Positive Solutions for a Coupled System of Nonlinear Semipositone Fractional Boundary Value Problems
In this paper, we consider a four-point coupled boundary value problem for system of the nonlinear semipositone fractional differential equation , , , where the coefficients are real positive constants, , are the standard Riemann-Liouville derivatives. Values of the parameters and are determined for which boundary value problem has positive solution by utilizing a fixed point theorem on cone.
In recent years, fractional-order calculus has been one of the most rapidly developing areas of mathematical analysis. In fact, a natural phenomenon may depend not only on the time instant but also on the previous time history, which can be successfully modeled by fractional calculus. Fractional-order differential equations are naturally related to systems with memory, as fractional derivatives are usually nonlocal operators. Thus, fractional differential equations (FDEs) play an important role because of their applications in various fields of science, such as mathematics, physics, chemistry, optimal control theory, finance, biology, and engineering [1–6]. In particular, a great interest has been shown by many authors in the subject of fractional-order boundary value problems (BVPs), and a variety of results for BVPs equipped with different kinds of boundary conditions have been obtained; for instance, see [7–18] and the references cited therein.
We consider the four-point coupled system of nonlinear fractional differential equations:with the coupled boundary conditionswhere , and are the standard Riemann-Liouville derivatives, and are real positive constants.
Here we emphasize that our problem is new in the sense of nonseparated coupled boundary conditions introduced here. To the best of our knowledge, fractional-order coupled system (1) has yet to be studied with the boundary conditions (2). In consequence, our findings of the present work will be a useful contribution to the existing literature on the topic. The existence of positive solution results for the given problem is new, though they are proved by applying the well-known fixed point theorem.
We present intervals for parameters , and such that the above problem (1)-(2) has at least one positive solution. By a positive solution (1)-(2), we mean a pair of functions satisfying (1) and (2) with for all and
We use the following notations for our convenience:
Before stating our results, we make precise assumptions throughout the paper:(H1)The functions and there exist functions such that and for any and (H2) are positive constants such that (H3) for all .(H4)The functions , may be singular at andor , and there exist functions , such that for all , with (H5)There exists such that
The rest of the paper is organized as follows. In Section 2, we construct the Green functions for the associated linear fractional-order boundary value problems and estimate the bounds for these Green functions. In Section 3, we establish the existence of at least one positive solution of the boundary value problem (1)-(2) by applying fixed point theorem. Finally, as an application, we give an example to illustrate our result.
2. Green Functions and Bounds
In this section, we construct the Green functions for the associated linear fractional-order boundary value problems and estimate the bounds for these Green functions, which are needed to establish the main results.
Lemma 1. Let Then, the differential equation has a solution for some , where is the smallest integer greater than or equal to .
Lemma 2. Let Then, for some , where is the smallest integer greater than or equal to .
Lemma 3. Let and . Let be given functions. Then, the boundary value problem,has an integral representationwhere
Theorem 8. Let be a Banach space with closed and convex. Assume is a relatively open subset of with , and let be a completely continuous operator (continuous and compact). Then, either (i) has a fixed point in , or(ii)there exist and such that .
Theorem 9 (). Let be a Banach space, and let be a cone in . Assume that and are two bounded open subsets of with , and let be a completely continuous operator such that either (i), and , or(ii), and . Then, has a fixed point in .
3. Main Results
We consider the system of nonlinear fractional differential equationswith the boundary conditionswhere a modified function for any by Here withis solution of the system of fractional differential equationswith the boundary conditionsUnder the assumptions and or and , we have for all
We consider the Banach space with supremum norm and the Banach space with the norm We define the cone where .
Lemma 10. If and or and hold, then operator is a completely continuous operator.
Proof. The operators and are well defined. To prove this, let be fixed with Then we have If and hold, then we deduce easily that and for all If and hold, we deduce, for all :where ,
Thus, is well defined.
Next, we show that For any fixed , by Lemmas 6 and 7, we have Similarly, Therefore, Hence, This implies that According to the Ascoli-Arzela theorem, we can easily get that is completely continuous.
Proof. Let be fixed. From and , there exist such thatWe define We will show that, for any and , problem (12)-(13) has at least one positive solution.
So, let and be arbitrary but fixed for the moment. We define the set We suppose that there exist or and such that or .
We deduce that Then by Lemma 3, for all , we obtainHence, and Then,