On a novel impulsive boundary value pantograph problem under Caputo proportional fractional derivative operator with respect to another function

Abstract

<abstract><p>In this manuscript, we study the existence and Ulam's stability results for impulsive multi-order Caputo proportional fractional pantograph differential equations equipped with boundary and integral conditions with respect to another function. The uniqueness result is proved via Banach's fixed point theorem, and the existence results are based on Schaefer's fixed point theorem. In addition, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of the proposed problem are obtained by applying the nonlinear functional analysis technique. Finally, numerical examples are provided to supplement the applicability of the acquired theoretical results.</p></abstract>

Ground state solutions for the fractional impulsive differential system with ψ‐Caputo fractional derivative and ψ–Riemann–Liouville fractional integral

“…Using some standard fixed point theorems and the Picard iteration method, the existence and uniqueness results for problem (2) were established as well as results on the long-term behavior of solutions. Most studies on the solutions of Caputo-type FDEs were based on methods in the topological degree theory [13][14][15][16]20]. Recently, Khaliq et al first used the variational method to investigate the existence of solutions for the 𝜓-Caputo fractional boundary value problem [17] C D 𝛼,𝜓 (t) T − (𝜓 ′ (t) C D 𝛼,𝜓(t)…”

“…For system (1), an equivalent formulation can be given by multiplying the first and second equations in (1) with 𝜓 ′ (t)u(t) and 𝜓 ′ (t)v(t), respectively, and integrating on both sides from a to b, combining (19) and (20), Equation ( 15) can be acquired. □…”

“…Using some standard fixed point theorems and the Picard iteration method, the existence and uniqueness results for problem () were established as well as results on the long‐term behavior of solutions. Most studies on the solutions of Caputo‐type FDEs were based on methods in the topological degree theory [13–16, 20]. Recently, Khaliq et al first used the variational method to investigate the existence of solutions for the $$$ \psi $$$‐Caputo fractional boundary value problem [17] $$$$ {\displaystyle \begin{array}{ll}{}^C{D}_{T^{-}}^{\alpha, \psi (t)}\left({\psi}^{\prime }{(t)}^C{D}_{0^{+}}^{\alpha, \psi (t)}u(t)\right)& =\nabla G\left(t,u(t)\right),t\in \left[0,T\right],\\ {}u(0)& =0=u(T),\end{array}} $$$$ this work is novel.…”

New Multiplicity Results for a Boundary Value Problem Involving a ψ-Caputo Fractional Derivative of a Function with Respect to Another Function

“…Due to abundant forms of fractional operators [7][8][9], it is natural for people to put forward new general fractional differentiations and integrations to unify such forms as a single one. For this reason, the general type called thhe ψ-Caputo fractional operator is proposed in some related works [10][11][12], whose definitions contain a nonsingular kernel depending upon a function, and the classical fractional integrals and derivatives can be acquired by choosing special kernels. This new form can more accurately describe practical problems, for instance, ref.…”

“…Combining critical point theory and properties of fractional calculus of thee ψ-Caputo fractional integral and derivative, new multiplicity results of infinitely many solutions are established for the problem (6). Recently, some achievements available in the references discussed the existence and multiplicity results for ψ-Caputotype fractional boundary value problems via fixed point theorems [12,13], while few results were based on variational methods, even though variational methods are effective ways for studying the existence of solutions for fractional differential equations [14][15][16][17][18]. Moreover, some simple algebraic conditions are applied in the paper instead of the conventional asymptotic conditions used in previous articles because most nonlinear functions can not adapted for these asymptotic conditions.…”