For these purposes, we must establish the global existence of **solution** for 1.1-1.2. Several methods have been used to study the existence of solutions to nonlinear wave equation. Notable among them is the variational approach through the use of Faedo-Galerkin approximation combined with the method of compactness and monotonicity, see 7 . To prove the **uniqueness**, we need to derive the various estimates for assumed **solution** ut. For the decay property, like 1.5, we use the method recently introduced by Martinez 8 to study the decay rate of **solution** to the wave equation u tt − Δu gu t 0 in Ω × R , where Ω

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The **uniqueness** of the **solution** for the deﬁnite problem of a parabolic variational inequality is proved. The problem comes from the study of the optimal exercise strategies for the perpetual executive stock options with unrestricted exercise in ﬁnancial market. Because the variational inequality is degenerate and the obstacle condition contains the partial derivative of an unknown function, it makes the theoretical study of the deﬁnite problem of the variational inequality problem very diﬃcult. Firstly, the property which the value function satisﬁes is derived by applying the Jensen inequality. Then the **uniqueness** of the **solution** is proved by using this property and maximum principles.

We have obtained a simple necessary and sufficient condition on f for **uniqueness** of the trivial **solution** in a semilinear parabolic equation with con- tinuous, increasing nonlinearity f . There were several key structural proper- ties required to achieve this: (a) monotonicity of f ; (b) semilinearity of the governing evolution equation; (c) monotonicity of the semigroup S β (equiva-

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Abstract In this paper, the existence and **uniqueness** of mixed linear Volterra-Fredholm integral equations of the second kind will be studied under some conditions in the Banach space and Fixed-point theory. Also approximate **solution** is presented using fixed-point iteration method (FPM), and then the Aitken method is used to accelerate the convergence. For more illustration the method is applied on several examples and programs are written in the Matlab to compute the results. The absolute errors are computed to clarify the efficiency of the method.

Integro-differential equations play a fundamental role in various fields of applied mathematics. The solutions of many engineering problems in general and mechanics and physics in particular, lead to this kind of equations. This paper focuses on the fuzzy Volterra integro-differential equation of n − th order of the second-kind with nonlinear fuzzy kernel and initial values. This equation is transformed to a nonlinear fuzzy Volterra integral equation in multi-integrals by application of a certain analytic **solution** adapted on fuzzy n − th order derivation under generalized Hakuhara derivative. The derived integral equations are solvable, the solutions of which are unique under certain conditions. The existence and **uniqueness** of the solutions are investigated in a theorem and an upper boundary is found for solutions. An easily-followed algorithm is provided to illustrate the process. The application of the proposed method helps solving the equation on the basis of the Adomian decomposition method under generalized H-derivation. Comparison of the exact and approximated solutions shows the least error.

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In this article, we established the existence and **uniqueness** of the **solution** for a generalized class of fractional order differential equations involving the Riemann- Liouville differential operator on unbounded domain [0, + ∞ ). The contraction principle has been used to obtain the results in this article.

BSDEs were known to be useful tools in the study of stochastic optimal control prob- lems; see, for example, [, ]. The optimal control problems for stochastic systems in in- ﬁnite dimensions have been considered in [–]. Using Malliavin calculus and BSDEs, Fuhrman and Tessitore [] showed that there exists a unique mild **solution** of nonlinear Kolmogorov equations and found that the mild **solution** coincides with the value function of the control problem. In Fuhrman and Tessitore [], the existence and **uniqueness** of the mild **solution** for semilinear elliptic diﬀerential equations in Hilbert spaces were obtained by means of inﬁnite horizon BSDEs in Hilbert spaces and Malliavin calculus, moreover, the existence of optimal control is proved by the feedback law. Fuhrman et al. [] consid- ered the optimal control problems for stochastic diﬀerential equations with delay and the associated Kolmogorov equations, the existence and **uniqueness** of the mild **solution** for the Kolmogorov equations was proved and the existence of optimal control was obtained. In Fuhrman et al. [], the optimal ergodic control of a Banach valued stochastic evolution equation was studied and the optimal ergodic control was obtained by the ergodic BSDEs. The main result of this paper is the proof of existence and **uniqueness** of the mild so- lution of (.) and (.). Some authors considered the Kolmogorov equations associated with stochastic evolution equations (see [, ]) and with stochastic delay diﬀerential equations (see []). However, as far as we know, there are few authors who concentrated on (.) and (.), for example, [] and [] for nonlinear parabolic partial diﬀerential equations. In this paper we want to extend the results of [] to stochastic delay evolution equations in Hilbert spaces. Thanks to Lemma . and Lemma ., we can consider the optimal control problem of (.) and the associated nonlinear Kolmogorov equations (.) and (.).

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The objective of this paper is to attempt to apply the theoretical techniques of probabilistic functional analysis to answer the question of existence and **Uniqueness** of a Random **Solution** to Itô Stochastic Integral Equation. Another type of stochastic integral equation which has been of considerable importance to applied mathematicians and engineers is that involving the Itô or Itô-Doob form of stochastic integrals.

