Positive solutions for a system of semipositone coupled fractional boundary value problems

Boundary Value Problems, Mar 2016

We study the existence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with sign-changing nonlinearities, subject to coupled integral boundary conditions. MSC: 34A08, 45G15.

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Positive solutions for a system of semipositone coupled fractional boundary value problems

Henderson and Luca Boundary Value Problems Positive solutions for a system of semipositone coupled 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 of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with sign-changing nonlinearities, subject to coupled integral boundary conditions. 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 [-]). Integral boundary conditions arise in thermal conduction problems, semiconductor problems and hydrodynamic problems. We consider the system of nonlinear fractional differential equations Riemann-Liouville fractional differential equations; coupled integral boundary conditions; positive solutions; sign-changing nonlinearities 1 Introduction (S) 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 coupled integral boundary conditions (BC) u() = u () = · · · = u(n–)() = , v() = v () = · · · = v(m–)() = , u() =  v(s) dH(s), v() =  u(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, the integrals from (BC) are Riemann-Stieltjes integrals, and f , g are sign-changing continuous functions (that is, we have a so-called system of semipositone boundary value problems). These functions may be nonsingular or singular at t =  and/or t = . The boundary conditions above include multi-point and integral boundary conditions and sum of these in a single framework. We present intervals for parameters λ and μ such that the above problem (S)-(BC) has at least one positive solution. 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 u(t) > , v(t) >  for all t ∈ (, ). In the case when f and g are nonnegative, problem (S)-(BC) has been investigated in [] by using the Guo-Krasnosel’skii fixed point theorem, and in [] where λ = μ =  and f (t, u, v) and g(t, u, v) are replaced by f˜(t, v) and g˜(t, u), respectively (denoted by (S)). In [], the authors study two cases: f and g are nonsingular and singular functions and they used some theorems from the fixed point index theory and the Guo-Krasnosel’skii fixed point theorem. The systems (S) and (S) with uncoupled boundary conditions (BC) u() = u () = · · · = u(n–)() = , v() = v () = · · · = v(m–)() = , u() =  u(s) dH(s), v() =  v(s) dK (s), were investigated in [] (problem (S)-(BC) with f , g nonnegative), in [] (problem (S)(BC) with f , g nonnegative, singular or not), and in [] (problem (S)-(BC) with f , g signchanging functions). We also mention paper [], where the authors studied the existence and multiplicity of positive solutions for system (S) with α = β, λ = μ, and the boundary conditions u(i)() = v(i)() = , i = , . . . , n – , u() = av(ξ ), v() = bu(η), ξ , η ∈ (, ), with ξ , η ∈ (, ),  < abξ η < , and f , g are sign-changing nonsingular or singular functions. The paper is organized as follows. Section  contains some preliminaries and lemmas. The main results are presented in Section , and finally in Section  some examples are given to support the new results. 