative terms, Appl. Math. Lett.,97:114– 120, 2019, https://doi.org/10.1016/j.aml.2019.05.032.

9. D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988, https://doi.org/10.1016/C2013-0-10750-7.

10. J. Henderson, R. Luca, Boundary Value Problems for Systems of Differential, Difference and Fractional Equations. Positive solutions, Elsevier, Amsterdam, 2016.

11. J. Henderson, R. Luca, Existence of positive solutions for a singular fractional boundary value problem, Nonlinear Anal. Model. Control, 22(1):99–114, 2017, https://doi.org/10. 15388/NA.2017.1.7.

12. J. Henderson, R. Luca, A. Tudorache, On a system of fractional differential equations with coupled integral boundary conditions, Fract. Calcul. Appl. Anal., 18(2):361–386, 2015, https://doi.org/10.1515/fca-2015-0024.

13. J. Henderson, R. Luca, A. Tudorache, Existence and nonexistence of positive solutions for coupled Riemann–Liouville fractional boundary value problems, Discrete Dyn. Nature Soc., 2016:2823971, 2016, https://doi.org/10.1155/2016/2823971.

14. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud., Vol. 204, Elsevier, Amsterdam, 2006.

15. J. Klafter, S.C. Lim, R. Metzler (Eds.), Fractional Dynamics: Recent Advances, World Scientific, Singapore, 2011, https://doi.org/10.1142/8087.

16. M.A. Krasnosel’skii, P.P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, New York, 1984.

17. L. Liu, H. Li, C. Liu, Y. Wu, Existence and uniqueness of positive solutions for singular fractional differential systems with coupled integral boundary value problems, J. Nonlinear Sci. Appl., 10:243–262, 2017, https://doi.org/10.22436/jnsa.010.01.24.

18. L. Liu, X. Zhang, Y. Wu, Positive solutions of fourth-order nonlinear singular Sturm–Liouville eigenvalue problems, J. Math. Anal. Appl., 326:1212–1224, 2007, https://doi.org/ 10.1016/j.jmaa.2006.03.029.

19. S. Liu, J. Liu, Q. Dai, H. Li, Uniqueness results for nonlinear fractional differential equations with infinite-point integral boundary conditions, J. Nonlinear Sci. Appl., 10:1281–1288, 2017, https://doi.org/10.22436/jnsa.010.03.37.

20. R. Luca, On a class of nonlinear singular Riemann–Liouville fractional differential equations, Results Math., 73:125, 2018, https://doi.org/10.1007/s00025-018-0887-5.

21. R. Luca, A. Tudorache, Positive solutions to a system of semipositone fractional boundary value problems, Adv. Difference Equ., 2014:179, 2014, https://doi.org/10.1186/ 1687-1847-2014-179.

22. R. Metzler, J. Klafter, The random walks guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339:1–77, 2000, https://doi.org/10.1016/S0370- 1573(00)00070-3.

23. M. Ostoja-Starzewski, Towards thermoelasticity of fractal media, J. Therm. Stress., 30:889– 896, 2007, https://doi.org/10.1080/01495730701495618.

24. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999.

25. R. Pu, X. Zhang, Y. Cui, P. Li, W.Wang, Positive solutions for singular semipositone fractional differential equation subject to multipoint boundary conditions, J. Funct. Spaces, 2017:5892616, 2017, https://doi.org/10.1155/2017/5892616.

26. J. Sabatier, O.P. Agrawal, J.A.T. Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007, https://doi.org/10.1007/978-1-4020-6042-7.

27. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.

28. C. Shen, H. Zhou, L. Yang, Positive solution of a system of integral equations with applications to boundary value problems of differential equations, Adv. Difference Equ., 2016:260, 2016, https://doi.org/10.1186/s13662-016-0953-9.

29. I.M. Sokolov, J. Klafter, A. Blumen, A fractional kinetics, Phys. Today, 55:48–54, 2002, https://doi.org/10.1063/1.1535007.

30. F. Wang, L. Liu, Y. Wu, Iterative unique positive solutions for a new class of nonlinear singular higher order fractional differential equations with mixed-type boundary value conditions, J. Inequal. Appl., 2019:210, 2019, https://doi.org/10.1186/s13660-019- 2164-x.

31. J. Xu, Z. Wei, Positive solutions for a class of fractional boundary value problems, Nonlinear Anal. Model. Control, 21:1–17, 2016, https://doi.org/10.15388/NA.2016.1.1.

32. K. Zhang, Nontrivial solutions of fourth-order singular boundary value problems with sign- changing nonlinear terms, Topol. Methods Nonlinear Anal., 40:53–70, 2012.

33. L. Zhang, Z. Sun, X. Hao, Positive solutions for a singular fractional nonlocal boundary value problem, Adv. Difference Equ., 2018:381, 2018, https://doi.org/10.1186/ s13662-018-1844-z.

34. X. Zhang, Positive solutions for a class of singular fractional differential equation with infnite- point boundary conditions, Appl. Math. Lett.,39:22–27, 2015, https://doi.org/10. 1016/j.aml.2014.08.008.

35. X. Zhang, Q. Zhong, Triple positive solutions for nonlocal fractional differential equations with singularities both on time and space variables, Appl. Math. Lett.,80:12–19, 2018, https://doi.org/10.1016/j.aml.2017.12.022.

36. Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014,https://doi.org/10.1142/9069.

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