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We investigate the Cauchy problem for the nonlinear damped wave equation $u_{tt}-\Delta u +u_t=|u|^p+|\nabla u|^q +w(x)$, where $N\geq 1$, $p,q>1$, $w\in L^1_{loc}(\mathbb{R}^N)$, $w\geq 0$ and $w\not\equiv 0$. Namely, we first obtain the Fujita critical exponent for the considered problem. Next, we determine its second critical exponent in the sense of Lee and Ni. In particular, we show that the nonlinear gradient term $|\nabla u|^q$ induces a phenomenon of discontinuity of the Fujita critical exponent.

Abstract In the present work, a numerical technique for solving a general form of nonlinear fractional order integro-differential equations (GNFIDEs) with linear functional arguments using Chebyshev series is presented. The recommended equation with its linear functional argument produces a general form of delay, proportional delay, and advanced non-linear arbitrary order Fredholm–Volterra integro-differential equations. Spectral collocation method is extended to study this problem as a matrix discretization scheme, where the fractional derivatives are characterized in the Caputo sense. The collocation method transforms the given equation and conditions to an algebraic nonlinear system of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. The introduced operational matrix of derivatives includes arbitrary order derivatives and the operational matrix of ordinary derivative as a special case. To the best of authors’ knowledge, there is no other work discussing this point. Numerical test examples are given, and the achieved results show that the recommended method is very effective and convenient.

Abstract In this article, we suggest a new notion of fractional derivative involving two singular kernels. Some properties related to this new operator are established and some examples are provided. We also present some applications to fractional differential equations and propose a numerical algorithm based on a Picard iteration for approximating the solutions. Finally, an application to a heat conduction problem is given.

Abstract We first establish some results involving Riemann–Liouville fractional integrals for partially differentiable functions. Then we obtain some fractional Ostrowski type inequalities for functions in class of functions L p $L_{p}$ , L ∞ $L_{\infty }$ and L 1 $L_{1}$ , respectively. We also give some midpoint type inequalities as special cases of our main results.

Abstract This paper is concentrated on a class of difference equations with a Weyl-like fractional difference in a Banach space X forms like △αx(n)=Ax(n+1)+F(n,x(n)),n∈Z, $$ \triangle ^{\alpha }x(n)=Ax(n+1)+F\bigl(n, x(n)\bigr), \quad n\in \mathbb{Z}, $$ where α∈(0,1) $\alpha \in (0, 1)$, the operator A generates a C0 $C_{0}$-semigroup on X, △α $\triangle ^{\alpha }$ denotes the Weyl-like fractional difference operator, F(n,x):Z×X→X $F(n, x): \mathbb{Z}\times X\rightarrow X$ is a nonlinear function. Some existence theorems for asymptotically almost periodic mild solutions to this system are obtained with the nonlinear perturbation F being of Lipschitz type or non-Lipschitz type. The results are a consequence of applications of the Banach contraction mapping theory, the Leray–Schauder alternative theorem, and Matkowski’s fixed point theorem. As an application, an example is provided to show the feasibility of the theoretical results.

We consider a higher-order evolution differential inequality in an exterior domain of ℝN{\mathbb{R}^{N}}, N≥3{N\geq 3}, with Dirichlet and Neumann boundary conditions. Using a unified approach, we obtain the critical exponents in the sense of Fujita for the considered problems. Moreover, the behavior of the solutions with respect to the initial data is discussed.

Abstract A Lyapunov-type inequality is derived for a nonlinear fractional boundary value problem involving Caputo-type fractional derivative. The obtained inequality provides a necessary condition for the existence of nontrivial solutions to the considered problem. Next, we extend our study to the case of systems.

In this study, considering the advantages of parallel fixed point algorithms arising from their symmetrical behavior, new types of parallel algorithms have been defined. Strong convergence of these algorithms for certain mappings with altering points has been analyzed, and it has been observed that their convergence behavior is better than existing algorithms with non-simple samples. In addition, the concept of data dependency for these algorithms has been examined for the first time in this study. Finally, it has been proven that the solution of a variational inequality system can be obtained using newly defined parallel algorithms under suitable conditions.

Abstract Recently, many generalizations and extensions of well-known inequalities were obtained via different kinds of fractional integrals. In this paper, we show that most of those results are particular cases of (or equivalent to) existing inequalities from the literature. As consequence, such results are not real generalizations.