2 Auxiliary results We present here the definitions of Riemann-Liouville fractional integral and RiemannLiouville fractional derivative and then some auxiliary results that will be used to prove our main results. Definition . The (left-sided) fractional integral of order α >  of a function f : (, ∞) → R is given by α I+f (t) =  (α)  t (t – s)α–f (s) ds, t > , provided the right-hand side is pointwise defined on (, ∞), where (α) is the Euler gamma function defined by (α) = ∞ tα–e–t dt, α > . Definition . The Riemann-Liouville fractional derivative of order α ≥  for a function f : (, ∞) → R is given by Dα+f (t) = d n dt In+–αf (t) =  d (n – α) dt n t f (s)  (t – s)α–n+ ds, t > , where n = α + , provided that the right-hand side is pointwise defined on (, ∞). The notation α stands for the largest integer not greater than α. 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 > . We consider now the fractional differential system Dα+u(t) + x˜(t) = , β D+v(t) + y˜(t) = , t ∈ (, ), n –  < α ≤ n, t ∈ (, ), m –  < β ≤ m, with the coupled integral boundary conditions where n, m ∈ N, n, m ≥ , and H, K : [, ] → R are functions of bounded variation. Lemma . ([]) If H, K : [, ] → R are functions of bounded variations, =  – (  τ α– dK (τ ))(  τ β– dH(τ )) =  and x˜, y˜ ∈ C(, ) ∩ L(, ), then the pair of functions (u, v) ∈ C([, ]) × C([, ]) given by   u(t) =  G(t, s)x˜(s) ds +  G(t, s)y˜(s) ds,   v(t) =  G(t, s)y˜(s) ds +  G(t, s)x˜(s) ds, t ∈ [, ], t ∈ [, ], where and ⎪⎧⎪⎪⎨⎪ GG((tt,,ss)) == gtα(–t, s)+gt(ατ–, s() dHτ β(τ–),dH(τ ))(  g(τ , s) dK (τ )), ⎪⎪⎩⎪⎪ GG((tt,, ss)) == gtβ–(t, s) +g(tβτ–,s() dKτ(ατ–),dK∀(τt,)s)(∈[g,(τ] , s) dH(τ )), ⎪⎪⎪⎪⎨⎧ g(t, s) = (α) ⎪⎩⎪⎪⎪ g(t, s) = (β) tα–( – s)α– – (t – s)α–, tα–( – s)α–, tβ–( – s)β– – (t – s)β–, tβ–( – s)β–,  ≤ s ≤ t ≤ ,  ≤ t ≤ s ≤ ,  ≤ s ≤ t ≤ ,  ≤ t ≤ s ≤ , is solution of problem ()-(). Lemma . The functions g, g given by () have the properties: (a) g, g : [, ] × [, ] → R+ are continuous functions, and g(t, s) > , g(t, s) >  for all (t, s) ∈ (, ) × (, ). (b) g(t, s) ≤ h(s), g(t, s) ≤ h(s) for all (t, s) ∈ [, ] × [, ], where h(s) = s(–s)α– (α–) and h(s) = s(–s)β– (β–) for all s ∈ [, ]. (c) g(t, s) ≥ k(t)h(s), g(t, s) ≥ k(t)h(s) for all (t, s) ∈ [, ] × [, ], where k(t) = min k(t) = min ( – t)tα– tα– α –  , α –  ( – t)tβ– tβ– β –  , β –  = = tα– α– , (–t)tα– α– , tβ– β– , (–t)tβ– β– ,   ≤ t ≤  ,  ≤ t ≤ ,   ≤ t ≤  ,  ≤ t ≤ . () () () () () (d) For any (t, s) ∈ [, ] × [, ], we have g(t, s) ≤ ( – t)tα– (α – ) ≤ tα– (α – ) , g(t, s) ≤ ( – t)tβ– (β – ) ≤ tβ– (β – ) . For the proof of Lemma .(a) and (b) see [], for the proof of Lemma .(c) see [], and the proof of Lemma .(d) is based on the relations g(t, s) = g( – s,  – t), g(t, s) = g( – s,  – t), and relations (b) above. Lemma . ([]) If H, K : [, ] → R are nondecreasing functions, and > , then Gi, i = , . . . ,  given by () are continuous functions on [, ] × [, ] and satisfy Gi(t, s) ≥  for all (t, s) ∈ [, ] × [, ], i = , . . . , . Moreover, if x˜, y˜ ∈ C(, ) ∩ L(, ) satisfy x˜(t) ≥ , y˜(t) ≥  for all t ∈ (, ), then the solution (u, v) of problem ()-() given by () satisfies u(t) ≥ , v(t) ≥  for all t ∈ [, ]. Lemma . Assume that H, K : [, ] → R are nondecreasing functions, > ,  τ α–( – τ ) dK (τ ) > ,  τ β–( – τ ) dH(τ ) > . Then the functions Gi, i = , . . . ,  satisfy the inequalities: (a) G(t, s) ≤ σh(s), ∀(t, s) ∈ [, ] × [, ], where σ =  +  K () – K ()   τ β– dH(τ ) > . (a) G(t, s) ≤ δtα–, ∀(t, s) ∈ [, ] × [, ], where δ = (α– )  +    τ β– dH(τ )   (a) G(t, s) ≥ tα–h(s), (t, s) ∈ [, ] × [, ], where  =    τ β– dH(τ )   k(τ ) dK (τ ) > . (b) G(t, s) ≤ σh(s), ∀(t, s) ∈ [, ] × [, ], where σ =  (H() – H()) > . (b) G(t, s) ≤ δtα–, ∀(t, s) ∈ [, ] × [, ], where δ = (β–) ( – τ )τ β– dH(τ ) > . (b) G(t, s) ≥ tα–h(s), ∀(t, s) ∈ [, ] × [, ], where  =   k(τ ) dH(τ ) > . (c) G(t, s) ≤ σh(s), ∀(t, s) ∈ [, ] × [, ], where σ =  +  H() – H()   τ α– dK (τ ) > . (c) G(t, s) ≤ δtβ–, ∀(t, s) ∈ [, ] × [, ], where δ = (β– )  +    τ α– dK (τ )   ( – τ )τ β– dH(τ ) > . (c) G(t, s) ≥ tβ–h(s), ∀(t, s) ∈ [, ] × [, ], where  =   (d) G(t, s) ≤ σh(s), ∀(t, s) ∈ [, ] × [, ], where σ =  (K () – K ()) > . (d) G(t, s) ≤ δtβ–, ∀(t, s) ∈ [, ] × [, ], where δ = (α–) ( – τ )τ α– dK (τ ) > . (d) G(t, s) ≥ tβ–h(s), ∀(t, s) ∈ [, ] × [, ], where  =   k(τ ) dK (τ ) > . Proof From the assumptions of this lemma, we obtain                 τ α– dK (τ ) ≥ τ α–( – τ ) dK (τ ) > ,       ( – τ )τ α– dK (τ ) ≥ ( – τ )τ α– dK (τ ) > , k(τ ) dK (τ ) ≥ α –   τ α–( – τ ) dK (τ ) > , τ β– dH(τ ) ≥ τ β–( – τ ) dH(τ ) > , ( – τ )τ β– dH(τ ) ≥ ( – τ )τ β– dH(τ ) > , k(τ ) dH(τ ) ≥ β –   τ β–( – τ ) dH(τ ) > , K () – K () = H() – H() =     dK (τ ) ≥ dH(τ ) ≥     τ α–( – τ ) dK (τ ) > , τ β–( – τ ) dH(τ ) > . By using Lemma ., we deduce, for all (t, s) ∈ [, ] × [, ]: (a) tα–     G(t, s) = g(t, s) + τ β– dH(τ ) g(τ , s) dK (τ ) ≤ h(s) +  = h(s)  + τ β– dH(τ ) k(τ ) dK (τ ) = tα–h(s). G(t, s) = G(t, s) ≤ tα– tα–   (b) (b) (b) (c) (c) (c) (d) (d) G(t, s) ≥ G(t, s) = k(τ ) dH(τ ) = tβ–h(s). G(t, s) = g(t, s) + tβ– tβ–   ( –(ατ )–τα)– dK (τ ) = δtβ–. tα–  g(τ , s) dH(τ ) ≤   =  H() – H() h(s) = σh(s). h(s) dH(τ )   ( –(βτ )–τβ)– dH(τ ) = δtα–. G(t, s) ≥ k(τ )h(s) dH(τ ) = tα–  h(s)  k(τ ) dH(τ ) = tα–h(s). G(t, s) ≤ ( –(βt)–tβ–) + ≤ (tββ–– )  +  tβ–   ( –(βτ )–τβ)– dH(τ ) ( – τ )τ β– dH(τ ) = δtβ–. tβ–  = tβ–h(s)  G(t, s) ≥ tβ–   k(τ )h(s) dK (τ ) = tβ–h(s)  k(s) dK (τ ) = tβ–h(s).   G t , s x˜(s) ds + σ  h(s)x˜(s) ds +  h(s)y˜(s) ds ≥ tα– min  ,  σ σ = γtα–u t , ∀t, t ∈ [, ], where γ = min  ,  σ σ > .  G t , s y˜(s) ds   G t , s x˜(s) ds + G t , s y˜(s) ds In a similar way, we deduce v(t) = G(t, s)y˜(s) ds + G(t, s)x˜(s) ds ≥ γtβ–v t , ∀t, t ∈ [, ], where γ = min σ , σ > . In the proof of our main results we shall use the nonlinear alternative of Leray-Schauder type and the Guo-Krasnosel’skii fixed point theorem presented below (see [, ]). Theorem . Let X be a Banach space with ⊂ X closed and convex. Assume U is a relatively open subset of with  ∈ U, and let S : U¯ → be a completely continuous operator (continuous and compact). Then either () S has a fixed point in U¯ , or () there exist u ∈ ∂U and ν ∈ (, ) such that u = νSu. Theorem . Let X be a Banach space and let C ⊂ X be a cone in X. Assume  and  are bounded open subsets of X with  ∈  ⊂ ¯  ⊂  and let A : C ∩ ( ¯  \ ) → C be a completely continuous operator such that either (i) Au ≤ u , u ∈ C ∩ ∂ , and Au ≥ u , u ∈ C ∩ ∂ , or (ii) Au ≥ u , u ∈ C ∩ ∂ , and Au ≤ u , u ∈ C ∩ ∂ . Then A has a fixed point in C ∩ ( ¯  \ ). () () () () 3 Main results In this section, we investigate the existence and multiplicity of positive solutions for our problem (S)-(BC). We present now the assumptions that we shall use in the sequel. (H) H, K : [, ] → R are nondecreasing functions, =  – (  τ α– dK (τ )) × (  τ β– dH(τ )) > , and  τ α–( – τ ) dK (τ ) > ,  τ β–( – τ ) dH(τ ) > . (H) The functions f , g ∈ C([, ] × [, ∞) × [, ∞), (–∞, +∞)) and there exist functions