Positive solutions for fractional boundary value problems with integral boundary conditions and parameter dependence

Positive solutions for fractional boundary value problems with integral boundary conditions and parameter dependence Hafida Abbas∗, Mohammed Belmekki∗∗ and Alberto Cabada∗∗∗ ∗Department of Mathematics, University of Sa¨ıda, BP 138, 20000 Sa¨ıda, Algeria. e-mail: [email protected] ∗∗ High School of Applied Sciences, BP 165 RP. Bel Horizon, Tlemcen, Algeria. e-mail: [email protected] ∗∗∗ Departamento de Estat´ıstica, An´alise Matem´atica e Optimizaci´on, Facultade de Matem´aticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain. e-mail: alb[email protected] Abstract We study the existence of positive solutions for a Riemann fractional boundary value problem with integral boundary conditions and parameter dependence. To state our results, we use Guo-Krasnoselskii fixed point theorem. Some examples are showed to point out the applicability of the obtained results. Key words. Fractional differential equation, Integral boundary conditions, Positive solutions, Green’s function, Fixed point theorem. AMS (MOS) subject classification: 26A33, 34B18 1 Introduction Fractional calculus appears in many fields of engineering and sciences as rheology, viscoelasticity, electrochemistry, electromagnetism, and so forth. Many different books and monographs are devoted to the development of fractional calculus. See for instance [5, 9, 10, 11, 14, 15, 16, 18]. The interest of the study of fractional order differential equations lies in the fact that there are more degrees of freedom in the fractional-order models. Furthermore, fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. Recent results on fractional differential equations can be seen in [6, 7, 8, 12]. Integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, population dynamics and so forth, for more details see [1, 4, 6, 17]. As in dynamic of populations, many fields of engineering and sciences focus their interest on existence of positive solutions. We mention the works [2, 3, 4, 13]. Motivated by this and the above cited works, in this paper we investigate the existence of positive solutions of the following fractional differential equation with integral boundary conditions. Dδu(t) + f(t, u(t)) = 0,0< t < 1,1< δ ≤2,(1) This version of the article has been accepted for publication, after peer review and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s40314-021-01546-y Fractional Boundary Value Problems 2 u(0) = 0, u(1) = λZ1 0h(r)u(r)dr. (2) Where Dδis the Riemann-Liouville fractional derivative and fis a given function. The boundary conditions (2) can be thought as a mechanism putted at the end point of an oscillator, which is characterized by the weighted function hand the parameter λ, that controls its displacement according to the feedback from devices measuring the displacements along different parts of the oscillator. This paper is organized as follows. In section 2, we recall some definitions concerning the fractional integral and derivative, and related basic properties which will be used in the sequel. We consider an auxiliary problem to derive the Green’s function. Our main existence results are given in Section 3. Some examples are introduced in the last section. 