Semipositone Boundary Value Problems Research Articles

Existence of positive solutions for third-order semipositone boundary value problems on time scales

In this paper, we consider the existence of positive solutions for a semipositone third-order nonlinear ordinary differential equation on time scales. In suitable growth conditions, by considering the properties on time scales and establishing a special cone, some new results on the existence of positive solutions are established when the nonlinearity is semipositone.

Existence of positive solutions for a nonlinear semipositone boundary value problems on a time scale

In this paper, we are concerned with the existence of positive solution of the following semipositone boundary value problem on time scales: \begin{equation*} (\psi(t)y^\Delta (t))^\nabla + \lambda_1 g(t, \,y(t)) + \lambda_2 h(t,\,y(t)) = 0, \,t \in [\rho(c), \,\sigma(d)]_\mathbb{T}, \end{equation*} with mixed boundary conditions \begin{equation*} \begin{split} \alpha y(\rho(c))-\beta \psi(\rho(c)) y^\Delta(\rho(c))=0,\\ \gamma y(\sigma(d))+\delta \psi(d) y^\Delta(d)=0, \end{split} \end{equation*} where $\psi:C[\rho(c),\, \sigma(d)]_\mathbb{T}$, $\psi(t)>0$ for all $t \in [\rho(c),\,\sigma(d)]_\mathbb{T}$; both $g$ and $h : [\rho(c),\,\sigma(d)]_\mathbb{T} \times [0,\,\infty) \to \mathbb{R}$ are continuous and semipositone. We have established the existence of at least one positive solution or multiple positive solutions of the above boundary value problem by using fixed point theorem on a cone in a Banach space, when $g$ and $h$ are both superlinear or sublinear or one is superlinear and the other is sublinear for $\lambda_i>0;\,i=1,\,2$ are sufficiently small.

Convergence, stability analysis, and solvers for approximating sublinear positone and semipositone boundary value problems using finite difference methods

Positone and semipositone boundary value problems are semilinear elliptic partial differential equations (PDEs) that arise in reaction–diffusion models in mathematical biology and the theory of nonlinear heat generation. Under certain conditions, the problems may have multiple positive solutions or even nonexistence of a positive solution. We develop analytic techniques for proving admissibility, stability, and convergence results for simple finite difference approximations of positive solutions to sublinear problems. We also develop guaranteed solvers that can detect nonuniqueness for positone problems and nonexistence for semipositone problems. The admissibility and stability results are based on adapting the method of sub- and supersolutions typically used to analyze the underlying PDEs. The new convergence analysis technique directly shows that all pointwise limits of finite difference approximations are solutions to the boundary value problem eliminating the possibility of false algebraic solutions plaguing the convergence of the methods. Most known approximation methods for positone and semipositone boundary value problems rely upon shooting techniques; hence, they are restricted to one-dimensional problems and/or radial solutions. The results in this paper will serve as a foundation for approximating positone and semipositone boundary value problems in higher dimensions and on more general domains using simple approximation methods. Numerical tests for known applied problems with multiple positive solutions are provided. The tests focus on approximating certain positive solutions as well as generating discrete bifurcation curves that support the known existence and uniqueness results for the PDE problem.

Multiple solutions for singular semipositone boundary value problems of fourth-order differential systems with parameters

The aim of this paper is to establish some results about the existence of multiple solutions for the following singular semipositone boundary value problem of fourth-order differential systems with parameters: {u(4)(t)+β1u″(t)−α1u(t)=f1(t,u(t),v(t)),0<t<1;v(4)(t)+β2v″(t)−α2v(t)=f2(t,u(t),v(t)),0<t<1;u(0)=u(1)=u″(0)=u″(1)=0;v(0)=v(1)=v″(0)=v″(1)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} u^{(4)}(t)+\\beta _{1}u''(t)-\\alpha _{1}u(t)=f_{1}(t,u(t),v(t)),\\quad 0< t< 1; \\\\ v^{(4)}(t)+\\beta _{2}v''(t)-\\alpha _{2}v(t)=f_{2}(t,u(t),v(t)),\\quad 0< t< 1; \\\\ u(0)=u(1)=u''(0)=u''(1)=0; \\\\ v(0)=v(1)=v''(0)=v''(1)=0, \\end{cases} $$\\end{document} where f_{1},f_{2}in C[(0,1)times mathbb{R}^{+}_{0}times mathbb{R}, mathbb{R}], mathbb{R}_{0}^{+}=(0,+infty ). By constructing a special cone and applying fixed point index theory, some new existence results of multiple solutions for the considered system are obtained under some suitable assumptions. Finally, an example is worked out to illustrate the main results.

