Stability and existence results for a class of nonlinear fractional differential equations with singularity

Abstract

In this paper, we are dealing with an analytical study of a singular fractional order nonlinear differential equation with fractional integral and differential boundary conditions and p -operator, for existence and stability results. Our problem is based on two types of fractional order derivatives, that is, Caputo factional derivative of order and Riemann-Liouville derivative of order , where m − 1 < , ≤ m, and m ∈ {3, 4, 5, … }. The suggested problem will be converted into an equivalent integral form by the help of Green function. After the proofs for these properties, some classical fixed point theorems are employed for the existence of positive solution (EPS). For application of the results, an expressive example is included. KEYWORDS Caputo fractional derivative, existence of positive solution, Hyers-Ulam stability, Riemann-Liouville fractional derivative, singular fractional differential equations c  u(t) = u(t) + (t, u(t)), u ′ (t) = 0, u(0) = h(u(T)), where 0 < T < +∞,  ∈ R + , , h are given functions and c  is Caputo derivative of fractional order ∈ (1, 2). Baleanu et al 15 studied EUS for the following equation: Math Meth Appl Sci. 2018;41:9321-9334.wileyonlinelibrary.com/journal/mma existence results for a class of nonlinear fractional differential equations with singularity. Math Meth Appl Sci. 2018;41:9321-9334.

Existence of solutions for some functional integrodifferential equations with nonlocal conditions

“…Recent research focus to use fractional calculus to solve many problems in various fields. [24][25][26][27][28][29] In this paper, we approach a new technique, that is, g is not compactness and using iterative methods based on the Sadovskii-Krasnosel'skii type of fixed point theorem technique in Ezzinbi et al 30 and to show that immediate norm continuity of R 1 (t 1 ) associated to (1.1) is to the semigroup T(t) is immediate norm continuity. Motivated by the above mentioned articles, 6,7,[16][17][18]30 the purpose of this paper is to study the existence of solutions for some functional integrodifferential equations with nonlocal conditions (1.1) and (1.2).…”

“…Recently, Lizama 23 proved the existence of mild solution of the equation of the form () when g is compact and R 1 ( t 1 ) is norm continuous. Recent research focus to use fractional calculus to solve many problems in various fields 24‐29 …”

Stability analysis of a nonlinear coupled implicit switched singular fractional differential system with p-Laplacian

“…where D α , c D σ denote the Riemann-Liouville and Caputo fractional derivatives of order α and σ , respectively, and φ(s) = |s| p-2 , p > 1. The function ϕ satisfies ϕ : In [8], Alkhazzan et al studied the existence and stability results for a class of nonlinear fractional differential equations with singularity of the form:…”

“…The objective of this paper is to use the concepts mentioned in [8] to examine the existence, uniqueness as well as different kinds of Hyers-Ulam stability for the solution of the nonlinear coupled implicit switched singular system of fractional differential equations with singularities of the form:…”

Stability analysis of nonlinear implicit fractional Langevin equation with noninstantaneous impulses

“…|x(s)| 8+e s +s 2 ) ds, t ∈ (0, 1] ∪(2,3],x(t) = x(t) (3+t 2 )(1+|x(t)|) , t ∈ (1, 2], |y(s)| 8+e s +s 2 ) ds| ≤ e t , t ∈ (0, 1] ∪ (2, 3], |y(t) -y(t) (3+t 2 )(1+|x(t)|) | ≤ 1, t ∈ (1, 2]. Let J = [0, 3], α = β =1 2 , r = 1 3 , = -2.70, θ =5 6 , p =4 3 , η = 1 4 and 0 = t0 < s 0 = 1 < t 1 = 2 < s 1 = τ = 3.…”