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Abstract: Our aim is to point out an error in the proof of Lemma 2 in the paper “Fixed point theory in symmetric spaces with applications to probabilistic spaces” by T.L. Hicks, B.E. Rhoades [Nonlinear Analysis 36 (1999) 331–344], and to indicate a way to repair it. [Copyright &y& Elsevier]

We extend some of main results of Abasov et al., 2016, Fotiy et al., (2020), Mykhaylyuk et al., (2015), Pliev and Fang, (2017), Pliev and Popov, (2014) to the setting of orthogonally additive operators on lattice-normed spaces. We introduce a new class of C -complete lattice-normed spaces which strictly includes a class of Banach–Kantorovich spaces. The first main result of the paper asserts that every laterally-to-norm continuous C -compact orthogonally additive operator T : X → Y from an atomless C -complete lattice-normed space X to a Banach space Y is narrow. As a non expecting consequence of the first main result we obtain necessary and sufficient conditions for a nonlinear superposition operator T N : E (X) → E (X) to be C -compact. We also show that the sum of two orthogonally additive operators G and T , where G is narrow and T is laterally-to-norm continuous and C -compact, is a narrow operator. In the last part of the article we investigate dominated orthogonally additive operators. In particular we show the narrowness of dominated operators taking values in a Banach sequence space Y or in a separable symmetric operator space C E. [ABSTRACT FROM AUTHOR]

In this paper we obtain symmetry and monotonicity of positive solutions for the systems involving fully nonlinear nonlocal operators in a domain (bounded or unbounded) in R n via using a direct method of moving planes. This extends the results in Wang and Niu (2017) and also is the first result of the symmetry for a fully nonlinear nonlocal system containing gradient terms with different order. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of W 2 , p (n < p < ∞) solutions of a double obstacle problem. Moreover, we show the optimal regularity of the solution and the local C 1 regularity of the free boundary under a thickness assumption at the free boundary point on the intersection of two free boundaries. In the study of the regularity of the free boundary, we deal with a general problem, the no-sign reduced double obstacle problem with an upper obstacle ψ , F (D 2 u , x) = f χ Ω (u) ∩ { u < ψ } + F (D 2 ψ , x) χ Ω (u) ∩ { u = ψ } , u ≤ ψ in B 1 , where Ω (u) = B 1 ∖ { u = 0 } ∩ { ∇ u = 0 } . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using the cone theory and Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions. The results in this paper are applied to a class of abstract semilinear evolution equations with noncompact semigroup in Banach spaces and the initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. The results obtained here improve and generalize many known results. [Copyright &y& Elsevier]

Abstract: In this paper, we establish the well-posedness for a class of nonlinear parabolic equations with strong degeneracy. By introducing a new definition of entropy solutions, we prove the uniqueness of weak solutions and this is the novel feature of the paper. [Copyright &y& Elsevier]

In this paper, we prove existence, uniqueness and special properties of the solution to a parabolic–elliptic reaction–diffusion system with cross-diffusion in an open bounded domain in R d , as well as in R d , d = 1 , 3 , for various classes of the system parameters. In the case of a bounded domain and of Neumann boundary conditions we develop an approach based on a semigroup technique in the dual space (H 1 (Ω)) ′. This gives the possibility of considering singular initial data, distributions or functionals that may represent relevant physical data. The influence of the system parameters upon the qualitative behavior of the solution is investigated and some sufficient conditions for the existence of global or local solutions, and for global solutions in the case of small initial data are provided. In the case of the whole space, the problem is studied for a singular diffusion expressed by a multivalued operator. Existence is proved by passing to the limit in an approximating problem for which existence follows via a semigroup approach in H − 1 (R d) , d = 1 , 3. An argument for the uniqueness is developed in Beppo Levi spaces. Finally, the results are illustrated by a model exhibiting a self-organized criticality. [ABSTRACT FROM AUTHOR]

