53412025
Fixed point theorems for a class of nonlinear sum-type operators and application in a fractional differential equation
In this paper, we consider the fixed point for a class of nonlinear sum-type operators ‘A+B+C$A+B+C$’ on an ordered Banach space, where A, B are two mixed monotone operators, C is an increasing operator. Without assuming the existence of upper-lower solutions or compactness or continuity conditions, we prove the unique existence of a positive fixed point and also construct two iterative schemes to approximate it. As applications, we research a nonlinear fractional differential equation with multi-point fractional boundary conditions. By using the obtained fixed point theorems of sum-type operator, we get the sufficient conditions which guarantee the existence and uniqueness of positive solutions. At last, a specific example is provided to illustrate our result.
119463564
On the number of solutions of nonlinear equations in ordered Banach spaces
Abstract Let E be an ordered Banach space and A a continuous operator mapping some bounded order interval [ v , w ] ⊂ E into itself. This paper is concerned with the number of fixed points of A on [ v , w ]. There are given conditions on A and the ordering which guarantee the existence of no fixed point, precisely one, two, and more than two, distinct fixed points. The nonexistence and uniqueness theorems are completely elementary. The multiplicity results are based on the fixed-point index for α-set contractions. All of these results have applications to nonlinear integral equations and to mildly nonlinear elliptic boundary-value problems.
12846867
A fixed point theorem for convex and decreasing operators
Abstract In this paper, we present a new fixed point theorem for noncompact, convex and decreasing operators, which extends the existing corresponding results. As a sample, we give an application of the fixed point theorem to the two-point boundary value problem for a second-order differential equation.
121292725
Fixed points of generalized e-concave (generalized e-convex) operators and their applications
In this paper, we present the definitions of generalized e-concave operators and generalized e-convex operators, which are the generalizations of e-concave operators and e-convex operators, respectively. Without compactness or continuity assumption of generalized e-concave operators and generalized e-convex operators, we have proved the existence, uniqueness and monotone iterative techniques of their fixed points. Our results are even new to e-concave operators and e-convex operators. Finally, we apply the results to the singular boundary value problems for second order differential equations.
33225344
Positive solutions of operator equations on ordered Banach spaces and applications
This paper is concerned with an operator equation on ordered Banach spaces. The existence and uniqueness of its' positive solutions is obtained by using the properties of cones and monotone iterative technique. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for two classes of integral equations.
88511166
Fixed point theorems for τ – φ-concave operators and applications
In this paper, by introducing τ–φ-concave operators and using the properties of cones and monotone iterative technique, some new existence and uniqueness theorems of fixed points for increasing operators with such concavity are established under more extensive conditions in ordered Banach spaces. As an application, we study the existence and uniqueness of positive solutions of generalized three-point boundary value problems for second order differential equations. The results obtained here improve and generalize many known results. © 2009 Elsevier Ltd. All rights reserved.
120627592
New fixed point theorems for mixed monotone operators and local existence–uniqueness of positive solutions for nonlinear boundary value problems☆
Abstract In this article we present a new fixed point theorem for a class of general mixed monotone operators, which extends the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for mixed monotone operators. Based on them the local existence–uniqueness of positive solutions for nonlinear boundary value problems which include Neumann boundary value problems, three-point boundary value problems and elliptic boundary value problems for Lane–Emden–Fowler equations is proved. The theorems for nonlinear boundary value problems obtained here are very general.
125802066
φ − ( h , e )-concave operators and applications
Abstract In this article, by introducing a new set and a new concept of φ − ( h , e ) -concave operators, and by using the cone theory and monotone iterative method, we present some new existence and uniqueness theorems of fixed points for increasing φ − ( h , e ) -concave operators without requiring the existence of upper and lower solutions. As an application, we establish the existence and uniqueness of a nontrivial solution for a new form of fractional differential equation with integral boundary conditions. The main results of this paper improve and extend some known results, and present a new method to study nonlinear equation problems.
120379922
Positive solutions for some boundary value problems via a new fixed point theorem
In this paper, we establish some new fixed point theorems of mixed monotone operator with a perturbation. Moreover, we prove the existence and the uniqueness of positive solutions of a second order Neumann boundary value problem, a second order Sturm Liouville boudary value problem and a nonlinear elliptic boundary value problem for the Lane–Emden–Fowler equation.
