Positive solutions to a system of semipositone fractional boundary value problems

Advances in Difference Equations, Jul 2014

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

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Positive solutions to a system of semipositone fractional boundary value problems

Rodica Luca 0 Alexandru Tudorache 1 0 Department of Mathematics, Gh. Asachi Technical University , Iasi, 700506 , Romania 1 Faculty of Computer Engineering and Automatic Control, Gh. Asachi Technical University , Iasi, 700050 , Romania We study the existence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with sign-changing nonlinearities, subject to integral boundary conditions. MSC: 34A08; 45G15 - with the integral boundary conditions D+u(t) + f (t, u(t), v(t)) = , t (, ), n < n, D+v(t) + g(t, u(t), v(t)) = , t (, ), m < m, u() = u () = = u(n)() = , v() = v () = = v(m)() = , where n, m N, n, m , D+, and D+ denote the Riemann-Liouville derivatives of orders and , 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 boundary conditions include multi-point and integral boundary conditions and the sum of these in a single framework. 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. By using a nonlinear alternative of Leray-Schauder type, we present intervals for parameters and 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, the above problem (S)-(BC) has been investigated in [] by using the Guo-Krasnoselskii fixed point theorem. The system (S) with = = , and with f (t, u, v) and g(t, u, v) replaced by f (t, v) and g(t, u), respectively, with the boundary conditions (BC), was studied in []. In [], the authors obtained the existence and multiplicity of positive solutions (u(t) , v(t) for all t [, ], supt[,] u(t) > , supt[,] v(t) > ) by applying some theorems from the fixed point index theory. We would also like to mention the paper [], where the authors investigated the existence and multiplicity of positive solutions of the semipositone system (S) with = and the boundary conditions u(i)() = v(i)() = , i = , . . . , n , u() = av( ), v() = bv(), , (, ), and < ab < . The paper is organized as follows. Section contains some preliminaries and lemmas. The main theorem is presented in Section and, finally, in Section , two examples are given to support the new result. 2 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 theorem. Definition . The (left-sided) fractional integral of order > of a function f : (, ) R is given by provided the right-hand side is pointwise defined on (, ), where () is the Euler gamma function defined by () = tet dt, > . Definition . The Riemann-Liouville fractional derivative of order for a function f : (, ) R is given by The notation stands for the largest integer not greater than . We also denote the Riemann-Liouville fractional derivative of f by D+f (t). If = m N then Dm+f (t) = f (m)(t) for t > , and if = then D+f (t) = f (t) for t > . Lemma . ([]) Let > and n = + for / N and n = for N; that is, n is the smallest integer greater than or equal to . Then the solutions of the fractional differential equation D+u(t) = , < t < , are u(t) = ct + ct + + cntn, < t < , where c, c, . . . , cn are arbitrary real constants. Lemma . ([, ]) Let > , n be the smallest integer greater than or equal to (n < n) and y L(, ). The solutions of the fractional equation D+u(t) + y(t) = , < t < , u(t) = where c, c, . . . , cn are arbitrary real constants. We consider now the fractional differential equation with the integral boundary conditions u() = u () = = u(n)() = , where n N, n , and H : [, ] R is a function of the bounded variation. By using Lemma ., after some computations, we obtain the following lemma. Lemma . ([]) If H : [, ] R is a function of bounded variation, = s dH(s) = and z C([, ]), then the solution of problem ()-() is u(t) = G(t, s) z(s) ds, where G(t, s) = g(t, s) + (t, s) [, ] [, ], g(t, s) = Lemma . The function g given by () has the properties (a) g : [, ] [, ] R+ is a continuous function, g(t, s) for all (t, s) [, ] [, ] and g(t, s) > for all (t, s) (, ) (, ). (b) g(t, s) h(s) for all (t, s) [, ] [, ], where h(s) = s(s) () . (c) g(t, s) k(t)h(s) for all (t, s) [, ] [, ], where k(t) = min Proof The first part (a) is evident. For the second part (b), see []. For part (c), for s t, we obtain g(t, s) = g(t, s) = t s( s) t () t( s) s ( ) Lemma . ([]) If H : [, ] R is a nondecreasing function and > , then the Greens function G of problem ()-() given by () is continuous on [, ] [, ] and satisfies G(t, s) for all (t, s) [, ] [, ], G(t, s) > for all (t, s) (, ) (, ). Moreover, if z C([, ]) satisfies z(t) for all t [, ], then the unique solution u of problem ()() satisfies u(t) for all t [, ]. dH( ) = + H() H() . Proof (a) We have G(t, s) = g(t, s) + h(s) + (b) For the second inequality, we obtain G(t, s) k(t)h(s) + Lemma . Assume that H : [, ] R is a nondecreasing function, > and z C([, ]), z(t) for all t [, ]. Then the solution u(t), t [, ] of problem ()-() satisfies the inequality u(t) (t) maxt [,] u(t ) for all t [, ]. Proof For t [, ], we obtain u(t) = G(t, s)z(s) ds J(s)z(s) ds t, t [, ]. We can also formulate similar results as Lemmas .