We investigate a general sequential hybrid class of fractional differential equations in the Caputo and Atangana–Baleanu fractional senses of derivatives. We consider the existence and uniqueness of solutions and the Hyers–Ulam (H-U) stability for a general class. We use the Banach and Leray–Schauder alternative theorems for the existence criteria. With the help of nonnegative Green’s functions, the fractional-order class is turned into m-equivalent integral forms. As an application of our problem, a fractional-order smoking model in terms of the Atangana–Baleanu derivative is presented as a particular case.

We investigate the nonlinear Klein-Gordon equation with Caputo fractional derivative. The general series solution of the system is derived by using the composition of the double Laplace transform with the decomposition method. It is noted that the obtained solution converges to the exact solution of the model. The existence of the model in the presence of Caputo fractional derivative is performed. The validity and precision of the presented method are exhibited with particular examples with suitable subsidiary conditions, where good agreements are obtained. The error analysis and its corresponding surface plots are presented for each example. From the numerical solutions, we observe that the proposed system admits soliton solutions. It is noticed that the amplitude of the wave solution increases with deviations in time, that concludes the factor $ \omega $ considerably increases the amplitude and disrupts the dispersion/nonlinearity properties, as a result, may admit the excitation in the dynamical system. We have also depicted the physical behavior that states the advancement of localized mode excitations in the system.

In this paper, we present a Hermite-Hadamard-Fejér inequality for conformable fractional integrals by using symmetric preinvex functions. We also establish an identity associated with the right hand side of Hermite-Hadamard inequality for preinvex functions; then by using this identity and preinvexity of functions and some well-known inequalities, we find several new Hermite-Hadamard type inequalities for conformal fractional integrals.

In this article, firstly, we establish a novel definition of weighted interval-valued fractional integrals of a function $\breve{\mathsf{{\rm Y}}}$ using an another function $\vartheta \left(\dot{\zeta }\right)$. As an additional observation, it is noted that the new class of weighted interval-valued fractional integrals of a function $\breve{\mathsf{{\rm Y}}}$ by employing an additional function $\vartheta \left(\dot{\zeta }\right)$ characterizes a variety of new classes as special cases, which is a generalization of the previous class. Secondly, we prove a new version of the Hermite-Hadamard-Fejér type inequality for h-convex interval-valued functions using weighted interval-valued fractional integrals of a function $\breve{\mathsf{{\rm Y}}}$ according to another function $\vartheta \left(\dot{\zeta }\right).$ Finally, by using weighted interval-valued fractional integrals of a function $\breve{\mathsf{{\rm Y}}}$ according to another function $\vartheta \left(\dot{\zeta }\right)$, we are establishing a new Hermite-Hadamard-Fejér type inequality for harmonically h-convex interval-valued functions that is not previously known. Moreover, some examples are provided to demonstrate our results.

An interval-valued q-rung orthopair fuzzy set ($IV{q}^{r}OFS$) is a robust and esteemed model in managing imprecise information, utilizing a controllable parameter $q\ge 1$. Expanding its applicability, we propose a refinement to it, termed the interval-valued p,q-rung orthopair fuzzy set, denoted $IVp,q^{r}OFS$. This advancement enables the handling of scenarios where varied assessment levels for membership and non-membership grades are necessary, a challenge unaddressed by existing extensions of interval-valued intuitionistic fuzzy sets. We establish the fundamental mathematical operations for the $IVp,q^{r}OFS$ and explore their key properties. To compare interval-valued p,q-rung orthopair fuzzy ($IVp,q^{r}OF$) numbers, a novel score function is put forward and compared to the known $IV{q}^{r}OFS$-based score functions to determine its viability. Moreover, we present $IVp,q^{r}OF$ aggregation operators (AOs), accompanied by rigorous proof of their essential characteristics, including boundedness, monotonicity, and symmetry. This study introduces the interaction of $IVp,q^{r}OF$ criteria through the inter-criteria correlation (CRITIC) approach. This method determines criteria weights based on the proposed distance measurements and the score function. The CRITIC method is integrated with the exponential TODIM approach (exp-TODIM), resulting in an innovative decision-making framework that is less susceptible to parameter fluctuations and showcases a high stability. An illustrative example of selecting a suitable supplier is provided to demonstrate the practical application of the developed exp-TODIM approach. Finally, a sensitivity analysis is conducted to exhibit the method’s stability, and a comparative analysis is performed to elucidate its strengths and advantages.

In this article, we establish several integral majorization type and generalized Favard’s inequalities for the class of strongly convex functions. Our results generalize and improve the previous known results.