Foam-based composites are advanced materials that combine lightweight design with high mechanical strength and stiffness, making them attractive for diverse engineering applications. Their mechanical performance is strongly influenced by foam morphology, including strut geometry in open-cell foams and cell wall topology in closed-cell foams. This study investigates the effect of these morphological features on the mechanical behavior of foam-based composites, considering both open-cell and closed-cell foams as matrix phases. A finite element-based numerical homogenization approach evaluates mechanical performance over a wide range of phase volume fractions and stiffness contrasts. The results reveal significant differences in stress distribution, load transfer efficiency, and overall mechanical response between the two foam morphologies. Open-cell foam composites, particularly those with cylindrical struts, show superior load-bearing capacity under high stiffness contrast. In contrast, closed-cell composites develop stress localization unless optimized with thin walls and dense reinforcement. The findings indicate that the mechanical efficiency is governed by three main factors: (i) phase stiffness contrast, determining how morphology affects performance; (ii) phase connectivity or interpenetration, controlling stress transfer through the stiff phase; and (iii) strut geometry, where rounded struts aligned with the load enable direct stress transfer while minimizing matrix deformation.

Foam-based composites are advanced materials that combine lightweight design with high mechanical strength and stiffness, making them attractive for diverse engineering applications. Their mechanical performance is strongly influenced by foam morphology, including strut geometry in open-cell foams and cell wall topology in closed-cell foams. This study investigates the effect of these morphological features on the mechanical behavior of foam-based composites, considering both open-cell and closed-cell foams as matrix phases. A finite element-based numerical homogenization approach evaluates mechanical performance over a wide range of phase volume fractions and stiffness contrasts. The results reveal significant differences in stress distribution, load transfer efficiency, and overall mechanical response between the two foam morphologies. Open-cell foam composites, particularly those with cylindrical struts, show superior load-bearing capacity under high stiffness contrast. In contrast, closed-cell composites develop stress localization unless optimized with thin walls and dense reinforcement. The findings indicate that the mechanical efficiency is governed by three main factors: (i) phase stiffness contrast, determining how morphology affects performance; (ii) phase connectivity or interpenetration, controlling stress transfer through the stiff phase; and (iii) strut geometry, where rounded struts aligned with the load enable direct stress transfer while minimizing matrix deformation.

This study investigates how authentic audiovisual (AAV) materials, particularly TED Talks and YouTube videos, contribute to second-year EFL students’ perceived speaking development at Batna2 University. To obtain the required data, 130 second-year students completed a questionnaire that measured three constructs, including the profiled AAV engagement, instructional use in-class vs. homework, and perceived gains across five speaking dimensions, in addition to Semi-structured interviews with seven teachers, who explored classroom practices, constraints, and conditions for effectiveness. The findings indicate positive student perceptions of AAV input, as evidenced by higher engagement with AAV content. It is also reported that the strongest perceived gains were in confidence/anxiety reduction and pronunciation/intonation awareness, while perceived gains in fluency were comparatively lower. At the same time, Teachers' views converged on one key condition: AAV materials are perceived as most beneficial when viewing is systematically linked to structured speaking tasks and supported by pedagogical scaffolding.

This study investigates how authentic audiovisual (AAV) materials, particularly TED Talks and YouTube videos, contribute to second-year EFL students’ perceived speaking development at Batna2 University. To obtain the required data, 130 second-year students completed a questionnaire that measured three constructs, including the profiled AAV engagement, instructional use in-class vs. homework, and perceived gains across five speaking dimensions, in addition to Semi-structured interviews with seven teachers, who explored classroom practices, constraints, and conditions for effectiveness. The findings indicate positive student perceptions of AAV input, as evidenced by higher engagement with AAV content. It is also reported that the strongest perceived gains were in confidence/anxiety reduction and pronunciation/intonation awareness, while perceived gains in fluency were comparatively lower. At the same time, Teachers' views converged on one key condition: AAV materials are perceived as most beneficial when viewing is systematically linked to structured speaking tasks and supported by pedagogical scaffolding.

