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This paper is contributed to the structural stability of multi-wave configurations to Cauchy problem for the compressible non-isentropic Euler system with adiabatic exponent γ ∈ (1, 3]. Given some small BV perturbations of the initial state, the author employs a modified wave front tracking method, constructs a new Glimm functional, and proves its monotone decreasing based on the possible local wave interaction estimates, then establishes the global stability of the multi-wave configurations, consisting of a strong 1-shock wave, a strong 2-contact discontinuity, and a strong 3-shock wave, without restrictions on their strengths.

In this paper we study the principal spectral theory and asynchronous exponential growth for age-structured models with nonlocal diffusion of Neumann type. First, we provide two general sufficient conditions to guarantee existence of the principal eigenvalue of the age-structured operator with nonlocal diffusion. Then we show that such conditions are also enough to ensure that the semigroup generated by solutions of the age-structured model with nonlocal diffusion exhibits asynchronous exponential growth. Compared with previous studies, we prove that the semigroup is essentially compact instead of eventually compact, where the latter is usually obtained by showing the compactness of solution trajectories. Next, following the technique developed in Vo (Principal spectral theory of time-periodic nonlocal dispersal operators of Neumann type. arXiv:1911.06119 , 2019), we overcome the difficulty that the principal eigenvalue of a nonlocal Neumann operator is not monotone with respect to the domain and obtain some limit properties of the principal eigenvalue with respect to the diffusion rate and diffusion range. Finally, we establish the strong maximum principle for the age-structured operator with nonlocal diffusion.

We construct a continuous nowhere monotone and nondifferentiable function depending on a single parameter q0 ϵ (0; 1). For functions from this continual class, we describe their structural, variational, fractal, and integrodifferential properties.

An edge-ordered graph is a graph with a total ordering of its edges. A path $$P=v_1v_2\ldots v_k$$ in an edge-ordered graph is called increasing if $$(v_iv_{i+1}) < (v_{i+1}v_{i+2})$$ for all $$i = 1,\ldots,k-2$$ ; and it is called decreasing if $$(v_iv_{i+1}) > (v_{i+1}v_{i+2})$$ for all $$i = 1,\ldots,k-2$$ . We say that P is monotone if it is increasing or decreasing. A rooted tree T in an edge-ordered graph is called monotone if either every path from the root to a leaf is increasing or every path from the root to a leaf is decreasing. Let G be a graph. In a straight-line drawing D of G, its vertices are drawn as different points in the plane and its edges are straight line segments. Let $$\overline{\alpha}(G)$$ be the largest integer such that every edge-ordered straight-line drawing of G contains a monotone non-crossing path of length $$\overline{\alpha}(G)$$ . Let $$\overline{\tau}(G)$$ be the largest integer such that every edge-ordered straight-line drawing of G contains a monotone non-crossing complete binary tree of $$\overline{\tau}(G)$$ edges. In this paper we show that $$\overline \alpha(K_n) = \Omega(\log\log n)$$ , $$\overline \alpha(K_n) = O(\log n), \overline \tau(K_n) = \Omega(\log\log \log n)$$ and $$\overline \tau(K_n) = O(\sqrt{n \log n})$$ .

We consider a class of two-dimensional integral equations in ℝ2 with monotone nonlinear terms and prove the existence of a nonnegative bounded solution. We study the asymptotic behavior of this solution with respect to each variable. The result is illustrated by an example.

We study the nonlinear newsvendor problem concerning goods of a non-discrete nature, and a class of stochastic dynamic programs with several application areas such as supply chain management and economics. The class is characterized by continuous state and action spaces, either convex or monotone cost functions that are accessed via value oracles, and affine transition functions. We establish that these problems cannot be approximated to any degree of either relative or additive error, regardless of the scheme used. To circumvent these hardness results, we generalize the concept of fully polynomial-time approximation scheme allowing arbitrarily small additive and multiplicative error at the same time, while requiring a polynomial running time in the input size and the error parameters. We develop approximation schemes of this type for the classes of problems mentioned above. In light of our hardness results, such approximation schemes are “best possible”. A computational evaluation shows the promise of this approach.