121521423
Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems
Abstract The purpose of this paper is to present some new fixed point theorems for mixed monotone operators with perturbation by using the properties of cones and a fixed point theorem for mixed monotone operators. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.
8733423
A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations
Article history: This paper is concerned with an operator equation Ax + Bx + Cx = x on ordered Banach spaces, where A is an increasing α-concave operator, B is an increasing sub-homogeneous operator and C is a homogeneous operator. The existence and uniqueness of its positive solutions is obtained by using the properties of cones and a fixed point theorem for increasing general β-concave operators. As applications, we utilize the fixed point theorems obtained in this paper to study the existence and uniqueness of positive solutions for two classes nonlinear problems which include fourth-order two-point boundary value problems for elastic beam equations and elliptic value problems for Lane-Emden-Fowler equations.
56407122
Fixed point theorems for the sum of three classes of mixed monotone operators and applications
In this paper we develop various new fixed point theorems for a class of operator equations with three general mixed monotone operators,namely A(x,x)+B(x,x)+C(x,x)=x$A(x,x)+B(x,x) +C(x,x)=x$ on ordered Banach spaces, where A, B, C are the mixed monotone operators. A is such that for any t∈(0,1)$t\in(0,1)$, there exists φ(t)∈(t,1]$\varphi(t)\in(t,1]$ such that for all x,y∈P$x,y\in P$, A(tx,t−1y)≥φ(t)A(x,y)$A(tx,t^{-1}y)\geq\varphi(t)A(x,y)$; B is hypo-homogeneous, i.e.B satisfies that for any t∈(0,1)$t\in(0,1)$, x,y∈P$x,y\in P$, B(tx,t−1y)≥tB(x,y)$B(tx,t^{-1}y)\geq tB(x,y)$; C is concave-convex, i.e.C satisfies that for fixed y, C(⋅,y):P→P$C(\cdot,y):P\rightarrow P$ is concave; for fixed x, C(x,⋅)$C(x,\cdot)$: P→P$P\rightarrow P$ is convex. Also we study the solution of the nonlinear eigenvalue equation A(x,x)+B(x,x)+C(x,x)=λx$A(x,x)+B(x,x)+C(x,x)=\lambda x$ and discuss its dependency to the parameter. Our work extends many existing results in the field of study. As an application, we utilize the results obtained in this paper for the operator equation to study the existence and uniqueness of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions.
13044244
The unique solution of a class of sum mixed monotone operator equations and its application to fractional boundary value problems
In this paper we study a class of operator equations A(x, x) + B(x, x) = x in ordered Banach spaces, where A,B are two mixed monotone operators. Various theorems are established to guarantee the existence of a unique solution to the problem. In addition, associated iterative schemes have been established for finding the approximate solution converging to the fixed point of the problem. We also study the solution of the nonlinear eigenvalue equation A(x, x) + B(x, x) = λx and discuss its dependency to the parameter. Our results extend and improve many known results in this field of study. We have also successfully demonstrated the application of our results to the study of nonlinear fractional differential equations with two-point boundary conditions. c ©2016 All rights reserved.
56235376
The solution for a class of sum operator equation and its application to fractional differential equation boundary value problems
In this paper we study a class of sum operator equation Ax+Bx+C(x,x)=x$Ax+Bx+C(x,x)=x$ on ordered Banach spaces, where A is an increasing operator, B is a decreasing operator, and C is a mixed monotone operator. The existence and uniqueness of its positive solution are obtained by using the properties of cone and fixed point theorems for mixed monotone operators. As an application, we utilize the obtained results to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.
56096092
Uniqueness of positive solutions for several classes of sum operator equations and applications
In this article we study several classes of sum operator equations on ordered Banach spaces and present some new existence and uniqueness results of positive solutions, which extend the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for several classes of sum operator equations. As applications, we utilize the main results obtained in this paper to study two classes nonlinear problems; one is the integral equation u(t)=λ∫abG(t,s)f(s,u(s))ds, where f and G are both nonnegative, λ>0 is a parameter; the other is the elliptic boundary value problem for the Lane-Emden-Fowler equation −Δu=λf(x,u), u(x)>0 in Ω, u(x)=0 on ∂ Ω, where Ω is a bounded domain with smooth boundary in RN (N≥1), λ>0 and f(x,u) is allowed to be singular on ∂ Ω. The new results on the existence and uniqueness of positive solutions for these problems are given, which complement the existing results of positive solutions for these problems in the literature.MSC:47H10, 47H07, 45G15, 35J60, 35J65.
