Nonlinear Alternative Research Articles

Darboux problem for impulsive partial hyperbolic differential equations of fractional order with variable times and infinite delay

This paper deals with the existence of solutions to impulsive partial functional differential equations with impulses at variable times and infinite delay, involving the Caputo fractional derivative. Our works will be considered by using the nonlinear alternative of Leray–Schauder type.

Existence of solutions for nonlinear functional integrodifferential evolution equations with nonlocal conditions

In this paper, we prove the existence of mild solutions for nonlinear functional integrodifferential evolution equations with nonlocal conditions in Banach spaces. The results are obtained by using the theory of resolvent operators and Leray–Schauder nonlinear alternative.

Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order

We investigate the existence of nontrivial solutions for a multi-point boundary value problem for fractional differential equations. Under certain growth conditions on the nonlinearity, several sufficient conditions for the existence of nontrivial solution are obtained by using Leray–Schauder nonlinear alternative. As an application, some examples to illustrate our results are given.

Existence Results for Fractional Order Semilinear Integro-Differential Evolution Equations with Infinite Delay

This paper is concerned with existence results of mild solutions for fractional order semilinear integro-differential evolution equations (FSIDEEs) and semilinear neutral integro-differential evolution equations (FSNIDEEs in short) with infinite delay in α-norm. Our tools include the Banach contraction principle, the nonlinear alternative of Leray–Schauder type and the Krasnoselskii–Schaefer type fixed point theorem.

Positive solutions for third order multi-point singular boundary value problems

We study a third order singular boundary value problem with multi-point boundary conditions. Sufficient conditions are obtained for the existence of positive solutions of the problem. Recent results in the literature are significantly extended and improved. Our analysis is mainly based on a nonlinear alternative of Leray-Schauder.

NONTRIVIAL SOLUTIONS FOR BOUNDARY-VALUE PROBLEMS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem(BVP) <TEX>$-D_0^{\alpha}+u(t)=\lambda[f(t, u(t))+q(t)]$</TEX>, 0 < t < 1 u(0) = u(1) = 0, where <TEX>$\lambda$</TEX> > 0 is a parameter, 1 < <TEX>$\alpha$</TEX> <TEX>$\leq$</TEX> 2, <TEX>$D_{0+}^{\alpha}$</TEX> is the standard Riemann-Liouville differentiation, f : [0, 1] <TEX>${\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$</TEX> is continuous, and q(t) : (0, 1) <TEX>$\rightarrow$</TEX> [0, <TEX>$+\infty$</TEX>] is Lebesgue integrable. We obtain serval sufficient conditions of the existence and uniqueness of nontrivial solution of BVP when <TEX>$\lambda$</TEX> in some interval. Our approach is based on Leray-Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity which was essential for the technique used in almost all existed literature on f.

Evidence of nonlinear mean reversion in the real interest rate

This article utilizes tests for a unit root that have power against nonlinear alternatives to provide empirical evidence on the time series properties of the ex-post real interest rate in the G7 countries. We find that the unit root hypothesis can be rejected in the presence of a nonlinear alternative motivated by theoretical literature on optimal monetary policy rules. This represents a reversal of the results obtained using standard linear unit-root and cointegration tests. Tests for linearity reject this hypothesis for Canada, France, Italy and Japan for which we estimate nonlinear models capturing the dynamics of the interest rate. For these countries, ex-post real interest rates follow a nonlinear model characterized by mean reversion and provide statistical evidence for the Fisher effect.

Positive Solutions for System of Nonlinear Fractional Differential Equations in Two Dimensions with Delay

We investigate the existence and uniqueness of positive solution for system of nonlinear fractional differential equations in two dimensions with delay. Our analysis relies on a nonlinear alternative of Leray‐Schauder type and Krasnoselskii′s fixed point theorem in a cone.

Some existence results for boundary value problems of fractional differential inclusions with non-separated boundary conditions

In this paper, we study the existence of solutions for a boundary value problem of differential inclusions of order q ∈ (1,2] with non-separated boundary conditions involving convex and non-convex multivalued maps. Our results are based on the nonlinear alternative of Leray Schauder type and some suitable theorems of fixed point theory.

Existence of positive solutions for boundary value problems of fractional functional differential equations

This paper deals with the existence of positive solutions for a boundary value problem involving a nonlinear functional differential equation of fractional ordergiven by Du(t) + f(t,ut) = 0, t 2 (0,1), 2 < � � 3, u ' (0) = 0, u ' (1) = bu ' (�), u0 = �. Our results are based on the nonlinear alternative of Leray-Schauder type and Krasnosel'skii fixed point theorem.

Existence and Localization Results for "Equation missing" -Laplacian via Topological Methods

AbstractWe show the existence of a week solution in "Equation missing" to a Dirichlet problem for "Equation missing" in "Equation missing", and its localization. This approach is based on the nonlinear alternative of Leray-Schauder.

Global existence and controllability to a stochastic integro-differential equation

In this paper, we are focused upon the global uniqueness results for a stochastic integro-differential equation in Frechet spaces. The main results are proved by using the resolvent operators combined with a nonlinear alternative of Leray-Schauder type in Frechet spaces due to Frigon and Granas. As an application, a controllability result with one parameter is given to illustrate

Existence and Uniqueness of Solutions for Coupled Systems of Higher-Order Nonlinear Fractional Differential Equations

AbstractWe study an initial value problem for a coupled Caputo type nonlinear fractional differential system of higher order. As a first problem, the nonhomogeneous terms in the coupled fractional differential system depend on the fractional derivatives of lower orders only. Then the nonhomogeneous terms in the fractional differential system are allowed to depend on the unknown functions together with the fractional derivative of lower orders. Our method of analysis is based on the reduction of the given system to an equivalent system of integral equations. Applying the nonlinear alternative of Leray-Schauder, we prove the existence of solutions of the fractional differential system. The uniqueness of solutions of the fractional differential system is established by using the Banach contraction principle. An illustrative example is also presented.

