EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6796 ISSN 1307-5543 – ejpam.com Published by New York Business Global An Inverse Problem for a Parabolic Equation with Nonlocal Boundary Conditions and Two-Point Overdetermination Elvin I. Azizbayov1,∗, Aynur N. Safarova2 1 Department of Intelligent Systems Management, The Academy of Public Administration Under the President of the Republic of Azerbaijan, Baku, Azerbaijan 2 Department of Functional Analysis, Institute of Mathematics and Mechanics, Baku, Azerbaijan Abstract. This paper is devoted to the study of an inverse boundary value problem for a parabolic equation with nonlocal boundary conditions and two-point overdetermination. To analyze the solvability of the problem, we first consider an associated auxiliary inverse boundary value problem. By applying the Fourier method, the solution of the auxiliary problem is reduced to a system of integral equations. The existence and uniqueness of the solution to the auxiliary problem are then established using the contraction mapping principle in an appropriate functional space. Finally, by employing the equivalence between the original and auxiliary formulations, the existence and uniqueness of the classical solution to the initial nonlinear inverse boundary value problem are proved. 2020 Mathematics Subject Classifications: 35R30, 35K10, 35A09, 35A01, 35A02 Key Words and Phrases: Inverse problem, parabolic equation, nonlocal conditions, classical solution, existence, uniqueness 1. Introduction It is known that mathematical modeling of various real-world processes in the natural sciences often leads to the study of inverse problems. An inverse problem generally refers to a situation where the goal is to deduce the causes or parameters of a system based on observed effects or data. One of the most widely studied types of inverse problems is the class of inverse boundary value problems. In the theory of mathematical physics, an inverse boundary value problem involves the simultaneous determination of unknown coefficients and/or the right-hand side of partial differential equations, using additional measurements. Inverse problems arise when the characteristics of an object of interest cannot be observed ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6796 Email addresses: eazizbayov@dia.edu.az (E. I. Azizbayov), safarova-aynur@bk.ru (A. N. Safarova) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 2 of 19 directly. For example, this includes restoring the characteristics of field sources based on their known values at specific points, as well as recovering or interpreting the original signal from a known output signal. The practical importance of inverse problems is substantial, as they arise in diverse fields including seismology, biology, medicine, mineral exploration, seawater desalination, fluid movement in porous media, and etc. As a result, they present some of the most pressing challenges in modern mathematics. The foundations of the theory and practice of investigating inverse problems of mathematical physics were established and developed in the seminal works of distinguished mathematicians such as Tikhonov [1], Lavrent’ev [2], Ivanov [3], and their followers. The practical significance of inverse problems has attracted considerable attention from researchers, resulting in the publication of numerous articles and monographs on the subject in recent decades (see, for example, [4–15] and references therein). It should be noted that one of the most notable problems among inverse boundary value problems are inverse problems in which the boundary conditions include nonlocal condi- tions. In the literature, the term “nonlocal boundary value problems” refers to problems that involve conditions relating the values of the solution and/or its derivatives either at different points on the boundary or at boundary points and certain interior points. The term “nonlocal conditions” and their classification were introduced by A.A. Dezin in [16]. Nonlocal conditions arise in situations where the boundary of the domain of a real process is not accessible for direct measurements, but additional information about the phenomenon under study can be obtained at interior points of the domain. Often, this information is provided in the form of average values of the desired solution. In mathe- matical