Abstract We establish Lyapunov-type inequalities for a system involving one-dimensional ( p i , q i ) $(p_{i},q_{i})$ -Laplacian operators ( i = 1 , 2 $i=1,2$ ). Next, the obtained inequalities are used to derive some geometric properties of the generalized spectrum associated to the considered problem.

We provide sufficient conditions for the blow-up of solutions to certain second-order differential inequalities and systems with advanced and delayed arguments. Our proofs are based on the test function method.

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 ∫RNμ(ω)dω>0, then for all q>1, the considered problem has no global weak solution.

In this paper, we introduce the concept of d*-complete topological spaces, which include earlier defined classes of complete metric spaces and quasi b-metric spaces. Further, we prove some fixed point results for mappings defined on d*-complete topological spaces, generalizing earlier results of Tasković, Ćirić and Prešić, Prešić, Bryant, Marjanović, Yen, Caccioppoli, Reich and Bianchini.

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 nonexistence theorem, we adopt the Pohozaev nonlinear capacity method, and consider the combined effects of absorption and dispersion terms. Further, we discuss in details some special cases of coefficient functions a 1 , a 2 ∈ L l o c 1 ( [ 0 , ∞ ) , R ) , and provide two illustrative examples.

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 R N , N ≥ 2 , D c = R N ∖ D , p > 1 , κ ∈ C , κ ≠ 0 , λ ∈ L 1 ( ∂ D , C ) is a nontrivial complex valued function, and ∂ ν is the outward unit normal vector on ∂ D , relative to D c . Namely, under a certain condition imposed on ( κ , λ ) , we show that if N ≥ 3 and p < p c , where p c = N N − 2 , then the considered problem admits no global weak solutions. However, if N = 2 , then for all p > 1 , the problem admits no global weak solutions. The proof is based on the test function method introduced by Mitidieri and Pohozaev, and an adequate choice of the test function.

We establish a Fujita-type theorem for the blow-up of nonnegative solutions to a certain class of parabolic inequalities in the Heisenberg group. Our proof is based on a duality argument.

In this article we study three differential inequalities of Sobolev type. Using Pokhozhaev's nonlinear capacity method, we provide sufficient conditions for the nonexistence of global nontrivial weak solutions to these problems.

Recently, Samet et al [24] introduced the concept of alpha-psi contractive mappings and studied the existence of fixed points for such mappings. In this article, we prove three fixed point theorems for this class of operators in complete metric spaces. Our results extend the results in [24] and well known fixed point theorems due to Banach, Kannan, Chatterjea, Zamfirescu, Berinde, Suzuki, Ciric, Nieto, Lopez, and many others. We prove that alpha-psi contractions unify large classes of contractive type operators, whose fixed points can be obtained by means of the Picard iteration. Finally, we utilize our results to discuss the existence and uniqueness of solutions to a class of quadratic integral equations.

In this study, we investigate the existence of chaos in the global cryptocurrency market. Specifically, we analyze parameters of chaotic order, nonlinearity, sensitivity to the initial conditions, monofractality, and multifractality. For this purpose, we conduct a comprehensive series of tests, including Brock–Dechert–Scheinkman (BDS) test, largest Lyapunov exponent, box-counting, and monogram analysis for fractal dimension, and multiple tests for long-range dependence (Aggregated Variances, Peng, Higuchi, R/S Analysis, and Multifractal Detrended Fluctuation Analysis (MFDFA)). All tests are performed over a variety of major cryptocurrencies: Bitcoin, Litecoin, Ethereum, and Ripple. The empirical results support the existence of chaos in the cryptocurrency market. Accordingly, cryptocurrency returns are not random and follow a chaotic order. Therefore, long term predictions are not possible, contrary to most of the discussions ongoing in the media and the public.

Recently, Azam, Arshad and Beg [ Banach contraction principle on cone rectangular metric spaces, Appl. Anal. Discrete Math. 2009] introduced the notion of cone rectangular metric spaces by replacing the triangular inequality of a cone metric space by a rectangular inequality. In this paper, we introduce the notion of c-chainable cone rectangular metric space and we establish a fixed point theorem for uniformly locally contractive mappings in such spaces. An example is given to illustrate our obtained result.

A class of nonlinear multipoint boundary value problems for singular fractional differential equations is considered. By means of a coupled fixed point theorem on ordered sets, some results on the existence and uniqueness of positive solutions are obtained.