21663278
Positive solutions for a class of fractional boundary value problems with fractional boundary conditions
In this paper, we study the solvability of a nonlinear fractional differential equation under fractional integral boundary conditions. Via a mixed monotone operator method, some new results on the existence and uniqueness of a positive solution for the considered model are obtained. Moreover, we provide iterative sequences for approximating the solution. Some examples are also presented in order to illustrate the obtained result.
206393308
Existence of positive solutions for the nonlinear elastic beam equation via a mixed monotone operator
Abstract In this paper, we use a mixed monotone operator method to investigate the existence and uniqueness of positive solution to a nonlinear fourth-order boundary value problem which describes the deflection of an elastic beam with the left extreme fixed and the right extreme is attached to a bearing device given by a known function. Moreover, we present a concrete example illustrating the result obtained.
125180443
Existence and uniqueness of positive solutions for a class of fractional differential equation with integral boundary conditions
The purpose of this paper is to investigate the existence and uniqueness of positive solutions for a class of fractional differential equation with integral boundary conditions. Our analysis relies on two fixed point theorems of a sum operator in partial ordering Banach space. The main results obtained can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it.
62841243
Existence and uniqueness of positive solutions for singular fractional differential systems with coupled integral boundary conditions
In this paper, we study the existence and uniqueness of positive solutions for a class of singular fractional differential systems with coupled integral boundary conditions. By using the properties of the Green function, the mixed monotone method and the fixed point theory in cones, we obtain the existence and uniqueness results for the problem. The results obtained herein generalize and improve some known results including singular and non-singular cases. c ©2017 All rights reserved.
3618777
Uniqueness of positive solutions for the singular fractional differential equations involving integral boundary value conditions
This paper is concerned with the uniqueness of positive solutions for a class of singular fractional differential equations with integral boundary conditions. The nonlinear term and boundary conditions of fractional differential equation contain the fractional order derivatives. The uniqueness of positive solutions is derived by the fixed point theorem of mixed monotone operator. An example is given to demonstrate the validity of our main results.
122051792
Basic theory of fractional differential equations
Abstract In this paper, the basic theory for the initial value problem of fractional differential equations involving Riemann–Liouville differential operators is discussed employing the classical approach. The theory of inequalities, local existence, extremal solutions, comparison result and global existence of solutions are considered.
118445850
Theory and Applications of Fractional Differential Equations
1. Preliminaries. 2. Fractional Integrals and Fractional Derivatives. 3. Ordinary Fractional Differential Equations. Existence and Uniqueness Theorems. 4. Methods for Explicitly solving Fractional Differential Equations. 5. Integral Transform Methods for Explicit Solutions to Fractional Differential Equations. 6. Partial Fractional Differential Equations. 7. Sequential Linear Differential Equations of Fractional Order. 8. Further Applications of Fractional Models. Bibliography Subject Index
124039638
Unbounded solutions of third order three-point boundary value problems on a half-line
Abstract We consider the following third order three-point boundary value problem on a half-line: x'''(t)+q(t)f(t,x(t),x'(t),x''(t)) = 0, t ∈ (0,+∞), x'(0) = A, x(η) = B, x''(+∞) = C, where η ∈ (0,+∞), but fixed, and f : [0,+∞) × ℝ3 → ℝ satisfies Nagumo's condition. We apply Schauder's fixed point theorem, the upper and lower solution method, and topological degree theory, to establish existence theory for at least one unbounded solution, and at least three unbounded solutions. To demonstrate the usefulness of our results, we illustrate two examples.
8115194
Existence of positive solutions for a singular fractional boundary value problem
We study the existence of positive solutions for a nonlinear Riemann–Liouville fractional differential equation with a sign-changing nonlinearity, subject to multi-point fractional boundary conditions.