Aseptic loosening driven by stress shielding remains the principal long-term failure mode of total hip arthroplasty (THA), affecting 10-15% of implants within 15 years and generating annual revision costs exceeding USD 8 billion globally. This study presents an integrated multiscale computational framework for the design of functionally graded femoral stems using Ti-6Al-4V/SiC whisker-reinforced composite materials. The framework systematically couples seven sequential modules : (M1) Halpin-Tsai micromechanical characterisation of three material systems Ti-6Al-4V, Ti/SiC 20 vol%, and Ti/SiC 40 vol%; (M2) analytical beam stress analysis under ISO 7206-4 oblique cantilever loading; (M3) SIMP topology optimisation at V* = 0.70 achieving 30% mass reduction (C/C₀= 0.37), (M4) TPMS lattice homogenisation for four architectures (Gyroid, Diamond, Cubic, Octet) with a corrected stress shielding index (SSI_Gyroid = +11.1%) representing an 87% reduction from monolithic Ti-6Al-4V (SSI = 82.5%), (M5) multiaxial fatigue analysis combining Rainflow cycle counting, modified Goodman criterion, and dual Miner damage reporting (D_ISO = 0.001-0.018; D_full = 0.117-1.426. The Ti/SiC 20 vol% composite in a functionally graded Gyroid architecture emerges as the optimal clinical configuration, uniquely satisfying all biomechanical, osseointegration, and SLM fabricability requirements simultaneously.Keywords:total hip arthroplasty; stress shielding; SIMP topology optimisation; TPMS lattice; Ti-6Al-4V/SiC composite; Goodman criterion; Miner damage; functionally graded implant.

Aseptic loosening driven by stress shielding remains the principal long-term failure mode of total hip arthroplasty (THA), affecting 10-15% of implants within 15 years and generating annual revision costs exceeding USD 8 billion globally. This study presents an integrated multiscale computational framework for the design of functionally graded femoral stems using Ti-6Al-4V/SiC whisker-reinforced composite materials. The framework systematically couples seven sequential modules : (M1) Halpin-Tsai micromechanical characterisation of three material systems Ti-6Al-4V, Ti/SiC 20 vol%, and Ti/SiC 40 vol%; (M2) analytical beam stress analysis under ISO 7206-4 oblique cantilever loading; (M3) SIMP topology optimisation at V* = 0.70 achieving 30% mass reduction (C/C₀= 0.37), (M4) TPMS lattice homogenisation for four architectures (Gyroid, Diamond, Cubic, Octet) with a corrected stress shielding index (SSI_Gyroid = +11.1%) representing an 87% reduction from monolithic Ti-6Al-4V (SSI = 82.5%), (M5) multiaxial fatigue analysis combining Rainflow cycle counting, modified Goodman criterion, and dual Miner damage reporting (D_ISO = 0.001-0.018; D_full = 0.117-1.426. The Ti/SiC 20 vol% composite in a functionally graded Gyroid architecture emerges as the optimal clinical configuration, uniquely satisfying all biomechanical, osseointegration, and SLM fabricability requirements simultaneously.Keywords:total hip arthroplasty; stress shielding; SIMP topology optimisation; TPMS lattice; Ti-6Al-4V/SiC composite; Goodman criterion; Miner damage; functionally graded implant.

Aseptic loosening driven by stress shielding remains the principal long-term failure mode of total hip arthroplasty (THA), affecting 10-15% of implants within 15 years and generating annual revision costs exceeding USD 8 billion globally. This study presents an integrated multiscale computational framework for the design of functionally graded femoral stems using Ti-6Al-4V/SiC whisker-reinforced composite materials. The framework systematically couples seven sequential modules : (M1) Halpin-Tsai micromechanical characterisation of three material systems Ti-6Al-4V, Ti/SiC 20 vol%, and Ti/SiC 40 vol%; (M2) analytical beam stress analysis under ISO 7206-4 oblique cantilever loading; (M3) SIMP topology optimisation at V* = 0.70 achieving 30% mass reduction (C/C₀= 0.37), (M4) TPMS lattice homogenisation for four architectures (Gyroid, Diamond, Cubic, Octet) with a corrected stress shielding index (SSI_Gyroid = +11.1%) representing an 87% reduction from monolithic Ti-6Al-4V (SSI = 82.5%), (M5) multiaxial fatigue analysis combining Rainflow cycle counting, modified Goodman criterion, and dual Miner damage reporting (D_ISO = 0.001-0.018; D_full = 0.117-1.426. The Ti/SiC 20 vol% composite in a functionally graded Gyroid architecture emerges as the optimal clinical configuration, uniquely satisfying all biomechanical, osseointegration, and SLM fabricability requirements simultaneously.Keywords:total hip arthroplasty; stress shielding; SIMP topology optimisation; TPMS lattice; Ti-6Al-4V/SiC composite; Goodman criterion; Miner damage; functionally graded implant.