We establish fixed-point results for mappings and cyclic mappings satisfying a generalized contractive condition in a complete gauge space. Our theorems generalize and extend some fixed-point results in the literature. We apply our obtained results to the study of existence and uniqueness of solution to a second-order nonlinear initial-value problem.

We extend the concept of α-ψ-contractive mappings introduced recently by Samet et al. (2012) to the setting of gauge spaces. New fixed point results are established on such spaces, and some applications to nonlinear integral equations on the half-line are presented.

We introduce new classes of cyclic mappings and we study the existence and uniqueness of fixed points for such mappings. The presented theorems generalize and improve several existing results in the literature.

Abstract In this study, we firstly derive a Wirtinger-type result, which gives the connection in between the integral of square of a function and the integral of square of its Caputo fractional derivatives with the help of left-sided and right-sided fractional Taylor’s Formulas. Afterward, we provide a more general inequality involving Caputo fractional derivatives for L r $L_{r}$ norm with r > 1 $r>1$ via Hölder’s inequality. Also, similar inequalities for Riemann–Liouville fractional derivatives are presented by means of a relation between Caputo fractional derivatives and Riemann–Liouville fractional derivatives.

A nonlinear time-fractional cable equation posed on the interval $ (0, 1) $ under a homogeneous Dirichlet boundary condition is investigated in this work. The considered equation reflects the anomalous electro-diffusion in nerve cells. Using nonlinear capacity estimates specifically adapted to the considered problem, we establish sufficient conditions for the nonexistence of weak solutions.

In this article, we establish Hermite-Hadamard-type inequalities for the two classes of functions X±λ(Ω)={f∈C2(Ω):Δf±λf≥0}{X}_{\pm \lambda }\left(\Omega )=\{f\in {C}^{2}\left(\Omega ):\Delta f\pm \lambda f\ge 0\}, where λ>0\lambda \gt 0 and Ω\Omega is an open subset of R2{{\mathbb{R}}}^{2}. We also obtain a characterization of the set X−λ(Ω){X}_{-\lambda }\left(\Omega ). Notice that in the one-dimensional case, if Ω=I\Omega =I (an open interval of R{\mathbb{R}}) and λ=ρ2\lambda ={\rho }^{2}, ρ>0\rho \gt 0, then Xλ(Ω){X}_{\lambda }\left(\Omega ) (resp. X−λ(Ω){X}_{-\lambda }\left(\Omega )) reduces to the class of functions f∈C2(I)f\in {C}^{2}\left(I) such that ff is trigonometrically ρ\rho -convex (resp. hyperbolic ρ\rho -convex) on II.

Nonexistence theorems constitute an important part of the theory of differential and partial differential equations. Motivated by the numerous applications of fractional differential equations in diverse fields, in this paper, we studied sufficient conditions for the nonexistence of solutions (or, equivalently, necessary conditions for the existence of solutions) for nonlinear fractional differential inequalities and systems in the sense of Erdélyi-Kober. Our approach is based on nonlinear capacity estimates specifically adapted to the Erdélyi-Kober fractional operators and some integral inequalities.

For a given sequence a=(a1,…,an)∈Rna=\left({a}_{1},\ldots ,{a}_{n})\in {{\mathbb{R}}}^{n}, our aim is to obtain an estimate of En≔a1+an2−1n∑i=1nai{E}_{n}:= \left|\hspace{-0.33em},\frac{{a}_{1}+{a}_{n}}{2}-\frac{1}{n}{\sum }_{i=1}^{n}{a}_{i},\hspace{-0.33em}\right|. Several classes of sequences are studied. For each class, an estimate of En{E}_{n} is obtained. We also introduce the class of convex matrices, which is a discrete version of the class of convex functions on the coordinates. For this set of matrices, new discrete Hermite-Hadamard-type inequalities are proved. Our obtained results are extensions of known results from the continuous case to the discrete case.

For μ∈C1(I)\mu \in {C}^{1}\left(I), μ>0\mu \gt 0, and λ∈C(I)\lambda \in C\left(I), where II is an open interval of R{\mathbb{R}}, we consider the set of functions f∈C2(I)f\in {C}^{2}\left(I) satisfying the second-order differential inequality ddtμdfdt+λf≥0\frac{{\rm{d}}}{{\rm{d}}t}\left(\phantom{\rule[-0.75em]{}{0ex}},\mu \frac{{\rm{d}}f}{{\rm{d}}t}\right)+\lambda f\ge 0 in II. The considered set includes several classes of generalized convex functions from the literature. In particular, if μ≡1\mu \equiv 1 and λ=k2\lambda ={k}^{2}, k>0k\gt 0, we obtain the class of trigonometrically kk-convex functions, while if μ≡1\mu \equiv 1 and λ=−k2\lambda =-{k}^{2}, k>0k\gt 0, we obtain the class of hyperbolic kk-convex functions. In this article, we establish a Fejér-type inequality for the introduced set of functions without any symmetry condition imposed on the weight function and discuss some special cases of weight functions. Moreover, we provide characterizations of the classes of trigonometrically and hyperbolic kk-convex functions.

