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One of the fundamental inequalities which is used in many inequities is Jensen inequality. In fact, it is a base of some inequality such as the arithmetic mean, harmonic mean inequality also in inequality with respect to entropies and information theory. The purpose of this research paper is to give a new interesting refinement of Jensen inequality for two particular finite sequences by using uniformly convex function. Also, we give some applications of this in information theory. [ABSTRACT FROM AUTHOR]

This paper aims to establish a general stability result for a one-dimensional linear swelling porous-elastic system with infinite memory, irrespective of the wave speeds of the system. The proof is based on the multiplier method and some properties of convex functions. The kernel in our memory term is more general and of a broader class. Our output extends and improves some of the available results on swelling porous media in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider a viscoelastic equation with nonlinear feedback localized on a part of the boundary and in the presence of infinite-memory term. With imposing a more general condition on the relaxation function, we establish a more general stability result that generalizes and improves many earlier results in the literature. Our results are obtained without imposing any restrictive growth assumption on the damping term and without using any assumption on the boundedness of initial data used in many earlier papers in the literature. [ABSTRACT FROM AUTHOR]

This paper deals with approximate solutions of a nonsmooth semi-infinite programming with multiple interval-valued objective functions. We first introduce four types of approximate quasi Pareto solutions of the considered problem by considering the lower-upper interval order relation and then apply some advanced tools of variational analysis and generalized differentiation to establish necessary optimality conditions for these approximate solutions. Sufficient conditions for approximate quasi Pareto solutions of such a problem are also provided by means of introducing the concepts of approximate (strictly) pseudo-quasi generalized convex functions defined in terms of the limiting subdifferential of locally Lipschitz functions. Finally, a Mond–Weir type dual model in approximate form is formulated, and weak, strong and converse-like duality relations are proposed. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following partially dissipative viscoelastic Timoshenko system { ρ 1 ϕ t t − κ (ϕ x + ψ) x + κ ∫ 0 t g (t − s) (ϕ x + ψ) x d s = 0 in (0 , L) × R + , ρ 2 ψ t t − b ψ x x + κ (φ x + ψ) − κ ∫ 0 t g (t − s) (ϕ x + ψ) d s = 0 in (0 , L) × R + , with damping mechanism acting only on the shear force, and with Dirichlet boundary conditions. We consider a very general relaxation function g ′ (t) ≤ − ξ (t) H (g (t)) , ∀ t ≥ 0. Under appropriate conditions on ξ and H, we establish a general stability result. The result is obtained under the assumption of equal speed of wave propagation. This work extends and generalizes many results in literature such as Alves et al. [On modeling and uniform stability of a partially dissipative viscoelastic Timoshenko system. SIAM J Math Anal. 2019;51(6):4520–4543]. [ABSTRACT FROM AUTHOR]

This paper deals with a nonlinear viscoelastic equation. The aim is to expand the class of the function of relaxation h (t) that ensuring a general decay. We adopt the following commonly condition on relaxation function h (t) : h ′ (t) ≤ − ξ (t) χ (h (t)) , where ξ is a nonincreasing function and χ is an increasing and convex function on the whole [ 0 , ∞) instead of is convex only near the origin in the literature. [ABSTRACT FROM AUTHOR]

For the first time in this paper, the dual problem is constructed for the problem with generalized first order partial differential inclusions, the duality theorem is proved. For discrete problems, necessary and sufficient optimality conditions are derived in the form of the Euler–Lagrange type inclusion. Thus, it is possible to construct dual problems for problems with partial differential inclusions on the basis of dual operations of addition and infimal convolution of convex functions. To pass from the dual problem to the discrete-approximation problem, important equivalence theorems are proved, without which it is unlikely that certain success can be achieved along this path. Hence, we believe that this method of constructing dual problems can serve as the only possible method for studying duality for a wide class of problems with partial/ordinary differential inclusions. The results obtained are demonstrated on some linear problem and on a problem with first-order polyhedral partial differential inclusions. [ABSTRACT FROM AUTHOR]

The paper deals with the optimal control of second-order viability problems for differential inclusions with endpoint constraint and duality. Based on the concept of infimal convolution and new approach to convex duality functions, we construct dual problems for discrete and differential inclusions and prove the duality results. It seems that the Euler–Lagrange type inclusions are 'duality relations' for both primary and dual problems. Finally, some special cases show the applicability of the general approach; duality in the control problem with second-order polyhedral DFIs and endpoint constraints defined by a polyhedral cone is considered. [ABSTRACT FROM AUTHOR]

In the previous work, we established a conjugate duality for the constrained vector optimization, in which the concept of a set-valued topical function introduced there played a key role. This is a map with good features and can be interpreted as an abstract convex function in some sense. This paper is devoted to the further study for the properties and characterizations of the set-valued topical function, within the framework of abstract convexity, especially about the subdifferentials. As applications, we also investigate some set-valued DC-type optimization problems. Some optimal conditions and dual results are obtained by virtue of the conjugation and subdifferentials. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Moore–Gibson–Thompson equation with a memory term in the subcritical case, which arises in high-frequency ultrasound applications accounting for thermal flux and molecular relaxation times. For a class of relaxation functions satisfying g ′ (t) ≤ − ξ (t) H (g (t)) for H to be increasing and convex function near the origin and ξ (t) to be a nonincreasing function, we establish the optimal explicit and general energy decay result, from which we can recover the earlier exponential rate in the special case. [ABSTRACT FROM AUTHOR]

