Nonlocal boundary value problem for fractional differential equations with p‐Laplacian

In this paper, we are concerned with the existence of positive solutions for the following nonlocal BVP of fractional DEs with p‐Laplacian operator By using the fixed point theorem in a cone, multiplicity solutions of the BVP are obtained. An example is also given to show the effectiveness of the obtained result. Copyright © 2013 John Wiley & Sons, Ltd.     

On stable iterative solutions for a class of boundary value problem of nonlinear fractional order differential equations

In this article, sufficient conditions for the existence of extremal solutions to nonlinear boundary value problem (BVP) of fractional order differential equations (FDEs) are provided. By using the method of monotone iterative technique together with upper and lower solutions, conditions for the existence and approximation of minimal and maximal solutions to the BVP under consideration are constructed. Some adequate results for different kinds of Ulam stability are investigated. Maximum error estimates for the corresponding solutions are given as well. Two examples are provided to illustrate the results.     

Positive solutions for multi‐point boundary value problems of fractional differential equations with p‐Laplacian

This paper investigates the existence of solutions for multi‐point boundary value problems of higher‐order nonlinear Caputo fractional differential equations with p‐Laplacian. Using the five functionals fixed‐point theorem, the existence of multiple positive solutions is proved. An example is also given to illustrate the effectiveness of ourmain result. Copyright © 2015 John Wiley & Sons, Ltd.     

Positive radial solutions of p‐Laplacian equation with sign changing nonlinear sources

This paper is concerned with the p‐Laplacian equation with singular sources which is allowed to change sign in a ball. The singularity may occur at the zero points of solutions and on the boundary. Using the upper and lower solutions method, we establish the existence of positive radial solutions for the problem considered. Copyright © 2006 John Wiley & Sons, Ltd.     

Existence of positive solutions for nonlinear singular boundary value problem with p‐Laplacian

In this paper, we study the existence of positive solutions for the following Sturm–Liouville‐like four‐point singular boundary value problem (BVP) with p‐Laplacian where ?_p(s)=|s|^p?2 s, p>1, f is a lower semi‐continuous function. Using the fixed‐point theorem of cone expansion and compression of norm type, the existence of positive solution and infinitely many positive solutions for Sturm–Liouville‐like singular BVP with p‐Laplacian are obtained. Copyright © 2009 John Wiley & Sons, Ltd.     

Positive solutions for a fourth order discrete p‐Laplacian boundary value problem

In this paper, we study the existence and multiplicity of positive solutions for the following fourth order nonlinear discrete p‐Laplacian boundary value problem where φ_p(s) = | s | ^{p ? 2}s, p > 1, is continuous, T is an integer with T ≥ 5 and . By virtue of Jensen's discrete inequalities, we use fixed point index theory to establish our main results. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence and multiplicity of nontrivial solutions for nonlinear fractional differential systems with p‐Laplacian via critical point theory

In this paper, the existence and multiplicity of nontrivial solutions are obtained for nonlinear fractional differential systems with p‐Laplacian by combining the properties of fractional calculus with critical point theory. Firstly, we present a result that a class of p‐Laplacian fractional differential systems exists infinitely many solutions under the famous Ambrosetti‐Rabinowitz condition. Then, a criterion is given to guarantee that the fractional systems exist at least 1 nontrivial solution without satisfying Ambrosetti‐Rabinowitz condition. Our results generalize some existing results in the literature.     

Asymptotical stability analysis of Riemann‐Liouville q‐fractional neutral systems with mixed delays

This paper investigates the asymptotical stability of Riemann‐Liouville q‐fractional neutral systems with mixed delays (constant time delay and distributed delay). By constructing some appropriate Lyapunov‐Kravsovskii functionals, some sufficient conditions on delay‐dependent and delay‐independent asymptotical stability are obtained in terms of linear matrix inequality (LMI). Our employed method is based on the direct calculation of quantum derivatives of the Lyapunov‐Kravsovskii functionals. Finally, two examples are presented to demonstrate the availability of our obtained results.     

Monotone method for initial value problem for fractional diffusion equation

Using the method of upper and lower solutions and its associated monotone iterative, consider the existence and uniqueness of solution of an initial value problem for the nonlinear fractional diffusion equation.     

Extremal solutions to fourth order discontinuous functional boundary value problems

In this paper, given a L¹‐Carathéodory function, it is considered the functional fourth order equation together with the nonlinear functional boundary conditions Here , , satisfy some adequate monotonicity assumptions and are not necessarily continuous functions. It will be proved an existence and location result in presence of non ordered lower and upper solutions.     