We prove boundary higher integrability for the (spatial) gradient of very weak solutions of quasilinear parabolic equations of the form u t − div A (x , t , ∇ u) = 0 on Ω × (− T , T) , u = 0 on ∂ Ω × (− T , T) , where the non-linear structure A (x , t , ∇ u) is modeled after the variable exponent p (x , t) -Laplace operator given by | ∇ u | p (x , t) − 2 ∇ u. To this end, we prove that the gradients satisfy a reverse Hölder inequality near the boundary by constructing a suitable test function which is Lipschitz continuous and preserves the boundary values. In the interior case, such a result was proved in Verena Bögelein and Qifan Li (2014) provided p (x , t) ≥ p − ≥ 2 holds and was then extended to the singular case 2 n n + 2 < p − ≤ p (x , t) ≤ p + ≤ 2 in Qifan Li (2017). This restriction was necessary because the intrinsic scalings for quasilinear parabolic problems are different in the case p + ≤ 2 and p − ≥ 2. In this paper, we develop a new approach, using which, we are able to extend the results of Verena Bögelein and Qifan Li (2014), Qifan Li (2017) to the full range 2 n n + 2 < p − ≤ p (x , t) ≤ p + < ∞ and also obtain analogous results up to the boundary. The main novelty of this paper is that we make use of a unified intrinsic scaling using which our methods are able to handle both the singular case and degenerate case simultaneously. Our techniques improve and simplify many aspects of the method of parabolic Lipschitz truncation (even in the constant exponent case) studied extensively in existing literature. To simplify the exposition, we will only prove the higher integrability result near the boundary, provided the domain Ω satisfies a uniform measure density condition and are non perturbative in nature, hence we make no regularity assumptions for the coefficients of the nonlinear operator. Our techniques are also applicable to higher order equations as well as systems. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear Kirchhoff type problem with steep potential well: − a ∫ R 3 ∇ u 2 d x + b Δ u + λ V (x) u = f x | u | p − 2 u in R 3 , u ∈ H 1 (R 3) , where a , b , λ > 0 , 2 < p < 4 , V ∈ C (R 3 , R +) and f ∈ L ∞ (R 3 , R). Such problem cannot be studied by applying variational methods in a standard way, even by restricting its corresponding energy functional on the Nehari manifold, because Palais–Smale sequences may not be bounded. In this paper, we introduce a novel constraint method to prove the existence of one and two positive solutions under the different assumptions on V , respectively. We conclude that steep potential well V may help Kirchhoff type equations to generate multiple solutions, which has never been involved before. [ABSTRACT FROM AUTHOR]

Abstract: Consider in a real Hilbert space the differential equation (E), for a.a. , with the condition , where is a (possibly set-valued) maximal monotone operator, with (or, more generally, ); , with and either or . Recall that equation (E) in the case , subject to and , was investigated in the early 1970s by V. Barbu, who derived in particular from his results a definition for the square root of the nonlinear operator . Subsequently H. Brézis, N.H. Pavel, L. Véron and others have paid attention to equation (E). In this paper we prove the existence and uniqueness of the solution to equation (E) subject to in the weighted space , where , , under our weak assumptions on and (see above) and . For we prove the existence of a generalized solution. This is a classic solution if , . If , the solutions give rise to a nonlinear semigroup of contractions. If is linear its infinitesimal generator is given by . [Copyright &y& Elsevier]

Abstract: In this paper we present a simple and unified approach to the fixed point results on cone symmetric spaces and metric type spaces based on symmetric spaces fixed point theory. We also give a new characterization of semi-metric spaces with open balls. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to discuss the existence and uniqueness of fixed point in a partially ordered Banach space. Based on the characterizations of reproducing cones, some fixed point theorems for nonlinear operators are proved. As an application, the existence and uniqueness of periodical solution for a first order differential equation is discussed. [Copyright &y& Elsevier]

Abstract: This paper is devoted to providing a sufficient condition for the maximality of the sum of subdifferential operators defined on reflexive Banach spaces and proving the maximal monotonicity in of the nonlinear elliptic operator with a maximal monotone graph . [ABSTRACT FROM AUTHOR]

