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In this paper, we study the following nonlinear first order partial differential equation: \[f(t,x,u,\partial_t u,\partial_x u)=0\quad\text{with}\quad u(0,x)\equiv 0.\] The purpose of this paper is to determine the estimate of Gevrey order under the condition that the equation is singular of a totally characteristic type. The Gevrey order is indicated by the rate of divergence of a formal power series. This paper is a continuation of the previous papers [Convergence of formal solutions of singular first order nonlinear partial differential equations of totally characteristic type, Funkcial. Ekvac. 45 (2002), 187-208] and [Maillet type theorem for singular first order nonlinear partial differential equations of totally characteristic type, Surikaiseki Kenkyujo Kokyuroku, Kyoto University 1431 (2005), 94-106]. Especially the last-mentioned paper is regarded as part I of this paper.

This note is concerned with the oscillation of third order nonlinear delay differential equations of the form \[\label{*} \left( r_{2}(t)\left( r_{1}(t)y^{\prime}(t)\right)^{\prime}\right)^{\prime}+p(t)y^{\prime}(t)+q(t)f(y(g(t)))=0.\tag{\(\ast\)}\] In the papers [A. Tiryaki, M. F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007), 54-68] and [M. F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear functional differential equations, Applied Math. Letters 23 (2010), 756-762], the authors established some sufficient conditions which insure that any solution of equation (\(\ast\)) oscillates or converges to zero, provided that the second order equation \[\left( r_{2}(t)z^{\prime }(t)\right)^{\prime}+\left(p(t)/r_{1}(t)\right) z(t)=0\tag{\(\ast\ast\)}\] is nonoscillatory. Here, we shall improve and unify the results given in the above mentioned papers and present some new sufficient conditions which insure that any solution of equation (\(\ast\)) oscillates if equation (\(\ast\ast\)) is nonoscillatory. We also establish results for the oscillation of equation (\(\ast\)) when equation (\(\ast\ast\)) is oscillatory.

In this paper, we are concerned with the existence of positive solutions of the following nonlinear elliptic system involving critical Hardy-Sobolev exponent \begin{equation*}\label{eq:1}(*) \left\{ \begin{array}{lll} -\Delta u= \frac{2\alpha}{\alpha+\beta}\frac{u^{\alpha-1}v^\beta}{|x|^s}-\lambda u^p, & \quad {\rm in}\quad \Omega,\\[2mm] -\Delta v= \frac{2\beta}{\alpha+\beta}\frac{u^\alpha v^{\beta-1}}{|x|^s}-\lambda v^p, & \quad {\rm in}\quad \Omega,\\[2mm] u\gt 0, v\gt 0, &\quad {\rm in}\quad \Omega,\\[2mm] u=v=0, &\quad {\rm on}\quad \partial\Omega, \end{array} \right. \end{equation*} where \(N\geq 4\) and \(\Omega\) is a \(C^1\) bounded domain in \(\mathbb{R}^N\) with \(0\in\partial\Omega\). \(0\lt s \lt 2\), \(\alpha+\beta=2^*(s)=\frac{2(N-s)}{N-2}\), \(\alpha,\beta\gt 1\), \(\lambda\gt 0\) and \(1 \lt p\lt \frac{N+2}{N-2}\). The case when 0 belongs to the boundary of \(\Omega\) is closely related to the mean curvature at the origin on the boundary. We show in this paper that problem \((*)\) possesses at least a positive solution.

