Existence and multiplicity of positive solutions for a system of fractional boundary value problems

Boundary Value Problems, Mar 2014

We study the existence and multiplicity of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations, subject to integral boundary conditions. The nonsingular and singular cases for the nonlinearities are investigated. MSC: 34A08, 45G15.

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Existence and multiplicity of positive solutions for a system of fractional boundary value problems

Boundary Value Problems Existence and multiplicity of positive solutions for a system of fractional boundary value problems Johnny Henderson 1 Rodica Luca 0 0 Department of Mathematics, Gh. Asachi Technical University , Iasi, 700506 , Romania 1 Department of Mathematics, Baylor University , Waco, TX 76798-7328 , USA We study the existence and multiplicity of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations, subject to integral boundary conditions. The nonsingular and singular cases for the nonlinearities are investigated. MSC: 34A08; 45G15 We consider the system of nonlinear ordinary fractional differential equations Riemann-Liouville fractional differential equation; integral boundary conditions; positive solutions 1 Introduction Dα+u(t) + f (t, v(t)) = , t ∈ (, ), n –  < α ≤ n, β D+v(t) + g(t, u(t)) = , t ∈ (, ), m –  < β ≤ m, with the integral boundary conditions u() = u () = · · · = u(n–)() = , v() = v () = · · · = v(m–)() = , u() =  u(s) dH(s), v() =  v(s) dK (s), where n, m ∈ N, n, m ≥ , Dα+ and Dβ+ denote the Riemann-Liouville derivatives of orders α and β, respectively, and the integrals from (BC) are Riemann-Stieltjes integrals. Under sufficient conditions on functions f and g, which can be nonsingular or singular in the points t =  and/or t = , we study the existence and multiplicity of positive solutions of problem (S)-(BC). We use the Guo-Krasnosel’skii fixed point theorem (see []) and some theorems from the fixed point index theory (from [] and []). By a positive solution of problem (S)-(BC) we mean a pair of functions (u, v) ∈ C([, ]) × C([, ]) satisfying (S) and (BC) with u(t) ≥ , v(t) ≥  for all t ∈ [, ] and supt∈[,] u(t) > , supt∈[,] v(t) > . The system (S) with α = n, β = m and the boundary conditions (BC) where H and K are scale functions (that is, multi-point boundary conditions) has been investigated in [] (the nonsingular case) and [] (the singular case). In [], the authors give sufficient conditions for λ, μ, f , and g such that the system Dα+u(t) + λf (t, u(t), v(t)) = , t ∈ (, ), n –  < α ≤ n, β D+v(t) + μg(t, u(t), v(t)) = , t ∈ (, ), m –  < β ≤ m, with the boundary conditions (BC) with H and K scale functions, has positive solutions (u(t) ≥ , v(t) ≥  for all t ∈ [, ], and (u, v) = (, )). Fractional differential equations describe many phenomena in various fields of engineering and scientific disciplines such as physics, biophysics, chemistry, biology, economics, control theory, signal and image processing, aerodynamics, viscoelasticity, electromagnetics, and so on (see [–]). In Section , we present the necessary definitions and properties from the fractional calculus theory and some auxiliary results dealing with a nonlocal boundary value problem for fractional differential equations. In Section , we give some existence and multiplicity results for positive solutions with respect to a cone for our problem (S)-(BC), where f and g are nonsingular functions. The case when f and g are singular at t =  and/or t =  is studied in Section . Finally, in Section , we present two examples which illustrate our main results. 