We analyze nonlocal Kirchhoff systems in fractional Musielak–Sobolev spaces. By means of Ricceri’s theorem, it is established that the system possesses weak solutions under fairly general conditions for the modular, the Kirchhoff functions, and the nonlinearities. Additionally, a case of growth of the logarithmic type is presented to visualize the outcome.

We study a class of nonlocal Kirchhoff-type equations driven by a fractional integro-differential operator acting in fractional Musielak–Sobolev spaces. The equation is considered with homogeneous Dirichlet boundary conditions. The model combines three different sources of nonlocality: a fractional operator, a Kirchhoff coefficient depending on a nonlocal energy, and a Musielak-type modular depending on the points of the domain. We cast the problem in a variational setting and apply a three critical points theorem due to Ricceri. Under suitable assumptions on the Kirchhoff function and on the nonlinear terms, we prove the existence of at least three weak solutions. The result extends previous multiplicity theorems for Kirchhoff-type and fractional problems to a more general framework with nonstandard growth.

. This study aims to determine the source term within a subdiffusion model using an artificial neural network approach, leveraging additional data. The core strategy involves replacing the unknown source term with a neural network, thereby converting the inverse source problem into an optimal control problem. By employing this approach, we establish the existence of an optimal solution for the control problem and compute the associated optimality conditions using the alternating direction multiplier method. Numerical solutions are derived via this proposed methodology, showcasing its effectiveness through a series of numerical tests on both regular and singular examples. Our results demonstrate the efficacy of the artificial neural network method, particularly evident in its comparison with established techniques like gradient descent and alternating direction multiplier method. This comparative analysis reinforces the strength and reliability of the artificial neural network-based approach in solving the source term determination problem within the sub diffusion model.

We establish the existence of a weak solution for a Dirichlet fractional Kirchhoff–type problem driven by a nonlocal Musielak operator with genuinely (x, y)–dependent growth on a bounded Lipschitz domain Ω⊂ℝ^N . The Kirchhoff coefficient depends on the global fractional Musielak modular, while the source term is merely Carathéodory with subcritical growth and is not assumed to be of potential type. The proof is nonvariational: we construct Faedo–Galerkin approximations in the nonlocal Dirichlet space W^s_0L_Φ _x,y(Q) and handle the σ –finite interaction set Q by a Young–measure representation on a finite–measure exhaustion combined with a diagonal extraction. Assuming uniform convexity of the Musielak density, we derive an (S_+) –type criterion, obtain strong convergence in the energy space, and pass to the Kirchhoff factor while identifying the nonlocal nonlinear term. The result applies in particular to variable–exponent, double–phase, and logarithmically perturbed kernels.

In this paper, we introduce and study a new class of fractional modular function spaces, called Fractional Anisotropic Musielak–Sobolev Spaces, which generalize both the fractional Anisotropic Orlicz–Sobolev spaces and the Anisotropic fractional Sobolev spaces with variable exponent. These spaces are designed to handle anisotropic and heterogeneous behaviors that naturally arise in nonlocal and nonlinear models. We develop their fundamental properties and embedding results, establishing a solid variational framework. As an application, we investigate a class of nonlocal anisotropic eigenvalue problems involving variable growth and direction-dependent fractional integro-differential operators. We prove the existence of eigenvalues by means of critical point theory and modular analysis. Our results extend and unify several existing models in the theory of nonlocal partial differential equations.

This paper addresses an inverse problem concerning the identification of an unknown initial value in a one-dimensional fractional diffusion equation. The initial value is reconstructed from the final noisy data. To address this inverse problem, we first examine the well-posedness of the direct problem. Then, the inverse problem is reformulated as a regularized optimal control problem by employing the least squares method. After that, we demonstrate the existence and stability of solutions to the optimal control problem. In the meanwhile, we prove the Fréchet differentiability of the cost function and establish its convexity. Furthermore, we reconstruct the initial value via using the conjugate gradient method. Finally, to illustrate the effectiveness of the proposed method, we present results from a series of numerical experiments.

Using the Nehari manifold, we establish the existence of two non-negative weak solutions for a fractional type problem driven by a non-local operator of the elliptic type in fractional Orlicz-Sobolev spaces. We show how the existence of solutions depends on the properties of the Nehari manifold. Moreover, under some suitable assumptions, continuous and compact embeddings results are established.

In this paper, we study the inverse problem of identifying the pa-rameters in a nonlinear subdiffusion model from an observation defined in the given S2T subset of S2. The nonlinear subdiffusion model involves a Caputo fractional derivative of order alpha is an element of (0,1) in time. To address our model, we first examine the regularity of the solution for the direct problem using the Mittag-Leffler function. To investigate our inverse parameter problem, we re-formulate first it in to Least-Squares optimization problem. Then, we establish the existence of the optimal solution and prove the convexity of the considered cost function by using its first derivative. To solve this problem numerically, we adapt a recent method in the literature known as the alternating direction method of multiplier (ADMM) which we establish its convergence. In order to show the effectiveness of the proposed method we present some numerical experiments.

The aim of this study is to determine a time-dependent source problem in the time-fractional diffusion equation from the additional measurement data at an inner point. By transforming the inverse problem into an minimization problem, we employ two distinct alternative approaches to solve the latter control problem. To assess the effectiveness of these approaches, we test them under conditions with and without noise. The results obtained are very interesting and encouraging, indicating the potential success of the proposed methods.