The theorems on the estimates of solutions for nonlinear second-order partial differential functional equations of parabolic type with Dirichlet's condition and for suitable implicit finite difference functional schemes are proved. The proofs are based on the com- parison technique. The convergent and stable difference method is considered without the assumption of the global generalized Perron condition posed on the functional variable but with the local one only. It is a consequence of our estimates theorems. In particular, these results cover quasi-linear equations. However, such equations are also treated separately. The functional dependence is of the Volterra type. The aim of the paper is to prove theorems on the estimates of solutions for non- linear second-order partial differential functional equations of parabolic type with Dirichlet's condition and for generated by them implicit finite difference functional schemes. We also give the applications of the results. More precisely, we prove the theorem on the convergence of a difference method to a classical solution for the differential functional problem, which by the given estimates, may be treated in the subspace C (,R) ⊂ C (,R), where R ⊂ R is an interval. It is a new idea in area of nonlinear implicit difference methods which was studied for explicit methods by K. Kropielnicka and L. Sapa (14). This considerably extends the class of problems which are solvable by the described method. Therefore, the Lipschitz, Perron or generalized Perron conditions posed onf with respect toz need not be global, inC (,R), as in the papers due to M. Malec, Cz. Mączka, W. Voigt, M. Rosati and L. Sapa (15-19,24,25)

In this paper we consider the semilinear wave equation with the multiplication of logarithmic and polynomial nonlinearities. We establish the global existence and finite time blow up of solutions at three different energy levels (\(E(0)\lt d\), \(E(0)=d\) and \(E(0)\gt 0\)) using potential well method. The results in this article shed some light on using potential wells to classify the solutions of the semilinear wave equation with the product of polynomial and logarithmic nonlinearity.

In this paper, we study semicircular elements and circular elements in a certain Banach \(*\)-probability space \((\mathfrak{LS},\tau ^{0})\) induced by analysis on the \(p\)-adic number fields \(\mathbb{Q}_{p}\) over primes \(p\). In particular, by truncating the set \(\mathcal{P}\) of all primes for given suitable real numbers \(t\lt s\) in \(\mathbb{R}\), two different types of truncated linear functionals \(\tau_{t_{1}\lt t_{2}}\), and \(\tau_{t_{1}\lt t_{2}}^{+}\) are constructed on the Banach \(*\)-algebra \(\mathfrak{LS}\). We show how original free distributional data (with respect to \(\tau ^{0}\)) are distorted by the truncations on \(\mathcal{P}\) (with respect to \(\tau_{t\lt s}\), and \(\tau_{t\lt s}^{+}\)). As application, distorted free distributions of the semicircular law, and those of the circular law are characterized up to truncation.

Global well-posedness and finite time blow up issues for some strongly damped nonlinear wave equation are investigated in the present paper. For subcritical initial energy by employing the concavity method we show a finite time blow up result of the solution. And for critical initial energy we present the global existence, asymptotic behavior and finite time blow up of the solution in the framework of the potential well. Further for supercritical initial energy we give a sufficient condition on the initial data such that the solution blows up in finite time.

In this paper, sufficient conditions for oscillation of the second order differential equations with several sublinear neutral terms are established. The results obtained generalize and extend those reported in the literature. Several examples are included to illustrate the importance and novelty of the presented results.

We study the existence and uniqueness of the backward stochastic variational inequalities driven by \(m\)-dimensional fractional Brownian motion with Hurst parameters \(H_k\) (\(k=1,\ldots m\)) greater than \(1/2\). The stochastic integral used throughout the paper is the divergence type integral.

In this paper we prove the stability of the zero equilibria of two systems of difference equations of exponential type, which are some extensions of an one-dimensional biological model. The stability of these systems is investigated in the special case when one of the eigenvalues is equal to -1 and the other eigenvalue has absolute value less than 1, using centre manifold theory. In addition, we study the existence and uniqueness of positive equilibria, the attractivity and the global asymptotic stability of these equilibria of some related systems of difference equations.

The paper deals with investigating of the first mixed problem for a fifth-order nonlinear evolutional equation which generalizes well known equation of the vibrations theory. We obtain sufficient conditions of nonexistence of a global solution in time variable.

In this paper, we investigate the existence of three generalized solutions for fourth-order Kirchhoff-type problems with a perturbed nonlinear term depending on two real parameters. Our approach is based on variational methods.

In this paper, we study a linear-quadratic optimal control problem with a fractional control system containing a Caputo derivative of unknown function. First, we derive the formulas for the differential and gradient of the cost functional under given constraints. Next, we prove an existence result and derive a maximum principle. Finally, we describe the gradient and projection of the gradient methods for the problem under consideration.