Accurate detection of built-up areas in semi-arid regions is vital for urban planning and environmental monitoring. However, built-up surfaces and bare soils often produce very similar spectral responses. As a result, this similarity causes confusion in satellite image classification. Additionally, spectral overlap among urban materials, bare soil, and sparse vegetation further complicates detection. This study evaluates several spectral indices, including DBSI, NDTI, NDVI, BRBA, and BSI, combined with Principal Component Analysis (PCA) to enhance built-up area extraction from Sentinel-2A imagery. Images captured during the driest season were selected to maximize spectral contrast. Three classification schemes based on Support Vector Machine (SVM) were tested. The first scheme used DBSI, NDTI, and NDVI. The second used BRBA, NDTI, and NDVI. The third relied on PCA-derived components. The results indicate that the PCA-based approach achieved the highest classification accuracy at 95%. In comparison, the DBSI/NDTI/NDVI combination reached 93%, while the BRBA/NDTI/NDVI scheme achieved 92%. Therefore, PCA helps reduce spectral confusion and enhances the identification of built-up areas in semi-arid environments. Overall, combining multiple spectral indices with dimensionality reduction offers a reliable method for urban analysis using Sentinel-2 imagery.

Accurate detection of built-up areas in semi-arid regions is vital for urban planning and environmental monitoring. However, built-up surfaces and bare soils often produce very similar spectral responses. As a result, this similarity causes confusion in satellite image classification. Additionally, spectral overlap among urban materials, bare soil, and sparse vegetation further complicates detection. This study evaluates several spectral indices, including DBSI, NDTI, NDVI, BRBA, and BSI, combined with Principal Component Analysis (PCA) to enhance built-up area extraction from Sentinel-2A imagery. Images captured during the driest season were selected to maximize spectral contrast. Three classification schemes based on Support Vector Machine (SVM) were tested. The first scheme used DBSI, NDTI, and NDVI. The second used BRBA, NDTI, and NDVI. The third relied on PCA-derived components. The results indicate that the PCA-based approach achieved the highest classification accuracy at 95%. In comparison, the DBSI/NDTI/NDVI combination reached 93%, while the BRBA/NDTI/NDVI scheme achieved 92%. Therefore, PCA helps reduce spectral confusion and enhances the identification of built-up areas in semi-arid environments. Overall, combining multiple spectral indices with dimensionality reduction offers a reliable method for urban analysis using Sentinel-2 imagery.

This study investigates the potential for enhancing the geotechnical performance of sabkha soil from Ain M’Lila (Algeria) through cement-lime stabilization, with optimization of stabilizer dosages conducted using Response Surface Methodology (RSM). Twelve mixtures with varying contents of soil (90–100%), lime (0–10%), and cement (0–10%) were evaluated using a central composite design. Experimental testing included Proctor compaction, unconfined compressive strength (UCS), direct shear, California Bearing Ratio (CBR), and electrical conductivity (Ec), with data interpreted via ANOVA and RSM modeling. The untreated sabkha soil displayed unfavorable properties, such as 60% fines, an electrical conductivity of 16.33 mS/cm, 3585 mg/L of chlorides, and 4704 mg/L of sulfates. Nevertheless, RSM identified an optimal formulation comprising 6% cement and 4% lime, which led to remarkable improvements: UCS increased by 1007% (1583 kPa at 90 days), the CBR index rose by 442.89% (83.83), cohesion reached 38.2 kPa, and the pH stabilized at a highly alkaline value of 12.3, promoting pozzolanic activity. ANOVA results confirmed the significance of the models (p < 0.05), with strong coefficients of determination (R² values of 87.13%, 91.39%, 90.85%, 80.47%, and 89.03%) corresponding to maximum dry density, cohesion, internal friction angle, CBR, and UCS, respectively. Overall, the findings confirm that cement-lime treatment can effectively convert problematic sabkha soil into a reliable subgrade material for road infrastructure. Moreover, the RSM-based optimization approach provides a cost-effective and technically sound solution for soil stabilization in arid and semi-arid environments.

