Abstract
This paper investigates the existence of denumerably many positive solutions and two infinite families of positive solutions for the n-dimensional higher-order fractional differential system mathbf{D}_{0^{+}}^{alpha}mathbf{x}(t)+lambda mathbf{g}(t)mathbf{f}(t,mathbf{x}(t))=0, 0< t<1. The vector-valued function x is defined by mathbf {x}=[x_{1},x_{2},dots,x_{n}]^{top}, mathbf{g}(t)=operatorname{diag}[g_{1}(t), g_{2}(t), ldots, g_{n}(t)], where g_{i}in L^{p}[0,1] for some pgeq 1, i=1,2,ldots, n, and has infinitely many singularities in [0,frac{1}{2}). Our methods employ the fixed point theorems combined with the partially ordered structure of a Banach space.
Highlights
Fractional differential equations, which provide a natural description of memory and hereditary properties of various materials and processes, are regarded as an important mathematical tool for better understanding of many real world problems in applied sciences, such as physics, chemistry, aerodynamics, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits and so on
This paper investigates the existence of denumerably many positive solutions and two infinite families of positive solutions for the n-dimensional higher-order fractional differential system Dα0+ x(t) + λg(t)f(t, x(t)) = 0, 0 < t < 1
1 Introduction Fractional differential equations, which provide a natural description of memory and hereditary properties of various materials and processes, are regarded as an important mathematical tool for better understanding of many real world problems in applied sciences, such as physics, chemistry, aerodynamics, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits and so on
Summary
Fractional differential equations, which provide a natural description of memory and hereditary properties of various materials and processes, are regarded as an important mathematical tool for better understanding of many real world problems in applied sciences, such as physics, chemistry, aerodynamics, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits and so on. Remark 3.2 It is well known that Lemma 3.3 and Lemma 3.4 have been instrumental in proving the existence of positive solutions to various boundary value problems for integerorder or fractional-order differential equations-for details; see Sect.
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