WATANABE, TOSHIKAZU

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RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:NOTE ON THE SOLVABILITY OF THE FULLYFOURTH ORDER NONLINEAR BOUNDARYVALUE PROBLEMS (New Developments onMathematical Decision Making UnderUncertainty)WATANABE, TOSHIKAZUWATANABE, TOSHIKAZU. NOTE ON THE SOLVABILITY OF THE FULLY FOURTH ORDER NONLINEAR BOUNDARY VALUEPROBLEMS (New Developments on Mathematical Decision Making Under Uncertainty). 数理解析研究所講究録 2022,2220: 114-1202022-05http://hdl.handle.net/2433/277153114NOTE ON THE SOLVABILITY OF THE FULLY FOURTH ORDER NONLINEAR BOUNDARY VALUE PROBLEMS TOKYO UNIVERSITY OF INFORMATION SCIENCES TOSHIKAZU WATANABE 1. INTRODUCTION For the existence and uniqueness of solutions for the fourth order two point boundary value problems, many researchers have studied, see [1, 8, 11, 14, 15, 16, 21, 22, 23, 19]. The proof is carried out using the method of Leray?-Schauder degree theory [18] or the Schauder fixed point theorem on the base of using the method of lower and upper solution [3, 5, 6, 17] or Fourier analysis [13]. It should be noticed that the following boundary value problems for the fully nonlinear fourth order equation with boundary conditions were studied in several authors in recent years, see [12, 10, 7, 17, 4], {u""(t) = f(t, u(t), u'(t), u"(t), u"'(t)), (1.1) u(0) = u'(0) = u"(l) = u"'(l) = 0, where f is a continuous nonlinear mapping of [0, 1] x JR4 into JR. It should be emphasized that in all the above works there is an essential assumption that the function f satisfies a Nagumo-type condition. In [20], this condition be freed by them due to considering f only in a bounded domain. In paticular, authors in [20] show that some examples demonstrate the applicability of the proposed ap-proach and iterative method. As the examples of boundary value problem (1.1) where f is a continuous nonlinear mapping of [0, 1] x JRi x ]R_ into lR+, they give 1. f t f(t ) - _llE_ + vv2 + 1 + JL :!!! ( + + - ) + 1 non mear unc ons .u, y, v, z 1152 576 4 54 , 6 u Y v z , Hu2+~tiv2+z2 + e-u2, ? ~ + uv + * + ½, which satisfy Theorem 2.2, 2.3 in [20]. Moreover they give also the function J(t.u, y, v, z) = u2+IYI ½ +lvl ½ +lzl ½ +1rsin('rrt), which does not satisfy Theorem 2.2, 2.3 in [20]. In this paper we proposed an one way of investigstion of method to obtain the positivity solution of boundary value problem (1.1). In particular we show that method of this paper is appricable for the function f(t, u, y, v, z) = u 2 + IYI ½ + lvl ½ + lzl ½ + 1rsin(1rt) where f is a continuous nonlinear mapping of [0, 1] x JRi x ]R_ into lR+. The paper is organized as follows. In Section 2, we consider the existence and uniqueness of a solution of the problem (1.1) with its positivity. Section 3 is devoted to some examples, including ones to which the condition of theorems in [20] cannot be applied, for demonstrating the applicability of our approach. Finally, Section 4 is conclusion. Throughout this paper, we denote by JR the set of all real numbers. A function u E C[0, 1] is called a solution of problem (1. 1) if u"" E C[0, 1], u"' E £ 1 [0, 1], u" E £ 1 [0, 1], u' E £ 1 [0, 1], and u satisfies the boundary conditions and equality in (1.1) a.e. on [0, 1]. The 2010 Mathematics Subject Classification. Primary 46A40, 47B50, 47Hl0. Key words and phrases. Fixed point theorem, fractional boundary value problem. 