Lyapunov-type inequalities for an anti-periodic fractional boundary value problem involving \u03c8-Caputo fractional derivative

Bessem Samet,Hassen Aydi

doi:10.1186/s13660-018-1850-4

Abstract

A Lyapunov-type inequality is established for the anti-periodic fractional boundary value problem \t\t\t(CDaα,ψu)(x)+f(x,u(x))=0,a<x<b,u(a)+u(b)=0,u′(a)+u′(b)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} & \\bigl({}^{C}D_{a}^{\\alpha,\\psi}u \\bigr) (x)+f \\bigl(x,u(x) \\bigr)=0,\\quad a< x< b, \\\\ &u(a)+u(b)=0,\\qquad u'(a)+u'(b)=0, \\end{aligned}$$ \\end{document} where (a,b)inmathbb{R}^{2}, a< b, 1<alpha<2, psiin C^{2}([a,b]), psi'(x)>0, xin[a,b], {}^{C}D_{a}^{alpha,psi} is the ψ-Caputo fractional derivative of order α, and f: [a,b]timesmathbb{R}tomathbb{R} is a given function. Next, we give an application of the obtained inequality to the corresponding eigenvalue problem.

Highlights

1 Introduction In this paper, we are concerned with the anti-periodic fractional boundary value problem
As an application of the obtained inequality, an upper bound of possible eigenvalues of the corresponding problem is obtained
From the proof of Theorem 3.1, the function v given by (3.2) satisfies (3.3) and the boundary conditions ψ (b) v(A) = v(B) = 0 and v (A) +

Summary

Introduction

We are concerned with the anti-periodic fractional boundary value problem. U(a) + u(b) = 0, u (a) + u (b) = 0, where (a, b) ∈ R2, a < b, 1 < α < 2, ψ ∈ C2([a, b]), ψ (x) > 0, x ∈ [a, b], CDaα,ψ is the ψ-Caputo fractional derivative of order α, and f : [a, b] × R → R is a given function. A Lyapunov-type inequality is derived for problem (1.1). As an application of the obtained inequality, an upper bound of possible eigenvalues of the corresponding problem is obtained. Suppose that u ∈ C2([a, b]), (a, b) ∈ R2, a < b, is a nontrivial solution to the boundary value problem u (x) + w(x)u(x) = 0, a < x < b, (1.2).

Methods

Results

Conclusion