We consider the class of functions $ u\in C^2((0, \infty)) $ satisfying second-order differential inequalities in the form $ u''(x)+\frac{k}{x^2}u(x)\geq 0 $ for all $ x > 0 $. For this class of functions, we establish Hermite-Hadamard-type inequalities in both cases ($ k=\frac{1}{4} $ and $ 0 < k < \frac{1}{4} $). We next extend our obtained results to the two-dimensional case. In the limit case $ k\rightarrow 0^+ $ we deriver some existing results from the literature that are related to convex functions and convex functions on the coordinates. In our approach, we make use of some tools from ordinary differential equations.

We unified several kinds of convexity by introducing the class $ \mathcal{A}_{\zeta, w}([0, 1]\times I^2) $ of $ (\zeta, w) $-admissible functions $ F: [0, 1]\times I\times I\to \mathbb{R} $. Namely, we proved that most types of convexity from the literature generate functions $ F\in \mathcal{A}_{\zeta, w}([0, 1]\times I^2) $ for some $ \zeta\in C([0, 1]) $ and $ w\in C^1(I) $ with $ w(I)\subset I $ and $ w' > 0 $. We also studied some properties of $ (\zeta, w) $-admissible functions and established some integral inequalities that unify various Hermite-Hadamard-type inequalities from the literature.

We introduce the class of $ \psi $-convex functions $ f:[0, \infty)\to \mathbb{R} $, where $ \psi\in C([0, 1]) $ satisfies $ \psi\geq 0 $ and $ \psi(0)\neq \psi(1) $. This class includes several types of convex functions introduced in previous works. We first study some properties of such functions. Next, we establish a double Hermite-Hadamard-type inequality involving $ \psi $-convex functions and a Simpson-type inequality for functions $ f\in C^1([0, \infty)) $ such that $ |f'| $ is $ \psi $-convex. Our obtained results are new and recover several existing results from the literature.

We establish weighted Hermite-Hadamard-type inequalities for some classes of differentiable functions without assuming any symmetry property on the weight function. Next, we apply our obtained results to the approximation of some classes of weighted integrals.

A class of nonlinear integral equations with delay, related to infectious diseases, is studied. Making use of some tools from operators theory, we deal with the well-posedness in an adequate functional space, approximation of solution, estimates of lower/upper solutions and the data dependence of solutions.

We establish necessary conditions for the existence of solutions to various systems of partial differential inequalities in the plane. The obtained conditions provide new Hermite-Hadamard-type inequalities for differentiable functions in the plane. In particular, we obtain a refinement of an inequality due to Dragomir [On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwan. J. Math. 5 (2001), no. 4, 775–788] for convex functions on the coordinates.

In this paper, we consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case.

We first construct new Hermite-Hadamard type inequalities which include generalized fractional integrals for convex functions by using an operator which generates some significant fractional integrals such as Riemann-Liouville fractional and the Hadamard fractional integrals. Afterwards, Trapezoid and Midpoint type results involving generalized fractional integrals for functions whose the derivatives in modulus and their certain powers are convex are established. We also recapture the previous results in the particular situations of the inequalities which are given in the earlier works.

We are concerned with the study of existence and nonexistence of weak solutions for a class of hyperbolic inequalities with a Hardy potential singular on the boundary $ \partial B_1 $ of the annulus $ A = \left\{x\in \mathbb{R}^3: 1 < |x|\leq 2\right\} $, where $ \partial B_1 = \left\{x\in \mathbb{R}^3: |x| = 1\right\} $. A singular potential function of the form $ (|x|-1)^{-\rho} $, $ \rho\geq 0 $, is considered in front of the power nonlinearity. Two types of inhomogeneous boundary conditions on $ (0, \infty)\times \partial B_2 $, $ \partial B_2 = \left\{x\in \mathbb{R}^3: |x| = 2\right\} $, are studied: Dirichlet and Neumann. We use a unified approach to show the optimal criteria of Fujita-type for each case.

