Calogero Vetro

Università degli Studi di Palermo | UNIPA · Dipartimento di Matematica e Informatica

We consider a double phase (unbalanced growth) Dirichlet problem with a Carathéodory reaction f ( z , x ) which is superlinear in x but without satisfying the AR-condition. Using the symmetric mountain pass theorem, we produce a whole sequence of distinct bounded solutions which diverge to infinity.

We consider a semilinear Dirichlet problem with a reaction which exhibits an asymmetric behaviour as x→±∞ (it is superlinear as x→+∞ and resonant as x→−∞). Using variational tools from the critical point theory, together with truncation and comparison techniques and critical groups, we prove two multiplicity theorems producing two and three nontriv...

We study the large-time behavior of solutions for the inhomogeneous nonlinear Schrödinger equation $$\begin{aligned} iu_t+\Delta u=\lambda |u|^p+\mu |\nabla u|^q+w(x),\quad t>0,\, x\in {\mathbb {R}}^N, \end{aligned}$$ i u t + Δ u = λ | u | p + μ | ∇ u | q + w ( x ) , t > 0 , x ∈ R N , where $$N\ge 1$$ N ≥ 1 , $$p,q>1$$ p , q > 1 , $$\lambda ,\mu \i...

We consider the Dirichlet problem \begin{equation*} - \Delta^{K_p}_{p(x)} u(x) - \Delta^{K_q}_{q(x)} u(x) = f(x,u(x), \nabla u(x)) \quad \mbox{in }\Omega, \quad u\big{|}_{\partial \Omega}=0, \end{equation*} driven by the sum of a $p(x)$-Laplacian operator and of a $q(x)$-Laplacian operator, both of them weighted by indefinite (sign-changing) Kir...

We consider the Navier problem −Δk,p2u(x)=f(x,u(x),∇u(x),Δu(x))inΩ,u∂Ω=Δu∂Ω=0,$$ -{\Delta}_{k,p}^2u(x)=f\left(x,u(x),\nabla u(x),\Delta u(x)\right)\kern0.30em \mathrm{in}\kern0.5em \Omega, \kern0.30em u{\left|{}_{\mathrm{\partial \Omega }}=\Delta u\right|}_{\mathrm{\partial \Omega }}=0, $$ driven by the sign‐changing (degenerate) Kirchhoff type p(x...

We establish sufficient conditions for the nonexistence of nontrivial solutions to higher order evolution inequalities, with respect to the time variable. We consider a nonlocal source term, and work on complete noncompact Riemannian manifolds. The obtained conditions depend on the parameters of the problem and the geometry of the manifold. Our mai...

We are concerned with the existence and nonexistence of global weak solutions for a certain class of time-fractional inhomogeneous pseudo-parabolic-type equations involving a nonlinearity of the form |u|p+ι|∇u|q, where p,q>1, and ι≥0 is a constant. The cases ι=0 and ι>0 are discussed separately. For each case, the critical exponent in the Fujita se...

We consider a parametric double phase Dirichlet problem. In the reaction there is a superlinear perturbation term which satisfies a weak Nehari-type monotonicity condition. Using the Nehari manifold method, we show that for all parameters below a critical value, the problem has at least three nontrivial solutions all with sign information. The crit...

In this paper, we consider local Dirichlet problems driven by the (r(u),s(u))-Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents r,s are real continuous functions and we have dependence on the solution u. The main contributions of this article are obtained in respec...

We study the behavior of solutions for the parametric equation \begin{document}$ -\Delta_{p}^{a_1} u(z)-\Delta_{q}^{a_2} u(z) = \lambda |u(z)|^{q-2} u(z)+f(z,u(z)) \quad \mbox{in } \Omega,\, \lambda >0, $\end{document} under Dirichlet condition, where $ \Omega \subseteq \mathbb{R}^N $ is a bounded domain with a $ C^2 $-boundary $ \partial \Omega $,...