115physical meaning of the derivatives of the deformation function u(t) is as follows: u"" is the load density stiffness, u'" is the shear force stiffness, u" is the bending moment stiffness, and u' is the slope; see [1, 2, 8, 9]. 2. MAIN RESULT In this paper we consider the boundary value problem (1.1). For u E C[O, 1], we set cp(t) = f(t, u(t), u'(t), u"(t), u"'(t)). The problem becomes {u""(t) = cp(t),O < t < 1, u(O) = u'(O) = u"(l) = u111 (l) = 0. It has a unique solution u(t) = 11 G(t,s)cp(s)ds, where G(t s) = { ¼ (-s 3 + 3ts2), if OS s St, ' ¼ ( -t3 + 3st2 ) , if t S s < 1. It is easy to see that Also we have and where and u'(t) = 11 G1(t, s)cp(s)ds, G ( ) - aG(t,s) -{½s2 , ifO S s St, 1 ts --~~-' at ½ (-t2 + 2st), if t <:'. s < 1,. G ( ) - aG1(t,s) -{O, ifOSsSt, 2ts -----' at s - t, if t s s < 1, -G ( )- aG2(t,s) -{O, ifO S s St, 3 ts--~~-' at 1, if t s s < 1. It is easy verify that these functions are nonnegative and 11 1 11 1 max G(t, s)ds = -, max G 1 (t, s)ds = -, O:S:t:Sl o 8 O:S:t:Sl o 6 11 1 11 max G2(t, s)ds = -, max -G3(t, s)ds = 1. O:S:t:S:1 o 2 O:S:t:Sl o (2.1) 116We give the following settings. For <.p E C4 [0, 1], we set (2.2) y(t) = u'(t), v(t) = u"(t), z(t) = u111 (t). Clearly, the solutions u, y, v and z of the above problems depend on <.p, that is, u(t) = u'P(t), y(t) = y'P(t), v(t) = v'P(t), z(t) = z'P(t). Therefore, for <.p we have the equation <.p=A<.p, where A is a nonlinear operator defined by (2.3) (A<.p)(x) = f(x, u(t), y(t), v(t), z(t)). We shall prove that under some conditions A is contractive operator. Let M > 0 and t0 with O :::; t0 < ½- For the space of continuous functions C[O, 1], define the norm 11'-PII = max:099 l'-PI for <.p E C[O, 1] and consider the integral value ft~ <.p(s)ds. We assume that <.p(t) > 0 for any t E [O, 1]. We also assume that for M > 0 and t0 , there exists D = D(M, to) such that ii :::; ft~ <.p(s)ds :::; 11'-PII :::; M, in particular if t0 = 0, then we assume that D = 0. By [12], 82;2 :::; G(t, s) for any s, t E [O, 1] we have t4 D t411 t411 _Q__:::; _Q_ <.p(s)ds:::; _Q_ <.p(s)ds (2.4) 3 M 3 to 3 to :::; 11 G(t,s)<.p(s)ds = u(t):::; 11 G(t,s)dsll'-P(s)II:::; ~11'-P(s)II:::;; for any t E [O, 1]. By the definiton of G1 (s, t), we have t2 D t211 11 s2 _Q__:::; _Q_ <.p(s)ds:::; -<.p(s)ds 2 M 2 to to 2 (2.5) :::; 11 G1 (t,s)<.p(s)ds=y(t) (1 1 M :::; lo G1(t,s)dsll'-P(s)II:::; 611'-P(s)II:::; 6 for any t E [O, 1]. By the definiton of G2(s, t), since to< ½ and <.p(t) > 0 for any t E [O, 1], we have to D to 11 11 -- :::; - <.p(s)ds:::; (s - to)<.p(s)ds 2 M 2 to to (2.6) :::; 11 G2 (t, s)<.p(s)ds = v(t) (1 1 M :::; lo G2(t,s)dsll'-P(s)II:::; 211'-P(s)II:::; 2 for any t E [O, l]. Also by the definiton of G3 (s, t), D 11 11 M :::; <.p(s)ds:::; -G3(t, s)dsll'-P(s)II :::; M, to 0 117then we have (2.7) for any t E [0, 1]. Under these settings, we can define sn+ (t ) o:::; t:::; 1, lKJ":::; u:::; sM, l 115 :::; Y:::; 5M, { I t 4 D 1 t 2 D 1 } M= ,u,y,v,z toi'.Sv:S½M,-M:Sz'.S-M ' and we denote SBM = { cp E C[O, 1] IO< cp(t) :SM for any t E [O, 1], ! :S { 1 cp(t)dt}. Remark 1. If to= 0, then D = 0 so SDt_ and SBM coincide the following nt_ and SM in [20], respectively; + {( )I O:St:Sl,O:Su:S½M,O:Sy:S½M,} DM = t,u,y,v,z O '.S v '.S ½M,-M :S z '.S 0 and SM= {cp E C[O, 1] IO< cp(t) :SM for any t E [O, 1]}. In this case our setting coinside the one in [20]. Under these settings, we have the following lemma. Lemma 1. Assume that there exists M > 0, to with O < to < ½ and D D(M, t0 ) > 0 such that (2.8) 0 '.S f(t, u, y, v, z) '.S M, (2.9) D 11 M '.S f (t, u, y, v, z)dt, to for any (t, u, y, v, z) E SDt_ and also there exist C1, C2, C3 , C4 ~ 0 such that 4 (2.10) lf(t, x1, x2, X3, X4,) - f(t, Y1, Y2, Y3, Y4)1 :SL Cilxi - Yil i=l for any (t, x1, x2, x 3 , x4), (t, Y1, Y2, y3 , y4) E SDt_. Then the operator A defined by (2.3) maps the SBM into itself. Moreover, if (2.11) C1 C2 C3 q := - + - + - + C4 < 1 8 6 2 ' then A is contractive operator in SB M. 118Proof. Let cp E BM with cp(t) > 0 for any t E [0, l]. By (2.4), (2.5), (2.6) and (2.7), (2.12) t4 D (1 1 M --f Ms u(t) s lo G(t,s)dsllcp(s)II S sllcp(s)II S 8' t2 D (1 1 M -"§:MS y(t) S lo G1(t,s)dsllcp(s)II S 6ilcp(s)II S 6 , t0 D (1 1 M 2 MS v(t) S lo G2(t,s)dsllcp(s)II S 2 llcp(s)II S 2 , 11 D - MS z(t) S -cp(s)ds S - M' to Therefore (t, u, y, v, z) E SD"t fort E [0, 1]. In view of (2.3) and the condition (2.8), we have Acp E SBM, i.e. the operator maps SBM into itself. Now, let cp1, cp2 E SBM and u1, u2 be the solutions of the problem (1.1) for cp1, cp2 respectively. We also denote Yi= u~, Vi= u/, Zi = ut(i = 1, 2). Then, as induced above (t, ui, Yi, vi, zi) E SBM (i = 1, 2) fort E [0, 1]. Due to the estimates (2.12) we have 1 1 llu2 - u1II s 8llcp2 - cp1II, IIY2 -yill s 6llcp2 - 'Pill, 1 llv2 - viii s 2llcp2 - 'Pill, llz2 - zill s llcp2 - 'Pill-(2.13) Now from (2.3) and (2.10) it follows that IIAcp2 - Acp1 II = IIJ(t, u2, Y2, v2, z2) - f (t, u1, Y1, v1, z1) II S Collu2 -u1II + C1IIY2 -y1II + C2llv2 -viii+ C3llz2 - z1II-Using the estimate (2.13) we obtain IIAcp2-Acp1II s (~1 + ~ 2 + ~ 3 +c4) llcp2-cp1II-Therefore, A is a contractive operator in SBM provided the condition (2.11) is satisfied. The lemma is proved. D Theorem 2. Under the assumptions of Lemma 1, the problem {1.1} has a unique nonnegative solution u and there hold the estimates tt D II II M tl D II 'II M to D II "II M D II 111 11 - - < u < - - - < u < - -- < u < - - < u < M. 3M- - 8'2M- - 6'2M- - 2'M- -Proof. The proof is same as that of Theorem 2.3 in [20]. □ Remark 2. By the Remark 1, Theorem 2 includs the Theorem 2.3 in [20]. 3. EXAMPLES We consider the folllowing example of nonlinear function f for demonstrating the applicability of the obtained theoretical result, which does not satisfies the conditions in Theorem 2.2 in [20] and Theorem 3.1, 3.2 in [12]. Example 3. We consider the following boundary value problem: {u""(t) = u2 + lu'(t)l 1/2 + lu"(t)l 1/2 + lu111 (t)l 1/ 2 +7rsin(7rt),0 < t < 1, u(0) = u'(0) = u"(l) = u111 (l) = 0. 119In this example f(t, u, Y, v, z) = u2 + IYl 112 + lvl 112 + lzl 112 + 1r sin(1rt), which maps [0, 1] x JRt x l!L --+ lR+. In this case, by (2.12), if 14.435 < M < 35.718, then f(t,u,y,v,z) ::; M. Thus (2.8) is satisfied. We take t0 with O ::;?t0 < ½-Since f is continuous and positive, we take L(M, to) with O < L(M, to) ::; M and (M - L(M, to)) ::; ft: J(t, u, y, v, z). If we take D = M(M - L(M, to)), then we have i < ft: f(t, u, y, v, z). In particular if to = 0, then we take L(M, to) = M, that is D = 0. We take C1 = lful,C2 = lfyl,C3 = lfvl, and C4 = lfzl• Then we have 1 1 1 q = 8C1 + 6C2 + 2C3 + C4 1 1 1 = 81ful + 61fyl + 21fvl + lfzl 1 11 1 11 1 1 1 ::; 8 * 2 * lul + 6 * 2 * IYl1;2 + 2 * 2 * lvll/2 + 2 * lzll/2 < !