integral and differential of fractional order are important notion that is often used in dealing with Frechet geometry. Based on pseudo-operations given by monotone and continuous function g, we study pseudo-derivative and pseudo-integral of fractional order through this paper.

In this paper, we express and prove  Stolarsky, Feng Qi and Markov type inequalities for two classes of pseudo-integrals. One of them concerning the pseudo-integrals based on a function reduces on the g-integral where pseudo-operations are defined by a monotone and continuous function $g$. The other one concerns the pseudo-integrals based on a semiring $( [a, b], \max, \odot )$, where $\odot$ is generated.  The integral  inequalities are  appling in multivariate approximation theory and probability theory and etc.

Conventional experimental techniques like, quasi-static, effective force and shake-table techniques are generally being adopted to evaluate the seismic response of a civil engineering structure under earthquake loading. Among these techniques, shake-table technique has a merit over the other techniques due the fact that it realistically simulates all the three basic dynamic force parameters namely inertial, damping and elastic forces in the test structure. However this technique needs a sophisticated shake-table driven by servo hydraulic actuators with excellent control electronics. In the absence of such an expensive shake-table, it is possible to simulate the three dynamic force parameters using a static actuator through application of an equivalent pseudo-dynamic force system by computation of inertial forces in the back-ground. Such a hybrid pseudo-dynamic (PsD) technique needs a specialized algorithm based on an appropriate mathematical model for the off-line time integration and computation of inertial forces such that the dynamic displacements/forces are applied statically through static actuators. Restoring forces offered by the structure are experimentally measured on-line at each time step and reflects the actual in-elastic and energy dissipation characteristics of the tested structure. The paper presents the mathematical formulation of a 'predictor-corrector' method using Newmark implicit relations and its implementation in PsD technique for seismic response evaluation of structures. In the proposed method displacement iterations are made in the corrector phase in achieving the implicit displacement which is an improvement over the conventional method where displacement iterations are made to achieve the explicit displacement resulting in lesser accuracy. To experimentally verify this improved PsD technique, the seismic response quantities including base shear, roof displacement and energy dissipation of a model steel frame structure subjected to a simulated earthquake predicted using PsD technique were calibrated with seismic responses evaluated using standard shake-table technique. The paper also presents the causes for the deviation in the predicted seismic responses using PsD technique and highlights its merits and demerits over shake-table technique.

In recent years, pseudo-dynamic (PsD) technique is being adopted as an alternate to conventional shake-table technique to evaluate the seismic performance of structures. The shake-table technique has the merit of simulating all the three force parameters namely inertial, damping and elastic forces in the tested structure realistically; however the technique needs sophisticated shake-table driven by servo controlled actuators with appropriate control electronics. On contrary, PsD technique simulates the three force parameters by using a static actuator through application of an equivalent pseudo-dynamic force system with computation of inertial forces in the back-ground. Such a hybrid technique needs specialized algorithm based on an appropriate mathematical model for the off-line time integration and computation of inertial forces. Several time integrals have been proposed for application in PsD testing and majority of them are derived from Newmark-b family of algorithms. The traditional PsD testing uses constant acceleration version of Newmark time integral in explicit form for mathematical simplicity. This simplified explicit formulation results in numerical damping leading to considerable amplitude error in PsD testing, limiting its application to simple structures. However, for complicated structures improvement is needed in the time integral form leading to unconditional stability and zero numerical damping. This paper presents an improved form of Newmark implicit time integral for PsD testing. The improvement is based on the inclusion of an additional term in displacement predictor, which not only renders the algorithm more consistent, but also eliminates numerical damping and makes the algorithm unconditionally stable. The paper presents the analytical study carried out on the stability and energy dissipation properties of the improved time integral by evaluating its spectral characteristics for verifying its suitability in PsD testing.

We define three new operators which‎, ‎in exceptional cases‎, ‎are reduced to\linebreak $q$-integral/derivative $q_a$-integral/derivative and ${}^bq$-integral/derivative operators‎. ‎Fundamental properties emerge among these specific $q$-operators‎, ‎like fractional calculus‎. By utilizing these three newly introduced operators‎, ‎we can prove various inequalities‎, ‎the most notable of which are the Chebyshev and Hermite-Hadamard types‎. ‎Consequently‎, ‎these new operators provide a generalized approach to many problems in classical inequalities using classical fractional calculus‎.

In this paper, we introduce the concept of pseudo-continuous semicopula. We show its relationship with continuity in the second variable and provide a characterization of all semicopulas S such that the smallest semicopula-based universal integral is S-homogeneous. This completely solves Open problem 2. 29 proposed by J. Borzov´ a-Moln´ arov´ a et al. in the paper [2].

In this paper, we prove  Bellman type inequality for Sugeno integral and generalize it for pseudo-integrals. Furthermore, we generalize the Bellman's inequality in abstract spaces. The strengthened version of this inequality is demonstrated in pseudo-integrals and abstract spaces. Also, we illustrate theorems with some examples.