In the last years, deep learning models have achieved remarkable generalization capability on computer vision tasks, obtaining excellent results in fine-grained classification problems. Sophisticated approaches based-on discriminative feature learning via patches have been proposed in the literature, boosting the model performances and achieving th...

We consider a parametric nonlinear Robin problem driven by the p-Laplacian and with a reaction having the competing effects of two terms. One is a parametric (p−1)-sublinear term (concave nonlinearity) and the other is a (p−1)-superlinear term (convex nonlinearity). We assume that the weight of the concave term is indefinite (that is, sign-changing...

We study the behavior of solutions for the parametric equation $$-\Delta_{p}^{a_1} u(z)-\Delta_{q}^{a_2} u(z)=\lambda |u(z)|^{q-2} u(z)+f(z,u(z)) \quad \mbox{in } \Omega,\, \lambda >0,$$ under Dirichlet condition, where $\Omega \subseteq \mathbb{R}^N$ is a bounded domain with a $C^2$-boundary $\partial \Omega$, $a_1,a_2 \in L^\infty(\Omega)$ with $...

We consider a inhomogeneous semilinear wave equation on a noncompact complete Riemannian manifold (M,g) of dimension N≥3, without boundary. The reaction exhibits the combined effects of a critical term and of a forcing term. Using a rescaled test function argument together with appropriate estimates, we show that the equation admits no global solut...

We study a nonlinear p(x)-Kirchhoff type problem with Dirichlet boundary condition, in the case of a reaction term depending also on the gradient (convection). Using a topological approach based on the Galerkin method, we discuss the existence of two notions of solutions: strong generalized solution and weak solution. Strengthening the bound on the...

We study the existence of positive solutions for a class of double phase Dirichlet equations which have the combined effects of a singular term and of a parametric superlinear term. The differential operator of the equation is the sum of a p-Laplacian and of a weighted q-Laplacian (q<p) with discontinuous weight. Using the Nehari method, we show th...

We consider a nonlinear Dirichlet problem driven by a (p, q)-Laplace differential operator (1 < q < p). The reaction is (p − 1)-linear near ±∞ and the problem is noncoercive. Using variational tools and truncation and comparison techniques together with critical groups, we produce five nontrivial smooth solutions all with sign information and order...

We consider an anisotropic (p,2)-equation, with a parametric and superlinear reaction term. We show that for all small values of the parameter the problem has at least five nontrivial smooth solutions, four with constant sign and the fifth nodal (sign-changing). The proofs use tools from critical point theory, truncation and comparison techniques,...

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the...

We prove the existence of at least three weak solutions to a mixed Dirichlet–Neumann boundary value problem for equations driven by the p(z)‐Laplace operator in the principal part. Our approach is variational and use three critical points theorems.

We consider nonlinear systems driven by a general nonhomogeneous differential operator with various types of boundary conditions and with a reaction in which we have the combined effects of a maximal monotone term $A(x)$ A ( x ) and of a multivalued perturbation $F(t,x,y)$ F ( t , x , y ) which can be convex or nonconvex valued. We consider the cas...

We study the existence of positive solutions for a class of double phase Dirichlet equations which have the combined effects of a singular term and of a parametric superlinear term. The differential operator of the equation is the sum of a $p$-Laplacian and of a weighted $q$-Laplacian ($q<p$) with discontinuous weight. Using the Nehari method, we s...

We study the wave inequality with a Hardy potential ∂ t t u − Δ u + λ | x | 2 u ≥ | u | p in ( 0 , ∞ ) × Ω , $$\begin{array}{} \displaystyle \partial_{tt}u-{\it\Delta} u+\frac{\lambda}{|x|^2}u\geq |u|^p\quad \mbox{in } (0,\infty)\times {\it\Omega}, \end{array}$$ where Ω is the exterior of the unit ball in ℝ N , N ≥ 2, p > 1, and λ ≥ − N − 2 2 2 $\b...

