Existence and Stability Results for a Tripled System of the Caputo Type with Multi-Point and Integral Boundary Conditions

Abstract

In this paper, we introduce and investigate the existence and stability of a tripled system of sequential fractional differential equations (SFDEs) with multi-point and integral boundary conditions. The existence and uniqueness of the solutions are established by the principle of Banach’s contraction and the alternative of Leray–Schauder. The stability of the Hyer–Ulam solutions are investigated. A few examples are provided to identify the major results.

Coupled system of three sequential Caputo fractional differential equations: Existence and stability analysis

“…In recent times, researchers have given the existence, uniqueness, and Ulam stability of solutions for differential equations and systems of arbitrary order; see [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. In recent years, many scientific researchers have considered differential equations and systems containing sequential fractional derivatives of different types; for instance, see [37][38][39][40][41][42][43][44]. In [6], for differential equations the existence and uniqueness of solutions with Caputo fractional derivatives of different orders have been given as…”

“…In recent times, researchers have given the existence, uniqueness, and Ulam stability of solutions for differential equations and systems of arbitrary order; see [17–36]. In recent years, many scientific researchers have considered differential equations and systems containing sequential fractional derivatives of different types; for instance, see [37–44]. In [6], for differential equations the existence and uniqueness of solutions with Caputo fractional derivatives of different orders have been given as $$$$ \left\{\begin{array}{l}{D}^{\delta}\left[{D}^{\gamma }z(t)-f\left(t,z(t)\right)\right]=g\left(t,z(t)\right),\\ {}z(0)=0,{D}^{\vartheta }z(T)=\kappa {I}^{\beta }z(T),\\ {}0\le t\le T,0<\delta, \gamma, \vartheta <1,\kappa \ne \frac{\Gamma \left(\gamma \kern3pt +\kern3pt \beta +1\right)}{T^{\vartheta +\beta}\Gamma \left(\beta \kern3pt -\kern3pt \vartheta +1\right)},\end{array}\right.$$…”

Applicability of Darbo’s Fixed Point Theorem on the Existence of a Solution to Fractional Differential Equations of Sequential Type

“…Te study of boundary value problems for equations with nonlinear fractional diferentials has a prominent and important role in the theory of fractional calculus and in the study of physical phenomena through the physical interpretation of boundary conditions. To pass quickly to the practical applications of fractional derivatives in various applied sciences, some valuable works in this feld can be found in [11][12][13][14][15][16][17][18][19].…”

Caputo-hadamard Fractional Differential Equations: Integral Boundary Condition System

“…The challenge of studying the existence and uniqueness solution is the study of its stability. Ulam-Hyers stability (UHS) is the most important types of stability is used in this field ( see [4,10,17,27,28,44], [7,25], [19]) Su [43], studied the existence of solutions for a coupled system of fractional differential equations:…”