Existence and uniqueness of positive solutions for a class of nonlinear fractional differential equations

Abstract

In the article, the existence and uniqueness of positive solutions for a class of fractional differential equation with nonlinear boundary conditions are discussed. By applying some fixed point theorems on cone, we gain a unique positive solution and construct two iterative sequences to approximate the solution. Moreover, as applications of our main results, some examples are also presented.

Fixed points of mixed non-monotone tripled operators in ordered Banach spaces and applications

“…The two-side restriction conditions (2.1) and (2.19)-(2.21) with respect to the ordering relations of nonlinear operators in Theorems 2.1 and 2.4 are a kind of more general conditions than those in [6,10,11,13,16], from which we can not only deduce the existence but also the uniqueness of fixed points. …”

“…Because the conditions in Theorems 2.1 and 2.4 of this paper are more general, their applications to nonlinear differential and integral equations must be extensive under the two side ordering relations conditions. Moreover, this also means that our results can not be obtained by the proof methods and results in [6,10,11,13,16].…”

“…The interesting point is that we do not require the upper-lower solutions. Therefore, we extend essentially the corresponding results in [6,10,11,13,16]. Finally, the results in this paper are applied to the system of nonlinear Volterra integral equations in Banach spaces.…”

“…The existence and uniqueness of solutions for (system of) mixed non-monotone binary operator equations were stated in [9,11,16]. However, fewer papers dealt with non-monotone tripled operators in ordered Banach spaces via the generating and normal cone theory; see for instance [6].…”

Nonlocal q-fractional boundary value problem with Stieltjes integral conditions

“…More recently, in connection with broad research on the mathematical modeling of systems, the description of hereditary properties of various materials and the optimal control theory, it has become necessary to investigate boundary value problems of fractional differential equations as the nonlocal characteristics of the corresponding fractional-order operators [5,6,[8][9][10]18,20,[24][25][26][27]29,31,32]. Moreover, these equations are always completely controllable, meanwhile the research on fractional q-differential equation boundary value problems is in a stage of rapid development; one can see [2,3,7,14,18,23,29,30,33] and the references cited therein.…”