We study the existence and nonexistence of weak solutions to a semilinear higher order (in time) evolution inequality involving a convection term in the exterior of the half-ball, under Dirichlet-type boundary conditions. A weight function of the form $ |x|^a $ is allowed in front of the power nonlinearity. When $ a > -2 $, we show that the dividing line with respect to existence or nonexistence is given by a critical exponent (Fujita critical exponent), which depends on the parameters of the problem, but independent of the order of the time-derivative. Our study yields naturally optimal nonexistence results for the corresponding stationary problem.

We study a time-fractional parabolic inequality posed on a bounded interval and involving a wight function W, where the fractional derivative is considered in the Caputo sense. We establish a general condition ensuring that the set of weak solutions is empty. Next, some particular cases of the weight function W are discussed.

We consider a fractional differential inequality involving $ \psi $-Caputo fractional derivatives of different orders, with a polynomial nonlinearity and a singular potential term. Using the test function method and some integral inequalities, we establish nonexistence criteria of global solutions in both cases: $ \lim\limits_{t\to \infty}\psi(t)=\infty $ and $ \lim\limits_{t\to \infty}\psi(t)

Several phenomena from natural sciences can be described by partial differential equations of Sobolev-type. On the other hand, it was shown that in many cases, the use of fractional derivatives provides a more realistic model than the use of standard derivatives. The goal of this paper is to study the nonexistence of weak solutions to a time-fractional differential inequality of Sobolev-type. Namely, we give sufficient conditions for the nonexistence or equivalently necessary conditions for the existence. Our method makes use of the nonlinear capacity method, which consists in making an appropriate choice of test functions in the weak formulation of the problem. This technique has been employed in previous papers for some classes of time-fractional differential inequalities of Sobolev-type posed on the whole space RN. The originality of this work is that the considered problem is posed on an annulus domain, which leads to some difficulties concerning the choice of adequate test functions.

We let υ be a convex function on an interval [ι1,ι2]⊂R. If ζ∈C([ι1,ι2]), ζ≥0 and ζ is symmetric with respect to ι1+ι22, then υ12∑j=12ιj∫ι1ι2ζ(s)ds≤∫ι1ι2υ(s)ζ(s)ds≤12∑j=12υ(ιj)∫ι1ι2ζ(s)ds. The above estimates were obtained by Fejér in 1906 as a generalization of the Hermite–Hadamard inequality (the above inequality with ζ≡1). This work is focused on the study of right-side Fejér-type inequalities in one- and two-dimensional cases for new classes of differentiable functions υ. In the one-dimensional case, the obtained results hold without any symmetry condition imposed on the weight function ζ. In the two-dimensional case, the right side of Fejer’s inequality is extended to the class of subharmonic functions υ on a disk.

In this paper, we consider a two-times nonlinear fractional differential inequality involving both Hadamard-Caputo and Caputo fractional derivatives of different orders, with a singular potential term. We obtain sufficient criteria depending on the parameters of the problem, for which a global solution does not exist. Some examples are provided to support our main results.

First of all, we establish an identity for higher-order differentiable functions. Then, we prove some integral inequalities for mappings that have continuous derivatives up to the order $n-1$ with $n\geq 1$ and whose n-th derivatives are the element of $L_{1},~L_{r}$, and $L_{\infty }.$ In addition, estimates of new composite quadrature rules are examined. Finally, natural applications for exponential and logarithmic functions are given.

Abstract In this paper, first we obtain a new identity for quantum integrals, the result is then used to prove midpoint type inequalities for differentiable coordinated convex mappings. The outcomes provided in this article are an extension of the comparable consequences in the literature on the midpoint inequalities for differentiable coordinated convex mappings.

Since the celebrated Brouwer’s fixed point theorem and Banach contraction principle were established, the rapid growth of fixed point theory and its applications have led to a number of scholarly essays studying the importance of its promotion and application in nonlinear analysis, applied mathematical analysis, economics, game theory, integral and differential equations and inclusions, dynamic systems theory, signal and image processing, etc [...]

A Schrödinger equation with a time-fractional derivative, posed in (0,∞)×I, where I=]a,b], is investigated in this paper. The equation involves a singular Hardy potential of the form λ(x−a)2, where the parameter λ belongs to a certain range, and a nonlinearity of the form μ(x−a)−ρ|u|p, where ρ≥0. Using some a priori estimates, necessary conditions for the existence of weak solutions are obtained.