A Cauchy Problem for Some Local Fractional Abstract Differential Equation with Fractal Conditions

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1 From the SelectedWorks of Xiao-Jun Yang 2013 A Cauchy Problem for Some Local Fractional Abstract Differential Equation with Fractal Conditions Yang Xiaojun Zhong Weiping Gao Feng Available at:

2 J. APPLIED FUNCTIONAL ANALYSIS, VOL. 8, NO. 1, 92-99, COPYRIGHT 2013 EUDOXUS PRESS, LLC A CAUCHY PROBLEM FOR SOME LOCAL FRACTIONAL ABSTRACT DIFFERENTIAL EQUATION WITH FRACTAL CONDITIONS WEIPING ZHONG, XIAOJUN YANG, AND FENG GAO Abstract. Fractional calculus is an important method for mathematics and engineering. In this paper, we review the existence and uniqueness of solutions to the Cauchy problem for the local fractional di erential equation with fractal conditions D x (t) = f (t; x (t)) ; t 2 [0; T ] ; x (t 0 ) = x 0 ; where 0 < 1 in a generalized Banach space. We use some new tools from Local Fractional Functional Analysis [25, 26] to obtain the results. 1. Introduction In this paper, the some properties of the solution of the local fractional abstract di erential equation d x (1.1) dt = f (t; x) ; x (t 0 ) = x 0 where 2 (0 ; 1], d dt is the local fractional operator [25,26], f (t; x) is a given function and both f (t; x) and x (t) are a non-di erential function, have been the subject many investigation. Local fractional calculus has revealed as one of useful tools in areas ranging from fundamental science to engineering [25-55]. It has gained importance and popularity during the past more than ten years, due to dealing with the fractal and continuously non-di erentiable functions in the real world. The theory of local fractional integrals and derivatives was successfully applied in fractal elasticity [40-41], local fractional Fokker Planck equation [34], local fractional transient heat conduction equation [42], local fractional di usion equation [42], relaxation equation in fractal space [42], local fractional Laplace equation [45], fractal-time dynamical systems [31], local fractional partial di erential equation [45], fractal signals [43,50], fractional Brownian motion in local fractional derivatives sense [39], fractal wave equation [53], Yang-Fourier transform [43,45,51,52], Yang-Laplace transform [45,47,51,53], discrete Yang-Fourier transform [46, 54], fast Yang-Fourier transform [48], local fractional Stieltjes transform in fractal space [44], local fractional Z transform in fractal space [51], local fractional short time transforms [25,26], local fractional wavelet transform [25, 26], and local fractional functional analysis [25,26,49]. Key words and phrases. Fractional analysis, local fractional di erential equation, generalized Banach space, local fractional functional analysis AMS Math. Subject Classi cation. 26A33; 28A80; 34G

3 2 W. ZHONG, X. YANG, AND F. GAO Based on the generalized Banach space [25, 26], the main aim of this paper is to show the existence and uniqueness of solutions to the Cauchy problem for the local fractional di erential equation with fractal conditions. The organization of this paper is as follows. In section 2, the preliminary results on the local fractional calculus and the generalized spaces are discussed. The existence and uniqueness of solutions to the Cauchy problem for the local fractional di erential equation with fractal conditions is investigated in section 3. Conclusions are in section Preliminaries 2.1. Local fractional continuity of functions. De nition 2.1. If there exists [25,26,47,49,50] (2.1) jf (x) f (x 0 )j < " with jx x 0 j <,for "; > 0 and "; 2 R, nowf (x) is called local fractional continuous at x = x 0, denote by lim f (x) = f (x 0 ) : x!x 0 Then f (x) is called local fractional continuous on the interval (a; b), denoted by (2.2) f (x) 2 C (a; b) : 2.2. Local fractional integrals. De nition 2.2. Let f (x) 2 C (a; b). Local fractional integral of f (x) of order in the interval [a; b] is given [25; 26; 47; 49; 50] (2.3) ai () b f (x) = 1 (1+) = 1 (1+) R b a j=n 1 lim t!0 j=0 f (t) (dt) ; P f (t j ) (t j ) where t j = t j+1 t j,t = max ft 1 ; t 2 ; t j ; :::g and [t j ; t j+1 ], j = 0; :::; N 1,t 0 = a; t N = b, is a partition of the interval [a; b]. For convenience, we assume that ai a () f (x) = 0 if a = b and a I () b f (x) = bi a () f (x) if a < b. For any x 2 (a; b), we get ai x () f (x) ; denoted by f (x) 2 I x () (a; b) : Remark 2.1. If I x () (a; b), we have that f (x) 2 C (a; b) : Theorem 2.3. (See [25; 26]) Suppose that f (x) 2 C [a; b], then there is a function y (x) = a I () x f (x), the function has its derivative with respect to (dx), (2.4) d y (x) dx = f (x) ; a < x < b: 93