In this paper, different sufficient conditions for exact controllability of semilinear systems with a single constant point delay in control are established in infinite dimensional space. The existence and uniqueness of mild solution is also proved under suitable assumptions. In particular, local Lipschitz continuity of a nonlinear function is used. To illustrate the developed theory some examples are given.

In this paper we study the Borel summability of formal solutions with a parameter of first order semilinear system of partial differential equations with \(n\) independent variables. In [Singular perturbation of linear systems with a regular singularity, J. Dynam. Control. Syst. 8 (2002), 313-322], Balser and Kostov proved the Borel summability of formal solutions with respect to a singular perturbation parameter for a linear equation with one independent variable. We shall extend their results to a semilinear system of equations with general independent variables.

This paper is concerned with positive solutions of the semilinear polyharmonic equation \((-\Delta)^{m} u = a(x){u}^{\alpha}\) on \(\mathbb{R}^{n}\), where \(m\) and \(n\) are positive integers with \(n\gt 2m\), \(\alpha\in (-1,1)\). The coefficient \(a\) is assumed to satisfy \[a(x)\approx{(1+|x|)}^{-\lambda}L(1+|x|)\quad \text{for}\quad x\in \mathbb{R}^{n},\] where \(\lambda\in [2m,\infty)\) and \(L\in C^{1}([1,\infty))\) is positive with \(\frac{tL'(t)}{L(t)}\longrightarrow 0\) as \(t\longrightarrow \infty\); if \(\lambda=2m\), one also assumes that \(\int_{1}^{\infty}t^{-1}L(t)dt\lt \infty\). We prove the existence of a positive solution \(u\) such that \[u(x)\approx{(1+|x|)}^{-\widetilde{\lambda}}\widetilde{L}(1+|x|) \quad\text{for}\quad x\in \mathbb{R}^{n},\] with \(\widetilde{\lambda}:=\min(n-2m,\frac{\lambda-2m}{1-\alpha})\) and a function \(\widetilde{L}\), given explicitly in terms of \(L\) and satisfying the same condition at infinity. (Given positive functions \(f\) and \(g\) on \(\mathbb{R}^{n}\), \(f\approx g\) means that \(c^{-1}g\leq f\leq cg\) for some constant \(c\gt 1\).)

In this paper, we establish existence and asymptotic behavior of a positive classical solution to the following semilinear boundary value problem: \[-\Delta u=q(x)u^{\sigma }\;\text{in}\;\Omega,\quad u_{|\partial\Omega}=0.\] Here \(\Omega\) is an annulus in \(\mathbb{R}^{n}\), \(n\geq 3\), \(\sigma \lt 1\) and \(q\) is a positive function in \(\mathcal{C}_{loc}^{\gamma }(\Omega )\), \(0\lt\gamma \lt 1\), satisfying some appropriate assumptions related to Karamata regular variation theory. Our arguments combine a method of sub- and supersolutions with Karamata regular variation theory.

In this paper, we prove the existence of weak heteroclinic solutions for a family of anisotropic difference equations under competition phenomena between parameters.

In this paper we study the following nonlinear boundary-value problem \[-\Delta_{p(x)} u=\lambda f(x,u) \quad \text{ in } \Omega,\] \[|\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}+\beta(x)|u|^{p(x)-2}u=\mu g(x,u) \quad \text{ on } \partial\Omega,\] where \(\Omega\subset\mathbb{R}^N\) is a bounded domain with smooth boundary \(\partial\Omega\), \(\frac{\partial u}{\partial\nu}\) is the outer unit normal derivative on \(\partial\Omega\), \(\lambda, \mu\) are two real numbers such that \(\lambda^{2}+\mu^{2}\neq0\), \(p\) is a continuous function on \(\overline{\Omega}\) with \(\inf_{x\in \overline{\Omega}} p(x)\gt 1\), \(\beta\in L^{\infty}(\partial\Omega)\) with \(\beta^{-}:=\inf_{x\in \partial\Omega}\beta(x)\gt 0\) and \(f : \Omega\times\mathbb{R}\rightarrow \mathbb{R}\), \(g : \partial\Omega\times\mathbb{R}\rightarrow \mathbb{R}\) are continuous functions. Under appropriate assumptions on \(f\) and \(g\), we obtain the existence and multiplicity of solutions using the variational method. The positive solution of the problem is also considered.