modeling, it is convenient to express such information as an integral. Note that problems with nonlocal integral conditions occur in the study of processes such as those in turbulent plasma, heat propagation, diffusion theory, and in the modeling of certain technological processes. Now, let us examine the content of some relevant works dedicated to inverse bound- ary value problems for parabolic equations. In the article by Durdiev and Rashidov [17], the inverse problem of determining the multidimensional kernel of the integral term in a second-order parabolic equation is considered, and a local existence and uniqueness the- orem for the inverse problem is proven. The studies [18], [19], [20], and [21] focus on investigating the classical solvability of inverse parabolic problems with various nonlocal boundary conditions. The existence and uniqueness conditions for the solution of the inverse problem of a parabolic equation with nonlocal boundary conditions and integral overdetermination are established in the paper by Ivanchov and Pabyrivs’ka [22]. Kamynin [23] investigated an inverse problem concerned with the simultaneous determination of the right-hand side and the coefficient of the lowest-order derivative in a parabolic equation, subject to an integral observation condition. In the article by Kerimov and Ismailov [24], under certain regularity and compatibility conditions on the given data, the existence, uniqueness, and continuous dependence of the solution on the data were established for an inverse problem with nonlocal boundary conditions and an integral overdetermination condition. A nonlocal inverse boundary value problems for mixed-type partial differential E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 3 of 19 equations are studied, and a criterion for the uniqueness of the solution to the consid- ered problem is established in the work of Martemyanova [25]. In the article by Prilepko, Kamynin, and Kostin [26], the inverse problem of determining the source in a non-uniform parabolic equation under the condition of integral observation is considered, and sufficient conditions for the unique solvability of the inverse problem are obtained. The numerical aspects of inverse problems for parabolic equations with various bound- ary conditions were studied in [27–31], and the references therein. The distinctive contribution of the present study lies in the application of a novel methodological approach to the analysis of an inverse problem for a parabolic equation with nonlocal boundary conditions that depend on both spatial and temporal variables. The paper is organized as follows. Section 1 establishes the relevance of the article’s topic, formulates its purpose, and offers a comprehensive review of the relevant literature with rigorous comparisons to previous works. In Section 2, the mathematical formulation of the inverse problem is presented and a lemma on the reduction of the original prob- lem to an auxiliary one are introduced. Section 3 presents some auxiliary results from spectral theory and introduces special functional spaces. Section 4 examines the existence and uniqueness of the classical solution to the considered inverse boundary value problem. Section 5 summarizes the key results of the investigation. 2. Mathematical formulation of the problem Let T > 0 be a fixed time moment, and let DT denote the rectangular domain in the xt-plane bounded by the inequalities 0 ≤ x ≤ 1 and 0 ≤ t ≤ T . Consider the problem of determining the unknown functions u(x, t) ∈ C2,1(DT ), a(t) ∈ C[0, T ], and b(t) ∈ C[0, T ] such that the triple {u(x, t), a(t), b(t)} satisfies the following parabolic equation a1(t)ut(x, t) + a(t)u(x, t) = uxx(x, t) + b(t)g(x, t) + f(x, t) (x, t) ∈ DT , (1) with the time-nonlocal condition u(x, 0) + δu(x, T ) = φ(x), 0 ≤ x ≤ 1, (2) the Neumann boundary condition ux(0, t) = 0, 0 ≤ t ≤ T, (3) nonlocal integral condition of the first kind 1∫ 0 (x− 1)u(x, t)dx = 0, 0 ≤ t ≤ T, (4) and the overdetermination conditions u(xi, t) = hi(t), i = 1, 2; 0 < x1, x2 < 1, x1 ̸= x2, 0 ≤ t ≤ T, (5) where δ ≥ 0 is given number, a1(t) > 0, f(x, t), g(x, t), φ(x), h1(t), and h2(t) are known functions. E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 4 of 19 Definition 1. The triple {u(x, t), a(t), b(t)} is said to be a classical solution to problem (1)–(5) if all three