Existence of positive solutions for a semipositone boundary value problem with sequential fractional derivatives

We investigate the existence and multiplicity of positive solutions for a system of Riemann‐Liouville fractional differential equations with sequential derivatives, positive parameters and sign‐changing singular nonlinearities, subject to nonlocal uncoupled boundary conditions which contain various fractional derivatives and Riemann‐Stieltjes integrals. We apply the nonlinear alternative of Leray‐Schauder type and the Guo‐Krasnosel'skii fixed point theorem in the proof of our main existence results.

Existence results for a class of p-q Laplacian semipositone boundary value problems

Existence results for a class of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math> Laplacian semipositone boundary value problems

Multiple positive solutions to singular positone and semipositone m-point boundary value problems of nonlinear fractional differential equations

In this paper, we consider the properties of Green’s function for the nonlinear fractional differential equation boundary value problem \t\t\tD0+αu(t)+f(t,u(t))=0,0<t<1,u(0)=0,u(1)=∑i=1m−2βiu(ηi),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} &\\mathbf{D}_{0+}^{\\alpha}u(t)+f\\bigl(t,u(t)\\bigr)=0,\\quad 0< t< 1, \\\\ &u(0)=0, \\qquad u(1)= {\\sum_{i=1}^{m-2}} \\beta_{i} u(\\eta_{i}), \\end{aligned}$$ \\end{document} where 1<alpha<2, 0<beta_{i}<1, i=1,2,ldots, m-2, 0<eta_{1}<eta _{2}<cdots<eta_{m-2}<1, sum_{i=1}^{m-2}beta_{i}eta_{i}^{alpha-1}<1, mathbf{D}_{0+}^{alpha} is the standard Riemann–Liouville derivative. Here our nonlinearity f may be singular at u=0. As an application of Green’s function, we give some multiple positive solutions for singular positone and semipositone boundary value problems by means of the Leray–Schauder nonlinear alternative and a fixed point theorem on cones.

Positive Solutions for a System of Nonlinear Semipositone Boundary Value Problems with Riemann-Liouville Fractional Derivatives

We study the existence of positive solutions for the system of nonlinear semipositone boundary value problems with Riemann-Liouville fractional derivatives D0+αD0+αu=f1t,u,u′,v,v′, 0&lt;t&lt;1, D0+αD0+αv=f2(t,u,u′,v,v′), 0&lt;t&lt;1, u0=u′0=u′(1)=D0+αu(0)=D0+α+1u(0)=D0+α+1u(1)=0, and v(0)=v′(0)=v′(1)=D0+αv(0)=D0+α+1v(0)=D0+α+1v(1)=0, where α∈(2,3] is a real number and D0+α is the standard Riemann-Liouville fractional derivative of order α. Under some appropriate conditions for semipositone nonlinearities, we use the fixed point index to establish two existence theorems. Moreover, nonnegative concave and convex functions are used to depict the coupling behavior of our nonlinearities.

Twin solutions to semipositone boundary value problems for fractional differential equations with coupled integral boundary conditions

Twin solutions to semipositone boundary value problems for fractional differential equations with coupled integral boundary conditions

Existence of Positive Solutions to a Singular Semipositone Boundary Value Problem of Nonlinear Fractional Differential Systems

In this paper, we consider the existence of positive solutions to a singular semipositone boundary value problem of nonlinear fractional differential equations. By applying the fixed point index theorem, some new results for the existence of positive solutions are obtained. In addition, an example is presented to demonstrate the application of our main results.

Existence of symmetric positive solutions for a semipositone problem on time scales

This paper studies the existence of symmetric positive solutions for a second order nonlinear semipositone boundary value problem with integral boundary conditions by applying the Krasnoselskii fixed point theorem. Emphasis is put on the fact that the nonlinear term f may take negative value. An example is presented to demonstrate the application of our main result.

