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D and( p, , c) – uniformly recurrent, ,where := k : k N. By Proposition four(iv), the set BCD(k);c (R : X) Epoxomicin Protocol equipped together with the metric d( := – is Shogaol custom synthesis actually a full metric space. Suppose now that a mapping F : X Y satisfies the estimate (15). We say that a continuous function u : R X can be a mild option of your semilinear Cauchy inclusion Dt, u(t) Au(t) F (t; u(t)), t R, if and only ift(22)u(t) =-R (t – s) F s; u(s) ds,t R.Keeping in thoughts Proposition 7 and Theorem 2, we are able to merely prove the following analogue of [12] (Theorem 3.1): Theorem three. Suppose that the above needs hold also as that the function F : R X X satisfies that for every single bounded subset B of X there exists a finite true constant MB 0 such that suptR supx B F (t; x) MB . If there exists a finite genuine quantity L 0 such that: (14) holds, and there exists an integer m N such that: Mm 1, where Mm := Lm supt 0 m t xm- -x-R (t – xm) R ( xi – xi-1) dx1 dx2 dxm ,i =then the abstract semilinear fractional Cauchy inclusion (22) includes a exclusive bounded Doss-( p, , c)uniformly recurrent solution which belongs towards the space BCD(k);c (R : X). 3. In this situation, we continue our evaluation with the popular d’Alembert formula. Let a 0; then we know that the common solution of your wave equation utt = a2 u xx in domain ( x, t) : x R, t 0, equipped with the initial circumstances u( x, 0) = f ( x) C2 (R) and ut ( x, 0) = g( x) C1 (R), is given by the d’Alembert formulaMathematics 2021, 9,23 ofu( x, t) =1 1 f ( x – at) f ( x at) two 2ax at x – atg(s) ds,x R, t 0.Suppose now that the function x ( f ( x), g[1] ( x)), x R is Doss-( p, c)-almost periodic for some p [1,) and c C, where: g[1] ( 0 g(s) ds. Clearly, the answer u( x, t) might be extended for the whole actual line in the time variable; we’ll prove that the option u( x, t) is Doss-( p, c)-almost periodic in ( x, t) R2 . In actual truth, we’ve (x, t, 1 , 2 R): u x 1 , t 2 – cu( x, t) 1 f ( x – at) (1 – a2) – c f ( x – at) 2 1 f ( x at) (1 a2) – c f ([ x at (1 a2)] – (1 a2)) two 1 [1] g ( x – at) (1 – a2) – cg[1] ( x – at) 2a 1 [1] g ( x at) – (1 – a2) – cg[1] ( x at) . 2a(23)If 1 – a2 satisfies that lim supl (1/l)l -l| f (v 1 – a2) – f (v)| p dvl -lp,then dvthere exists a finite actual quantity l0 ( , 1 , 2) 0 such that p l, l l ( , ,) and consequently: 0 1|( x,t)|l| f (v 1 – a2) -f (v)| pf ( x – at) (1 – a2) – c f ( x – at)pdx dtp= =ll-l -ll lf ( x – at) (1 – a2) – c f ( x – at)dx dtp-l-llf ( x – at) (1 – a2) – c f ( x – at)x al x – aldt dx1 a 1 a 1 a-llf v (1 – a2) – c f (v)pdv dxpl (1 a)-lp- l (1 a)lf v (1 – a2) – c f (v) dx = 1 apdv dxl (1 a)-ll (1 a),l (1 a)-1 l0 ( , 1 , 2),where we’ve applied the Fubini theorem inside the third line of computation. The remaining 3 addends in (23) can be estimated similarly, in order that the final conclusion basically follows as inside the final part of [12] (Example 1.2). 4. In [7], we’ve not too long ago the existence and uniqueness of c-almost periodic form options of the wave equations in R3 : utt (t, x) = d2 x u(t, x), x R3 , t 0; u(0, x) = g( x), ut (0, x) = h( x), (24)where d 0, g C3 (R3 : R) and h C2 (R3 : R). Let us recall that the renowned Kirchhoff formula (see e.g., [31] (Theorem 5.4, pp. 27778); we’ll use the same notion and notation) says that the function:Mathematics 2021, 9,24 ofu(t, x) :=1 t 4d2 tB1 (0) B1 (0)Bdt ( x)g d 1 4d2 tB1 (0)Bdt ( x)g d1 4 t =g( x dt) d h( x dt) ddtg( x dt) d:= u1 (t, x) u2 (t, x) u3 (t, x),t 0, x R3 ,is usually a one of a kind answer of dilemma (24) which belongs for the.

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