2 Preliminaries Here we present some basic knowledge and definitions for fractional calculus which will be used in the sequel. Definition 2.1 ([15, 18]). The Riemann-Liouville fractional primitive of order δ > 0 of a function f: (0,1] →Ris given by Iδ 0f(t) = 1 Γ(δ)Zt 0(t−τ)δ−1f(τ)dτ, provided that the right side is pointwise defined on (0,1], and where Γis the gamma function. Definition 2.2 ([15, 18]). For a continuous function f: (0,1] →R, The RiemannLiouville derivative of fractional order δ > 0is given by Dδf(t) = 1 Γ(n−δ) dn dtnZt 0(t−τ)n−δ−1f(τ)dτ, n = [δ]+1, where [δ] denotes the integer part of the real number δ. Lemma 2.1 ([15, 18]). Let δ > 0, then the solutions of the fractional differential equation Dδu(t)=0 are given by the following expression u(t) = c1tδ−1+c2tδ−2+... +cntδ−n, ci∈R, i = 1, ..., n, n = [δ]+1. From Lemma 2.1 we deduce the following result. Fractional Boundary Value Problems 3 Lemma 2.2 ([15, 18]). Let δ > 0, then IδDδu(t)=u(t) + c1tδ−1+c2tδ−2+... +cntδ−n, ci∈R, i = 1, ..., n, n = [δ]+1. In order to get the expression for the Green’s function of boundary value problem (1) −(2), we start by solving the following auxiliary problem: Dδu(t) + σ(t) = 0,0<t<1,1< δ ≤2,(3) u(0) = 0, u(1) = λZ1 0h(r)u(r)dr. (4) Lemma 2.3 Let 1< δ ≤2. Suppose that 1−λR1 0h(r)rδ−1dr 6= 0. A function u∈C[0,1] is a solution of the linear boundary value problem (3)-(4) if and only if it satisfies the integral equation u(t) = Z1 0G(t, s)σ(s)ds, where G(t, s)is the Green’s function given by G(t, s) = G1(t, s) + G2(t, s) with G1(t, s) =      tδ−1(1−s)δ−1−(t−s)δ−1 Γ(δ),0≤s≤t≤1; tδ−1(1−s)δ−1 Γ(δ),0≤t≤s≤1. (5) and G2(t, s) = λtδ−1 1−λR1 0h(r)rδ−1dr Z1 0h(r)G1(r, s)dr (6) Proof. By Lemma 2.2 we have that the uis a solution of the linear equation (3) if and only if it satisfies u(t) = −Zt 0 (t−s)δ−1 Γ(δ)σ(s)ds +c1tδ−1+c2tδ−2. Condition u(0) = 0 implies necessarily that c2= 0. Since u(1) = λR1 0h(r)u(r)dr, we deduce that c1=Z1 0 (1 −s)δ−1 Γ(δ)σ(s)ds +λc1Z1 0h(s)sδ−1ds −λ Γ(δ)Z1 0h(r)Zr 0(r−s)δ−1σ(s)dsdr. Now, since 1 −λR1 0h(r)rδ−1dr 6= 0, we have c1=1 Γ(δ)(1 −λR1 0h(r)rδ−1dr)Z1 0(1 −s)δ−1σ(s)ds −λZ1 0h(r)Zr 0(r−s)δ−1σ(s)dsdr Fractional Boundary Value Problems 4 Finally, we have the expression u(t) = −Zt 0 (t−s)δ−1 Γ(δ)σ(s)ds +tδ−1 Γ(δ)(1 −λR1 0h(r)rδ−1dr)Z1 0(1 −s)δ−1σ(s)ds −λtδ−1 Γ(δ)(1 −λR1 0h(r)rδ−1dr)Z1 0h(r)Zr 0(r−s)δ−1σ(s)dsdr =−Zt 0 (t−s)δ−1 Γ(δ)σ(s)ds +tδ−1(1 −λR1 0h(r)rδ−1dr +λR1 0h(r)rδ−1dr) Γ(δ)(1 −λR1 0h(r)rδ−1dr)Z1 0(1 −s)δ−1σ(s)ds −λtδ−1 Γ(δ)(1 −λR1 0h(r)rδ−1dr)Z1 0h(r)·Zr 0(r−s)δ−1σ(s)dsdr =−Zt 0 (t−s)δ−1 Γ(δ)σ(s)ds +tδ−1 Γ(δ)Z1 0(1 −s)δ−1σ(s)ds +λtδ−1R1 0h(r)rδ−1dr Γ(δ)(1 −λR1 0h(r)rδ−1dr)Z1 0(1 −s)δ−1σ(s)ds −λtδ−1 Γ(δ)(1 −λR1 0h(r)rδ−1dr)Z1 0h(r)·Zr 0(r−s)δ−1σ(s)dsdr =−Zt 0 (t−s)δ−1 Γ(δ)σ(s)ds +tδ−1 Γ(δ)Zt 0(1 −s)δ−1σ(s)ds +tδ−1 Γ(δ)Z1 t(1 −s)δ−1σ(s)ds +λtδ−1 Γ(δ)(1 −λR1 0h(r)rδ−1dr)Z1 0h(r)rδ−1dr ·Z1 0(1 −s)δ−1σ(s)ds −λtδ−1 Γ(δ)(1 −λR1 0h(r)rδ−1dr)Z1 0h(r)·Zr 0(r−s)δ−1σ(s)dsdr =−Zt 0 (t−s)δ−1 Γ(δ)σ(s)ds +tδ−1 Γ(δ)Zt 0(1 −s)δ−1σ(s)ds +tδ−1 Γ(δ)Z1 t(1 −s)δ−1σ(s)ds +λtδ−1 Γ(δ)(1 −λR1 0h(r)rδ−1dr)Z1 0h(r)·Z1 0rδ−1(1 −s)δ−1σ(s)dsdr −λtδ−1 Γ(δ)(1 −λR1 0h(r)rδ−1dr)Z1 0h(r)·Zr 0(r−s)δ−1σ(s)dsdr Fractional Boundary Value Problems 5 =−Zt 0 (t−s)δ−1 Γ(δ)σ(s)ds +tδ−1 Γ(δ)Zt 0(1 −s)δ−1σ(s)ds +tδ−1 Γ(δ)Z1 t(1 −s)δ−1σ(s)ds +λtδ−1 Γ(δ)(1 −λR1 0h(r)rδ−1dr)Z1 0h(r)·Zr 0rδ−1(1 −s)δ−1σ(s)dsdr −λtδ−1 Γ(δ)(1 −λR1 0h(r)rδ−1dr)Z1 0h(r)·Zr 0(r−s)δ−1σ(s)dsdr +λtδ−1 Γ(δ)(1 −λR1 0h(r)rδ−1dr)Z1 0h(r)·Z1 rrδ−1(1 −s)δ−1σ(s)dsdr =Z1 0G1(t, s)σ(s)ds +λtδ−1 Γ(δ)(1 −λR1 0h(r)rδ−1dr)Z1 0h(r)·Z1 0G1(r, s)σ(s)dsdr =Z1 0G1(t, s)σ(s)ds +Z1 0 λtδ−1 Γ(δ)(1 −λR1 0h(r)rδ−1dr)Z1 0h(r)G1(r, s)drσ(s)ds =Z1 0G1(t, s)σ(s)ds +Z1 0G2(t, s)σ(s)ds. ut As a direct consequence of the previous result, we deduce the following properties. Lemma 2.4 The function G1(t, s)defined in Lemma 2.3 has the following properties: 1. G1(t, s)∈ C ([0,1] ×[0,1]). 2. G1(t, s)>0for (t, s)∈(0,1) ×(0,1) and G1(0, s) = 0 = G1(1, s)for s∈[0,1]. 3. G1(t, s) = G1(1 −s, 1−t),∀t, s ∈[0,1]. In next result, we deduce two inequalities that, as we will see, will be fundamental to ensure the existence of the solutions of the nonlinear problem (1)-(2). Lemma 2.5 Let the function G1(t, s)be defined in Lemma 2.3 and fix t0∈(0,1), then G1satisfies the following inequalities: G1(t, s)≤sδ−1(1 −s)δ−1 Γ(δ),∀t∈[0,1], s ∈[0,1] (7) and sδ−1(1 −s)δ−1k(t, t0)≤G1(t, s),∀t∈[0,1], s ∈[t0,1],(8) with k(t, t0) :=          tδ−1 Γ(δ)if 0≤t≤t0<1 min tδ−1 Γ(δ),tδ−1(1−t0)δ−1−(t−t0)δ−1 Γ(δ)tδ−1 0(1−t0)δ−1if 0< t0< t ≤1 . Fractional Boundary Value Problems 6 Proof. For s>t,∂G1 ∂t (t, s) = δ−1 Γ(δ)(1 −s)δ−1tδ−2>0. For s < t, since 1 < δ ≤2, we have ∂G1 ∂t (t, s) = δ−1 Γ(δ)(1 −s)δ−1tδ−2−(t−s)δ−2≤δ−1 Γ(δ)tδ−2−(t−s)δ−2<0. As a consequence, it is fulfilled that G1(t, s)≤G1(s, s) = sδ−1(1 −s)δ−1 Γ(δ)∀t, s ∈[0,1] and inequality (7) holds. By using the third property on Lemma 2.4, we deduce that ∂G1 ∂s (t, s)>0 for 0 ≤s<t≤1 and ∂G1 ∂s (t, s)<0 if 0 ≤t<s≤1. Now, we introduce the following function: F1(t, s) = G1(t, s) sδ−1(1 −s)δ−1,(t, s)∈[0,1] ×(0,1), as a direct consequence of previous arguments, we deduce that ∂F1 ∂t (t, s)<0 for 0 ≤s<t≤1 and ∂F1 ∂t (t, s)>0 if 0 ≤t < s ≤1. As a consequence, we have that G1(t, s) sδ−1(1 −s)δ−1≤G1(s, s) sδ−1(1 −s)δ−1=1 Γ(δ). By the other hand, ∂F1 ∂s (t, s) =          −tδ−1 Γ(δ−1) sδ0≤t<s≤1, (δ−1)(t(s2−2st+t)(t−s)δ−(t−s)2tδ(1−s)δ) tΓ(δ)(t−s)2(1−s)δsδ0≤s<t≤1. Fractional Boundary Value Problems 7 As a direct consequence, we deduce that ∂F1 ∂s (t, s)<0 for 0 ≤t<s≤1. On the other hand, for the case 0 ≤s < t ≤1 we have that ∂F1 ∂s (t, s)>0 if and only if h1(t, s, δ) := (1 −s)δtδ−1(t−s)2−δ< s2−2s t +t=: h2(t, s). Now, since ∂h1 ∂δ (t, s, δ) = (1 −s)δtδ−1(t−s)2−δlog t−t s t−s, we have that h1is strictly increasing on the δinterval [1,2] for any 0 ≤s<t≤1 given. Thus, since h2(t, s)−h1(t, s, 2) = (1 −t)s2>0, we conclude that ∂F1 ∂s (t, s)>0 for all 0 <s<t<1. So, for any t0∈(0,1) fixed, we have that G1(t, s) sδ−1(1 −s)δ−1≥min (lim s→1− G1(t, s) s(1 −s)δ−1,G1(t, t0) tδ−1 0(1 −t0)δ−1) = min (tδ−1 Γ(δ),G1(t, t0) tδ−1 0(1 −t0)δ−1)=: k(t, t0),∀t∈[0,1], s ∈[t0,1], and the result is concluded. ut By virtue of this lemma, we can give now the main result of this section. Lemma 2.6 Let t0∈(0,1) be fixed and hintroduced at the boundary conditions (2). Denote by A=R1 0h(r)rδ−1dr,B=R1 0h(r)dr and C0=R1 t0k(r, t0)h(r)dr. Assume that h≥0on [0,1] and 1−λ A > 0. Then the Green’s function G(t, s)defined in Lemma 2.3 satisfies the inequalities λ C0tδ−1 1−λA sδ−1(1 −s)δ−1≤G(t, s)≤1 Γ(δ) 1 + λB 1−λA!sδ−1(1 −s)δ−1,∀t, s ∈[0,1]. (9) Proof. From the definition of G, the inequality (7) and the fact that 1 < δ ≤2, we have the following inequalities for all t, s ∈[0,1]: G(t, s)≤1 Γ(δ)sδ−1(1 −s)δ−1+λtδ−1 1−λA Z1 0 1 Γ(δ)sδ−1(1 −s)δ−1h(r)dr ≤1 Γ(δ) 1 + λB 1−λA!sδ−1(1 −s)δ−1. Fractional Boundary Value Problems 8 On the other hand, by Lemma 2.4 (2) and (8), we have for all t, s ∈[0,1]: G(t, s) = G1(t, s) + G2(t, s) ≥λtδ−1 1−λR1 0h(r)rδ−1dr Z1 t0 h(r)G1(r, s)dr ≥λtδ−1 1−λA C0sδ−1(1 −s)δ−1, as we want to prove. ut As a direct consequence, we deduce the following Corollary: Corollary 2.1 If h≥0on [0,1] and 1−λ A > 0then the Green’s function G(t, s) defined in Lemma 2.3 satisfies the inequalities λ t 1−λA C0sδ−1(1−s)δ−1≤t2−δG(t, s)≤1 Γ(δ) 1 + λB 1−λA!sδ−1(1−s)δ−1,∀t, s ∈[0,1] 3 Main Results Now for any u: (0,1] →R, we define function u: [0,1] →Ras follows: u(t) = (t2−δu(t) if t∈(0,1], limt→0+t2−δu(t) if t= 0, provided that such limit exists. Consider the Banach space E=Cδ[0,1] := {¯u: [0,1] →R,is a continuous function in [0,1]} endowed with the maximum norm kuk= max 0≤t≤1|¯u(t)|and define the cone P0⊂Eby P0={u∈E, ¯u(t)≥t2−δp(t, t0)kuk, for all t ∈[0,1]}, where p(t, t0) = Γ(δ)λtδ−1 1−λA C0, 1 + λB 1−λA!, t ∈[0,1], with t0∈(0,1) fixed, and A,Band C0introduced in Lemma 2.6. Notice that, provided that h≥0 on [0,1] and 1 −λ A > 0, we deduce from (9) that 0≤p(t, t0)≤1 for all t∈[0,1] and t0∈(0,1). Now, we assume the following hypothesis on the nonlinear part of the equation: (H1) Function f: [0,1] ×R→[0,∞) is continuous. Fractional Boundary Value Problems 9 So, we define the operator T:P0→Eby (Tu)(t) = Z1 0G(t, s)f(s, ¯u(s))ds, t ∈[0,1] (10) Lemma 3.1 T:P0→P0is completely continuous. Proof: Let us prove in first that T(P0)⊂P0. Notice from the definition of Tand Corollary 2.1 that for u∈P0, Tu(t)≥0 for all t∈[0,1] and t2−δ(Tu)(t) = Z1 0t2−δG(t, s)f(s, ¯u(s))ds ≥Z1 0t2−δλtδ−1 1−λA C0sδ−1(1 −s)δ−1f(s, ¯u(s))ds =t2−δΓ(δ)λtδ−1 1−λA C0 1 + λB 1−λA Z1 01 + λB 1−λA  Γ(δ)sδ−1(1 −s)δ−1f(s, ¯u(s))ds ≥t2−δp(t, t0)Z1 0max 0≤t≤1nt2−δG(t, s)of(s, ¯u(s))ds ≥t2−δp(t, t0)max 0≤t≤1Z1 0t2−δG(t, s)f(s, ¯u(s))ds =t2−δp(t, t0)kTuk. Thus, T(P0)⊂P0. In addition, since fis a continuous function it follows that Tis a continuous operator. Next, we show that Tis uniformly bounded. Let D⊂Pbe a bounded set, i.e. there exists a constant L > 0 such that kuk ≤ L, for all u∈D. Set M= max 0≤s≤1,0≤u≤L{f(s, ¯u(s))}. Then, from Lemma 2.6, and for all u∈D, we have |t2−δTu(t)|=Z1 0t2−δG(t, s)f(s, ¯u(s))ds ≤M Γ(δ) 1 + λB 1−λA!Z1 0sδ−1(1 −s)δ−1ds =M 1 + λB 1−λA!