The aim of this paper is to establish the existence of at least two non-zero homoclinic solutions for a nonlinear Laplacian difference equation without using Ambrosetti-Rabinowitz type-conditions. The main tools are mountain pass theorem and Palais-Smale compactness condition involving suitable functionals.

We investigate the large-time behavior of solutions for a class of inhomogeneous semilinear wave equations involving double damping and potential terms. Namely, we first establish a general criterium for the absence of global weak solutions. Next, some special cases of potential and inhomogeneous terms are studied. In particular, when the inhomogen...

We consider a parametric Dirichlet problem driven by the (p,q)-Laplacian and a reaction which is gradient dependent (convection) and the competing effects of two more terms, one a parametric singular term and a locally defined perturbation. We show that for all small values of the parameter the problem has a positive smooth solution.

This book collects papers on major topics in fixed point theory and its applications. Each chapter is accompanied by basic notions, mathematical preliminaries and proofs of the main results. The book discusses common fixed point theory, convergence theorems, split variational inclusion problems and fixed point problems for asymptotically nonexpansi...

We consider a nonlinear Dirichlet problem driven by the sum of a p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p$\end{document}-Laplacian and of a q\documentclass[12p...

We consider a parametric nonlinear Robin problem driven by the negative p-Laplacian plus an indefinite potential. The equation can be thought as a perturbation of the usual eigenvalue problem. We consider the case where the perturbation f(z,·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{a...

We investigate the large time behavior for the inhomogeneous damped wave equation with nonlinear memory ϕtt(t,ω)−Δϕ(t,ω)+ϕt(t,ω)=1Γ(1−ρ)∫0t(t−σ)−ρ|ϕ(σ,ω)|qdσ+μ(ω),t>0, ω∈RN imposing the condition (ϕ(0,ω),ϕt(0,ω))=(ϕ0(ω),ϕ1(ω))inRN, where N≥1, q>1, 0<ρ<1, ϕi∈Lloc1(RN), i=0,1, μ∈Lloc1(RN) and μ≢0. Namely, it is shown that, if ϕ0,ϕ1≥0, μ∈L1(RN) and ∫R...

We consider a nonlinear parametric Neumann problem driven by the sum of a p-Laplacian and of a q-Laplacian and exhibiting in the reaction the competing effects of a singular term and of a resonant term. Using variational methods together with suitable truncation and comparison techniques, we show that for small values of the parameter the problem h...

We consider a nonlinear elliptic Dirichlet problem driven by the (p,q)-Laplacian and a reaction consisting of a parametric singular term plus a Carathéodory perturbation f(z,x) which is (p−1)-linear as x→+∞. First we prove a bifurcation-type theorem describing in an exact way the changes in the set of positive solutions as the parameter λ>0 moves....

This paper is concerned with the nonexistence of global solutions to fractional in time nonlinear Schrödinger equations of the form i α ∂ t α ω ( t , z ) + a 1 ( t ) Δ ω ( t , z ) + i α a 2 ( t ) ω ( t , z ) = ξ | ω ( t , z ) | p , ( t , z ) ∈ ( 0 , ∞ ) × R N , where N ≥ 1 , ξ ∈ C ∖ { 0 } and p > 1 , under suitable initial data. To establish our no...

We consider a Neumann problem driven by the (p, q)-Laplacian under the Landesman-Lazer type condition. Using the classical saddle point theorem and other classical results of the calculus of variations, we show that the problem has at least one nontrivial weak solution.

We consider a nonlinear control system involving a maximal monotone map and with a priori feedback. We assume that the control constraint multifunction U(t,x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\od...

We consider a parametric Robin problem driven by a nonlinear, nonhomogeneous differential operator which includes as special cases the p-Laplacian and the (p,q)-Laplacian. The source term is parametric and only locally defined (that is, in a neighborhood of zero). Using suitable cut-off techniques together with variational tools and comparison prin...

