Existence of Positive Solution for the Eighth-Order Boundary Value Problem Using Classical Version of Leray–Schauder Alternative Fixed Point Theorem

Abstract

In this work, we investigate the existence of solutions for the particular type of the eighth-order boundary value problem. We prove our results using classical version of Leray–Schauder nonlinear alternative fixed point theorem. Also we produce a few examples to illustrate our results.

Two efficient methods for solving fractional Lane–Emden equations with conformable fractional derivative

“…Problem 1 Set the initial conditions y(0) = 1 and y (0) = 0 into Eq. (16), then the general fractional solution of Eq. (16) in terms of Adomian polynomials is given by Eqs.…”

“…(16), then the general fractional solution of Eq. (16) in terms of Adomian polynomials is given by Eqs. (11), (12), and (14), respectively.…”

“…Problem 4 Set k = 2 into Eq. (16) and by using the initial conditions y(0) = 1 and y (0) = 0, then we have:…”

“…Furthermore, they do not satisfy the non-linear derivative rules as product, quotient, and chain rules. More recently, a new definition of fractional derivatives which is the so-called conformable fractional derivative has been introduced and attracted the attention of many researchers because it contains many characteristics that correspond to the usual derivative, particularly the Leibniz rules [11][12][13][14][15][16]. This definition is very simple and more welcome than other fractional definitions since it has been receiving a lot of attention, many applications and phenomena can be modeled based on the CFDs, and it contains many interesting advantages such as the following: it is a local derivative that simulates the normal derivative because it depends on the limit in its formulation and it generalizes all concepts of ordinary calculus and can solve different fractional differential equations with all cases.…”

On the Uniqueness Classes of Solutions of Boundary Value Problems for Third-Order Equations of the Pseudo-Elliptic Type

“…Boundary value problems have applications in fluid dynamics, astrophysics, hydrodynamic, hydromagnetic stability, astronomy, beam and long wave theory, induction motors, engineering, and applied physics. Boundary value problems of higher order is studied in papers [16,17]. An overview of some results on the class of functions with subharmonic behaviour and their invariance properties under conformal and quasiconformal mappings is presented in [18].…”