of the following conditions are satisfied for all functions u(x, t), a(t), and b(t): i) The function u(x, t) and its derivatives ut(x, t) and uxx(x, t) are continuous in the rectangle DT . ii) The functions a(t) and b(t) are continuous on the interval [0, T ]. iii) Equation (1) and conditions (2)–(5) are satisfied in the classical (usual) sense. The following theorem is proved using a method similar to that presented in [32]. Theorem 1. Suppose that δ ≥ 0, 0 < a1(t) ∈ C[0, T ], f(x, t), g(x, t) ∈ C(DT ), φ(x) ∈ C[0, 1], 1∫ 0 (x− 1)f(x, t)dx = 1∫ 0 (x− 1)g(x, t)dx = 0, 0 ≤ t ≤ T , hi(t) ∈ C1[0, T ] (i = 1, 2), h(t) ≡ h2(t)g(x1, t)− h1(t)g(x2, t) ̸= 0, 0 ≤ t ≤ T , and the compatibility conditions 1∫ 0 (x− 1)φ(x)dx = 0, φ(xi) = hi(0) + δhi(T ), i = 1, 2, hold. Then the problem of finding a classical solution of (1)–(5) is equivalent to the problem of determining the functions u(x, t) ∈ C2,1(DT ), a(t) ∈ C[0, T ], and b(t) ∈ C[0, T ] satisfying (1)–(3), and the conditions u(0, t) = u(1, t), 0 ≤ t ≤ T, (6) a1(t)h ′ i(t) + a(t)hi(t) = uxx(xi, t) + a(t)g(xi, t) + f(xi, t), i = 1, 2; 0 ≤ t ≤ T. (7) 3. Some auxiliary results from spectral theory and the introduction of special functional spaces Let us consider the following sequences of functions X0(x) = 2, ..., X2k−1(x) = 4x sinλkx, X2k(x) = 4 cosλkx, (8) Y0(x) = 1− x, ..., Y2k−1(x) = sinλkx, Y2k(x) = (1− x) cosλkx. (9) According to the Keldysh theorem [33], the system of root functions (9) is complete in L2(0, 1). Moreover, the system of functions (8) and (9) form a biorthogonal system and satisfy the necessary and sufficient conditions to be a basis in the space L2(0, 1), for λk = 2kπ (k = 1, 2, ...) as first established by V.A. Il’in [34]. Then an arbitrary function v(x) ∈ L2(0, 1) can be expanded into a biorthogonal series: v(x) = v0X0(x) + ∞∑ k=1 v2k−1X2k−1(x) + ∞∑ k=1 v2kX2k(x), E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 5 of 19 where the coefficients v0, v2k−1, and v2k are computed according to the formulas v0 = 1∫ 0 v(x)Y0(x)dx, v2k−1 = 1∫ 0 v(x)Y2k−1(x)dx, v2k = 1∫ 0 v(x)Y2k(x)dx. It is easy to see that |v0| ≤ ∥v(x)(1− x)∥L2(0,1) ,( ∞∑ k=1 |v2k−1|2 ) 1 2 ≤ 1√ 2 ∥v(x)∥L2(0,1) , ( ∞∑ k=1 |v2k|2 ) 1 2 ≤ 1√ 2 ∥v(x)(1− x)∥L2(0,1) . (10) The following items are true: 1. For certain function v(x) with the properties v(x) ∈ C[0, 1], v′(x) ∈ L2(0, 1), v(0) = v(1) the following estimates are valid( ∞∑ k=1 (λk |v2k−1|)2 ) 1 2 ≤ 1√ 2 ∥v′(x)∥L2(0,1) ,( ∞∑ k=1 (λk |v2k|)2 ) 1 2 ≤ 1√ 2 ∥v′(x)(1− x)− v(x)∥L2(0,1) . (11) 2. If the function v(x) satisfies conditions v(x), v′(x) ∈ C[0, 1], v′′(x) ∈ L2(0, 1), v(0) = v(1), v′(0) = 0, then ( ∞∑ k=1 (λ2 k |v2k−1|)2 ) 1 2 ≤ 1√ 2 ∥v′′(x)∥L2(0,1) ,( ∞∑ k=1 (λ2 k |v2k|)2 ) 1 2 ≤ 1√ 2 ∥v′′(x)(1− x)− 2v′(x)∥L2(0,1) . (12) 3. Under conditions v(x), v′(x) ∈ C[0, 1], v′′(x) ∈ L2(0, 1), v(0) = v(1), v′(0) = 0, v′′(0) = v′′(1), it can be stated that the following estimates are valid:( ∞∑ k=1 (λ3 k |v2k−1|)2 ) 1 2 ≤ 1√ 2 ∥v′′′(x)∥L2(0,1) ,( ∞∑ k=1 (λ3 k |v2k|)2 ) 1 2 ≤ 1√ 2 ∥v′′′(x)(1− x)− 3v′′(x)∥L2(0,1) . (13) E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 6 of 19 In order to study the problem (1)–(3), (6), (7), we consider the following special functional spaces: Let B3 2,T [35] denote the set of all functions of the form u(x, t) = ∞∑ k=0 uk(t)Xk(x), considered in domain DT . Moreover, the functions uk(t) (k = 0, 1, ...) contained in last sum are continuous on the interval [0, T ], and JT (u) ≡ ∥u0(t)∥C[0,T ]+ ( ∞∑ k=1 (λ3 k ∥u2k−1(t)∥C[0,T ]) 2 ) 1 2 + ( ∞∑ k=1 (λ3 k ∥u2k(t)∥C[0,T ]) 2 ) 1 2 < ∞. The norm in the space B3 2,T is defined as follows: ∥u(x, t)∥B3 2,T = JT (u). Furthermore, let E3 T denote the space consisting of the topological product B3 2,T × C[0, T ]× C[0, T ] which is the norm of the element z = {u, a, b} defined by the formula ∥z∥E3 T = ∥u(x, t)∥B3 2,T + ∥a(t)∥C[0,T ] + ∥b(t)∥C[0,T ] . It is clear that the spaces B3 2,T and E3 T are Banach spaces [35]. 