We give sufficient conditions for the existence of weak solutions to quasilinear elliptic Dirichlet problem driven by the A-Laplace operator in a bounded domain Ω. The techniques, based on a variant of the symmetric mountain pass theorem, exploit variational methods. We also provide information about the asymptotic behavior of the solutions as a su...

We consider a parametric nonlinear elliptic problem driven by the sum of a p-Laplacian and of a q-Laplacian (a (Formula presented.) -equation) with a singular and (Formula presented.) -superlinear reaction and a Robin boundary condition with (Formula presented.) -sublinear boundary term (Formula presented.). So, the problem has the combined effects...

We consider a nonlinear Robin problem driven by a nonhomogeneous differential operator, with reaction which exhibits the competition of two Carath\'eodory terms. One is parametric, $(p-1)$-sublinear with a partially concave nonlinearity near zero. The other is $(p-1)$-superlinear and has almost critical growth. Exploiting the special geometry of th...

We consider a nonlinear Robin problem driven by a nonhomogeneous differential operator, with reaction which exhibits the competition of two Carathéodory terms. One is parametric, \((p-1)\)-sublinear with a partially concave nonlinearity near zero. The other is \((p-1)\)-superlinear and has almost critical growth. Exploiting the special geometry of...

We study the large-time behavior of solutions to the nonlinear exterior problem L u ( t , x ) = κ | u ( t , x ) | p , ( t , x ) ∈ ( 0 , ∞ ) × D c under the nonhomegeneous Neumann boundary condition ∂ u ∂ ν ( t , x ) = λ ( x ) , ( t , x ) ∈ ( 0 , ∞ ) × ∂ D , where L : = i ∂ t + Δ is the Schrödinger operator, D = B ( 0 , 1 ) is the open unit ball in...

We consider a parametric nonlinear Dirichlet problem driven by the sum of a p-Laplacian and of a Laplacian (a (p,2)-equation) and with a reaction which has the competing effects of two distinct nonlinearities. A parametric term which is (p−1)-superlinear (convex term) and a perturbation which is (p−1)-sublinear (concave term). First we show that fo...

We consider a parametric Dirichlet problem driven by the sum of a p-Laplacian (\(p>2\)) and a Laplacian (a (p, 2)-equation). The reaction consists of an asymmetric \((p-1)\)-linear term which is resonant as \(x \rightarrow - \infty \), plus a concave term. However, in this case the concave term enters with a negative sign. Using variational tools t...

We consider an elliptic equation driven by a p-Laplacian-like operator, on an n-dimensional Riemannian manifold. The growth condition on the right-hand side of the equation depends on the geometry of the manifold. We produce a nontrivial solution by using a Palais–Smale compactness condition and a mountain pass geometry.

We consider a parametric double phase Dirichlet problem. Using variational tools together with suitable truncation and comparison techniques, we show that for all parametric values [Formula: see text] the problem has at least three nontrivial solutions, two of which have constant sign. Also, we identify the critical parameter [Formula: see text] pr...

We consider a parametric double phase problem with Robin boundary condition. We prove two existence theorems. In the first the reaction is ( p − 1 )-superlinear and the solutions produced are asymptotically big as λ → 0 + . In the second the conditions on the reaction are essentially local at zero and the solutions produced are asymptotically small...

Our objective is to study a new type of Dirichlet boundary value problem consisting of a system of equations with parameters, where the reaction terms depend on both the solution and its gradient (i.e., they are convection terms) and incorporate the effects of convolutions. We present results on existence, uniqueness and dependence of solutions wit...

We consider a parametric nonlinear Robin problem driven by a nonlinear nonhomogeneous differential operator plus an indefinite potential. The reaction term is \((p-1)\)-superlinear but need not satisfy the usual Ambrosetti–Rabinowitz condition. We look for positive solutions and prove a bifurcation-type result for the set of positive solutions as t...

