WATANABE, TOSHIKAZU

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RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:SOLUTIONS FOR A FRACTIONAL-ORDER DIFFERENTIAL EQUATIONWITH BOUNDARY CONDITIONS(Study on Nonlinear Analysis andConvex Analysis)WATANABE, TOSHIKAZUWATANABE, TOSHIKAZU. SOLUTIONS FOR A FRACTIONAL-ORDER DIFFERENTIAL EQUATION WITH BOUNDARYCONDITIONS (Study on Nonlinear Analysis and Convex Analysis). 数理解析研究所講究録 2021, 2194: 178-1822021-08http://hdl.handle.net/2433/265734178SOLUTIONS FOR A FRACTIONAL-ORDER DIFFERENTIAL EQUATION WITH BOUNDARY CONDITIONS MEIJI UNIVERSITY TOSHIKAZU WATANABE 1. INTRODUCTION In [5], the authors considered the fractional-order boundary value problem, we consider existence and uniqueness of solutions of the fractional-order boundary value problem (1.1) {Dtf+u(t) = f(t, u(t)), 0 < t < 1, u(0) = u'(0) = u"(l) = u"'(l) = 0. Inspiring the result of Ma and Silva [4], we consider the problem (1.2). The problem (1.1) is the case that g = 0 of the problem (1.2). Due to the restriction of g, results in [5] cannot deal with the problem (1.3). where 3 < a :S 4, f is a continuous function of [0, 1] x ~ into ~, g is a function of ~ into itself and Dtf+ denotes the Riemann-Liouville fractional derivative of order a which is defined in Section 2. A function u E C[0, 1], where C[0, 1] is the set of all continuous functions of [0, 1] into~, is called a solution of the problem (1.1) if Dtf+u E C[0, 1] and u satisfies (1.1). (1.2) {Dtf+u(t) = f(t, u(t)), 0 < t < 1, u(0) = u'(0) = u"(l) = 0, u111 (1) = g(u(l)), A function u E C[0, 1], where C[0, 1] is the set of all continuous functions of [0, 1] into ~, is called a solution of the problem (1.2) if Dtf+u E C[0, 1] and u satisfies (1.2). When a = 4, the problem (1.1) is the boundary value problems for cantilever beam equations and the problem (1.2) is the following fourth-order boundary value problem (1.3) {u""(t) = f(t, u(t)), 0 < t < 1, u(0) = u'(0) = u"(l) = 0, u111 (l) = g(u(l)). In the problem (1.3), since u" represents the shear force at t = 1, the sonditions u"(l) = 0, u111 (1) = g(u(l)) means that the vertical force is equal to, which denotes a relation, possibly nonlinear, between the vertical force and the displacement u. Furthermore, since u"(l) = 0 indicates that there is no bending moment at t = 1, the beam is resting on the bearing g. Existence and iterative schemes to solve the problem (1.3) were studied by Ma and da Silva [4]. The purpose of this article is to follow the results of the fractional-order problems (1.1) and (1.2). 2010 Mathematics Subject Classification. Primary 46A40, 47B50, 47H10. Key words and phrases. Fixed point theorem, fractional boundary value problem. 1792. PRELIMINARIES In this section, we introduce preliminary facts. Especially, we construct the Green function G(s, t) for the boundary value problems (1.1) and (1.2), and we discuss some properties of the function. We start with the definition of the Riemann-Liouville fractional integral and fractional derivative. Let a > 0 and u be a continuous function of [O, 1] into R The Riemann-Liouville fractional integral of order a of u, denoted Iff+u, is defined by for O ::; t ::; 1. The Riemann-Liouville fractional derivative of order a of u, denoted Dif+ u, is defined by a ( ) 1 dn ft ( )n-a-1 ( )d Do+U t = r(n - a) dtn lo t - s u s s for O ::; t ::; 1, where n denotes a positive integer such that n - l < a ::; n. For a 2: 0 and f3 > -l, we have see [3]. For the case of (1.2), let h be a continuous mapping of [O, 1] into R Let 3 < a ::; 4. Then the unique solution of the boundary value problem is where (2.1) G(t, s) u(t) = fo 1 G(t, s)h(s)ds, see [5]. Also for the case of (1.2) {Dif+u(t) = h(t), 0 < t < 1, u(O) = u'(O) = u"(l) = 0, u111 (l) = 1 if and only if u is a solution of the integral equation [1 ,ta-1 ,ta-2 (2.2) u(t) = lo G(t, s)h(s)ds + (a_ l)(a _ 2) - (a_ 2)(a _ 3) for O::; t::; 1, where G(t, s) is defined by (2.1); see [6]. 1803. MAIN RESULTS In this section, we consider the boundary value problems (1.1) and (1.2). By the Banach fixed point theorem, we obtain a sufficient condition for uniqueness and existence of solutions of the problems. Theorem 1. Let 3 < a ::::'. 