4. Existence and uniqueness of the classical solution Since the system (8) forms a Riesz basis in L2(0, 1), each solution to problem (1)–(3), (6), (7) can be sought in the form: u(x, t) = ∞∑ k=0 uk(t)Xk(x), (14) where uk(t) = 1∫ 0 u(x, t)Yk(x)dx (k = 0, 1, ...), (15) and the functions Xk(x), Yk(x) (k = 0, 1, ...) are correspondingly defined by relations (8) and (9). Using the method of separation of variables, from (1), (2), we have a1(t)u ′ 0(t) + a(t)u0(t) = f0(t) + b(t)g0(t), (16) a1(t)u ′ 2k−1(t) + a(t)u2k−1(t) + λ2 ku2k−1(t) = f2k−1(t) + b(t)g2k−1(t), k = 1, 2, ..., (17) a1(t)u ′ 2k(t) + a(t)u2k(t) + λ2 ku2k(t) = f2k(t) + b(t)g2k(t)− 2λku2k−1(t), k = 1, 2, ..., (18) E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 7 of 19 uk(0) + δuk(T ) = φk, k = 0, 1, ..., (19) where fk(t) = 1∫ 0 f(x, t)Yk(x)dx, gk(t) = 1∫ 0 g(x, t)Yk(x)dx, φk = 1∫ 0 φ(x)Yk(x)dx, k = 0, 1, ...., λk = 2kπ, k = 1, 2, .... Solving problem (16)–(19), we get u0(t) = (1 + δ)−1 φ0 − δ T∫ 0 1 a1(τ) F0(τ ;u, a, b)dτ + t∫ 0 1 a1(τ) F0(τ ;u, a, b)dτ, (20) u2k−1(t) = e − t∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds φ2k−1 + t∫ 0 1 a1(τ) F2k−1(τ ;u, a, b)e − t∫ τ λ2k a1(s) ds dτ − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(τ) F2k−1(τ ;u, a, b)e − t∫ τ λ2k a1(s) ds dτ, k = 1, 2, ..., (21) u2k(t) = e − t∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds φ2k + t∫ 0 1 a1(τ) F2k(τ ;u, a, b)e − t∫ τ λ2k a1(s) ds dτ − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(τ) F2k(τ ;u, a, b)e − t∫ τ λ2k a1(s) ds dτ + 2λke − t∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds  t∫ 0 dτ a1(τ) − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 dτ a1(τ) φ2k−1 +2λk  t∫ 0 1 a1(τ)  τ∫ 0 1 a1(ξ) F2k−1(ξ;u, a, b)e − t∫ ξ λ2k a1(s) ds dξ  dτ − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(τ)  τ∫ 0 1 a1(ξ) F2k−1(ξ;u, a, b)e − t∫ ξ λ2k a1(s) ds dξ dτ  E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 8 of 19 − 2λkδe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(ξ) F2k−1(ξ;u, a, b)e − t∫ τ λ2k a1(s) ds dξ ×  t∫ 0 1 a1(τ) dτ − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(τ) dτ  , k = 1, 2, ..., (22) where Fk(t;u, a, b) = −a(t)uk(t) + b(t)gk(t) + fk(t), k = 0, 1, .... Substituting the expressions of (20), (21), and (22) into (14), we get the following formula for the component u(x, t) of the classical solution to problem (1)–(3), (6), (7): u(x, t) = (1 + δ)−1 φ0 − δ T∫ 0 1 a1(τ) F0(τ ;u, a, b)dτ  + t∫ 0 1 a1(τ) F0(τ ;u, a, b)dτ X0(x) + ∞∑ k=1  e − t∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds φ2k−1 + t∫ 0 1 a1(τ) F2k−1(τ ;u, a, b)e − t∫ τ λ2k a1(s) ds dτ − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(τ) F2k−1(τ ;u, a, b)e − t∫ τ λ2k a1(s) ds dτ X2k−1(x) + ∞∑ k=1  e − t∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds φ2k + t∫ 0 1 a1(τ) F2k(τ ;u, a, b)e − t∫ τ λ2k a1(s) ds dτ − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(τ) F2k(τ ;u, a, b)e − t∫ τ λ2k a1(s) ds dτ − 2λke − t∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds  t∫ 0 dτ a1(τ) − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 dτ a1(τ) φ2k−1 +2λk  t∫ 0 1 a1(τ)  τ∫ 0 1 a1(ξ) F2k−1(ξ;u, a, b)e − t∫ ξ λ2k a1(s) ds dξ  dτ E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 9 of 19 − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(τ)  τ∫ 0 1 a1(ξ) F2k−1(ξ;u, a, b)e − t∫ ξ λ2k a1(s) ds dξ  dτ  − 2λkδe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(ξ) F2k−1(ξ;u, a, b)e − t∫ ξ λ2k a1(s) ds dξ ×  t∫ 0 1 a1(τ) dτ − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(τ) dτ  X2k(x). (23) Now from (7), taking into account (14), we have a(t) = [h(t)]−1{g(x1, t)(f(x2, t)− a1(t)h ′ 2(t))− g(x2, t)(f(x1, t)− a1(t)h ′ 1(t)) − ∞∑ k=1 λ2 ku2k−1(t)(g(x1, t)X2k−1(x2)− g(x2, t)X2k−1(x1)) − ∞∑ k=1 λ2 ku2k(t)(g(x1, t)X2k(x2)− g(x2, t)X2k(x1)) } , (24) b(t) = [h(t)]−1{h1(t)(f(x2, t)− a1(t)h ′ 2(t))− h2(t)(f(x1, t)− a1(t)h ′ 1(t)) − ∞∑ k=1 λ2 ku2k−1(t)(h1(t)X2k−1(x2)− h2(t)X2k−1(x1)) − ∞∑ k=1 λ2 ku2k(t)(h1(t)X2k(x2)− h2(t)cX2k(x1)) } . (25) Next, substituting the expressions for u0(t), u2k−1(t), and u2k(t) from (20), (21), and (22) into (24) and (25), respectively, we obtain: a(t) = [h(t)]−1{g(x1, t)(f(x2, t)− a1(t)h ′ 2(t))− g(x2, t)(f(x1, t)− a1(t)h ′ 1(t)) − ∞∑ k=1 λ2 k  e − t∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds φ2k−1 + t∫ 0 1 a1(τ) F2k−1(τ ;u, a, b)e − t∫ τ λ2k a1(s) ds dτ − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(τ) F2k−1(τ ;u, a, b)e − t∫ τ λ2k a1(s) ds dτ  E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 10 of 19 ×(g(x1, t)X2k−1(x2)− g(x2, t)X2k−1(x1)) − ∞∑ k=1 λ2 k  e − t∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds φ2k + t∫ 0 1 a1(τ) F2k(τ ;u, a, b)e − t∫ τ λ2k a1(s) ds dτ − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(τ) F2k(τ ;u, a, b)e − t∫ τ λ2k a1(s) ds dτ + 2λke − t∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds  t∫ 0 dτ a1(τ) − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 dτ a1(τ) φ2k−1 +2λk  t∫ 0 1 a1(τ)  τ∫ 0 1 a1(ξ) F2k−1(ξ;u, a, b)e − t∫ ξ λ2k a1(s) ds dξ  dτ − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(τ)  τ∫ 0 1 a1(ξ) F2k−1(ξ;u, a, b)e − t∫ ξ λ2k a1(s) ds dξ dτ  − 2λkδe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(ξ) F2k−1(ξ;u, a, b)e − t∫ τ λ2k a1(s) ds dξ ×  t∫ 0 1 a1(τ) dτ − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(τ) dτ   ×(g(x1, t)X2k(x2)− g(x2, t)X2k(x1))}, (26) b(t) = [h(t)]−1{h1(t)(f(x2, t)− a1(t)h ′ 2(t))− h2(t)(f(x1, t)− a1(t)h ′ 1(t)) − ∞∑ k=1 λ2 k  e − t∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds φ2k−1 + t∫ 0 1 a1(τ) F2k−1(τ ;u, a, b)e − t∫ τ λ2k a1(s) ds dτ − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(τ) F2k−1(τ ;u, a, b)e − t∫ τ λ2k a1(s) ds dτ  E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 11 of 19 ×(h1(t)X2k−1(x2)− h2(t)cX2k−1(x1)) − ∞∑ k=1 λ2 k  e − t∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds φ2k + t∫ 0 1 a1(τ) F2k(τ ;u, a, b)e − t∫ τ λ2k a1(s) ds dτ − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(τ) F2k(τ ;u, a, b)e − t∫ τ λ2k a1(s) ds dτ + 2λke − t∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds  t∫ 0 dτ a1(τ) − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 dτ a1(τ) φ2k−1 +2λk  t∫ 0 1 a1(τ)  τ∫ 0 1 a1(ξ) F2k−1(ξ;u, a, b)e − t∫ ξ λ2k a1(s) ds dξ  dτ − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(τ)  τ∫ 0 1 a1(ξ) F2k−1(ξ;u, a, b)e − t∫ ξ λ2k a1(s) ds dξ dτ  − 2λkδe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(ξ) F2k−1(ξ;u, a, b)e − t∫ τ λ2k a1(s) ds dξ ×  t∫ 0 1 a1(τ) dτ − δe − T∫ 0 λ2k a1(s) ds 1 + δe − T∫ 0 λ2 k a1(s) ds T∫ 0 1 a1(τ) dτ   ×(h1(t)X2k(x2)− h2(t)cX2k(x1))}, (27) where h(t) ≡ h2(t)g(x1, t)− h1(t)g(x2, t) ̸= 0. Thus the solution of problem (1)–(3), (6), (7) was reduced to the solution of systems (23), (26), (27) with respect to unknown functions u(x, t), a(t) and b(t). We state the following lemma without proof. Lemma 1. If {u(x, t), a(t), b(t)} is any solution to problem (1)–(3), (6), (7), then the functions uk(t) = 1∫ 0 u(x, t)Yk(x)dx (k = 0, 1, ...), E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 12 of 19 satisfy the system (20)–(22) on an interval [0, T ]. To study the uniqueness of the solution to problem (1)–(3), (6), (7), the following corollary plays an important role. Corollary 1. Assume that the system (23), (26), (27) has a unique solution. Then the problem (1)–(3), (6), (7) has at most one solution, i.e., if the problem (1)–(3), (6), (7) has a solution, then it is unique. Let us now consider the operator Φ(u, a, b) = {Φ1(u, a, b),Φ2(u, a, b),Φ3(u, a, b)}, in the space E3 T , where Φ1(u, a, b) = ũ(x, t) ≡ ∞∑ k=0 ũk(t)Xk(x), Φ2(u, a, b) = ã(t), Φ3(u, a, b) = b̃(t), and the functions ũ0(t), ũ2k−1(t), ũ2k(t) (k = 1, 2, ...), ã(t), and b̃(t) are equal to the right- hand sides of (20), (21), (22), (26), and (27), respectively. Assume that the data for the problem (1)–(3), (6), (7) satisfy the following conditions: C1) φ(x) ∈ C2[0, 1], φ′′′(x) ∈ L2(0, 1), and φ(0) = φ(1), φ′(0) = 0, φ′′(0) = φ′′(1); C2) f(x, t), fx(x, t), fxx(x, t) ∈ C(DT ), fxxx(x, t) ∈ L2(DT ), and f(0, t) = f(1, t), fx(0, t) = 0, fxx(0, t) = fxx(1, t), 0 ≤ t ≤ T ; C3) g(x, t), gx(x, t), gxx(x, t) ∈ C(DT ), gxxx(x, t) ∈ L2(DT ), and g(0, t) = g(1, t), gx(0, t) = 0, gxx(0, t) = gxx(1, t), 0 ≤ t ≤ T ; C4) δ ≥ 0, 0 < a1(t) ∈ C[0, T ], hi(t) ∈ C1[0, T ] (i = 1, 2), h(t) ≡ h2(t)g(x1, t)− h1(t)g(x2, t) ̸= 0, 0 ≤ t ≤ T. Then, by applying simple transformations, we obtain: ∥ũ(x, t)∥B3 2,T ≤ A1(T ) +B1(T ) ∥a(t)∥C[0,T ] ∥u(x, t)∥B3 2,T +D1(T ) ∥b(t)∥C[0,T ] , (28) ∥ã(t)∥C[0,T ] ≤ A2(T ) +B2(T ) ∥a(t)∥C[0,T ] ∥u(x, t)∥B3 2,T +D2(T ) ∥b(t)∥C[0,T ] , (29)∥∥∥b̃(t)∥∥∥ C[0,T ] ≤ A3(T ) +B3(T ) ∥a(t)∥C[0,T ] ∥u(x, t)∥B3 2,T +D3(T ) ∥b(t)∥C[0,T ] , (30) where A1(T ) = (1 + δ)−1 ∥φ(x)(1− x)∥L2(0,1) + 1 m (1 + δ(1 + δ)−1)[ √ T ∥f(x, t)(1− x)∥L2(DT ) + √ 2ρ(T ) ∥∥φ′′′(x) ∥∥ L2(0,1) + √ 2T m (1 + δρ(T )) ∥fxxx(x, t)∥L2(DT ) E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 13 of 19 + 3√ 2 ρ(T ) ∥∥φ′′′(x)(1− x)− 3φ′′(x) ∥∥ L2(0,1) + 3 √ T m √ 2 (1 + δρ(T )) ∥fxxx(x, t)(1− x)− 3fxx(x, t)∥L2(DT ) + 2 √ 2MT m (1 + δρ(T )) ∥∥φ′′′(x) ∥∥ L2(0,1) + 2 √ 2M m2 (1 + δρ(T ))2T ∥fxxx(x, t)∥L2(DT ) , B1(T ) = 1 m (1 + δ(1 + δ)−1)T + 5T m √ 2 (1 + δρ(T )) + 2 √ 2M m2 (1 + δρ(T ))2T √ T , D1(T ) = 1 m (1 + δ(1 + δ)−1)T ∥g(x, t)(1− x)∥L2(DT ) + √ 2 m (1 + δρ(T )) √ T ( 1 + 2 √ TM m (1 + δρ(T ))2 ) ∥gxxx(x, t)∥L2(DT ) + 3 m √ 2 (1 + δρ(T )) √ T ∥gxxx(x, t)(1− x)− 3gxx(x, t)∥L2(DT ) , A2(T ) = ∥∥[h(t)]−1 ∥∥ C[0,T ] ×{ ∥∥h1(t)(f(x2, t)− a1(t)h ′ 2(t))− h2(t)(f(x1, t)− a1(t)h ′ 1(t)) ∥∥ C[0,T ] +4 ( ∞∑ k=1 λ−2 k ) 1 2 ∥|g(x1, t)|+ |g(x2, t)|∥C[0,T ] [ √ 2ρ(T ) ∥∥φ′′′(x) ∥∥ L2(0,1) + 2 m (1 + δρ(T )) √ T 2 ∥fxxx(x, t)∥L2(DT ) + 3√ 2 ρ(T ) ∥∥φ′′′(x)(1− x)− 3φ′′(x) ∥∥ L2(0,1) + 3 m (1 + δρ(T )) √ T 2 ∥fxxx(x, t)(1− x)− 3fxx(x, t)∥L2(DT ) + 4 √ MT m (1 + δρ(T )) ∥∥φ′′′(x) ∥∥ L2(0,1) + 4 √ M m2 (1 + δρ(T ))2T ∥fxxx(x, t)∥L2(DT ) , B2(T ) = 4 ∥∥[h(t)]−1 ∥∥ C[0,T ] ( ∞∑ k=1 λ−2 k ) 1 2 ∥|g(x1, t)|+ |g(x2, t)|∥C[0,T ] T × ( 1 + (1 + δρ(T )) m ( 3 + 4 √ TM m (1 + δρ(T )) )) , D2(T ) = 4 ∥∥[h(t)]−1 ∥∥ C[0,T ] ( ∞∑ k=1 λ−2 k ) 1 2 ∥|g(x1, t)|+ |g(x2, t)|∥C[0,T ] E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 14 of 19 × [ 2 m (1 + δρ(T )) √ T ( 1√ 2 + 2 √ TM m2 (1 + δρ(T )) ) ∥gxxx(x, t)∥L2(DT ) + 3 m (1 + δρ(T )) √ T 2 ∥gxxx(x, t)(1− x)− 3gxx(x, t)∥L2(DT ) ] , A3(T ) = ∥∥[h(t)]−1 ∥∥ C[0,T ] ×{ ∥∥h1(t) (f(x2, t)− a1(t)h ′ 2(t) ) − h2(t) ( f(x1, t)− a1(t)h ′ 1(t) )∥∥ C[0,T ] +4 ( ∞∑ k=1 λ−2 k ) 1 2 ∥|h1(t)|+ |h2(t)|∥C[0,T ] [ √ 2ρ(T ) ∥∥φ′′′(x) ∥∥ L2(0,1) + 2 m (1 + δρ(T )) √ T 2 ∥fxxx(x, t)∥L2(DT ) + 3√ 2 ρ(T ) ∥∥φ′′′(x)(1− x)− 3φ′′(x) ∥∥ L2(0,1) + 3 m (1 + δρ(T )) √ T 2 ∥fxxx(x, t)(1− x)− 3fxx(x, t)∥L2(DT ) + 4 √ MT m (1 + δρ(T )) ∥∥φ′′′(x) ∥∥ L2(0,1) + 4 √ M m2 (1 + δρ(T ))2T ∥fxxx(x, t)∥L2(DT ) , B3(T ) = 4 ∥∥[h(t)]−1 ∥∥ C[0,T ] ( ∞∑ k=1 λ−2 k ) 1 2 ×∥|h1(t)|+ |h2(t)|∥C[0,T ] T ( 1 + (1 + δρ(T )) m ( 3 + 4 √ TM m (1 + δρ(T )) )) , D2(T ) = 4 ∥∥[h(t)]−1 ∥∥ C[0,T ] ( ∞∑ k=1 λ−2 k ) 1 2 ∥|h1(t)|+ |h2(t)|∥C[0,T ] × [ 2 m (1 + δρ(T )) √ T ( 1√ 2 + 2 √ TM m2 (1 + δρ(T )) ) ∥gxxx(x, t)∥L2(DT ) + 3 m (1 + δρ(T )) √ T 2 ∥gxxx(x, t)(1− x)− 3gxx(x, t)∥L2(DT ) ] From inequalities (28)–(30) we conclude ∥ũ(x, t)∥B3 2,T + ∥ã(t)∥C[0,T ] + ∥∥∥b̃(t)∥∥∥ C[0,T ] ≤ A(T ) +B(T ) ∥a(t)∥C[0,T ] ∥u(x, t)∥B3 2,T +D(T ) ∥b(t)∥C[0,T ] , (31) where A(T ) = A1(T ) +A2(T ) +A3(T ), B(T ) = B1(T ) +B2(T ) +B3(T ), D(T ) = D1(T ) +D2(T ) +D3(T ). Now, let us prove the following theorem. E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 15 of 19 Theorem 2. Let the conditions C1)− C4) and the condition (B(T )(A(T ) + 2) +D(T ))(A(T ) + 2) < 1, (32) be fulfilled. Then, problem (1)–(3), (6), (7) has a unique solution in the ball K = KR (∥z∥E3 T ≤ R = A(T ) + 2) of space E3 T . Remark 1. Inequality (32) is satisfied for sufficiently small values of T . Proof. Let’s consider in the space E3 T , the operator equation z = Φz, (33) where z = {u, a, b}. The components Φi(u, a, b) (i = 1, 2, 3) of operator Φ(u, a, b) defined by the right side of equations (23), (26), and (27), respectively. Now, consider the operator Φ(u, a, b) in the ball K = KR of the space E3 T . Analogously to (31), we obtain that for any z, z1, z2 ∈ KR, the following estimates hold: ∥Φz∥E3 T ≤ A(T ) +B(T ) ∥a(t)∥C[0,T ] ∥u(x, t)∥B3 2,T +D(T ) ∥b(t)∥C[0,T ] ≤ A(T ) +B(T )R2 +D(T )R ≤ A(T ) +B(T )(A(T ) + 2)2 +D(T )(A(T ) + 2), (34) ∥Φz1 − Φz2∥E3 T ≤ (B(T )R+D(T )) ×(∥a1(t)− a2(t)∥C[0,T ] + ∥b1(t)− b2(t)∥C[0,T ] + ∥u1(x, t)− u2(x, t)∥B3 2,T ) ≤ (B(T ) (A(T ) + 2) +D(T )) ∥z1 − z2∥E3 T . (35) Then by (32), from estimates (34) and (35) it is clear that the operator Φ acts in a ball K = KR and satisfy the assertion of the contraction mapping principle. Therefore the operator Φ has a unique fixed point {u, a, b} in the ball K = KR, which is a unique solution of equation (33); i.e. {u, a, b} is a unique solution of the systems (23), (26), (27) in the ball K = KR. Thus, we obtain that the function u(x, t) as an element of the space B3 2,T is continuous and has continuous derivatives ux(x, t) and uxx(x, t) in DT . Analogously [20] it can be show that the derivative ut(x, t) is also continuous in the region DT . It is easy to verify that Equation (1) and conditions (2), (3), (6), and (7) are satisfied in the ordinary sense. Consequently, {u(x, t), a(t), b(t)} is a solution of problem (1)–(3), (6), (7) and by Lemma 1 this solution is unique in the ball K = KR. Hence, from Theorem 2, by virtue of Theorem 1, it follows that the original problem (1)–(5) has a unique classical solution, i.e. the following theorem is valid. E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 16 of 19 Theorem 3. Assume that all the conditions of Theorem 2 are satisfied and 1∫ 0 (x− 1)f(x, t)dx = 1∫ 0 (x− 1)g(x, t)dx = 0, 0 ≤ t ≤ T, 1∫ 0 (x− 1)φ(x)dx = 0, φ(xi) = hi(0) + δhi(T ), i = 1, 2. Then problem (1)–(5) has a unique classical solution in the ball K = KR of the space E3 T . 