Assume that the function $f:[0,\infty )\rightarrow \mathbb {R}$ is operator monotone in $[0,\infty )$ . We can define the perspective $\mathcal {P}_{f}\left (B,A\right ) $ by setting $$ \mathcal{P}_{f}\left( B,A\right) :=A^{1/2}f\left( A^{-1/2}BA^{-1/2}\right) A^{1/2}, $$ where A, B > 0. In this paper, we show among others that, if σ ≥ C ≥ ρ > 0, D > 0, ς ≥ Q ≥ τ > 0 and 0 < n ≤ D − C ≤ N for some constants ρ, σ, ς, τ, n, N, then $$ \begin{array}{@{}rcl@{}} 0& \le& \frac{n}{N{\varsigma}^{2}}\left[ \mathcal{P}_{f}\left( {\varsigma} ,N+\sigma \right) -\mathcal{P}_{f}\left( {\varsigma} ,\sigma \right) \right] Q^{2} \\ & \leq& \mathcal{P}_{f}\left( Q,D\right) -\mathcal{P}_{f}\left( Q,C\right) \\ & \leq& \frac{N}{n\tau^{2}}\left[ \mathcal{P}_{f}\left( \tau ,n+\rho \right) -\mathcal{P}_{f}\left( \tau ,\rho \right) \right] Q^{2}. \end{array} $$ Applications for the weighted operator geometric mean and the perspective $$ \mathcal{P}_{\ln \left( \cdot +1\right) }\left( B,A\right) :=A^{1/2}\ln \left( A^{-1/2}BA^{-1/2}+1\right) A^{1/2},~ A,B>0 $$ are also provided.

In this paper, we introduce two algorithms for finding a common solution of the monotone inclusion problem and the fixed point problem for a relatively nonexpansive mapping in reflexive Banach spaces. The weak convergence results for both algorithms are established without the prior knowledge of the Lipschitz constant of the mapping. An application to the variational inequality problem is considered. Finally, some numerical experiments of the proposed algorithms including comparisons with other algorithms are provided.

We investigate qualitative properties of solutions of special classes of convolution type nonlinear integral equations on the whole line. We study the asymptotic properties, continuity, and monotonicity of arbitrary nontrivial bounded solutions. Depending on the properties of the kernel of the equation, we find out whether there exist nontrivial bounded solutions with a finite limit at ±∞. Based on the obtained results, we establish uniqueness theorems for large classes of bounded functions. The results obtained are illustrated by examples from applications.

In this article, the notions of partially continuous mappings, partially bounded, partially compact and partially closed sets of an ordered E-metric space are defined. These notions are used to introduce partial E-measure of noncompactness and prove the fixed point results of monotone mappings and sum of two monotone mappings. In this way we generalize many results including the well known results of Schauder, Darbo and Krasnoselskii, in the settings of ordered E-metric and ordered Banach spaces. We also provide nontrivial examples and existence results for a class of integral equations to validate the significance of our theory and results.

We prove that if $$X^{*}$$ is strictly convex, a convex function $$f$$ is coercive and b-Lipschitzian iff there exist two convex function sequences $$\{f_{n}\}_{n=1}^{\infty}$$ and $$\{g_{n}\}_{n=1}^{\infty}$$ such that (1) $$f_{n}\leq f_{n+1}\leq f$$ and $$f\leq g_{n+1}\leq g_{n}$$ for all integers $$n \geq 1$$ ; (2) $$f_{n}$$ and $$g_{n}$$ are continuous and Gâteaux differentiable on $$X$$ ; (3) $$f_n \to f$$ and $${g_n \to f}$$ uniformly on $$X$$ ; (4) $$f_n$$ and $$g_n$$ are coercive and b-Lipschitzian. Moreover, we also prove that if $$X^{*}$$ is strictly convex, then a convex function f is Lipschitzian iff conditions (1)-(3) are true and there exists $$m>0$$ such that $$|f_{n}(x)-f_{n}(y)|\leq m\|x-y\|$$ and $$|g_{n}(x)-g_{n}(y)|\leq m\|x-y\|$$ whenever $$x, y \in X$$ and $$n\in N$$ .