This study investigates the potential for enhancing the geotechnical performance of sabkha soil from Ain M’Lila (Algeria) through cement-lime stabilization, with optimization of stabilizer dosages conducted using Response Surface Methodology (RSM). Twelve mixtures with varying contents of soil (90–100%), lime (0–10%), and cement (0–10%) were evaluated using a central composite design. Experimental testing included Proctor compaction, unconfined compressive strength (UCS), direct shear, California Bearing Ratio (CBR), and electrical conductivity (Ec), with data interpreted via ANOVA and RSM modeling. The untreated sabkha soil displayed unfavorable properties, such as 60% fines, an electrical conductivity of 16.33 mS/cm, 3585 mg/L of chlorides, and 4704 mg/L of sulfates. Nevertheless, RSM identified an optimal formulation comprising 6% cement and 4% lime, which led to remarkable improvements: UCS increased by 1007% (1583 kPa at 90 days), the CBR index rose by 442.89% (83.83), cohesion reached 38.2 kPa, and the pH stabilized at a highly alkaline value of 12.3, promoting pozzolanic activity. ANOVA results confirmed the significance of the models (p < 0.05), with strong coefficients of determination (R² values of 87.13%, 91.39%, 90.85%, 80.47%, and 89.03%) corresponding to maximum dry density, cohesion, internal friction angle, CBR, and UCS, respectively. Overall, the findings confirm that cement-lime treatment can effectively convert problematic sabkha soil into a reliable subgrade material for road infrastructure. Moreover, the RSM-based optimization approach provides a cost-effective and technically sound solution for soil stabilization in arid and semi-arid environments.

This study investigates the potential for enhancing the geotechnical performance of sabkha soil from Ain M’Lila (Algeria) through cement-lime stabilization, with optimization of stabilizer dosages conducted using Response Surface Methodology (RSM). Twelve mixtures with varying contents of soil (90–100%), lime (0–10%), and cement (0–10%) were evaluated using a central composite design. Experimental testing included Proctor compaction, unconfined compressive strength (UCS), direct shear, California Bearing Ratio (CBR), and electrical conductivity (Ec), with data interpreted via ANOVA and RSM modeling. The untreated sabkha soil displayed unfavorable properties, such as 60% fines, an electrical conductivity of 16.33 mS/cm, 3585 mg/L of chlorides, and 4704 mg/L of sulfates. Nevertheless, RSM identified an optimal formulation comprising 6% cement and 4% lime, which led to remarkable improvements: UCS increased by 1007% (1583 kPa at 90 days), the CBR index rose by 442.89% (83.83), cohesion reached 38.2 kPa, and the pH stabilized at a highly alkaline value of 12.3, promoting pozzolanic activity. ANOVA results confirmed the significance of the models (p < 0.05), with strong coefficients of determination (R² values of 87.13%, 91.39%, 90.85%, 80.47%, and 89.03%) corresponding to maximum dry density, cohesion, internal friction angle, CBR, and UCS, respectively. Overall, the findings confirm that cement-lime treatment can effectively convert problematic sabkha soil into a reliable subgrade material for road infrastructure. Moreover, the RSM-based optimization approach provides a cost-effective and technically sound solution for soil stabilization in arid and semi-arid environments.

This study investigates the potential for enhancing the geotechnical performance of sabkha soil from Ain M’Lila (Algeria) through cement-lime stabilization, with optimization of stabilizer dosages conducted using Response Surface Methodology (RSM). Twelve mixtures with varying contents of soil (90–100%), lime (0–10%), and cement (0–10%) were evaluated using a central composite design. Experimental testing included Proctor compaction, unconfined compressive strength (UCS), direct shear, California Bearing Ratio (CBR), and electrical conductivity (Ec), with data interpreted via ANOVA and RSM modeling. The untreated sabkha soil displayed unfavorable properties, such as 60% fines, an electrical conductivity of 16.33 mS/cm, 3585 mg/L of chlorides, and 4704 mg/L of sulfates. Nevertheless, RSM identified an optimal formulation comprising 6% cement and 4% lime, which led to remarkable improvements: UCS increased by 1007% (1583 kPa at 90 days), the CBR index rose by 442.89% (83.83), cohesion reached 38.2 kPa, and the pH stabilized at a highly alkaline value of 12.3, promoting pozzolanic activity. ANOVA results confirmed the significance of the models (p < 0.05), with strong coefficients of determination (R² values of 87.13%, 91.39%, 90.85%, 80.47%, and 89.03%) corresponding to maximum dry density, cohesion, internal friction angle, CBR, and UCS, respectively. Overall, the findings confirm that cement-lime treatment can effectively convert problematic sabkha soil into a reliable subgrade material for road infrastructure. Moreover, the RSM-based optimization approach provides a cost-effective and technically sound solution for soil stabilization in arid and semi-arid environments.

