On a Periodic Boundary Value Problem for Fractional Quasilinear Differential Equations with a Self-Adjoint Positive Operator in Hilbert Spaces

Abstract

In this paper we study the existence of a mild solution of a periodic boundary value problem for fractional quasilinear differential equations in a Hilbert spaces. We assume that a linear part in equations is a self-adjoint positive operator with dense domain in Hilbert space and a nonlinear part is a map obeying Carathéodory type conditions. We find the mild solution of this problem in the form of a series in a Hilbert space. In the space of continuous functions, we construct the corresponding resolving operator, and for it, by using Schauder theorem, we prove the existence of a fixed point. At the end of the paper, we give an example for a boundary value problem for a diffusion type equation.

Three-Point Boundary Value Problems of Coupled Nonlocal Laplacian Equations

“…(3) A definition of mild solution to boundary value problems (BVPs) (2) and ( 3) is introduced and the existence and uniqueness of mild solution are investigated by using some fixedpoint theorems. (4) Theorems 2 and 3 of this article achieve some easily verifiable conditions compared with the previous works in [6,7,9,11,22,26].…”

“…Remark 3. The condition on eigenvalues is much weaker than the condition (A) in [11] which required the inverse operator A −1 of self-adjoint operator A is bounded. The existence of solutions is investigated in [6] by the method of Laplace transform, this paper further considers the uniqueness of solutions.…”

“…Based on the properties (E1)-(E5) of Mittag-Leffler functions, some vital estimates of Green functions are introduced, which abolishes the restriction of eigenvalues sequence by condition (A) in Ref. [11]. In addition, the flexibility of Green functions enhances the difficulty of this article.…”

“…If s 1 = s 2 = 1, then Equation ( 2) is turned into Caputo-type fractional derivative in the literature [6,9,11,23,24]. If all the orders equal 1, then Equation (2) is changed into the classical parabolic equation [17,21].…”

“…Byszewski [2], Byszewski and Akca [3] then went on to study the existence and uniqueness of mild solutions for different kinds of evolution equations in Banach spaces. Moreover, various existence results for evolution equations and evolution inclusions applying the fixed-point theory and semigroup theory have been achieved [4][5][6][7][8][9][10][11]. For instance, Li et al [9] learned the following fractional evolution equation in ordered Banach spaces c D q 0 u(t) + Au(t) = f (t, u(t)), t > 0, u(0) = u 0 ,…”