In this paper, we present some new results on the existence of solutions of generalized variational inequalities for set-valued mappings in reflexive Banach spaces with Frechet differentiable norms. Moreover, the structure of the solution sets is investigated. The result obtained in this paper improves and extends the ones announced by Fang and Peterson [S. C. Fang and E. L. Peterson, Generalized Variational Inequalities, J. Optim. Theory Appl., 38 (1982), 363-383] and others.

In this paper, we study an extension of the split equality equilibrium problem called the extended split equality equilibrium problem. We give an iterative algorithm for approximating a solution of extended split equality equilibrium and fixed point problems and obtained a strong convergence result in a real Hilbert space. We further applied our result to solve extended split equality monotone variational inclusion and equilibrium problems. The result of this paper complements and extends results on split equality equilibrium problems in the literature.

International audience; Existence of solutions of reaction-diffusion systems of equations in unbounded domains is studied by the Leray-Schauder (LS) method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces. We identify some reaction-diffusion systems for which there exist two subclasses of solutions separated in the function space, monotone and nonmonotone solutions. A priori estimates and existence of solutions are obtained for monotone solutions allowing to prove their existence by the LS method. Various applications of this method are given.

This paper focuses on the problem of convex constraint nonlinear equations involving monotone operators in Euclidean space. A Fletcher and Reeves type derivative-free conjugate gradient method is proposed. The proposed method is designed to ensure the descent property of the search direction at each iteration. Furthermore, the convergence of the proposed method is proved under the assumption that the underlying operator is monotone and Lipschitz continuous. The numerical results show that the method is efficient for the given test problems.

In this paper we characterize the functions that map the positive general monotone sequences to themselves, i.e., we obtain a necessary and a sufficient condition for the functions φ to satisfy $${\sum\limits_{k = n}^{2n} {\left| {{a_{k + 1}} - {a_k}} \right| \le C{a_n} \Rightarrow \sum\limits_{k = n}^{2n} {\left| {\phi \left( {{a_{k + 1}}} \right)} \right| \le {C^\prime }\phi \left( {{a_n}} \right).} } }$$ for all $${\left\{ {{a_n}} \right\}_{n = 1}^\infty \subset {\mathbb{R}^ + }}$$ .

In this paper, we establish the convergence theorems for two projection algorithms for finding a null point of the sum of two monotone operators in Hilbert spaces. Our algorithms are the combination the inertial forward–backward with the shrinking of hybrid projection methods. To clarify the acceleration, effectiveness, and performance of proposed algorithms, numerical contributions have been incorporated.

In this paper we consider the following nonlinear nonlocal Choquard equation $$\begin{aligned} {{\mathcal {F}}} _{\alpha }\left( u(x)\right) +\omega u(x) = C_{n,2s} \left( |x|^{2s-n}*u^q(x)\right) u^r(x), ~ x\in {\mathbb {R}}^n, \end{aligned}$$ where $$0

This paper focuses on two-point boundary value problem for first order causal difference equations. We will start with two new comparison theorems. Then, by utilizing these theorems and fixed point theorems, we obtain the existence of solutions for the corresponding linear problem. By applying monotone iterative technique, sufficient conditions for the existence of extremal solutions are also established. The results of this paper extend some existing results in the literature. Finally, two examples to show the usefulness of our results are exhibited.

We study the sums of nondegenerate sine series with monotone coefficients and consider the sets of positivity of such functions. We obtain the sharp lower estimate of the measure of such a set on $$[\pi/2, \pi]$$ and a new lower bound on its measure on $$[0,\pi]$$ . It is shown that the latter measure is at least $$\pi/2 + 0.24$$ and in the case of fulfilling special conditions it is at least $$2\pi/3$$ , which is an unimprovable estimate.