p, p ∈ C([, ], [, ∞)) such that f (t, u, v) ≥ –p(t) and g(t, u, v) ≥ –p(t) for any t ∈ [, ] and u, v ∈ [, ∞). (H) f (t, , ) > , g(t, , ) >  for all t ∈ [, ]. (H) The functions f , g ∈ C((, ) × [, ∞) × [, ∞), (–∞, +∞)), f , g may be singular at t =  and/or t = , and there exist functions p, p ∈ C((, ), [, ∞)), α, α ∈ C((, ), [, ∞)), β, β ∈ C([, ] × [, ∞) × [, ∞), [, ∞)) such that f∞ = u+lvi→m∞ t∈m[c,in–c] f (ut,+u,vv) = ∞ or g∞ = u+lvi→m∞ t∈m[c,in–c] g(ut,+u,vv) = ∞. (H) βi∞ = limu+v→∞ maxt∈[,] βi(t,u,v) = , i = , . u+v We consider the system of nonlinear fractional differential equations Dα+x(t) + λ(f (t, [x(t) – q(t)]∗, [y(t) – q(t)]∗) + p(t)) = , β D+y(t) + μ(g(t, [x(t) – q(t)]∗, [y(t) – q(t)]∗) + p(t)) = ,  < t < ,  < t < ,     with the integral boundary conditions x() = x () = · · · = x(n–)() = , y() = y () = · · · = y(m–)() = , x() =  y(s) dH(s), y() =  x(s) dK (s), where z(t)∗ = z(t) if z(t) ≥ , and z(t)∗ =  if z(t) < . Here (q, q) with q(t) = λ is solution of the system of fractional differential equations Dα+q(t) + λp(t) = , β D+q(t) + μp(t) = ,  < t < ,  < t < , with the integral boundary conditions q() = q() = · · · = q(n–)() = , q() = q() = · · · = q(m–)() = , q() =  q(s) dH(s), q() =  q(s) dK (s). Under the assumptions (H) and (H), or (H) and (H), we have q(t) ≥ , q(t) ≥  for all t ∈ [, ]. We shall prove that there exists a solution (x, y) for the boundary value problem ()-() with x(t) ≥ q(t) and y(t) ≥ q(t) on [, ], x(t) > q(t), y(t) > q(t) on (, ). In this case (u, v) with u(t) = x(t) – q(t) and v(t) = y(t) – q(t), t ∈ [, ] represents a positive solution of boundary value problem (S)-(BC). By using Lemma . (relations ()), a solution of the system ⎨⎪⎪⎪⎧ x(t) = λ+ μ G(Gt,s()t(,fs()s(,g[(xs(,s[)x–(s)q–(sq)](∗s,)[]y∗(,s[)y(–s)q–(qs)](∗s))]+∗)p+(sp))(ds)s) ds,  ⎪⎩⎪⎪ y(t) = μ+ λ G G(t,(st),(sg)((sf,([sx, ([sx)(s–) q–q(s)(]s∗),][∗y,([sy)(s–) q–q(s)(]s∗))]∗+) p+p(s)()s)d)sds, t ∈ [, ], t ∈ [, ], is a solution for problem ()-(). We consider the Banach space X = C([, ]) with the supremum norm · , and the Banach space Y = X × X with the norm (u, v) Y = u + v . We define the cones P = x ∈ X, x(t) ≥ γtα– x , ∀t ∈ [, ] , P = y ∈ X, y(t) ≥ γtβ– y , ∀t ∈ [, ] , where γ, γ are defined in Section  (Lemma .), and P = P × P ⊂ Y . For λ, μ > , we introduce the operators Q, Q : Y → X and Q : Y → Y defined by Q(x, y) = (Q(x, y), Q(x, y)), (x, y) ∈ Y with Q(x, y)(t) = λ If (H) and (H) hold, then we deduce easily that Q(x, y)(t) < ∞ and Q(x, y)(t) < ∞ for all t ∈ [, ]. If (H) and (H) hold, we deduce, for all t ∈ [, ], Q(x, y)(t) ≤ λσ where M = max{maxt∈[,],u,v∈[,L] β(t, u, v), maxt∈[,],u,v∈[,L] β(t, u, v), }. Besides, by Lemma ., we conclude that Q(x, y)(t) ≥ γtα– Q(x, y) , Q(x, y)(t) ≥ γtβ– Q(x, y) , and so Q(x, y), Q(x, y) ∈ P. By using standard arguments, we deduce that operator Q : P → P is a completely continuous operator (a compact operator, that is, one that maps bounded sets into relatively compact sets and is continuous). Theorem . Assume that (H)-(H) hold. Then there exist constants λ >  and μ >  such that, for any λ ∈ (, λ] and μ ∈ (, μ], the boundary value problem (S)-(BC) has at least one positive solution. Proof Let δ ∈ (, ) be fixed. From (H) and (H), there exists R ∈ (, ] such that f (t, u, v) ≥ δf (t, , ), g(t, u, v) ≥ δg(t, , ), ∀t ∈ [, ], u, v ∈ [, R]. () max t∈[,],u,v∈[,R] f (t, u, v) + p(t) ≥ max δf (t, , ) + p(t) > , t∈[,] g(t, u, v) + p(t) ≥ max