We consider a parametric nonlinear Robin problem driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation). The reaction term is \((p-1)\)-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques and critical groups, we show that for...

The aim of this paper is to prove the existence of at least one nontrivial weak solution for equations involving the (p(·),q(·))-Laplace operator. The approach is variational and based on the critical point theory.

We consider a nonlinear Neumann problem driven by a nonhomogeneous differential operator with a reaction term that exhibits strong resonance at infinity. Using variational tools based on the critical point theory, we prove the existence of two nontrivial smooth solutions.

We propose an iterative algorithm to interpolate graph signals from only a partial set of samples. Our method is derived from the well known Papoulis-Gerchberg algorithm by considering the optimal value of a constant involved in the iteration step. Compared with existing graph signal reconstruction algorithms, the proposed method achieves similar o...

We establish a best proximity pair theorem for noncyclic φ-condensing operators in strictly convex Banach spaces by using a measure of noncompactness. We also obtain a counterpart result for cyclic φ-condensing operators in Banach spaces to guarantee the existence of best proximity points, and so, an extension of Darbo’s fixed point theorem will be...

We consider a two phase eigenvalue problem driven by the (p,q)-Laplacian plus an indefinite and unbounded potential, and Robin boundary condition. Using a modification of the Nehari manifold method, we show that there exists a nontrivial open interval I⊆R such that every λ∈I is an eigenvalue with positive eigenfunctions. When we impose additional r...

We present two iterative algorithms to interpolate graph signals from only a partial set of samples. Our methods are derived from classical iterative schemes in presence of irregular samples and compared with existing graph signal reconstruction algorithms in order to study the rate of convergence and the computational efficiency. The experimental...

We consider nonlinear, nonhomogeneous Robin problems with a (p − 1)-superlinear reaction term, which need not satisfy the Ambrosetti-Rabinowitz condition. We look for positive solutions and prove existence and multiplicity theorems. For the particular case of the p-Laplacian, we prove existence results under a different geometry near the origin.

We consider nonlinear elliptic equations driven by a nonhomogeneous differential operator plus an indefinite potential. The boundary condition is either Dirichlet or Robin (including as a special case the Neumann problem). First we present the corresponding regularity theory (up to the boundary). Then we develop the nonlinear maximum principle and...

We consider a nonlinear Dirichlet equation driven by the sum of a p-Laplacian and of a Laplacian (a (p, 2)-equation). The hypotheses on the reaction f(z, x) are minimal and make the energy (Euler) functional of the problem coercive. We prove two multiplicity theorems producing three and four nontrivial smooth solutions, respectively, all with sign...

We consider a semilinear Neumann problem with convection. We assume that the drift coefficient is indefinite. Using the theory of nonlinear operators of monotone type, together with truncation and comparison techniques and flow invariance arguments, we prove a multiplicity theorem producing three nontrivial smooth solutions (positive, negative and...

We consider the notion of generalized c-distance in the setting of ordered cone b-metric spaces and obtain some new fixed point results. Our results provide a more general statement, under which can be unified some theorems of the existing literature. In particular, we refer to the results of Sintunavarat et al. [W. Sintunavarat, Y.J. Cho, P. Kumam...

We consider a nonlinear nonparametric Dirichlet problem driven by the sum of a p-Laplacian and of a Laplacian (a (p, 2)-equation) and a reaction which involves a singular term and a (p-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepa...

We investigate the existence of fixed points of self-mappings via simulation functions and measure of noncompactness. We use different classes of additional functions to get some general contractive inequalities. As an application of our main conclusions, we survey the existence of a solution for a class of integral equations under some new conditi...

We study a semilinear Robin problem driven by the Laplacian plus an indefinite potential. We consider the case where the reaction term f is a Carathéodory function exhibiting linear growth near ±∞. So, we establish the existence of at least two solutions, by using the Lyapunov-Schmidt reduction method together with variational tools.

