Lyapunov-type inequalities for mixed non-linear forced differential equations within conformable derivatives

Abstract

We state and prove new generalized Lyapunov-type and Hartman-type inequalities for a conformable boundary value problem of order with mixed non-linearities of the form satisfying the Dirichlet boundary conditions , where , , and g are real-valued integrable functions, and the non-linearities satisfy the conditions . Moreover, Lyapunov-type and Hartman-type inequalities are obtained when the conformable derivative is replaced by a sequential conformable derivative , . The potential functions , as well as the forcing term g require no sign restrictions. The obtained inequalities generalize some existing results in the literature.

On stable iterative solutions for a class of boundary value problem of nonlinear fractional order differential equations

“…Obviously in Equation 21, the sequence e * n → 0 as n → ∞, which implies the following conclusion. According to the Lemma 11, the iterative sequences for BVP 1 are constructed as…”

“…[9][10][11][12][13][14][15][16][17] In last few years, new kinds of fractional derivatives including nonsingular and Mittag-Leffler type kernels have been introduced. Also, some authors have given attention to study conformable fractional order derivatives and their applications, for detail see the studies of Abdeljawad et al [18][19][20][21] The aforesaid results were based on classical fixed point theorems and nonlinear functional analysis. As far, we know the aforesaid area has been greatly developed.…”

Conformable Euler's theorem on homogeneous functions

“…Commonly, these definitions are not known to many researchers and, until recent years, it has been used in a purely mathematical context only. However, during the last decades, these kinds of derivative operators have been applied to many science context due, in part, to their frequent appearance in various applications in the fields of viscoelasticity, fluid mechanic, biology, physic entropy theory, and engineering . Nowadays, many definitions of fractional derivative have been introduced, but most of them are in integral form, which is more complicated to manage.…”

“…However, during the last decades, these kinds of derivative operators have been applied to many science context due, in part, to their frequent appearance in various applications in the fields of viscoelasticity, fluid mechanic, biology, physic entropy theory, and engineering. [1][2][3][4][5][6][7][8] Nowadays, many definitions of fractional derivative have been introduced, but most of them are in integral form, which is more complicated to manage. The most known are the Riemann-Liouville definition and the Caputo definition, those are both defined using integral forms as follows.…”

Generalization of Hermite-Hadamard Type Inequalities via Conformable Fractional Integrals

“…Recently, the conformable integrals and derivatives have attracted the attention of many researchers, and many remarkable properties and inequalities for the conformable integrals and derivatives can be found in the literature [17][18][19][20][21][22][23][24]. Anderson [14] found the conformable integral version of the Hermite-Hadamard inequality as follows.…”