Xinguang Zhang

The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium

Abstract In this paper, we establish the uniqueness of positive solution for a fractional model of turbulent flow in a porous medium by using the fixed point theorem of the mixed monotone operator. An example is also given to illustrate the application of the main result.

The eigenvalue for a class of singular p-laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition

In this paper, we are concerned with the eigenvalue problem of a class of singular p-Laplacian fractional differential equations involving the Riemann–Stieltjes integral boundary condition. The conditions for the existence of at least one positive solution is established together with the estimates of the lower and upper bounds of the solution at any instant of time. Our results are derived based on the method of upper and lower solutions and the Schauder fixed point theorem.

Positive Solutions of Eigenvalue Problems for a Class of Fractional Differential Equations with Derivatives

By establishing a maximal principle and constructing upper and lower solutions, the existence of positive solutions for the eigenvalue problem of a class of fractional differential equations is discussed. Some sufficient conditions for the existence of positive solutions are established.

The spectral analysis for a singular fractional differential equation with a signed measure

In this paper, by using the spectral analysis of the relevant linear operator and Gelfands formula, we obtain some properties of the first eigenvalue of a fractional differential equation. Based on these properties, the fixed point index of the nonlinear operator is calculated explicitly and some sufficient conditions for the existence of positive solutions are established.

Variational structure and multiple solutions for a fractional advection-dispersion equation

By establishing a variational structure and using the critical point theory, we investigate the existence of multiple solutions for a class of fractional advection-dispersion equations arising from a symmetric transition of the mass flux. Several criteria for the existence of multiple nonzero solutions are established under certain assumptions.

Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection

The abstract fractional dynamics model is based on a class of bioprocesses of HIV infection.The nonlinear terms and boundary conditions all depend on fractional derivatives of unknown functions.The system is singular and semipositone.The system involves some uncertain parametrical variations λ . Fractional order derivative is nonlocal which exhibits a long time memory behavior. With advantage of these, fractional order dynamic system models are more accurate than integer order ones in understanding the dynamic behavior of bioprocesses such as HIV infection. In this paper, we systematically study the existence of positive solutions of an abstract fractional semipositone differential system involving integral boundary conditions arising from the study of HIV infection models. By using the fixed point theorem in cone, some new results are established and an example is given to demonstrate the application of our main results.

Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion

Abstract In this paper, we consider the existence of nontrivial solutions for a class of fractional advection–dispersion equations. A new existence result is established by introducing a suitable fractional derivative Sobolev space and using the critical point theorem.

Entire blow-up solutions for a quasilinear p-Laplacian Schrödinger equation with a non-square diffusion term

Abstract In this paper, we study the entire blow-up solutions for a quasilinear p -Laplacian Schrodinger elliptic equation with a non-square diffusion term. By using the dual approach and some new iterative techniques, the difficulty due to the non-square diffusion term and the p -Laplacian operator is overcome and the nonexistence and existence of entire blow-up solutions are established.

The existence and uniqueness theorem of the solution to a class of nonlinear fractional order system with time delay

Abstract In this paper, we investigate the existence and uniqueness of the solution to a class of nonlinear fractional order system with delay. The estimate value of the above solution is also obtained by using the generalized Gronwall inequality.

Positive Solutions for (n - 1, 1)-Type Singular Fractional Differential System with Coupled Integral Boundary Conditions

We study the positive solutions of the -type fractional differential system with coupled integral boundary conditions. The conditions for the existence of positive solutions to the system are established. In addition, we derive explicit formulae for the estimation of the positive solutions and obtain the unique positive solution when certain additional conditions hold. An example is then given to demonstrate the validity of our main results.