Existence of positive S-asymptotically periodic solutions of the fractional evolution equations in ordered Banach spaces

Abstract

In this paper, we discuss the asymptotically periodic problem for the abstract fractional evolution equation under order conditions and growth conditions. Without assuming the existence of upper and lower solutions, some new results on the existence of the positive S-asymptotically ω-periodic mild solutions are obtained by using monotone iterative method and fixed point theorem. It is worth noting that Lipschitz condition is no longer needed, which makes our results more widely applicable.

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].…”

“…By means of the method of eigenvalues' expansions, the differential equation is converted to the integral form. Taking Equations (6) and (7) into consideration, it demonstrates that the solution of Equation (9) is the weak solution of Equation (2).…”

“…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 ,…”

“…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 ,…”