The aim of this note is to analyse the results in the paper by D.Y. Gao and G. Strang [Geometric nonlinearity: Potential energy, complementary energy, and the gap function, Quart. Appl. Math. 47 (1989), pp. 487-504] where, according to D.Y. Gao, the complementary gap function was first introduced, and several flaws, mistakes, and inconsistencies are present, and to discuss the mathematical manner in which the mentioned paper presents itself, especially from the point of view of convex analysis. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear viscoelastic kirchhoff equation in a bounded domain with a time varying delay in the weakly nonlinear internal feedback is considered, where the global existence of solutions in suitable Sobolev spaces by means of the energy method combined with Faedo–Galarkin procedure is proved with respect to the condition of the weight of the delay term in the feedback and the weight of the term without delay and the speed of delay. Furthermore, a general stability estimate using some properties of convex functions is given. [ABSTRACT FROM AUTHOR]

In this paper, we consider a viscoelastic equation of variable coefficients in the presence of infinite memory (past history) with nonlinear damping term and nonlinear delay term in the boundary feedback and acoustic boundary conditions. Under suitable assumptions, two arbitrary decay results of the energy solution are established via suitable Lyapunov functionals and some properties of the convex functions. The first stability result is given with a relation between the damping term and relaxation function and the second result is given without imposing any restrictive growth assumption on the damping term and the kernel function g. Our result extends the decay result obtained for problems with finite history to those with infinite history. [ABSTRACT FROM AUTHOR]

In this paper, we consider a viscoelastic equation with a nonlinear frictional damping and in the presence of an infinite-memory term. We prove an explicit and general decay result using some properties of the convex functions. Our approach allows a wider class of kernels, from which those of exponential decay type are only special cases. [ABSTRACT FROM PUBLISHER]

In the present paper, the method of guiding functions is applied to study the periodic problem for a differential inclusion with a causal multioperator. At first we consider the case when the multioperator is closed and convex-valued. Then the case of a non-convex-valued and lower semicontinuous right-hand part is considered. Thereafter, the theory is extended to the case of non-smooth guiding functions. [ABSTRACT FROM AUTHOR]

In this paper, a general integral identity for twice differentiable functions is derived. By using of this identity, the author establishes some new estimates on Hermite-Hadamard type and Simpson type inequalities fors-convex via Riemann–Liouville fractional integral. [ABSTRACT FROM AUTHOR]

Motivated by computing medians and means, two proximal splitting methods, backward-backward scheme introduced by Passty [Ergodic convergence to a zero of the sum of monotone operators in Hilbert spaces. J Math Anal Appl. 1979;72:383–390.] and barycentric method introduced by Lehdili and Lemaire [Metric spaces of non-positive curvature. Berlin: Springer; 1999.] are investigated in Hadamard space setting. A weak ergodic convergence theorem and some strong convergene results for both schemes are studied. [ABSTRACT FROM AUTHOR]

The stationary Stokes equations for a generalized Newtonian fluid with nonlinear unilateral, and slip and leak boundary conditions are investigated. Boundary conditions include the generalized Clarke gradient and the convex subdifferential, and the variational formulation of the problem is the variational–hemivariational inequality for the velocity field. Existence and uniqueness result for weak solution is proved by using a surjectivity theorem for a pseudomonotone perturbation of a maximal monotone operator. [ABSTRACT FROM AUTHOR]

We consider the asymptotic behaviour of integral energies with convex integrands defined on one-dimensional networks contained in a region of the three-dimensional space with a fast-oscillating boundary as the period of the oscillation tends to zero, keeping the oscillation themselves of fixed size. The limit energy, obtained as a -limit with respect to an appropriate convergence, is defined in a 'stratified' Sobolev space and is written as an integral functional depending on all, two or just one derivative, depending on the connectedness properties of the sublevels of the function describing the profile of the oscillations. In the three cases, the energy function is characterized through an usual homogenization formula for p-connected networks, a homogenization formula for thin-film networks and a homogenization formula for thin-rod networks, respectively. [ABSTRACT FROM AUTHOR]

We prove a quantitative estimate on the -harmonic approximation of Sobolev functions. This improves a previous qualitative estimate of the second authors, Stroffolini and Verde. [ABSTRACT FROM AUTHOR]

In this article, we considern-dimensional thermoelastic system with a nonlinear weak frictional damping. We establish an explicit and general decay rate result, using some properties of the convex functions. Our result is obtained without imposing any restrictive growth assumption on the damping term. [ABSTRACT FROM PUBLISHER]

In this article, we consider the well-known following shape optimization problem: [image omitted] where λ2(Ω) denotes the second eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions in Ω ⊂ 2, and |Ω| is the area of Ω. We prove, under some technical assumptions, that any optimal shape Ω* is [image omitted] and is not C1,α for any [image omitted]. We also derive from our strategy some more general regularity results, in the framework of partially overdetermined boundary value problems, and we apply these results to some other shape optimization problems. [ABSTRACT FROM AUTHOR]