δg(t, , ) + p(t) > , t∈[,] f¯(R) = g¯(R) = c = σ c = σ λ = max max t∈[,],u,v∈[,R]  h(s) ds, h(s) ds,    c = σ c = σ h(s) ds, h(s) ds,  R , R cf¯(R) cf¯(R) μ = max R , R cg¯(R) cg¯(R) We will show that, for any λ ∈ (, λ] and μ ∈ (, μ], problem ()-() has at least one positive solution. So, let λ ∈ (, λ] and μ ∈ (, μ] be arbitrary, but fixed for the moment. We define the set U = {(x, y) ∈ P, (u, v) Y < R}. We suppose that there exist (x, y) ∈ ∂U ( (x, y) Y = R or x + y = R) and ν ∈ (, ) such that (x, y) = νQ(x, y) or x = νQ(x, y), y = νQ(x, y). We deduce that x(t) – q(t) ∗ = x(t) – q(t) ≤ x(t) ≤ R, if x(t) – q(t) ≥ , x(t) – q(t) ∗ = , for x(t) – q(t) < , ∀t ∈ [, ], y(t) – q(t) ∗ = y(t) – q(t) ≤ y(t) ≤ R, if y(t) – q(t) ≥ , y(t) – q(t) ∗ = , for y(t) – q(t) < , ∀t ∈ [, ]. Then by Lemma ., for all t ∈ [, ], we obtain x(t) = νQ(x, y)(t) ≤ Q(x, y)(t)     ≤ λσ h(s)f¯(R) ds + μσ h(s)g¯(R) ds ≤ λcf¯(R) + μcg¯(R) ≤ R + R = R , y(t) = νQ(x, y)(t) ≤ Q(x, y)(t) ≤ μσ h(s)g¯(R) ds + λσ h(s)f¯(R) ds ≤ μcg¯(R) + λcf¯(R) ≤ R + R = R . Hence x ≤ R and y ≤ R . Then R = (x, y) Y = x + y ≤ R + R = R , which is a contradiction. Therefore, by Theorem . (with = P), we deduce that Q has a fixed point (x, y) ∈ U¯ ∩ P. That is, (x, y) = Q(x, y) or x = Q(x, y), y = Q(x, y), and x + y ≤ R with x(t) ≥ γtα– x and y(t) ≥ γtβ– y for all t ∈ [, ]. Moreover, by (), we conclude x(t) = Q(x, y)(t) ≥ λ ≥ λ ≥ μ         G(t, s) δf (t, , ) + p(s) ds + μ G(t, s) δg(t, , ) + p(s) ds G(t, s) δg(t, , ) + p(s) ds + λ G(t, s) δf (t, , ) + p(s) ds                     ≥ μ Therefore x(t) ≥ q(t), y(t) ≥ q(t) for all t ∈ [, ], and x(t) > q(t), y(t) > q(t) for all t ∈ (, ). Let u(t) = x(t) – q(t) and v(t) = y(t) – q(t) for all t ∈ [, ]. Then u(t) ≥ , v(t) ≥  for all t ∈ [, ], u(t) > , v(t) >  for all t ∈ (, ). Therefore (u, v) is a positive solution of (S)-(BC). Theorem . Assume that (H), (H), and (H) hold. Then there exist λ∗ >  and μ∗ >  such that, for any λ ∈ (, λ∗] and μ ∈ (, μ∗], the boundary value problem (S)-(BC) has at least one positive solution. Proof We choose a positive number R > max , γ     γγ  γγ    , – , with M = max M = max max t∈[,] u,v≥,u+v≤R max t∈[,] u,v≥,u+v≤R β(t, u, v),  , β(t, u, v),  . x(s) – q(s) ∗ ≤ x(s) ≤ x ≤ R, y(s) – q(s) ∗ ≤ y(s) ≤ y ≤ R. Let λ ∈ (, λ∗] and μ ∈ (, μ∗]. Then, for any (x, y) ∈ P ∩ ∂  and s ∈ [, ], we have Then, for any (x, y) ∈ P ∩ ∂ , we obtain Q(x, y) ≤ λσ Therefore On the other hand, we choose a constant L >  such that λL γc(α–) μL γc(α–) –c –c c c h(s) ds ≥ , λL γc(β–) h(s) ds ≥ , μL γc(β–) –c –c c c h(s) ds ≥ , h(s) ds ≥ . From (H), we deduce that there exists a constant M >  such that Q(x, y) Y = Q(x, y) + Q(x, y) ≤ (x, y) Y , ∀(x, y) ∈ P ∩ ∂ . () f (t, u, v) ≥ L(u + v) or g(t, u, v) ≥ L(u + v), ∀t ∈ [c,  – c], u, v ≥ , u + v ≥ M. () Now we define R = max R, γMcα– , γMcβ– , γ   and let  = {(x, y) ∈ P, (x, y) Y < R}. We suppose that f∞ = ∞, that is, f (t, u, v) ≥ L(u + v) for all t ∈ [c,  – c] and u, v ≥ , u + v ≥ M. Then, for any (x, y) ∈ P ∩ ∂ , we have (x, y) Y = R or x + y = R. We deduce that x ≥ R or y ≥ R . We suppose that x R . Then, for any (x, y) ∈ P ∩ ∂ , we obtain ≥  x(t) – q(t) = x(t) – λ Therefore, we conclude x(t) – q(t) ∗ = x(t) – q(t) ≥  x(t) ≥  γtα– x Hence   ≥  γtα–R ≥  γcα–R ≥ M, ∀t ∈ [c,  – c]. x(t) – q(t) ∗ + y(t) – q(t) ∗ ≥ x(t) – q(t) ∗ = x(t) – q(t) ≥ M, ∀t ∈ [c,  – c]. () Then, for any (x, y) ∈ P ∩ ∂  and t ∈ [c,  – c], by () and (), we deduce f