A new generalized American option under $\psi$-Caputo fractional-order derivative Heston (AO$\psi$CFDH) model is investigated. A new numerical implicit scheme method (NISM) has been developed for solving the AO$\psi$CFDH model. The stability and convergence of the NISM are analyzed. Two numerical examples are proposed in order to show the robustness and the efficiency of both the model AO$\psi$CFDH and the NISM.

We state and prove an integral representation formula of the “variation-of-constant” type for continuous solutions of linear nonautonomous difference-delay systems, in terms of a Lebesgue–Stieltjes integral involving a fundamental solution and the initial data of the system. This gives a precise and correct version of several formulations appearing in the literature, and extends them to the time-varying case. This is of importance for further stability studies of various kinds of delay systems.

We study two semi-analytical methods: the successive approximation method, that is, the Picard method (PM), the Adomian decomposition method (ADM), and one numerical technique (NT) method for finding the approximate solution of generalized nonlinear functional integral equations (GNFIE). The existence and uniqueness results are proved by applying Banach’s contraction theorem. In some cases, it is difficult to find the integral when we use the ADM to find the approximate solution of certain nonlinear integral equations. To overcome this problem, some numerical techniques are applied based on GNFIE. Our existence results contain many functional integral equations as a special case. Finally, we discuss some examples and compare the methods along with the error analysis.

We study the existence of a unique bounded continuous solution of a Caputo fractional differential equation. The results are obtained from an equivalent Volterra integral equation which is derived by inverting the fractional differential equation. The kernel function of this integral equation is weakly singular, and hence the standard techniques that are normally used for Volterra integral equations do not apply here. This hurdle is overcome by using a resolvent equation, applying some known properties of the resolvent, and then employing the contraction mapping principle.

The main purpose of this paper is to investigate some existence results for pseudo $S$-asymptotically ($\omega ,c$)-periodic sequential solutions to a semilinear difference equation of convolution type and a semilinear Weyl-like fractional difference equation in Banach spaces. For this purpose, we first give the definition of the pseudo $S$-asymptotically $(\omega ,c)$-periodic sequence and prove the completeness, convolution and superposition theorems for such a sequence in abstract spaces. We show some existence and uniqueness of pseudo $S$-asymptotically $(\omega ,c)$-periodic sequential solutions under some different Lipschitz type conditions of the nonlinear force term with its second variable. We also consider the existence of pseudo $S$-asymptotically $(\omega ,c)$-periodic sequential solutions under a non-Lipschitz growth condition.

We consider the numerical analysis for a partial integrodifferential equation of hyperbolic type. The central difference formula and the second-order convolution quadrature are applied to construct the numerical scheme, where the convolution quadrature method could accommodate the complicated case that the explicit form of the memory kernel is not available. We propose a novel analysis to prove the finite-time stability of the numerical solutions and specify the condition that ensures the long time stability. We also prove error estimates for the numerical scheme based on a newly developed approximate result of convolution quadrature for convolution of nonsmooth functions. Finally, we extend the developed methods to construct and analyze a numerical scheme for the corresponding nonlinear problems. Numerical experiments are performed to substantiate the theoretical findings.

We investigate the existence of positive solutions for a fourth-order beam equation

with sign-changing Green’s function. Combine a priori estimates of Green’s function, we apply the Leray–Schauder fixed point theorem to obtain the existence of positive solutions. We give the Ulam–Hyers stability result of the solution for the equation.