分数阶脉冲半线性发展方程和一类分数阶p-Laplacian奇异边值问题解的研究

发布时间：2018-07-11 17:41

本文选题：半线性发展方程 + 积分-微分方程　；参考：《曲阜师范大学》2017年硕士论文

【摘要】：近年来,分数阶微分方程被广泛应用于光学和热学系统,电磁学,控制和机器人等诸多领域,已经引起国内外数学及自然科学界的高度重视.非线性分数阶微分方程解的存在性研究是国际热点研究方向之一.非线性分数阶微分方程是数学中的一个既有深刻理论意义又有广泛应用价值的研究方向.本文主要利用非线性泛函分析理论和方法研究分数阶脉冲半线性发展方程和一类p-Laplacian奇异边值问题解的存在性和解的性质.本文共分为以下三章:第一章,我们运用C_0半群和Banach压缩映射原理研究Bauach空间(E，‖·‖)上的分数阶脉冲半线性发展方程适度解的存在性其中0 q 1, 是Caputo型分数阶导数,A是C0半群(G(t))t≥0的无穷小生成元,0 t1 t2 … tm T0, f ∈ C(J × E×E, E), Ik ∈ C(E, E) (k = 1,2, …m), u0 ∈ E.T是线性算子(Tu)(t)=∫0t k(t,s)u(s)ds,t∈J,其中 kC ∈ (-∞,+∞),D={(t,s)∈J×J:t≥s}.△u|t=tk=u(tk+) -u(tk-), 其中u(tk-)和u(tk+)分别代表u(t)在t=tk处的左极限和右极限.第二章,我们研究Banach空间(E,‖·‖)上的混合型分数阶脉冲半线性积分-微分方程非局部问题适度解的存在性其中0 q 1,是Caputo型分数阶导数,A是C0半群(G(t))t0的无穷小生成元,0 tx t2 … tm T0, f ∈ C(J × E × E × E, E), Ik ∈ C(E, E) (kk = 1, 2, …m),g∈ (J,E], E),u0 ∈ E. T 和 S 是线性算子(Tu)(t)=∫0t k(t,s)u(s)ds,(Su)(t)=∫0T0h(t, s)u(s)ds, t∈J,其中 k∈ ∈C(D,(-∞,+∞)),D= {{(t,s∈ J × J : t ≥s }, ∈C(J ×Jt, (-∞,+∞O).△u|t=tk=u(tk+)-u(tk),其中u(tk-)和u(tk+)分别代表u(t)在t= k处的左极限和右极限.第三章,我们运用上下解方法和不动点理论研究一类分数阶p-Laplacian奇异边值问题解的存在性其中 α ∈ (1,2], β∈(3,4],D_(0+)~α,D_(0+)~β是 Riemann-Liouville型分数阶导数,f ∈C((0,1)×(0, +∞),[0, +∞)),f(t,u)不仅允许在t=0和/或t = 1奇异,而且允许在u = 0处奇异,Φ_p(s) = |s|p-2s, p1, Φ_p~(-1)=Φ_q,1/p + 1/q = 1, η ∈ (0,1), b ∈ (0,η(1-α/p-1)).
[Abstract]:In recent years, fractional differential equations have been widely used in many fields such as optical and thermal systems, electromagnetics, control and robots, which have aroused great attention in the field of mathematics and natural sciences at home and abroad. The existence of solutions of nonlinear fractional differential equations is one of the international hot research directions. The number of nonlinear fractional differential equations is the number of the nonlinear fractional differential equations In this paper, we mainly use the theory and method of nonlinear functional analysis to study the existence and conciliatory properties of the fractional pulse semilinear development equation and the solution of a class of p-Laplacian singular boundary value problems. This paper is divided into the following three chapters: Chapter 1, we use C_0 The principle of semigroups and Banach compression mapping studies the existence of the moderate solution of the fractional pulse semilinear development equation in the Bauach space (E), which is 0 Q 1, Caputo type fractional derivative, A is the infinitesimal generator of C0 Semigroups (G (T)) t > 0, 0 T1 T2... TM T0, f C (J x E * E, E), Ik C C (E, E) (= = =,... M), U0 E.T is a linear operator (T) = 0t K (T, s) U (s) ds. The existence of a moderate solution for the nonlocal problem of a partial differential equation is 0 Q 1, a Caputo type fractional derivative, and A is the infinitesimal fraction of the C0 semigroup (G (T)) T0, and 0 TX T2... TM T0, f f C (J x E * E * E, E), Ik C C (E), (= = 1, 2,... And G sum and sum sum and sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sums sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum sum the left and right poles are represented respectively In the third chapter, we use the upper and lower solutions and the fixed point theory to study the existence of the solution of a class of fractional p-Laplacian singular boundary value problems, in which 1,2], 3,4], D_ (0+) ~ (0+), D_ (0+) ~ beta are Riemann-Liouville type fractional derivatives, f C ((0,1) * (0, + infinity), [0, + infinity). It is allowed to be singular at u = 0, _p (s) = |s|p-2s, P1, _p~ (-1) = _q, 1/p + 1/q = 1, ETA (0,1), 1/q (0, 1/p).
【学位授予单位】：曲阜师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8

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