Positive solutions for a boundary-value problem with Riemann–Liouville fractional derivative*

In this work, we are mainly concerned with the existence of positive solutions for the fractional boundary-value problem $$ \left\{ {\begin{array}{*{20}{c}} {D_{0+}^{\alpha }D_{0+}^{\alpha }u=f\left( {t,u,{u}^{\prime},-D_{0+}^{\alpha }u} \right),\quad t\in \left[ {0,1} \right],} \hfill \\ {u(0)={u}^{\prime}(0)={u}^{\prime}(1)=D_{0+}^{\alpha }u(0)=D_{0+}^{{\alpha +1}}u(0)=D_{0+}^{{\alpha +1}}u(1)=0.} \hfill \\ \end{array}} \right. $$ Here ?? ?? (2, 3] is a real number, $ D_{0+}^{\alpha } $ is the standard Riemann?CLiouville fractional derivative of order ??. By virtue of some inequalities associated with the fractional Green function for the above problem, without the assumption of the nonnegativity of f, we utilize the Krasnoselskii?CZabreiko fixed-point theorem to establish our main results. The interesting point lies in the fact that the nonlinear term is allowed to depend on u, u??, and $ -D_{0+}^{\alpha } $ .     

Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives

In this paper, we are concerned with Cauchy problems of fractional differential equations with Riemann–Liouville fractional derivatives in infinite-dimensional Banach spaces. We introduce the notion of fractional resolvent, obtain some its properties, and present a generation theorem for exponentially bounded fractional resolvents. Moreover, we prove that a homogeneous α-order Cauchy problem is well posed if and only if its coefficient operator is the generator of an α-order fractional resolvent, and we give sufficient conditions to guarantee the existence and uniqueness of weak solutions and strong solutions of an inhomogeneous α-order Cauchy problem.     

Existence and regularity of solutions for evolution equations with Riemann–Liouville fractional derivatives

In this paper, we derive the existence and uniqueness of mild solutions for inhomogeneous fractional evolution equations in Banach spaces by means of the method of fractional resolvent. Furthermore, we give the necessary and sufficient conditions for the existence of strong solutions. An example of the fractional diffusion equation is also presented to illustrate our theory.     

Eigenvalue problem of Sturm–Liouville systems with separated boundary conditions

Let \(\lambda _j\) be the jth eigenvalue of Sturm–Liouville systems with separated boundary conditions, we build up the Hill-type formula, which represent \(\prod \nolimits _{j}(1-\lambda _j^{-1})\) as a determinant of finite matrix. Consequently, we get the Krein-type trace formula based on the Hill-type formula, which express \(\sum \nolimits _{j}{1\over \lambda _j^m}\) as trace of finite matrices. The trace formula can be used to estimate the conjugate point alone a geodesic in Riemannian manifold and to get some infinite sum identities.     

Positive steady-state solutions for predator–prey systems with prey-taxis and Dirichlet conditions

This paper is concerned with positive steady-state solutions of a class of cross-diffusion systems which arise in the study of the predator–prey systems with prey-taxis. Under homogeneous Dirichlet boundary conditions, we use the theory of fixed point index in positive cones to establish the existence of positive steady-state solutions. By analyzing two related eigenvalues, we further obtain the coexistence region with respect to the growth rates of two species and characterize the differences of coexistence region if different predator–prey interactions are adopted. Additionally, we investigate the limiting behavior of positive steady-state solutions as some parameter tends to infinity. Our results not only generalize the previously known one, but also present some new conclusions.     

Probabilistic solutions to nonlinear fractional differential equations of generalized Caputo and Riemann–Liouville type

This paper provides well-posedness and integral representations of the solutions to nonlinear equations involving generalized Caputo and Riemann–Liouville type fractional derivatives. As particular cases, we study the linear equation with non constant coefficients and the generalized composite fractional relaxation equation. Our approach relies on the probabilistic representation of the solution to the generalized linear problem recently obtained by the authors. These results encompass some known cases in the context of classical fractional derivatives, as well as their far reaching extensions including various mixed derivatives.     

Existence of solutions for fractional differential equations of order q∈(2,3] with anti-periodic boundary conditions

In this paper, we prove the existence of solutions for an anti-periodic boundary value problem of fractional differential equations of order q∈(2,3]. The contraction mapping principle and Krasnoselskii’s fixed point theorem are applied to establish the results.     

Riemann–Liouville abstract fractional Cauchy problem with damping

This paper is concerned with Riemann–Liouville abstract fractional Cauchy Problems with damping. The notion of Riemann–Liouville fractional (α,β,c)$(\alpha ,\beta ,c)$ resolvent is developed, where 0<β<α≤1$0<\beta <\alpha \le 1$. Some of its properties are obtained. By combining such properties with the properties of general Mittag-Leffler functions, existence and uniqueness results of the strong solution of Riemann–Liouville abstract fractional Cauchy Problems with damping are established. As an application, a fractional diffusion equation with damping is presented.     

Analytical solutions for the multi-term time–space fractional advection–diffusion equations with mixed boundary conditions

In this paper, we consider the analytical solutions of multi-term time–space fractional advection–diffusion equations with mixed boundary conditions on a finite domain. The technique of spectral representation of the fractional Laplacian operator is used to convert the multi-term time–space fractional advection–diffusion equations into multi-term time fractional ordinary differential equations. By applying Luchko’s theorem to the resulting fractional ordinary differential equations, the desired analytical solutions are obtained. Our results are applied to derive the analytical solutions of some special cases to demonstrate their practical applications.     

Fourth order singular p-Laplacian differential equations with integral boundary conditions

In this paper, we investigate the existence of positive solutions for fourth order singular p-Laplacian differential equations with integral boundary conditions and non-monotonic function terms. Firstly, we establish a comparison theorem, then we define a partial ordering in E ₀ and construct lower and upper solutions to give a necessary and sufficient condition for the existence of C ²[0,1] as well as pseudo-C ³[0,1] positive solutions. Our nonlinearity f(t,x,y) may be singular at x=0, y=0, t=0 and t=1. Finally, we give some the dual results for the other cases of fourth order singular integral boundary value problems and an example to demonstrate the corresponding main results.     

On a boundary value problem for an equation of mixed type with a Riemann–Liouville fractional partial derivative

For an equation of mixed type with a Riemann–Liouville fractional partial derivative, we prove the uniqueness and existence of a solution of a nonlocal problem whose boundary condition contains a linear combination of generalized fractional integro-differentiation operators with the Gauss hypergeometric function in the kernel. A closed-form solution of the problem is presented.     

Nonlinear discrete Sturm–Liouville problems with global boundary conditions

This paper is devoted to the study of nonlinear difference equations subject to global nonlinear boundary conditions. We provide sufficient conditions for the existence of solutions based on properties of the nonlinearities and the eigenvalues of an associated linear Sturm–Liouville problem.