Semipositone boundary value problems with nonlocal, nonlinear boundary conditions

We demonstrate the existence of at least one positive solution to \begin{equation} \begin{split} -y''(t)& =\lambda f(t,y(t))\text{, }t\in(0,1)\\ y(0)& =H(\varphi(y))\text{, }y(1)=0,\notag \end{split} \end{equation} where $H : \mathbb{R}\rightarrow\mathbb{R}$ is a continuous function and $\varphi : \mathcal{C}([0,1])\rightarrow\mathbb{R}$ is a linear functional so that the boundary condition at $t=0$ may be both nonlocal and nonlinear. Since the continuous function $f : [0,1]\times\mathbb{R}\rightarrow\mathbb{R}$ may assume negative values, our results apply to semipositone problems. The classical Leray-Schauder degree is utilized to derive the existence result, which we obtain in the case where $\lambda$ is small and which permits $f$ to be negative on its entire domain. The result is illustrated by an example.

Positive solutions for nonlinear semipositone fractional q-difference system with coupled integral boundary conditions

This paper investigates the coupled integral boundary value problem for systems of nonlinear fractional q-difference equations. The nonlinear terms are continuous and semipositone. We firstly give the corresponding Green’s function for the boundary value problem and some of its properties. Moreover, by applying the nonlinear alternative of Leray–Schauder type and Krasnoselskii’s fixed point theorems, we derive an interval of λ such that any λ lying in this interval, the semipositone boundary value problem has one or multiple positive solutions. As applications, some interesting examples are presented to illustrate the main results. Finally, some related boundary value problems are also discussed.

A note on semipositone boundary value problems with nonlocal, nonlinear boundary conditions

We consider the existence of at least one positive solution to a semipositone boundary value problem with nonlocal, nonlinear boundary conditions, which can be quite general since the nonlinearity is realized as a Stieltjes integral. By assuming that the associated Stieltjes measure decomposes in a certain way, the classical Leray-Schauder degree is utilized to derive the existence result.

Positive solutions for a class of fractional differential equation multi-point boundary value problems with changing sign nonlinearity

In this paper, we consider the multi-point boundary value problem of nonlinear fractional differential equation $$\begin{aligned} \left\{ \begin{array}{lcl} D_{0{+}}^{\alpha }u(t)+{\lambda }f(t,u(t))=0, \quad 0<t<1,\\ u(0)=u^{\prime }(0)=\cdots =u^{(n-2)}(0)=0,\\ u^{(i)}(1)=\sum \nolimits _{j=0}^{m-2}{\eta }_{j}u^{\prime }(\xi _{j}), \end{array} \right. \end{aligned}$$ where \(\lambda \) is a parameter, \(\alpha \ge 2\), \(n-1 0\), $$\begin{aligned} \Delta = \left\{ \begin{array}{ll} {1}, &{} \quad {i = 0}; \\ {(\alpha - 1)(\alpha - 2) \cdots (\alpha - i)}, &{} \quad {i \ge 1}. \\ \end{array} \right. \end{aligned}$$ \(D_{0{+}}^{\alpha }\)is the Riemann–Liouville’s fractional derivative, \(f\) may change sign and may be singular at \(t=0,1\). We give the corresponding Green’s function for the boundary value problem and its some properties. Moreover, we derive an interval of \(\lambda \) such that for any \(\lambda \) lying in this interval, the semipositone boundary value problem has positive solutions.

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

We consider the existence of a positive solution to the first-order dynamic equation y?(t)+p(t)y?(t) = ?f (t, y?(t)), t?(a, b)T, subject to the boundary condition y(a) = y(b) + ?T1,T2 F(s, y(s)) ?s for ?1,?2 ? [a,b]T. In this setting, we allow f to take negative values for some (t; y). Our results generalize some recent results for this class of problems, and because we treat the problem on a general time scale T we provide new results for this problem in the case of differential, difference, and q-difference equations. We also provide some discussion of the applicability of our results.

Positive solutions for singular semipositone boundary value problems on infinite intervals

By using the fixed point theory on a cone with a special norm and space, we discuss the existence of positive solutions for a class of semipositone boundary value problems on infinite intervals. The work improves many known results including singular and non-singular cases.

Bicortical anchors, a new principle: Mechanical study in cadavers

By using the fixed point theory on a cone with a special norm and space, we discuss the existence of positive solutions for a class of semipositone boundary value problems on infinite intervals. The work improves many known results including singular and non-singular cases.

Multiple positive solutions of a semipositone singular boundary value problem on time scales

In this paper, we study the following singular semipositone boundary value problem on time scales:

On a first-order semipositone discrete fractional boundary value problem

In this paper we demonstrate the existence of at least one positive solution to a discrete fractional semipositone boundary value problem. Our results extend existing results not only due to the fact that the problem we treat here is of fractional order but also due to the fact that the boundary condition we study is potentially very general being as it can be both nonlocal and nonlinear.