Multiple nonnegative solutions of systems with coupled nonlinear boundary conditions

Abstract

Using the theory of fixed point index, we discuss the existence and multiplicity of nonnegative solutions of a wide class of boundary value problems with coupled nonlinear boundary conditions. Our approach is fairly general and covers a variety of situations. We illustrate in an example that all the constants that occur in our theory can be computed. Copyright © 2013 John Wiley & Sons, Ltd.

On a first-order semipositone boundary value problem on a time scale

“…such that T y 0 = y 0 , with y 0 a positive solution of the modified problem (25). Now define x : [a, b] T → R by x(t) := y 0 (t) − w(t).…”

“…Note that in both (25) and the sequel the function w is the unique solution to the auxiliary problem (4).…”

“…BVPs possessing either nonlinear and/or nonlocal BCs have seen a great deal of study recently, spanning such areas as first-and second-order problems, coupled systems of second-order problems, and higher-order problems -see, for example, recent works by Franco et al [10], Goodrich [11][12][13][14][15][16][17], Graef et al [18], Infante et al [22][23][24][25], Kang et al [26], Webb and Infante [31], Yang [32,33], and the references therein. In addition, the study of BVPs with specifically integral boundary conditions has also seen much attention in recent years, and one may consult [5,19,27] and the references therein for some recent examples.…”

Unique solutions for a new coupled system of fractional differential equations

“…First, we list the following hypotheses A lot of boundary value problems of coupled systems involving fractional differential equations have been investigated extensively, see the works and the references therein. Different boundary conditions of coupled systems can be found in the discussions of some problems such as Sturm-Liouville problems and some reaction-diffusion equations (see [26,27]), and they have some applications in many fields such as mathematical biology (see [28,29]), natural sciences and engineering; for example, we can see beam deformation and steady-state heat flow [30,31] and heat equations [14,32,33]. So nonlinear coupled systems subject to different boundary conditions have been paid much attention to, and the existence or multiplicity of solutions for the systems has been given in literature, see [4][5][6][7][8][9][10][11][12][13][14][16][17][18][19][20][21][22][23][24][25] for example.…”

Coupled lower and upper solution approach for the existence of Solutions of nonlinear coupled system with nonlinear coupled boundary conditions

“…The study of ordinary differential systems (ODSs) with coupled and non-coupled boundary conditions has also attracted many authors. The reader can study [4,5,6,7,24,25] and references therein. A second order ordinary differential system (ODS) firstly appeared from the study of chemical reactors [8].…”