Existence and uniqueness results for nonlinear system of mixed ψ-Riemann-Liouville and φ-Caputo fractional derivatives with mixed boundary conditions

This study investigates necessary and sufficient conditions to existence and uniqueness results for nonlinear system of mixed ?-Riemann-Liouville and ?-Caputo fractional derivatives for the sum of three component functions with coefficients of continuous functions. We use the properties of mixed monotone fixed points theorem to derive these results, which are supported by Banach fixed point theorems. The novelty of our work is that, in this paper, we study coupled system of mixed ?-Riemann-Liouville and ?-Caputo fractional derivatives involving novel results of mixed monotone fixed points theorem {cases} {}^{Dα_1,Ψ_1₀₊}({}CDα_2,Φ_1₀₊)ϑ(t) = μ_1(t)f_1(t,ϑ(t),(t)) + ν_1(t)h_1(t,ϑ(t),(t)), \\[1em] {}CDα_3,Ψ_2₀₊({}^{Dα_4,Φ_2₀₊})(t) = μ_2(t)f_2(t,ϑ(t),(t)) + ν_2(t)h_2(t,ϑ(t),(t)), {cases} , with mixed boundary conditions {cases}β_1 ϑ(0) = β_2 ϑ(1) = β_3 {}CD₀₊α_2,Φ_1 ϑ(0) = β_4 {}CD₀₊α_2,Φ_1 ϑ(1) = 0, \\[6pt]β_1^* (0) = β_2^* (1) = β_3^* {}^{D₀₊α_4,Φ_2} (0) = β_4^* {}^{D₀₊α_4,Φ_2} (1) = 0.{cases}