English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this manuscript, we obtain sufficient conditions required for the existence of solution to the following coupled system of nonlinear fractional order differential equations: D γ ω ( ℓ ) = F ( ℓ , ω ( λ ℓ ) , υ ( λ ℓ ) ) , D δ υ ( ℓ ) = F ‾ ( ℓ , ω ( λ ℓ ) , υ ( λ ℓ ) ) , $$ \begin{gathered} D^{\gamma}\omega(\ell)= \mathcal{F} \bigl( \ell,\omega(\lambda\ell), \upsilon(\lambda\ell) \bigr), \\ D^{\delta}\upsilon(\ell)=\mathcal{\overline{F}} \bigl(\ell,\omega ( \lambda\ell), \upsilon(\lambda\ell) \bigr), \end{gathered} $$ with fractional integral boundary conditions a 1 ω ( 0 ) − b 1 ω ( η ) − c 1 ω ( 1 ) = 1 Γ ( γ ) ∫ 0 1 ( 1 − ρ ) γ − 1 ϕ ( ρ , ω ( ρ ) ) d ρ and a 2 υ ( 0 ) − b 2 υ ( ξ ) − c 2 υ ( 1 ) = 1 Γ ( δ ) ∫ 0 1 ( 1 − ρ ) δ − 1 ψ ( ρ , υ ( ρ ) ) d ρ , $$ \begin{gathered} \mathfrak{a}_{1}\omega(0)- \mathfrak{b}_{1}\omega(\eta)-\mathfrak {c}_{1}\omega(1)= \frac{1}{\varGamma(\gamma)} \int_{0}^{1}(1-\rho )^{\gamma-1} \phi \bigl( \rho, \omega(\rho) \bigr)\, d\rho\quad\text{and} \\ \mathfrak{a}_{2}\upsilon(0)-\mathfrak{b}_{2} \upsilon (\xi)-\mathfrak{c}_{2}\upsilon(1)=\frac{1}{\varGamma(\delta)} \int _{0}^{1}(1-\rho)^{\delta-1} \psi \bigl( \rho, \upsilon(\rho) \bigr) \,d\rho, \end{gathered} $$ where ℓ ∈ Z = [ 0 , 1 ] $\ell\in\mathfrak{Z}=[0,1]$ , γ , δ ∈ ( 0 , 1 ] $\gamma, \delta\in(0,1]$ , 0 < λ < 1 $0<\lambda<1$ , D denotes the Caputo fractional derivative (in short CFD), F , F ‾ : Z × R × R → R $\mathcal{F}, \mathcal{\overline{F}}: \mathfrak{Z}\times \mathfrak{R}\times\mathfrak{R} \rightarrow\mathfrak{R}$ and ϕ , ψ : Z × R → R $\phi , \psi:\mathfrak{Z}\times\mathfrak{R}\rightarrow\mathfrak{R}$ are continuous functions. The parameters η , ξ are such that 0 < η , ξ < 1 $0<\eta, \xi<1$ , and a i , b i , c i $\mathfrak{a}_{i}, \mathfrak{b}_{i}, \mathfrak {c}_{i}$ ( i = 1 , 2 $i=1, 2$ ) are real numbers with a i ≠ b i + c i $\mathfrak{a}_{i}\neq\mathfrak {b}_{i}+\mathfrak{c}_{i}$ ( i = 1 , 2 $i=1, 2$ ). Using topological degree theory, sufficient results are constructed for the existence of at least one and unique solution to the concerned problem. For the validity of our result, an appropriate example is presented in the end.

advances in difference equations

Existence and uniqueness of solutions for coupled system of fractional differential equations involving proportional delay by means of topological degree theory