Abstract: By using Lim’s inequalities, Nadler’s results, the new parametric resolvent operator technique associated with -maximal monotone operators, in this paper, the existence theorem for a new class of generalized nonlinear parametric -maximal monotone operator inclusion systems with relaxed cocoercive type operators in Hilbert spaces is analyzed and established. Our results generalize sensitivity analysis results of other recent works on strongly monotone quasi-variational inclusions, nonlinear implicit quasi-variational inclusions and nonlinear mixed quasi-variational inclusion systems in Hilbert spaces. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study hybrid iterative schemes of Halpern types for a semigroup of relatively nonexpansive mappings on a closed and convex subset of a Banach space with respect to a sequence of asymptotically left invariant means defined on an appropriate invariant subspace of . We prove that given a certain sequence in , , we can generate an iterative sequence which converges strongly to where is the generalized projection from onto the fixed point set . Our main result is even new for the case of a Hilbert space. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce and consider a new generalized system of nonconvex variational inequalities with different nonlinear operators. We establish the equivalence between the generalized system of nonconvex variational inequalities and the fixed point problems using the projection technique. This equivalent alternative formulation is used to suggest and analyze a general explicit projection method for solving the generalized system of nonconvex variational inequalities. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities. [Copyright &y& Elsevier]

Abstract: In the paper, the new concepts of bifurcation points and asymptotic bifurcation points of the compact continuous operator are introduced in M-PN spaces. Some sufficient conditions for the existence of bifurcation points and asymptotic bifurcation points are obtained, and some theorems on the existence of intrinsic values are obtained. Meanwhile some theorems are generalized. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss some problems for nonlinear operators in the Menger PN space. Meanwhile, we prove existence theorems for the solution of an operator equation, and obtain some new results by means of the theory of topological degree for the Z-P-S space. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study the existence of fixed points by using measures of weak noncompactness. Later on, we provide an existence principle for solutions for a nonlinear integral equation. [Copyright &y& Elsevier]

Abstract: This paper deals with the eigenvalue problems for a class of positive nonlinear operators. The projective metric technique is used to prove the existence of a unique fixed point of the operator. We then show how to apply our results to the study of nonlinear elliptic problems. [Copyright &y& Elsevier]

Abstract: This paper presents some new methods of computation of the fixed point index for cone mapping. As applications of our main results, the existence of positive solutions for the fourth-order boundary value problem both in scale spaces and in abstract spaces is considered. [Copyright &y& Elsevier]

Abstract: Many dynamical systems in physics, chemistry, biology and engineering sciences have impulsive dynamical behaviors due to abrupt jumps at certain instants during the dynamical processes. The mathematical models of such processes are called differential systems with impulse effect. This paper studies the exact controllability issue of certain types of second order nonlinear impulsive control differential systems. Sufficient conditions are formulated and proved for the exact controllability of such systems. Without imposing a compactness condition on the cosine family of operators, we establish controllability results by using a fixed point analysis approach. Finally, an example is presented to illustrate the utility of the proposed result. The results improve some recent results. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the existence of at least three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations. The associated Green’s function for the boundary value problems is first given. The proofs of our main results are based upon the Leggett–Williams fixed point theorem. Finally, we give an example to demonstrate our result. [Copyright &y& Elsevier]

Abstract: This paper is devoted to using the Leggett–Williams fixed point theorem to derive some sufficient conditions for the existence of triple positive solutions to a type of -Laplacian boundary value problems on time scales. The interesting point is that the nonlinear term depends on the first-order delta derivative explicitly. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce two iterative sequences for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a Banach space. Then we study the strong and weak convergence of the sequences. [Copyright &y& Elsevier]

Abstract: In this paper we are concerned with the existence of symmetric positive solutions of the following singular second order three-point boundary value problem where , is symmetric on and may be singular at and . First, the Green’s function for associated linear boundary value problem is constructed, and some useful properties of the Green’s function are obtained. Then by applying the fixed-point index theory, we establish some optimal criteria for the existence of one or two symmetric positive solutions which involve the principal eigenvalue of a related linear operator. Finally we illustrate our results by several examples, none of which can be handled using the existing results. [Copyright &y& Elsevier]

Abstract: In this paper, we present a common fixed point theorem for a family of set-valued -generalized contraction mappings on gauge spaces, which generalizes the main result from Espínola and Kirk [R. Espínola, W.A. Kirk, Set-valued contractions and fixed points, Nonlinear Anal. 54 (2003) 485–494]. [Copyright &y& Elsevier]

Abstract: In this paper, by using fixed point theory, under quite general conditions on the nonlinear term, we obtain an existence result on bounded positive solutions of certain semilinear elliptic equations in . [Copyright &y& Elsevier]