5. Conclusions In this work, we have investigated the classical solvability of a nonlinear inverse bound- ary value problem for a parabolic equation subject to nonlocal boundary conditions. The analysis begins with a transformation of the original inverse problem into an equivalent auxiliary inverse boundary value problem with trivial data, which simplifies the subse- quent analytical treatment. By employing the Fourier method, the auxiliary problem is reduced to a system of nonlinear integral equations whose properties can be rigorously analyzed within an appropriate functional framework. The existence and uniqueness of the solution to the auxiliary problem have been estab- lished by means of the contraction mapping (Banach fixed-point) principle in a suitably defined Banach space. This approach provides a constructive framework for demonstrat- ing the well-posedness of the problem and ensures the stability of the solution with respect to the given data. Owing to the established equivalence between the original and auxiliary formulations, the existence and uniqueness of a classical solution to the initial nonlinear inverse boundary value problem have consequently been proved. The theoretical results obtained in this study contribute to the broader theory of in- verse problems for parabolic equations with nonlocal conditions, which are often encoun- tered in mathematical models of diffusion and heat conduction processes with memory or spatial interaction effects. Future research may extend the present analysis to fractional- order parabolic operators, multidimensional domains, or problems involving noisy data and regularization techniques. Acknowledgements The authors express their sincere gratitude to the anonymous reviewers for their careful reading, insightful comments, and constructive suggestions, which have greatly contributed to improving the quality and clarity of this manuscript. E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 17 of 19 References [1] AN Tikhonov. On stability of inverse problems. Doklady Akademii Nauk SSSR, 39(5):195–198, 1943. [2] MM Lavrent’ev. On Some Ill-posed Problems of Mathematical Physics. Nauka, Novosibirsk, Russia, 1962. [3] VK Ivanov. On linear ill-posed problems. Doklady Akademii Nauk SSSR, 145(2):270– 272, 1962. [4] EI Azizbayov and YT Mehraliyev. A boundary value problem for the equation of motion of a homogeneous bar with periodic conditions. American Journal of Applied Mathematics and Statistics, 3(6):252–256, 2015. [5] AM Denisov. Elements of the Theory of Inverse Problems. De Gruyter, Berlin, Germany, 1999. [6] MJ Huntul and I Tekin. An inverse problem of identifying the time-dependent po- tential and source terms in a two-dimensional parabolic equation. Hacettepe Journal of Mathematics and Statistics, 52(6):1578–1599, 2023. [7] MS Hussein, S Gani, and TE Dyhoum. Timewise-dependent coefficients identification problems for third-order pseudo-parabolic equations from nonlocal extra conditions. Ibn AL-Haitham Journal For Pure and Applied Sciences, 38(1):465–492, 2025. [8] MI Ismailov and F Kanca. An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions. Mathematical Methods in the Applied Sciences, 34(6):692–702, 2011. [9] MI Ivanchov. Inverse Problem for Equations of Parabolic Type. Lviv: VNTL Pub- lishers, Monograph Series, Lviv, Ukraine, 2003. [10] SI Kabanikhin. Inverse and Ill-posed Problems: Theory and Applications. De Gruyter, Berlin, Germany, 2012. [11] UK Koilyshov, MA Sadybekov, and KA Beisenbayeva. Solution of nonlocal boundary value problems for the heat equation with discontinuous coefficients, in the case of two discontinuity points. Bulletin of the Karaganda University. Mathematics Series, 117(1):81–91, 2025. [12] AI Kozhanov. Composite Type Equations and Inverse Problems, Inverse and ill-posed problems series. De Gruyter, Berlin, Germany, 1999. [13] D Lesnic. Inverse Problems with Applications in Science and Engineering. Chapman and Hall/CRC, London, United Kingdom, 