t, x(t) – q(t) ∗, y(t) – q(t) ∗ ≥ L x(t) – q(t) ∗ + y(t) – q(t) ∗ ≥ L x(t) – q(t) ∗ L ≥  x(t), ∀t ∈ [c,  – c]. It follows that, for any (x, y) ∈ P ∩ ∂ , t ∈ [c,  – c], we obtain If y ≥ R , then by a similar approach, we obtain again relation (). We suppose now that g∞ = ∞, that is, g(t, u, v) ≥ L(u + v), for all t ∈ [c,  – c] and u, v ≥ , u + v ≥ M. Then, for any (x, y) ∈ P ∩ ∂ , we have (x, y) Y = R. Hence x ≥ R or y ≥ R . If x ≥ R , then for any (x, y) ∈ P ∩ ∂  we deduce in a similar manner as above that x(t) – q(t) ≥  x(t) for all t ∈ [, ] and Q(x, y)(t) ≥ μ G(t, s)L x(s) – q(s) ∗ ds ≥ μ c –c G(t, s)  Lγcα–R ds  tα–h(s)  Lγcα–R ds ≥ μc(α–)    LγR ∀t ∈ [c,  – c]. c –c Hence we obtain relation (). If y ≥ R , then in a similar way as above, we deduce again relation (). Therefore, by Theorem ., relations () and (), we conclude that Q has a fixed point (x, y) ∈ P ∩ ( ¯  \ ), that is, R ≤ (x, y) Y ≤ R. Since (x, y) Y ≥ R, then x ≥ R or y ≥ R . We suppose first that x ≥ R . Then we deduce x(t) – q(t) = x(t) – λ G(t, s)p(s) ds – μ G(t, s)p(s) ds   ≥ x(t) – tα– δ ≥ x(t) – γx(xt)  ≥  – γR   ≥  – γR    R  – ≥   γR   = tα–,           and so x(t) ≥ q(t) + tα– for all t ∈ [, ], where  = γR – (δp(s) + δp(s)) ds > . Then y() =  x(s) dK (s) ≥   sα– dK (s) >  and     γR ≥  Therefore, we obtain y(t) – q(t) = y(t) – μ G(t, s)p(s) ds – λ G(t, s)p(s) ds ≥ y(t) – tβ– ≥ y(t) – ≥ y(t)  –     y(t) γ y   –   ≥ = ≥ γtβ– y  – γγR tβ–     γγR –   –   where  = γγR  sα– dK (s) – (δp(s) + δp(s)) ds > .   Hence y(t) ≥ q(t) + tβ– for all t ∈ [, ]. If y ≥ R , then by a similar approach, we deduce that y(t) ≥ q(t) + tβ– and x(t) ≥ q(t) + tα– for all t ∈ [, ], where  = γR – (δp(s) + δp(s)) ds >  and  = γγR  sβ– dH(s) – (δp(s) + δp(s)) ds > . Let u(t) = x(t) – q(t) and v(t) = y(t) – q(t) for all t ∈ [, ]. Then (u, v) is a positive solution of (S)-(BC) with u(t) ≥ tα– and v(t) ≥ tβ– for all t ∈ [, ], where  = min{ , } and  = min{ , }. This completes the proof of Theorem .. Theorem . Assume that (H), (H), (H), and (H ) The functions f , g ∈ C([, ] × [, ∞) × [, ∞), (–∞, +∞)) and there exist functions p, p, α, α ∈ C([, ], [, ∞)), β, β ∈ C([, ] × [, ∞) × [, ∞), [, ∞)) such that for all t ∈ [, ], u, v ∈ [, ∞), with  pi(s) ds > , i = , , hold. Then the boundary value problem (S)-(BC) has at least two positive solutions for λ >  and μ >  sufficiently small. Proof Because assumption (H ) implies assumptions (H) and (H), we can apply Theorems . and .. Therefore, we deduce that, for  < λ ≤ min{λ, λ∗} and  < μ ≤ min{μ, μ∗}, problem (S)-(BC) has at least two positive solutions (u, v) and (u, v) with (u + q, v + q) Y ≤  and (u + q, v + q) Y > . Theorem . Assume that λ = μ, and (H), (H), and (H) hold. In addition if (H) there exists c ∈ (, /) such that δp(s) + δp(s) ds, γ   –  , then there exists λ∗ >  such that for any λ ≥ λ∗ problem (S)-(BC) (with λ = μ) has at least one positive solution. Proof By (H) we conclude that there exists M >  such that f (t, u, v) ≥ L or g(t, u, v) ≥ L, ∀t ∈ [c,  – c], u, v ≥ , u + v ≥ M. We define λ∗ = max We assume now λ ≥ λ∗. Let R = max λ γ   M cα–   .  