Symmetric-periodic solutions for some types of generalized neutral equations

The existence of symmetric-periodic outcomes for a class of fractional differential equations has been increasingly studied. Such study has used various methods such as fixed point theory, critical point theory, and approximation theory. In this work, we study the m-pseudo almost automorphic (m-P$$\Lambda \Lambda$$ΛΛ) outcomes for a category of fractional neutral differential equations. To satisfy this aim, we introduce composition results under suitable conditions and employ them to establish some extant outcomes using interpolation theory mixed with fixed point technique. Examples are illustrated.


Introduction
The symmetry in the field of differential equations is a transformation that preserves its domestic of results invariant. Symmetry study can be utilized to resolve some classes of ordinary, partial, fractional differential equations, though defining the symmetries can be computationally concentrated like other mathematical methods. The best method for symmetry is by finding the periodic solution of the differential equation.
In 1962, Bochner [1] introduced the concept of almost automorphy, which is an important generalization of almost periodicity. The concept of almost periodic functions was introduced by Bohr [2]. It was named as PAA functions because they originally presented themselves, in their work in differential geometry, as scalars and tensors on manifolds with (discrete) groups of automorphisms [3]. Later, PKK function has become one of the most attractive topics in the qualitative theory of evolution equations, and there have been several interesting, natural and powerful generalizations of the classical PKK functions [4][5][6][7][8][9]. Recently, Digana et al., studied the concept for different classes of ODF and PDE (see [10][11][12][13][14]). Xiao et al. [15] introduced the concept of PKK functions for a natural and a significant extension of PKK functions. Moreover, they proved that the space of PKK functions is complete; so they solve a key fundamental problem on this issue and pave the road to further study the applications of PKK functions. They investigated the existence of PKK to u 0 ðtÞ ¼ KðtÞuðtÞ þ f ðtÞ and u 0 ðtÞ ¼ KðtÞuðtÞ þ f ðt; uðtÞÞ in a Banach space. Chang and Luos [16] presented a composition theorem for m-PKK function, which was proved under appropriate conditions. They applied this theorem to investigate whether the m-PKK solutions exist in the neutral differential equation as follows: Periodic motion is a very important and special phenomena not only in natural science, but also in social science, such as climate, food supplement, insecticide population and sustainable development. Periodic solutions are desired property in differential equations, constituting one of the most important research directions in the theory of differential equations. The existence of periodic solutions is often a desired property in dynamical systems, constituting one of the most important research directions in the theory of dynamical systems, with applications ranging from celestial mechanics to biology and finance. Fractional differential equations (FDEs) are the most important generalizations of the field of ODE [17][18][19][20][21]. Recent investigations in physics, engineering, biological sciences and other fields have demonstrated that the dynamics of many systems are described more accurately using FDEs, and that FDE with delay are often more realistic to describe natural phenomena than those without delay. Periodic solution fractional differential equations have been studied by many researchers. They studied periodic solutions of the equation (see [17,18]) where A and B are closed linear operators defined on a complex Banach space X with domains D(A) and D(B), respectively, 0 b\a 2. The aim of this paper is to study the existence of periodic solutions for the following FDE: D l tðtÞ þ uðt; tðtÞÞ ¼ KtðtÞ þ #ðt; tðtÞÞ; t 2 R ð1Þ 8l 2 ð0; 1; where K : domðKÞ & v ! v is considered the operator of a hyperbolic analytic semigroup TðtÞ t ! 0 ; and u : R Â v ! v d ð\k\d\Þ; # R Â v ! v are appropriate continuous functions; v d refers to the appropriate interpolation space and D l is the Riemann-Liouville fractional differential operator (R-L operator). This paper is classified as follows. In ''Setting'', we present some basic definitions, lemmas, and setting results which will be used in this study. In ''Findings'', we introduce some existence results of almost-periodic and mild solutions of the fractional neutral differential equation. Examples are illustrated in the sequel.

