Nonlinear Fractional Boundary Value Problem Research Articles

The Multiplicity of Solutions for a Class of Nonlinear Fractional Dirichlet Boundary Value Problems with p-Laplacian Type via Variational Approach

Abstract In this paper, by using variational methods and a critical point theorem due to Bonanno and Marano, the existence of at least three weak solutions is obtained for a class of p-Laplacian type nonlinear fractional coupled systems depending on two parameters. Two examples are given to illustrate the applications of our main results.

Nonlinear sum operator equations and applications to elastic beam equation and fractional differential equation

In this paper, by studying the solutions of the abstract operator equation A(x,x)+B(x,x)+e=x on ordered Banach spaces, where A, B are two mixed monotone operators and ein P with theta leq eleq h, we prove a class of boundary value problems on elastic beam equation to have a unique solution. Furthermore, we also apply our abstract result to establish the existence and uniqueness theorem of nontrivial solutions for nonlinear fractional boundary value problems. The iterative sequences to approximate unique solutions for the above two classes of problems are also obtained.

Existence results for nonlinear fractional boundary value problem involving generalized proportional derivative

We introduce nonlinear fractional BVPs including a generalized proportional derivatives with nonlocal multipoint and substrip boundary conditions. The nonlinearities are defined on the Orlicz space and depend on the unknown function and its generalized derivative. Existence results for a nonlinear boundary value problem involving a proportional fractional derivative by utilizing some fixed point theorems are presented. The obtained results are new and are well illustrated with an example.

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 D0+αu(t)+λf(t,u(t),v(t))=0, 0<t<1, D0+αv(t)+μg(t,u(t),v(t))=0, 0<t<1, u(0)=v(0)=0, a1D0+βu(1)=b1D0+βv(ξ), a2D0+βv(1)=b2D0+βu(η), η,ξ∈(0,1), where the coefficients ai,bi,i=1,2 are real positive constants, α∈(1,2],β∈(0,1],D0+α, D0+β are the standard Riemann-Liouville derivatives. Values of the parameters λ and μ are determined for which boundary value problem has positive solution by utilizing a fixed point theorem on cone.

Lyapunov-type inequalities for nonlinear fractional differential equations and systems involving Caputo-type fractional derivatives

A Lyapunov-type inequality is derived for a nonlinear fractional boundary value problem involving Caputo-type fractional derivative. The obtained inequality provides a necessary condition for the existence of nontrivial solutions to the considered problem. Next, we extend our study to the case of systems.

Lyapunov Type Inequality for a Nonlinear Fractional Hybrid Boundary Value Problem

In this paper, we present a Lyapunov type inequality for a nonlinear fractional hybrid boundary value problem. We illustrate the main result through a series of examples.

Iterative Analysis of the Unique Positive Solution for a Class of Singular Nonlinear Boundary Value Problems Involving Two Types of Fractional Derivatives with p‐Laplacian Operator

This article is concerned with a class of singular nonlinear fractional boundary value problems with p‐Laplacian operator, which contains Riemann–Liouville fractional derivative and Caputo fractional derivative. The boundary conditions are made up of two kinds of Riemann–Stieltjes integral boundary conditions and nonlocal infinite‐point boundary conditions, and different fractional orders are involved in the boundary conditions and the nonlinear term, respectively. Based on the method of reducing the order of fractional derivative, some properties of the corresponding Green’s function, and the fixed point theorem of mixed monotone operator, an interesting result on the iterative sequence of the uniqueness of positive solutions is obtained under the assumption that the nonlinear term may be singular in regard to both the time variable and space variables. And we obtain the dependence of solution upon parameter. Furthermore, two numerical examples are presented to illustrate the application of our main results.

Positive solutions for a singular coupled system of nonlinear higher-order fractional q-difference boundary value problems with two parameters

Positive solutions for a singular coupled system of nonlinear higher-order fractional q-difference boundary value problems with two parameters

POSITIVE SOLUTIONS FOR A NONLINEAR DISCRETE FRACTIONAL BOUNDARY VALUE PROBLEM WITH A <inline-formula><tex-math id="M1">$ P $</tex-math></inline-formula>-LAPLACIAN OPERATOR

In this paper using the monotone iterative technique we establish the existence and uniqueness of positive solutions for a nonlinear discrete fractional boundary value problem with a $p$-Laplacian operator. Also we discuss an iterative sequence which yields the approximate solution for this problem.