4 A CAUCHY PROBLEM FOR SOME LOCAL FRACTIONAL ABSTRACT DIFFERENTIAL EQUATION3 Theorem 2.4. (Existence Theorem) Let f(x; y) be local fractional continuous and bounded in the strip T = f(x; y) : jx x 0 j a; kf (x; y) f (x; y 0 )k L ky y 0 k ; L > 0g : Then the Cauchy value problem (1) has at least one solution injx x 0 j a Local fractional derivative. De nition 2.5. Let f (x) 2 C (a; b). Local fractional derivative of f (x) of order at x = x 0 is given [25,26,47,49,50] (2.5) f () (x 0 ) = d f (x) (f (x) f (x 0 )) dx j x=x0 = lim x!x 0 (x x 0 ) ; where (f (x) f (x 0 )) = (1 + ) (f (x) f (x 0 )). For any x 2 (a; b), there exists f () (x) = D x () f (x) ; denoted by f (x) 2 D x () (a; b) : 2.4. Generalized Banach spaces. De nition 2.6. (Generalized Banach space) (See [25; 26]) Let X be a generalized normed linear space. Since X is complete, the Cauchy sequence fx ng 1 n=1 is convergent; ie for each " > 0 there exists a positive integer N such that (2.6) kx n x mk < " whenever m; n N. This is equivalent to the requirement that (2.7) lim m;n!1 kx n x mk = 0: A complete generalized normed linear space is called a generalized Banach space. There is an open ball in a generalized Banach space X: B (x 0 ; r) = fx 2 X : kx x 0 k < r g with r > 0. De nition 2.7. (Boundary of the fractal domain) (See [25; 26]) A set F in a generalized Banach space X is bounded if F is contained in some ball B (x 0 ; r) with r > 0. De nition 2.8. (Local fractional continuity) (See [25; 26]) The function f (x) with domain D is local fractional continuous at a if (i) the point a is in an open interval I contained in D, and (ii) for each positive number " there is a positive number such that jf (x) f (x 0 )j < " whenever jx x 0 j < and 0 < 1. If a function f (x) is said in the space C [a; b] if f (x) is called local fractional continuous at [a; b]. De nition 2.9. (Local fractional uniform continuity) (See [25; 26]) A function f (x) with domain D is said to be local fractional uniformly continuous on D if for each positive number " there is a positive number such that jf (x 1 ) f (x 2 )j < " whenever jx 1 x 2 j <, x 1 ; x 2 2 D and 0 < 1. De nition (Convergence in fractal set) (See [25; 26]) A sequence fx ng of fractal setf of fractal dimension,0 < 1, is said to converge to x, if given any neighborhood of x, there exists an integer m, such that x n 2 F whenever n m. 94

5 4 W. ZHONG, X. YANG, AND F. GAO De nition (Cauchy sequence in fractal set) (See [25; 26]) A sequence fx ng in a generalized Banach space X is a Cauchy sequence if for every " > 0 there is a positive integer N such thatkx n x mk < " whenever n; m > N Generalized linear operators. To begin with we give the de nition of a generalized linear operator (See [25; 26]). De nition (Generalized linear operator)(see [25; 26]) Let X and Y be generalized linear spaces over a eld F and let T : X! Y. If (2.8) T (ax + by ) = at (x ) + bt (y ) ; 8x ; y 2 X; 8a; b 2 F: We say T is a generalized linear operator or a generalized linear transformation from X into Y. Also, we write (2.9) T (X) = ft (x ) : x 2 Xg :: The local fractional di erential operator D is a generalized linear operator [25, 26]: (2.10) D (1 + ) [f (x) f (x 0 )] f (x) = lim x!x 0 (x x 0 ) : The local fractional integral operator I is a generalized linear operator [25, 26]: (2.11) I f (x) = 1 (1 + ) Z x a f (x) (dx) : 2.6. Contraction mapping on a generalized Banach space. De nition (Contraction mapping on a generalized Banach space) (See [25; 26]) Let X be a generalized Banach space, and let T : X! X. If there exists a number 2 (0; 1) such that (2.12) kt (x ) T (y )k kx y k for all x ; y 2 X. We say that T is a contraction mapping on a generalized Banach space X. It is remarked that the above de nition is equal to [25,26], which is referred to fractional set theory [26,55]. Theorem (See [25; 26]) Let X be a generalized Banach space. A convergent sequence in X may have more than one limit in X : Theorem (Contraction Mapping Theorem in Generalized Banach Space) (See [25; 26]) A contraction mapping T de ned on a complete generalized Banach space X has a unique xed point. Theorem (Generalized Contraction Mapping Theorem in Generalized Banach Space) Suppose that T : X! X is a map on a generalized Banach space X such that for some m 1,T m is a contraction, ie., kt m (y ) T m (x )k kx y k for allx ; y 2 X; 2 (0; 1). Then T has a unique xed point. 95