This study investigates the potential for enhancing the geotechnical performance of sabkha soil from Ain M’Lila (Algeria) through cement-lime stabilization, with optimization of stabilizer dosages conducted using Response Surface Methodology (RSM). Twelve mixtures with varying contents of soil (90–100%), lime (0–10%), and cement (0–10%) were evaluated using a central composite design. Experimental testing included Proctor compaction, unconfined compressive strength (UCS), direct shear, California Bearing Ratio (CBR), and electrical conductivity (Ec), with data interpreted via ANOVA and RSM modeling. The untreated sabkha soil displayed unfavorable properties, such as 60% fines, an electrical conductivity of 16.33 mS/cm, 3585 mg/L of chlorides, and 4704 mg/L of sulfates. Nevertheless, RSM identified an optimal formulation comprising 6% cement and 4% lime, which led to remarkable improvements: UCS increased by 1007% (1583 kPa at 90 days), the CBR index rose by 442.89% (83.83), cohesion reached 38.2 kPa, and the pH stabilized at a highly alkaline value of 12.3, promoting pozzolanic activity. ANOVA results confirmed the significance of the models (p < 0.05), with strong coefficients of determination (R² values of 87.13%, 91.39%, 90.85%, 80.47%, and 89.03%) corresponding to maximum dry density, cohesion, internal friction angle, CBR, and UCS, respectively. Overall, the findings confirm that cement-lime treatment can effectively convert problematic sabkha soil into a reliable subgrade material for road infrastructure. Moreover, the RSM-based optimization approach provides a cost-effective and technically sound solution for soil stabilization in arid and semi-arid environments.

Modern factories aim to boost production to increase profits and remain competitive amid intense market pressure. This goal is pursued through production planning, which not only improves process efficiency but also ensures product quality and long-term market success. Among the key tools that support this objective is production forecasting, which plays a central role in facilitating manufacturing processes. In this study, we focus on enhancing the efficiency of industrial systems by developing smart predictive models based on Deep Reinforcement Learning (DRL). Specifically, time-series data from the rotary cement kiln at the SCIMAT factory (Batna, Algeria) were used to predict clinker production. The proposed model, based on the DDQN network, outperformed both the traditional Q-learning network and the DQN. The DDQN achieved a Mean Squared Error (MSE) of 0.0010, a Root Mean Squared Error (RMSE) of 0.0316, and a coefficient of determination (R2) of 0.9484. In contrast, the DQN achieved an MSE of 0.0014, an RMSE of 0.0374 and an R2 of 0.9374. The traditional Q-learning network achieved an MSE of 0.0052, an RMSE of 0.0725 and an R2 of 0.8865. The LSTM algorithm achieved an MSE of 0.0069, an RMSE of 0.0830 and an R2 of 0.6570. The developed model is designed to be integrated into the plant’s control system via Programmable Logic Controllers (PLCs), enabling automated operation without direct human intervention. The DQN model demonstrated strong predictive accuracy, leading to improved resource management and operational efficiency. Its application can also be extended to other industrial domains.

Modern factories aim to boost production to increase profits and remain competitive amid intense market pressure. This goal is pursued through production planning, which not only improves process efficiency but also ensures product quality and long-term market success. Among the key tools that support this objective is production forecasting, which plays a central role in facilitating manufacturing processes. In this study, we focus on enhancing the efficiency of industrial systems by developing smart predictive models based on Deep Reinforcement Learning (DRL). Specifically, time-series data from the rotary cement kiln at the SCIMAT factory (Batna, Algeria) were used to predict clinker production. The proposed model, based on the DDQN network, outperformed both the traditional Q-learning network and the DQN. The DDQN achieved a Mean Squared Error (MSE) of 0.0010, a Root Mean Squared Error (RMSE) of 0.0316, and a coefficient of determination (R2) of 0.9484. In contrast, the DQN achieved an MSE of 0.0014, an RMSE of 0.0374 and an R2 of 0.9374. The traditional Q-learning network achieved an MSE of 0.0052, an RMSE of 0.0725 and an R2 of 0.8865. The LSTM algorithm achieved an MSE of 0.0069, an RMSE of 0.0830 and an R2 of 0.6570. The developed model is designed to be integrated into the plant’s control system via Programmable Logic Controllers (PLCs), enabling automated operation without direct human intervention. The DQN model demonstrated strong predictive accuracy, leading to improved resource management and operational efficiency. Its application can also be extended to other industrial domains.