This work is concerned with the classical problem of finding a zero of a sum of maximal monotone operators. For the projective splitting framework recently proposed by Combettes and Eckstein, we show how to replace the fundamental subproblem calculation using a backward step with one based on two forward steps. The resulting algorithms have the same kind of coordination procedure and can be implemented in the same block-iterative and highly flexible manner, but may perform backward steps on some operators and forward steps on others. Prior algorithms in the projective splitting family have used only backward steps. Forward steps can be used for any Lipschitz-continuous operators provided the stepsize is bounded by the inverse of the Lipschitz constant. If the Lipschitz constant is unknown, a simple backtracking linesearch procedure may be used. For affine operators, the stepsize can be chosen adaptively without knowledge of the Lipschitz constant and without any additional forward steps. We close the paper by empirically studying the performance of several kinds of splitting algorithms on a large-scale rare feature selection problem.

In this paper, we introduce a new Viscosity algorithm for approximating an element in the intersection of the set of common solutions of monotone inclusion problems and common fixed points of family of asymptotically quasi-non expansive mappings in a complete CAT(0) space. Strong convergence theorem is proved which improved and generalized recently announced results in the literature.

In this paper, we study the split equality problem for systems of monotone variational inclusions and fixed point problems of set-valued demi-contractive mappings in real Hilbert spaces. A new viscosity algorithm for solving this problem is introduced along with its strong convergence theorem. Several known theoretical applications, such as, split common null point problem for systems of monotone variational inclusions and fixed point problems, split equality saddle-point and fixed point problem are given. Two primary numerical examples which illustrate and compare the behavior of the new scheme, suggest that the method has a potential applicable value besides its theoretical generalization. Our work extends and generalizes some existing works in the literature as well as provide some new direction for future work.

Let E be a real reflexive Banach space with its dual $$E^*$$ and f be a proper, convex and lower-semi-continuous function on E. The purpose of this paper is to introduce and study a new class of mappings from E into $$E^*$$ called f-pseudocontractive mappings with the notion of f-fixed points. In the case that E is a real reflexive Banach space and f is a strongly coercive, bounded and uniformly Frechet differentiable Legendre function which is strongly convex on bounded subsets of E, a sequence is constructed which converges strongly to a common f-fixed point of two f-pseudocontractive mappings. As a consequence, we obtain a scheme which converges strongly to a common zero of monotone mappings. Furthermore, this analog is applied to approximate solutions to convex optimization problems. Our results improve and generalize many of the results in the literature.

In this paper, we prove that continuous non-PM functions with non-monotonicity height equal to 1 need not be strictly monotone on its range, unlike PM functions. An existence theorem is obtained for the iterative roots of such functions. We also discuss the Hyers–Ulam stability for the functional equation of the iterative root problem.

Using the hyper-exponential recurrence criterion, a large deviation principle for the occupation measure is derived for a class of non-linear monotone stochastic partial differential equations. The main results are applied to many concrete SPDEs such as stochastic $p$-Laplace equation, stochastic porous medium equation, stochastic fast-diffusion equation, and even stochastic real Ginzburg-Landau equation driven by $\alpha$-stable noises., Comment: This paper generalizes the idea in our NOT published paper arXiv:1510.03522. There is a substantial overlap with arXiv:1510.03522

By introducing for monotone recurrence relations pseudo solutions, which are analogues of pseudo orbits of dynamical systems, we show that for general monotone recurrence relations the rotation set is closed, and each element in the rotation set is realized by a Birkhoff orbit. Moreover, if there is an orbit without rotation number, then the system has positive topological entropy, and we can construct orbits shadowing different rotation numbers.