Ulam stability to a toppled systems of nonlinear implicit fractional order boundary value problem

In this manuscript, we give some sufficient conditions for existence, uniqueness and various kinds of Ulam stability for a toppled system of fractional order boundary value problems involving the Riemann–Liouville fractional derivative. Applying the Banach contraction principle and the Leray–Schauder result of cone type, uniqueness and existence results are proved for the proposed toppled system. Stability is investigated by using the classical technique of nonlinear functional analysis. The results obtained are well illustrated with the aid of an example.

Higher order multi-point fractional boundary value problems with integral boundary conditions

In this paper, we concerned with positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions. We establish the criteria for the existence of at least one, two and three positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions by using a result from the theory of fixed point index, Avery-Henderson fixed point theorem and the Legget-Williams fixed point theorem, respectively.

Existence results for a coupled system of nonlinear fractional multi-point boundary value problems at resonance

By using the coincidence degree theory, we present an existence result for a coupled system of nonlinear fractional differential equations with multi-point boundary conditions at resonance.

Existence results for fractional order differential equation with nonlocal Erd\xe9lyi\u2013Kober and generalized Riemann\u2013Liouville type integral boundary conditions at resonance

In this paper, we discuss a nonlinear fractional order boundary value problem with nonlocal Erdélyi–Kober and generalized Riemann–Liouville type integral boundary conditions. By using Mawhin continuation theorem, we investigate the existence of solutions of this boundary value problem at resonance. It is shown that the boundary value problem \t\t\tDqcx(t)=f(t,x(t),x′(t)),t∈[0,T],1<q≤2,x(0)=αIηγ,δx(ζ),x(T)=βρIpx(ξ),\\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}$$\\begin{gathered} {}^{c}D^{q}x(t)=f \\bigl(t, x(t),x'(t) \\bigr),\\quad t \\in[0,T], 1< q\\leq2, \\\\ x(0)=\\alpha I^{\\gamma,\\delta}_{\\eta}x(\\zeta),\\qquad x(T)=\\beta{}^{\\rho }I^{p}x( \\xi),\\end{gathered} $$\\end{document} has at least one solution under some suitable conditions, where alpha, betainmathbb{R}, 0<zeta, xi<T.

Existence of solutions for a nonlocal fractional boundary value problem

We study a nonlinear fractional boundary value problem with nonlocal boundary conditions. An associated Green's function is constructed as a series of functions by the perturbation approach. Criteria for the existence of solutions are obtained based upon it.

Upper and Lower Solution Method for Nonlinear Fractional Differential Equations Boundary Value Problem

Upper and Lower Solution Method for Nonlinear Fractional Differential Equations Boundary Value Problem

A Legendre collocation method for distributed-order fractional optimal control problems

In many dynamic processes, the fractional differential operators not only appear as discrete fractional, but they also possess a continuous nature in a sense that their order is distributed over a given range. This paper is concerned with the optimization of systems whose governing equations contain a fractional derivative of distributed order, in the Caputo sense. By using the Lagrange multiplier within the calculus of variations and by applying the fractional integration-by-parts formula, the necessary optimality conditions are derived in terms of a nonlinear two-point distributed-order fractional boundary value problem. A Legendre spectral collocation method is developed for solving such problem. The solution method involves the use of three-term recurrence relations for both the left- and right-sided fractional integrals of the shifted Legendre polynomials. The convergence of the proposed method is discussed. The optimal profiles show the performance of the numerical solution and the effect of the fractional derivatives in the optimal results.

Existence of nonnegative solutions for a nonlinear fractional boundary value problem

Abstract This paper deals with the existence of solutions for a class of boundary value problem (BVP) of fractional differential equation with three point conditions via Leray-Schauder nonlinear alternative. Moreover, the existence of nonnegative solutions is discussed.

On linear and nonlinear fractional Hadamard boundary value problems

On linear and nonlinear fractional Hadamard boundary value problems

On analytical solution of system of nonlinear fractional boundary value problems associated with obstacle

In this paper, we use the modified variation of parameters method (MVPM), an elegant coupling of variation of parameters method (VPM) and Adomian's decomposition method (ADM), for finding the analytical solution of system of nonlinear fractional boundary value problems associated with obstacle. Caputo sense of fractional derivative is used to coup up with fractional term. The results are calculated in terms of series with easily computable components. The used technique is quite easy and convenient for such type problems because it has been previously applied over several nonlinear obstacle systems.

Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator

In this paper, we investigate the existence of positive solutions for a system of nonlinear fractional differential equations nonlocal boundary value problems with parameters and p-Laplacian operator. Under different combinations of superlinearity and sublinearity of the nonlinearities, various existence results for positive solutions are derived in terms of different values of parameters via the Guo-Krasnosel’skii fixed point theorem.