An Analysis on the Positive Solutions for a Fractional Configuration of the Caputo Multiterm Semilinear Differential Equation

In this paper, we consider a multiterm semilinear fractional boundary value problem involving Caputo fractional derivatives and investigate the existence of positive solutions by terms of different given conditions. To do this, we first study the properties of Green’s function, and then by defining two lower and upper control functions and using the wellknown Schauder’s fixed-point theorem, we obtain the desired existence criteria. At the end of the paper, we provide a numerical example based on the given boundary value problem and obtain its upper and lower solutions, and finally, we compare these positive solutions with exact solution graphically.

Hereditary Evolution Processes Under Impulsive Effects

In this note, we deal with a model of population dynamics with memory effects subject to instantaneous external actions. We interpret the model as an impulsive Cauchy problem driven by a semilinear differential equation with functional delay. The existence of delayed impulsive solutions to the Cauchy problem leads to the presence of hereditary impulsive dynamics for the model. Furthermore, using the same procedure we study a nonlinear reaction–diffusion model.

Antiperiodic Boundary Value Problem for a Semilinear Differential Equation of Fractional Order

The present paper is concerned with an antiperiodic boundary value problem for a semilinear differential equation with Caputo fractional derivative of order q ∈ (1, 2) considered in a separable Banach space. To prove the existence of a solution to our problem, we construct the Green’s function corresponding to the problem employing the theory of fractional analysis and properties of the Mittag-Leffler function . Then, we reduce the original problem to the problem on existence of fixed points of a resolving integral operator. To prove the existence of fixed points of this operator we investigate its properties based on topological degree theory for condensing mappings and use a generalized B.N. Sadovskii-type fixed point theorem.

Weighted Pseudo Almost Automorphic Solutions to Nonautonomous Semilinear Differential Equations

We consider the existence and uniqueness of Weighted Pseudo almost automorphic solutionsto the non-autonomous semilinear differential equation in a Banach space X :( ) = ( ) ( ) ( , ( )), ' u t A t u t f t u t t Rwhere A(t), t R, generates an exponentially stable evolution family {U(t, s)} andf :R X X satisfies a Lipschitz condition with respect to the second argument.MSC 2010: 43A60; 34G20, 47Dxx

Existence of the Mild Solution for Impulsive Semilinear Differential Equation

We study the existence of solutions of impulsive semilinear differential equation in a Banach space X in which impulsive condition is not instantaneous. We establish the existence of a mild solution by using the Hausdorff measure of noncompactness and a fixed point theorem for the convex power condensing operator.

LINEAR ASYMPTOTIC BEHAVIOUR OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

We study the semilinear differential equation u″ + F(t,u,u′)=0 on a half-line. Under different growth conditions on the function F, equations with globally defined solutions asymptotic to lines are characterized. Both fixed initial data and fixed asymptote are considered.

The maximum principle for a type of hereditary semilinear differential equation

An optimal control problem governed by a class of delay semilinear differential equations is studied. The existence of an optimal control is proven, and the maximum principle and approximating schemes are found. As applications, three examples are discussed.