2 Preliminaries and auxiliary results We present here the definitions, some lemmas from the theory of fractional calculus and some auxiliary results that will be used to prove our main theorems. provided the right-hand side is pointwise defined on (, ∞), where (α) is the Euler gamma function defined by (α) = ∞ tα–e–t dt, α > . The notation α stands for the largest integer not greater than α. We also denote the Riemann-Liouville fractional derivative of f by Dα+f (t). If α = m ∈ N then Dm+f (t) = f (m)(t) for t > , and if α =  then D+f (t) = f (t) for t > . Lemma . ([]) Let α >  and n = α +  for α ∈/ N and n = α for α ∈ N; that is, n is the smallest integer greater than or equal to α. Then the solutions of the fractional differential equation Dα+u(t) = ,  < t < , are u(t) = ctα– + ctα– + · · · + cntα–n,  < t < , where c, c, . . . , cn are arbitrary real constants. Lemma . ([, ]) Let α > , n be the smallest integer greater than or equal to α (n –  < α ≤ n) and y ∈ L(, ). The solutions of the fractional equation Dα+u(t) + y(t) = ,  < t < , are where c, c, . . . , cn are arbitrary real constants. We consider now the fractional differential equation  < t < , n –  < α ≤ n, with the integral boundary conditions u() = u () = · · · = u(n–)() = , u() = Lemma . If H : [, ] → R is a function of bounded variation,  =  –  sα– dH(s) =  and y ∈ C([, ]), then the solution of problem ()-() is given by Proof By Lemma ., the solutions of equation () are  ≤ t ≤ . So, we obtain c = Therefore, we get the expression () for the solution of problem ()-(). value problem ()-() is given by G(t, s) = g(t, s) + (t, s) ∈ [, ] × [, ], g(t, s) =  ≤ s ≤ t ≤ ,  ≤ t ≤ s ≤ . Proof By Lemma . and relation (), we conclude u(t) = G(t, s)y(s) ds,  where g and G are given in () and (), respectively. Hence u(t) =  G(t, s)y(s) ds for all t ∈ [, ]. for all s ∈ [, ], Proof By using the assumptions of this lemma, we have G(t, s) ≥  for all (t, s) ∈ [, ] × [, ], and so u(t) ≥  for all t ∈ [, ]. Lemma . Assume that H : [, ] → R is a nondecreasing function and Green’s function G of the problem ()-() satisfies the inequalities: (a) G(t, s) ≤ J(s), ∀(t, s) ∈ [, ] × [, ], where s ∈ [, ]. (b) For every c ∈ (, /), we have G(t, s) ≥ γJ(s) ≥ γG t , s , ∀t , s ∈ [, ].  > . Then the G(t, s) ≥ cα–g θ(s), s + g(τ , s) dH(τ ) = γJ(s) ≥ γG t , s . Therefore, we obtain the inequalities (b) of this lemma. Lemma . Assume that H : [, ] → R is a nondecreasing function and  > , c ∈ (, /), and y ∈ C([, ]), y(t) ≥  for all t ∈ [, ]. Then the solution u(t), t ∈ [, ] of problem ()-() satisfies the inequality mint∈[c,–c] u(t) ≥ γ maxt ∈[,] u(t ). u(t) = G t , s y(s) ds = γu t . Then we deduce the conclusion of this lemma.  < t < , m –  < β ≤ m, with the integral boundary conditions v() = v () = · · · = v(m–)() = , v() = 3 The nonsingular case We consider the Banach space X = C([, ]) with supremum norm cone P ⊂ X by P = {u ∈ X, u(t) ≥ , ∀t ∈ [, ]}. We also define the operators A : P → X by · and define the (Au)(t) = G(t, s)f s, G(s, τ )g τ , u(τ ) dτ ds, t ∈ [, ], and B : P → X, C : P → X by (Bu)(t) = G(t, s)u(s) ds, (Cu)(t) = G(t, s)u(s) ds, t ∈ [, ]. Under the assumptions (H) and (H), using also Lemma ., it is easy to see that A, B, and C are completely continuous from P to P. Thus the existence and multiplicity of positive solutions of the system (S)-(BC) are equivalent to the existence and multiplicity of fixed points of the operator A. (i) f ∞i = lim inf inf f (t, u) u→∞ t∈[c,–c] up ∈ (, ∞]; (ii) g∞i = lim inf inf g(t, u) u→∞ t∈[c,–c] u/p = ∞. (i) fs = lim sup sup f (t, u) ∈ [, ∞); u→+ t∈[,] uq (ii) gs = lim sup sup g(t, u) u→+ t∈[,] u/q = , Proof Because