In this paper, we investigate the existence of constant-sign solutions for a nonlinear Neumann boundary value problem involving the discrete \(p\)-Laplacian. Our approach is based on an abstract local minimum theorem and truncation techniques.

In this paper, we establish some new criteria on the asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations on time scales.

In the paper, we study a linear-quadratic optimal control problem of Mayer type given by a fractional control system. First, we prove a theorem on the existence of a solution to such a problem. Next, using the local implicit function theorem, we derive a formula for the gradient of a cost functional under constraints given by a control system and prove a maximum principle in the case of a control constraint set. The formula for the gradient is used to implement the gradient methods for the problem under consideration.

In this paper we study a class of two-point boundary value systems. Using very recent critical points theorems, we establish the existence of one non-trivial solution and infinitely many solutions of this problem, respectively.

We consider the following second order differential equation with delay \[\begin{cases} (Lx)(t)\equiv{x''(t)+\sum_{j=1}^p {b_{j}(t)x(t-\theta_{j}(t))}}=f(t), \quad t\in[0,\omega],\\ x(t_j)=\gamma_{j}x(t_j-0), x'(t_j)=\delta_{j}x'(t_j-0), \quad j=1,2,\ldots,r. \end{cases}\] In this paper we find necessary and sufficient conditions of positivity of Green's functions for this impulsive equation coupled with one or two-point boundary conditions in the form of theorems about differential inequalities. By choosing the test function in these theorems, we obtain simple sufficient conditions. For example, the inequality \(\sum_{i=1}^p{b_i(t)\left(\frac{1}{4}+r\right)}\lt \frac{2}{\omega^2}\) is a basic one, implying negativity of Green's function of two-point problem for this impulsive equation in the case \(0\lt \gamma_i\leq{1}\), \(0\lt \delta_i\leq{1}\) for \(i=1,\ldots ,p\).

In this paper we study a class of one-dimensional Dirichlet boundary value prob- lems involving the Caputo fractional derivatives. The existence of infinitely many solutions for this equations is obtained by exploiting a recent abstract result. Concrete examples of applications are presented.

A generalized Cauchy problem for quasilinear hyperbolic functional differential systems is considered. A theorem on the local existence of weak solutions is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. The existence of solutions for this system is proved by using a method of successive approximations. We show a theorem on the differentiability of solutions with respect to initial functions which is the main result of the paper.

The paper deals with the difference inequalities generated by initial boundary value problems for hyperbolic nonlinear differential functional systems. We apply this result to investigate the stability of constructed difference schemes. The proof of the convergence of the difference method is based on the comparison technique, and the result for difference functional inequalities is used. Numerical examples are presented.

In this paper, we are motivated by uncertainty problems in volatility. We prove the equivalent theorem of Wiener chaos with respect to \(G\)-Brownian motion in the framework of a sublinear expectation space. Moreover, we establish some relationship between Hermite polynomials and \(G\)-stochastic multiple integrals. An equivalent of the orthogonality of Wiener chaos was found.

In this paper we show some construction of nonseparable compactly supported bivariate wavelets. We deal with the dilation matrix \(A = \tiny{\left[\begin{matrix}0 & 2 \cr 1 & 0 \cr \end{matrix} \right]}\) and some three-row coefficient mask; that is a scaling function satisfies a dilation equation with scaling coefficients which can be contained in the set \(\{c_{n}\}_{n \in\mathcal{S}},\) where \(\mathcal{S}=S_{1} \times \{0,1,2\},\) \(S_{1} \subset \mathbb{N},\) \(\sharp S_{1} \lt \infty.\)

In this paper we introduce a unified representation of starlike and convex harmonic functions with negative coefficients, related to uniformly starlike and uniformly convex analytic functions. We obtain extreme points, distortion bounds, convolution conditions and convex combinations for this family.