Unbounded Solutions of Second-Order Multipoint Boundary Value Problem on the Half-Line

This paper investigates the second-order multipoint boundary value problem on the half-line ,, , , , where , , , , and is continuous. We establish sufficient conditions to guarantee the existence of unbounded solution in a special function space by using nonlinear alternative of Leray-Schauder type. Under the condition that is nonnegative, the existence and uniqueness of unbounded positive solution are obtained based upon the fixed point index theory and Banach contraction mapping principle. Examples are also given to illustrate the main results.

Global uniqueness results for partial functional and neutral functional evolution equations with infinite delay

In this paper, we provide sufficient conditions for the existence of a unique mild solution on a semi-infinite interval for two classes of first-order partial functional and neutral functional differential evolution equations with infinite delay using a recent nonlinear alternative of Leray Schauder type due to Frigon and Granas for contractions maps in Fréchet spaces, combined with semigroup theory.

Nontrivial periodic solutions of nonlinear functional differential systems with feedback control

This paper examines the existence of nontrivial periodic solutions for the nonlinear functional differential system with feedback control: \{\aligned x'(t)=x(t)a(t)-\big[\sum_{i=1}^n a_i(t)\int_0^{+\infty} f(t, x(t-\theta)) d}\varphi_i(\theta) +\sum_{j=1}^m b_j(t) \int_0^{+\infty} f(t,x'(t-\theta))\,d}\phi_j(\theta)+\sum_{\mu=1}^p c_\mu(t) \int_0^{\infty} u(t-\theta)\,d}\delta_\mu(\theta)\big], u'(t)=-\rho(t)u(t)+\sum_{\nu=1}^q \beta_\nu(t) \int_0^{\infty} f(t, x(t-\theta))\,d}\psi_\nu(\theta).\endaligned Under certain growth conditions on the nonlinearity f, several sufficient conditions for the existence of nontrivial solution are obtained by using Leray-Schauder nonlinear alternative.

Existence of positive solution for singular fractional differential equation

In this paper, we establish the existence of a positive solution to a singular boundary value problem of nonlinear fractional differential equation. Our analysis rely on nonlinear alternative of Leray–Schauder type and Krasnoselskii’s fixed point theorem in a cone.

Existence results for fractional order semilinear functional differential equations with nondense domain

In this paper, we establish sufficient conditions for existence and uniqueness of solutions for some nondensely defined semilinear functional differential equations involving the Riemann–Liouville derivative. Our approach is based on integrated semigroup theory, the Banach contraction principle, and the nonlinear alternative of Leray–Schauder type.

Pairwise Tests of Purchasing Power Parity

Given nominal exchange rates and price data on N + 1 countries indexed by i = 0,1,2,…, N, the standard procedure for testing purchasing power parity (PPP) is to apply unit root or stationarity tests to N real exchange rates all measured relative to a base country, 0, often taken to be the U.S. Such a procedure is sensitive to the choice of base country, ignores the information in all the other cross-rates and is subject to a high degree of cross-section dependence which has adverse effects on estimation and inference. In this article, we conduct a variety of unit root tests on all possible N(N + 1)/2 real rates between pairs of the N + 1 countries and estimate the proportion of the pairs that are stationary. This proportion can be consistently estimated even in the presence of cross-section dependence. We estimate this proportion using quarterly data on the real exchange rate for 50 countries over the period 1957–2001. The main substantive conclusion is that to reject the null of no adjustment to PPP requires sufficiently large disequilibria to move the real rate out of the band of inaction set by trade costs. In such cases, one can reject the null of no adjustment to PPP up to 90% of the time as compared to around 40% in the whole sample using a linear alternative and almost 60% using a nonlinear alternative.

Impulsive differential inclusions with fractional order

In this paper, we first present an impulsive version of the Filippov–Ważewski theorem and a continuous version of the Filippov theorem for fractional differential inclusions of the form D∗αy(t)∈F(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=I¯k(y(tk−)),k=1,…,m,y(0)=a,y′(0)=c, where J=[0,b],D∗α denotes the Caputo fractional derivative, and F is a set-valued map. The functions Ik,I¯k characterize the jump of the solutions at impulse points tk(k=1,…,m). Additional existence results are obtained under both convexity and nonconvexity conditions on the multivalued right-hand side. The proofs rely on the nonlinear alternative of Leray–Schauder type, a Bressan–Colombo selection theorem, and Covitz and Nadler’s fixed point theorem for multivalued contractions. The compactness of the solution set is also investigated. Finally, some geometric properties of solution sets, Rδ sets, acyclicity and contractibility, corresponding to Aronszajn–Browder–Gupta type results, are obtained. We also consider the impulsive fractional differential equations D∗αy(t)=f(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=Īk(y(tk−)),k=1,…,m,y(0)=a,y′(0)=c, and D∗αy(t)=f(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(0,1],y(tk+)=Ik(y(tk−)),k=1,…,m,y(0)=a, where f:J×R→R is a single map. Finally, we extend the existence result for impulsive fractional differential inclusions with periodic conditions, D∗αy(t)∈φ(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=I¯k(y(tk−)),k=1,…,m,y(0)=y(b),y′(0)=y′(b), where φ:J×R→P(R) is a multivalued map. The study of the above problems use an approach based on the topological degree combined with a Poincaré operator.