Standing waves for a class of fractional p‐Laplacian equations with a general critical nonlinearity

In this paper, we concern with the following fractional p‐Laplacian equation with critical Sobolev exponent $$\left\{\begin{array}{l}{\epsilon }^{ps}{\left\{?\mathrm{\Delta }\right.}_p^{s}u+V(x){\left|u\right|}^{p?2}u=\lambda f(x){\left|u\right|}^{q?2}u+{\left|u\right|}^{p_s^{?}?2}u\text{in}{?}^N,\\ u\in W^{s,p}\left\{{?}^N\right.,u>0,\end{array}\right.$$ where ε > 0 is a small parameter,  λ > 0 , N is a positive integer, and N > ps with s ∈ (0, 1) fixed, $1<q\le p,p_s^{?}:=Np/\left(N?ps\right)$. Since the nonlinearity $h(x,u):=\lambda f(x){\left|u\right|}^{q?2}u+{\left|u\right|}^{p_s^{?}?2}u$ does not satisfy the following Ambrosetti‐Rabinowitz condition: $$0<\mu H(x,u):=\mu {\int }_0^{u}h(x,t)dt\le h(x,u)u,x\in {?}^N,0\ne u\in ?,$$ with μ > p , it is difficult to obtain the boundedness of Palais‐Smale sequence, which is important to prove the existence of positive solutions. In order to overcome the above difficulty, we introduce a penalization method of fractional p‐Laplacian type.     

Existence of periodic solutions for the Newtonian equation of motion with p‐Laplacian operator

In this paper, we study the existence of periodic solutions for the Newtonian equation of motion with p ‐Laplacian operator by asymptotic behavior of potential function, establish some new sufficient criteria of existence of periodic solutions for the differential system under the frame of Fuc?ik spectrum, generalize and improve some known works, and give an example to illustrate the application of the theorems. Copyright © 2017 John Wiley & Sons, Ltd.     

Limit behaviour of solutions to equivalued surface boundary value problem for p‐Laplacian equations

In this paper, we discuss the limit behaviour of solutions to boundary value problem with equivalued surface for p‐Laplacian equations when the equivalued surface boundary shrinks to a point in certain way. Copyright © 2000 John Wiley & Sons, Ltd.     

Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative

In this paper, we shall discuss the properties of the well-known Mittag-Leffler function, and consider the existence and uniqueness of solution of the initial value problem for fractional differential equation involving Riemann-Liouville sequential fractional derivative by using monotone iterative method.     

Some orthogonal polynomials on the finite interval and Gaussian quadrature rules for fractional Riemann‐Liouville integrals

Inspired with papers by Bokhari, Qadir, and Al‐Attas (2010) and by Rapai?, ?ekara, and Govedarica (2014), in this paper we investigate a few types of orthogonal polynomials on finite intervals and derive the corresponding quadrature formulas of Gaussian type for efficient numerical computation of the left and right fractional Riemann‐Liouville integrals. Several numerical examples are included to demonstrate the numerical efficiency of the proposed procedure.     

Studies on anti‐periodic boundary value problems for two classes of special second order impulsive differential equations

We consider anti‐periodic boundary value problems for two classes of special second order impulsive differential equations. On the basis of several important impulsive differential inequalities, by using the monotone iterative technique coupled with lower and upper solutions, we obtain sufficient conditions to guarantee the existence and uniqueness of solutions for such problems. Further, we give two examples to illustrate our conclusions. Copyright © 2013 John Wiley & Sons, Ltd.     

Two classes of conformable fractional Sturm‐Liouville problems: Theory and applications

In this paper, we investigate in more detail some useful theorems related to conformable fractional derivative (CFD) and integral and introduce two classes of conformable fractional Sturm‐Liouville problems (CFSLPs): namely, regular and singular CFSLPs. For both classes, we study some of the basic properties of the Sturm‐Liouville theory. In the class of r‐CFSLPs, we discuss two types of CFSLPs which include left‐ and right‐sided CFDs, each of order α∈(n,n+1], and prove properties of the eigenvalues and the eigenfunctions in a certain Hilbert space. Also, we apply a fixed‐point theorem for proving the existence and uniqueness of the eigenfunctions. As an operator for the class of s‐CFSLPs, we first derive two fractional types of the hypergeometric differential equations of order α∈(0,1] and obtain their analytical eigensolutions as Gauss hypergeometric functions. Afterwards, we define the conformable fractional Legendre polynomial/functions (CFLP/Fs) as Jacobi polynomial and investigate their basic properties. Moreover, the conformable fractional integral Legendre transforms (CFILTs) based on CFLP/Fs‐I and ‐II are introduced, and using these new transforms, an effective procedure for solving explicitly certain ordinary and partial conformable fractional differential equations (CFDEs) are given. Finally, as a theoretical application, some fractional diffusion equations are solved.     

Fundamental solutions for discrete dynamical systems involving the fractional Laplacian

We prove representation results for solutions of a time‐fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example, chain processes in chemistry and radioactivity. Our focus is in the problem , where 0<β ≤ 2, 0<α ≤ 1, , (?Δ_d)^α is the discrete fractional Laplacian, and is the Caputo fractional derivative of order β. We discuss important special cases as consequences of the representations obtained.     

Limit behaviour of solutions to equivalued surface boundary value problem for p‐Laplacian equations: II

In this paper (which is a continuation of Part‐I), we discuss the limit behaviour of solutions to boundary value problem with equivalued surface for p‐Laplacian equations in the case of 1<p?2?1/N when the equivalued surface boundary shrinks to a point in certain way. Copyright © 2001 John Wiley & Sons, Ltd.