In this paper, using the three critical points theorem we obtain the existence of three weak solutions for a Kirchhoff type problem driven by a nonlocal operator of the elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions.

This paper tackles new kind of double phase problems characterized by variable exponent matrices diffusion and nonlinear boundary conditions. Using the Nehari manifold technique, we investigate the existence and multiplicity results of nonnegative weak solutions within the Musielak-Orlicz framework. Our methodology primarily integrates variational methods with novel technical estimates.

In this paper, we aim to study an inverse problem for determining two time-independent coefficients in a fractional diffusion system from the final measurements. First, we prove the well-posedness of the state problem, and then we show some regularity results for the solution of the direct system using the Mittag-Leffler function. Then, we reformulate our inverse problem into an optimal control one. Afterwards, we establish the existence of the minimizer and prove the stability estimate for two coefficients with respect to the final data. The descent method is proposed as a numerical one based on the gradient calculus via the adjoint state and we compare it with the conjugate gradient method. Finally, we will present some numerical tests that shows the efficiency of the proposed methods.

In this paper, we develop some properties of the a_x,y(·)-Neumann derivative for the nonlocal s(·,·)-order operator in fractional Musielak-Sobolev spaces with variable s(·,·)-order. Therefore we prove the basic proprieties of the correspondent function spaces. In the second part of this paper, by means of Ekeland's variational principal and direct variational approach, we prove the existence of weak solutions to the following double phase Neumann and Robin problem with variable s(·,·)-order: {[ -Δ)^s_1(x,·)_a^1_(x,·) u+(-Δ)^s_2(x,·)_a^2_(x,·) u +a^1_x(|u|)u+a^2_x(|u|)u = λ f(x,u) in Ω,; 𝒩^s_1(x,·)_a^1(x,·)u+𝒩^s_2(x,·)_a^2(x,·)u+β(x)( a^1_x(|u|)u+a^2_x(|u|)u ) = 0 in ℝ^N∖Ω, ]. where (-Δ)^s_i(x,·)_a^i_(x,·) and 𝒩^s_i(x,·)_a^i(x,·) denote the variable s_i(·,·)-order fractional Laplace operator and the nonlocal normal a_i(·,·)-derivative of s_i(·,·)-order, respectively.

In this paper, we introduce the s (., .)-fractional Musielak–Sobolev spaces W^s(x,y)L_ _x,y(Ω ) . Then, we show that there exists λ _*>0 such that any λ∈ (0, λ _*) is an eigenvalue for the following problem, by means of Ekeland’s variational principle (𝒫_a) {[ ( -Δ) ^s(x,.)_a_(x,.) u = λ |u|^q(x)-2u in Ω ,; ; u = 0 in ℝ ^N∖Ω , ]. where Ω is a bounded open subset of ℝ ^N with C^0,1 -regularity and bounded boundary.

In this paper, we develop some properties of the ax,y(.)-Neumann derivative for the fractional ax,y(.)-Laplacian operator. Therefore we prove the basic proprieties of the correspondent function spaces. In the second part of this paper, by means of Ekeland’s variational principal and direct variational approach, we prove the existence of weak solutions for a nonlocal problem with nonhomogeneous Neumann and Robin boundary condition.

This paper is concerned with a class of fractional p -Laplace type problems with Dirichlet boundary data of the following form (P_s) {[ M(||u||^p)( (-Δ )^s_p u +|u|^p-2u) = λ f(x,u) in Ω; ; u = 0 in ℝ^N∖Ω . ]. By means of Ekeland’s variational principle and a direct variational approach, we investigate the existence of nontrivial weak solution for the above problem.

In this paper, we are concerned with some qualitative properties of the new fractional Musielak-Sobolev spaces (WL)-L-s phi(x, y) such that the generalized Poincare type inequality and some continuous and compact embedding results. Moreover, we prove that any function in (WL)-L-s phi(x, y) (Omega) may be extended to a function in (WL)-L-s phi(x, y) (R-N), with Omega subset of R-N is a bounded domain of class C-0,C-1. In addition, we establish a result that relates to the complemented subspace in (WL)-L-s phi(x, y) (R-N). As an application, using the mountain pass theorem and some variational methods, we investigate the existence of a nontrivial weak solution for a class of nonlocal fractional type problems with Dirichlet boundary data.

In this work, we investigate an inverse source problem for determining the unknown source term in one and two dimensional space of a linear elliptic equation. First, the inverse problem is formulated into an optimization problem with the Tikhonov regularization method. Then, the existence and uniqueness of the solution for the direct problem are proved. Second, the existence of the optimal solution is proved. Then, the convexity of the optimization problem is shown in order to ensure the uniqueness of the optimal solution. Moreover, the conjugate gradient method is applied to reconstruct the source term. Finally, to show the efficiency of the suggested approach, we give some numerical results in one and two dimensional space.

The purpose of this work is to estimate a source term in the time-fractional diffusion equation from additional measurements. After recasting the inverse source problem as an nonsmooth optimization problem. Two different alternative methods are applied to find the solution of the control problem. The accuracy of the proposed methods is tested in the cases of with and without noise, and the findings are very promising and encouraging.

We are interested in the multiplicity of weak solutions for a binonlocal fractional p (x, .)-Kirchhoff type problems. Our technical approach is based on the general three critical points theorem obtained by B. Ricceri.