6 A CAUCHY PROBLEM FOR SOME LOCAL FRACTIONAL ABSTRACT DIFFERENTIAL EQUATION5 Proof. By Theorem 4, T m has a unique xed point x 0. Take into account kt x 0 x 0 k = T (2.13) m+1 x 0 T m x 0 = kt m (T x 0 ) T m x 0 k kt x 0 x 0 k Hence kt x 0 x 0 k = 0 and thus x 0 is a xed point of T. If x 0;0; x 0;1 are xed points of T, they are xed points of T m and so x 0;0 = x 0;1. (3.1) 3. Existence and uniqueness solution to the local fractional abstract differential equation For the given equation d x dt = f (t; x) x (t 0 ) = x 0 form Theorem 1 and Theorem 2 we have that Z 1 t (3.2) x = x 0 + f (t; x) (dt) ; (1 + ) t 0 where kf (x 1 ; t) f (x 0 ; t)k k kx 1 x 0 k. Hence, by Theorem 2.4. we give the existence of solution to the local fractional abstract di erential equation. Furthermore, we suppose that the map T : X! X de ned by (3.3) T (x (t)) = x 0 + We claim that for all n, 1 (1 + ) Z t t 0 f (x; t) (dt) (3.4) kt n (x 1 (t)) T n (x 0 (t))k k n jt t 0j n The proof is by induction on n. The case n = 0 is trivial. When n = 1, we have that (3.5) kt (x 1 (t)) T (x 0 (t))k k jt t 0j (1 + n) kx 1 x 0 k : (1 + ) kx 1 x 0 k : The induction step is as follows: T n+1 (x 1 (t)) T n+1 (x 0 (t)) R 1 t = (1+) t 0 f (t; T n x 1 (t)) f (t; T n x 0 (t)) (dt) R 1 t (1+) t (3.6) 0 k kf (t; T n x 1 (t)) f (t; T n x 0 (t))k (dt) R 1 t k (n+1) jt t 0j n (1+) t 0 (1+n) kx 1 x 0 k (dt) R t t 0j n x 0 k (dt) We have k (n+1) jt 1 (1+) k (n+1) jt t 0j (n+1) t 0 k (n+1) jt (1+n) kx 1 t 0j (n+1) (1+(n+1)) kx 1 x 0 k (1+(n+1)) kx 1 x 0 k! 0 as n! 0. So far n su ciently large, (3.7) 0 < k (n+1) jt t 0j (n+1) (1 + (n + 1) ) < 1 96

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9 8 W. ZHONG, X. YANG, AND F. GAO [51] Y. Guo, Local fractional Z transform in fractal space, Adv. Digital Multimedia, 1(2), (2012). [52] W.P. Zhong, F. Gao, X.M. Shen, Applications of Yang-Fourier transform to local Fractional equations with local fractional derivative and local fractional integral, Adv. Mat. Res., 416, (2012). [53] W. P. Zhong, F. Gao, Application of the Yang-Laplace transforms to solution to nonlinear fractional wave equation with local fractional derivative. In: Proc. of the rd International Conference on Computer Technology and Development, ASME, 2011, pp [54] X.J. Yang, A new viewpoint to the discrete approximation discrete Yang-Fourier transforms of discrete-time fractal signal, ArXiv: v1[math-ph], [55] G. S. Chen, A generalized Young inequality and some new results on fractal Space, Adv. Comput. Math. Appl., 1(1), 56 59(2012). (W. Zhong) State Key Laboratory for GeoMechanics and Deep Underground Engineering, China University of Mining & Technology, Jiangsu, P.R. China School of Mechanics & civil Engineering, China University of Mining & Technology, Jiangsu, P.R. China address: wpzhong@cumt.edu.cn (X. Yang) Department of Mathematics & Mechanics, China University of Mining & Technology, Xuzhou Campus, Xuzhou, Jiangsu, P. R. China Shanghai YinTing Metal Product Co. Ltd, Minfa Road No. 698, Songjiang district, Shanghai, P. R. China address: dyangxiaojun@163.com (F. Gao) School of Mechanics & civil Engineering, China University of Mining & Technology, Jiangsu, P.R. China State Key Laboratory for GeoMechanics and Deep Underground Engineering, China University of Mining & Technology, Jiangsu, P.R. China address: fgao@cumt.edu.cn 99