Γ(δ) Γ(2δ). Hence, T(D) is bounded. Finally, we show that Tis equicontinuous, as follows. For all  > 0 and for each u∈P, let t1, t2∈[0,1], be such that t1< t2. Fractional Boundary Value Problems 16 References [1] Z. Bai, and H. L¨u, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005), 495–505. [2] A. Cabada, and Z. Hamdi, Nonlinear fractional differential equations with integral boundary value conditions, Appl. Math. Comput. 228 (2014) 251–257. [3] A. Cabada, S. Dimitrijevic, T. Tomovic and S. Alecsic, The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions, Math. Meth. Appl. Sci (2017),40, 1880–1891. [4] A. Cabada, and G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl. 389 (2012) 403–411. [5] K. Diethelm, and A. D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, in “Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties” (F. Keil, W. Mackens, H. Voss, and J. Werther, Eds), pp 217–224, Springer-Verlag, Heidelberg, 1999. [6] M. Feng, X. Zhang, and W. Ge, New existence results for higher-order nonlinear fractional differential equations with integral boundary conditions, Bound. Value Probl. 2011 (2011). ART. ID 720702. [7] W. G. Gl¨ockle, and T. F. Nonnenmacher, A fractional calculus approach of selfsimilar protein dynamics, Biophys. J. 68 (1995), 46–53. [8] N. Heymans, and I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheol. Acta 45(5) (2006), 765–772. [9] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. [10] V. Kiryakova, Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics Series, 301. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1994. [11] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in “Fractals and Fractional Calculus in Continuum Mechanics” (A. Carpinteri, and F. Mainardi, Eds), pp. 291–348, Springer-Verlag, Wien, 1997. Fractional Boundary Value Problems 17 [12] F. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys. 103 (1995), 7180–7186. [13] J. J. Nieto and J. Pimentel. Positive solutions of a fractional thermostat model. Bound. Value Probl. 2013, 2013:5, 11 pp. [14] K. S. Miller, and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. [15] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [16] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5 (2002), 367–386. [17] H. A. H. Salem. Fractional order boundary value problem with integral boundary conditions involving Pettis integral, Acta Math. Sci. Ser. B (Engl. Ed.) 31(2) (2011) 661–672. [18] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993. View publication stats