Abstract
In this paper, we study the nonlinear coupled system of equations with fractional integral boundary conditions involving the Caputo fractional derivative of ordersθ1 and θ2and Riemann–Liouville derivative of ordersϱ1 and ϱ2with thep-Laplacian operator, wheren−1<θ1,θ2,ϱ1,ϱ2≤n, andn≥3. With the help of two Green’s functionsGϱ1w,ℑ,Gϱ2w,ℑ, the considered coupled system is changed to an integral system. Since topological degree theory is more applicable in nonlinear dynamical problems, the existence and uniqueness of the suggested coupled system are treated using this technique, and we find appropriate conditions for positive solutions to the proposed problem. Moreover, necessary conditions are highlighted for the Hyer–Ulam stability of the solution for the specified fractional differential problems. To confirm the theoretical analysis, we provide an example at the end.
Highlights
PreliminariesE family of each bounded set of P(Ω) symbolized by B
E topological degree theory is a useful tool in nonlinear analysis with numerous applications to operatorial equations, optimization theory, fractal theory, and other topics
Where ρi, θi ∈
Summary
E family of each bounded set of P(Ω) symbolized by B. For Z(w): (0, +∞) ⟶ R, the Caputo fractional derivative of noninteger order ρ > 0 is known by cDρZ(w) w. For Z(w): (0, +∞) ⟶ R, the Riemann–Liouville fractional derivative of noninteger order ρ > 0 is known by RDρZ(w). We indicate that the class of each θ− condensing mappings F: Ψ ⟶ Π by Cθ(Ψ) and the class of each strict θ− contractions F: Ψ ⟶ Π by ζCθ(Ψ). Let F, G: Ψ ⟶ Π be θ− Lipschitz operators with constants η1 and η2, respectively, F + G: Ψ ⟶ Π is θ− Lipschitz with constants η1 + η2. E above theorem that we mentioned plays a substantial role in obtaining our main results
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