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 \begin{cases} {}^{D^{\alpha_1,\Psi_1}_{0+}}\bigl({}^{C}D^{\alpha_2,\Phi_1}_{0+}\bigr)\vartheta(t) = \mu_1(t)f_1(t,\vartheta(t),\wp(t)) + \nu_1(t)h_1(t,\vartheta(t),\wp(t)), \\[1em] {}^{C}D^{\alpha_3,\Psi_2}_{0+}\bigl({}^{D^{\alpha_4,\Phi_2}_{0+}}\bigr)\wp(t) = \mu_2(t)f_2(t,\vartheta(t),\wp(t)) + \nu_2(t)h_2(t,\vartheta(t),\wp(t)), \end{cases} , with mixed boundary conditions \begin{cases}\beta_1 \vartheta(0) = \beta_2 \vartheta(1) = \beta_3 {}^{C}D_{0+}^{\alpha_2,\Phi_1} \vartheta(0) = \beta_4 {}^{C}D_{0+}^{\alpha_2,\Phi_1} \vartheta(1) = 0, \\[6pt]\beta_1^* \wp(0) = \beta_2^* \wp(1) = \beta_3^* {}^{D_{0+}^{\alpha_4,\Phi_2}} \wp(0) = \beta_4^* {}^{D_{0+}^{\alpha_4,\Phi_2}} \wp(1) = 0.\end{cases}