On the Integral of Almost Periodic Functions. of Several Variables

Transcription

1 Applied Mathematical Sciences, Vol. 6, 212, no. 73, On the Integral of Almost Periodic Functions of Several Variables Saud M. A. Alsulami Department of Mathematics King Abdulaziz University Jeddah, Saudi Arabia Abstract As a tool to investigate almost periodic functions of several variables which is important for the study of asymptotic behavior of differential equations, we give answer to a question raised by Basit in 1971, which is an extension of the classical Bohl-Bohr theorem (to almost periodic functions with two or more variables). We also extend a theorem due to Loomis (who obtained it for functions with values in a finite dimensional space) to functions with values in Banach space E, under conditions introduced by Kadets. As an application, we obtain a result on the almost periodicity of a double integral of an almost periodic function defined on the plane. 1 Introduction A classical result of Bohl and Bohr for almost periodic functions states that if f(t) is a scalar-valued almost periodic function defined on R and F (t) = t f(s) ds, then F is almost periodic if and only if it is bounded over R. This question plays an important role in the study of asymptotic behavior of differential equation : u (t) =Au(t)+f(t). See e.g. [1],[2],[4], [6],[7], [8] and

2 3616 S. M. A. Alsulam [9]. For applications to differential equations in Banach space, it is important to have generalizations of the Bohl-Bohr theorem to functions with values in a Banach space. However, a direct extension, without any further condition on the function f or the underlying Banach space E, is not valid. An example can be found in ( [3],[12, p.179]). Since boundedness of the integral of a vector-valued almost periodic function does not imply, in general, its almost periodicity, it would be desirable to clarify under what conditions (besides boundedness of the integral) the almost periodicity of the integral would follow. There have been obtained numerous results with conditions either on the space E or on the function F. We will mention here that Kadets discovered that the Bohl-Bohr Theorem for the vector-valued functions is valid under either one of the following conditions: (C1) The Banach space does not contain a subspace isomorphic to c [11]. (C2) F has weakly relatively compact range [1]. For sake of simplicity we restrict ourselves to the case of time dimension two, but we note that the results remain true for arbitrary finite time dimension. 2 On Basit s question In 1971, Basit (see [5]) showed that for a complex valued almost periodic function f(x, y) defined on the plane, the boundedness of the integral F (x, y) = x f(t, y) dt does not imply its almost periodicity. In this section, we impose a condition on f that ensures the almost periodicity of F. Some facts about the almost periodic functions with several variables are straightforward from the definition and will be used implicitly in our proofs and among such: If f(s, t) is an almost periodic function in the plane, then f(s, t ) is almost periodic in R for fixed t R. Also, if g(s) and h(t) are almost periodic in R, then g(s) +h(t) is an almost periodic function in the plane. Moreover, by letting h(t), we get: if g(s) is an almost periodic function in R, then G(s, t) :=g(s) is an almost periodic function in the plane.

3 On the integral of almost periodic functions 3617 We also need the following which is a result of Loomis for scalar-valued functions in [13]. Theorem 2.1 (Loomis Theorem )[13, p.366] For the finite dimensional Euclidean group : If f(x) :=f(x 1,..., x n ) is bounded scalar-valued function and f x i is almost periodic for every i, then f is almost periodic. Theorem 2.2 Let f(s, t) be an almost periodic function in the plane, F (x, y) = x f(t, y) dt be bounded and suppose that there exists a scalar-valued almost periodic function g(x, y) on the plane such that f y g (x, y) = (x, y) x Then, F (x, y) is a scalar-valued almost periodic function. Proof: Let Note that F x Since F (x, y) = x f(t, y) dt (= f(x, y)) is almost periodic on the plane. F x y = f (t, y) dt y = x g (t, y) dt t = g(x, y) g(,y) is almost periodic in the plane, it follows by Loomis theorem 2.1 that F is almost periodic in the plane. Our next objective is to generalize the above theorem to functions with values in a Banach space E. As an intermediate step, we extend, in the next section, Loomis theorem to the vector-valued case.

4 3618 S. M. A. Alsulam 3 Loomis theorem for vector-valued almost periodic functions We observe that if f(s, t) is almost periodic in the plane, then for fixed (a, b) R 2, a f(s + v, t + b) dv is almost periodic in the plane. Let us recall the following theorem of Basit, [5]: Theorem 3.1 If the vector-valued function f(t) satisfies the difference equation f(tγ) f(t) =g γ (t), where for each γ G (G is a group) the function g γ (t) is almost periodic, then under one of the conditions (C1)-(C2) the function f(t) is almost periodic. Theorem 3.2 Let u(s,t) and u(s,t) be vector-valued almost periodic functions s t with values in a Banach space E. If one of the conditions (C1)-(C2) is satisfied, then u(s, t) is almost periodic. Proof: In view of Basit s theorem (when G = R 2 ),we only have to show that for each (a, b) R 2, u(s + a, t + b) u(s, t) is an almost periodic function. Fix (a, b) R 2. integral. Then, we can express the difference as a sum of line u(s + a, t + b) u(s, t) = a u b u (s + v, t + b) dv + (s, t + w) dw v w Thus, a u (s + v, t+ b) dv and b u (s, t + w) dw are almost periodic functions in the plane. Therefore, their sum is almost v w periodic. The following theorem extends theorem 2.2 for vector-valued functions under the conditions of Kadets. By using theorem 3.2, the proof is similar to that of theorem 2.2 and therefore is omitted.