Setting
The researchers allocated this section to investigate some results required in the sequel. In this paper, the notations ðv; k kÞ and ð!; k k ! Þ denote the two Banach spaces, whereas BCðR; vÞ refers to the Banach space of all bounded continuous functions from R to v, qualified with the supremum norm kuk 1 ¼ sup t2R kuðtÞk: Let v k be a space mediated between domðKÞ and v:BðR; v k Þ for k 2 ð0; 1Þ refers to Banach space of all bounded continuous functions r : R ! v k when supported with the k À sup norm: krk k;1 :¼ sup t2R krðtÞk k for r 2 BCðR; v k Þ: Throughout this paper, } denotes the Lebesgue field of R and @ the set of all positive measures m on } satisfying mðRÞ ¼ þ1 and mð½a; bÞ\ þ 1; for all a; b 2 Rða\bÞ: Definition 2.1 [3] A continuous function u : R ! v is referred to as automorphic in the case that every sequence of real numbers ð1 g Þ g2N has a subsequence ð1 0 Likely, qKK 0 ðR Â v; vÞ is defined as the gathering of combined continuous functions u : The space of all such functions is denoted as fðR; v; mÞ and ðfðR; v; mÞ; k k 1 Þ is a Banach space (see [19], Proposition 2.13]). The word ergodic (work) is employed to explain the dynamical system which has the same behavior averaged during the item time as averaged over the phase space.
Definition 2.4 [19] Let m 2 @. A continuous function u : In the sequel, we need some notions and properties of intermediate spaces and hyperbolic semi groups. Let v and Z be Banach spaces, with norms k k v ; k k Z ; respectively, and assume that Z is continuously embedded in v, that is, Z,!v.
Definition 2.8 The Riemann-Liouville fractional integral is defined as follows: where C denotes the gamma function (see [22,23]).

Definition 2.9
The Riemann-Liouville fractional derivative is defined as follows: Definition 2.10 [8, Definition 2.5] A semi group ðTðtÞÞ t ! 0 on v is stated to be hyperbolic if there is a projection q and constants @; S [ 0 such that each T(t) commutes with q; Kerq is invariant with respect to TðtÞ; TðtÞ : ImQ ! ImQ is invertible and for every x 2 v kTðtÞqxk M. Àst kxk; fort ! 0; ð2Þ kTðtÞQxk M. Àst kxk; fort 0; where Q :¼ I À q and, for t\0; TðtÞ ¼ TðÀtÞ À1 : For the problem (1), we list the following assumptions: (H1) If 0 k\d\1; then we let k 1 be the bound of the embedding v k ,!v; that is ktk k 1 ktk k for t 2 v k : (H2) Let 0 k\d\1 and the function u : R Â v ! v d belongs to qKKðR; v d ; mÞ; while # : R Â v ! v belongs to qKKðR; v; mÞ: Moreover, the functions u; # are uniformly Lipschitz in rotation to the second following argument: there exist K [ 0 such that kuðt; tÞ À uðt; tÞk d Kkt À mk and k#ðt; tÞ À #ðt; tÞk Kkt À mk for all t; m 2 v and t 2 R.

Findings
In the present section, a composition theorem is proved for m-PKK functions under appropriate conditions. Then, we apply this composition theorem to obtain some results regarding Eq. (1).

À1
KT l ðÀ1Þq½Uð1 þ t þ 1 g À 1 n Þ À Uð1 þ tÞ CðlÞ k d1: Hence, by (6) and the fact that kKðT l Þk kKðTÞk; we receive The result comes from Eq. (11)  for every 1 2 R: The proof is completed by applying the Lebesgue's dominated convergence theorem and similarly for c l 2 t using (7). h mÀPKKM outcomes The rest of this section is conducted to find the existence of m-PKK mild solutions (m-PKKM) of Eq. (1). Recently, Ibrahim et al. studied the mild solution of a class of FDE, by utilizing the fractional resolvent concept (see [24,25]).
Definition 3.4 Let k 2 ð0; 1Þ: A bounded continuous function t : R ! v k is stated to be a mild solution to (1) indicate that the function 1 ! KT l ðtÀ1q CðlÞ uð1; tð1ÞÞ is integrable on ðÀ1; tÞ; 1 ! AT l ðt À 1ÞQuð1; tð1ÞÞ is integrable on ðt; 1Þ and tðtÞ ¼ Àuðt; tðtÞÞ À Proof Consider the fractional integral operator^: qKKðR; v k ; mÞ À! qKKðR; v k ; mÞ such that ^tðtÞ :¼ Àuðt; tðtÞÞ À proved a composition theorem for m-PKK functions under appropriate conditions. Our technique is based on interpolation theory and Banach's fixed point theorem. Therefore, the solution, in this case, is unique. Moreover, we investigated the mild solution, for such a class by illustrating a new fractional resolvent concept. This functional is constructed to keep the periodicity of the solution and consequently its symmetry.

Author contribution
The authors jointly worked on deriving the results and approved the final manuscript. There is no conflict of interests regarding the publication of this article.