We consider a nonlinear nonhomogeneous Dirichlet problem driven by the sum of a p-Laplacian and a Laplacian and a reaction term which is (p− 1)-linear near ±∞ and resonant with respect to any nonprincipal variational eigenvalue of (−∆p, W01,p(Ω)). Using variational tools together with truncation and comparison techniques and Morse Theory (critical...

We consider a parametric nonlinear Robin problem driven by a nonhomogeneous differential operator. The reaction is a Carathéodory function which is only locally defined (that is, the hypotheses concern only its behaviour near zero). The conditions on the reaction are minimal. Using variational tools together with truncation, perturbation and compar...

We consider a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential and a Caratheodory reaction term which is resonant both at zero and ±∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgre...

We consider a nonlinear parametric Dirichlet problem driven by the p-Laplacian and a reaction which exhibits the competing effects of a singular term and of a resonant perturbation. Using variational methods together with suitable truncation and comparison techniques, we prove a bifurcation-type theorem describing the dependence on the parameter of...

We prove the existence of at least two positive homoclinic solutions for a discrete boundary value problem of equations driven by the (p,q)-Laplace operator. The properties of the nonlinearity ensure that the energy functional, corresponding to the problem, satisfies a mountain pass geometry and a Palais–Smale compactness condition.

We consider a nonlinear nonhomogeneous Robin equation driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation) plus an indefinite potential term and a parametric reaction of logistic type (superdiffusive case). We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter \(\...

We consider a semilinear Robin problem with an indefinite linear part and a superlinear reaction term, which does not satisfy the usual in such cases AR condition. Using variational methods, together with truncation–perturbation techniques and Morse theory (critical groups), we establish the existence of three nontrivial solutions. Our result exten...

In this paper, a reduction technique is used to show that some recent results on the existence of best proximity points for various classes of proximal contractions can be concluded from the corresponding results in fixed point theory.

We consider nonlinear elliptic Dirichlet problems driven by the sum of a p-Laplacian and a Laplacian (a -equation). The reaction term at is resonant with respect to any variational eigenvalue of the p-Laplacian. We prove two multiplicity theorems for such equations.

We consider a multivalued nonlinear Duffing system driven by a nonlinear nonhomogeneous differential operator. We prove existence theorems for both the convex and nonconvex problems (according to whether the multivalued perturbation is convex valued or not). Also, we show that the solutions of the nonconvex problem are dense in those of the convex...

We consider relatively Meir–Keeler condensing operators to study the existence of best proximity points (pairs) by using the notion of measure of noncompactness, and extend a result of Aghajani et al. [‘Fixed point theorems for Meir–Keeler condensing operators via measure of noncompactness’, Acta Math. Sci. Ser. B35 (2015), 552–566]. As an applicat...

We consider differential systems in RN driven by a nonlinear nonhomogeneous second order differential operator, a maximal monotone term and a multivalued perturbation F(t,u,u′). For periodic systems we prove the existence of extremal trajectories, that is solutions of the system in which F(t,u,u′) is replaced by extF(t,u,u′) (= the extreme points o...

We consider a multivalued nonlinear Duffing system driven by a nonlinear nonhomogeneous differential operator. We prove existence theorems for both the convex and nonconvex problems (according to whether the multivalued perturbation is convex valued or not). Also, we show that the solutions of the nonconvex problem are dense in those of the convex...

We consider a nonlinear elliptic problem driven by the Dirichlet $p$-Laplacian and a reaction term which depends also on the gradient (convection). No growth condition is imposed on the reaction term $f(z, \cdot,y)$. Using topological tools and the asymptotic analysis of a family of perturbed problems, we prove the existence of a positive smooth so...

We consider nonlinear (driven by the p-Laplacian) and semilinear Robin problems with indefinite potential and strong resonance with respect to the principal eigenvalue. Using variational methods and critical groups, we prove four multiplicity theorems producing up to four nontrivial smooth solutions.