5 On the integral of almost periodic functions 3619 Theorem 3.3 Let f(x, y) be a vector-valued almost periodic function and F (x, y) = x f(t, y) dt be bounded. Assume that there exists a vector-valued almost periodic function g(x, y) on the plane such that f y g (x, y) = (x, y) x If one of the conditions (C1)-(C2) is satisfied, then F (x, y) is a vector-valued almost periodic function. Remark 3.4 An almost automorphic version of Loomis theorem also hold ( with the same proof). As a consequence, the almost automorphic version of Theorem 3.2 also holds. 4 An alternative form of the theorem on the integral of almost periodic functions Let f(s, t) be an almost periodic function with values in a Banach space E. In this section, we formulate conditions which guarantee the function F (s, t) = f(x, y) dx dy is almost periodic. t s Theorem 4.1 Let f(s, t) be an integrable almost periodic function. Assume that there exists an almost periodic function f 1 (s, t) such that f = f 1 s t and an almost periodic function f 2 (s, t) such that f = f 2. If F (s, t) = t s s t f(v, w) dw dv is bounded, then F is almost periodic, provided one of the conditions (C1)-(C2) holds. Proof : We show that F and F are (vector-valued) almost periodic s t functions, which implies, by theorem 3.2, that F (s, t) is almost periodic. Differentiating F s with respect to s and with respect to t, we have 2 F t s = f (s, w) dw 2 s

6 362 S. M. A. Alsulam = t f 1 (s, w) dw w 2 F t s = f 1 (s, t) f 1 (s, ) Thus, 2 F is a vector-valued almost periodic function. On the other hand, s 2 = f(s, t) is a vector-valued almost periodic function on the plane. Hence, by theorem 3.2, F is a vector-valued almost periodic function. Analogously, s F is a vector-valued almost periodic function. Therefore, applying theorem t 3.2 again, we obtain that F (s, t) is a vector-valued almost periodic function. By using theorem 2.1 and proceeding as above, we obtain a scalar-valued version of the above theorem, namely: Theorem 4.2 Let f(s, t) be an integrable scalar-valued almost periodic function. Assume that there exists an almost periodic function f 1 (s, t) such that f = f 1 s t F (s, t) = s and an almost periodic function f 2 (s, t) such that f = f 2. Then, t s f(v, w) dw dv is almost periodic if (and only if) F is bounded. t Acknowledgements The author would like to thank Ohio University for the facilities that given to him and the Deanship of Scientific Research at King Abdulaziz University, Jeddah, for the financial support through the project No. (1431 \ 13 \ 329) to writing the research project. References [1] Alsulami, S. M. On Evolution Equations In Banach Spaces And Commuting Semigroups. Ph.D. dissertation, Ohio University. June (25). [2] Alsulami, S. M. and Vu, Q.P. On Mild solutions and almost periodic solutions of differential equations with multi-time. AMS meeting at Bowling Green, Kentucky. March 18-19, (25).

7 On the integral of almost periodic functions 3621 [3] Amerio, L. Sull integrazione delle funzioni quasi-periodiche astratte. Ann. Mat. Pura Appl., (4) 53 (1961), [4] Amerio, L. and Prouse, G. Almost periodic functions and functional equations. Van Nostrand, R.C. (1971). [5] Basit, B. Generalization of two theorems of M. I. Kadets concerning the indefinite integral of abstract almost periodic functions. Mathematical Notes, 9 (1971), [6] Basit, B. Harmonic Analysis and Asymptotic Behavior of Solutions to the Abstract Cauchy Problem. Semigroup Forum, Vol. 54 (1997), [7] Bochner, S. A new approach to almost periodicity. Proc. of the National Academy of Sciences of the U.S.A., V.48, no. 12 (1962), [8] Corduneau, C. Almost periodic functions. Interscience, New York (1968). [9] Gunzler, H. Integration of almost periodic functions. Math. Z., V. 12 (1967), [1] Kadets, M. I. The method of equivalent norms in the theory of abstract almost periodic functions. Studia Mathematica, 31 (1968), [11] Kadets, M. I. The integration of almost periodic functions with values in a Banach space. Functional Analysis and its Applications, 3 (1969), [12] Levitan, B. M. Integration of almost periodic functions with values in Banach spaces. (in Russian) Izv. Akad. nauk SSSR, 3 (1966). Translated in Amer. Math. Trans., [13] Loomis, L. H. The spectral characterization of a class of almost periodic functions. Ann. of Math., V.72 no.2 (196),

8 3622 S. M. A. Alsulam Received: March, 212