An n-dimensional quantum observable in quantum structures is a kind of a σ-homomorphism defined on the Borel σ-algebra of $\mathbb R^{n}$ with values in a monotone σ-complete effect algebra or in a σ-complete MV-algebra. It defines an n-dimensional spectral resolution that is a mapping from $\mathbb R^{n}$ into the quantum structure which is a monotone, left-continuous mapping with non-negative increments and which is going to 0 if one variable goes to $-\infty $ and it goes to 1 if all variables go to $+\infty $ . The basic question is to show when an n-dimensional spectral resolution entails an n-dimensional quantum observable. We show cases when this is possible and we apply the result to existence of three different kinds of joint observables.

We study the single projection algorithm of Tseng for solving a variational inequality problem in a 2-uniformly convex Banach space. The underline cost function of the variational inequality is assumed to be monotone and Lipschitz continuous. A weak convergence result is obtained under reasonable assumptions on the variable step-sizes. We also give the strong convergence result for when the underline cost function is strongly monotone and Lipchitz continuous. For this strong convergence case, the proposed method does not require prior knowledge of the modulus of strong monotonicity and the Lipschitz constant of the cost function as input parameters, rather, the variable step-sizes are diminishing and non-summable. The asymptotic estimate of the convergence rate for the strong convergence case is also given. For completeness, we give another strong convergence result using the idea of Halpern’s iteration when the cost function is monotone and Lipschitz continuous and the variable step-sizes are bounded by the inverse of the Lipschitz constant of the cost function. Finally, we give an example of a contact problem where our proposed method can be applied.

We establish strong and weak Hardy–Littlewood–Sobolev inequalities for the subadditive operators majorized by operators from a certain class of integral convolutions of the Riesz-potential type with almost monotone kernels generated both by operators of ordinary shift and by operators of generalized shift associated with the differential Laplace–Bessel operator.

In this paper, we propose a new algorithm for solving a split equilibrium problem involving nonmonotone and monotone equilibrium bifunctions in real Hilbert spaces by using a shrinking projection method with a general Armijo line search rule on the e-subdifferential. We obtain a strong convergence theorem for the new algorithm.

This paper is devoted to the study of sensitivity to perturbation of parametrized variational inclusions involving maximally monotone operators in a Hilbert space. The perturbation of all the data involved in the problem is taken into account. Using the concept of proto-differentiability of a multifunction and the notion of semi-differentiability of a single-valued map, we establish the differentiability of the solution of a parametrized monotone inclusion. We also give an exact formula of the proto-derivative of the resolvent operator associated to the maximally monotone parameterized variational inclusion. This shows that the derivative of the solution of the parametrized variational inclusion obeys the same pattern by being itself a solution of a variational inclusion involving the semi-derivative and the proto-derivative of the associated maps. An application to the study of the sensitivity analysis of a parametrized primal-dual composite monotone inclusion is given. Under some sufficient conditions on the data, it is shown that the primal and the dual solutions are differentiable and their derivatives belong to the derivative of the associated Kuhn–Tucker set.

This work aims to prove the large deviation principle for a class of stochastic partial differential equations with locally monotone coefficients under the extended variational framework, which generalizes many previous works. Using stochastic control and the weak convergence approach, we prove the Laplace principle, which is equivalent to the large deviation principle in our framework. Instead of assuming compactness of the embedding in the corresponding Gelfand triple or finite dimensional approximation of the diffusion coefficient in some existing works, we only assume some temporal regularity in the diffusion coefficient.

The purpose of this paper is to study and analyze an iterative method for split feasibility problem and variational inclusion problem (also known as the problem of finding a zero of the sum of two monotone operators) in the framework of real Banach spaces. By combining Mann’s and Halpern’s approximation methods, we propose an iterative algorithm for approximating a common solution of the aforementioned problems. Furthermore, we derive the strong convergence of the proposed algorithm under appropriate conditions. In all our results, we use the new way introduced by Suanti et al. to select the step-size which ensures the convergence of the sequences generated by our scheme. We also gave an application of our results and a numerical example of the proposed algorithm in comparison with the algorithm of Suanti et al. to show the efficiency and advantage of our algorithm. Our results extend and complement many known related results in the literature.