4. Let f be a continuous function of [O, 1] x "JR. into R Let g be a Lipschitz continuous function of "JR. into itself with a nonnegative constant L. Assume that there exists a nonnegative constant >. with >.A+ 2L < 1 (a - l)(a - 2)(a - 3) such that for any O :::; t :::; 1 and u1, u2 E "JR., where A is the constant A= sup [1 IG(t, s)lds. O:St:Sl lo Then the boundary value problem (1.2) has a unique solution. Proof. See [6, Theorem 1]. □ For the case that g = 0 in Theorem 1, we have the following; see [5, Theorem 3.1]. Corollary 2. Let 3 < a ::::'. 4. Let f be a continuous mapping of [O, 1] x "JR. into R Assume that there exists >. E [ 0, ¾) such that for any u, v E [O, oo) and t E [O, 1], where lf(t, u) - f(t, v)I ::::'. >.iu - vl, A= sup [1 IG(t, s)lds o:<,;t9 Jo and G is the function given by (2.1). Then the boundary value problem represented by (1.1) has a unique solution. For the case that a= 4 in Theorem 1, we have the following; see Theorem 1 in [4]. Corollary 3. Let f be a continuous function of [O, 1] x "JR. into "JR. with bounded partial derivative with respect to the second variable. Let g be a Lipschitz continuous function of "JR. into itself with a nonnegative constant L. Let >. = max -(t,u) . 18f I (t,u)E[O,l]x!R OU If >. L 8 + 3 < l, then the boundary value problem (1.3) has a unique solution. 181Proof. By the mean value theorem, we have for any 0 :::; t ::=; 1 and u1, u2 E IR, lf(t, u1) - f(t, u2)I ::=:: -Alu1 - u2I-In the case that a= 4, the function G(t, s) reduces to Since G(t s) = {½s2(3t- s) (s < t), ' ½t2(3s - t) (t:::; s). A= sup [1 IG(t, s)lds = -81 , o:s;t::;i lo we obtain the conclusion by Theorem 1. Remark 4. Let 3 <a::=; 4. Since the function G(t, s) satisfies (3.1) [1 1 (1 a ) lo IG(t, s)lds:::; r(a) -; + a - 3 for all 0:::; t ::=; 1, SUPo:s;t::;i f01 IG(t, s)lds is finite. The function G(t, s) satisfies l(t, s):::; G(t, s):::; m(t, s) for 0 :::; t ::=; 1 and 0 :::; s < 1. where and see [6]. {- 1-t"'-2(1- s)°'-4 (2s + st - t) l(t, s) = ;~)2 --t°'-2(1 - s)°'-4s r(a) (s < t), (t:::; s) □ To conclude the paper, we present an example demonstrating an application of Theorem 1. Example 5. Let us consider the boundary value problem (3.2) {DgJu(t) = (54et+1)(Hlu(t)I)' 0 < t < 1, u(0) = u'(0) = u"(l) = 0, u111 (1) = g(u(l)), where 1 . g(t) = l00 smt. By (3.1), the constant A in Theorem 1 satisfies A < _l_ (~ + -°'-) = 14.2530 · · · < 15. - f(a) a a-3 Moreover we have, for any 0 :::; t ::=; 1 and u1, u2 E IR, 5 where 3 lf(t,u1)-f(t,u2)I ::=:: 55 lu1 -u2I, 3 f(t, u) = (54et + 1)(1 + lul) 182for 0 ~ t ~ 1 and u E IR; see Section 4 in [2]. Since the constants >. L = l6o in Theorem 1, we have l5 and 2£ 3 2 X 1010 . . >.A+ ------- < - x 15 + ------"-"-"--- = 0.904761 < 1. (a - l)(a - 2)(0: - 3) 55 2.1 x 1.1 x 0.1 It follows from Theorem 1 that the problem (3.2) has a unique solution. Example 6. We also consider the following. (3.3) where { D5Ju(t) = 50e•(i+u2 ), 0 < t < 1, u(0) = u'(0) = u"(l) = 0, u111 (1) = g(u(l)), 1 . g(t) = l00 smt. For any 0 ~ t ~ 1 and u 1,u2 E IR, where 3 IJ(t,u1)-f(t,u2)I ~ 50 iu1 -u2I, 3 f(t, u) = 50et(l + u2) for 0 ~ t ~ 1 and u E R In this case we also have 2£ 3 2x-1->.A + -------- < - x 15 + lOO :::::,; 0.9866 < 1. (a - l)(a - 2)(0: - 3) - 50 2.1 x 1.1 x 0.1 REFERENCES [1] Z. Bai and H. Lii, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311, No.2 (2005), 495-505. [2] M. Benchohra, N. Hamidi and J. Henderson, Fractional differential equations with anti-periodic boundary condition, Numer. Funct. Anal. Optim. 34, No.4 (2013), 404?-414. [3] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, In North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006. [4] T. F. Ma and J. da Silva, Iterative solutions for a beam equation with nonlinear boundary conditions of third order, Appl. Math. Comput. 159, No.1 (2004), 11 ?-18. [5] M. Toyoda and T. Watanabe, Note on solutions of boundary value problems involving a fractional differential equation, Linear Nonlinear Anal. 3, No.3 (2017), 449?-455. [6] M. Toyoda and T. Watanabe, Solutions for a fractional-order differential equation with bound-ary conditions of third order, International Journal of Applied Mathematics Volume 33 No. 3 (2020), 403-412. (Toshikazu Watanabe) MEIJI UNIVERSITY Email address: wa-toshi©mti. biglobe .ne .jp 

SOLUTIONS FOR A FRACTIONAL-ORDER DIFFERENTIAL EQUATION WITH BOUNDARY CONDITIONS (Study on Nonlinear Analysis and Convex Analysis)

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SOLUTIONS FOR A FRACTIONAL-ORDER DIFFERENTIAL EQUATION WITH BOUNDARY CONDITIONS (Study on Nonlinear Analysis and Convex Analysis)

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