Abstract: In this paper, the global exponential stability and periodicity are investigated for a class generalized high-order neural networks with time delays and impulses; some sufficient conditions are derived for checking the global exponential stability and existence of periodic solutions for this system using the generalized Halanay inequality, mathematical induction and a fixed point theorem. The criteria given are easily verified and possess many adjustable parameters, which provides flexibility for the design and analysis of the system. Finally, an example with numerical simulation is given to show the effectiveness of the proposed method and results. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the impulsive differential equation By using a fixed point theorem, some criteria are established for the existence of a positive -periodic solution of the above equation. Our work extends and improves some known results. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the following system on time scale : where and . By using the theory of the fixed point index, we investigate the effect of on the existence and nonexistence of positive solution for the above system in sublinear cases. The results obtained are even new for the special cases of difference system and differential system, as well as in the general time scale setting. [Copyright &y& Elsevier]

Abstract: In this paper, we established strong convergence theorems for a common fixed point of two asymptotically nonexpansive mappings and for a common fixed point of two asymptotically nonexpansive semigroups by using the hybrid method in a Hilbert space. Moreover, we also proved a strong convergence theorem for a common fixed point of two nonexpansive mappings. Our results extend and improve the recent ones announced by Kim and Xu [T.W. Kim, H.W. Xu, Strong convergence of modified Mann iteration for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006) 1140–1152], Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379], and many others. [Copyright &y& Elsevier]

Abstract: In this paper, we prove two strong convergence theorems of modified Ishikawa iteration and modified Halpern iteration for relatively nonexpansive mappings in a Banach space. Our results extend and improve the recent ones announced by Nakajo, Takahashi, Kim, Martinez-Yanes, Xu and some others. [Copyright &y& Elsevier]

Abstract: In this paper, we first introduce generalized -correspondence and generalized -majorized mappings without open lower sections in the nonconvexity setting of topological space. Some existence theorems of maximal elements for generalized -correspondence and generalized -majorized mappings are obtained in topological spaces without convexity. As applications, we establish new equilibrium existence theorems for qualitative games and generalized games with infinite sets of players and generalized -majorized preference correspondences in topological spaces without convexity. Our results generalize and improve the corresponding results in the recent literature. [Copyright &y& Elsevier]

Abstract: In this paper, we first give a generalized Takahashi’s existence theorem. From the existence theorem, we establish some equivalence relations between generalized Caristi’s fixed point theorems and generalized Ekeland’s variational principles. Some applications to the existence theorems of weak sharp minima and global error bounds for lower semicontinuous functions are also given. [Copyright &y& Elsevier]

Abstract: In this paper, by using the fixed point theorem and operator spectral theorem, the author establishes the theorem on the existence of positive solutions for the fourth-order boundary value problem with variable parameters as follows: where and is continuous. [Copyright &y& Elsevier]

Abstract: In this paper we study the existence of non-trivial periodic solutions for a periodic -Laplacian with nonlocal terms based on the theory of Leray–Schauder degree. The key step is dealing with the degeneracy of the -Laplacian and the logistic-type terms arising in the right hand side of the equation. [Copyright &y& Elsevier]

Abstract: In this paper, the boundary stabilization for a Kirchhoff-type nonlinear beam with one end fixed and control at the other end is considered. A gain adaptive controller is designed in terms of measured end velocity. The existence and uniqueness of the classical solution of the closed-loop system are justified. The exponential stability of the system is obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce the concept of the strict Kakutani multimap (KKM) property and investigate the fixed point problem for multimaps having this property on almost -convex subsets of locally -convex uniform spaces. Our new fixed point theorem generalizes the well-known Fan–Glicksberg fixed point theorem and partially extends the Himmelberg fixed point theorem. Its application to minimax theorem is also given. [Copyright &y& Elsevier]

Abstract: In this paper we consider the Lasota–Wazewska modelBy using a fixed point theorem, some criteria are established for the existence of the unique positive -periodic solution of the above equation. In particular, we not only give the conclusion of convergence of to , where is a successive sequence, but also show that is a global attractor of all other positive solutions. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the problem of the existence of positive solutions for non-linear third-order periodic boundary value problemHere, is a positive constant and our non-linearity may be singular at . The proof relies on a non-linear alternative of Leray–Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones. [Copyright &y& Elsevier]

In this paper, an existence result of sign-changing solution for some three-point boundary value problems is obtained by using the fixed-point index method. [Copyright &y& Elsevier]

In this paper, we discuss the necessary and sufficient condition for common fixed points of one-parameter nonexpansive semigroups in Banach spaces with the Opial property. We next prove the convergence theorem to a common fixed point. Finally, we discuss the existence of nonexpansive retraction onto the set of common fixed points. In these theorems, we may not assume that the strict (uniform) convexity of a Banach space. [Copyright &y& Elsevier]

In this paper the existence results of positive solutions are obtained for fourth-order periodic boundary value problemu(4)−βu″+αu=f(t, u), 0&les;t&les;1,u(i)(0)=u(i)(1), i=0,1,2,3,where f : [0, 1]×R+→R+ is continuous, α, β∈R and satisfy 0<α<(β/2+2π2)2,β>−2π2,α/π4+β/π2+1>0. The discussion is based on a new maximum principle for operator L4 u=u(4)−βu″+αu in periodic boundary condition and fixed point index theory in cones. [Copyright &y& Elsevier]

The aim of this paper is to present several properties of the nonnegative weak solutions to a class of very singular equations whose prototype is u t = div ( u m − 1 | D u | p − 2 D u) , p > 1 and 3 − p < m + p < 2. Namely, we prove L loc r and L loc r − L loc ∞ estimates and Harnack estimates. Note that 3 − p = m + p is a critical value: under this threshold the energy estimates hold with a reverse sign. [ABSTRACT FROM AUTHOR]

This paper deals with the following nonlinear Schrödinger–Bopp–Podolsky system − Δ u + V (x) u + ϕ u = μ f (u) + u 5 − 2 Δ ϕ + a 2 Δ 2 ϕ = 4 π u 2 in R 3 , where a > 0 , V ∈ C (R 3 , [ 0 , ∞)) with V ∞ ≔ lim | y | → ∞ V (y) ≥ sup x ∈ R 3 V (x) > 0 , and f ∈ C (R , R) satisfying ∫ 0 t f (s) d s ≥ t p with p ∈ (2 , 6) for all t ≥ 0. By using some new analytic techniques and new inequalities, we prove that the above system admits ground state solutions for all μ > 0 if p ∈ (4 , 6) ; for all μ > μ 0 if p ∈ (2 , 4 ] , where μ 0 is a positive constant determined by a , V ∞ and p. Our results extend and complement the existence results in d'Avenia and G. Siciliano (2019) which only dealt with the subcritical case. [ABSTRACT FROM AUTHOR]

We consider a nonlinear eigenvalue problem for some elliptic equations governed by general operators including the p -Laplacian. The natural framework in which we consider such equations is that of Orlicz–Sobolev spaces. We exhibit two positive constants λ 0 and λ 1 with λ 0 ≤ λ 1 such that λ 1 is an eigenvalue of the problem while any value λ < λ 0 cannot be so. By means of Harnack-type inequalities and a strong maximum principle, we prove the isolation of λ 1 on the right side. We emphasize that throughout the paper no Δ 2 -condition is needed. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a nonlinear problem driven by the fractional M -Laplacian and with a nonlinear nonhomogeneous reaction term. First, we give some further properties for the new fractional Orlicz–Sobolev space. Then, using the theory of nonlinear operators of monotone-type, we show the existence of a nonnegative solution. Our problem is nonvariational in nature. [ABSTRACT FROM AUTHOR]

Abstract: This is a summary of some results from recent papers of the author. Here we consider strongly nonlinear parabolic systems determined by nonlinear, monotone, hemicontinuous operator valued measures. We prove existence of solutions and their regularity properties and then consider optimal control problems, proving existence of optimal controls from the space of vector measures. Also we consider necessary conditions of optimality with or without state constraints. The paper is concluded with an example. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a class of doubly nonlinear quasivariational evolution inequalities for a class of reaction–diffusion models with time-derivative constraint. We prove an existence result by using the fixed point argument with the monotonicity technique in the nonlinear operator theory. [Copyright &y& Elsevier]