We study and solve one class of discrete matrix equations with cubic nonlinearity. The existence of a two-parameter family of monotone and bounded solutions is proved. Under certain additional conditions, we determine the asymptotic behavior of the constructed solutions. The obtained results are extended to the corresponding inhomogeneous discrete matrix equations and to some more general cases of nonlinearity.

In this paper, we introduce a new algorithm with self adaptive step-size for finding a common solution of a split feasibility problem and a fixed point problem in real Hilbert spaces. Motivated by the self adaptive step-size method, we incorporate the self adaptive step-size to overcome the difficulty of having to compute the operator norm in the proposed method. Under standard and mild assumption on the control sequences, we establish the strong convergence of the algorithm, obtain a common element in the solution set of a split feasibility problem for sum of two monotone operators and fixed point problem of a demimetric mapping. Numerical examples are presented to illustrate the performance and the behavior of our method. Our result extends, improves and unifies other results in the literature.

In this paper, we introduce and study strong convergence of a proximal point algorithm for approximating common solution of a finite family of monotone inclusion problems and fixed point problem of a generalized demimetric mapping in complete CAT(0) spaces. An illustrative example is given to validate theoretical result obtained herein. Our results improve and generalize some well-known results in the literature.

The correspondence between the monotonicity of a (possibly) set-valued operator and the firm nonexpansiveness of its resolvent is a key ingredient in the convergence analysis of many optimization algorithms. Firmly nonexpansive operators form a proper subclass of the more general - but still pleasant from an algorithmic perspective - class of averaged operators. In this paper, we introduce the new notion of conically nonexpansive operators which generalize nonexpansive mappings. We characterize averaged operators as being resolvents of comonotone operators under appropriate scaling. As a consequence, we characterize the proximal point mappings associated with hypoconvex functions as cocoercive operators, or equivalently; as displacement mappings of conically nonexpansive operators. Several examples illustrate our analysis and demonstrate tightness of our results.

We present a new smoothing Newton method for the symmetric cone complementarity problem with the Cartesian $$P_0$$ -property. The new method is based on a new smoothing function and a nonmonotone line search which contains a monotone line search as a special case. It is proved that the new method is globally and locally superlinearly/quadratically convergent under mild conditions. Preliminary numerical results are also reported which indicate the proposed method is promising.

We propose a method for solving the split variational inequality problem (SVIP) involving Lipschitz continuous and pseudomonotone mappings. The proposed method is inspired by the Halpern subgradient extragradient method for solving the monotone variational inequality problem with a simple step size. A strong convergence theorem for an algorithm for solving such a SVIP is proved without the knowledge of the Lipschitz constants of the mappings. As a consequence, we get a strongly convergent algorithm for finding the solution of the split feasibility problem (SFP), which requires only two projections at each iteration step. A simple numerical example is given to illustrate the proposed algorithm.

In this paper, we introduce an iterative process for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of the variational inequality for a Lipschitz-continuous, monotone mapping in a Banach space. We obtain a strong convergence theorem for three sequences generated by this process. Our results improve and extend the corresponding results announced by many others. A simple numerical example is given to support our theoretical results.

We establish conditions for the existence of a class of monotone solutions of the second-order differential equations with regularly and rapidly varying nonlinearities and the asymptotic representations of these solutions as t ↑ ω (ω ≤ + ∞).

We find a one to one mapping between genuinely incoherent operations and special one-way local operations and classical communication(LOCC) for density matrices with full rank. We also define “generalized entanglement monotones” and “genuinely coherence monotones” under special one-way LOCC and genuinely incoherent operations respectively. Any entanglement monotone proposed by Vidal et al. is a generalized entanglement monotone. Any coherence monotone under incoherent operations is a genuinely coherence monotone. Furthermore, we clarify the relationship between generalized entanglement monotones and genuinely coherence monotones. We demonstrate that any generalized entanglement monotone of bipartite pure state is the lower bound of a suitable genuinely coherence monotone; any genuinely coherence monotone of a quantum state is the generalized entanglement monotone of the corresponding maximally correlated state.