ThispaperinvestigatesseveralformsofSylvester-typeoperatorequationsininfinite-dimensional Hilbertspaces, focusingonboththeclassicalequation AX−XB = CanditsgeneralizedversionAX−YB = C, which involves two unknowns. We establish new necessary and sufficient conditions for the existence of solutions by employing generalized inverses under novel structuralassumptions. Specialattentionisgiven to the behavior of these equations when restricted to subspaces such asker(A+I)andker(B+I), andtocases involving two distinct subspaces. The study highlights how operator properties-such as involution and pseudo-inverses-govern solvability and solution structure. The results offer a unified theoretical frame work that encompasses both classical and generalized operator equations, with potential applications in control theory, perturbation analysis, and related areas. Illustrative examples are provided to demonstrate the applicability and relevance of the theoretical developments.

ThispaperinvestigatesseveralformsofSylvester-typeoperatorequationsininfinite-dimensional Hilbertspaces, focusingonboththeclassicalequation AX−XB = CanditsgeneralizedversionAX−YB = C, which involves two unknowns. We establish new necessary and sufficient conditions for the existence of solutions by employing generalized inverses under novel structuralassumptions. Specialattentionisgiven to the behavior of these equations when restricted to subspaces such asker(A+I)andker(B+I), andtocases involving two distinct subspaces. The study highlights how operator properties-such as involution and pseudo-inverses-govern solvability and solution structure. The results offer a unified theoretical frame work that encompasses both classical and generalized operator equations, with potential applications in control theory, perturbation analysis, and related areas. Illustrative examples are provided to demonstrate the applicability and relevance of the theoretical developments.

Fixed point theory stands as a fundamental pillar in nonlinear functional anal ysis, being essential for proving the existence of solutions for nonlinear dif ferential and integral equations. Krasnoselskii’s hybrid fixed point theorem, which combines the Banach contraction principle with Schauder’s theorem, is a pivotal contribution. Recent efforts have focused on refining and relaxing the conditions of this theorem. This study aims to extend the theoretical frame work of Krasnoselskii-type fixed point theorems to address a broader and more general class of nonlinear operator equations within a Banach algebra setting. It also seeks to establish rigorous conditions for the existence (and uniqueness, where possible) of solutions. The approach involves developing local variants of the classic Krasnoselskii fixed point theorems. We performed a comparative analysis of previous studies, introduced modifications to the operator equations to relax restrictive assumptions, and theoretically generalized the theorems to accommodate a complex structure involving four operators. To validate the results, they were applied to a nonlinear functional integral equation within the Banach space C[0,1]. We successfully generalized existing results by in corporating the H¨older continuity condition, which is less restrictive than the standard Lipschitz condition. The unified theoretical framework led to the es tablishment of a comprehensive set of theorems and corollaries covering a wide class of operator equations such as : Ax(ρ2)Bxρ1 + Cx(ρ3)Dxρ1 = x. Our re sults provide less restrictive local existence conditions and wider applicability in the analysis of complex mathematical systems.

Fixed point theory stands as a fundamental pillar in nonlinear functional anal ysis, being essential for proving the existence of solutions for nonlinear dif ferential and integral equations. Krasnoselskii’s hybrid fixed point theorem, which combines the Banach contraction principle with Schauder’s theorem, is a pivotal contribution. Recent efforts have focused on refining and relaxing the conditions of this theorem. This study aims to extend the theoretical frame work of Krasnoselskii-type fixed point theorems to address a broader and more general class of nonlinear operator equations within a Banach algebra setting. It also seeks to establish rigorous conditions for the existence (and uniqueness, where possible) of solutions. The approach involves developing local variants of the classic Krasnoselskii fixed point theorems. We performed a comparative analysis of previous studies, introduced modifications to the operator equations to relax restrictive assumptions, and theoretically generalized the theorems to accommodate a complex structure involving four operators. To validate the results, they were applied to a nonlinear functional integral equation within the Banach space C[0,1]. We successfully generalized existing results by in corporating the H¨older continuity condition, which is less restrictive than the standard Lipschitz condition. The unified theoretical framework led to the es tablishment of a comprehensive set of theorems and corollaries covering a wide class of operator equations such as : Ax(ρ2)Bxρ1 + Cx(ρ3)Dxρ1 = x. Our re sults provide less restrictive local existence conditions and wider applicability in the analysis of complex mathematical systems.