λ δp(s) + δp(s) ds, γ   –  x(t) – q(t) ≥ γtα– x – λtα–δ p(s) ds – λtα–δ         δp(s) + δp(s) ds – λ δp(s) + δp(s) ds = tα–λ ≥ tα–λ∗     δp(s) + δp(s) ds ≥ cMα– tα– ≥ . Therefore, for any (x, y) ∈ P ∩ ∂  and t ∈ [c,  – c], we have Hence, for any (x, y) ∈ P ∩ ∂  and t ∈ [c,  – c], we conclude     c –c c –c Therefore we obtain Q(x, y) ≥ R for all (x, y) ∈ P ∩ ∂ , and so Q(x, y) Y ≥ R = (x, y) Y , ∀(x, y) ∈ P ∩ ∂ . () If y ≥ R/, then by a similar approach we deduce again relation (). We suppose now that g∞i > L, that is, g(t, u, v) ≥ L for all t ∈ [c,  – c] and u, v ≥ , u + v ≥ M. Let (x, y) ∈ P ∩ ∂ . Then (x, y) Y = R, so x ≥ R/ or y ≥ R/. If x ≥ R/, then we obtain in a similar manner as in the first case above (f ∞i > L) that x(t) – q(t) ≥ cMα– tα– ≥  for all t ∈ [, ]. Therefore, for any (x, y) ∈ P ∩ ∂  and t ∈ [c,  – c], we deduce inequalities (). Hence, for any (x, y) ∈ P ∩ ∂  and t ∈ [c,  – c], we conclude Therefore we obtain Q(x, y) ≥ R, and so P ∩ ∂ , that is, we have relation (). By a similar approach we obtain relation () if y ≥ R/. On the other hand, we consider the positive number ε = min  λσ  λσ     –   – h(s)α(s) ds –  , λσ   h(s)α(s) ds – Q(x, y) Y ≥ R = (x, y) Y for all (x, y) ∈ Then by (H) we deduce that there exists M >  such that βi(t, u, v) ≤ ε(u + v), ∀t ∈ [, ], u, v ≥ , u + v ≥ M, i = , . Therefore we obtain βi(t, u, v) ≤ M + ε(u + v), ∀t ∈ [, ], u, v ≥ , i = , , where M = maxi=,{maxt∈[,],u,v≥,u+v≤M βi(t, u, v)}. We define now R = max R, λσ max{M, } h(s) α(s) + p(s) ds,       λσ max{M, } h(s) α(s) + p(s) ds, λσ max{M, } h(s) α(s) + p(s) ds, λσ max{M, } h(s) α(s) + p(s) ds , and let  = {(x, y) ∈ P, (x, y) Y < R}. For any (x, y) ∈ P ∩ ∂ , we have Q(x, y)(t) ≤ λ σh(s) α(s)β s, x(s) – q(s) ∗, y(s) – q(s) ∗ + p(s) ds            + λ σh(s) α(s)β s, x(s) – q(s) ∗, y(s) – q(s) ∗ + p(s) ds ≤ λσ h(s) α(s) M + ε x(s) – q(s) ∗ + y(s) – q(s) ∗ + p(s) ds + λσ h(s) α(s) M + ε x(s) – q(s) ∗ + y(s) – q(s) ∗ + p(s) ds ≤ λσ max{M, } h(s) α(s) + p(s) ds + λσεR    ≤ R + R + R + R = R = (x,y) Y , and so Q(x, y) ≤ (x,y) Y for all (x, y) ∈ P ∩ ∂ .  In a similar way we obtain Q(x, y)(t) ≤ (x,y) Y for all t ∈ [, ], and so Q(x, y) ≤ (x,y) Y   for all (x, y) ∈ P ∩ ∂ . Therefore, we deduce Q(x, y) Y ≤ (x, y) Y , ∀(x, y) ∈ P ∩ ∂ . () By Theorem ., (), and (), we conclude that Q has a fixed point (x, y) ∈ P ∩ ( ¯  \ ). Since (x, y) ≥ R then x ≥ R/ or y ≥ R/. ≥ λtβ– ≥ λ∗tβ–             sα– dK (s) – λtβ– Therefore, we deduce that, for all t ∈ [, ], y(t) – q(t) ≥ y(t) – λδ tβ–p(s) ds – λδ tβ–p(s) ds ≥ γtβ– y – λtβ– If y ≥ R/, then by a similar approach we conclude again that x(t) – q(t) ≥ cMα– tα– and y(t) – q(t) ≥ cMβ– tβ– for all t ∈ [, ]. Let u(t) = x(t) – q(t) and v(t) = y(t) – q(t) for all t ∈ [, ]. Then u(t) ≥ tα– and v(t) ≥ tβ– for all t ∈ [, ], where  = cMα– ,  = cMβ– . Hence we deduce that (u, v) is a positive solution of (S)-(BC), which completes the proof of Theorem .. In a similar manner as we proved Theorem ., we obtain the following theorems. Theorem . Assume that λ = μ, and (H), (H), and (H) hold. In addition if (H ) there exists c ∈ (, /) such that f ∞i = lim inf min f (t, u, v) > L or g∞i = lim inf min g(t, u, v) > L, u+u,vv→≥∞ t∈[c,–c] u+u,vv→≥∞ t∈[c,–c] where     Theorem . Assume that λ = μ, and (H), (H), and (H) hold. In addition if (H) there exists c ∈ (, /) such that fˆ∞ = u+ul,viv→m≥∞ t∈m[c,in–c] f (t, u, v) = ∞ or gˆ∞ = u+ul,viv→m≥∞ t∈m[c,in–c] g(t, u, v) = ∞, then there exists λ˜∗ >  such that for any λ ≥ λ˜∗ problem (S)-(BC) (with λ = μ) has at least one positive solution. 