In this paper, we present the existence and uniqueness of random solution of a random integral equation of Volterra type on time scales. We also study the asymptotic properties of the unique random solution.

In this paper we study the nonlinear elliptic problem with p(x)-Laplacian (hemi- variational inequality). We prove the existence of a nontrivial solution. Our approach is based on critical point theory for locally Lipschitz functionals due to Chang (J. Math. Anal. Appl. 80 (1981), 102-129). p(x)-Laplacian ( p(x)u ju(x)j p(x) 2 u(x)2 @j(x;u(x)) a.e. in

The paper investigates the existence and uniqueness of weak solutions for a non-linear boundary value problem involving the weighted \(p\)-Laplacian. Our approach is based on variational principles and representation properties of the associated spaces.

Let \(J\) and \(R\) be anti-commuting fundamental symmetries in a Hilbert space \(\mathfrak{H}\). The operators \(J\) and \(R\) can be interpreted as basis (generating) elements of the complex Clifford algebra \(Cl_2(J,R):=\text{span}\{I,J,R,iJR\}\). An arbitrary non-trivial fundamental symmetry from \(Cl_2(J,R)\) is determined by the formula \(J_{\vec{\alpha}}=\alpha_1 J +\alpha_2 R+\alpha_3 iJR\), where \(\vec{\alpha} \in \mathbb{S}^2\). Let \(S\) be a symmetric operator that commutes with \(Cl_2(J,R)\). The purpose of this paper is to study the sets \(\Sigma_{J_{\vec{\alpha}}}\) (\(\forall \vec{\alpha} \in \mathbb{S}^2\)) of self-adjoint extensions of \(S\) in Krein spaces generated by fundamental symmetries \(J_{\vec{\alpha}}\) (\(J_{\vec{\alpha}}\)-self-adjoint extensions). We show that the sets \(\Sigma_{J_{\vec{\alpha}}}\) and \(\Sigma_{J_{\vec{\beta}}}\) are unitarily equivalent for different \(\vec{\alpha}, \vec{\beta} \in \mathbb{S}^2\) and describe in detail the structure of operators \(A \in \Sigma_{J_{\vec{\alpha}}}\) with empty resolvent set.

The aim of this paper is to investigate a class of boundary value problems for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of weak noncompactness.

In this paper, we discuss the existence of solutions for a four-point integral boundary value problem of \(n\)-th order differential inclusions involving convex and non-convex multivalued maps. The existence results are obtained by applying the nonlinear alternative of Leray Schauder type and some suitable theorems of fixed point theory.

In this paper we show some construction of compactly supported multi-wavelets in \(L^2(\mathbb{R}^d)\), \(d \geq 2\) which is based on the one-dimensional case, when \(d=1\). We also demonstrate that some methods, which are useful in the construction of wavelets with a compact support at \(d=1\), can be adapted to higher-dimensional cases if \(A \in M_{d \times d}(\mathbb{Z})\) is an expansive matrix of a special form.

In this paper we introduce a radial version of the Kontorovich-Lebedev transform in the unit ball. Mapping properties and an inversion formula are proved in \(L_p\)

A presentation of numerical ranges for rectangular matrices is undertaken in this paper, introducing two different definitions and elaborating basic properties. Further, we extend to the \(q\)-numerical range.

In this paper, we discuss a four-point boundary value problem for a nonlinear differential equation of fractional order. The differential operator is the Riemann-Liouville derivative and the inhomogeneous term depends on the fractional derivative of lower order. We obtain the existence of at least one solution for the problem by using the Schauder fixed-point theorem. Our analysis relies on the reduction of the problem considered to the equivalent Fredholm integral equation.

