Functional Coupled Systems with Generalized Impulsive Conditions and Application to a SIRS-Type Model

In this paper, we consider a first-order coupled impulsive system of equations with functional boundary conditions, subject to the generalized impulsive effects. It is pointed out that this problem generalizes the classical boundary assumptions, allowing two-point or multipoint conditions, nonlocal and integrodifferential ones, or global arguments, as maxima or minima, among others. Our method is based on lower and upper solution technique together with the fixed point theory. The main theorem is applied to a SIRS model where to the best of our knowledge, for the first time, it includes impulsive effects combined with global, local, and the asymptotic behavior of the unknown functions.

An investigation of fractional Bagley-Torvik equation

In this paper the authors prove the existence as well as approximations of the solutions for the Bagley-Torvik equation admitting only the existence of a lower (coupled lower and upper) solution. Our results rely on an appropriate fixed point theorem in partially ordered normed linear spaces. Illustrative examples are included to demonstrate the validity and applicability of our technique.

Antiperiodic Boundary Value Problems for Impulsive Fractional Functional Differential Equations via Conformable Derivative

In this paper, by using the lower and upper solution method and the monotone iterative technique, we investigate the existence of solutions to antiperiodic boundary value problems for impulsive fractional functional equations via a recent novel concept of conformable fractional derivative. An example is given to illustrate our theoretical results.

Boundary value problems for fractional differential equation with causal operators

In this paper, we study the two-point boundary value problems for fractional differential equation with causal operator. By lower and upper solution method and the monotone iterative technique, some results for the extremal solution and quasisolutions are obtained. At last, an example is given to demonstrate the validity of assumptions and theoretical results.

An Existence Result of Positive Solutions for Fully Second-Order Boundary Value Problems

An existence result of positive solutions is obtained for the fully second-order boundary value problem -u′′(t)=f(t,u(t),u′(t)), t∈[0,1], u(0)=u(1)=0,wheref:[0,1]×R+×R→Ris continuous. The nonlinearityf(t,x,y)may be sign-changing and superlinear growth onxandy. Our discussion is based on the method of lower and upper solution.

The Existence and Uniqueness of a Class of Fractional Differential Equations

By using inequalities, fixed point theorems, and lower and upper solution method, the existence and uniqueness of a class of fractional initial value problems,D0+qx(t)=f(t,x(t), D0+q-1x(t)), t∈(0,T), x(0)=0, D0+q-1x(0)=x0, are discussed, wheref∈C([0,T]×R2,R),D0+qx(t)is the standard Riemann-Liouville fractional derivative,1<q<2. Some mistakes in the literature are pointed out and some new inequalities and existence and uniqueness results are obtained.

On the existence of blow up solutions for a class of fractional differential equations

In this paper, by using fixed-point theorems, and lower and upper solution method, the existence for a class of fractional initial value problem (FIVP) $\begin{gathered} D_{0 + }^\alpha u(t) = f(t,u(t)),t \in (0,h), \hfill \\ t^{2 - \alpha } u(t)|_{t = 0} = b_1 D_{0 + }^{\alpha - 1} u(t) = |_{t = 0} = b_2 , \hfill \\ \end{gathered} $ is discussed, where f ∈ C([0, h]×R,R), D 0+α u(t) is the standard Riemann-Liouville fractional derivative, 1 < α < 2. Some hidden confusion and fallacy in the literature are commented. A new condition on the nonlinear term is given to guarantee the equivalence between the solution of the FIVP and the fixed-point of the operator. Based on the new condition, some new existence results are obtained and presented as example.