the proof of the theorem is similar to that of Theorem . from [], we will sketch some parts of it. From assumption (i) of (H), we deduce that there exist C, C >  such that f (t, u) ≥ Cup – C, ∀(t, u) ∈ [c,  – c] × [, ∞). (Au)(t) ≥ C where C = C c–c J(s) ds. ∀t ∈ [, ], () For c given in (H), we define the cone P = {u ∈ P, inft∈[c,–c] u(t) ≥ γ u }, where γ = min{γ, γ}. From our assumptions and Lemma ., for any y ∈ P, we can easily show that u = By ∈ P and v = Cy ∈ P, that is, B(P) ⊂ P and C(P) ⊂ P. We now consider the function u(t) =  G(t, s) ds = (By)(t) ≥ , t ∈ [, ], with y(t) =  for all t ∈ [, ]. We define the set u(t) = (Au)(t) + λu(t) ≥ (Au)(t) –c –c where C = C + CCmmγγp > . Hence, inft∈[c,–c] u(t) ≥  inft∈[c,–c] u(t) – C, and so u(t) ≤ C, ∀u ∈ M. Now from relations () and (), one obtains u ≤ (inft∈[c,–c] u(t))/γ ≤ C/γ , for all u ∈ M, that is, M is a bounded subset of X. u ≤ γ t∈[icn,f–c] u(t), ∀u ∈ M. g(t, u) p ≥ εpu – C, ∀(t, u) ∈ [c,  – c] × [, ∞), i(A, BL ∩ P, P) = . f (t, u) ≤ Muq, ∀(t, u) ∈ [, ] × [, ]; where ε = min{/M, (/(MMMq))/q} > , M =  J(s) ds > , M =  J(s) ds > . Hence, for any u ∈ Bδ ∩ P and t ∈ [, ], we obtain Besides, there exists a sufficiently large L >  such that J(s) u(s) /q ds ≤ εM u /q ≤ . (Au)(t) ≤ M J(s) ds = MεqMMq u ≤  u . This implies that Au ≤ u / for all u ∈ ∂Bδ ∩ P. From [], we conclude that the fixed point index of the operator A over Bδ ∩ P with respect to P is Combining () and (), we obtain Using similar arguments as those used in the proofs of Theorem . and Theorem . in [], we also obtain the following results for our problem (S)-(BC). (ii) g∞s = lim sup sup g(t, u) u→∞ t∈[,] u/r = . (ii) gi = lim inf inf u→+ t∈[c,–c] ∈ (, ∞]; = ∞, ∀t ∈ [, ], where m = max{K, K}, K = maxs∈[,] J(s), K = maxs∈[,] J(s), and J, J are defined in Section , then the problem (S)-(BC) has at least two positive solutions (u(t), v(t)), (u(t), v(t)), t ∈ [, ]. 4 The singular case f (t, x) ≤ p(t)q(x), g(t, x) ≤ p(t)q(x), ∀t ∈ (, ), x ∈ R+. (Au)(t) = G(t, s)f s, (Au)(t) = G(t, s)f s,   g(t, s) + Proof We denote by α =  J(s)p(s) ds and β =  J(s)p(s) ds. Using (H), we deduce that  < α < ∞ and  < β < ∞. By Lemma . and the corresponding lemma for G, we see that A maps P into P. We shall prove that A maps bounded sets into relatively compact sets. Suppose D ⊂ P is an arbitrary bounded set. Then there exists M >  such that u ≤ M for all u ∈ D. By using (H) and Lemma ., we obtain Au ≤ αM for all u ∈ D, where M = supx∈[,βM] q(x), and M = supx∈[,M] q(x). In what follows, we shall prove that A(D) is equicontinuous. By using Lemma ., we have ∀t ∈ [, ]. (Au) (t) = So, for any t ∈ (, ), we deduce (Au) (t) ≤ h(t) = ≤ M g(τ , s) dH(τ ) p(s) ds, t ∈ (, ). h(t) dt = h(t) dt +    ( – s)α–p(s) + H() – H() (Au)(t) – (Au)(t) = (Au) (t) dt ≤ M Therefore A is a compact operator. Besides, we can easily show that A is continuous on P. Hence A : P → P is completely continuous. (i) q∞ = lim sup q(x) s x→∞ xα ∈ [, ∞); (ii) qs∞ = lixm→s∞up qxα(x) = . (i) fi = lixm→in+f t∈[icn,f–c] f (xtβ,x) ∈ (, ∞]; (ii) gi = lixm→in+f t∈[icn,f–c] g(xtβ,x) = ∞, Proof Because the proof of this theorem is similar to that of Theorem  in [], we will sketch some parts of it. For c given in (H), we consider the cone P = {u ∈ X, u(t) ≥ , ∀t ∈ [, ], mint∈[c,–c] u(t) ≥ γ u }, where γ = min{γ, γ}. Under assumptions (H)(H), we obtain A(P) ⊂ P. By (H), we deduce that there exist C, C, C >  and ε ∈ (, (α Cαβα )–/α ) such that ∀x ∈ [, ∞). (Au)(t) ≤ G(t, s)p(s)q ≤ C G(t, s)p(s) ≤ C ≤ C J(s)p(s) ds J(s)p(s) J(s)p(s) ds   ≤ Cα εα αβα u αα + Cα αβα Cα + αC, ∀t ∈ [, ]. Au ≤ u , ∀u ∈ ∂BR ∩ P. ∀(t, x) ∈ [c,  – c] × [, x], J(τ )p(τ )q u(τ ) dτ ≤ x. (Au)(t) ≥ C ≥ C By (), (), and the Guo-Krasnosel’skii fixed point theorem, we deduce that A has at least one fixed point u ∈ (BR \ Bε ) ∩ P. Then our problem (S)-(BC) has at least one positive solution (u, v) ∈ P × P where v(t) =  G(t, s)g(s, u(s)) ds. The proof of Theorem . is completed. Using similar arguments as those used in the proof of Theorem  in [] (see also [] for a particular case of the problem studied in []), we also obtain the following result for our problem (S)-(BC). (i) qs = lim sup q(x) x→+ xr ∈ [, ∞); (ii) g∞i = lim inf inf g(t, x) x→∞ t∈[c,–c] xl = ∞, and K (t) = t for all t ∈ [, ]. Then   sv(s) ds. We consider the system of fractional differential equations  u(s) dH(s) = u(  ) + u(  ) and  v(s) dK (s) = with the boundary conditions D/+u(t) + f (t, v(t)) = , D/+v(t) + g(t, u(t)) = , u() = u () = , u() = u(  ) + u(  ), v() = , v() =   sv(s) ds. Then we obtain  =  –  sα– dH(s) =  – (  )/ – (  )/ = (–√) ≈ . > ,  –  sβ– dK (s) =  –   s/ ds =  = . > . We also deduce  =  θ(s) = –s+s and θ(s) = s for all s ∈ [, ]. For the functions J and J, we obtain J(s) = g θ(s), s +  = g  ⎪⎪⎧ √π { (s–(–s+s)s/)/ + (–√) [(( – s)/ – ( – s)/) = ⎨⎪⎪⎪⎪⎪ √π+{((s–√(–s+s()s/)–/ s+)/(––√() –[(s)–/s))]}/, ⎪⎪⎪⎪⎪⎪⎪⎩ √π+[((s–√(–s+s)s(/)–/ s+)/(+–√((–)√(––)s)s/)]/,)]},  ≤ s <  , J(s) = g θ(s), s +  = g(s, s) +  Example  We consider the functions f (t, u) = a uα + uβ , g(t, u) = b uγ + uδ , t ∈ [, ], u ≥ , f ∞i = ul→im∞ a(uαu+/uβ ) = ∞, fi = ul→im+ a(uαu+ uβ ) = ∞, g∞i = ul→im∞ b(uγu+ uδ ) = ∞, gi = ul→im+ b(uγu+ uδ ) = ∞. Example  We consider the functions We have  <  p(s) ds < ∞,  <  p(s) ds < ∞. In (H), for r < a, r < b and rr ≥ , we obtain lixm→su+p qx(rx) = , lixm→su+p qx(rx) = . lixm→i∞nf t∈[icn,f–c] f (xtl,x) = ∞, lixm→i∞nf t∈[icn,f–c] g(xtl,x) = ∞. q(x) = xa, q(x) = xb. For example, if a = /, b = , r = , r = /, l = , l = /, the above conditions are satisfied. Then, by Theorem ., we deduce that the problem (S)-(BC) has at least one positive solution. Acknowledgements The work of R Luca was supported by the CNCS grant PN-II-ID-PCE-2011-3-0557, Romania. 1. Guo , D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, New York ( 1988 ) 2. Amann , H: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces . SIAM Rev . 18 , 620 - 709 ( 1976 ) 3. Zhou , Y, Xu , Y : Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations . J. Math. Anal. Appl . 320 , 578 - 590 ( 2006 ) 4. Henderson , J, Luca, R : Existence and multiplicity for positive solutions of a system of higher-order multi-point boundary value problems . Nonlinear Differ. Equ. Appl . 20 ( 3 ), 1035 - 1054 ( 2013 ) 5. Henderson , J, Luca, R : Positive solutions for singular systems of higher-order multi-point boundary value problems . Math. Model. 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Sabatier , J, Agrawal, OP, Machado, JAT (eds.) : Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering . Springer, Dordrecht ( 2007 ) 13. Samko , SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives: Theory and Applications . Gordon & Breach, Yverdon ( 1993 ) 14. Liu , B, Liu , L, Wu, Y : Positive solutions for singular systems of three-point boundary value problems . Comput. Math. Appl. 53 , 1429 - 1438 ( 2007 ) Cite this article as: Henderson and Luca: Existence and multiplicity of positive solutions for a system of fractional boundary value problems . Boundary Value Problems 2014 , 2014 : 60


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Johnny Henderson, Rodica Luca. Existence and multiplicity of positive solutions for a system of fractional boundary value problems, Boundary Value Problems, 2014, 60,