-. above for the fractional differential equation with the integral boundary conditions v() = v () = = v(m)() = , where m N, m , K : [, ] R is a nondecreasing function and z C([, ]). We denote by , g, G, h, k, , J, and the corresponding constants and functions for problem ()-() defined in a similar manner as , g, G, h, k, , J, and , respectively. In the proof of our main result we shall use the following nonlinear alternative of LeraySchauder type (see []). 3 Main result D+x(t) + (f (t, [x(t) q(t)], [y(t) q(t)]) + p(t)) = , D+y(t) + (g(t, [x(t) q(t)], [y(t) q(t)]) + p(t)) = , with the integral boundary conditions z(t) = x() = x () = = x(n)() = , y() = y () = = y(m)() = , and (q, q) with q(t) = G(t, s)p(s) ds, q(t) = G(t, s)p(s) ds is the solution of the system of fractional differential equations with the integral boundary conditions q() = q() = = q(n)() = , q() = q() = = q(m)() = , By (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) for all t [, ]. In this case, the functions u(t) = x(t) q(t) and v(t) = y(t) q(t), t [, ], represent a nonnegative solution, positive on (, ) of the boundary value problem (S)-(BC). Indeed, by ()-() and ()-(), we have t (, ), D+v(t) = D+y(t) D+q(t) = g t, x(t) q(t) , y(t) q(t) t (, ), u() = x() q() = , v() = y() q() = , u(n)() = x(n)() q(n)() = , v(m)() = y(m)() q(m)() = , u() = x() q() = x(s) dH(s) q(s) dH(s) = v() = y() q() = y(s) dK (s) q(s) dK (s) = Therefore, in what follows, we shall investigate the boundary value problem ()-(). By using Lemma ., the system ()-() is equivalent to the system x(t) = G(t, s)(f (s, [x(s) q(s)], [y(s) q(s)]) + p(s)) ds, y(t) = G(t, s)(g(s, [x(s) q(s)], [y(s) q(s)]) + p(s)) ds, We consider the Banach space X = C([, ]) with supremum norm and the Banach space Y = X X with the norm (x , y) Y = x + y . We also define the cones P = x X, x(t) (t)x , t [, ] X, P = y X, y(t) (t)y , t [, ] X, and P = P P Y . t , t . Lemma . If (H) and (H) hold, then the operator Q : P P is a completely continuous operator. (t)J(s) g s, x(s) q(s) , y(s) q(s) + p(s) ds (t)J(s) f s, x(s) q(s) , y(s) q(s) + p(s) ds for all t [, ], and for all t, t [, ]. Therefore, we obtain t [, ], It is clear that (x, y) P is a solution of problem ()-() if and only if (x, y) is a fixed point of Q. Theorem . Assume that (H)-(H) hold. Then there exist constants > and > such that for any (, ] and (, ], the boundary value problem (S)-(BC) has at least one positive solution. t [, ], u, v [, R]. f(R) = g(R) = c = f (t, u, v) + p(t) max f (t, , ) + p(t) > , t g(t, u, v) + p(t) max g(t, , ) + p(t) > , t J(s) ds > , c = J(s) ds > , then, for all t [, ], we obtain Hence x R/ and y R /. Then R = (x , y) = x is a contradiction. + y R + R = R , which G(t, s)p(s) ds = q(t), t (, ), G(t, s)p(s) ds = q(t), t (, ). Therefore, x(t) > q(t) > and y(t) > q(t) > for all t (, ). Let u(t) = x(t) q(t) and v(t) = y(t) q(t) for all t [, ], with u(t) > , v(t) > on (, ). Then (u, v) is a positive solution of the boundary value problem (S)-(BC). 4 Examples H(t) = ,, tt [[/,,//),), /, t [/, ], and K (t) = t for all t [, ]. Then sv(s) ds. We consider the system of fractional differential equations u(s) dH(s) = u( ) + u( ) and v(s) dK (s) = D/+u(t) + f (t, u(t), v(t)) = , D+/v(t) + g(t, u(t), v(t)) = , with the boundary conditions u() = u () = , u() = u( ) + u( ), v() = v () = v () = , v() = sv(s) ds. Then we obtain = s/ dH(s) = ( )/ ( )/ = . > , s/ dK (s) = s/ ds = . > . We also deduce = g(t, s) = Example We consider the functions t [, ], u, v , where b > a > , d > c > , , > . There exists M > such that f (t, u, v) + M , g(t, u, v) + M (p(t) = p(t) = M, t [, ]) for all t [, ], u, v . Indeed, M = max{ (ba) + , (dc) + } satisfies the above inequalities. Let = min{ a((abb+a)) , c(dcdc) } < and R = min{ a , c , , }. Then for all t [, ], u, v [, R]. Besides, f(R) = g(R) = f (t, u, v) + p(t) = ab + + M, Example We consider the functions t [, ], u, v , where a, b, , > . There exists M > (M = ) such that f (t, u, v) + M , g(t, u, v) + M (p(t) = p(t) = M, t [, ]) for all t [, ], u, v . Let = < and R = min{ , }. Then f (t, u, v) f (t, , ) = , g(t, u, v) g(t, , ) = , t [, ], u, v [, R]. f(R) = g(R) = f (t, u, v) + p(t) = Ra + , g(t, u, v) + p(t) = Rb + . + , = (+)c ., and = (+)c has a positive solution (u, v), with (u , v) Competing interests The authors declare that they have no competing interests. Authors contributions Both authors contributed equally to this paper. Both authors read and approved the final manuscript. Acknowledgements This work was supported by the CNCS grant PN-II-ID-PCE-2011-3-0557, Romania.


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Rodica Luca, Alexandru Tudorache. Positive solutions to a system of semipositone fractional boundary value problems, Advances in Difference Equations, 2014, 179,