Existence Of Solutions To Boundary Value Problem Of Caputo Fractional Differential Equation

In science, engineering, chemistry, physics, biology and other fields, fractional order differential equation are not only given reasonable description but also application is very extensive. Such as microphone feedback and analysis, field of heat transfer and fluid, etc. So the fractional order equation boundary value problem has significance. The main research works of this dissertation are summarized as follows:1. We consider the nonlinear fractional differential equation boundary value problem where f:[0,+∞)×Râ†'[0,+∞) is continuous, α∈(2,3], and c D0+α is the standard Caputo differentiation. By using fixed point theorems on cone, we give some existence results concerning positive solutions. Here the solutions especially are the interior points of cone. Finally, three examples are presented to show the effectiveness of the proposed approach. 2. We discuss a system of Caputo fractional difference equations with nonlocal boundary value conditions Preliminaries Where t∈[0,b+1]N0:={0,1,…, b+1}, b>3,2<vj≤3, fj:Râ†'R are given continuous functions, φj,ψi,φj:Rb+4â†'R are given continuous functions for each j(j=1,2,…,n). â–³Cvcy(t) is the standard Caputo difference. We use the Banach fixed point theorem and the Brouwer fixed point theorem to deduce the existence of solutions, and an example is given to illustrate the main result.3. We investigate a Caputo fractional difference equation boundary value problem where t∈{0,1,...,b+1}:=[0,b+1]N0,b>5 is an integer.f:[v-2,b+v]Nv-2 ×Râ†'R is continuous and f is not identically zero,2<v≤3> and â–³ACvy(t)is the standard Caputo difference. We use Schauder fixed point theorem and the theory of discrete fractional calculus to deduce the existence of solutions. Finally, a example are presented to show the effectiveness of the proposed approach.