4 Examples Let α = / (n = ), β = / (m = ), H(t) = t, K (t) = t. Then  u(s) dK (s) =   su(s) ds and  v(s) dH(s) =   sv(s) ds. We consider the system of fractional differential equations (S) (BC) with the boundary conditions D/+u(t) + λf (t, u(t), v(t)) = , D/+v(t) + μg(t, u(t), v(t)) = , t ∈ (, ), t ∈ (, ), u() = u () = , v() = v () = , u() =   sv(s) ds, v() =   su(s) ds. Then we obtain =  – (  sα– dK (s))(  sβ– dH(s)) =  > ,  τ α–( – τ ) dK (τ ) =  > ,  τ β–( – τ ) dH(τ ) =  > . The functions H and K are nondecreasing, and so assumption (H) is satisfied. Besides, we deduce  g(t, s) = √π t/( – s)/ – (t – s)/, t/( – s)/,  ≤ s ≤ t ≤ ,  ≤ t ≤ s ≤ , g(t, s) =  (/) t/( – s)/ – (t – s)/, t/( – s)/, G(t, s) = g(t, s) + t/ τ g(τ , s) ds, G(t, s) = g(t, s) +  t/      ≤ s ≤ t ≤ ,  ≤ t ≤ s ≤ , G(t, s) =  t/ We also obtain h(s) = √π s( – s)/, h(s) = (/) s( – s)/,  k(t) =  t/,  ≤ t ≤ /,  ( – t)t/, / ≤ t ≤ , k(t) =  t/,  ≤ t ≤ /,  ( – t)t/, / ≤ t ≤ . In addition, we have σ = , δ = √π ,  = √√– , σ =  , δ = √(/) ,  = √– , σ =  , δ =  (/) ,  = √√– , σ =  , δ = √π ,  = √ √– , γ = √– ≈ .√, γ = (√ √–) ≈ .. τ g(τ , s)dτ , G(t, s) = t/ τ g(τ , s) dτ .     τ g(τ , s) dτ , Example  We consider the functions g(t, u, v) = + ln( – t), t ∈ (, ), u, v ≥ . We have p(t) = – ln t, p(t) = – ln( – t), α(t) = α(t) = √t(–t) for all t ∈ (, ), β(t, u, v) = (u + v), β(t, u, v) =  + sin(u + v) for all t ∈ [, ], u, v ≥ ,  p(t) dt = ,  p(t) dt = ,    αi(t) dt = π , i = , . Therefore, assumption (H) is satisfied. In addition, for c ∈ (, /) fixed, assumption (H) is also satisfied (f∞ = ∞). After some computations, we deduce (δp(s) + δp(s)) ds ≈ ., (δp(s) + δp(s)) ds ≈ .,  h(s)(α(s) + p(s)) ds ≈ .,  h(s)(α(s) + p(s)) ds ≈ .. We choose R = , which satisfies the condition from the beginning of the proof of Theorem .. Then M = R, M = , λ∗ ≈ . · –, and μ∗ = . By Theorem ., we conclude that (S)-(BC) has at least one positive solution for any λ ∈ (, λ∗] and μ ∈ (, μ∗]. Example  We consider the functions f (t, u, v) = (u + v) + cos u, g(t, u, v) = (u + v)/ + cos v, t ∈ [, ], u, v ≥ . We have p(t) = p(t) =  for all t ∈ [, ], and then assumption (H) is satisfied. Besides, assumption (H) is also satisfied, because f (t, , ) =  and g(t, , ) =  for all t ∈ [, ]. Let δ =  <  and R = . Then  f (t, u, v) ≥ δf (t, , ) =  ,  g(t, u, v) ≥ δg(t, , ) =  , ∀t ∈ [, ], u, v ∈ [, ]. In addition, f¯(R) = f¯() = g¯(R) = g¯() = We also obtain c ≈ ., c ≈ ., c ≈ ., c ≈ ., R R R R and then λ = max{ cf¯(R) , cf¯(R) } ≈ . and μ = max{ cg¯(R) , cg¯(R) } ≈ .. By Theorem ., for any λ ∈ (, λ] and μ ∈ (, μ], we deduce that problem (S)-(BC) has at least one positive solution. Because assumption (H ) is satisfied (α(t) = α(t) = , β(t, u, v) = (u+v) +, β(t, u, v) = (u + v)/ +  for all t ∈ [, ], u, v ≥ ) and assumption (H) is also satisfied (f∞ = ∞), by Theorem . we conclude that problem (S)-(BC) has at least two positive solutions for λ and μ sufficiently small. Example  We consider λ = μ and the functions (u + v)a  f (t, u, v) =  t( – t) – √t , g(t, u, v) = ln( + u + v)  t( – t) – √   – t , t ∈ (, ), u, v ≥ , where a ∈ (, ). Here we have p(t) = √t , p(t) = √–t , α(t) = √t(–t) , α(t) = √t(–t) for all t ∈ (, ), the assumptions (H), (H), and (H) are satisfied (βi∞ =  for i = ,  and fˆ∞ = ∞). Then by Theorem ., we deduce that there exists λ˜∗ >  such that for any λ ≥ λ˜∗ our problem (S)-(BC) (with λ = μ) has at least one positive solution. 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Johnny Henderson, Rodica Luca. Positive solutions for a system of semipositone coupled fractional boundary value problems, Boundary Value Problems, 2016, 61, DOI: 10.1186/s13661-016-0569-8