In this paper, we investigate the existence of positive solutions for Hadamard type fractional differential system with coupled nonlocal fractional integral boundary conditions on an infinite domain. Our analysis relies on Guo-Krasnoselskii’s and Leggett-Williams fixed point theorems. The obtained results are well illustrated with the aid of examples.

The Sharma–Mittal holographic dark energy model is investigated in this paper using the Chern–Simons modified gravity theory. We investigate several cosmic parameters, including the deceleration, equation of state, square of sound speed, and energy density. According to the deceleration parameter, the universe is in an decelerating and expanding phase known as de Sitter expansion. The Sharma–Mittal HDE model supports a deceleration to acceleration transition that is compatible with the observational data. The EoS depicts the universe’s dominance era through a number of components, such as $\omega =0$, $\frac{1}{3}$, 1, which indicate that the universe is influenced by dust, radiation, and stiff fluid, while $-1<\omega <\frac{1}{3}$, $\omega =-1$, and $\omega <-1$ are conditions for quintessence DE, $\mathrm{\Lambda }$CDM, and Phantom era dominance. Our findings indicate that the universe is in an accelerated expansion phase, and this is similar to the observational data.

In this study, first we establish a $\left(p,q\right)$-integral identity involving the second $\left(p,q\right)$-derivative, and then, we use this result to prove some new midpoint-type inequalities for twice-$\left(p,q\right)$-differentiable convex functions. It is also shown that the newly established results are the refinements of the comparable results in the literature.

In this manuscript, we discuss fractional fuzzy Goursat problems with Caputo’s $gH$-differentiability. The second-order mixed derivative term in Goursat problems and two types of Caputo’s $gH$-differentiability pose challenges to dealing with Goursat problems. Therefore, in this study, we convert Goursat problems to equivalent systems fuzzy integral equations to deal properly with the mixed derivative term and two types of Caputo’s $gH$-differentiability. In this study, we utilize the concept of metric fixed point theory to discuss the existence of a unique solution of fractional fuzzy Goursat problems. For the useability of established theoretical work, we provide some numerical problems. We also discuss the solutions to numerical problems by conformable double Laplace transform. To show the validity of the solutions we provide 3D plots. We discuss, as an application, why fractional partial fuzzy differential equations are the generalization of usual partial fuzzy differential equations by providing a suitable reason. Moreover, we show the advantages of the proposed fractional transform over the usual Laplace transform.

In this paper, we establish a generalized fractional integrals identity involving some parameters and differentiable functions. Then, we use the newly established identity and prove different generalized fractional integrals inequalities like midpoint inequalities, trapezoidal inequalities and Simpson’s inequalities for differentiable convex functions. Finally, we give some applications of newly established inequalities in the context of quadrature formulas.

In this paper, we study the existence and uniqueness of solutions for impulsive Atangana-Baleanu-Caputo A B C fractional integro-differential equations with boundary conditions. Schaefer’s fixed point theorem and Banach contraction principle are used to prove the existence and uniqueness results. An example is presented to illustrate the results.