In this work, we prove the existence , **uniqueness** and stability **Solution** . for another system of non-linear integro-differential equations of Volterra type Consider the following system of non-linear integro-differential equations which has the form :

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has a **solution** provided f is continuous and satisﬁes a Lipschitz condition by C. Corduneanu [2]. The deﬁnition given here generalizes that of Aumann [1] for set- valued mappings. Kaleva [3] discussed the properties of diﬀerentiable fuzzy set-valued mappings and gave the existence and **uniqueness** theorem for a **solution** of the fuzzy diﬀerential equation x (t) = f (t,x(t)) when f satisﬁes the Lipschitz condition. Also,

Recently, some excellent work on nonlocal integral boundary condition for fractional diﬀerential equation and system was done by Zhang et al. [] and Ahmad and Nieto []. In [], by establishing some comparison results and combining with a monotone iterative method, the existence of an extremal **solution** for a nonlinear system involving the right- handed Riemann-Liouville fractional derivative with nonlocal coupled integral boundary conditions was obtained. Ahmad and Nieto [] employed standard ﬁxed point theorems to study the **uniqueness** and existence of **solution** for a class of Riemann-Liouville frac- tional diﬀerential equations with fractional boundary conditions. Some new existence and **uniqueness** results are obtained. Here we also refer the reader to some recent work on frac- tional diﬀerential equation (see [–]).

Much attention has been paid to the existence and **uniqueness** of the solutions of frac- tional dynamic systems [–] on account of the fact that existence is the fundamental problem and a necessary condition for considering some other properties for fractional dynamic systems, such as controllability, stability, etc. Besides, one has to take into ac- count the peculiarity of the kernel to obtain explicit results in practice not only to reduce such an equation to an integral equation. Moreover, many authors have considered the multi-term fractional order systems [–] due to their successful applications in me- chanical system, the dynamics of certain gases, the dynamics of sphere. And the existence and **uniqueness** of the solutions of this multi-term fractional order systems becomes more complicated than one-term fractional order systems which have been considered before by many authors. In this paper, motivated by the above, we will consider the existence and **uniqueness** of the **solution** of the following fractional damped dynamical systems:

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fractional nonlinear diﬀusion equations. For the one nonlinear fractional diﬀusion equa- tion of type () with unknown function, many authors have considered the initial boundary value problem. For example, Gaﬁychuk and Datsko investigated possible scenarios of pat- tern formations in fractional reaction-diﬀusion systems with initial boundary (Neumann) conditions []. In [], Luchko established the existence and **uniqueness** of the **solution** for the initial boundary (Dirichlet) value problem. In this paper, motivated by the above discussion, we extend it to a more involved and complex system and study the existence of the **solution**.

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By using the Manásevich-Mawhin continuation theorem and some analysis skills, we establish some suﬃcient condition for the existence and **uniqueness** of positive T-periodic solutions for a generalized Rayleigh type φ -Laplacian operator equation. The results of this paper are new and they complement previous known results. MSC: 34K13; 34C25

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By topological degree theory and some analysis skills, we consider a class of generalized Li´enard type p-Laplacian equations. Upon some suitable assumptions, the existence and **uniqueness** of periodic solutions for the generalized Li´enard type p-Laplacian diﬀerential equations are obtained. It is significant that the nonlinear term contains two variables.

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To our knowledge, the only study regarding this problem is due to Russo and Trutnau [16] where they investigate a stochastic equation like (16) (which is the stochastic analog of (1)) but in space dimension one. The authors proceed by freez- ing the realization of the noise for each ω and overcome the problem of defining the product between a function and a distribution by means of a probabilistic represen- tation: They express the parabolic PDE probabilistically through the associated diffusion which is the **solution** of a stochastic differential equation with generalized drift.

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3 J. S. W. Wong, “On the generalized Emden-Fowler equation,” SIAM Review, vol. 17, pp. 339–360, 1975. 4 R. P. Agarwal, D. O’Regan, V. Lakshmikantham, and S. Leela, “An upper and lower **solution** theory for singular Emden-Fowler equations,” Nonlinear Analysis: Real World Applications, vol. 3, no. 2, pp. 275–291, 2002.

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In this paper, we present some new existence and **uniqueness** results for nonlinear fractional diﬀerential equations with a kind of general irregular boundary condition in Banach space by using a ﬁxed-point theorem and contraction mapping principle. Moreover, the boundary condition is extended, therefore, some conclusions from other references are special cases of our results.

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To conclude this section, we would like to point out that after this paper had been submitted for publication, it came to our attention that a similar (**uniqueness**) re- sult had been given in [4]. However, there is no further discussion in [4] as we do in Section 4.

In [5], Song et al. studied the existence and **uniqueness** of the global genera- lized **solution** and the global classical for the initial boundary value problem of Equation (1.1). In [6], Song et al. also studied the nonexistence of the global so- lutions for the initial boundary value problem of Equation (1.1).

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