This paper is devoted to the study of efficient elements for set-valued maps. We propose two new notions of relative weak $$\epsilon $$ -efficient element and strict relative weak $$\epsilon $$ -efficient element of set-valued maps and provide new necessary optimality conditions for the proposed concepts. We provide existence results for efficient elements. The critical ingredients for the existence results for efficient elements are the well-known separation arguments and Fan’s lemma. As an application of the existence results, we derive relationships between the efficiency concepts and the local optimizers of certain optimization problems.

Funding for open access charge: Universidad de Granada / CBUA., The authors are grateful to Margarita Arias, Jos´e Antonio G´alvez and Francisco Martín for helpful comments during the preparation of this manuscript., In this paper, we study phi-minimal surfaces in R-3 when the function phi is invariant under a two-parametric group of translations. Particularly those which are complete graphs over domains in R-2. We describe a full classification of complete flat-embedded phi-minimal surfaces if phi is strictly monotone and characterize rotational phi-minimal surfaces by its behavior at infinity when phi has a quadratic growth., Universidad de Granada / CBUA

In this paper, first we prove the Crandall–Liggett exponential theorem in nonlinear semigroup theory on Hadamard manifolds. This theorem states that a semigroup of contractions can be constructed by the resolvent of a monotone vector field on Hadamard manifolds. Then, we show that the generated semigroup satisfies the evolution equation governed by the monotone vector field. The results of this paper are extensions of the classical results of Crandall and Liggett (Am J Math 93:265–298, 1971) and Brezis and Pazy (Israel J Math 8:367–383, 1970) to Hadamard manifolds. Some examples are also presented in the last part of the paper.

We present techniques, inspired by monodromy considerations, for constructing compact monotone Lagrangians in certain affine hypersurfaces, chiefly of Brieskorn-Pham type. We focus on dimensions 2 and 3, though the constructions generalise to higher ones. The techniques give significant latitude in controlling the homology class, Maslov class and monotonicity constant of the Lagrangian, and a range of possible diffeomorphism types; they are also explicit enough to be amenable to calculations of pseudo-holomorphic curve invariants. Applications include infinite families of monotone Lagrangian $S^1 \times \Sigma_g$ in $\mathbb{C}^3$, distinguished by soft invariants for any genus $g \geq 2$; and, for fixed soft invariants, a range of infinite families of Lagrangians in Brieskorn-Pham hypersurfaces. These are generally distinct up to Hamiltonian isotopy. In specific cases, we also set up well-defined counts of Maslov zero holomorphic annuli, which distinguish the Lagrangians up to compactly supported symplectomorphisms. Inter alia, these give families of exact monotone Lagrangian tori which are related neither by geometric mutation nor by compactly supported symplectomorphisms., Comment: accepted version

In this research work, we consider the second-order viscoelastic equation with a weak internal damping, a time-varying delay term and nonlinear weights $$\begin{aligned} u_{tt}(t) + {\mathcal {A}} u(t) - \int _0^t g (t-s) {\mathcal {A}} u(s) ds + \mu _1 (t) u_t (t)+ \mu _2(t) u_t(t-\tau (t)) =0 \forall t>0 , \end{aligned}$$ together with suitable initial conditions. We first prove the existence of a unique global weak solution by means of the classical Faedo–Galerkin method. Then, by assuming the general condition: $$\begin{aligned} g'(t) \le - \xi (t) H(g(t)), \forall t\ge 0, \end{aligned}$$ where H is a positive increasing and convex function and $$\xi $$ is a positive function which is not necessarily monotone, we establish optimal explicit and general stability estimates which rely on the well-known multipliers method and some properties of convex functions. This study generalizes and improves many earlier ones in the existing literature.