*!M+~* ((2* (~)2) ( M ))1;2 - 4 8 12 to M(M - L(M, to)) + ~ * ( ( 2 * ~) * M(M -~(M,to))) 112 1 ( M ) 1/2 + 2 * M(M - L(M, to)) M l ( 2 ) 1/2 l ( 2 ) 1/2 l ( l ) 1/2 ::; 32 + 12 * t5(M - L(M, t0 )) + 4 * t0 (M - L(M, t0 )) + 2 * M - L(M, t0 ) Moreover if we take M = 16, t0 = ½, and L(M, t0 ) = 7, in this case D = 16 * 9, then we have D = 9::; J1 f(t,u,y,v,z)dt < 16. M 1/3 In fact if f(t, u, y, v, z) = 15, then we have ft: f(t, u, y, v, z)dt = 10 > 9 = ~- Thus (2.9) is satisfied. Moreover by the mean value theorem, we have (2.10). Finally we have 1 1 1 q = 81ful + 61fyl + 21fzl + lfwl < 16 +~* (2*32)1/2 +!* (2*3)1/2 +!* (!)1/2 - 32 12 9 4 9 2 9 ;::, 0.988642 < 1. 4. CONCLUSION In this paper we have proposed a method for investigating the solvability of solution of the cantilever beam equation with fully nonlinear term. In particuler for the contraction mapping principle to an operator equation for the function J(t,u(t),u'(t),u"(t),u"'(t)) = u2 + lu'(t)l 1/2 + lu"(t)l 1/2 + lu"'(t)1 1/ 2 + 1rsin(1rt) in [20], we have proposed a one way of investigation of method to apply to this 120function f. The proposed approach can be used for some other nonlinear boundary value problems for ordinary and partial differential equations. REFERENCES [1] A. R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value prob-lems, J. Math. Anal. Appl. 116 (1986) 415-426. [2] R.P. Agarwal, Boundary Value Problems for Higher Order Differential Equations, World Scientific, Singapore, 1986. [3] Z. Bai, The upper and lower solution method for some fourth-order boundary value problems, Nonlinear Anal. 67 (2007) 1704-1709. [4] A. Cabada, J. Fialho, M. 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Chang, Existence of solutions for a fully nonlinear fourth-order two-point bound-ary value problem, J. Appl. Math. Comput. 37 (2011) 287-295 [19] M. A. Del Pino, R. F. Manasevich, Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition, Proc. Amer. Math. Soc. 112 (1991) 81-86. [20] D. Quang A., N. Quyb, Existence results and iterative method for solving the cantilever beam equation with fully nonlinear term, Nonlinear Anal:Real World Applications, 36 (2017) 56-68. [21] R. A. Usmani, A uniqueness theorem for a boundary value problems, Proc. Amer. Math. Soc. 77 (1979) 329-335. [22] D. -B. Wang, J. -P. Sun Existence of a solution and a positive solution of a boundary value problem for a nonlinear fourth-order dynamic equation, Nonlinear Anal. 69 (2008) 1817-1823. [23] Y. Yang, Fourth-order two-point boundary value problems, Proc. Amer. Math. Soc. 104 (1988) 175-180. (Toshikazu Watanabe) MEIJI UNIVERSITY Email address: wa-toshi©mti. biglobe .ne. jp 

NOTE ON THE SOLVABILITY OF THE FULLY FOURTH ORDER NONLINEAR BOUNDARY VALUE PROBLEMS (New Developments on Mathematical Decision Making Under Uncertainty)

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NOTE ON THE SOLVABILITY OF THE FULLY FOURTH ORDER NONLINEAR BOUNDARY VALUE PROBLEMS (New Developments on Mathematical Decision Making Under Uncertainty)

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