We apply fixed point iterative schemes to variational inequality problems, via admissible perturbations of projection operators in real Hilbert spaces. Then, we prove some convergence theorems, extending and complementing the results in the existing literature. In particular, we deal with the class of α-co-coercive operators with application to gen...

We consider a nonlinear Robin problem driven by the p-Laplacian, with unilateral constraints and a reaction term depending also on the gradient (convection term). Using a topological approach based on fixed point theory (the Leray-Schauder alternative principle) and approximating the original problem using the Moreau-Yosida approximations of the su...

We consider a two-point boundary value problem of second-order random differential equation. Using a variant of the α-ψ-contractive type mapping theorem in metric spaces, we show the existence of at least one solution.

We consider Dirichlet boundary value problems for equations involving the (p(z), q(z))-Laplacian operator in the principal part and prove the existence of one and three nontrivial weak solutions, respectively. Here, the nonlinearity in the reaction term is allowed to depend on the solution, but does not satisfy the Ambrosetti–Rabinowitz condition....

The purpose of this article is to compute a global minimizer of the function x⟶d(x,Tx)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\longrightarrow d(x, Tx)$$\end{do...

For the homogeneous Dirichlet problem involving a system of equations driven by (p, q)-Laplacian operators and general gradient dependence we prove the existence of solutions in the ordered rectangle determined by a subsolution-supersolution. This extends the preceding results based on the method of subsolution-supersolution for systems of elliptic...

We consider a discrete Dirichlet boundary value problem of equations with the (p, q)-Laplacian operator in the principal part and prove the existence of at least two positive solutions. The assumptions on the reaction term ensure that the Euler-Lagrange functional, corresponding to the problem, satisfies an abstract two critical points result.

We consider an approximate multivalued fixed point problem under two constraint inequalities, for which we provide sufficient conditions for the existence of at least one solution. Then, we present some consequences and related results.

We consider a semilinear Robin problem driven by Laplacian plus an indefinite and unbounded potential. The reaction function contains a concave term and a perturbation of arbitrary growth. Using a variant of the symmetric mountain pass theorem, we show the existence of smooth nodal solutions which converge to zero in C¹(Ω). If the coefficient of th...

The main concern of this study is to introduce the notion of \(\varphi \)-best proximity points and establish the existence and uniqueness of \(\varphi \)-best proximity point for non-self mappings satisfying \((F,\varphi )\)-proximal and \((F,\varphi )\)-weak proximal contraction conditions in the context of complete metric spaces. Some examples a...

We consider a class of nonlinear initial value problems with delay. Using an abstract fixed point theorem, we prove an existence result producing a unique bounded solution.

In this chapter we review a class of contraction conditions, which are largely used to obtain interesting generalizations of the Banach fixed-point theorem in various abstract settings. We also present a new fixed-point existence result obtained by considering such a kind of contraction condition and a measure of noncompactness. Moreover, we show t...

This book offers a comprehensive treatment of the theory of measures of noncompactness. It discusses various applications of the theory of measures of noncompactness, in particular, by addressing the results and methods of fixed-point theory. The concept of a measure of noncompactness is very useful for the mathematical community working in nonline...

In this article, we use fixed-point methods and measure of noncompactness theory to focus on the problem of establishing the existence of at least a solution for the following functional integral equation u(t) = g (t, u(t)) + ∫0t G(t, s, u(s)) ds, t ∈ [0, +∞[, in the space of all bounded and continuous real functions on ℝ+, under suitable assumptio...

We introduce a contractive condition involving new auxiliary function and prove a fixed point theorem for closed multivalued maps on complete metric spaces. An example and an application to integral equation are given in support of our findings.

We consider a semilinear elliptic problem with Robin boundary condition and an indefinite and unbounded potential. The reaction term is a Carathéodory function exhibiting linear growth near ±∞. We assume that double resonance occurs with respect to any positive spectral interval. Using variational tools and critical groups, we show that the problem...