This paper concerns the oscillation of solutions of the differential eq. \[ \left[ r\left( t\right) \psi \left(x\left( t\right) \right) f\text{ }( \overset{\cdot }{x}(t))\right]^{\cdot }+q\left( t\right) \varphi (g\left( x\left( t\right) \right), r\left( t\right) \psi \left( x\left( t\right) \right) f(\overset{\cdot }{x}(t)))=0,\] where \(u\varphi \left( u,v\right) \gt 0\) for all \(u\neq 0\), \(xg\left( x\right) \gt 0\), \(xf\left( x\right)\gt 0\) for all \(x\neq 0\), \(\psi \left( x\right) \gt 0\) for all \(x\in \mathbb{R}\), \(r\left( t\right) \gt 0\) for \(t\geq t_{0}\gt 0\) and \(q\) is of arbitrary sign. Our results complement the results in [A.G. Kartsatos, On oscillation of nonlinear equations of second order, J. Math. Anal. Appl. 24 (1968), 665–668], and improve a number of existing oscillation criteria. Our main results are illustrated with examples.

In this paper, we investigate the existence and uniqueness of positive solutions to arbitrary order nonlinear fractional differential equations with advanced arguments. By applying some known fixed point theorems, sufficient conditions for the existence and uniqueness of positive solutions are established.

In this paper, we study the existence and uniqueness of solutions to the nonlocal problems for the fractional differential equation in Banach spaces. New sufficient conditions for the existence and uniqueness of solutions are established by means of fractional calculus and fixed point method under some suitable conditions. Two examples are given to illustrate the results.

In this paper, we study a class of integrodifferential evolution equations with nonlocal initial conditions in Banach spaces. Existence results of mild solutions are proved for a class of integrodifferential evolution equations with nonlocal initial conditions in Banach spaces. The main results are obtained by using the Schaefer fixed point theorem and semigroup theory. Finally, an example is given for demonstration.

This paper is concerned with ground state solutions for the Hénon type equation \(-\Delta u(x)=|y|^{\alpha} u^{p-1}(x)\) in \(\Omega\), where \(\Omega=B^k(0,1)\times B^{n-k}(0,1)\subset \mathbb{R}^n\) and \(x=(y,z) \in \mathbb{R}^k \times \mathbb{R}^{n-k}\). We study the existence of cylindrically symmetric and non-cylindrically symmetric ground state solutions for the problem. We also investigate asymptotic behavior of the ground state solution when \(p\) tends to the critical exponent \(2^*=\frac {2n}{n-2}\) if \(n\geq 3\).

In this paper, we study the steady-state of a semilinear pulse-width sampler controlled system on infinite dimensional spaces. Firstly, by virtue of Schauder's fixed point theorem, the existence of periodic solutions is given. Secondly, utilizing a generalized Gronwall inequality given by us and the Banach fixed point theorem, the existence and stabilizability of a steady-state for the semilinear control system with pulse-width sampler is also obtained. At last, an example is given for demonstration.

The aim of this paper is to study the spectrum of the one-dimensional discrete Schrodinger operator with a perturbed periodic potential. We obtain natural conditions under which this perturbation preserves the essential spectrum of the considered operator. Conditions on the number of isolated eigenvalues are given.

This paper deals with some impulsive fractional differential equations in Banach spaces. Utilizing the Leray-Schauder fixed point theorem and the impulsive nonlinear singular version of the Gronwall inequality, the existence of \(PC\)-mild solutions for some fractional differential equations with impulses are obtained under some easily checked conditions. At last, an example is given for demonstration.

In this paper, we study the existence and uniqueness of the \(PC\)-mild solution for a class of impulsive fractional differential equations with time-varying generating operators and nonlocal conditions. By means of the generalized Ascoli-Arzela Theorem given by us and the fixed point theorem, some existence and uniqueness results are obtained. Finally, an example is given to illustrate the theory.

In this paper, we discuss how changes in the coefficients matrix of piecewise linear fractional programming problems affect the non-degenerate optimal solution. We consider separate cases when changes occur in the coefficients of the basic and non-basic variables and derive bounds for each perturbation, while the optimal solution is invariant. We explain that this analysis is a generalization of the sensitivity analysis for \(LP\), \(LFP\) and \(PLP\). Finally, the results are described by some numerical examples.