2021. [14] GK Namazov. Inverse Problems of the Theory of Equations of Mathematical Physics. Elm, Baku, Azerbaijan, 1984. [15] AG Ramm. Inverse Problems. Springer, Berlin, Germany, 2005. [16] AA Dezin. The simplest solvable extensions of ultrahyperbolic and pseudoparabolic operators. Doklady Akademii Nauk SSSR, 148(5):1013–1016, 1963. [17] DK Durdiev and AS Rashidov. Inverse problem of determining the kernel in an integro-differential equation of parabolic type. Differential Equations, 50(1):110–116, 2020. [18] EI Azizbayov. The nonlocal inverse problem of the identification of the lowest co- E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 18 of 19 efficient and the right-hand side in a second-order parabolic equation with integral conditions. Boundary Value Problems, 2019(11):1–19, 2019. [19] EI Azizbayov and YT Mehraliyev. Inverse problem for a parabolic equation in a rectangle domain with integral conditions. European Journal of Pure and Applied Mathematics, 10(5):981–994, 2017. [20] EI Azizbayov and YT Mehraliyev. Solvability of nonlocal inverse boundary-value problem for a second-order parabolic equation with integral conditions. Electronic Journal of Differential Equations, 2017(125):1–4, 2017. [21] EI Azizbayov and YT Mehraliyev. Nonlocal inverse problem for determination of time derivative coefficient in a second-order parabolic equation. Advances in Differential Equations and Control Processes, 19(1):15–36, 2018. [22] MI Ivanchov and NV Pabyrivska. Simultaneous determination of two coefficients of a parabolic equation in the case of nonlocal and integral conditions. Ukrainian Mathematical Journal, 53:674–684, 2001. [23] VL Kamynin. Inverse problem of simultaneously determining the right-hand side and the coefficient of a lower order derivative for a parabolic equation on the plane. Differential Equations, 50(6):792–804, 2014. [24] MB Kerimov and MI Ismailov. An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions. Journal of Mathematical Analysis and Applications, 396(2):546–554, 2012. [25] NV Martemyanova. Inverse problem for the equation of the mixed type with non- local boundary condition. Vestnik of Samara University, Natural Science Series, (6(80)):27–38, 2010. [26] AI Prilepko, VL Kamynin, and AB Kostin. Inverse source problem for parabolic equation with the condition of integral observation in time. Journal of Inverse and Ill-posed Problems, 26(4):523–539, 2018. [27] C Ashyralyyev and P Akkan. Source identification problem for a parabolic equation with multipoint nonlocal boundary condition. Numerical Functional Analysis and Optimization, 41(16):1913–1935, 2020. [28] MJ Huntul, TE Oussaeif, M Tamsir, and MA Aiyashi. Unique solvability for an inverse problem of a nonlinear parabolic pde with nonlocal integral overdetermination condition. Open Mathematics, 20(1):1407–1431, 2022. [29] YT Mehraliyev, MJ Huntul, and EI Azizbayov. Simultaneous identification of the right-hand side and time-dependent coefficients in a two-dimensional parabolic equa- tion. Mathematical Modelling and Analysis, 29(1):90–108, 2024. [30] K Rashedi, H Adibi, and M Dehghan. Determination of space-time-dependent heat source in a parabolic inverse problem via the ritz-galerkin technique. Inverse Problems in Science and Engineering, 22(7):1077–1108, 2014. [31] L Yang, M Dehghan, JN Yu, and GW Luo. Inverse problem of time-dependent heat sources numerical reconstruction. Mathematics and Computers in Simulation, 81(8):1656–1672, 2011. [32] YaT Mehraliev and AN Safarova. On one nonlocal inverse boundary problem for the second-order parabolic equation. Vestnik Yuzhno-Ural’skogo Gosudarstvennogo E. I. Azizbayov, A. N. Safarova / Eur. J. Pure Appl. Math, 18 (4) (2025), 6796 19 of 19 Universiteta. Seriya Matematika. Mekhanika. Fizika., 9(2):13–21, 2017. [33] V Keldysh. On eigenvalues and eigenfunctions of some classes of non-self-adjoint equations. Doklady Akademii Nauk SSSR, 77(1):11–14, 1951. [34] VA Il’in. Existence of a reduced system of eigen-and associated functions for a non- selfadjoint ordinary differential operator. Proceedings of the Steklov Institute of Math- ematics, 142:148–155, 1976. [35] KI Khudaverdiyev and AA Veliyev. Investigation of One-dimensional Mixed Problem for a Class of Pseudohyperbolic Equations of Third Order with Nonlinear Operator Right Side. Chashyoghly, Baku, Azerbaijan, 2010.